Quinta Essentia - Part 4

QUINTA ESSENTIA A Practical Guide to Space-Time Engineering

PART 4 “THE DERIVATION OF THE HUBBLE CONSTANT & THE COSMIC MICROWAVE BACKGROUND RADIATION (CMBR) TEMPERATURE” “To my parents” RESEARCH NOTES Key Words: Big-Bang, CMBR, Cosmological Evolution / Expansion / History / Inflation, Dark Energy / Matter, Gravitation, Hubble constant.

2nd Edition Project Initiated: October 13, 2005 Project Completed: April 14, 2007 Revised: Thursday, 24 November 2011 RICCARDO C. STORTI1

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[email protected] © Copyright 2011: Delta Group Engineering (dgE): All rights reserved.

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Preface This text is a compilation of research notes and a companion to “Quinta Essentia - Part 3”, extending the principles of Electro-Gravi-Magnetics (EGM) to two important aspects of Cosmology [i.e. the present value of the Hubble constant “H0” and Cosmic Microwave Background Radiation (CMBR) temperature “T0”]. Subsequently, the reader is actively encouraged to review “Part 3”, to obtain a full appreciation of the EGM method. “Part 3” develops an equation facilitating the harmonic representation of all fundamental particles relative to an arbitrarily chosen base particle. It is demonstrated, for example, that the EGM wavefunction frequency of an Electron “ωΩ(rε,me)” is twice that of the Proton “ωΩ(rπ,mp)”, and the harmonic relationship between them “Stω” has a value of “2”. Hence, a table of fundamental particle harmonics was formulated. This resulted in a relationship between the mass-energy and size of fundamental particles based upon Zero-Point-Field (ZPF) equilibria. Although the EGM harmonic representation is an approximation derived from basic engineering principles, it produces experimentally verified results substantially beyond the current abilities of the Standard Model (SM) of Particle-Physics to do so, to at least four orders of magnitude. “Part 4” utilises the principles of mass-energy distribution and similitude by ZPF equilibria developed in “Part 3”, to derive “H0” and “T0”. It is demonstrated that a mathematical relationship exists between the Hubble constant and CMBR temperature such that “T0” is derived from “H0”. Consequently, this enables the complete and precise specification of the thermodynamic, inflationary and expansive history of the Universe, from the “Big-Bang” to the present day. Astonishingly, the application of the EGM construct to Cosmology produces “Black-BodyRadiation” curve characteristics, without the application of the “Black-Body-Law”, further reinforcing the validity of the “H0” and “T0” formulations of approximately “67.0843(km/s/Mpc)” and “2.7248(K)” respectively. Considering that the experimental tolerance of the CMBR temperature is presently “2.725 ± 0.001(K)”, it is obvious that any determination within such a tight tolerance should be given serious consideration. The derivation of “H0” and “T0” is possible assuming that, instantaneously prior to the “BigBang”, the Primordial Universe was analogous to a homogeneous Planck scale particle of maximum permissible energy density, characterised by a single EGM wavefunction. Simultaneously, we represent the “Milky-Way” as a Planck scale object of equivalent total Galactic mass “MG”, acting as a Galactic Reference Particle (GRP) characterised by a large number of EGM wavefunctions with respect to the solar distance from the Galactic centre “Ro”. This facilitates a comparative analysis between the Primordial and Galactic particle representations utilising the harmonic equation derived in “Part 3”, yielding “H0” in terms of “Ro” and “MG”. Moreover, we extend the analysis by determining the theoretical frequency shift of a fictitious EGM wavefunction being radiated from the Primordial particle, yielding “T0” in terms of “H0”. Consequently, by utilising the measured value of “T0”, we derive improved estimates for “Ro” and “MG” as being approximately “8.1072(kpc)” and “6.3142 x1011(solar-masses)” respectively. Because the value of “H0” is still widely debated and the associated experimental tolerance is much broader than “T0”, the EGM construct implies that the observed “accelerated expansion” of the Universe is attributable to the determination of the ZPF energy density threshold “UZPF” being “< -2.52 x10-13(Pa)”. Moreover, it is graphically illustrated that the gradient of the Hubble constant in the time domain is presently positive. Subsequently, it is demonstrated that the majority of what is currently conjectured to constitute “Dark Matter / Energy” by the scientific community, is nothing more than Photons due to the definition of a Graviton under the EGM construct. In addition, it is mathematically shown that the magnitude of the impact of “Dark Matter / Energy” on the value of the Hubble constant and CMBR temperature is “< 1(%)” such that the Universe is composed of: • “> 94.4(%) Photons”, “< 1(%) Dark Matter / Energy” and “4.6(%) Atoms”. 3

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Table of Contents Preface ........................................................................................................................................... 3 Scientific Achievements .............................................................................................................. 23 1

Introduction......................................................................................................................... 25

1.1 The natural philosophy of the Cosmos ............................................................................... 25 1.1.1 Objectives and scope................................................................................................... 25 1.1.2 How are these objectives achieved? ............................................................................ 25 1.1.2.1 Synopsis................................................................................................................. 25 1.1.2.2 Derivation process.................................................................................................. 26 1.1.2.2.1 Hubble constant “HU” ...................................................................................... 26 1.1.2.2.2 CMBR temperature “TU” ................................................................................. 27 1.1.2.2.3 “HU → HU2, TU → TU2 → TU3” ........................................................................ 28 1.1.2.2.4 Rate of change “dHdt” ...................................................................................... 29 1.1.3 Sample results............................................................................................................. 30 1.1.3.1 Numerical evaluation and analysis.......................................................................... 30 1.1.3.1.1 Cosmological properties................................................................................... 30 1.1.3.1.2 Significant temporal ordinates .......................................................................... 32 1.1.3.2 Graphical evaluation and analysis........................................................................... 33 1.1.3.2.1 Average Cosmological temperature vs. age ...................................................... 33 1.1.3.2.2 Magnitude of the Hubble constant vs. Cosmological age .................................. 34 1.1.3.2.3 Cosmological evolution process ....................................................................... 35 1.1.4 History of the Universe according to EGM ................................................................. 36 1.1.5 Discussion .................................................................................................................. 38 1.1.5.1 Conceptualization................................................................................................... 38 1.1.5.1.1 “λx”.................................................................................................................. 38 1.1.5.1.2 “TL” ................................................................................................................. 38 1.1.5.1.3 “CΩ_J” .............................................................................................................. 38 1.1.5.1.4 “Stω” ................................................................................................................ 39 1.1.5.2 Dynamic, kinematic and geometric similarity......................................................... 39 1.1.5.2.1 “HU” ................................................................................................................ 39 1.1.5.2.2 “TU”................................................................................................................. 40 1.1.6 Concluding remarks.................................................................................................... 40 1.2 Fundamentals .................................................................................................................... 42 1.2.1 General Relativity (GR) .............................................................................................. 42 1.2.2 Black-Holes (BH’s) .................................................................................................... 43 1.2.3 Quantum Mechanics (QM).......................................................................................... 44 1.2.4 Particle-Physics .......................................................................................................... 45 1.2.4.1 Synopsis................................................................................................................. 45 1.2.4.2 Subatomic particles ................................................................................................ 45 1.2.4.3 History ................................................................................................................... 45 1.2.4.4 Standard Model (SM)............................................................................................. 46 1.2.4.5 Experiment............................................................................................................. 46 1.2.4.6 Theory.................................................................................................................... 46 1.2.5 Zero-Point-Field (ZPF) Theory ................................................................................... 48 1.2.5.1 Synopsis................................................................................................................. 48 1.2.5.1.1 Zero-Point-Energy (ZPE) ................................................................................. 48 5

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1.2.5.1.1.1 General...................................................................................................... 48 1.2.5.1.1.2 Elementary particles .................................................................................. 48 1.2.5.1.1.3 Implications............................................................................................... 48 1.2.5.1.2 History ............................................................................................................. 49 1.2.5.1.3 Foundational Physics........................................................................................ 49 1.2.5.1.4 Varieties of ZPE............................................................................................... 50 1.2.5.1.5 Experimental evidence ..................................................................................... 50 1.2.5.1.6 Gravitation and Cosmology.............................................................................. 50 1.2.5.1.7 Propulsion theories........................................................................................... 50 1.2.5.1.8 Popular culture................................................................................................. 51 1.2.5.2 Spectral Energy Density (SED) .............................................................................. 51 1.2.6 The Polarisable Vacuum (PV) model of gravity .......................................................... 51 1.2.7 Dimensional Analysis Techniques and Buckingham’s “Π” (Pi) Theory ...................... 52 1.2.7.1 The principles......................................................................................................... 52 1.2.7.2 The atomic bomb.................................................................................................... 53 1.2.7.3 The birth and foundations of Electro-Gravi-Magnetics (EGM) ............................... 54 1.2.8 EGM: the natural philosophy of fundamental particles ................................................ 57 1.2.8.1 How was it derived? ............................................................................................... 57 1.2.8.2 Poynting Vector “Sω” ............................................................................................. 61 1.2.8.3 The size of the Proton, Neutron and Electron (radii: “rπ”, “rν”, “rε”) ....................... 62 1.2.8.4 The harmonic representation of fundamental particles ............................................ 64 1.2.8.4.1 Establishing the foundations............................................................................. 64 1.2.8.4.2 Improving accuracy.......................................................................................... 64 1.2.8.4.3 Formulating an hypothesis ............................................................................... 65 1.2.8.5 Identifying a mathematical pattern.......................................................................... 65 1.2.8.6 Results ................................................................................................................... 66 1.2.8.6.1 Harmonic evidence of unification..................................................................... 66 1.2.8.6.2 Recent developments........................................................................................ 67 1.2.8.6.2.1 PDG mass-energy ranges........................................................................... 67 1.2.8.6.2.2 Electron Neutrino and Up / Down / Bottom Quark mass............................ 68 1.2.8.6.2.3 Top Quark mass ........................................................................................ 68 1.2.8.6.2.3.1 The dilemma....................................................................................... 68 1.2.8.6.2.3.2 The resolution..................................................................................... 68 1.2.8.7 Discussion.............................................................................................................. 69 1.2.8.7.1 Experimental evidence of unification ............................................................... 69 1.2.8.7.2 The answers to some important questions......................................................... 70 1.2.8.7.2.1 What causes harmonic patterns to form?.................................................... 70 1.2.8.7.2.1.1 ZPF equilibrium.................................................................................. 70 1.2.8.7.2.1.2 Inherent quantum characteristics......................................................... 70 1.2.8.7.2.2 Why haven’t the “new” particles been experimentally detected?................ 71 1.2.8.7.2.3 Why can all fundamental particles be described in harmonic terms? .......... 71 1.2.8.7.2.4 Why is EGM a method and not a theory?................................................... 72 1.2.8.7.2.5 What would one need to do, in order to disprove EGM? ............................ 72 1.2.8.7.2.6 Why does EGM produce current and not constituent Quark masses? ......... 72 1.2.8.7.2.7 Why does EGM yield only the three observed families? ............................ 73 1.2.8.8 What may the periodic table of elementary particles look like under EGM? ........... 73 1.2.8.9 What are the most important results determined by the EGM construct?................. 74 1.2.8.9.1 PV and ZPF ..................................................................................................... 74 1.2.8.9.1.1 Gravitational amplitude spectrum “CPV”.................................................... 74 1.2.8.9.1.2 Gravitational frequency spectrum “ωPV”.................................................... 74 1.2.8.9.1.3 Harmonic cut-off frequency “ωΩ”.............................................................. 74 1.2.8.9.2 Photons, Gravitons and Euler's Constant .......................................................... 74 6

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1.2.8.9.2.1 The mass-energy of a Graviton “mgg” ........................................................ 74 1.2.8.9.2.2 The mass-energy of a Photon “mγγ” ........................................................... 74 1.2.8.9.2.3 The radius of a Photon “rγγ” ....................................................................... 75 1.2.8.9.2.4 The radius of a Graviton “rgg”.................................................................... 75 1.2.8.9.3 All Other Particles............................................................................................ 75 1.2.8.9.3.1 The Fine Structure Constant “α”................................................................ 75 1.2.8.9.3.2 Harmonic cut-off frequency ratio (the ratio of two particle spectra) “Stω”.. 75 1.2.8.9.3.3 Neutron Magnetic Radius “rνM”................................................................. 75 1.2.8.9.3.4 Proton Electric Radius “rπE” ...................................................................... 75 1.2.8.9.3.5 Proton Magnetic Radius “rπM” ................................................................... 75 1.2.8.9.3.6 Classical Proton Root Mean Square Charge Radius “rp” ............................ 76 1.2.8.9.3.7 The first term of the Hydrogen Spectrum (Balmer Series) “λA” ................. 76 1.2.8.10 Graphical representation of fundamental particles under EGM ............................... 76 1.2.8.11 Concluding remarks about EGM ............................................................................ 78 1.2.9 The Hubble Constant “H0”.......................................................................................... 79 1.2.9.1 Description............................................................................................................. 79 1.2.9.2 Discovery............................................................................................................... 79 1.2.9.3 Interpretation.......................................................................................................... 80 1.2.9.4 Olbers’ paradox...................................................................................................... 81 1.2.9.5 Measuring the Hubble constant .............................................................................. 82 1.2.10 CMBR temperature..................................................................................................... 83 1.2.10.1 Description............................................................................................................. 84 1.2.10.2 Features.................................................................................................................. 84 1.2.10.3 Relationship to the “Big-Bang” .............................................................................. 85 1.2.10.3.1 General .......................................................................................................... 85 1.2.10.3.2 Temperature................................................................................................... 85 1.2.10.3.3 Primary anisotropy ......................................................................................... 86 1.2.10.3.4 Late time anisotropy....................................................................................... 87 1.2.10.3.5 Polarisation .................................................................................................... 88 1.2.10.4 Microwave background observations...................................................................... 88 2 2.1

Definition of Terms ............................................................................................................. 93 Numbering conventions..................................................................................................... 93

2.2 Quinta Essentia – Part 3..................................................................................................... 93 2.2.1 Alpha Forms “αx” ....................................................................................................... 93 2.2.2 Amplitude Spectrum ................................................................................................... 93 2.2.3 Background Field........................................................................................................ 93 2.2.4 Bandwidth Ratio “∆ωR” .............................................................................................. 93 2.2.5 Beta Forms “βx”.......................................................................................................... 93 2.2.6 Buckingham Π Theory (BPT) ..................................................................................... 93 2.2.7 Casimir Force “FPP” .................................................................................................... 93 2.2.8 Change in the Number of Modes “∆nS”....................................................................... 93 2.2.9 Compton Frequency “ωCx”.......................................................................................... 93 2.2.10 Cosmological Constant ............................................................................................... 94 2.2.11 Critical Boundary “ωβ” ............................................................................................... 94 2.2.12 Critical Factor “KC” .................................................................................................... 94 2.2.13 Critical Field Strengths “EC and BC” ........................................................................... 94 2.2.14 Critical Frequency “ωC”.............................................................................................. 94 2.2.15 Critical Harmonic Operator “KR H” ............................................................................. 94 7

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2.2.16 2.2.17 2.2.18 2.2.19 2.2.20 2.2.21 2.2.22 2.2.23 2.2.24 2.2.25 2.2.26 2.2.27 2.2.28 2.2.29 2.2.30 2.2.31 2.2.32 2.2.33 2.2.34 2.2.35 2.2.36 2.2.37 2.2.38 2.2.39 2.2.40 2.2.41 2.2.42 2.2.43 2.2.44 2.2.45 2.2.46 2.2.47 2.2.48 2.2.49 2.2.50 2.2.51 2.2.52 2.2.53 2.2.54 2.2.55 2.2.56 2.2.57 2.2.58 2.2.59 2.2.60 2.2.61 2.2.62 2.2.63 2.2.64 2.2.65 2.2.66

Critical Mode “NC”..................................................................................................... 94 Critical Phase Variance “φC”....................................................................................... 94 Critical Ratio “KR”...................................................................................................... 94 Curl ............................................................................................................................ 94 DC-Offsets ................................................................................................................. 94 Dimensional Analysis Techniques (DAT's)................................................................. 95 Divergence ................................................................................................................. 95 Dominant Bandwidth .................................................................................................. 95 Electro-Gravi-Magnetics (EGM)................................................................................. 95 Electro-Gravi-Magnetics (EGM) Spectrum ................................................................. 95 Energy Density (General) ........................................................................................... 95 Engineered Metric ...................................................................................................... 95 Engineered Refractive Index “KEGM” .......................................................................... 95 Engineered Relationship Function “∆K0(ω,X)” ........................................................... 95 Experimental Prototype (EP)....................................................................................... 95 Experimental Relationship Function “K0(ω,X)” .......................................................... 95 Fourier Spectrum ........................................................................................................ 95 Frequency Bandwidth “∆ωPV” .................................................................................... 96 Frequency Spectrum ................................................................................................... 96 Fundamental Beat Frequency “∆ωδr(1,r,∆r,M)”........................................................... 96 Fundamental Harmonic Frequency “ωPV(1,r,M)” ........................................................ 96 General Modelling Equations (GMEx) ........................................................................ 96 General Relativity (GR) .............................................................................................. 96 General Similarity Equations (GSEx) .......................................................................... 96 Gravitons “γg”............................................................................................................. 96 Graviton Mass-Energy Threshold “mγg”...................................................................... 96 Group Velocity ........................................................................................................... 96 Harmonic Cut-Off Frequency “ωΩ”............................................................................. 96 Harmonic Cut-Off Function “Ω”................................................................................. 97 Harmonic Cut-Off Mode “nΩ” .................................................................................... 97 Harmonic Inflection Mode “NX”................................................................................. 97 Harmonic Inflection Frequency “ωX”.......................................................................... 97 Harmonic Inflection Wavelength “λX” ........................................................................ 97 Harmonic Similarity Equations (HSEx) ....................................................................... 97 IFF.............................................................................................................................. 97 Impedance Function.................................................................................................... 97 Kinetic Spectrum ........................................................................................................ 97 Mode Bandwidth ........................................................................................................ 97 Mode Number (Critical Boundary Mode) “nβ”............................................................ 97 Number of Permissible Modes “N∆r”........................................................................... 98 Phenomena of Beats.................................................................................................... 98 Photon Mass-Energy Threshold “mγ”.......................................................................... 98 Polarisable Vacuum (PV)............................................................................................ 98 Polarisable Vacuum (PV) Beat Bandwidth “∆ωΩ”....................................................... 98 Polarisable Vacuum (PV) Spectrum ............................................................................ 98 Potential Spectrum...................................................................................................... 98 Poynting Vector.......................................................................................................... 98 Precipitations .............................................................................................................. 98 Primary Precipitant ..................................................................................................... 98 Radii Calculations by Electro-Gravi-Magnetics (EGM) .............................................. 99 Range Factor “Stα” ..................................................................................................... 99 8

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2.2.67 2.2.68 2.2.69 2.2.70 2.2.71 2.2.72 2.2.73 2.2.74 2.2.75 2.2.76 2.2.77 2.2.78 2.2.79 2.2.80 2.2.81 2.2.82 2.2.83 2.2.84 2.2.85 2.2.86 2.2.87 2.2.88 2.2.89 2.2.90 2.2.91 2.2.92 2.2.93

Reduced Average Harmonic Similarity Equations (HSExA R) ...................................... 99 Reduced Harmonic Similarity Equations (HSEx R) ...................................................... 99 Refractive Index “KPV” ............................................................................................... 99 Representation Error “RError”....................................................................................... 99 RMS Charge Radii (General) ...................................................................................... 99 RMS Charge Radius of the Neutron “rν” ..................................................................... 99 Similarity Bandwidth “∆ωS” ....................................................................................... 99 Spectral Energy Density “ρ0(ω)”............................................................................... 100 Spectral Similarity Equations (SSEx) ........................................................................ 100 Subordinate Bandwidth............................................................................................. 100 Unit Amplitude Spectrum ......................................................................................... 100 Zero-Point-Energy (ZPE).......................................................................................... 100 Zero-Point-Field (ZPF) ............................................................................................. 100 Zero-Point-Field (ZPF) Spectrum ............................................................................. 100 Zero-Point-Field (ZPF) Beat Bandwidth “∆ωZPF”...................................................... 100 Zero-Point-Field (ZPF) Beat Cut-Off Frequency “ωΩ ZPF”......................................... 100 Zero-Point-Field (ZPF) Beat Cut-Off Mode “nΩ ZPF”................................................. 100 1st Sense Check “Stβ”................................................................................................ 100 2nd Reduction of the Harmonic Similarity Equations (HSExA R)................................. 100 2nd Sense Check “Stγ” ............................................................................................... 100 3rd Sense Check “Stδ”................................................................................................ 101 4th Sense Check “Stε”................................................................................................ 101 5th Sense Check “Stη” ............................................................................................... 101 6th Sense Check “Stθ”................................................................................................ 101 Physical Constants .................................................................................................... 101 Mathematical Constants and Symbols ....................................................................... 102 Solar System Statistics .............................................................................................. 102

2.3 Quinta Essentia – Part 4................................................................................................... 103 2.3.1 “Big-Bang”............................................................................................................... 103 2.3.2 Black-Hole “BH”...................................................................................................... 103 2.3.3 Broadband Propagation............................................................................................. 103 2.3.4 Buoyancy Point......................................................................................................... 103 2.3.5 CMBR Temperature “T0” ......................................................................................... 103 2.3.6 EGM-CMBR Temperature “TU” ............................................................................... 103 2.3.7 EGM Flux Intensity “CΩ_J” ....................................................................................... 103 2.3.8 EGM Hubble constant “HU”...................................................................................... 103 2.3.9 Event Horizon “RBH” ................................................................................................ 103 2.3.10 Galactic Reference Particle “GRP” ........................................................................... 103 2.3.11 Gravitational Interference ......................................................................................... 103 2.3.12 Gravitational Propagation ......................................................................................... 103 2.3.13 Hubble Constant “H0”............................................................................................... 104 2.3.14 Narrowband Propagation .......................................................................................... 104 2.3.15 Non-Physical ............................................................................................................ 104 2.3.16 Physical .................................................................................................................... 104 2.3.17 Primordial Universe.................................................................................................. 104 2.3.18 Schwarzschild-Black-Hole “SBH” ............................................................................ 104 2.3.19 Schwarzschild-Planck-Black-Hole “SPBH” .............................................................. 104 2.3.20 Schwarzschild-Planck-Particle .................................................................................. 104 2.3.21 Singularity ................................................................................................................ 104 2.3.22 Singularity Radius “rS” ............................................................................................. 104 2.3.23 Solar Mass ................................................................................................................ 104 9

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2.3.24 2.3.25 2.3.26 3

Super-Massive-Black-Hole “SMBH”........................................................................ 105 Total Mass-Energy.................................................................................................... 105 Astronomical / Cosmological statistics...................................................................... 105

Glossary of Terms ............................................................................................................. 107

3.1 Quinta Essentia – Part 3................................................................................................... 107 3.1.1 Acronyms ................................................................................................................. 107 3.1.2 Symbols in alphabetical order ................................................................................... 109 3.2 Quinta Essentia – Part 4................................................................................................... 116 3.2.1 Acronyms ................................................................................................................. 116 3.2.2 Symbols by chapter................................................................................................... 117 3.2.3 Symbols in alphabetical order ................................................................................... 121 4

Derivation Processes.......................................................................................................... 125

4.1 Main sequence................................................................................................................. 125 4.1.1 Characterisation of the gravitational spectrum........................................................... 125 4.1.2 Derivation of “Planck-Particle” and “Schwarzschild-Black-Hole” characteristics ..... 125 4.1.3 Fundamental Cosmology .......................................................................................... 125 4.1.4 Advanced Cosmology............................................................................................... 126 4.1.5 Gravitational Cosmology .......................................................................................... 126 4.1.6 Particle Cosmology................................................................................................... 126 4.2 The Hubble sequence....................................................................................................... 127 4.2.1 Preconditions ............................................................................................................ 127 4.2.2 Assumptions ............................................................................................................. 127 4.2.3 Simplified sequence.................................................................................................. 127 4.3 The CMBR temperature sequence ................................................................................... 128 4.3.1 Preconditions ............................................................................................................ 128 4.3.2 Assumptions ............................................................................................................. 128 4.3.3 Simplified sequence.................................................................................................. 128 5

Characterisation of the Gravitational Spectrum ............................................................. 129

Abstract ...................................................................................................................................... 129 5.1 Simplification of the EGM equations............................................................................... 130 5.1.1 “Ω → Ω1”, “nΩ → nΩ_1” and “ωΩ → ωΩ_1” ............................................................... 130 5.1.2 Computing errors ...................................................................................................... 130 5.2 Derivation of gravitational acceleration in terms of “ωΩ”................................................. 131 5.2.1 Transformation: “ωΩ_1 → ωΩ_2”................................................................................ 131 5.2.1.1 Simplification....................................................................................................... 131 5.2.1.2 Computing errors ................................................................................................. 132 5.2.1.2.1 “ωΩ_1 → ωΩ_2” ............................................................................................... 132 5.2.1.2.2 “g” ................................................................................................................. 132 5.2.1.3 Error analysis ....................................................................................................... 132 5.2.2 Transformation: “ωΩ_1 → ωΩ_3”................................................................................ 133 5.2.2.1 Simplification....................................................................................................... 133 10

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5.2.2.2

Computing errors ................................................................................................. 133

5.3

Formulation of a generalised cubic frequency expression in terms of “g”: “g → ωPV3” .... 134

5.4

Determination of the gravitationally dominant EGM frequency: “SωΩ → c⋅Um”............... 134

5.5 Derivation of EGM Flux Intensity “CΩ_J” ........................................................................ 135 5.5.1 Simplification ........................................................................................................... 135 5.5.2 Computing errors ...................................................................................................... 135 5.5.3 Error analysis............................................................................................................ 135 6

Derivation of “Planck-Particle” and “Schwarzschild-Black-Hole” Characteristics ...... 137

Abstract ...................................................................................................................................... 137 6.1 Derivation of the minimum physical “Schwarzschild-Planck-Particle” mass and radius... 138 6.1.1 What does “physical” mean?..................................................................................... 138 6.1.1.1 Conceptualisation................................................................................................. 138 6.1.1.2 Assumptions......................................................................................................... 138 6.1.1.3 Definitions ........................................................................................................... 139 6.1.1.3.1 Matter ............................................................................................................ 139 6.1.1.3.2 Energy density ............................................................................................... 139 6.1.1.3.3 Planck scale properties ................................................................................... 139 6.1.2 What is a “Schwarzschild-Planck-Particle”? ............................................................. 140 6.1.2.1 Conceptualisation................................................................................................. 140 6.1.2.2 Assumptions......................................................................................................... 140 6.1.2.3 Definition............................................................................................................. 140 6.1.3 Construct .................................................................................................................. 141 6.1.4 Computing errors ...................................................................................................... 143 6.1.5 Convergent and divergent spectra ............................................................................. 144 6.1.6 Honourable mention ................................................................................................. 144 6.1.7 Concluding remarks.................................................................................................. 144 6.1.7.1 Characteristics of a physical SPBH....................................................................... 144 6.1.7.2 Characteristics of a non-physical “Planck-Particle” .............................................. 144 6.1.7.3 Physicality of the EGM adjusted Planck Length ................................................... 145 6.2 Derivation of the value of the “KPV” at the event horizon of a SPBH ............................... 146 6.2.1 Synopsis ................................................................................................................... 146 6.2.2 Construct .................................................................................................................. 146 6.2.2.1 1st Formulation ..................................................................................................... 146 6.2.2.2 2nd Formulation .................................................................................................... 148 6.2.3 Concluding remarks.................................................................................................. 148 6.3 Derivation of “ωΩ” at the event horizon of a SPBH ......................................................... 149 6.3.1 Synopsis ................................................................................................................... 149 6.3.2 Assumptions ............................................................................................................. 149 6.3.3 Construct .................................................................................................................. 149 6.3.4 “ωPV(1,λxλh,mxmh)” .................................................................................................. 150 6.3.5 Honourable mention ................................................................................................. 151 6.3.6 Concluding remarks.................................................................................................. 151 6.4 Derivation of “ωΩ” at the event horizon of a SBH............................................................ 152 6.4.1 Synopsis ................................................................................................................... 152 11

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6.4.2 6.4.3 6.4.4 6.4.5

Assumptions ............................................................................................................. 152 Construct .................................................................................................................. 152 Sample calculations .................................................................................................. 152 Concluding remarks.................................................................................................. 152

6.5 Derivation of “rS” ............................................................................................................ 154 6.5.1 Synopsis ................................................................................................................... 154 6.5.2 Assumptions ............................................................................................................. 154 6.5.3 Construct .................................................................................................................. 154 6.5.3.1 1st Formulation ..................................................................................................... 154 6.5.3.2 2nd Formulation .................................................................................................... 155 6.5.3.3 3rd Formulation..................................................................................................... 155 6.5.4 Sample calculations .................................................................................................. 155 6.5.5 Honourable mention ................................................................................................. 156 6.5.6 Concluding remarks.................................................................................................. 156 6.6 “nΩ” and “ωΩ” profiles (as “r → RBH”) of SBH’s............................................................. 157 6.6.1 “nΩ”.......................................................................................................................... 157 6.6.1.1 Synopsis............................................................................................................... 157 6.6.1.2 Assumptions......................................................................................................... 157 6.6.1.3 Construct.............................................................................................................. 157 6.6.1.4 Sample calculations.............................................................................................. 157 6.6.1.5 Sample plots (log vs. log) ..................................................................................... 158 6.6.2 “ωΩ” ......................................................................................................................... 158 6.6.2.1 Synopsis............................................................................................................... 158 6.6.2.2 Assumptions......................................................................................................... 158 6.6.2.3 Construct.............................................................................................................. 159 6.6.2.4 Sample calculations.............................................................................................. 159 6.6.2.5 Sample plots (log vs. log) ..................................................................................... 160 6.6.3 “ωPV(1,r,MBH)” ......................................................................................................... 160 6.6.3.1 Synopsis............................................................................................................... 160 6.6.3.2 Assumptions......................................................................................................... 160 6.6.3.3 Construct.............................................................................................................. 161 6.6.3.4 Sample calculations.............................................................................................. 161 6.6.3.5 Sample plots (log vs. log) ..................................................................................... 161 6.6.3.6 Honourable mention............................................................................................. 162 6.6.4 Concluding remarks.................................................................................................. 162 6.7 Derivation of the minimum gravitational lifetime of matter “TL” ..................................... 163 6.7.1 Synopsis ................................................................................................................... 163 6.7.1.1 Fundamentals ....................................................................................................... 163 6.7.1.2 Assumptions......................................................................................................... 164 6.7.1.3 Sample calculations.............................................................................................. 164 6.7.2 Construct .................................................................................................................. 165 6.7.2.1 Reconciliation ...................................................................................................... 165 6.7.2.1.1 Dilemma ........................................................................................................ 165 6.7.2.1.2 Resolution...................................................................................................... 165 6.7.2.1.2.1 Uncertainty.............................................................................................. 165 6.7.2.1.2.2 Quasi-Uncertainty ................................................................................... 165 6.7.2.2 “TL” ..................................................................................................................... 165 6.7.2.2.1 Fundamentals ................................................................................................. 165 6.7.2.2.2 Sample calculations........................................................................................ 166 12

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6.7.2.2.3 Simplifications ............................................................................................... 167 6.7.3 Concluding remarks.................................................................................................. 167 6.8 Derivation of the average emission frequency per Graviton “ωg”..................................... 168 6.8.1 Synopsis ................................................................................................................... 168 6.8.2 Assumptions ............................................................................................................. 168 6.8.3 Construct .................................................................................................................. 168 6.8.4 Sample calculations .................................................................................................. 169 6.8.5 Concluding remarks.................................................................................................. 169 6.9 Why can't we observe BH’s? ........................................................................................... 170 6.9.1 Synopsis ................................................................................................................... 170 6.9.2 Assumptions ............................................................................................................. 170 6.9.3 Construct .................................................................................................................. 171 6.9.4 Sample calculations .................................................................................................. 171 6.9.4.1 SBH’s .................................................................................................................. 171 6.9.4.2 SPBH’s ................................................................................................................ 172 6.9.5 Concluding remarks.................................................................................................. 173 7

Fundamental Cosmology................................................................................................... 175

Abstract ...................................................................................................................................... 175 7.1 Derivation of the primordial and present Hubble constants “Hα, HU”............................... 176 7.1.1 Synopsis ................................................................................................................... 176 7.1.2 Assumptions ............................................................................................................. 176 7.1.3 Construct .................................................................................................................. 177 7.1.3.1 “AU, RU, HU”........................................................................................................ 177 7.1.3.2 “Hα”..................................................................................................................... 178 7.1.3.3 “ρU” ..................................................................................................................... 179 7.1.3.4 “MU” .................................................................................................................... 179 7.1.4 Concluding remarks.................................................................................................. 179 7.2 Derivation of the CMBR temperature “TU”...................................................................... 180 7.2.1 Synopsis ................................................................................................................... 180 7.2.2 Assumptions ............................................................................................................. 180 7.2.3 Construct .................................................................................................................. 181 7.2.4 Concluding remarks.................................................................................................. 181 7.3 Numerical solutions for “Hα, AU, RU, ρU, MU, HU” and “TU” ........................................... 182 7.3.1 “r2, r3, M2, M3” ......................................................................................................... 182 7.3.2 Computational results ............................................................................................... 183 7.3.3 Honourable mention ................................................................................................. 183 7.3.4 Concluding remarks.................................................................................................. 184 7.4 Determination of the impact of “Dark Matter / Energy” on “HU” and “TU”...................... 185 7.4.1 Synopsis ................................................................................................................... 185 7.4.2 Assumptions ............................................................................................................. 185 7.4.3 Construct .................................................................................................................. 185 7.4.4 Concluding remarks.................................................................................................. 185 7.5

“TU” as a function of a generalised Hubble constant “TU → TU2” .................................... 186 13

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7.6 Derivation of “Ro”, “MG”, “HU2” and “ρU2” from “TU2”................................................... 187 7.6.1 Synopsis ................................................................................................................... 187 7.6.2 Assumptions ............................................................................................................. 187 7.6.3 Construct .................................................................................................................. 188 7.6.3.1 “Ro” or “MG”........................................................................................................ 188 7.6.3.2 “Ro” and “MG” ..................................................................................................... 189 7.6.3.3 “Ro”, “MG”, “HU2” and “ρU2” ............................................................................... 190 7.6.3.3.1 “Ro” and “MG” ............................................................................................... 190 7.6.3.3.2 “HU2” and “ρU2” ............................................................................................. 190 7.6.4 Concluding remarks.................................................................................................. 191 7.7 Experimentally implicit derivation of the ZPF energy density threshold “UZPF”............... 192 7.7.1 Synopsis ................................................................................................................... 192 7.7.2 Assumptions ............................................................................................................. 192 7.7.3 Construct .................................................................................................................. 192 7.7.4 Concluding remarks.................................................................................................. 193 8

Advanced Cosmology ........................................................................................................ 195

Abstract ...................................................................................................................................... 195 8.1 Time dependent CMBR temperature “TU2 → TU3” .......................................................... 196 8.1.1 Synopsis ................................................................................................................... 196 8.1.2 Assumptions ............................................................................................................. 196 8.1.3 Construct .................................................................................................................. 196 8.1.4 Sample calculations .................................................................................................. 197 8.1.5 Sample plots ............................................................................................................. 197 8.1.5.1 “TU3 vs. Hβ”: Figure 4.22 ..................................................................................... 198 8.1.5.2 “TU3 vs. t = (HβHα)-1” (i): Figure 4.23................................................................... 199 8.1.6 Honourable mention ................................................................................................. 200 8.1.7 Concluding remarks.................................................................................................. 200 8.2 Rates of change of CMBR temperature “TU3 → TU4 → d1,2,3TU4/dt1,2,3” ........................... 201 8.2.1 Synopsis ................................................................................................................... 201 8.2.2 Assumptions ............................................................................................................. 201 8.2.3 Construct .................................................................................................................. 201 8.2.4 Sample calculations .................................................................................................. 202 8.2.5 Sample plots ............................................................................................................. 203 8.2.5.1 “TU3 vs. t = (HβHα)-1” (ii): Figure 4.24.................................................................. 204 8.2.5.2 “TU3 vs. t = (HβHα)-1” (iii): Figure 4.25................................................................. 205 8.2.5.3 “TU3 vs. H = (HβHα)” (i): Figure 4.26 ................................................................... 206 8.2.5.4 “TU3 vs. H = (HβHα)” (ii): Figure 4.27 .................................................................. 207 8.2.5.5 “TU3 vs. r = (HβHα)-1c” (i): Figure 4.28................................................................. 208 8.2.5.6 “TU3 vs. r = (HβHα)-1c” (ii): Figure 4.29................................................................ 209 8.2.5.7 “dTU4/dt vs. t = (HβHα)-1” (i): Figure 4.30............................................................. 210 8.2.5.8 “dTU4/dt vs. t = (HβHα)-1” (ii): Figure 4.31............................................................ 211 8.2.5.9 “d2TU4/dt2 vs. t = (HβHα)-1” (i): Figure 4.32 .......................................................... 212 8.2.5.10 “d2TU4/dt2 vs. t = (HβHα)-1” (ii): Figure 4.33 ......................................................... 213 8.2.5.11 “|d3TU4/dt3| vs. t = (HβHα)-1” (i): Figure 4.34......................................................... 214 8.2.5.12 “|d3TU4/dt3| vs. t = (HβHα)-1” (ii): Figure 4.35 ....................................................... 215 8.2.6 Concluding remarks.................................................................................................. 216 14

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8.3 Rates of change of the Hubble constant “d1,2H/dt1,2”........................................................ 217 8.3.1 Synopsis ................................................................................................................... 217 8.3.2 Assumptions ............................................................................................................. 217 8.3.3 Construct .................................................................................................................. 217 8.3.4 Sample calculations .................................................................................................. 220 8.3.5 Construct errors ........................................................................................................ 222 8.3.5.1 How can they be determined?............................................................................... 222 8.3.5.2 Analytical............................................................................................................. 222 8.3.5.3 Graphical ............................................................................................................. 223 8.3.5.4 Numerical ............................................................................................................ 223 8.3.5.4.1 General case................................................................................................... 223 8.3.5.4.2 Specific case .................................................................................................. 223 8.3.6 Sample plots ............................................................................................................. 224 8.3.6.1 “dH/dt vs. (HβηHα)-1” (i): Figure 4.36 ................................................................... 225 8.3.6.2 “dH/dt vs. (HβηHα)-1” (ii): Figure 4.37.................................................................. 226 8.3.6.3 “dH/dt vs. (HβηHα)-1” (iii): Figure 4.38................................................................. 227 8.3.6.4 “dH/dt vs. (HβηHα)-1” (iv): Figure 4.39 ................................................................. 228 8.3.6.5 “d2H/dt2 vs. (HβηHα)-1” (i): Figure 4.40 ................................................................ 229 8.3.6.6 “d2H/dt2 vs. (HβηHα)-1” (ii): Figure 4.41 ............................................................... 230 8.3.6.7 “d2H/dt2 vs. (HβηHα)-1” (iii): Figure 4.42 .............................................................. 231 8.3.6.8 “d2H/dt2 vs. (HβηHα)-1” (iv): Figure 4.43............................................................... 232 8.3.6.9 “d2H/dt2 vs. (HβηHα)-1” (v): Figure 4.44................................................................ 233 8.3.6.10 “d2H/dt2 vs. (HβηHα)-1” (vi): Figure 4.45............................................................... 234 8.3.6.11 “|H| vs. (HβηHα)-1” (i): Figure 4.46 ....................................................................... 235 8.3.6.12 “|H| vs. (HβηHα)-1” (ii): Figure 4.47 ...................................................................... 236 8.3.6.13 “TU2,3 vs. |H|”: Figure 4.48 ................................................................................... 237 8.3.6.14 “TU2 vs. |H|”: Figure 4.49 ..................................................................................... 238 8.3.7 Concluding remarks.................................................................................................. 239 8.4

Cosmological evolution process ...................................................................................... 241

8.5 History of the Universe.................................................................................................... 243 8.5.1 According to the Standard Model (SM)..................................................................... 243 8.5.1.1 Graphical representation (i) .................................................................................. 243 8.5.1.2 Graphical representation (ii) ................................................................................. 244 8.5.1.3 Graphical representation (iii) ................................................................................ 245 8.5.1.4 Graphical representation (iv) ................................................................................ 246 8.5.2 According to EGM.................................................................................................... 247 8.6 EGM Cosmological construct limitations ........................................................................ 248 8.6.1 Synopsis ................................................................................................................... 248 8.6.2 Assumptions ............................................................................................................. 248 8.6.3 Construct .................................................................................................................. 248 8.6.3.1 The mass limit “ML” ............................................................................................ 248 8.6.3.2 The size limit “rL” ................................................................................................ 248 8.6.3.3 The age limit “tL” ................................................................................................. 249 8.6.4 Boundary ratio .......................................................................................................... 249 8.6.5 Sample calculations .................................................................................................. 249 8.6.6 Concluding remarks.................................................................................................. 249 8.7 Are conventional radio telescopes, practical tools for gravitational astronomy? ............... 250 8.7.1 Synopsis ................................................................................................................... 250 15

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8.7.2 8.7.3 8.7.4 8.7.5 9

Assumptions ............................................................................................................. 250 Construct .................................................................................................................. 250 Sample calculations .................................................................................................. 251 Concluding remarks.................................................................................................. 251

Gravitational Cosmology .................................................................................................. 253

Abstract ...................................................................................................................................... 253 9.1 Gravitational propagation: the mechanism for interaction ................................................ 254 9.1.1 Synopsis ................................................................................................................... 254 9.1.2 Construct .................................................................................................................. 255 9.1.2.1 Broadband............................................................................................................ 255 9.1.2.2 Narrowband ......................................................................................................... 255 9.1.3 Testing...................................................................................................................... 257 9.1.3.1 Newtonian............................................................................................................ 257 9.1.3.2 Relativistic ........................................................................................................... 257 9.1.3.3 PV........................................................................................................................ 258 9.1.4 Concluding remarks.................................................................................................. 258 9.2 Gravitational interference: the mechanism of interaction ................................................. 259 9.2.1 Synopsis ................................................................................................................... 259 9.2.2 Construct .................................................................................................................. 259 9.2.2.1 Broadband............................................................................................................ 259 9.2.2.2 Narrowband ......................................................................................................... 260 9.2.3 Concluding remarks.................................................................................................. 262 9.2.3.1 Broadband............................................................................................................ 262 9.2.3.2 Narrowband ......................................................................................................... 262 10

Particle Cosmology........................................................................................................ 263

Abstract ...................................................................................................................................... 263 10.1

Derivation of the Photon and Graviton mass-energies lower limit.................................... 264

10.2

Derivation of the Photon and Graviton RMS charge radii lower limit .............................. 264

10.3

Derivation of the Photon charge threshold ....................................................................... 264

10.4

Derivation of the Photon charge upper limit..................................................................... 265

10.5

Derivation of the Photon charge lower limit..................................................................... 266

10.6

Other useful relationships ................................................................................................ 266

11

Equation Summary ....................................................................................................... 269

11.1 Gravitation ...................................................................................................................... 269 11.1.1 “Stg” ......................................................................................................................... 269 11.1.2 “ωΩ_2”....................................................................................................................... 269 11.1.3 “aEGM_ωΩ” ................................................................................................................. 269 11.1.4 “StG”......................................................................................................................... 269 11.1.5 “ωΩ_3”....................................................................................................................... 269 16

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11.1.6 11.1.7 11.1.8 11.1.9 11.1.10 11.1.11 11.1.12 11.1.13 11.1.14 11.1.15 11.1.16 11.1.17 11.1.18 11.1.19 11.1.20

“λΩ_3” ....................................................................................................................... 269 “G” ........................................................................................................................... 269 “ωPV(nPV,r,M)3” ........................................................................................................ 269 “StJ”.......................................................................................................................... 269 “CΩ_J1, CΩ_Jω” ........................................................................................................... 269 “nΩ_2” ....................................................................................................................... 270 “KDepp” ..................................................................................................................... 270 “KPV”........................................................................................................................ 270 “TL”.......................................................................................................................... 270 “ωg”.......................................................................................................................... 270 “ngg” ......................................................................................................................... 270 “rω”........................................................................................................................... 270 “aPV”......................................................................................................................... 270 “ag”........................................................................................................................... 270 “gav” ......................................................................................................................... 270

11.2 Planck-Particles............................................................................................................... 270 11.2.1 “mx”.......................................................................................................................... 270 11.2.2 “λx” .......................................................................................................................... 270 11.2.3 “ρm, ρS” .................................................................................................................... 271 11.2.4 “r3, M3”..................................................................................................................... 271 11.3 SBH’s.............................................................................................................................. 271 11.3.1 “StBH” ....................................................................................................................... 271 11.3.2 “ωΩ_4”....................................................................................................................... 271 11.3.3 “rS” ........................................................................................................................... 271 11.3.4 “nΩ_4” ....................................................................................................................... 271 11.3.5 “nΩ_5” ....................................................................................................................... 271 11.3.6 “nBH” ........................................................................................................................ 271 11.3.7 “ωΩ_5”....................................................................................................................... 271 11.3.8 “ωBH”........................................................................................................................ 271 11.3.9 “ωΩ_6”....................................................................................................................... 271 11.3.10 “ωΩ_7”....................................................................................................................... 271 11.3.11 “ωPV_1” ..................................................................................................................... 272 11.3.12 “ng”........................................................................................................................... 272 11.4 Cosmology ...................................................................................................................... 272 11.4.1 “r2, M2”..................................................................................................................... 272 11.4.2 “λy” .......................................................................................................................... 272 11.4.3 “KU” ......................................................................................................................... 272 11.4.4 “AU” ......................................................................................................................... 272 11.4.5 “RU” ......................................................................................................................... 272 11.4.6 “HU, HU2, HU5, |H|” ................................................................................................... 272 11.4.7 “Hα” ......................................................................................................................... 272 11.4.8 “ρU, ρU2”................................................................................................................... 273 11.4.9 “MU”......................................................................................................................... 273 11.4.10 “KT” ......................................................................................................................... 273 11.4.11 “TW” ......................................................................................................................... 273 11.4.12 “StT” ......................................................................................................................... 273 11.4.13 “TU, TU2, TU3, TU4, TU5” ............................................................................................ 273 11.4.14 “dTdt, dT2dt2, dT3dt3”................................................................................................. 274 17

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11.4.15 “dHdt, dH2dt2” ........................................................................................................... 274 11.4.16 “t1, t2, t3, t4, t5” .......................................................................................................... 274 11.5 ZPF ................................................................................................................................. 275 11.5.1 “ΩEGM” ..................................................................................................................... 275 11.5.2 “ΩZPF” ...................................................................................................................... 275 11.5.3 “UZPF”....................................................................................................................... 275 11.6 EGM Construct limits...................................................................................................... 275 11.6.1 “ML”......................................................................................................................... 275 11.6.2 “rL” ........................................................................................................................... 275 11.6.3 “tL” ........................................................................................................................... 275 11.6.4 “ML / rL = MEGM / REGM = tL / tEGM” .......................................................................... 275 11.7 Particle-Physics ............................................................................................................... 275 11.7.1 “mγγ2” ....................................................................................................................... 275 11.7.2 “mgg2” ....................................................................................................................... 275 11.7.3 “rγγ2” ......................................................................................................................... 275 11.7.4 “rgg2”......................................................................................................................... 276 11.7.5 “Nγ” .......................................................................................................................... 276 11.7.6 “Qγ” .......................................................................................................................... 276 11.7.7 “Qγγ”......................................................................................................................... 276 11.7.8 “Qγγ2”........................................................................................................................ 276 11.7.9 “tL / TL = mγγ / mγγ2 = Qγγ / Qγγ2”................................................................................ 276 11.8

Other useful relationships ................................................................................................ 276

11.9

Quick symbol guide......................................................................................................... 277

APPENDIX 4.A......................................................................................................................... 281 Thermodynamic “Π” Groupings of BH’s .................................................................................... 281 Conventional calculation of SPBH temperature “TBH” ................................................................ 281 “TU2 : TBH”.................................................................................................................................. 282 Approximations of “TU2(t1-1)” ..................................................................................................... 282 • “1st” Form................................................................................................................. 282 • “2nd” Form ................................................................................................................ 284 Approximation of “λx” in terms of physical constants ................................................................. 284 Physical interpretation of “λx”..................................................................................................... 285 Bibliography 4........................................................................................................................... 287 APPENDIX 4.B ......................................................................................................................... 291 Quinta Essentia – Part 3 .............................................................................................................. 291 • MathCad 8 Professional: calculation engine .............................................................. 291 a. Computational environment ........................................................................................ 291 18

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b. c. e. f. g. h. i. j. k. l. m. n. o. p. q. r. • a. b. c. d. e.

Units of measure (definitions) ..................................................................................... 291 Constants (definitions) ................................................................................................ 292 Planck characteristics (definitions) .............................................................................. 293 Astronomical statistics ................................................................................................ 293 Other........................................................................................................................... 293 Arbitrary values for illustrational purposes.................................................................. 293 PV / ZPF equations ..................................................................................................... 294 Casimir equations........................................................................................................ 295 Fundamental particle equations ................................................................................... 296 Particle summary matrix 3.1........................................................................................ 300 Particle summary matrix 3.2........................................................................................ 301 Particle summary matrix 3.3........................................................................................ 302 Particle summary matrix 3.4........................................................................................ 303 Similarity equations .................................................................................................... 304 Calculation results....................................................................................................... 305 Resonant Casimir cavity design specifications (experimental)..................................... 313 MathCad 12: High precision calculation results ........................................................ 315 Computational environment ........................................................................................ 315 Particle summary matrix 3.1........................................................................................ 315 Particle summary matrix 3.2........................................................................................ 316 Particle summary matrix 3.3........................................................................................ 317 Particle summary matrix 3.4........................................................................................ 318

Quinta Essentia – Part 4 .............................................................................................................. 321 • MathCad 8 Professional ............................................................................................ 321 a. Complete simulation ................................................................................................... 321 i. Computational environment.................................................................................... 321 ii. Units of measure (definitions)................................................................................. 321 iii. Constants (definitions)............................................................................................ 321 iv. Astronomical statistics............................................................................................ 321 v. Characterisation of the gravitational spectrum ........................................................ 321 1. “Ω → Ω1, nΩ → nΩ_1, ωΩ → ωΩ_1”....................................................................... 321 2. “g → ωΩ” ............................................................................................................. 323 i. “ωΩ_1 → ωΩ_2” ................................................................................................. 323 ii. “ωΩ_1 → ωΩ_3” ................................................................................................. 325 3. “g → ωPV3” .......................................................................................................... 325 4. “SωΩ → c⋅Um” ...................................................................................................... 326 5. “CΩ_J” .................................................................................................................. 326 vi. Derivation of “Planck-Particle” and SBH characteristics ........................................ 327 1. “λx, mx”................................................................................................................ 327 2. “ρm(λxλh,mxmh), Um(λxλh,mxmh)”......................................................................... 329 3. Physicality of “Kλ”............................................................................................... 329 4. “KPV @ λxλh”....................................................................................................... 329 i. “KPV = Undefined”........................................................................................... 329 ii. “KDepp = KPV”................................................................................................... 331 5. “ωΩ_3” .................................................................................................................. 332 6. “ωΩ_4” .................................................................................................................. 333 7. “rS”....................................................................................................................... 334 i. “rS(λxλh)” ......................................................................................................... 334 ii. “rS(ΜΒΗ), rS(RΒΗ)” ........................................................................................... 334 iii. “MBH(rS)” ........................................................................................................ 335 19

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“r → RBH” ............................................................................................................ 336 i. “nΩ → nΩ_4, nΩ_5, nBH” ..................................................................................... 336 ii. “ωΩ → ωΩ_5, ωBH” ........................................................................................... 337 iii. “ωΩ_6, ωΩ_7, ωPV_1” .......................................................................................... 338 9. “TL” ..................................................................................................................... 339 10. “ωg, ngg”............................................................................................................... 340 11. BH’s..................................................................................................................... 341 vii. Fundamental Cosmology ........................................................................................ 343 1. “Hα, HU” .............................................................................................................. 343 i. “AU, RU, HU”.................................................................................................... 343 ii. “Hα”................................................................................................................. 344 iii. “ρU”................................................................................................................. 344 iv. “MU”................................................................................................................ 345 2. “TU” ..................................................................................................................... 345 3. “TU → TU2”.......................................................................................................... 346 4. “TU2 → Ro, MG, HU2, ρU2” .................................................................................... 347 5. “UZPF” .................................................................................................................. 349 viii. Advanced Cosmology............................................................................................. 349 1. “nΩ_2 → nΩ_6”....................................................................................................... 349 2. “KU2 → KU3”........................................................................................................ 350 3. “HU2 → HU3, TU2 → TU3” ..................................................................................... 350 4. “HU3 → HU4, TU3 → TU4” ..................................................................................... 350 5. “HU4 → HU5, TU4 → TU5” ..................................................................................... 350 6. “HU3, HU4, HU5, TU3, TU4, TU5” ............................................................................. 351 7. Time dependent characteristics............................................................................. 351 8. History of the Universe ........................................................................................ 360 9. “ML, rL, tL, tEGM” .................................................................................................. 361 10. Radio astronomy .................................................................................................. 362 ix. Gravitational Cosmology........................................................................................ 363 x. Particle Cosmology ................................................................................................ 365 b. Calculation engine ...................................................................................................... 367 i. Computational environment.................................................................................... 367 ii. Standard relationships............................................................................................. 367 iii. Derived constants ................................................................................................... 367 iv. Base approximations / simplifications..................................................................... 368 v. SBH mass and radius.............................................................................................. 369 vi. “nΩ”........................................................................................................................ 370 vii. “ωΩ, TΩ, λΩ”........................................................................................................... 371 viii. Gravitation ............................................................................................................. 373 ix. Flux intensity.......................................................................................................... 374 x. Photon and Graviton populations ............................................................................ 376 xi. Hubble constant and CMBR temperature................................................................ 377 xii. SBH temperature .................................................................................................... 384 xiii. ZPF ........................................................................................................................ 385 xiv. Cosmological limits................................................................................................ 386 xv. Particle Cosmology ................................................................................................ 386 • MathCad 12 .............................................................................................................. 389 c. High precision calculation engine................................................................................ 389 i. Computational environment.................................................................................... 389 ii. Astronomical statistics............................................................................................ 389 iii. Derived constants ................................................................................................... 389 8.

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iv. d.

Algorithm............................................................................................................... 389 Various forms of the derived constants........................................................................ 390

Periodic Table of the Elements ................................................................................................. 391 Notes

22, 24, 56, 78, 92, 102, 105, 106, 108, 115, 116, 120, 124, 127, 128, 136, 145, 151, 153, 167, 173, 174, 179, 184, 191, 194, 200, 216, 240, 242, 251, 252, 267, 268, 279, 280, 286, 290, 314, 319, 320, 366, 388, 390, 392 ERRATA

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Scientific Achievements The physical characteristics derived herein (from 1st principles), based upon a single paradigm [i.e. the application of Buckingham “Π” Theory (BPT) and Dimensional Analysis Techniques (DAT’s)] may be articulated as follows (many of which are experimentally verified or implied), • Astro-Physics 1. Derivation of the minimum physical “Schwarzschild-Black-Hole” (SBH) mass and radius. 2. Derivation of maximum permissible energy density. 3. Derivation of the harmonic mode and frequency characteristics and profiles of a SBH. 4. Derivation of the SBH singularity radius. • Cosmology • General 5. Derivation of the minimum gravitational lifetime of matter. 6. Derivation of the present Cosmological age. 7. Derivation of the present Cosmological size. 8. Derivation of the total Cosmological mass. 9. Derivation of the present Cosmological mass-density. • Hubble constant 10. Derivation of the Hubble constant at the instant of the “Big-Bang”. 11. Derivation of the maximum Hubble constant since the “Big-Bang”. 12. Derivation of the present Hubble constant within experimental tolerance. 13. Derivation of the Hubble constant in the time domain. 14. Derivation of the rates of change of the Hubble constant in the time domain. • Cosmic Microwave Background Radiation (CMBR) temperature 15. Derivation of the CMBR temperature at the instant of the “Big-Bang”. 16. Derivation of the maximum Cosmological temperature since the “Big-Bang”. 17. Derivation of the present CMBR temperature within experimental tolerance. 18. Derivation of the CMBR temperature in the time domain. 19. Derivation of the rates of change of the CMBR temperature in the time domain. • Evolutionary processes 20. Categorisation of the Cosmological evolution process into two regimes: comprised of four distinct periods. 21. Determination of the impact of “Dark Matter / Energy” on the Hubble constant and CMBR temperature. 22. Articulation of the precise history of the Universe. • Cosmological constant 23. Experimentally implicit derivation of the Zero-Point-Field (ZPF) energy density threshold, yielding an insight into the Cosmological constant. • Particle-Physics 24. Derivation of the Photon and Graviton mass-energies lower limit. 25. Derivation of the Photon and Graviton Root-Mean-Square (RMS) charge radii lower limit. 26. Derivation of the Photon charge threshold. 27. Derivation of the Photon charge upper and lower limits. Note: where possible, calculated results have been compared to physical measurement. Cognisant of experimental uncertainty, key predictions herein may be considered to be exact.

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1 Introduction 1.1 The natural philosophy of the Cosmos 1.1.1 Objectives and scope2 The present Hubble constant “H0” and Cosmic Microwave Background Radiation (CMBR) temperature “T0” denote two of the most important Cosmological phenomenon to have been identified in the last hundred years and may hold significant insight into the natural philosophy of the Cosmos. Experimental measurements of “H0” and “T0” are advancing dramatically and have raised some important aspects regarding the nature of the Cosmological evolution process. This text is a companion to “Quinta Essentia – Part 3”, applying a method termed ElectroGravi-Magnetics (EGM). [1-19] Storti et. Al. derived the EGM construct, utilising Dimensional Analysis Techniques (DAT’s) and Buckingham “Π” Theory (BPT)3, to represent fundamental particles in harmonic form to high computational precision in favourable agreement with the Standard Model (SM) of Particle-Physics and experimental measurement. One of the key findings was that, at a fundamental physical level, mass-energy is distributed over space-time in only one manner4. The EGM construct has been re-applied to Cosmology with the following derivational objectives (within experimental tolerance where applicable): i. The Hubble constant (see: Ch. 7.1, 7.3, 7.6, 8.3). ii. The CMBR temperature (see: Ch. 7.2, 7.3, 7.5, 8.1, 8.2). iii. The ZPF energy density threshold (see: Ch. 7.7). iv. The Cosmological evolution process (see: Ch. 8.4). v. The history of the Universe (see: Ch. 8.5). 1.1.2 How are these objectives achieved? 1.1.2.1 Synopsis The primary tool employed to achieve our objectives is similitude5, subject to the following simplified constraints (see: Ch. 6.1, 7.1 – 7.3), i. The Cosmos at an instant prior to the “Big-Bang” is termed the “Primordial Universe”. It was characterised by a single wavefunction with maximum permissible energy density distributed homogeneously, analogous to a Planck scale particle of radius “λxλh” and mass “mxmh” such that it was dynamically, kinematically and geometrically similar to a “Schwarzschild-Black-Hole” (SBH). ii. The relationship between the “Primordial Universe” and its present visible size obeys the EGM harmonic representation of fundamental particles. iii. The “Milky-Way” (MW) Galaxy may be represented as a Planck scale particle of homogeneous energy density and equivalent total mass. This configuration has been termed the Galactic Reference Particle (GRP), such that dynamic, kinematic and geometric similarity exists between the “Primordial Universe” and the GRP.

2

At the time of commencement of formulation of this text, only the National Institute of Standards and Technology (NIST) 2002 data was available. Subsequently, the NIST 2006 values for physical constants may differ slightly, but do not change any computed or predicted results herein, or in “Quinta Essentia – Part 3”, by any significant measure. 3 Refer to the many standard texts relating to DAT’s and BPT. 4 In accordance with Zero-Point-Field equilibria. 5 A reference to DAT’s and BPT. 25

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1.1.2.2 Derivation process 1.1.2.2.1 Hubble constant “HU” i. Utilising harmonic cut-off frequency in “ωΩ_3” form (see: Ch. 5.2.2), derive an expression for EGM Flux Intensity “CΩ_J1”: (see: Ch. 5.5), Output: St J

C Ω_J1( r , M )

9

. M

2

5

8

r

r

(4.52)

where, 9 .c . St G 4 .π 4

St J

St G

• • •

3.

3 .ω h 4 .π .h

2 9

(4.51) 2

. c 2

9

(4.35)

“c = 299792458(m/s)”. “h = 6.6260693 x10-34(Js)”. “ωh = 1 / th = 1 / √(Gh/c5), G = 6.6742 x10-11(m3kg-1s-2)”.

ii. Derive an expression for the minimum gravitational lifetime of matter “TL”: (see: Ch. 6.7.2.2), Output: TL

h m γγ

(4.196)

where, • “mγγ” denotes the mass-energy of a Photon defined in [8,10]. • “mγγ = 3.195095 x10-45(eV)”. iii. Derive an expression for the EGM Hubble constant “HU” utilising the EGM harmonic representation of fundamental particles: (see: Ch. 7.1), Output: λ y r 2, M 2

K U r 2, r 3, M 2, M 3

ln

1 ln n Ω_2 r 2 , M 2

(4.229)

λ y r 2, M 2 .M C Ω_J1 λ y r 2 , M 2 .r 3 , 3 2 C Ω_J1 r 2 , M 2

A U r 2, r 3, M 2, M 3

H U r 2, r 3, M 2, M 3

(4.231)

TL K U r 2, r 3, M 2, M 3

5

(4.233)

1 A U r 2, r 3, M 2, M 3

(4.235)

where, “nΩ_2” denotes the non-refractive form of “nΩ” defined in [4]. 26

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1.1.2.2.2 CMBR temperature “TU” iv. Derive an expression for the average number of Gravitons “ng” radiated by a SBH at frequency “ω”: (see: Ch. 6.7.1.1), Output: n g ω , M BH

E M BH E g( ω )

(4.177)

where, “MBH” denotes SBH mass. v. Derive an expression for the value of the EGM Hubble constant at the instant of the “BigBang”, termed the primordial Hubble constant “Hα”: (see: Ch. 7.1.3.2), Output: H α r 3, M 3

2.

2. . . π G ρ m r 3, M 3 3

(4.237)

vi. Derive an expansive scaling factor “KT” incorporating “ng”, “Hα” and “HU”: (see: Ch. 7.2.3), Output: K T r 2, r 3, M 2, M 3

n g ω Ω_3 r 3 , M 3 , M 3 .ln

H α r 3, M 3 H U r 2, r 3 , M 2, M 3

(4.240)

where, “ωΩ_3” has a generalised definition according to, 9

ω Ω_3( r , M )

2

M St G. 5 r

(4.36)

vii. Derive a thermodynamic scaling factor “TW” incorporating Wien’s displacement constant “KW” and EGM wavelength of the form “λΩ_3”: (see: Ch. 7.2.3), Output: T W r 2, r 3, M 2 , M 3

KW λ Ω_3 R U r 2 , r 3 , M 2 , M 3 , M 3

(4.241)

where, “ωΩ_3(r,M) → ωΩ_3(RU(r2,M2,r3,M3),M3)” “λΩ_3(RU(r2,M2,r3,M3),M3) = c / ωΩ_3(RU(r2,M2,r3,M3),M3)” R U r 2, r 3 , M 2, M 3

•

c .A U r 2 , r 3 , M 2 , M 3

(4.234)

“KW = 2.8977685 x10-3(mK)”.

viii. Derive an expression for EGM Cosmological temperature “TU” utilising “KT” and “TW”: (see: Ch. 7.2.3), Output: T U r 2, r 3, M 2, M 3

K T r 2 , r 3 , M 2 , M 3 .T W r 2 , r 3 , M 2 , M 3

27

(4.242)

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1.1.2.2.3 “HU → HU2, TU → TU2 → TU3” ix. Derive the minimum physical dimensions of mass and radius for a SBH with maximum permissible energy density at the Planck scale: (see: Ch. 6.1.3), Output: mx

λx

λx 2

(4.71)

4 . 2 π 3

6

(4.72)

Hence, the dimensions of a SBH at maximum permissible energy density at the Planck scale is given by, • “Mass = mxmh” when “mh = √(hc/G)”. • “Radius = λxλh” when “λh = √(Gh/c3)”. x. Assume that the “Primordial Universe” (i.e. the Universe instantaneously prior to the “Big-Bang”) is analogous to a SBH of Planck scale dimensions at a condition of maximum permissible energy density, with radius “r3 = λxλh” and mass “M3 = mxmh = λxmh / 2”: (see: Ch. 7.3.1), xi. Formulate generalised expressions for “r2” and “M2” incorporating the EGM adjusted Planck length and mass: (see: Ch. 7.3.1), Output: r2(r) = Kλ⋅r

(4.247)

M2(M) = Km⋅M = Kλ⋅M

(4.248)

where, “Kλ = Km = [π / 2](1 / 3) ≈ 1.162447” as defined in [13]. xii. Simplify “ng”: (see: Ch. 7.3.1, 7.6), Output: For “r3 = λxλh” and “M3 = mxmh = λxmh / 2”: “ng[ω,MBH] = ng[ωΩ_3(r3,M3),M3] = 8 / 3”. xiii. Simplify “Hα”: (see: Ch. 7.3.2), Output: H α λ x.λ h , m x.m h

ωh λx

(4.249)

For brevity in future applications, let: “Hα = ωh / λx”. xiv. Transform “HU” to “HU2”: (see: Ch. 7.6.1), Output: H U2( r , M )

H U K λ .r , λ x.λ h , K m.M , m x.m h

28

(4.276)

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xv. Transform “TU” to “TU2”: (see: Ch. 7.5), Output: K W .St T .ln

T U2( H )

ωh λ x.H

9

. H5

(4.275)

where, “H” denotes a generalised reference to Hubble constant and “StT” is a constant according to. 9

4. 3. 1 . λ x 3 4 c5 π .λ 2 h 3

St T

2

(4.274)

xvi. Transform “TU2” to “TU3”: (see: Ch. 8.1.3), Output: T U3 H β

K W .St T .ln

1 Hβ

. H .H β α

5 .µ

2

(4.318)

where, “µ = 1 / 3” and “Hβ” denotes a dimensionless range variable such that “1 ≥ Hβ > 0”. xvii. Select values of “r” and “M” for application to “r2(r), M2(M)” utilising the following measures: (see: Ch. 7.3.2): Input: i. ii. iii. iv. v. vi.

“r = Ro” denotes the mean distance from the Sun to the MW Galactic centre. “Ro = 8(kpc)” as defined by the PDG [20] (“kpc” = kilo-parsec). “M = MG” denotes the total mass (i.e. visible + dark) of the MW Galaxy. “MG ≈ 6 x1011” solar masses as defined by [21]. “H0 = 71(km/s/Mpc)” as defined by the PDG [22] (“Mpc” = Mega-parsec). “T0 = 2.725(K)” as defined by the PDG. [20]

1.1.2.2.4 Rate of change “dHdt” xviii. Derive a generalised expression for the rate of change of the EGM Hubble constant in the time domain “dHdt” as a function of the dimensionless range variable “Hγ” such that: “1 ≥ Hγ > 0” and “Hγ ∝ Hβη”: (see: Ch. 8.3.3), Output: dH dt H γ

2 H α .H γ . 5 .ln 1 .µ 2 2 Hγ . 5µ Hγ

1

(4.361)

Note: “dHdt” is alternative notation introduced to replace the typical differential form “dH/dt”, for application in the “MathCad 8 Professional” computational environment. xix. For solutions where the deceleration parameter is zero, derive an expression for the magnitude of the EGM Hubble constant “|H|” in the time domain6: (see: Ch. 8.3.3), Output: H

d H dt

(4.378)

6

This terminology is an abbreviated reference to “the square-root of the magnitude of the rate of change of the Hubble constant in the time domain”, as indicated by the equation. 29

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xx. Devise a numerical approximation method facilitating the graphical representation of “|H|” in terms of an indicial power “η” (see: Ch. 8.3.3) such that, Input: 1

t

H γ .H α

Hγ Hβ

(4.359)

η

(4.376)

xxi. For non-zero deceleration parameter solutions, derive the ZPF energy density threshold “UZPF” (see: Ch. 7.7), Output: 3 .c . H U2 R o , M G Ω ZPF . 8 .π .G 2

U ZPF

2

(4.315)

where, Ω ZPF

1

Ω EGM

(4.313)

ρ U2 r x5.R o , m g5 .M G

Ω EGM

ρ U2 R o , M G

(4.308)

xxii. Reduce the expression for the EGM Hubble constant and Cosmological temperature to their simplest functionally dependent forms: “HU5” and “TU5” respectively (see: App. 4.B: “MathCad 8 Professional – b. Calculation engine – xi”, “MathCad 12 – c. High precision calculation engine – iv”). Output: H U5( r , M )

1 . ln TL

T U5( r , M )

( 3 .π )

7 .µ .

256 KW c

µ

µ

32

2

µ

. µ m .ln ( 3 π ) . h 4 M

. .ln . 4µ H U5( r , M ) λ h Hα

2 .µ

7 .µ

2

. r λh .

1 π .H α

2 7 .µ

5

.

mh

5 .µ

2

. r λh

M 2 .µ

2

2 26 .µ

(4.529)

. 2

.H ( r , M ) 5 µ U5

(4.530)

1.1.3 Sample results 1.1.3.1 Numerical evaluation and analysis 1.1.3.1.1 Cosmological properties Evaluating “AU”, “RU”, “HU” and “TU” yields, 9 A U K λ .R o , λ x.λ h , K m.M G, m x.m h = 14.575885 10 .yr

(4.250)

9 R U K λ .R o , λ x.λ h , K m.M G, m x.m h = 14.575885 10 .Lyr

(4.251)

H U K λ .R o , λ x.λ h , K m.M G, m x.m h = 67.084304

km . s Mpc

T U K λ .R o , λ x.λ h , K m.M G, m x.m h = 2.724752 ( K )

30

(4.254) (4.255)

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The EGM construct error associated with “HU” and “TU” with respect to expert opinion and physical measurement is given by, 1 . H U K λ .R o , λ x.λ h , K m.M G, m x.m h H0 1 . T U K λ .R o , λ x.λ h , K m.M G, m x.m h

1 = 5.515064 ( % )

(4.256) . 1 = 9.08391310

T0

3

(%)

(4.257)

It is possible to calculate the value of “HU” and “TU” based upon the “visible mass only” of the MW Galaxy by a simple substitution of values (i.e. “M2 / 3 = KmMG / 3”) as follows, 1 km H U K λ .R o , λ x.λ h , .K m.M G, m x.m h = 67.753267 . 3 s Mpc

(4.262)

1 T U K λ .R o , λ x.λ h , .K m.M G, m x.m h = 2.739618 ( K ) 3

(4.263)

Hence, the magnitude of the impact of “Dark Matter / Energy” on the value of “HU” and “TU” is demonstrated to be “< 1(%)” when compared to the previously derived value according to, H U K λ .R o , λ x.λ h , K m.M G, m x.m h

1 = 0.987352 ( % )

1 H U K λ .R o , λ x.λ h , .K m.M G, m x.m h 3 T U K λ .R o , λ x.λ h , K m.M G, m x.m h

(4.264) 1 = 0.542607 ( % )

1 T U K λ .R o , λ x.λ h , .K m.M G, m x.m h 3

(4.265)

A simple test verifying “TU2” is demonstrated below. Since the computed value of “TU2[HU2(Ro,MG/3)]” based upon visible MW Galactic mass “MG/3” is exactly compliant with “TU” (i.e. “TU = TU2”), no technical error exists. Moreover, the result “TU2(H0) ≈ T0” agrees precisely with historical expectation (i.e. prior to measurement by satellite) of “T0”. 1 T U2 H U2 R o , .M G 3 T U2 H 0

=

2.739618 2.810842

( K)

(4.277)

Note: the validation of “TU = TU2” above, also verifies that “HU = HU2”. In addition, it is also demonstrated and numerically verified in “App. 4.B” that “HU2 = HU5”. The preceding results demonstrate that the impact of “Dark Matter / Energy” on “HU” and “TU” is very small. This implies that the constitution of the Universe under the EGM construct is quite different from current thinking. The contemporary view asserted in [23] is that the constitution of the Universe is, i. “72(%) Dark Energy”. ii. “23(%) Dark Matter”. iii. “4.6(%) Atoms”. However, the EGM construct generalises the constitution of the Universe as being, iv. “> 94.4(%) Photons”. v. “< 1(%) Dark Matter / Energy”. vi. “4.6(%) Atoms”.

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For solutions where the deceleration parameter is zero, “η” may be numerically approximated utilising the “Given” and “Find” commands within the “MathCad 8 Professional” computational environment, subject to the constraint that “dHdt” as a function of the present value of “Hβ” [i.e. “≈ HU2(Ro,MG) / Hα”] raised to an indicial power, is equal to the square of the present Hubble constant as determined by the EGM construct “HU2(Ro,MG)2” according to the following algorithm, Given dH dt

H U2 R o , M G

η

Hα

H U2 R o , M G

η

1

(4.379)

Find( η )

(4.380)

Hence, “η = 4.595349”. 1.1.3.1.2 Significant temporal ordinates (See: Ch. 8.3.4) Significant temporal ordinates of Cosmological evolutionary events (marked on the proceeding graphs) are given in matrix form as follows, 1

t1

e

5 .µ

10 .µ

t2

e

2

. 1 Hα

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

t3 e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

. 1 Hα

2.206287 2.206287 4.196153 4.196153

2 3

2

. 1 Hα

e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

t5 e

10

42 .

s

20.932666 20.932666 8.385263 8.385263

1

t4

= 6.205726 6.205726

1

4

2 2 2 5 .µ . 5 .µ . 5 .µ

. 1 Hα 2

1

2

. 1 Hα

(4.384)

where, “t5” denotes the temporal ordinate of the local minima of the “2nd” time derivative of the Hubble constant (see: Ch. 8.3.3, 8.3.6.10).

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1.1.3.2 Graphical evaluation and analysis 1.1.3.2.1 Average Cosmological temperature vs. age (See: Ch. 8.2.5.1, 8.2.5.2) Av. Cosmological Temperature vs. Age 1 Hα

Av. Cosmological Temperature (K)

T U3 H β

t1

3 .1031

1 T U3 e

T U3 e

T U3 e

5 .µ

2

2 .1031

2 10 .µ 1 2 2 5 .µ . 5 .µ 1 2 2 15 .µ . 5 .µ 2 2 2 2 2 . . . . . 2 5µ 5µ 5µ 3

1 .1031

1 .10

43

1 .10

42

1 .10

41

1 .10

40

1 .10 1

39

1 .10

38

1 .10

37

1 .10

36

H β .H α Cosmological Age (s)

Figure 4.24, Av. Cosmological Temperature vs. Age t2t3

3 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

T U3 e

T U3 e

5 .µ

2

2 .1031

2 1 10 .µ 2 2 5 .µ . 5 .µ 1 2 2 2 2 15 .µ . 5 .µ 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

1 .1031

1 .10

43

1 .10

42

1 .10

41

1 .10

40

1 .10 1

39

1 .10

38

1 .10

37

1 .10

36

H β .H α Cosmological Age (s)

Figure 4.25,

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1.1.3.2.2 Magnitude of the Hubble constant vs. Cosmological age (See: Ch. 8.3.6.11, 8.3.6.12) Mag. of Hubble Cons. vs. Cosm. Age 2.5 .10 dH dt H β

dH dt e

η

1 2 5 .µ

2 .10

(Hz) dH dt e

1 Hα

t1

42

1 1.5 .1042

1 dH dt e

42

2 2 5 .µ . 5 .µ

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 1

1 .1042

5 .10

41

0 43 42 41 40 39 38 37 36 35 34 33 32 31 30 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 η H β .H α Cosmological Age (s)

Figure 4.46, Mag. of Hubble Cons. vs. Cosm. Age 2.5 .10 dH dt H β

dH dt e

η

1 2 5 .µ

2 .10

(Hz) dH dt e

1 Hα

t4

42

1 1.5 .1042

1 dH dt e

42

2 2 5 .µ . 5 .µ

1

2 2 2 4 5 .µ . 5 .µ 2 2 2 2 5 .µ . 5 .µ . 5 .µ 1

1 .1042

5 .10

41

0 43 42 41 40 39 38 37 36 35 34 33 32 31 30 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 .10 1 η H β .H α Cosmological Age (s)

Figure 4.47,

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1.1.3.2.3 Cosmological evolution process Based upon the preceding graphical output, the Cosmological evolution process may be categorised into two regimes, comprised of four distinct periods (i.e. three inflationary and one expansive) as follows, Time Temperature Hubble Constant 0 < t < Hα-1 -∞ < TU2 < 0 +∞ > |H| > Hα -1 0 → Hα -∞ → 0 +∞ → Hα Hα-1 ≤ t < t1 0 ≤ TU2 < TU2(t1-1) Hα ≥ |H| > 0 Hα-1 → t1 0 → TU2(t1-1) Hα → 0 -1 t1 ≤ t < t4 TU2(t1 ) ≥ TU2 > TU2(t4-1) 0 ≤ |H| < √|dHdt[(t4Hα)-1]| t1 → t4 TU2(t1-1) → TU2(t4-1) 0 → √|dHdt[(t4Hα)-1]| -1 TU2(t4 ) ≥ TU2 ≥ TU2(HU2) √|dHdt[(t4Hα)-1]| ≥ |H| ≥ HU2 t4 ≤ t < AU t4 → AU TU2(t4-1) → TU2(HU2) √|dHdt[(t4Hα)-1]| → HU2 Description Primordial Inflation (prior to the “Big-Bang”): the Universe may be described as “inverted and non-physical” such that the interior of the Cosmos existed outside the exterior boundary “RBH” in accordance with the “Primordial Universe” model described in Ch. (7, 8) such that: 1. “TU2” increases from negative infinity to zero. 2. “dHdt” increases from negative infinity to “-Hα2”. 3. “|H|” decreases from positive infinity to “Hα”. Thermal Inflation: the period from the instant of the “Big-Bang” to the instant of maximum Cosmological temperature such that: 4. “TU2” increases from zero to its maximum value “TU2(t1-1)”. 5. “dHdt” increases from “-Hα2” to zero. 6. “|H|” decreases from “Hα” to zero. Hubble Inflation: the period from the instant of maximum Cosmological temperature to the instant of maximum post-primordial “|H|” such that: 7. “TU2” decreases from its maximum value to “TU2(t4-1)”. 8. “dHdt” increases from zero to its maximum physical value “dHdt[(t4Hα)-1]”. 9. “|H|” increases from zero to its maximum physical value “√|dHdt[(t4Hα)-1]|”. Hubble Expansion: the period from the maximum post-primordial “|H|” to the present day such that: 10. “TU2” decreases from “TU2(t4-1)” to “TU2(HU2)”. 11. “dHdt” decreases from its maximum physical value to “HU22”. 12. “|H|” decreases from its maximum physical value to “HU2”. Symbol Definition / Value The EGM Hubble constant at the instant of the “Big-Bang”: Hα ≈ 2.742004 x1042(Hz) ≈ 8.460941 x1061(km/s/Mpc) -Hα2 ≈ -7.518587 x1084(Hz2) ≈ -7.158752 x10123(km/s/Mpc)2 HU2 The present value of the EGM Hubble constant: = HU2(Ro,MG) ≈ 67.084304(km/s/Mpc) HU22 ≈ 4.500304 x103(km/s/Mpc)2 H0 The PDG Hubble constant: ≈ 71(km/s/Mpc) H02 ≈ 5.041 x103(km/s/Mpc)2 Hα-1 The instant of the “Big-Bang”: ≈ 3.646967 x10-43(s) t1 The instant of max. Cosmological temperature: ≈ 2.206287 x10-42(s)

Physical @ {RBH ≥ rS}

Non-Physical @ {RBH < rS}

Period Primordial Inflation Thermal Inflation Hubble Inflation Hubble Expansion Regime

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t4 AU TU2(Hα) TU2(t1-1) TU2(t4-1) TU2(HU2) T0 dHdt[(t4Hα)-1] √|dHdt[(t4Hα)-1]| RU 2 Hα ⋅(dHdt[(t4Hα)-1])-1 Hα⋅(√|dHdt[(t4Hα)-1]|)-1

The instant of maximum physical “|H|”: ≈ 2.093267 x10-41(s) The EGM Cosmological age: = HU2-1 ≈ 14.575885 x109(yr) The EGM Cosmological temperature at the instant of the “Big-Bang”: = 0(K) The Maximum EGM Cosmological temperature: ≈ 3.195518 x1031(K) The EGM Cosmological temperature at the instant of maximum physical “|H|”: ≈ 2.059945 x1031(K) The present EGM Cosmological temperature: = TU3(HU2Hα-1) ≈ 2.724752(K) The present experimentally measured CMBR temperature: ≈ 2.725(K) The approximated maximum rate of change of the physical EGM Hubble constant: ≈ 1.553518 x1084(Hz2) ≈ 1.479167 x10123(km/s/Mpc)2 The approximated maximum physical “|H|”: ≈ 1.246402 x1042(Hz) ≈ 3.845994 x1061(km/s/Mpc) The EGM Cosmological size: = c⋅HU2-1 ≈ 14.575885 x109(Lyr) ≈ 4.839718 ≈ 2.199936 Table 4.10,

Time 0 -1 Hα ≈ 3.646967 x10-43(s) t1 ≈ 2.206287 x10-42(s) t4 ≈ 2.093267 x10-41(s) AU ≈ 14.575885 x109(yr)

TU2 (K) dHdt (km/s/Mpc)2 -∞ -∞ 0 ≈ -7.158752 x10123 31 0 ≈ 3.195518 x10 31 ≈ 2.059945 x10 ≈ 1.479167 x10123 ≈ 2.724752 ≈ 4.500304 x103 Table 4.11,

|H|| (km/s/Mpc) +∞ ≈ 8.460941 x1061 0 ≈ 3.845994 x1061 ≈ 67.084304

1.1.4 History of the Universe according to EGM Utilising “TU2”, the history of the Universe may be articulated as follows, Epoch or Event

Time Domain t

Primordial epoch

Grand unification epoch

Electroweak / Quark Epoch

Lepton Epoch

Boundary Temperature Value

1

T U2 H α = 0 ( K )

Hα

1

< t 10

34 .

(s)

Hα

10-34 < t(s) ≤ 10-10

10-10 < t(s) ≤ 102

1

T U2 10

34 .

10

10 .

(s)

1

T U2

T U2

. 28 ( K ) = 1.92400510

. 15 ( K ) = 3.43308810

(s)

1 2.

. = 1.01325410 ( K) 9

10 ( s )

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Photon Epoch

Universe becomes transparent

102 < t(s) ≤ 1013 1013(s) ≈ 3 x105(yr)

T U2

3 x105 < t(yr) ≤ 109

T U2

1 13 .

10

= 978.724031 ( K )

(s)

1 9.

= 11.838588 ( K )

10 ( yr )

109 < t(yr) ≤ 5 x109

First Supernovae

T U2

5 x109 < t(yr) ≤ 14.58 x109

Present Epoch

1 9.

5 .10 ( yr )

= 4.898955 ( K )

T U2 H U2 R o , M G

= 2.724752 ( K )

Table 4.12, T U2

1

T U2 T U2

1 .( day ) 1 . 31 ( day )

T U2 T U2 T U2

1 1 .( s )

1 1 .( yr ) 1 2 10 .( yr )

1 3.

1 4.

10 ( yr )

5.

10 ( yr ) 1

T U2

6.

10 ( yr ) 1

T U2

7.

10 ( yr ) T U2 T U2 T U2

10 ( yr ) T U2

1

T U2

1

. 7 2.52413210

521.528169

. 3.86401510

147.71262

6

= 1.00307810 . 6

41.823796

1

. 4 8.07751510

11.838588

9 10 .( yr )

. 2.29089210

3.35005

1

. 6.49496110

0.947724

8 10 .( yr )

4 3

( K)

10 .

10 T U2

. 10 1.84076810 . 3 1.2497710

( yr )

1 11 .

10

( yr )

37

(4.405)

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1.1.5 Discussion 1.1.5.1 Conceptualization 1.1.5.1.1 “λx” A physical interpretation of “λx” is possible utilising the Stefan-Boltzmann Law by considering the energy flux emitted from a “Black-Body” and equating it to the peak average Cosmological temperature. “λx” is shown to be proportional to the “4th power-root” of the energy flux of the Universe at the peak average Cosmological temperature (see: App. 4.A). 1.1.5.1.2 “TL” The minimum gravitational lifetime of matter “TL” is a simple concept to embrace by considering all matter to represent a vast store of Gravitons within, being ejected at a uniform rate with an emission frequency of “ωg” (see: Ch. 6.7.2.2, 6.8). 1.1.5.1.3 “CΩ_J” The initial step in conceptualizing the method of solution for the derivation of the Hubble constant and CMBR temperature presented herein is to understand the nature of EGM Flux Intensity “CΩ_J”. The EGM construct represents gravitational fields as a spectrum of conjugate wavefunction pairs, each comprising of a population of Photons. The spectrum is gravitationally dominated by the energy of the population of conjugate Photon pairs at the harmonic cut-off frequency7 “ωΩ” (see: Ch. 5.4). Subsequently, all gravitational objects may be usefully represented by approximation as wavefunction radiators of a single population of conjugate Photon pairs (see: Ch. 9.2.2.2, 9.2.3.2). The EGM spectrum is derived from the application of Fourier Series Harmonics, involving the hybridization of “2” spectra (i.e. an amplitude spectrum and a frequency spectrum). The relationship between “CΩ_J” and harmonic cut-off mode “nΩ” (which also denotes the total number of modes in the PV spectrum8) is analogous to the relationship between the amplitude and frequency spectra inherent in Fourier Series Harmonics. Thus, i. “CΩ_J” decreases with Cosmological expansion and is analogous to the decrease in PV spectral amplitude as the distance to the subject increases (i.e. the gravitational influence decreases). ii. Instantaneously after the “Big-Bang”, there were no Galaxies and as the Universe expanded, energy condensed into matter and the EGM spectrum developed into its current form such that matter radiates a spectrum of conjugate wavefunction pairs, each comprising of a population of Photons. Therefore, a single frequency mode describing the “Primordial Universe” becomes “many modes” when describing matter in the present state of the Universe. Hence, “nΩ” increases with Cosmological expansion as the distance to the subject increases. iii. EGM finds the convergent solution relating “2” spectra of opposing gradient. That is, “CΩ_J” decreases and “nΩ” increases as the Universe expands. iv. For solutions to “ωΩ” where the Refractive Index “KPV” approaches unity9, it is demonstrated that “ωΩ → ωΩ_3” (see: Ch. 5.1, 5.2), consequently “CΩ_J” may be simplified 7

i.e. the high-end terminal spectral frequency. The PV spectrum is a bandwidth of the EGM spectrum. 9 The typical representation of “KPV” is an isomorphic weak field approximation to General Relativity (GR). 8

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to “CΩ_J1” (see: Ch. 5.5.1) and a definition stated as follows: EGM Flux Intensity is a representation of gravitational field strength (i.e. the gradient in the energy density of the space-time manifold) expressed in “Jansky’s” (Jy). v. The gravitational forces governing the formation of the “Milky-Way” Galaxy are equivalent to the gravitational forces responsible for the current state of the Universe as a whole. Subsequently, the average EGM Flux Intensity of the “Milky-Way” Galaxy is proportional to the average value of the present Universe and the peak value of the “Primordial Universe” instantaneously prior to the “Big-Bang”. This means that the EGM Flux Intensity of the “Milky-Way” Galaxy acts a baseline reference. 1.1.5.1.4 “Stω” The EGM harmonic representation of fundamental particles “Stω” demonstrates that the mass-energy distribution over the space-time manifold at the elementary level, utilising the condition of ZPF equilibria, occurs in only one manner. The significance of this is that it provokes an obvious question with respect to Cosmology. That is: “perhaps it applies on a Cosmological scale?” Simply described, the representation works by expressing the values of “ωΩ” of two fundamental particles10, as an integer ratio (i.e. a harmonic of the reference particle). Subsequently, it follows that “CΩ_J” may be expressed in a similar manner as it is derived utilising “ωΩ”. Thus, if the EGM harmonic representation of fundamental particles with respect to mass-energy distribution over the space-time manifold were universally valid, we would expect that in order to apply it cosmologically: i. The ratio of the presently observable Cosmological size “rf”, to the initial size “ri” of the “Primordial Universe” instantaneously prior to the “Big-Bang”, is proportional to the corresponding EGM Flux Intensity {i.e. “(rf / ri) ∝ [CΩ_J1(rf) / CΩ_J1(ri)]”}. ii. The value of “CΩ_J” at the periphery of the “Primordial Universe” (i.e. instantaneously prior to the “Big-Bang”) is substantially greater than the value at the edge of the presently observable Universe. That is, the gradient of the energy density of the “Primordial Universe”, instantaneously prior to the “Big-Bang”, was substantially greater than the gradient of the energy density at the periphery of the presently observable Universe. iii. Since the values of wavefunction amplitude in the EGM spectrum decrease inversely with “nΩ”, and “nΩ” increases with radial displacement, it follows that “some sort” of naturally logarithmic or exponential relationship should exist between the ratio of the sizes described above and the associated EGM Flux Intensities. iv. “Stω9” represents the harmonic relationship between the values of “ωΩ” of two dimensionally similar particles. Hence, recognising that the frequency and time domains are interchangeable, we may apply “Stω9” as the ratio of “TL” to the present “Hubble age” of the Universe by the EGM method “AU”. Hence, it follows that the ratio of the sizes described above is proportional to the ratio “TL : AU” (see: Ch. 6.7.2.2). 1.1.5.2 Dynamic, kinematic and geometric similarity 1.1.5.2.1 “HU” The “Primordial Universe” was analogous to a spherical particle on the Planck scale with radius “r1” and homogeneous mass distribution “M1”, described by a single wavefunction whereas the presently observable Universe is described by a spectrum of wavefunctions. The maximum EGM Flux Intensity measured by an observer at the edge of the “Primordial Universe” is given by “CΩ_J1(r1,M1)”. 10

One of them being an arbitrarily selected reference particle from which to compare all others. 39

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Matter radiates Gravitons11 at a spectrum of frequencies such that the Cosmological majority of it exists in Photonic form, resulting in an approximately homogeneous mass-energy distribution throughout the Universe whereby any Galactic formation is dynamically, kinematically and geometrically equivalent to a spherical particle of homogeneous mass distribution and may be represented as a Planck scale object to be utilised as a Galactic Reference Particle (GRP). The associated EGM Flux Intensity of the GRP is given by “CΩ_J1(r2,M2)” where, “r2” denotes the mean “H0” measurement distance12 to the Galactic centre and “M2” represents total Galactic mass (i.e. visible + dark). The definition of “r2” comes from the scientific requirement to compare calculation or prediction to measurement. Subsequently, one should also utilise parameters within the same frame of reference as the measurement, against which the construct is being tested. It is not known by physical validation that “H0” is measured as being the same from all locations in the Universe. It is believed to be the case by contemporary theory; however it is not factually known to be true. To verify it physically, one would be required to perform the “H0” measurement from a significantly different location in space. Thus, to minimise potential modelling errors, we shall confine “r2” to the same frame of reference13 as the measurement of “H0” (see: Ch. 7.1). 1.1.5.2.2 “TU” EGM defines the “Primordial Universe” as a single mode wavefunction, therefore any temperature calculation must be scaled accordingly for application to black-body radiation (i.e. black-bodies emit a spectrum of thermal frequencies, not just one). Hence, we would expect that the peak CMBR temperature since the “Big-Bang” is proportional to the average number of Gravitons being radiated per harmonic period by the “Primordial Universe” instantaneously prior to the “BigBang” (see: Ch. 7.2). 1.1.6 Concluding remarks ⇒ The CBMR temperature is a function of the Hubble constant. ⇒ The Hubble constant and CBMR temperature instantaneously prior to the “Big-Bang” is calculated to be: • Hα = ωh / λx ≈ 8.460941 x1061(km/s/Mpc). • TU2[Hα] = 0(K). ⇒ Physical Laws become real instantaneously after the “Big-Bang”. For example, the “2nd Law of Thermodynamics” is not violated at “TU2[H > Hα]” because “TU2 > 0(K)”. ⇒ The magnitude of the EGM Hubble constant14 at the instant of maximum EGM Cosmological temperature is graphically illustrated to be: • |H(t1)| = 0(km/s/Mpc). ⇒ The maximum EGM Cosmological temperature is calculated to be: • TU2(t1-1) ≈ 3.195518 x1031(K). 11

Coherent populations of conjugate Photon pairs for a minimum period of “TL”. i.e. the distance relative to the Galactic centre from where a physical measurement of “H0” is performed. 13 The solar system. 14 This terminology is an abbreviated reference to “the square-root of the magnitude of the rate of change of the Hubble constant in the time domain”. 12

40

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⇒ The magnitude of the maximum physical (i.e. post “Big-Bang”) EGM Hubble constant (abbreviated reference) is calculated to be: • |H(t4)| = √|dHdt[(t4Hα)-1]| ≈ 3.845994 x1061(km/s/Mpc). ⇒ The EGM Cosmological temperature at the instant of maximum physical EGM Hubble constant (abbreviated reference) is calculated to be: • TU2(t4-1) ≈ 2.059945 x1031(K). ⇒ The present EGM Hubble constant and average EGM Cosmological temperature is calculated to be: • HU2(Ro,MG) ≈ 67.084304(km/s/Mpc). • TU2[HU2(Ro,MG)] ≈ 2.724752(K). ⇒ The present CMBR temperature is measured to be: • T0 ≈ 2.725 ± 0.001(K). ⇒ The present Hubble constant is stated by the PDG15 to be: • H0 = 71, +1/-2(km/s/Mpc). ⇒ The EGM Cosmological temperature based upon the PDG Hubble constant is calculated to be: • TU2[H0] ≈ 2.810842(K). ⇒ The Universe is composed of: • “> 94.4(%) Photons”. • “< 1(%) Dark Matter / Energy”. • “4.6(%) Atoms”. ⇒ The magnitude of the impact of “Dark Matter / Energy” on the value of the Hubble constant and CMBR temperature is “< 1(%)”. ⇒ The EGM construct exhibits characteristics satisfying the observed phenomena of “accelerated Cosmological expansion” due to: • The ZPF energy density threshold value “UZPF < -2.52 x10-13(Pa)”. • The gradient of the Hubble constant in the time domain is presently positive. On a human scale, this translates to levels of ZPF energy according to, i. “< -252(yJ/mm3)”. On an astronomical scale, this becomes, ii. “< -0.252(mJ/km3)”. iii. “< -7.4 x1012(YJ/pc3)”. On a Cosmological scale, this becomes, iv. “< -6.6 x1041(YJ/RU3)”. The deceleration parameter, v. “ΩEGM” may be utilised to obtain non-zero deceleration parameter solutions. Note: although on the human scale the quantities of ZPF energy are extremely small, on the astronomical or Cosmological scales, they become extremely large when approaching the dimensions of the visible Universe according to “RU → RU(KλRo,λxλh,KmMG,mxmh)”. 15

http://pdg.lbl.gov/2006/reviews/hubblerpp.pdf (pg. 20 - “WMAP + All”). 41

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1.2 Fundamentals The following statements are verbatim quotations from [24]. 1.2.1 General Relativity (GR) General Relativity (GR) is a geometrical theory of gravitation published by Albert Einstein in 1915-16. It unifies special relativity and Sir Isaac Newton’s law of universal gravitation with the insight that gravitation is not due to a force but rather is a manifestation of curved space and time, with this curvature being produced by the mass-energy and momentum content of the space-time. GR is distinguished from other metric theories of gravitation by its use of the Einstein field equations to relate space-time content and space-time curvature. GR is currently the most successful gravitational theory, being almost universally accepted and well confirmed by observations. The first success of general relativity was in explaining the anomalous perihelion precession of Mercury. Then in 1919, Sir Arthur Eddington announced that observations of stars near the eclipsed Sun confirmed GR’s prediction that massive objects bend light. Since then, other observations and experiments have confirmed many of the predictions of GR, including gravitational time dilation and gravitational red-shift of light. In addition, numerous observations are interpreted as confirming the weirdest prediction of GR, the existence of Black-Holes (BH’s). In the mathematics of GR, the Einstein field equations become a set of simultaneous differential equations which are solved to produce metric tensors of space-time. These metric tensors describe the shape of the space-time manifold and are used to obtain predictions. The connections of the metric tensors specify the geodesic paths that objects follow when travelling inertially. Important solutions of the Einstein field equations include the Schwarzschild solution (for the space-time surrounding a spherically symmetric uncharged and non-rotating massive object), the Reissner-Nordström solution (for a charged spherically symmetric massive object), and the Kerr metric (for a rotating massive object). In spite of its overwhelming success, there is discomfort with GR in the scientific community due to its being incompatible with Quantum Mechanics (QM) and the reachable singularities of BH’s (at which the math of GR breaks down). Because of this, numerous other theories have been proposed as alternatives to GR. The most successful of these was Brans-Dicke theory, which appeared to have observational support in the 1960’s. However, those observations have since been refuted and modern measurements indicate that any Brans-Dicke type of deviation from GR must be very small if it exists at all. End of verbatim quotation.

Albert Einstein16,

16

http://nobelprize.org/physics/laureates/1921/index.html 42

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1.2.2 Black-Holes (BH’s) The following statements are verbatim quotations from [24]. A Black-Hole (BH) is an object predicted by GR, with a gravitational field so powerful that even ElectroMagnetic (EM) radiation (such as light) cannot escape its pull. A BH is defined to be a region of space-time where escape to the outside Universe is impossible. The outer boundary of this region is called the event horizon. Nothing can move from inside the event horizon to the outside, even briefly, due to the extreme gravitational field existing within the region. For the same reason, observers outside the event horizon cannot see any events which may be happening within the event horizon; thus any energy being radiated or events happening within the region are forever unable to be seen or detected from outside. Within the BH is a singularity, an anomalous place where matter is compressed to the degree that the known laws of Physics no longer apply to it. Theoretically, a BH can be of any size. Astrophysicists expect to find BH’s with masses ranging between roughly the mass of the Sun (“stellar-mass” BH’s) to many millions of times the mass of the Sun (i.e. SuperMassive BH’s). The existence of BH’s in the Universe is well supported by astronomical observation, particularly from studying X-ray emission from X-ray binaries and active Galactic nuclei. It has also been hypothesised that BH’s radiate an undetectably small amount of energy due to QM effects called Hawking radiation. End of verbatim quotation.

Figure 4.1: A Feeding SuperMassive “Black-Hole” - credit: Gabriel Pérez Díaz, SMM del IAC, 43

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1.2.3 Quantum Mechanics (QM) The following statements are verbatim quotations from [24]. Quantum Mechanics (QM) refers to a discrete unit that Quantum Theory assigns to certain physical quantities, such as the energy of an atom at rest. The discovery that waves could be measured in particle-like small packets of energy called quanta led to the branch of Physics that deals with atomic and subatomic systems which we today call QM. It is the underlying mathematical framework of many fields of Physics and Chemistry. The foundations of QM were established during the first half of the twentieth century by Werner Heisenberg, Max Planck, Louis de Broglie, Niels Bohr, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli and others. Some fundamental aspects of the theory are still actively studied. It is necessary to use QM to understand the behavior of systems at atomic length scales and smaller. For example, if Newtonian mechanics governed the workings of an atom, Electrons would rapidly travel towards and collide with the nucleus. However, in the natural world the Electrons normally remain in an unknown orbital path around the nucleus, defying classical ElectroMagnetism. QM was initially developed to explain the atom, especially the spectra of light emitted by different atomic species. The Quantum Theory of the atom developed as an explanation for an Electron remaining in its orbital, which could not be explained by Newton's laws of motion and by classical ElectroMagnetism. In the formalism of QM, the state of a system at a given time is described by a complex wave function (sometimes referred to as orbital’s in the case of atomic Electrons), and more generally, elements of a complex vector space. This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments. For example, it allows one to compute the probability of finding an Electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one cannot make simultaneous predictions of conjugate variables, such as position and momentum, with arbitrary accuracy. For instance, Electrons may be considered to be located somewhere within a region of space, but with their exact positions being unknown. Contours of constant probability, often referred to as “clouds”, may be drawn around the nucleus of an atom to conceptualise where the Electron might be located with the most probability. It should be stressed that the Electron itself is not spread out over such cloud regions. It is either in a particular region of space, or it is not. Heisenberg’s uncertainty principle quantifies the inability to precisely locate the particle. The other exemplar that led to QM was the study of EM waves such as light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or quanta, Albert Einstein exploited this idea to show that an EM wave such as light could be described by a particle called the Photon with discrete energy, dependent upon its frequency. This led to a theory of unity between subatomic particles and EM waves called waveparticle duality in which particles and waves were neither one nor the other, but had certain properties of both. While QM describes the world of the very small, it also is needed to explain certain “macroscopic quantum systems” such as superconductors and superfluids. Broadly speaking, QM incorporates four classes of phenomena that classical Physics cannot account for: (i) the quantisation (discretisation) of certain physical quantities, (ii) wave-particle duality, (iii) the uncertainty principle and (iv), quantum entanglement. End of verbatim quotation.

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1.2.4 Particle-Physics The following statements are verbatim quotations from [24]. 1.2.4.1 Synopsis Particle-Physics is a branch of Physics that studies the elementary constituents of matter and radiation, and the interactions between them. It is also called High Energy Physics (HEP), because many elementary particles do not occur under normal circumstances in nature, but can be created and detected during energetic collisions of other particles, as is done in particle accelerators. 1.2.4.2 Subatomic particles Modern Particle-Physics research is focused on subatomic particles, which have less structure than atoms. These include atomic constituents such as Electrons, Protons, and Neutrons (Protons and Neutrons are actually composite particles, made up of Quarks), particles produced by radiative and scattering processes such as Photons, Neutrinos and Muons, as well as a wide range of exotic particles. Strictly speaking, the term particle is a misnomer because the dynamics of Particle-Physics are governed by QM. As such, they exhibit wave-particle duality, displaying particle-like behavior under certain experimental conditions and wave-like behavior in others (more technically they are described by state vectors in Hilbert space). Particle Physicists use the term “elementary particles” to refer to objects such as Electrons and Photons, with the understanding that these “particles” display wave-like properties as well. All the particles and their interactions observed to date can be described by a Quantum Field Theory (QFT) called the Standard Model (SM). The SM has 40 species of elementary particles (24 Fermions, 12 Vector Bosons, and 4 Scalars), which can combine to form composite particles, accounting for the hundreds of other species of particles discovered since the 1960’s. The SM has been found to agree with almost all the experimental tests conducted to date. However, most particle Physicists believe that it is an incomplete description of Nature, and that a more fundamental theory awaits discovery. In recent years, measurements of Neutrino mass have provided the first experimental deviations from the SM. Particle-Physics has had a large impact on the philosophy of science. Some particle Physicists adhere to reductionism, a point of view that has been criticized and defended by philosophers and scientists. 1.2.4.3 History The idea that all matter is composed of elementary particles dates to at least the 6th century BC. The philosophical doctrine of atomism was studied by ancient Greek philosophers such as Leucippus, Democritus, and Epicurus. In the 19th century John Dalton, through his work on stoichiometry, concluded that each element of nature was composed of a single, unique type of particle. Dalton and his contemporaries believed these were the fundamental particles of nature and thus named them atoms, after the Greek word “atomos”, meaning “indivisible”. However, near the end of the century, Physicists discovered that atoms were not, in-fact, the fundamental particles of nature, but conglomerates of even smaller particles. The early 20th century explorations of Nuclear and Quantum-Physics culminated in proofs of Nuclear Fission in 1939 by Lise Meitner (based on experiments by Otto Hahn), and Nuclear Fusion by Hans Bethe in the same year. These discoveries gave rise to an active industry of generating one atom from another, even rendering possible (although not profitable) the transmutation of lead into gold. They also led to the development of Nuclear Weapons. 45

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Throughout the 1950’s and 1960’s, a bewildering variety of particles were found in scattering experiments. This was referred to as the “particle zoo”. This term was deprecated after the formulation of the SM during the 1970’s in which the large numbers of particles were explained as combinations of a (relatively) small number of fundamental particles. 1.2.4.4 Standard Model (SM) The current state of the classification of elementary particles in the SM describes the Strong, Weak and ElectroMagnetic fundamental forces utilising mediating Gauge Bosons. The species of Gauge Bosons are the Gluons, W-, W+, Z Bosons and Photons. The model also contains 24 fundamental particles which are the constituents of matter. Finally, it predicts the existence of the Higgs Boson, which has yet to be discovered. 1.2.4.5 Experiment The major laboratories researching Particle-Physics are (listed in alphabetical order): i. Brookhaven National Laboratory, located on Long Island, USA. Its main facility is the Relativistic Heavy Ion Collider, colliding heavy ions such as gold ions and Protons. ii. Budker Institute of Nuclear Physics (Novosibirsk, Russia). iii. CERN, located on the French-Swiss border near Geneva. Its main project is now the Large Hadron Collider (LHC). Earlier facilities include LEP, the Large Electron-Positron collider, which was stopped in 2001 and which is now dismantled to give way for LHC; and Super Proton Synchrotron (SPS). iv. DESY, located in Hamburg, Germany. Its main facility is HERA colliding Electrons, Positrons and Protons. v. Fermilab, located near Chicago, USA. Its main facility is the Tevatron, colliding Protons and Anti-Protons. vi. KEK the High Energy Accelerator Research Organization located in Tsukuba, Japan. It is the home of a number of interesting experiments such as K2K (a Neutrino oscillation experiment) and Belle (an experiment measuring the CP-Symmetry violation in the BMeson). vii. SLAC, located near Palo Alto, USA. Its main facility is PEP-II, colliding Electrons and Positrons. The techniques required to do modern experimental Particle-Physics are quite varied and complex, constituting a sub-specialty nearly completely distinct from the theoretical side of the field. 1.2.4.6 Theory Theoretical Particle-Physics attempts to develop the models, theoretical framework, and mathematical tools to understand current experiments and make predictions for future experiments. There are several major efforts in theoretical Particle-Physics today and each includes a range of different activities and the efforts in each area are interrelated. One of the major activities in theoretical Particle-Physics is the attempt to better understand the SM and its tests. Extracting the parameters of the SM from experiments with less uncertainty probes the limits of the SM and therefore expands our understanding of nature. These efforts are made challenging by the difficult nature of calculating many quantities in Quantum ChromoDynamics (QCD). The second major effort is in model building where scientists develop ideas for what Physics may lie beyond the SM (at higher energies or smaller distances). This work is often motivated by the hierarchy problem and is constrained by existing experimental data. It may involve work on 46

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supersymmetry, alternatives to the Higgs mechanism, extra spatial dimensions or other ideas. The third major effort in theoretical Particle-Physics is String Theory (ST). String theorists attempt to construct a unified description of QM and GR by building a theory based upon small strings and branes rather than particles. If the theory is successful, it may be considered a “Theory of Everything” (ToE). There are also other areas of work in theoretical Particle-Physics ranging from particle Cosmology to Loop-Quantum-Gravity (LQG). End of verbatim quotation.

Figure 4.2: credit: USDoE, http://pdg.lbl.gov/barnett/extradim.html,

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1.2.5 Zero-Point-Field (ZPF) Theory 1.2.5.1 Synopsis The following statements are verbatim quotations from [24]. 1.2.5.1.1 Zero-Point-Energy (ZPE) 1.2.5.1.1.1 General In Physics, Zero-Point-Energy (ZPE) is the lowest possible energy that a QM system may possess, representing the energy of the ground state. The field associated with ZPE is termed the Zero-Point-Field (ZPF). The concept of ZPE was proposed by Albert Einstein and Otto Stern in 1913, which they originally called “residual energy” or “Nullpunktsenergie”. All QM systems are associated with ZPE. The term arises commonly in reference to the ground state of the quantum harmonic oscillator and its null oscillations. In QFT, it is a synonym for vacuum energy, an amount of energy associated with the vacuum of empty space and is the underlying background energy that exists in space even when devoid of matter. In Cosmology, ZPE is taken to be the origin of the Cosmological constant. It is widely accepted that ZPE results in the existence of most (if not all) of the fundamental forces and the effects derived from them. They have been observed in various experiments such as the spontaneous emission of light, gamma radiation, Van-Der Waals bonds and the Lamb shift etc. Importantly, the ZPE of the vacuum leads directly to the Casimir effect and is directly observable in nanoscale devices. It is thought (but not yet demonstrated) to have consequences for the behavior of the Universe on a Cosmological scale. Because ZPE is the lowest possible energy a system can have, it cannot be removed. Despite the definition, the concept of ZPE has attracted the attention of amateur inventors with the prospect of extracting “free energy” from the vacuum. Numerous perpetual motion and other pseudoscientific devices, often called free energy devices exploiting the idea, have been proposed. As a result of this activity and its intriguing theoretical explanation, it has taken on a life of its own in popular culture, appearing in science fiction books, games and movies. 1.2.5.1.1.2 Elementary particles QFT, which describes interactions between elementary particles in terms of fields, allows a contribution to ZPE (even when no particles are present) via the ZPF. An example is the Casimir effect whereby two metal plates experience a small attractive force between them. This has been attributed to the dependence on the ZPF and the distance between the plates. This has important consequences on a Cosmological scale because ZPE is expected to contribute to the Cosmological constant, which affects the expansion of the Universe. The calculation of the ZPE in QFT, in terms of Feynman diagrams, may be considered as accounting for virtual particles (also known as vacuum fluctuations), which are created and destroyed out of the vacuum. Additional contributions to the ZPE come from spontaneous symmetry breaking in QFT. 1.2.5.1.1.3 Implications Vacuum energy has a number of consequences. For one, vacuum fluctuations are always created as particle / antiparticle pairs. The creation of these “virtual particles” near the event horizon of a Black-Hole (BH) has been hypothesised by Physicist Stephen Hawking to be a mechanism for the eventual “evaporation” of BH’s such that the net energy of the Universe remains zero so long as 48

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the particle pairs annihilate each other within Planck time. If one of the pair is pulled into the BH before this, then the other particle becomes “real” and energy / mass is radiated into space from the BH. The loss is cumulative and could result in the Black-Hole's disappearance over time. The time required is dependent upon the mass of the BH, but could be in the order of “10100” years for large solar-mass BH’s. Grand Unification Theory (GUT) predicts a non-zero Cosmological constant from the energy of vacuum fluctuations. Examining normal physical processes with knowledge of these field phenomena leads to interesting insights into ElectroDynamics (ED). 1.2.5.1.2 History In 1900, Max Planck derived the formula for the energy of a single “radiator” (i.e. a vibrating atomic unit) as: (W.1) In 1913, using this formula as a basis, Albert Einstein and Otto Stern published a paper of great significance in which they suggested for the first time the existence of energy that all oscillators have at absolute zero. They called this “residual energy” and then “Nullpunktsenergie” (in German), which later became translated as ZPE. They carried out an analysis of the specific heat of Hydrogen gas at low temperature and concluded that the data is best represented if the vibration energy is taken to have the form, (W.2) Thus, according to this expression - even at absolute zero, the energy of an atomic system has the value “½hν”. In 1934, Georges Lemaître used an unusual perfect-fluid equation of state to interpret the Cosmological constant as a result of ZPE. In 1973, Edward Tryon proposed that the Universe may be a large scale QM vacuum fluctuation where positive mass-energy is balanced by negative gravitational potential energy. During the 1980’s, many attempts were made to relate fields that generate vacuum energy to specific fields that were predicted by the GUT and to use observations of the Universe to confirm it. So far, these efforts have failed and the exact nature of the particles or fields that generate ZPE, with a density such as that required by inflation theory, remains a mystery. 1.2.5.1.3 Foundational Physics In classical Physics, the energy of a system is defined in relation to “some” given state (often called the reference state). Typically, one might associate a motionless system with zero energy, although doing so is purely arbitrary. However, in quantum Physics it is natural to associate the energy with the expectation value of a certain operator - the Hamiltonian of the system. For almost all QM systems, the lowest possible expectation value that this operator can obtain is not zero; the lowest possible value is called the ZPE (caveat: if we add an arbitrary constant to the Hamiltonian, we get another theory which is physically equivalent to the previous Hamiltonian - because of this, only relative energy is observable, not the absolute energy). The origin of non-zero minimal energy can be intuitively understood in terms of the Heisenberg uncertainty principle. This principle states that the position and momentum of a QM particle cannot both be known arbitrarily accurately. If the particle is confined to a potential well, then its position is at least partly known - it must be within the well. Thus, one may deduce that within the well, the particle cannot have zero momentum otherwise the uncertainty principle would be violated. Because the kinetic energy of a moving 49

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particle is proportional to the square of its velocity, it cannot be zero either. This example however, is not applicable to a free particle - the kinetic energy of which can be zero. 1.2.5.1.4 Varieties of ZPE The concept of ZPE occurs in a number of situations and it is important that they be distinguished. In ordinary QM, the ZPE is the energy associated with the ground state of the system. The most famous example is the energy associated with the ground state of the quantum harmonic oscillator. More precisely, the ZPE is the expectation value of the Hamiltonian of the system. In QFT, the fabric of space is visualised as consisting of fields and every point in space and time being a quantised simple harmonic oscillator, with neighboring oscillators interacting. In this case, one has a contribution of energy from every point in space, resulting in infinite ZPE. The ZPE is the expectation value of the Hamiltonian. In Quantum Perturbation Theory (QPT), it is sometimes stated that the contribution of oneloop and multi-loop Feynman diagrams, to elementary particle propagators, are the contributions of vacuum fluctuations (ZPE) to particle masses. 1.2.5.1.5 Experimental evidence The simplest experimental evidence for the existence of ZPE in QFT is the Casimir effect. This effect was proposed in 1948 by Dutch Physicist Hendrik B. G. Casimir, who considered the quantised EM field between a pair of grounded, neutral metal plates. A small force can be measured between the plates ascribable to a change of the ZPE of the EM field between the plates. Although the Casimir effect at first proved difficult to measure because its manifestation can be seen only at very small distances, it is taking on increasing importance in nanotechnology. The Casimir effect can be accurately measured in specially designed nanoscale devices, and increasingly needs to be taken into account in the design and manufacturing processes of small devices. It can exert significant forces and stress on nanoscale devices, causing them to bend, twist, stick or break. Other experimental evidence includes spontaneous emissions of light (Photons) by atoms and nuclei, the observed Lamb shift of positions of energy levels of atoms and the anomalous value of the Electron’s gyromagnetic ratio etc. 1.2.5.1.6 Gravitation and Cosmology In Cosmology, ZPE offers an intriguing possibility for explaining the speculative positive values of the proposed Cosmological constant. In brief, if the energy is “really there”, then it should exert a gravitational force. In General Relativity (GR), mass and energy are equivalent; either produces a gravitational field. One obvious difficulty with this association is that the ZPE of the vacuum is absurdly large. Naively, it is infinite, but one must argue that new Physics takes over at the Planck scale, and so its growth is cut off at that point. Even so, what remains is so large that it would visibly bend space, and thus, there seems to be a contradiction. There is no easy way out, and reconciling the seemingly huge ZPE of space with the observed zero or small Cosmological constant has become one of the important problems in theoretical Physics. Subsequently, it has become a criterion by which to judge a candidate “Theory of Everything” (ToE). 1.2.5.1.7 Propulsion theories Another area of research in the field of ZPE is how it could be used for propulsion. NASA and British Aerospace both have programs running to this end, though practical technology is still a long way off. For any success in this area, it would have to be possible to create repulsive effects in 50

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the quantum vacuum, which according to theory should be possible. Experiments to produce and measure these effects are planned for the future. Note: Haisch, Rueda and Puthoff have proposed that an accelerated massive object interacts with the ZPF to produce an EM drag force giving rise to the phenomenon of inertia. 1.2.5.1.8 Popular culture The Casimir effect has established ZPE as an uncontroversial and scientifically accepted phenomenon. However, the term ZPE has also become associated with a highly controversial area of human endeavour – i.e. so-called “free energy” devices, similar to perpetual motion machines of the past. These devices purport to “tap” the ZPF and somehow extract energy from it, thus providing an “inexhaustible”, cheap, and non-polluting energy source. Controversy arises when such devices are promoted without scientifically acceptable proof that they tap the energy sources claimed. Promoters of a device frequently demonstrate no understanding of how the device might do so; they may demonstrate misunderstanding of widely accepted scientific facts and methods, in development or communication of a theory concerning a device; and they generally have made no attempt to investigate simpler explanations for the claimed performance of a device. Any of these behaviours are liable to taint the reputations of those involved with such devices, and qualified researchers are therefore likely to be reluctant to make any attempt to verify or even seriously examine such a device unless its promoters demonstrate enough competence to be taken seriously. End of verbatim quotation. 1.2.5.2 Spectral Energy Density (SED) An extremely important development in ZPF Theory - utilised as a foundation for the EGM construct, is the concept of Spectral Energy Density (SED). This quantifies the spectral energy distribution of ZPE within the ZPF and may be described in terms of the energy density per frequency mode by, ρ 0( ω )

2 .h .ω c

3

3

(3.47)

where, “h” denotes Planck’s Constant [6.6260693 x10-34(Js)] and “ω” is in “(Hz)”. 1.2.6 The Polarisable Vacuum (PV) model of gravity The Polarizable Vacuum (PV) refers to an analogue of GR to describe gravity in optical terms offering the following, i. The potential to replace GR in terms of the propagation of light through an optical medium, with a scalar theory of gravitation featuring formal analogies with Maxwell's theory of ElectroMagnetism. ii. The potential to unify gravitation and ElectroMagnetism in a theory of Electro-Gravity. iii. The potential to provide a physical mechanism for how space-time “gets curved” in GR, suggesting the possibility of “metric engineering” for spacecraft propulsion etc. The Polarisable Vacuum (PV) model of gravity is an optical representation such that the curvature of the space-time manifold is expressed in terms of a Refractive Index “KPV”. The value of “KPV” of a solid spherical object with homogeneous mass-energy distribution (i.e. a weak field isomorphic approximation to GR) is given by, 51

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2.

K PV e

G .M 2 r .c

(3.55)

where, Variable KPV

e G M r c

Description Units Refractive Index of a gravitational field in the PV model of gravity, only contributing significantly when a large gravitational mass (i.e. a strong gravitational field) is None considered. For weak gravitational fields, the effect is approximated to KPV(r,M) = 1. Exponential function Gravitational constant m3kg-1s-2 Mass kg Magnitude of position vector from centre of mass m Velocity of light in a vacuum m/s Table 4.1,

1.2.7 Dimensional Analysis Techniques and Buckingham’s “Π” (Pi) Theory 1.2.7.1 The principles The following statement is a verbatim quotation from [24] Dimensional Analysis is a conceptual tool often applied in Physics, Chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. It is routinely used by physical scientists and engineers to check the plausibility of derived equations. Only like dimensioned quantities may be added, subtracted, compared, or equated. When like or unlike dimensioned quantities are multiplied or divided, their dimensions are likewise multiplied or divided. When dimensioned quantities are raised to a power or a power root, the same is done to the dimensions attached to those quantities. The dimensions of a physical quantity is associated with symbols such as “M, L, T” which represent mass, length and time, each raised to rational powers. For instance, the dimension of the physical variable speed is “distance / time (L/T)” and the dimension of force is “mass × distance / time² (ML/T²)”. In mechanics, every dimension can be expressed in terms of distance (which Physicists often call “length”), time and mass, or alternatively in terms of force, length and mass. Depending on the problem, it may be advantageous to choose one or another set of dimensions. In ElectroMagnetism, for example, it may be useful to use dimensions of “M, L, T, and Q”, where “Q” represents the quantity of electric charge. The units of a physical quantity are defined by convention, related to some standard; e.g. length may have units of meters, feet, inches, miles or micrometres; but length always has dimension “L” whether it is measured in meters, feet, inches, miles or micrometres. In the most primitive form, dimensional analysis may be used to check the “correctness” of physical equations: in every physically meaningful expression, only quantities of the same dimension can be added or subtracted. Moreover, the two sides of any equation must have the same dimensions. For example, the mass of a rat and the mass of a flea may be added, but the mass of a flea and the length of a rat cannot be added. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers. The logarithm of “3(kg)” is undefined, but the logarithm of “3” is “0.477”. It should be noted that very different physical quantities may have the same dimensions: work and torque, for example, have the same dimensions, “M L2T-2”. An equation with torque on one side and energy on the other would be dimensionally 52

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correct, but cannot be physically correct! However, torque multiplied by an angular twist measured in (dimensionless) radians is work or energy. The radian is the mathematically natural measure of an angle and is the ratio of arc of a circle swept by such an angle divided by the radius of the circle. The value of a dimensional physical quantity is written as the product of a unit within the dimension and a dimensionless numerical factor. When like dimensioned quantities are added, subtracted or compared, these dimensioned quantities must be expressed in consistent units so that the numerical values of these quantities may be directly added or subtracted. But, conceptually, there is no problem adding quantities of the same dimension expressed in different units. Buckingham “Π” (Pi) Theory (BPT) forms the basis of the central tool of Dimensional Analysis. This theorem describes how every physically meaningful equation involving “n” variables can be equivalently rewritten as an equation of “n-m” dimensionless parameters, where “m” is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables, even if the form of the equation is still unknown. BPT is a systematic method of Dimensional Analysis, whereby variables that are relevant to a particular situation are formed into dimensionless Π groups. The number of dimensionless groups equals the original number of variables minus the number of fundamental dimensions present in all the variables. This analysis reduces the degrees of freedom for a physical situation and can be used to guide experimental design programs. Proofs of BPT often begin by considering the space of fundamental and derived physical units as a vector space, with the fundamental units as basis vectors and with multiplication of physical units as the “vector addition” operation and raising to powers as the “scalar multiplication” operation. Making the physical units match across sets of physical equations can then be regarded as imposing linear constraints in the physical unit vector space. Two systems for which these parameters coincide are called similar; they are equivalent for the purposes of the equation and the experimentalist whom wishes to determine the form of the equation can choose the most convenient one. BPT uses linear algebra: the space of all possible physical units can be seen as a vector space over rational numbers if we represent a unit as the set of exponents needed for the fundamental units (with a power of zero if the particular fundamental unit is not present). Multiplication of physical units is then represented by vector addition within this vector space. The algorithm of BPT is essentially a Gauss-Jordan elimination carried out in this vector space. End of verbatim quotation. 1.2.7.2 The atomic bomb The following statement is a verbatim quotation from [24] In 1941, “Sir Geoffrey I. Taylor” used Dimensional Analysis to estimate the energy released in an atomic bomb explosion. The first atomic bomb was detonated in New Mexico on July 16, 1945. In 1947, movies of the explosion were declassified, allowing “Sir Taylor” to complete the analysis and estimate the energy released in the explosion, even though the energy release was still classified! The actual energy released was later declassified and its value was remarkably close to Taylor's estimate. Taylor supposed that the explosive process was adequately described by five physical quantities, the time “t” since the detonation, the energy “E” which is released at a single point in space at detonation, the radius “R” of the shock wave at time “t”, the ambient atmospheric pressure “p” and density “ρ”. There are only three fundamental physical units in this combination (MLT) which yield Taylor's equation. Once the radius of the explosion as a function of the time was known, the energy of the explosion was calculated. End of verbatim quotation. 53

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1.2.7.3 The birth and foundations of Electro-Gravi-Magnetics (EGM) Historically, Dimensional Analysis Techniques (DAT’s) and BPT has been used extensively in the engineering field to model, predict and optimise fluid flow and heat transfer. However, in principle, it may be applied to any system that is dynamically, kinematically and geometrically founded – such as the geometric space-time manifold. Typical examples of experimentally verified Π groupings in fluid mechanics are Froude, Mach, Reynolds and Weber numbers. Thermodynamic examples are Eckert, Grashof, Prandtl and Nusselt numbers. Moreover, the Planck Length commonly used in theories of Quantum Gravity shares its origins with the Dimensional Analysis Technique (the foundation of BPT). The application of BPT is not an attempt to answer fundamental physical questions but to apply universally accepted engineering design methodologies to real world problems. It is primarily an experimental process. It is not possible to derive system representations without involving experimental relationship functions. These functions incorporate all variables within the experimental environment that influence results and behaviour including parameters that might otherwise be neglected due to practical calculation limitations, in theoretical analysis. Once the Π groupings have been formed, they may be manipulated or simplified as required to test ideas and applied to determine experimental relationship functions. Ultimately, these functions validate the system equations developed. Ideally, experimental relationship functions possess values of unity relative to the distant observer. This indicates a loss-less relationship between an Experimental Prototype (EP) and the mathematical model utilised to describe the EP. Typically, due to viscous forces and energy loss / transformation effects, experimental relationship functions take extreme values of magnitude (i.e. large or small). If we consider the EP to be the ambient gravitational environment (i.e. local spacetime manifold) and the mathematical model to be the PV model of gravity, then we expect all experimental relationship functions to approach unity. The reasons for this are: i. The true nature of gravity is currently unknown to Physics. ii. The mathematical descriptions used to predict gravitational behaviour are constructed from observation of effects, not causes. iii. A mathematical description is nothing more than just that. It is a non-physical manifestation of human understanding. For example, GR is a Tensor based mathematical formulation only - there is no physical evidence to validate the contention that the true nature of space-time is physically geometric with Planck scale grid lines radiating from Cosmological objects. iv. There can be no physical losses between two mathematical representations of the same thing. BPT commences with the selection of significant parameters. There are no right or wrong choices with respect to the selection of these parameters. Often, the experience of the researcher exerts the greatest influence to the beginning of the process and the choice of significant parameters is validated (or refuted) by experimentation. When applying BPT, it is important to avoid repetition of dimensions. Subsequently, it is often desirable to select variables that may be formulated by the manipulation of simpler variables already chosen. The selected variables used in EGM are shown by Storti et. Al. in [1]. These parameters were selected to facilitate experiments utilising ElectroMagnetic (EM) fields and assume that there is a physical device to be tested, located on a laboratory test bench. The objective is to utilise a superposition of EM fields to reduce the weight of a test-mass when placed in the volume of space located directly above the device. Therefore, the significant parameters are those factors that may affect the acceleration of the test-mass within this volume. The selection of significant parameters involved the magnitude of vector quantities and scalars. This avoids unnecessary repetition of fundamental units in accordance with the application of BPT methodology. The significant vector magnitude parameters are acceleration, Magnetic field, 54

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Electric field and position. The scalar quantities are Electric charge and frequency. Since static charge on the device or the test-mass may also exert strong Lorentz forces and therefore accelerations, the scalar value of static charge is included to determine its contribution. If the device is small then the distance between the surface of the device and the test-mass suspended in the volume above it is trivial and that the magnitude of the position vector is usefully constant. Storti et. Al. utilise BPT to relate gravitational acceleration, EM acceleration by the superposition of applied fields, ZPF Theory and the PV model of gravity via Einstein’s equivalence principle. Dimensionally, there is no difference between gravitational and EM acceleration. The equivalence principle provides a well accepted vehicle for the logical application of BPT and DAT’s to gravity. Much of Thermodynamics and Fluid Mechanics is built form the application of BPT and DAT’s. BPT facilitates the ability to string together any number of variables in a way that permits one to test one’s own idea. So, it is really a mix between science and art. There is nothing wrong with any grouping formed utilising BPT, it is simply a question of how “well” a grouping tends to fit physical observation. To derive the PV spectrum, Storti et. Al. take the standard ZPF spectral energy density equation that describes the energy density in a region of space as a smooth cubic distribution and combine it with a Fourier distribution. This yields the beginning and endpoint of the spectrum. In other words, objects with mass polarise the ZPF which may be described as a Fourier distribution at the surface of the object. The surface is the equilibrium boundary between the energy contained within the object and the polarized state of the ZPF surrounding it. The difference between the EGM, PV and ZPF spectra is that the EGM spectrum commences incrementally above “0(Hz)” and approaches the Planck Frequency. The PV spectrum is mass specific and represents a bandwidth of the EGM spectrum commencing at a non-zero fundamental frequency. The EGM and PV spectra follow a Fourier distribution. The ZPF spectrum has the same frequency bandwidth as the EGM spectrum, but does not follow a Fourier distribution. So, the EGM spectrum is the polarized form of the ZPF spectrum, whilst the PV spectrum is an object specific subset of the EGM spectrum following a Fourier distribution. Note: the EGM spectrum is a simple, but extreme, extension of the EM spectrum. DAT’s and BPT bring to the research and design table, the following key elements17: • It helps to assess the reasonableness of a model and which variables it should contain. • It reduces the number of variables and parameters to a minimum. • It reduces the number of needed experiments, on computers as well as in the laboratory. • It provides the fundamental theory behind experiments on scale models. • It is a systematic method for the analysis of problems. • It forces you to make estimates and to understand the problem. • It helps you understand what is important and what is not. • It produces dimensionless equations with small (or large) parameters. • It facilitates a “reverse engineering” approach to gravity if a region of space-time on a laboratory test bench is considered to be the Experimental Prototype (EP) for the mathematical model produced by the application of DAT’s and BPT. Subsequently, the mathematical model can be applied to the EP for scaling purposes, leading to gravity control experiments. Note: DAT’s and BPT should be applied before numerical computations are done. 17

Norwegian University of Science and Technology, http://www.math.ntnu.no/~hanche/kurs/matmod/1998h/ http://www.math.ntnu.no/~hanche/notes/buckingham/ 55

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EGM develops a dynamic, kinematic and geometric equivalent of the ZPF, expressed in Fourier terms, which describes gravity at the surface of the Earth as a PV. The EGM spectrum is a simple, but extreme, extension of the EM spectrum. In the same way that radio waves, visible light, ultra violet, x-rays and gamma rays exist, gravitational waves exist as a spectrum of frequencies. The EGM spectrum is in fact the EM spectrum (subject to a Fourier distribution) but with an “end point” approaching the Planck Frequency at conditions of maximum permissible energy density. Typically, for the surface of the Earth for example, the vast majority of gravitational waves exist well above the Terahertz (THz) range. It is extremely important to note that gravity does not exist as a single wave; it exists as a spectrum of frequencies with a group propagation velocity of zero. EGM does not differentiate between EM and gravitational spectra but does predict the endpoint as being far above what we currently measure the EM spectrum to be. NOTES

56

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1.2.8 EGM: the natural philosophy of fundamental particles 1.2.8.1 How was it derived? To date, great strides have been made by GR to our understanding of gravity. It is an excellent tool that represents space-time as a geometric manifold of events, where gravitation manifests itself as a curvature of space-time and is described by a metric tensor. However, GR does not easily facilitate engineering solutions that may allow us to design electromechanical devices with which to affect the space-time metric. If mankind wishes to engineer the space-time metric, alternative tools must be developed to compliment those already available. Subsequently, the EGM methodology was derived to achieve this goal. EGM is defined as the modification of vacuum polarisability by applied EM fields. It provides a theoretical description of space-time as a PV derived from the superposition of EM fields. The PV representation of GR is a heuristic tool and is isomorphic to GR by weak field approximation. Utilising EGM, EM fields may be applied to affect the state of the PV and thereby facilitate interactions with the local gravitational field. BPT is a powerful tool that has been in existence, tried and experimentally proven for many years. It is an excellent tool that may be applied to the task of determining a practical relationship between gravitational acceleration and applied EM fields. The underlying principle of BPT is the preservation of dynamic, kinematic and geometric similarity between a mathematical model and an EP. EGM is a term describing a hypothetical harmonic relationship between Electricity, Gravity and Magnetism. The hypothesis may be mathematically articulated by the application of DAT’s and BPT, both being well established and thoroughly tested geometric engineering principles, via Fourier harmonics. The hypothesis may be tested by the correct derivation of experimentally verified fundamental properties not predicted within the Standard Model (SM) of Particle-Physics. To understand the way in which EGM was derived, one requires a basic knowledge of engineering principles. Primarily, EGM is a method of calculation (not a theory) based upon fundamental engineering principles and techniques. It does not compete with or contradict the SM of Particle-Physics in any manner. The creation and development of EGM was driven by necessity. A scan of contemporary approaches in gravitational Physics illustrates an obvious lack of mathematical tools facilitating engineering of the space-time manifold. Or rather, engineering possibilities are obvious, but require massive objects on a planetary, stellar or Galactic scale. Therefore, to facilitate gravity control, a new tool is required permitting engineering of the space-time manifold. To begin the process, we must first make some basic assumptions based upon the availability and practicality of existing tools by which we may construct further tools. We shall use one tool to build another. EGM is nothing more than an engineering tool constructed from other engineering tools and should be always regarded as such. Engineering is fundamentally a practical discipline that does not search for highly precise numerical or exact results. Instead, it aims to achieve physically meaningful quantitative solutions. Again, practicality and common sense must prevail and, by necessity, must commence with the assumption that any realistic attempt at gravity control must physically fit on a laboratory test bench. There is no benefit in developing a tool requiring non-practical scales of reality. Einstein brought forth the concept that mass and energy are interchangeable. This is trivially obvious by virtue of his now famous equation (E = mc2). This, combined with practical thinking, clearly suggests that EM radiation is the mechanism of choice. Hence, we have established the basic requirements going forward. That is, we are necessarily bounded in research and design terms by practical benchtop EM fields. The next step is to find a tool that facilitates the construction of relationships tying EM fields to acceleration. For an experienced engineer, the answer is obvious. In situations where little has been established previously, DAT's and BPT are solid first steps. In addition to being able to 57

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connect seemingly unrelated parameters, it also serves to minimise the number of experiments required to investigate physical behaviour. Mainstream understanding of gravity is based upon GR - a geometric approach. Assuming Einstein was correct and the enormous collective scientific effort since 1905 has not been a poor investment, it follows that any geometrically based engineering tool is an excellent starting point. Being geometric in nature makes it ideally suited to gravitational problems in keeping with BPT. However, a strict GR approach is unwieldy and a simpler description would be highly advantageous. Subsequently, Storti et. Al. utilise the PV model of gravity as a substitute to GR, which is isomorphic in the weak field, is conducive to engineering approaches and facilitates the development of the EGM construct. Thus far, we have established several of the baseline elements forming a skeletal EGM structure. To add flesh, we require a way to relate the geometric output of BPT to the PV model of gravity. The relationship between the two may be bridged by assuming the equivalence principle applies cross discipline. Considering the need for an EM mechanism, we shall assume that the PV model of gravity denotes a polarized state of the ZPF representing a sinusoidal manifestation of the space-time manifold by virtual particles, Photons or wavefunctions. Consequently, it follows that the representation of gravity at a mathematical point by Fourier Harmonics is a useful tool by which to represent the ZPF. Therefore, we may relate the logic of the preceding arguments in a solution algorithm constituting the EGM construct by five simple steps as follows, i. Apply DAT's, BPT and similarity principles to combine Electricity, Magnetism and resultant EM acceleration in the form of Π groupings. ii. Apply the equivalence principle to the Π groupings formed in (i). iii. Apply Fourier Harmonics to the equivalence principle. iv. Apply ZPF Theory to Fourier Harmonics. v. Apply the PV model of gravity to the ZPF. Hence, the complete EGM derivation process flow was constructed by Storti et. Al. in [1-19] as follows, Dimensional Analysis Techniques ↓ Buckingham Π Theory ↓ General Modelling Equations ↓ Amplitude and Frequency Spectra ↓ General Similarity Equations ↓ Harmonic Similarity Equations ↓ Reduced Harmonic Similarity Equations ↓ nd 2 Reduction of Harmonic Similarity Equations (Reduced Average Harmonic Similarity Equations) ↓ Spectral Similarity Equations ↓ Fundamental Particle Properties, the Hydrogen Atom Spectrum and the Casimir Force Table 4.2, 58

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It was shown that a theoretical representation of constant acceleration at a mathematical point in a gravitational field may be defined by a summation of trigonometric terms utilising modified complex Fourier series in exponential form, according to the harmonic distribution “nPV = -N, 2 - N ... N”, where “N” is an odd number harmonic. Hence, the magnitude of the gravitational acceleration vector “g” (via the equivalence principle) may be usefully represented by Eq. (3.63) as “|nPV| → ∞”, g( r , M )

G. M . 2

r

n PV

2 . i . π .n PV .ω e π . n PV

.. PV ( 1 , r , M ) t i

(3.63)

such that, the frequency spectrum of the harmonic gravitational field “ωPV” is given by Eq. (3.67), ω PV n PV, r , M

n PV 3 2 . c . G. M . . K ( r, M ) PV r π .r

(3.67)

where, Variable ωPV(1,r,M) KPV

nPV r M G

Description Units Fundamental spectral frequency Hz Refractive Index of a gravitational field in the Polarisable Vacuum (PV) model of gravity, only contributing significantly when a large gravitational mass (i.e. a strong gravitational field) is considered. For None all applications herein, the effect is approximated to KPV(r,M) = 1. Harmonic modes of the gravitational field Magnitude of position vector from centre of mass m Mass kg Gravitational constant m3kg-1s-2 Table 4.3,

Subsequently, the harmonic (Fourier) representation of the magnitude of the gravitational acceleration vector (in the time domain) at the surface of the Earth up to “N = 21” is graphically shown to be,

Gravitational Acceleration

g

Time

Figure 4.3: harmonic representation of gravitational acceleration, As “N → ∞”, the magnitude of the gravitational acceleration vector becomes measurably constant. Hence, Eq. (3.63, 3.67) illustrate that the Newtonian representation of “g” is easily harmonised over the Fourier domain, from geometrically based methods (i.e. DAT’s and BPT). Therefore, unifying (in principle) Newtonian, geometric (relativistic) and quantum (harmonised) models of gravity. Storti et. Al. showed that the spectrum defined by Eq. (3.67) is discrete and finite. The lower boundary value is given by “ωPV(1,r,M)”, whilst the upper boundary value “ωΩ” (also termed the harmonic cut-off frequency) is given by Eq. (3.73), ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

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(3.73) www.deltagroupengineering.com

supported by the following equation set, n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

1

(3.71)

3

Ω ( r, M )

108.

U m( r , M )

12. 768 81.

U ω( r , M )

U m( r , M )

U ω( r , M )

3 .M .c

U m( r , M )

2

U ω( r , M )

(3.72)

2

4 .π .r

(3.70)

h . 4 ω PV( 1 , r , M ) 3 2 .c

(3.69)

3

where, Variable nΩ Ω Um Uω h

Description Units None Harmonic cut-off mode [mode number at ωΩ] Harmonic cut-off function Mass-energy density of a solid spherical gravitational object Pa Energy density of mass induced gravitational field scaled to the fundamental spectral frequency Planck’s Constant [6.6260693 x10-34] Js Table 4.4,

Since the relationship between trigonometric terms, at each amplitude and corresponding frequency, is mathematically defined by the nature of Fourier series, the derivation of Eq. (3.71, 3.72) is based on the compression of energy density to one change in odd harmonic mode whilst preserving dynamic, kinematic and geometric similarity in accordance with BPT. The preservation of similarity across one change in odd mode is due to the mathematical properties of constant functions utilising Fourier series as discussed in [4]. The subsequent application of these results to Eq. (3.63) acts to decompress the energy density over the Fourier domain yielding a highly precise reciprocal harmonic representation of “g” whilst preserving dynamic, kinematic and geometric similarity to Newtonian gravity, identified by the “compression technique” stated above. Key gravitational characteristics for the Earth18 in the displacement domain may be graphically represented as follows,

Fundamental Frequency

RE

ω PV 1 , r , M E ω PV 1 , R E , M E

r Radial Distance

Figure19 3.7,

18 19

“RE” and “ME” denote the radius and mass of the Earth respectively. Fundamental frequency (|nPV| = 1) as a function of planetary radial displacement. 60

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RE n Ω R E, M E n Ω r, M E ω Ω r, M E ω Ω R E, M E

r Radial Distance

Cutoff Mode Cutoff Frequency

Figure20 3.8, 1.2.8.2 Poynting Vector “Sω” It was demonstrated by “Haisch, Puthoff and Rueda” in [25-28] that “inertia” may have ElectroMagnetic (EM) origins due to the ZPF of Quantum-Electro-Dynamics (QED), manifested by the Poynting Vector, via the equivalence principle. Hence, it follows that gravitational acceleration may also be EM in nature and the Polarizable Vacuum (PV) model of gravity is an EM polarized state of the ZPF with a Fourier distribution, assigning physical meaning to Eq. (3.63). Subsequently, it follows that the energy density of a mass induced gravitational field may be scaled to changes in odd harmonic mode numbers satisfying the mathematical properties of any constant function described in terms of Fourier series utilising Eq. (3.69) - such that, U ω n PV, r , M

U ω( r , M ) .

n PV

2

4

4

n PV

(3.68)

Therefore, the Poynting Vector21 of the polarized Zero-Point (ZP) gravitational field “Sω” surrounding a solid spherical object with homogeneous mass-energy distribution is given by, S ω n PV, r , M

c .U ω n PV, r , M

(3.74)

ZPF Poynting Vector

and may be graphically represented as follows,

S ω n PV , R E , M E

n PV Harmonic

Figure 3.9, Fig. (3.9) illustrates that the Poynting Vector of the ZP gravitational field increases with “nPV”. Further work by Storti et. Al. showed that “>>99.99(%)” of the effect in a gravitational field 20 21

Harmonic cut-off mode “nΩ” and frequency “ωΩ” as a function of planetary radial displacement. Per change in odd harmonic mode number. 61

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exists well above the “THz” range. Hence, it becomes apparent that “nΩ” and “ωΩ” are important characteristics of gravitational fields and were used to “quasi-unify” Particle-Physics in harmonic form. 1.2.8.3 The size of the Proton, Neutron and Electron (radii: “rπ”, “rν”, “rε”) In 2005, Storti et. Al. derived the mass-energy threshold of the Photon utilising “nΩ” and the classical Electron radius as shown in [8], to within “4.3(%)” of the Particle Data Group (PDG) value22 stated in [29], then proceeded to derive the mass-energies and radii of the Photon and Graviton in [10] by the consistent utilisation of “nΩ”. The method developed in [8] was re-applied in [9] to derive the sizes23 of the Electron, Proton and Neutron. The motivation for this was to test the hypothesis presented in Ch. 1.2.8.1 by direct comparison of the computed size values to experimentally measured fact. They believe that highly precise computational predictions’, agreeing with experimental evidence beyond the abilities of the SM to do so, is conclusive evidence of the validity of the harmonic method developed. To date, highly precise measurements have been made of the Root-Mean-Square (RMS) charge radius of the Proton by [30] and the Mean-Square (MS) charge radius of the Neutron as demonstrated in [31]. However, the calculations presented in [9] are considerably more accurate than the physical measurements articulated in [30,31], lending support for the harmonic representation of the magnitude of the gravitational acceleration vector stated in Eq. (3.63). The basic approach utilised in [9] was to determine the equilibrium position between the polarized state of the ZPF and the mass-energy of the fundamental particle inducing space-time curvature as would appear in General Relativity (GR). In other words, one may consider the curvature of the space-time manifold surrounding an object to be a “virtual fluid” in equilibrium with the object itself24. This concept is graphically represented in Fig. (4.4). A free fundamental particle with classical form factor is depicted in equilibrium with the surrounding space-time manifold. The ZPF is polarized by the presence of the particle in accordance with the PV model of gravity, which is (at least) isomorphic to GR in the weak field.

Figure 4.4: free fundamental particle with classical form factor, In the case of the Proton, the ZPF equilibrium radius coincides with the RMS charge radius “rπ” [Eq. (3.199)] producing the experimentally verified result “rp” by the SELEX Collaboration as stated in [30]25,

22

Consistent with experimental evidence and interpretation of data. From first principles and from a single paradigm. 24 The intention is not to suggest that the space-time manifold is actually a fluid, it is merely to present a method by which to solve a problem. 25 rπ = 0.8306(fm), rp = 0.8307 ± 0.012(fm). 23

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rπ

h .m e

5

4

. . m . 27 h c . e 2 3 4 .π .G m p 16.c .π .m p

(3.199)

where, “me” and “mp” denote Electron and Proton rest-mass respectively. In the case of the Neutron, the ZPF equilibrium radius coincides with the radial position of zero charge density “rν” [Eq. (3.200)] with respect to the Neutron charge distribution as illustrated in Fig. (4.5). It is shown in [14] that “rν” relates to the MS charge radius “KS” by a simple formula [Eq. (3.396)] producing the experimentally verified result “KX” as presented in [31]26, h .m e

5

4

. . m . 27 h c . e rν 2 3 4 .π .G m n 16.c .π .m n

(3.200)

where, “mn” denotes Neutron rest-mass. Neutron Charge Distribution

Charge Density

rν

r dr

ρ ch( r ) ρ ch r 0

r dr

5. 3

rν

ρ ch r dr

r Radius

Charge Density Maximum Charge Density Minimum Charge Density

Figure 4.5: Neutron charge distribution, KS

3. π .r ν 8

2

. (1

x) . x

1

x x

3

2

(3.396)

27

where, “x” is solved numerically within the “MathCad” environment by the following algorithm, Given 2

x

ln( x) . 2

x x

1

(3.398)

1 3

(3.399)

Find ( x)

Utilising “KS”, “KX” may be converted to determine an experimental zero charge density radial position value “rX” according to Eq. (3.418),

26 27

rν = 0.8269(fm), KS = -0.1133(fm2), KX = -0.113 ± 0.005(fm2). x = 0.6829, rX = 0.8256 ± 0.018(fm). 63

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rν

rX

KS

. K .K S X

(3.418)

In the case of the Electron (as with the Proton), the ZPF equilibrium radius coincides with the RMS charge radius “rε” [Eq. (3.203)] producing an experimentally implied result28 as stated in [32], 9

r ε r e.

1. 2

ln 2 .n Ω r e , m e

5

γ

(3.203)

where, “re” and “γ” [33] denote the classical Electron radius and Euler-Mascheroni constant respectively. 1.2.8.4 The harmonic representation of fundamental particles 1.2.8.4.1 Establishing the foundations Motivated by the physical validation of Eq. (3.199, 3.200), Storti et. Al. conducted thought experiments in [9] to investigate harmonic and trigonometric relationships by analysing various forms of radii combinations for the Electron, Proton and Neutron consistent with the DAT’s and BPT derivations in [1-8] – yielding the following useful approximations, ω Ω r ε, m e

ω Ω r ε, m e

ω Ω r π, m p

ω Ω r ν ,mn

rε rπ

α

2

(4.1)

π rν

rε

(3.214) 2

.e

3

rπ

(3.204)

where, i. ii.

iii. iv.

“α” and “e” denote the fine structure constant and exponential function respectively. Eq. (4.1) error: (a) Associated with “ωΩ(rε,me)/ωΩ(rπ,mp) = 2” is “8.876 x10-3(%)” (b) Associated with “ωΩ(rε,me)/ωΩ(rν,mn) = 2” is “0.266(%)”. Eq. (3.214) error is “2.823(%)”. Eq. (3.204) error is “0.042(%)”.

1.2.8.4.2 Improving accuracy Since the experimental value of the RMS charge radius of the Proton is considered by the scientific community to be precisely known29, the accuracy of Eq. (3.214, 3.204) may be improved by re-computing the value of “rν” and “rε”. This action further strengthens the validity of Eq. (4.1) by verifying trivial deviation utilising the re-computed values. Hence, it follows that numerical solutions for “rν” and “rε”, constrained by exact mathematical statements [Eq. (3.203, 3.204, 3.214, 4.1)], suggests that the gravitational relationship between the Electron and Proton, as inferred by the result “ωΩ(rε,me)/ωΩ(rπ,mp) = 2”, is harmonic. 28 29

rε ≥ 0.0118(fm), γ = 0.577215664901533. To a degree of accuracy significantly greater than the Electron or Neutron. 64

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The computational algorithm supporting this contention may be stated as follows, Given α

r ε ω Ω r ε, m e r e ω Ω r π, m p

rν rε

rε rπ

rε rν

9

2

.e

rπ

3

1. 2

ln 2 .n Ω r e , m e

5

γ

2 π

Find r ν , r ε

(4.2)

(4.3)

yields, rν

0.826838

rε

0.011802

.( fm)

(4.4)

where, i.

ii. iii.

Eq. (4.1) error: (a) Associated with “ωΩ(rε,me)/ωΩ(rπ,mp) = 2” is “4.493 x10-7(%)”. (b) Associated with “ωΩ(rε,me)/ωΩ(rν,mn) = 2” is “0.282(%)”. Eq. (3.214) error is “1.11 x10-13(%)”. Eq. (3.204) error is “0.026(%)”.

1.2.8.4.3 Formulating an hypothesis In the preceding calculations utilising known particle mass and radii as a reference, it was found that the harmonic cut-off frequency ratio of an Electron to a Proton was precisely “2”. This provokes the hypothesis that a simple harmonic pattern may exist describing the relationship of all fundamental particles relative to an arbitrarily chosen base particle according to, ω Ω r 1, M 1 ω Ω r 2, M 2

St ω

(3.230i)

Performing the appropriate substitutions utilising Eq.(3.69 – 3.73), Eq. (3.230i) may be simplified to, M1 M2

2

.

r2

5

r1

St ω

9

(3.230ii)

where, “Stω” represents the ratio of two particle spectra. Subsequently, “rε” may be simply calculated according to, 5

1 . me r ε r π. 9 2 mp

2

(3.231)

1.2.8.5 Identifying a mathematical pattern Utilising Eq. (3.230ii), Storti et. Al. identify mathematical patterns in [11-13] showing that “Stω” may be represented in terms of the Proton, Electron and Quark harmonic cut-off frequencies derived from the respective particle. Potentially, three new Leptons (L2, L3, L5 and associated Neutrino’s: ν2, ν3, ν5) and two new Quark / Boson’s (QB5 and QB6) are predicted, beyond the SM as shown in table (4.5).

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The EGM Harmonic Representation of Fundamental Particles (i.e. table (4.5)) is applicable to the size relationship between the Proton and Neutron (i.e. to calculate “rπ” from “rν” and viceversa utilising “Stω = 1”) as an approximation only. For precise calculations based upon similar forms, the reader should refer to Eq. (3.199, 3.200). Note: although the newly predicted Leptons are within the kinetic range30 and therefore “should have been experimentally detected”, there are substantial explanations discussed in the proceeding sections. Proton Electron Quark Harmonics Harmonics Harmonics Proton (p), Neutron (n) Stω = 1 Stω = 1/2 Stω = 1/14 2 1 1/7 Electron (e), Electron Neutrino (ν νe) 4 2 2/7 L2, ν2 (Theoretical Lepton, Neutrino) 6 3 3/7 L3, ν3 (Theoretical Lepton, Neutrino) 8 4 4/7 Muon (µ µ), Muon Neutrino (ν νµ) 10 5 5/7 L5, ν5 (Theoretical Lepton, Neutrino) 12 6 6/7 Tau (ττ), Tau Neutrino (ν ντ ) Up Quark (uq), Down Quark (dq) 14 7 1 Strange Quark (sq) 28 14 2 Charm Quark (cq) 42 21 3 Bottom Quark (bq) 56 28 4 QB5 (Theoretical Quark or Boson) 70 35 5 QB6 (Theoretical Quark or Boson) 84 42 6 W Boson 98 49 7 Z Boson 112 56 8 Higgs Boson (H) (Theoretical) 126 63 9 Top Quark (tq) 140 70 10 Table 4.5: harmonic representation of fundamental particles,

Existing and Theoretical Particles

1.2.8.6 Results 1.2.8.6.1 Harmonic evidence of unification Exploiting the mathematical pattern articulated in table (4.5), EGM predicts the RMS charge radius and mass-energy of less accurately known particles, comparing them to expert opinion. The values of “Stω” shown in table (4.5), predict possible particle mass and radii for all Leptons, Neutrinos, Quarks and Intermediate Vector Bosons (IVB’s), in complete agreement with the SM, PDG estimates and studies by Hirsch et. Al in [34] as shown in table (4.6), Particle Proton (p) Neutron (n) Electron (e) Muon (µ µ) Tau (ττ) Electron Neutrino (ν ν e) Muon Neutrino (ν νµ) 30

EGM Radii x10-16(cm) rπ = 830.5957 rν = 826.8379 rε = 11.8055 rµ = 8.2165 rτ = 12.2415 ren ≈ 0.0954 rµn ≈ 0.6556

EGM Mass-Energy (computed or utilised)

PDG Mass-Energy Range (2005 Values)

Mass-Energy precisely known, See: National Institute of Standards and Technology (NIST) [35] Note: δm = 10-100 men(eV) ≈ 3 - δm mµn(MeV) ≈ 0.19 - δm

men(eV) < 3 mµn(MeV) < 0.19

A region extensively explored in Particle-Physics experiments. 66

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Tau Neutrino (ν ν τ) rτn ≈ 1.9588 mτn(MeV) ≈ 18.2 - δm mτn(MeV) < 18.2 Up Quark (uq) 1.5 < muq(MeV) < 4 ruq ≈ 0.7682 muq(MeV) ≈ 3.5060 Down Quark (dq) 3 < mdq(MeV) < 8 rdq ≈ 1.0136 mdq(MeV) ≈ 7.0121 Strange Quark (sq) 80 < msq(MeV) < 130 rsq ≈ 0.8879 msq(MeV) ≈ 113.9460 1.15 < mcq(GeV) < 1.35 Charm Quark (cq) mcq(GeV) ≈ 1.1833 rcq ≈ 1.0913 Bottom Quark (bq) 4.1 < mbq(GeV) < 4.4 rbq ≈ 1.071 mbq(GeV) ≈ 4.1196 Top Quark (tq) 169.2 < mtq(GeV) < 179.4 rtq ≈ 0.9294 mtq(GeV) ≈ 178.4979 W Boson 80.387 < mW(GeV) < 80.463 rW ≈ 1.2839 mW(GeV) ≈ 80.425 Z Boson 91.1855 < mZ(GeV) < 91.1897 rZ ≈ 1.0616 mZ(GeV) ≈ 91.1876 Higgs Boson (H) mH(GeV) ≈ 114.4 + δm mH(GeV) > 114.4 rH ≈ 0.9403 Photon (γγ) rγγ = ½Kλλh mγγ ≈ 3.2 x10-45(eV) mγ < 6 x10-17(eV) No definitive commitment Graviton (γγg) rgg = 2(2/5)rγγ mgg = 2mγγ L2 (Lepton) mL(2) ≈ 9(MeV) rL ≈ 10.7518 mL(3) ≈ 57(MeV) L3 (Lepton) L5 (Lepton) mL(5) ≈ 566(MeV) ν2 (L2 Neutrino) rν2,ν3,ν5 mν2 ≈ men Not predicted or considered ≈ ν3 (L3 Neutrino) mν3 ≈ mµn ren,µn,τn ν5 (L5 Neutrino) mν5 ≈ mτn QB5 (Quark or Boson) rQB ≈ 1.0052 mQB(5) ≈ 10(GeV) QB6 (Quark or Boson) mQB(6) ≈ 22(GeV) Table 4.6: RMS charge radii and mass-energies of fundamental particles, where, i. “Kλ” denotes a Planck scaling factor, determined to be “(π/2)1/3” in [13]. ii. “λh” denotes Planck Length [4.05131993288926 x10-35(m)]. iii. “rL” and “rQB” denote the average radii of SM Leptons and Quark / Bosons (respectively) utilised to calculate the mass-energy of the proposed “new particles”. Note: iv. A formalism for the approximation of ν2, ν3 and ν5 mass-energy is shown in [15]. v. It is shown in [8,10,13] that the RMS charge diameters of a Photon and Graviton are “λh” and “1.5λh” respectively, in agreement with Quantum Mechanical (QM) models. 1.2.8.6.2 Recent developments 1.2.8.6.2.1 PDG mass-energy ranges The EGM construct was finalized by Storti et. Al. in 2004 and tested against published PDG data of the day [i.e. the 2005 values shown in table (4.6)]. Annually, as part of their “continuous improvement cycle”, the PDG reconciles its published values of particle properties against the latest experimental and theoretical evidence. The 2006 changes in PDG mass-energy range values not impacting EGM are as follows: i. Strange Quark = “70 < msq(MeV) < 120”. ii. Charm Quark = “1.16 < mcq(GeV) < 1.34”. iii. “W” Boson = “80.374 < mW(GeV) < 80.432”. iv. “Z” Boson = “91.1855 < mZ(GeV) < 91.1897”. Therefore, we may conclude that the EGM construct continues to predict experimentally verified results within the SM to high computational precision.

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1.2.8.6.2.2 Electron Neutrino and Up / Down / Bottom Quark mass Particle-Physics research is a highly dynamic field supporting a landscape of constantly changing hues. The EGM construct relates “mass to size” in harmonic terms. If one applies Eq. (3.230ii) and utilizes the Proton as the reference particle in accordance with table (4.5), one obtains a single expression with two unknowns, as implied by Eq. (3.231). Since contemporary Physics is currently incapable of specifying the mass and size of most fundamental particles precisely and concurrently, EGM is required to approximate values of either mass or radius to predict one or the other (i.e. mass or size). Subsequently, the EGM predictions articulated in table (4.6) denote values based upon estimates of either mass or radius. Hence, some of the results in table (4.6) are approximations and subject to revision as new experimental evidence regarding particle properties (particularly mass), come to light. The 2006 changes in PDG mass-energy values affecting table (4.6) are shown below. In this data set, the EGM radii are displayed as a range relating to its mass-energy influence. Note: the average value of EGM “Up + Down Quark” mass from table (4.6) [i.e. 5.2574(MeV)] remains within the 2006 average mass range specified by the PDG [i.e. 2.5 to 5.5(MeV)]. EGM Radii x10-16(cm)

PDG Mass-Energy Range (2006 Values) men(eV) < 2 Electron Neutrino (ν νe) ren < 0.0811 PDG Mass-Energy Up Quark (uq) 1.5 < muq(MeV) < 3 0.5469 < ruq < 0.7217 Range (2006 Values) Down Quark (dq) 3 < mdq(MeV) < 7 0.7217 < rdq < 1.0128 Bottom Quark (bq) 1.0719 > rbq > 1.0863 4.13 < mbq(GeV) < 4.27 Table 4.7: RMS charge radii and mass-energies of fundamental particles, Particle

EGM Mass-Energy (utilised)

The predicted radii ranges above demonstrate that no significant deviation from table (4.6) values exists. This emphasizes that the EGM harmonic representation of fundamental particles is a robust formulation and is insensitive to minor fluctuations in particle mass, particularly in the absence of experimentally determined RMS charge radii. Therefore, we may conclude that the EGM construct continues to predict experimentally verified results within the SM to high computational precision. 1.2.8.6.2.3 Top Quark mass 1.2.8.6.2.3.1 The dilemma The Collider Detector at Fermilab (CDF) and “D-ZERO” (D0) Collaborations have recently revised their world average value of “Top Quark” mass from “178.0(GeV/c2)” in 2004 [36] to, “172.0” in 2005 [37], “172.5” in early 2006, then “171.4” in July 2006. [38] Note: since the precise value of “mtq” is subject to frequent revision, we shall utilize the 2005 value in the resolution of the dilemma as it sits between the 2006 values. 1.2.8.6.2.3.2 The resolution The EGM method utilizes fundamental particle RMS charge radius to determine mass. Currently, Quark radii are not precisely known and approximations were applied in the formulation of “mtq” displayed in table (4.6). However, if one utilizes the revised experimental value of “mtq = 172.0(GeV/c2)” to calculate the RMS charge radius of the Top Quark “rtq”, based on Proton 68

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harmonics, it is immediately evident that a decrease in “rtq” of “< 1.508(%)” produces the new world average value precisely. The relevant calculations may be performed simply as follows, The revised “Top Quark” radius based upon the “new world average Top Quark” mass, 5

GeV 172. 2 1 . c r π. 9 mp 140

2

= 0.9156 10

16 .

cm

(4.5)

The decrease in “Top Quark” RMS charge radius [relative to the table (4.6) value] based upon the “new world average Top Quark” mass becomes, r tq

1 = 1.5076 ( % )

5

GeV 172. 2 1 . c r π. 9 mp 140

2

(4.6)

where, “rtq” denotes the RMS charge radius of the “Top Quark” from table (4.6). Therefore, since the change in “rtq” is so small and its experimental value is not precisely known, we may conclude the EGM construct continues to predict experimentally verified results within the SM to high computational precision. Note: the 2006 value for revised “mtq” modifies the error defined by Eq. (4.6) to “< 1.65(%)”. 1.2.8.7 Discussion 1.2.8.7.1 Experimental evidence of unification Table (4.5, 4.6, 4.7) display mathematical facts demonstrating that all fundamental particles may be represented as harmonics of an arbitrarily selected reference particle, in complete agreement with the SM. Considering that the EGM method is so radically different and quantifies the physical world beyond contemporary solutions, one becomes tempted to disregard table (4.5, 4.6, 4.7) in favour of concluding these to be “coincidental”. However, it is inconceivable that such precision from a single paradigm spanning the entire family of fundamental particles could be “coincidental”. The derivation of the “Top Quark” massenergy is in itself, an astonishing result which the SM is currently incapable of producing. Moreover, the derivation of (a), EM radii characteristics of the Proton and Neutron (rπE, rπM and rνM) (b), the classical RMS charge radius of the Proton (c), the 1st term of the Hydrogen atom spectrum “λA” and (d), the Bohr radius “rx”: all from the same paradigm, strengthens the harmonic case. Additionally, Storti et. Al. demonstrate in “Quinta Essentia, A Practical Guide to SpaceTime Engineering, Part 3: pg. 54 (see: Ref.)” that the probability of coincidence is “<< 10-38” based upon the results shown in table (4.8), Particle / Atom EGM Prediction Proton (p) rπ = 830.5957 x10-16(cm) [9] rπE = 848.5274 x10-16(cm) [14] rπM = 849.9334 x10-16(cm) [14] rp = 874.5944 x10-16(cm) [14] Neutron (n) rν = 826.8379 x10-16(cm) [9] 69

Experimental Measurement rπ = 830.6624 x10-16(cm) [30] rπE = 848 x10-16(cm) [39,40] rπM = 857 x10-16(cm) [39,40] rp = 875.0 x10-16(cm) [35] rX ≈ 825.6174 x10-16(cm) [14]

(%) Error < 0.008 < 0.062 < 0.825 < 0.046 < 0.148

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Neutron (n) Top Quark (tq) Hydrogen (H)

where, i. ii. iii. iv.

KS = -0.1133 x10-26(cm2) [14] KX = -0.113 x10-26(cm2) [31] rνM = 878.9719 x10-16(cm) [14] rνM = 879 x10-16(cm) [39,40] mtq(GeV) ≈ 172.0 [37] mtq(GeV) ≈ 178.4979 [12,17] λA = 657.3290(nm) [16] λB = 656.4696(nm) [41] rx = 0.0527(nm) [16] rBohr = 0.0529(nm) [35] Table 4.8: experimentally verified EGM predictions,

< 0.296 < 0.003 < 3.64 < 0.131 < 0.353

“rπE” and “rπM” denote the Electric and Magnetic radii of the Proton respectively. “rνM” denotes the Magnetic radius of the Neutron. “λA” and “λB” denote the first term of the Hydrogen atom spectrum (Balmer series). “rp = 875.0 x10-16(cm)” and “rBohr = 0.0529(nm)” are not experimental values, they denote the classical RMS charge radius of the Proton and the Bohr radius, i.e. the official values listed by NIST.

Note: numerical simulations generating all values in table (4.5, 4.6, 4.8) can be found in [17-19]. 1.2.8.7.2 The answers to some important questions 1.2.8.7.2.1 What causes harmonic patterns to form? 1.2.8.7.2.1.1 ZPF equilibrium A free fundamental particle is regarded by EGM as a “bubble” of energy equivalent mass. Nature always seeks the lowest energy state: so surely, the lowest state for a free fundamental particle “should be” to diffuse itself to “non-existence” in the absence of “something” acting to keep it contained? This provokes the suggestion that a free fundamental particle is kept contained by the surrounding space-time manifold. In other words, free fundamental particles are analogous to “neutrally buoyant bubbles” floating in a locally static fluid (the space-time manifold). EGM is an approximation method, developed by the application of standard engineering tools, which finds the ZPF equilibrium point between the mass-energy equivalence of the particle and the space-time manifold (the ZPF) surrounding it - as depicted by Fig. (4.4). 1.2.8.7.2.1.2 Inherent quantum characteristics If one assumes that the basic nature of the Universe is built upon quantum states of existence, it follows that ZPF equilibrium is a common and convenient feature amongst free fundamental particles by which to test this assumption. Relativity tells us that no absolute frames of reference exist, so a logical course of action is to define a datum as EGM is derived from a gravitational base. In our case, it is an arbitrary choice of fundamental particle. To be representative of the quantum realm, it follows that ZPF equilibrium between free fundamental particles should also be analogous to quantum and fractional quantum numbers – as one finds with the “Quantum Hall Effect”. Subsequently, the harmonic patterns of table (4.5) form because the determination of ZPF equilibrium is applied to inherently quantum characteristic objects – i.e. fundamental particles. Hence, it should be no surprise to the reader that comparing a set of inherently quantum characterised objects to each other, each of which may be described by a single wavefunction at its harmonic cut-off frequency, results in a globally harmonic description. That is, the EGM harmonic representation of fundamental particles is a quantum statement of ZPF equilibrium – as one would expect. In-fact, it would be alarming if table (4.5), or a suitable variation thereof, could not be formulated. Therefore, harmonic patterns form due to inherent quantum characteristics and ZPF equilibrium. 70

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1.2.8.7.2.2 Why haven’t the “new” particles been experimentally detected? EGM approaches the question of particle existence, not just by mass as in the SM, but by harmonic cut-off frequency “ωΩ” (i.e. by mass and ZPF equilibrium). Storti et. Al. showed in [5] that the bulk of the PV spectral energy31 at the surface of the Earth exists well above the “THz” range. Hence, generalizing this result to any mass implies that the harmonic cut-off period32 “TΩ” defines the minimum detection interval to confirm (or refute) the existence of the proposed “L2, L3, L5” Leptons and associated “ν2, ν3, ν5” Neutrinos. In other words, a particle exists for at least the period specified by “TΩ” – i.e. its minimum lifetime. Quantum Field Theory (QFT) approaches this question from a highly useful, but extremely limited perspective compared to the EGM construct. QFT utilizes particle mass to determine the minimum detection period (in terms of eV) to be designed into experiments. To date, this approach has been highly successful, but results in the conclusion that no new Leptons exist beyond the SM in the mass-energy range specified by the proposed Leptons. Whilst QFT is a highly useful yardstick, it is by no means a definitive benchmark to warrant termination of exploratory investigations for additional particles. Typically in the SM, short lived particles are seen as resonances in cross sections of data sets and many Hadrons in the data tables are revealed in this manner. Hence, the SM asserts that the more unstable particles are, the stronger the interaction and the greater the likelihood of detection. The EGM construct regards the existing Leptons of the SM as long-lived particles. It also asserts that the SM does not adequately address the existence or stability of the extremely shortlived Leptons proposed. This assertion is supported by the fact that detection of these particles is substantially beyond current capabilities due to: i. The minimum detection interval (with negligible experimental error) being “< 10-29(s)”. ii. The possibility that the proposed Leptons are transient (intermediate) states of particle production processes which decay before detection. For example, perhaps an Electron passes through an “L2” phase prior to stabilization to Electronic form (for an appropriate production process). Subsequently, this would be not be detected if the transition process is very rapid and the accelerator energies are too low. iii. The possibility of statistically low production events. Hence: iv. The proposed Leptons are too short-lived to appear as resonances in cross-sections. v. The SM assertion that the more unstable particles are, the stronger the interaction and the greater the likelihood of detection is invalid for the proposed Leptons. Therefore, contemporary particle experiments are incapable of detecting the proposed Leptons at the minimum accelerator energy levels required to refute the EGM construct. 1.2.8.7.2.3 Why can all fundamental particles be described in harmonic terms? Because of the precise experimental and mathematical evidence presented in table (4.5, 4.6, 4.8). These results were achieved by construction of a model based upon a single gravitational paradigm. Moreover, Storti et. Al. also derives the Casmir force in [7] from [1-6] utilising Eq. (3.63, 3.67, 3.73).

31 32

“>> 99.99(%)”. The inverse of “ωΩ”. 71

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1.2.8.7.2.4 Why is EGM a method and not a theory? EGM is a method and not a theory because: (i) it is an engineering approximation and (ii), the mass and size of most subatomic particles are not precisely known. It harmonizes all fundamental particles relative to an arbitrarily chosen reference particle by parameterising ZPF equilibrium in terms of harmonic cut-off frequency “ωΩ”. The formulation of table (4.5) is a robust approximation based upon PDG data. Other interpretations are possible, depending on the values utilised. For example, if one re-applies the method presented in [12] based upon other data; the values of “Stω” in table (4.5) might differ. However, in the absence of exact experimentally measured mass and size information, there is little motivation to postulate alternative harmonic sequences, particularly since the current formulation fits the available experimental evidence extremely well. If all mass and size values were exactly known by experimental measurement, the main sequence formulated in [12] (or a suitable variation thereof) will produce a precise harmonic representation of fundamental particles, invariant to interpretation. Table (4.5) values cannot be dismissed due to potential multiplicity before reconciling how: i. “ωΩ”, which is the basis of the table (4.5) construct, produces Eq. (3.199, 3.200) as derived in [9]. These generate radii values substantially more accurate than any other contemporary method. In-fact, it is a noteworthy result that EGM is capable of producing the Neutron MS charge radius as a positive quantity. Conventional techniques favour the non-intuitive form of a negative squared quantity. ii. “ωΩ” is capable of producing “a Top Quark” mass value – the SM cannot. iii. EGM produces the results defined in table (4.8). iv. Extremely short-lived Leptons [i.e. with lifetimes of “< 10-29(s)”] cannot exist, or do not exist for a plausible harmonic interpretation. v. Any other harmonic interpretation, in the absence of exact mass and size values determined experimentally, denote a superior formulation. Therefore, EGM is a method facilitating the harmonic representation of fundamental particles. 1.2.8.7.2.5 What would one need to do, in order to disprove EGM? Explain how experimental measurements of charge radii and mass-energy by international collaborations such as CDF, D0, L3, SELEX and ZEUS in [30,36-38,42-45], do not correlate to EGM calculations. 1.2.8.7.2.6 Why does EGM produce current and not constituent Quark masses? The EGM method is capable of producing current and constituent Quark masses, only current Quark masses are presented herein. This manuscript is limited to current Quark masses because it is the simplest example of ZPF equilibrium applicable whereby a particle is treated as “a system” and the equilibrium radius is calculated. Determination of the constituent Quark mass is a more complicated process, but the method of solution remains basically the same. For example, Storti et. Al. calculate an experimentally implicit value of the Bohr radius in [16] by treating the atom as “a system” in equilibrium with the polarized ZPF.

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1.2.8.7.2.7 Why does EGM yield only the three observed families? This occurs because it treats all objects with mass as a system (e.g. the Bohr atom) in equilibrium with the polarized ZPF (the objects own gravitational field). Therefore, since fundamental particles with classical form factor denote fundamental states (or systems: Quarks in the Proton and Neutron) of polarized ZPF equilibrium, it follows that only the three families will be predicted. 1.2.8.8 What may the periodic table of elementary particles look like under EGM? Assuming “QB5,6” to be Intermediate Vector Bosons (IVB's), we shall conjecture that the periodic table of elementary particles may be constructed as follows, Types of Matter Group II Group III Up 14 Charm 42 Top 140 +2/3,1/2,[R,G,B] +2/3,1/2,[R,G,B] +2/3,1/2,[R,G,B] uq cq tq 1.5 < muq(MeV) < 3 ≈ 1.1833(GeV) ≈ 172.0(GeV) 28 Bottom Down 14 Strange 56 -1/3,1/2,[R,G,B] -1/3,1/2,[R,G,B] -1/3,1/2,[R,G,B] dq sq bq (GeV) < 4.27 3 < mdq(MeV) < 7 4.13 < m ≈ 113.9460(MeV) bq Electron 2 Muon 8 Tau 12 -1,1/2 -1,1/2 -1,1/2 e µ τ = 0.5110(MeV) = 105.7(MeV) = 1.777(GeV) 8 Tau Neutrino 12 Electron Neutrino 2 Muon Neutrino 0,1/2 0,1/2 0,1/2 νe νµ ντ < 2(eV) < 0.19(MeV) < 18.2(MeV) L2 4 L3 6 L5 10 -1,1/2 -1,1/2 -1,1/2 L2 L3 L5 ≈ 9(MeV) ≈ 57(MeV) ≈ 566(MeV) L2 Neutrino 4 L3 Neutrino 6 L5 Neutrino 10 0,1/2 0,1/2 0,1/2 ν2 ν3 ν5 ≈ men ≈ mµn ≈ mτn Standard Model and EGM Bosons Photon N/A Gluon ? QB6 84 Z Boson 112 1,Colour,1 1,Weak Charge,10-6 1,Weak Charge,10-6 1,Charge,α gl Q B6 Z γ -45 < 10(MeV) ≈ 22(GeV) ≈ 91.1875(GeV) ≈ 3.2 x10 (eV) Graviton N/A QB5 70 W Boson 98 Higgs Boson 126 2,Energy,10-39 1,Weak Charge,10-6 1,Weak Charge,10-6 0,Higgs Field,? QB 5 W H γg ≈ 10(GeV) ≈ 80.27(GeV) > 114.4(GeV) = 2mγγ Table 4.9: predicted periodic table of elementary particles, E GM Leptons

Standard Model Leptons

Quarks

Group I

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Legend Leptons

Quarks

Bosons Stω Name Stω Name Stω Charge(e),Spin,Colour Charge(e),Spin Spin,Source,*SC Symbol Symbol Symbol Mass-Energy Mass-Energy Mass-Energy (i) *Where, “SC” denotes coupling strength at “1(GeV)”. [46] (ii) The values of “Stω” in table (4.9) utilize the Proton as the reference particle. This is due to its RMS charge radius and mass-energy being precisely known by physical measurement.

Name

Table 4.9: particle legend, 1.2.8.9 What are the most important results determined by the EGM construct? The most important results determined by the EGM construct may be categorised into five main areas as follows: i. Polarisable Vacuum and Zero-Point-Field. ii. Photons, Gravitons and Euler's Constant. iii. All other particles. iv. The Casimir Force. v. The Planck scale and the Bohr radius. 1.2.8.9.1 PV and ZPF 1.2.8.9.1.1 Gravitational amplitude spectrum “CPV” G.M .

C PV n PV, r , M

2

r

2 . π n PV

(3.64)

1.2.8.9.1.2 Gravitational frequency spectrum “ωPV” n PV 3 2 . c . G. M . . K ( r, M ) PV r π .r

ω PV n PV, r , M

(3.67)

1.2.8.9.1.3 Harmonic cut-off frequency “ωΩ” ω Ω ( r , M ) n Ω ( r , M ) .ω PV( 1 , r , M )

(3.73)

1.2.8.9.2 Photons, Gravitons and Euler's Constant Note: Euler's Constant “γ” may be calculated by: (i) physical measurement of “mγγ” and (ii), the assumption that “2 x rγγ” is precisely equal to the experimentally implicit value of the Planck Length characterised by “Kλ x λh”. 1.2.8.9.2.1 The mass-energy of a Graviton “mgg” mgg = 2mγγ

(3.216)

1.2.8.9.2.2 The mass-energy of a Photon “mγγ” 3

m γγ

h . re

3

π .r e 2 .c .G.m e

.

512.G.m e 2

c .π

74

2

.

n Ω r e, m e ln 2 .n Ω r e , m e

γ

2

(3.220)

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1.2.8.9.2.3 The radius of a Photon “rγγ” 5

2

m γγ

r γγ r e .

m e .c

2

(3.225)

G.h . r µ

r γγ K ω .

c

rτ

3

(3.274)

1.2.8.9.2.4 The radius of a Graviton “rgg” r gg

5

4 .r γγ

(3.227)

1.2.8.9.3 All Other Particles 1.2.8.9.3.1 The Fine Structure Constant “α” α

rε

2

.e

3

rπ

(3.204) rµ

rε

α

rτ

.e

rν

(3.236)

1.2.8.9.3.2 Harmonic cut-off frequency ratio (the ratio of two particle spectra) “Stω” 2

ω Ω r 1, M 1

M1

ω Ω r 2, M 2

M2

5

9

.

r2

9

r1

St ω

(3.230)

1.2.8.9.3.3 Neutron Magnetic Radius “rνM” r dr rν r ν . ρ ch r νM

ρ ch ( r ) d r rν

(3.420)

1.2.8.9.3.4 Proton Electric Radius “rπE” r dr r ν . ρ ch r πE

ρ ch ( r ) d r rν

(3.423)

1.2.8.9.3.5 Proton Magnetic Radius “rπM” ∞ r ν . ρ ch r πM

ρ ch ( r ) d r r dr rν

75

(3.426)

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1.2.8.9.3.6 Classical Proton Root Mean Square Charge Radius “rp” r P r πE

1. 2

r νM

rν

(3.429)

1.2.8.9.3.7 The first term of the Hydrogen Spectrum (Balmer Series) “λA” •

By EGM: utilising the Bohr radius “rBohr” and the fundamental PV wavelength “λPV” λA

λ PV 1 , K ω .r Bohr , m p 2 .n Ω K ω .r Bohr , m p

(3.457)

1.2.8.10 Graphical representation of fundamental particles under EGM Illustrational (only) wavefunction “ψ” [Eq. (3.458)] based on Proton harmonics, sin St ω .2 .π .ω Ω r π , m p .t

ψ St ω , t

(3.458)

1. T Ω r π ,m p 2

ψ( 1, t ) ψ( 2, t ) ψ( 4, t )

0

5 .10

29

1 .10

28

1.5 .10

28

2 .10

28

2.5 .10

28

3 .10

28

3.5 .10

28

ψ( 6, t )

t

Proton, Neutron Electron, Electron Neutrino L2, v2 L3, v3

Figure 3.44,

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1 . T Ω r π ,m p

16

ψ( 8,t) ψ ( 10 , t ) ψ ( 12 , t )

5 .10

0

30

1 .10

29

1.5 .10

29

2 .10

29

2.5 .10

29

3 .10

29

3.5 .10

29

4 .10

29

4.5 .10

29

ψ ( 14 , t )

t

Muon, Muon Neutrino L5, v5 Tau, Tau Neutrino Up and Down Quark

Figure 3.45, 1 . T Ω r π ,m p

56

ψ ( 28 , t ) ψ ( 42 , t ) ψ ( 56 , t )

0

1 .10

30

2 .10

30

3 .10

30

4 .10

30

5 .10

30

6 .10

30

30

7 .10

8 .10

30

9 .10

30

1 .10

29

1.1 .10

29

1.2 .10

29

1.3 .10

29

ψ ( 70 , t )

t

Strange Quark Charm Quark Bottom Quark QB5

Figure 3.46,

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1 . T Ω r π ,m p

168

ψ ( 84 , t ) ψ ( 98 , t ) ψ ( 112 , t ) ψ ( 126 , t )

0

5 .10

31

1 .10

30

1.5 .10

30

2 .10

30

2.5 .10

30

3 .10

30

3.5 .10

30

4 .10

30

4.5 .10

30

ψ ( 140 , t )

t

QB6 W Boson Z Boson Higgs Boson Top Quark

Figure 3.47, 1.2.8.11 Concluding remarks about EGM A concise mathematical relationship, based upon homogeneity concepts inherent in BPT, augmented with Fourier series, has been used to combine gravitational acceleration and ElectroMagnetism into a method producing fundamental particle properties to extraordinary precision. This also results in the representation of fundamental particles as harmonic forms of each other. Additionally, the representation predicts the existence of new fundamental particles not found within the Standard Model – suggesting the following: i. An exciting avenue for community exploration, beyond the Standard Model. ii. The potential for new Physics at higher accelerator energies. iii. The potential for unification of fundamental particles. iv. Physical limitations on the value of two extremely important mathematical constants [i.e. “π” and “γ”] at the QM level – subject to uncertainty principles. NOTES

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1.2.9 The Hubble Constant “H0” The following statements are verbatim quotations from [24]. 1.2.9.1 Description Hubble’s Law is the statement in physical Cosmology that the red-shift in light coming from distant Galaxies is proportional to their distance. The law was first formulated by Edwin Hubble and Milton Humason in 1929 after nearly a decade of observations. It is considered the first observational basis for the expanding space paradigm and today serves as one of the most often cited pieces of evidence in support of the “Big-Bang”. The most recent calculation of the constant, using the satellite WMAP began in 2003, yielding a value of “71 ± 4(km/s/Mpc)”. As of August 2006, the figure had been refined using data from NASA's orbital Chandra X-ray Observatory to “77(km/s/Mpc)” with an uncertainty of 15(%). 1.2.9.2 Discovery In the decade before Hubble made his observations, a number of Physicists and Mathematicians had established a consistent theory of the relationship between space and time by using Einstein's field equations of GR. Applying the most general principles to the question of the nature of the Universe yielded a dynamic solution that conflicted with the then prevailing notion of a static Universe. In 1922, Alexander Friedmann derived his famous equations from GR, showing that the Universe might expand at a calculable rate. The parameter used by Friedman is known today as the scale factor which can be considered as a scale invariant form of the proportionality constant of Hubble's Law. Georges Lemaître independently found a similar solution in 1927. From the Friedmann equations, the Friedmann-Lemaître-Robertson-Walker metric was derived for a fluid with a given density and pressure. This idea of expanding space-time would eventually lead to the “Big-Bang” theory of Cosmology. Before the advent of modern Cosmology, there was considerable talk as to the size and shape of the Universe. In 1920, a famous debate took place between Harlow Shapley and Heber D. Curtis over this very issue with Shapley arguing for a small Universe the size of our “Milky-Way” Galaxy and Curtis arguing that the Universe was much larger. The issue would be resolved in the coming decade with Hubble's improved observations. Edwin Hubble did most of his professional astronomical observation work at Mount Wilson observatory, the world's most powerful telescope at the time. His observations of Cepheid variable stars in spiral nebulae enabled him to calculate the distances to these objects. Surprisingly, these objects were discovered to be at distances which placed them well outside the “Milky-Way”. The nebulae were first described as “island Universes”, and it was only later that the term “Galaxy” would be applied to them. Combining his measurements of Galactic distances with Vesto Slipher's measurements of the red-shifts, Hubble discovered a rough proportionality of the objects’ distances with their red-shifts. Though there was considerable scatter (now known to be due to peculiar velocities), Hubble was able to plot a trend line from the 46 Galaxies he studied and obtained a value for the Hubble constant of 500(km/s/Mpc), which is much higher than the currently accepted value due to errors in his distance calibrations - a frequent problem even for modern astronomers.

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In 1958, the first good estimate of “H0”, 75(km/s/Mpc), was published by Allan Sandage. But it would be decades before a consensus was achieved. After Hubble’s discovery was published, Albert Einstein abandoned his work on the Cosmological constant which he had designed to allow for a static solution to his equations. He would later term this work his “greatest blunder” since the belief in a static Universe was what prevented him from predicting the expanding Universe. Einstein would make a famous trip to Mount Wilson in 1931 to thank Hubble for providing the observational basis for modern Cosmology. 1.2.9.3 Interpretation The discovery of the linear relationship between recessional velocity and distance yields a straightforward mathematical expression for Hubble’s Law as “v = H0D” where, “v” is the recessional velocity due to red-shift, typically expressed in “km/s”. “H0” is Hubble's constant and corresponds to the value of “H” (often termed the Hubble parameter which is time dependent) in the Friedmann equations taken at the moment of observation denoted by the subscript “0”. This value is the same throughout the Universe for a given conformal time. “D” is the proper distance that the light had traveled from the Galaxy in the rest frame of the observer, measured in “MegaParsecs” (Mpc). For relatively nearby Galaxies, the velocity “v” can be estimated from the Galaxy’s red-shift “z” using the formula “v = zc “where, “c” is the speed of light. For far away Galaxies, “v” can be determined from “z” by using the relativistic Doppler Effect. However, the best way to calculate the recessional velocity and its associated expansion rate of space-time is by considering the conformal time associated with the Photon travelling from the distant Galaxy. In very distant objects, “v” can be larger than “c”. This is not a violation of the special relativity however because a metric expansion is not associated with any physical object’s velocity. In using Hubble's law to determine distances, only the velocity due to the expansion of the Universe can be used. Since gravitationally interacting Galaxies move relative to each other, independent of the expansion of the Universe, these relative velocities, called peculiar velocities, need to be considered when applying Hubble's law. The Finger of God Effect is one result of this phenomenon discovered in 1938 by Benjamin Kenneally. In systems that are gravitationally bound, such as Galaxies or our planetary system, the expansion of space is (more than) annihilated by the attractive force of gravity. The mathematical derivation of an idealised Hubble’s Law for a uniformly expanding Universe is a fairly elementary theorem of geometry in 3-dimensional Cartesian / Newtonian coordinate space, which considered as a metric space, is entirely homogeneous and isotropic (properties do not vary with location or direction). Simply stated the theorem is this: “Any two points which are moving away from the origin, each along straight lines and with speed proportional to distance from the origin, will be moving away from each other with a speed proportional to their distance apart”. The ultimate fate and age of the Universe can both be determined by measuring the Hubble constant today and extrapolating with the observed value of the deceleration parameter, uniquely characterised by values of total density parameter (Ω, Ω0). 80

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A so-called “closed Universe” (Ω, Ω0 > 1) comes to an end in a “Big-Crunch” and is considerably younger than its Hubble age. An “open or flat Universe” (Ω, Ω0 ≤ 1) expands forever and has an age that is closer its Hubble age. For the accelerating Universe that we inhabit, the age of the Universe is coincidentally very close to the Hubble age.

Figure 4.6, The value of Hubble parameter changes over time either increasing or decreasing depending on the sign of the deceleration parameter “q” which is defined by,

(W.3) In a Universe with a deceleration parameter equal to zero, it follows that “H = 1 / t”, where “t” is the time since the “Big-Bang”. A non-zero, time-dependent value of “q” simply requires integration of the Friedmann equations backwards from the present time to the time when the comoving horizon size was zero. It was long thought that “q” was positive, indicating that the expansion is slowing down due to gravitational attraction. This would imply an age of the Universe less than “1 / H”, for instance, a value for “q = ½” (one theoretical possibility) would give the age of the Universe as “2/3⋅H-1”. The discovery in 1998 that “q” is apparently negative means that the Universe could actually be older than “1 / H”. In-fact, independent estimates of the age of the Universe come out fairly close to “1/H”. Note: we may define the “Hubble age” (also known as the “Hubble time” or “Hubble period”) of the Universe as “1 / H”. 1.2.9.4 Olbers’ paradox The expansion of space summarised by the “Big-Bang” interpretation of Hubble’s Law is relevant to the old conundrum known as Olbers' paradox: if the Universe were infinite, static, and filled with a uniform distribution of stars (notice that this also requires an infinite number of stars), then every line of sight in the sky would end on a star, and the sky would be as bright as the surface of a star. However, the night sky is largely dark. Since the 1600’s, astronomers and other thinkers have proposed many possible ways to resolve this paradox, but the currently accepted resolution 81

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depends in part upon the “Big-Bang” theory and in part upon the Hubble expansion. In a Universe that exists for a finite amount of time, only the light of finitely many stars has had a chance to reach us yet, and the paradox is resolved. Additionally, in an expanding Universe distant objects recede from us, causing the light emanating from them to be red-shifted and diminished in brightness. Both effects contribute (the red-shift being the more important of the two; remember the original paradox was couched in terms of a static Universe) to the darkness of the night sky, providing a kind of confirmation for the Hubble expansion of the Universe. 1.2.9.5 Measuring the Hubble constant For most of the second half of the 20th century the value of “H0” was estimated to be 5090(km/s/Mpc). The value of the Hubble constant was the topic of a long and rather bitter controversy between Gérard de Vaucouleurs who claimed the value was 80 and Allan Sandage who claimed the value was 40. In 1996, a debate moderated by John Bahcall between Gustav Tammann and Sidney van den Bergh was held in similar fashion to the earlier Shapley-Curtis debate over these two competing values. This difference was partially resolved with the introduction of the Lambda-CDM model of the Universe in the late 1990’s. With these model observations of high-red-shift clusters at X-ray and microwave wavelengths using the Sunyaev-Zel'dovich Effect, measurements of anisotropies in the Cosmic Microwave Background Radiation (CMBR), and optical surveys all gave a value of around 70 for the constant. In particular the Hubble Key Project (led by Dr. Wendy L. Freedman, Carnegie Observatories) gave the most accurate optical determination in May 2001 with its final estimate of “72 ± 8(km/s/Mpc)”, consistent with a measurement of “H0” based upon Sunyaev-Zel'dovich Effect observations of many Galaxy clusters having similar accuracy. The highest accuracy CMBR determinations were “71 ± 4(km/s/Mpc)” by WMAP in 2003, and “70 +2.4/-3.2(km/s/Mpc)” for measurements up to 2006. The consistency of the measurements from all three methods lends support to both the measured value of “H0” and the Lambda-CDM model. A value for “q” measured from standard candle observations of “Type Ia supernovae”, which was determined in 1998 to be negative, surprised many astronomers with the implication that the expansion of the Universe is currently “accelerating” (although the Hubble factor is still decreasing with time). In August 2006, using NASA's Chandra X-ray Observatory, a team from NASA's Marshall Space Flight Center (MSFC) found the Hubble constant to be “77(km/s/Mpc)”, with an uncertainty of about 15(%). End of verbatim quotation. Note: the Particle Data Group (PDG) estimate “H0” to be “71(km/s/Mpc)”.

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1.2.10 CMBR temperature

Figure 4.7: credit: http://map.gsfc.nasa.gov/, 83

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The following statements are verbatim quotations from [24]. 1.2.10.1 Description In Cosmology, the Cosmic Microwave Background Radiation (CMBR), also referred as relic radiation, is a form of ElectroMagnetic radiation discovered in 1965 that fills the entire Universe. It has a thermal “2.725(K)” Black-Body spectrum which peaks in the microwave range at a frequency of “160.4(GHz)”, corresponding to a wavelength of “1.9(mm)”. Most cosmologists consider this radiation to be the best evidence for the hot “Big-Bang” model of the Universe. 1.2.10.2 Features The Cosmic Microwave Background is isotropic to roughly one part in 100,000: the RootMean-Square (RMS) variations are only “18(µK)”. The Far-Infrared Absolute Spectrophotometer (FIRAS) instrument on the NASA Cosmic Background Explorer (COBE) satellite has carefully measured the spectrum of the CMBR. FIRAS compared the CMBR with a reference “Black-Body” and no difference could be seen in their spectra. Any deviations from the “Black-Body” form that might still remain undetected in the CMBR spectrum over the wavelength range from “0.5 - 5(mm)” must have a weighted RMS value of (at most) 50 parts per million [0.005(%)] of the CMBR peak brightness. This made the CMBR spectrum the most precisely measured “Black-Body” spectrum in nature. The CMBR is a prediction of “Big-Bang” theory such that the early Universe was made up of hot plasma of Photons, Electrons and Baryons. The Photons were constantly interacting with the plasma through Thomson scattering. As the Universe expanded, adiabatic cooling (of which the Cosmological red-shift is an on-going symptom) caused the plasma to cool until it became favourable for Electrons to combine with Protons and form Hydrogen atoms. This happened at around “3,000(K)” or when the Universe was approximately 380,000 years old. At this point, the Photons did not scatter off the neutral atoms and began to travel freely through space. This process is called recombination or decoupling (referring to Electrons combining with nuclei and to the decoupling of matter and radiation respectively). The Photons have continued cooling ever since; they have now reached “2.725(K)” and their temperature will continue to drop as long as the Universe continues expanding. Accordingly, the radiation from the sky we measure today comes from a spherical surface, called the surface of last scattering, from which the Photons that decoupled from interaction with matter in the early Universe, 13.7 Billion years ago, are just now reaching observers on Earth. The “Big-Bang” suggests that the CMBR fills all of observable space, and that most of the radiation energy in the Universe is in the Cosmic Microwave Background, which only makes up a small fraction of the total density of the Universe. Two of the greatest successes of the “Big-Bang” theory are its prediction of its almost perfect “Big-Bang” spectrum and its detailed prediction of the anisotropies in the cosmic microwave background. The recent Wilkinson Microwave Anisotropy Probe (WMAP) has precisely measured these anisotropies over the whole sky down to angular scales of “0.2°”. These can be used to estimate the parameters of the standard Lambda-CDM model of the “Big-Bang”. Some information, such as the shape of the Universe, can be obtained straightforwardly from the CMBR, while others, such as the Hubble constant, are not constrained and must be inferred from other measurements.

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1.2.10.3 Relationship to the “Big-Bang” 1.2.10.3.1 General The standard hot “Big-Bang” model of the Universe requires that the initial conditions for the Universe are a Gaussian random field with a nearly scale invariant or Harrison-Zel'dovich spectrum. This is, for example, a prediction of the cosmic inflation model. This means that the initial state of the Universe is random, but in a clearly specified way such that meaningful statements about the in-homogeneities in the Universe need to be statistical in nature. This leads to cosmic variance in which the uncertainties in the variance of the largest scale fluctuations observed in the Universe are difficult to accurately compare to theory. 1.2.10.3.2 Temperature The CMBR and the Cosmological red-shift are together regarded as the best available evidence for the “Big-Bang” theory. The discovery of the CMBR in the mid-1960s curtailed interest in alternatives such as the steady state theory. The CMBR gives a snapshot of the Universe when, according to standard Cosmology, the temperature dropped enough to allow Electrons and Protons to form Hydrogen atoms, thus making the Universe transparent to radiation. When it originated some 400,000 years after the “Big-Bang” – this time period is generally known as the “time of last scattering” or the period of recombination or decoupling – the temperature of the Universe was about “3,000(K)”. This corresponds to energy of about “0.25(eV)”, which is much less than the “13.6(eV)” ionization energy of Hydrogen. Since then, the temperature of the radiation has dropped by a factor of roughly 1100 due to the expansion of the Universe.

Figure 4.8: credit: http://map.gsfc.nasa.gov/, Note: as the Universe expands, the CMBR Photons are red-shifted, making the CMBR temperature inversely proportional to the Universe’s scale length. 85

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1.2.10.3.3 Primary anisotropy The anisotropy of the CMBR is divided into two sorts: primary anisotropy – which is due to effects which occur at the last scattering surface and before – and secondary anisotropy – which is due to effects such as interactions with hot gas or gravitational potentials, between the last scattering surface and the observer. The structure of the CMBR anisotropies is principally determined by two effects: acoustic oscillations and diffusion damping (also called collision-less damping or Silk damping). The acoustic oscillations arise because of a competition in the Photon-Baryon plasma in the early Universe. The pressure of the Photons tends to erase anisotropies, whereas the gravitational attraction of the Baryons, moving at speeds “<< c”, makes them tend to collapse to form dense haloes. These two effects compete to create acoustic oscillations which give the microwave background its characteristic peak structure. The peaks correspond, roughly, to resonances in which the Photons decouple when a particular mode is at its peak amplitude.

Figure 4.9: credit: http://pdg.lbl.gov/, The peaks contain interesting physical signatures. The angular scale of the first peak determines the curvature of the Universe (but not the topology of the Universe). The second peak – truly the ratio of the odd peaks to the even peaks – determines the reduced Baryon density. The third peak can be used to extract information about the dark matter density. The location of the peaks also gives important information about the nature of the primordial density perturbations. There are two fundamental types of density perturbations – called “adiabatic” and “isocurvature”. A general density perturbation is a mixture of these two types and different theories that purport to explain the primordial density perturbation spectrum predict different mixtures. For adiabatic density perturbations, the fractional over-density in each matter component (Baryons, Photons etc.) is the same. That is, if there is “1(%)” more energy in Baryons than average in one location, then with pure adiabatic density perturbations there is also “1(%)” more energy in Photons and Neutrinos, than average. Cosmic inflation predicts that the primordial perturbations are adiabatic. 86

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With isocurvature density perturbations, the sum of the fractional over-densities is zero. That is, a perturbation with “1(%)” more energy in Baryons and Photons than average, and “2(%)” lower energy in Neutrinos than average, would be a pure isocurvature perturbation. Cosmic strings would produce mostly isocurvature primordial perturbations. The CMBR spectrum is able to distinguish between these two types of perturbations because they produce different peak locations. Isocurvature density perturbations produce a series of peaks whose angular scales are roughly in the ratio “1 : 3 : 5 ...”, while adiabatic density perturbations produce peaks whose locations are in the ratio “1 : 2 : 3 ...”. Observations are consistent with the primordial density perturbations being entirely adiabatic, providing key support for inflation, and ruling out many models of structure formation involving, for example, cosmic strings. Collision-less damping is caused by two effects when the treatment of the primordial plasma as a fluid begins to break down: i. The increasing mean free path of the Photons as the primordial plasma becomes increasingly rarefied in an expanding Universe. ii. The finite thickness of the last scattering surface (LSS), which causes the mean free path to increase rapidly during decoupling, even while some Compton scattering is still occurring. Note: these effects contribute about equally to the suppression of anisotropies on small scales, and give rise to the characteristic exponential damping tail seen in the very small angular scale anisotropies. The thickness of the LSS refers to the fact that the decoupling of the Photons and Baryons does not happen instantaneously, but instead requires an appreciable fraction of the age of the Universe up to that era. One method to quantify exactly how long this process took uses the Photon Visibility Function (PVF). This function is defined so that, denoting the PVF by “P(t)”, the probability that a CMBR Photon last scattered between time “t” and “t + dt” is given by “P(t)dt”. The maximum of the PVF (the time where it is most likely that a given CMBR Photon last scattered) is known quite precisely. The first-year WMAP results put the time at which “P(t)” is maximum as “372 +/- 14(kyr)”. This is often taken as the “time” at which the CMBR formed. However, to figure out how long it took the Photons and Baryons to decouple, we need a measure of the width of the PVF. The WMAP team found that the PVF is greater than half of its maximum value (the “full width at half maximum”, or FWHM) over an interval of “115 +/- 5(kyr)”. By this measure, decoupling took place over roughly 115,000 years and when it was complete, the Universe was roughly 487,000 years old. 1.2.10.3.4 Late time anisotropy Since the “Big-Bang”, the CMBR was modified by several physical processes collectively referred to as late-time anisotropy or secondary anisotropy. After the establishment of the CMBR, ordinary matter in the Universe was mostly in the form of neutral Hydrogen and Helium atoms, but from observations of Galaxies it seems that most of the volume of the Inter-Galactic Medium (IGM) today consists of ionized material (since there are few absorption lines due to Hydrogen atoms). This implies a period of reionization in which the material of the Universe breaks down into Hydrogen ions. The CMBR Photons scatter off free charges such as Electrons that are not bound in atoms. In an ionized Universe, such Electrons have been liberated from neutral atoms by ionizing (ultraviolet) radiation. Today these free charges are at sufficiently low density in most of the volume of the Universe that they do not measurably affect the CMBR. However, if the IGM was ionized at very early times when the Universe was denser, then there are two main effects on the CMBR: 87

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i. Small scale anisotropies are erased (just as when looking at an object through fog, details of the object appear fuzzy). ii. The Physics of how Photons scatter off free Electrons (Thomson scattering) induces polarisation anisotropies on large angular scales. This large angle polarisation is correlated with the large angle temperature perturbation. Note: both of these effects have been observed by the WMAP satellite, providing evidence that the Universe was ionized at very early times. Other effects that occur between reionization and our observation of the Cosmic Microwave Background which cause anisotropies include the Sunyaev-Zel’dovich Effect, such that a cloud of high energy Electrons scatters the radiation, transferring some energy to the CMBR Photons and the Sachs-Wolfe effect, thus causing Photons from the Cosmic Microwave Background to be gravitationally red-shifted or blue-shifted due to changing gravitational fields. 1.2.10.3.5 Polarisation The cosmic microwave background is polarized at the level of a few “µK”. There are two types of polarisation, called E-modes and B-modes. This is in analogy to Electrostatics in which the Electric Field (E-field) has a vanishing curl and the Magnetic Field (B-field) has a vanishing divergence. The E-modes arise naturally from Thomson scattering in in-homogeneous plasma. The B-modes, which have not been measured and are thought to have an amplitude of (at most) “0.1(µK), are not produced from the plasma Physics alone. They are a signal from cosmic inflation and are determined by the density of primordial gravitational waves. Detecting the B-modes will be extremely difficult, particularly given that the degree of foreground contamination is unknown and weak gravitational lensing signal mixes the relatively strong E-mode signal with the B-mode signal. 1.2.10.4 Microwave background observations Subsequent to the discovery of the CMBR, hundreds of Cosmic Microwave Background experiments have been conducted to measure and characterise the signatures of the radiation. The most famous experiment is probably the NASA Cosmic Background Explorer (COBE) satellite that orbited in 1989–1996, which detected and quantified the large scale anisotropies at the limit of its detection capabilities. Inspired by the initial COBE results of an extremely isotropic and homogeneous background, a series of ground and balloon-based experiments quantified CMBR anisotropies on smaller angular scales over the next decade. The primary goal of these experiments was to measure the angular scale of the first acoustic peak, for which COBE did not have sufficient resolution. These measurements were able to rule out cosmic strings as the leading theory of cosmic structure formation, and suggested cosmic inflation was the right theory. During the 1990’s, the first peak was measured with increasing sensitivity and by 2000, the BOOMERanG experiment reported that the highest power fluctuations occur at scales of approximately “1°”. Together with other Cosmological data, these results implied that the geometry of the Universe is flat. A number of ground-based interferometers provided measurements of the fluctuations with higher accuracy over the next three years, including the Very Small Array, Degree Angular Scale Interferometer (DASI) and the Cosmic Background Imager (CBI). In June 2001, NASA launched a second CMBR space mission (WMAP) to make much more precise measurements of the large scale anisotropies over the full sky. The first results from this mission, disclosed in 2003, were detailed measurements of the angular power spectrum to below degree scales, tightly constraining various Cosmological parameters. The results are broadly consistent with those expected from cosmic inflation as well as various other competing theories, and are available in detail at NASA's data center for CMBR. 88

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Although WMAP provided very accurate measurements of the large angular-scale fluctuations in the CMBR (structures about as large in the sky as the moon), it did not have the angular resolution to measure the smaller scale fluctuations which had been observed using previous ground-based interferometers. End of verbatim quotation.

Figure 4.10: WMAP - credit: http://map.gsfc.nasa.gov/,

Figure 4.11: CMBR history - credit: http://map.gsfc.nasa.gov/, 89

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Figure 4.12: credit: http://map.gsfc.nasa.gov/,

Figure 4.13: credit: http://map.gsfc.nasa.gov/,

Figure 4.14: credit: http://map.gsfc.nasa.gov/,

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Figure 4.15: credit: http://map.gsfc.nasa.gov/,

Figure 4.16: credit: http://map.gsfc.nasa.gov/,

Figure 4.17: credit: http://map.gsfc.nasa.gov/,

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NOTES

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2 Definition of Terms 2.1

Numbering conventions • •

2.2

References of the form “3.*” refer to “Quinta Essentia – Part 3”. References of the form “4.*” refer to “Quinta Essentia – Part 4”. Quinta Essentia – Part 3

2.2.1 Alpha Forms “αx” • An inversely proportional description of how energy density may result in acceleration. 2.2.2 Amplitude Spectrum • A family of wavefunction amplitudes. • The amplitudes associated with a frequency spectrum. • See: Frequency Spectrum. 2.2.3 Background Field • Reference to the background (ambient) gravitational field. • Reference to the local gravitational field at the surface of the Earth. 2.2.4 Bandwidth Ratio “∆ωR” • The ratio of the bandwidth of the ZPF spectrum to the Fourier spectrum of the PV. 2.2.5 Beta Forms “βx” • A directly proportional description of how energy density may result in acceleration. 2.2.6 Buckingham Π Theory (BPT) • Arrangement of variables determined by DAT's into Π groupings. These groupings represent sub-systems of dimensional similarity for scale relationships. • Minimises the number of experiments required to investigate phenomena. • See: DAT's. 2.2.7 Casimir Force “FPP” • Attractive (non-gravitational) force between two parallel and neutrally charged mirrored plates of equal area. 2.2.8 Change in the Number of Modes “∆nS” • The difference between the ZPF beat cut-off mode and the Mode Number at the Critical Boundary as a function of the Critical Ratio. • See: Mode Number “nβ”. • See: Critical Ratio “KR”. 2.2.9 Compton Frequency “ωCx” • The generalised definition of Compton frequency applied globally herein is: ωCx = mxc2 / h-bar = 2πm 2π xc2/ h = 2πc 2π 2/ λCx. • This is the only equation in which the “h-bar” form of Planck's constant is used.

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2.2.10 Cosmological Constant • A constant introduced into the equations of GR to facilitate a steady state Cosmological solution. • See: General Relativity. 2.2.11 Critical Boundary “ωβ” • Represents the lower boundary (commencing at the ZPF beat cut-off frequency) of the ZPF spectrum yielding a specific proportional similarity value. • See: Zero-Point-Field Beat Cut-Off Frequency “ωΩ ZPF”. • See: Critical Ratio “KR”. 2.2.12 Critical Factor “KC” • A proportional measure of the applied EM field intensity (or magnitude of Poynting Vectors) within an experimental test volume. • The ratio of two experimentally determined relationship functions. 2.2.13 Critical Field Strengths “EC and BC” • RMS strength values of applied Electric and Magnetic fields for complete dynamic, kinematic and geometric similarity with the background gravitational field. • See: Background Field. 2.2.14 Critical Frequency “ωC” • The minimum frequency for the application of Maxwell's Equations within an experimental context. 2.2.15 Critical Harmonic Operator “KR H” • A representation of the Critical Ratio at ideal dynamic, kinematic and geometric similarity utilising a unit amplitude spectrum. 2.2.16 Critical Mode “NC” • The ratio of the critical frequency to the fundamental harmonic frequency of the PV. • See: Critical Frequency “ωC”. • See: Fundamental Harmonic Frequency “ωPV(1,r,M)” 2.2.17 Critical Phase Variance “φC” • The difference in phase between applied Electric and Magnetic fields for complete dynamic, kinematic and geometric similarity with the background gravitational field. • See: Background Field. 2.2.18 Critical Ratio “KR” • A proportional indication of anticipated experimental configurations by any suitable measure. Typically, this is the magnitude of the ratio of the applied EM experimental fields to the ambient background gravitational field. 2.2.19 Curl • The limiting value of circulation per unit area. 2.2.20 DC-Offsets • A proportional value of applied RMS Electric and / or Magnetic fields acting to offset the applied function/s. 94

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2.2.21 Dimensional Analysis Techniques (DAT's) • Formal experimentally based research methods facilitating the derivation, from first principles, of any number or combination of parameters considered important by an experimentalist. • See: BPT. 2.2.22 Divergence • The rate at which “density” exits a given region of space. 2.2.23 Dominant Bandwidth • The bandwidth of the EGM spectrum which dominates gravitational effects. • See: Electro-Gravi-Magnetics (EGM) Spectrum. 2.2.24 Electro-Gravi-Magnetics (EGM) • A method of calculation (not a theory) based upon energy density. • Being a calculation method, it does not favour or bias any particular theory in the Standard Model of particle physics. • Developed as a tool for engineers to modify gravity. • The modification of vacuum polarisability based upon the superposition of EM fields. 2.2.25 Electro-Gravi-Magnetics (EGM) Spectrum • A simple but extreme extension of the EM spectrum (including gravitational effects) based upon a Fourier distribution. 2.2.26 Energy Density (General) • Energy per unit volume. 2.2.27 Engineered Metric • A metric tensor line element utilising the Engineered Refractive Index. 2.2.28 Engineered Refractive Index “KEGM” • An EM based engineered representation of the Refractive Index. 2.2.29 Engineered Relationship Function “∆K0(ω,X)” • A change in the Experimental Relationship Function resulting from a modification in the local value of the magnitude of acceleration by similarity of applied EM fields to the background gravitational field. 2.2.30 Experimental Prototype (EP) • Reference to the gravitational acceleration through a practical benchtop volume of space-time in a laboratory at the surface of the Earth. 2.2.31 Experimental Relationship Function “K0(ω,X)” • A proportional scaling factor relating an experimental prototype (typically herein, it is the local gravitational field or ambient physical conditions) to a mathematical model. 2.2.32 Fourier Spectrum • Two spectra combined into one (an amplitude spectrum and a frequency spectrum) obeying a Fourier Series. • See: Amplitude Spectrum. • See: Frequency Spectrum. 95

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2.2.33 Frequency Bandwidth “∆ωPV” • The bandwidth of the Fourier spectrum describing the PV. • See: Fourier Spectrum. • See: Polarisable Vacuum (PV). 2.2.34 Frequency Spectrum • A family of wavefunction frequencies. • The frequencies associated with an amplitude spectrum. • See: Amplitude Spectrum. 2.2.35 Fundamental Beat Frequency “∆ωδr(1,r,∆r,M)” • The change in fundamental harmonic frequency of the PV across an elemental displacement. • See: Fundamental Harmonic Frequency “ωPV(1,r,M)”. • See: Polarisable Vacuum (PV). 2.2.36 Fundamental Harmonic Frequency “ωPV(1,r,M)” • The lowest frequency in the PV spectrum utilising Fourier harmonics. 2.2.37 General Modelling Equations (GMEx) • Proportional solutions to the Poisson and Lagrange equations resulting in acceleration. 2.2.38 General Relativity (GR) • The representation of space-time as a geometric manifold of events where gravitation manifests itself as a curvature of space-time and is described by a metric tensor. 2.2.39 General Similarity Equations (GSEx) • Combines General Modelling Equations with the Critical Ratio by utilisation of the Engineered Relationship Function. • See: Critical Ratio “KR”. 2.2.40 Gravitons “γg” • Conjugate Photon pairs responsible for gravitation. This is an inherent mathematical conclusion arising from similarity modelling utilising a Fourier distribution in Complex form and the PV model of gravity considerate of ZPF Theory (due to harmonic symmetry about the “0th” mode). 2.2.41 Graviton Mass-Energy Threshold “mγg” • The upper boundary value of the mass-energy of a Graviton as defined by the Particle Data Group. 2.2.42 Group Velocity • The velocity with which energy propagates. 2.2.43 Harmonic Cut-Off Frequency “ωΩ” • The terminating frequency of the Fourier spectrum of the PV. • See: Fourier Spectrum. • See: Polarisable Vacuum (PV).

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2.2.44 Harmonic Cut-Off Function “Ω” • A mathematical function associated with the harmonic cut-off mode and frequency. • See: Harmonic Cut-Off Mode “nΩ”. • See: Harmonic Cut-Off Frequency “ωΩ”. 2.2.45 Harmonic Cut-Off Mode “nΩ” • The terminating mode of the Fourier spectrum of the PV. • See: Fourier Spectrum. • See: Polarisable Vacuum (PV). 2.2.46 Harmonic Inflection Mode “NX” • The mode at which the phase variance between the Electric and Magnetic wavefunctions describing the PV in a classical Casimir experiment begins to alter dramatically. • A conjectured resonant mode of the PV in a classical Casimir experiment. • See: Casimir Force “FPP”. • See: Polarisable Vacuum (PV). 2.2.47 Harmonic Inflection Frequency “ωX” • The frequency associated with the harmonic inflection mode. • See: Harmonic Inflection Mode “NX”. 2.2.48 Harmonic Inflection Wavelength “λX” • The wavelength associated with the harmonic inflection frequency. 2.2.49 Harmonic Similarity Equations (HSEx) • A harmonic representation of General Similarity Equations utilising the Critical Harmonic Operator. • A family of equations defined by relating the experimental prototype to a mathematical model (General Similarity Equations). • See: Critical Harmonic Operator “KR H”. • See: General Similarity Equations (GSEx). 2.2.50 IFF • If and only if. 2.2.51 Impedance Function • A measure of the ratio of the permeability to the permittivity of a vacuum. • Resistance to energy transfer through a vacuum. 2.2.52 Kinetic Spectrum • Another term for the ZPF spectrum. • See: ZPF Spectrum. 2.2.53 Mode Bandwidth • The modes associated with a frequency bandwidth. 2.2.54 Mode Number (Critical Boundary Mode) “nβ” • The ratio of the Critical Boundary frequency to the fundamental frequency of the PV. • The harmonic mode associated with the Critical Boundary frequency.

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2.2.55 Number of Permissible Modes “N∆r” • The number of modes permitted for the application of Maxwell's Equations within an experimental context, based upon the harmonic cut-off frequency. • See: Harmonic Cut-Off Frequency “ωΩ”. 2.2.56 Phenomena of Beats • The interference between two waves of slightly different frequencies. 2.2.57 Photon Mass-Energy Threshold “mγ” • The upper boundary value of the mass-energy of a Photon as defined by the Particle Data Group. 2.2.58 Polarisable Vacuum (PV) • The polarized state of the Zero-Point-Field due to mass influence. • Characterised by a Refractive Index. • Obeys a Fourier distribution. • A bandwidth of the EGM Spectrum. • See: Electro-Gravi-Magnetics (EGM). • See: Electro-Gravi-Magnetics (EGM) Spectrum. 2.2.59 Polarisable Vacuum (PV) Beat Bandwidth “∆ωΩ” • The change in harmonic cut-off frequency across an elemental displacement. • See: Harmonic Cut-Off Frequency “ωΩ”. • See: Phenomena of Beats. • See: Polarisable Vacuum (PV). 2.2.60 Polarisable Vacuum (PV) Spectrum • Another term for the Fourier spectrum applied by EGM to describe the PV harmonically. • A bandwidth of the EGM Spectrum. • See: Electro-Gravi-Magnetics (EGM). • See. Fourier Spectrum. • See: Polarisable Vacuum (PV). 2.2.61 Potential Spectrum • Another term for the PV spectrum. • See: Polarisable Vacuum (PV) Spectrum. 2.2.62 Poynting Vector • Describes the direction and magnitude of EM energy flow. • The cross product of the Electric and Magnetic field. 2.2.63 Precipitations • Results driven by the application of limits. 2.2.64 Primary Precipitant • The frequency domain precipitation. • See: Precipitations.

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2.2.65 Radii Calculations by Electro-Gravi-Magnetics (EGM) • Radii calculations by EGM represent the radial position of energetic equilibrium between the energy density of a homogeneous spherical mass with the ZPF. • The radii predictions calculated by EGM coincide with the RMS charge radii of all charged fundamental particles. • See: Electro-Gravi-Magnetics (EGM). • See: Zero-Point-Field (ZPF). 2.2.66 Range Factor “Stα” • The product of the change in energy density and the Impedance Function. • An “at-a-glance” tool indicating the boundaries of the applied energy requirements for complete dynamic, kinematic and geometric similarity with the background field. • See: Energy Density. • See: Background Field. • See: Impedance Function. 2.2.67 Reduced Average Harmonic Similarity Equations (HSExA R) • See: 2nd Reduction of the Harmonic Similarity Equations. 2.2.68 Reduced Harmonic Similarity Equations (HSEx R) • A simplification of the Harmonic Similarity Equations by substitution of RMS expressions for the time varying representations of applied Electric and Magnetic field harmonics. • A simplification of the Harmonic Similarity Equations facilitating the investigation of the effects of phase variance [on a modal (per mode) basis]. 2.2.69 Refractive Index “KPV” • Characterisation value of the PV. 2.2.70 Representation Error “RError” • Error associated with the mathematical representation of a physical system. 2.2.71 RMS Charge Radii (General) • The RMS charge radius refers to the RMS value of the charge distribution curve. 2.2.72 RMS Charge Radius of the Neutron “rν” • The RMS charge radius of a Neutron “rν” is so termed by analogy to the Neutron Mean Square charge radius “KX” which is typically represented as a squared length quantity “fm2”. Therefore, the dimensional square root of “KX” represents “rν” by analogy. • “rν” represents the cross-over radius (the node) on the Neutron charge distribution curve. 2.2.73 Similarity Bandwidth “∆ωS” • The difference between the ZPF beat cut-off frequency and the critical boundary frequency. • A measure of similarity between the background gravitational field spectrum and the applied field frequencies (commencing at the ZPF beat cut-off frequency). • See: Background Field. • See: Critical Boundary “ωβ”. • See: Zero-Point-Field Beat Cut-Off Frequency “ωΩ ZPF”.

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2.2.74 Spectral Energy Density “ρ0(ω)” • Energy density per frequency mode. 2.2.75 Spectral Similarity Equations (SSEx) • A representation of the complete spectrum of the PV utilising the 2nd Reduction of the Harmonic Similarity Equations by application of similarity principles. 2.2.76 Subordinate Bandwidth • The EM spectrum. • See: Dominant Bandwidth. • See: Electro-Gravi-Magnetics (EGM) Spectrum. 2.2.77 Unit Amplitude Spectrum • A harmonic representation of unity (the number one) utilising the amplitude spectrum of a Fourier distribution. 2.2.78 Zero-Point-Energy (ZPE) • The lowest possible energy of the space-time manifold described in quantum terms. 2.2.79 Zero-Point-Field (ZPF) • The field associated with ZPE. 2.2.80 Zero-Point-Field (ZPF) Spectrum • The spectrum of amplitudes and frequencies associated with the ZPF. 2.2.81 Zero-Point-Field (ZPF) Beat Bandwidth “∆ωZPF” • The difference between the ZPF beat cut-off frequency and the fundamental beat frequency. • See: Fundamental Beat Frequency “∆ωδr(1,r,∆r,M)”. • See: Zero-Point-Field (ZPF) Beat Cut-Off Frequency “ωΩ ZPF”. 2.2.82 Zero-Point-Field (ZPF) Beat Cut-Off Frequency “ωΩ ZPF” • The terminating frequency of the ZPF spectrum across an elemental displacement. 2.2.83 Zero-Point-Field (ZPF) Beat Cut-Off Mode “nΩ ZPF” • The terminating mode of the ZPF spectrum across an elemental displacement. 2.2.84 1st Sense Check “Stβ” • A common sense test relating the ZPF beat bandwidth to the Compton frequency of an Electron. • See: Compton Frequency “ωCx”. • See: Zero-Point-Field (ZPF) Beat Bandwidth “∆ωZPF”. 2.2.85 2nd Reduction of the Harmonic Similarity Equations (HSExA R) • A time averaged simplification of the Reduced Harmonic Similarity Equations. 2.2.86 2nd Sense Check “Stγ” • A common sense test relating the PV beat bandwidth to the Compton frequency of an Electron. • See: Compton Frequency “ωCx”. • See: Polarisable Vacuum (PV) Beat Bandwidth “∆ωΩ”. 100

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2.2.87 3rd Sense Check “Stδ” • A common sense test relating the harmonic cut-off mode across an elemental displacement. • See: Harmonic Cut-Off Mode “nΩ”. 2.2.88 4th Sense Check “Stε” • A common sense test relating the representation error across an elemental displacement. • See: Representation Error “RError”. 2.2.89 5th Sense Check “Stη” • A common sense test relating the harmonic cut-off frequency of a Proton to the Compton frequency of a Proton. • See: Compton Frequency “ωCx”. 2.2.90 6th Sense Check “Stθ” • A common sense test relating the harmonic cut-off frequency of a Neutron to the Compton frequency of a Neutron. • See: Compton Frequency “ωCx”. 2.2.91 Physical Constants Symbol α c G ε0 µ0 h h-bar λCe λCP λCN λCµ λCτ me mp mn mµ mτ re rp λh mh th ωh eV

Description Fine Structure Constant Velocity of light in a vacuum Universal Gravitation Constant Permittivity of a vacuum Permeability of a vacuum Planck's Constant Planck's Constant (2π form) Electron Compton Wavelength Proton Compton Wavelength Neutron Compton Wavelength Muon Compton Wavelength Tau Compton Wavelength Electron rest mass Proton rest mass Neutron rest mass Muon rest mass Tau rest mass Classical Electron radius Classical Proton RMS charge radius Planck Length Planck Mass Planck Time Planck Frequency Electron Volt

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NIST value utilised by EGM 7.297352568 x10-3 299792458 6.6742 x10-11 8.854187817 x10-12 4π x10-7 6.6260693 x10-34 1.05457168 x10-34

Units None m/s m3kg-1s-2 F/m N/A2 Js

= h / (me,p,n,µ,τ c)

m

9.1093826 x10-31 1.67262171 x10-27 1.67492728 x10-27 1.88353140 x10-28 3.16777 x10-27 2.817940325 x10-15 0.8750 x10-15 = √(Gh/c3) = √(hc/G) = √(Gh/c5) = 1/th 1.60217653 x10-19

kg

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2.2.92 Mathematical Constants and Symbols • Euler-Mascheroni Constant (Euler's Constant) [33] “γ” = 0.5772156649015328 • “∩” Refers to an intersection. • “∪” Refers to a union. • “→” Or “↓” refers to a process: “leads to”. 2.2.93 Solar System Statistics Symbol MM ME MJ MS RM RE RJ RS

Description Mass of the Moon Mass of the Earth Mass of Jupiter Mass of the Sun Mean Radius of the Moon Mean Radius of the Earth Mean Radius of Jupiter Mean Radius of the Sun

Value utilised by EGM Units kg 7.35 x1022 5.977 x1024 1898.8 x1024 1.989 x1030 m 1.738 x106 6.37718 x106 7.1492 x107 6.96 x108

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2.3

Quinta Essentia – Part 4

2.3.1 “Big-Bang” • The moment of Cosmological creation. • The explosion of space-time at the moment of Cosmological creation. 2.3.2 Black-Hole “BH” • A massive gravitational body such that light cannot escape. • See: Super-Massive-Black-Hole “SMBH”. 2.3.3 Broadband Propagation • The propagation of the EGM wavefunctions in the PV spectrum, at a group velocity of zero. 2.3.4 Buoyancy Point • The point of gravitational acceleration balance (neutrality) between the Earth and the Moon. 2.3.5 CMBR Temperature “T0” • Cosmic Microwave Background Radiation (CMBR) temperature. • See: EGM-CMBR Temperature “TU”. 2.3.6 EGM-CMBR Temperature “TU” • Derivation of the CMBR temperature by the EGM method. • See: CMBR Temperature “T0”. 2.3.7 EGM Flux Intensity “CΩ_J” • The flux intensity of gravitational energy expressed in Jansky's. • Formulated by considering celestial objects as point radiation sources of a high frequency EGM wavefunction. 2.3.8 EGM Hubble constant “HU” • Derivation of the Hubble constant by the EGM method. • See: Hubble Constant “H0”. 2.3.9 • • •

Event Horizon “RBH” Refers to “RBH” of a SBH. The radial displacement from the centre of a SBH from which light cannot escape. See: Schwarzschild-Black-Hole “SBH”.

2.3.10 Galactic Reference Particle “GRP” • The total mass / energy of any Galactic formation represented as a particle with dimensions approaching the Planck scale. 2.3.11 Gravitational Interference • The formation of interference patterns from either broadband or narrowband EGM wavefunction propagation between two or more gravitational fields. 2.3.12 Gravitational Propagation • See: Broadband Propagation. • See: Narrowband Propagation.

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2.3.13 Hubble Constant “H0” • A measure of the rate of Cosmological expansion. • See: EGM Hubble constant “HU”. 2.3.14 Narrowband Propagation • An approximation of Broadband characteristics. • See: Broadband Propagation. 2.3.15 • • •

Non-Physical Situations where “nΩ” is less than unity. See: “nΩ”, “Quinta Essentia – Part 3”. See: Physical.

2.3.16 • • •

Physical Situations where “nΩ” is greater than or equal to unity. See: “nΩ”, “Quinta Essentia – Part 3”. See: Non-Physical.

2.3.17 Primordial Universe • The Universe prior to the “Big-Bang”. • The Universe at the instant prior to the “Big-Bang”. 2.3.18 Schwarzschild-Black-Hole “SBH” • A static Black-Hole. • The simplest form of Black-Hole. 2.3.19 Schwarzschild-Planck-Black-Hole “SPBH” • A SBH of maximum permissible energy density existing at the Planck scale such that the singularity and event horizon radii coincide. • The value of harmonic cut-off mode “nΩ” at the periphery is unity. • The minimum physical radius is “λxλh”. • The minimum physical mass is “mxmh”. • See: Schwarzschild-Black-Hole “SBH”. • See: “nΩ”, “Quinta Essentia – Part 3”. 2.3.20 Schwarzschild-Planck-Particle • Generalised reference to a SPBH. 2.3.21 Singularity • The maximum permissible energy density at the centre of a SBH, represented as a particle. • The particle representation (or mathematical point) at the centre of a SBH for which physical laws are unknown to apply. • See: Schwarzschild-Black-Hole “SBH”. 2.3.22 Singularity Radius “rS” • The radius of the singularity at the centre of a SBH. • See: Schwarzschild-Black-Hole “SBH”. 2.3.23 Solar Mass • The mass of the Sun. 104

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2.3.24 Super-Massive-Black-Hole “SMBH” • A BH of greater than “109” solar masses. • See: Black-Hole “BH”. 2.3.25 Total Mass-Energy • Refers to “visible + dark”. 2.3.26 Astronomical / Cosmological statistics Symbol DE2M H0 MG MG/3 MNS RNS Ro T0

Description Mean Earth-Moon distance Hubble constant (present value) Total Galactic mass Visible Galactic mass Mass of Neutron Star Mean Radius of a Neutron Star Mean distance to Galactic centre CMBR temperature (present value)

Value utilised by EGM 3.844 x108 71 6 x1011MS 2 x1011MS ≥ MS 20 8 2.725

Units m km/s/Mpc kg

km kpc K

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3 Glossary of Terms 3.1

Quinta Essentia – Part 3

3.1.1 Acronyms BNL BPT CCFR CERN CHARM-II D0C DAT DELPHI DONUT E734 EGM EM EP ERF FNAL FS GME1 GME2 GMEx GPE GR GSE1 GSE2 GSE3 GSE4 GSE5 GSEx HERA HSE1 HSE2 HSE3 HSE4 HSE5 HSEx IFF IHEP INFN LANL LEP LHS MCYT

Brookhaven National Laboratory Buckingham Π Theory Chicago Columbia Fermi-Lab Rochester European Organisation for Nuclear Research Experiment: study of Neutrino-Electron scattering at CERN D-Zero Collaboration: an international research effort of leading scientists utilising facilities at FNAL in Illinois, USA Dimensional Analysis Techniques Detector with Lepton, Photon and Hadron Identification Experiment: a search for direct evidence of the Tau Neutrino at Fermi-Lab Experiment: a measurement of the elastic scattering of Neutrino's from Electrons and Protons (at BNL) Electro-Gravi-Magnetics: a mathematical method based upon the modification of the vacuum polarisability by the superposition of EM fields ElectroMagnetic Experimental Prototype Experimental Relationship Function Fermi National Accelerator Laboratory (FERMI-LAB) Fourier Series General Modelling Equation One General Modelling Equation Two Generalised reference to GME1 and GME2 Gravitational Potential Energy General Relativity General Similarity Equation One General Similarity Equation Two General Similarity Equation Three General Similarity Equation Four General Similarity Equation Five Generalised reference to GSE1, GSE2, GSE3, GSE4 or GSE5 Hadron Electron Ring Accelerator in Hamburg, Germany Harmonic Similarity Equation One Harmonic Similarity Equation Two Harmonic Similarity Equation Three Harmonic Similarity Equation Four Harmonic Similarity Equation Five Generalised reference to HSE1, HSE2, HSE3, HSE4 or HSE5 If and only if Institute of High Energy Physics National Institute of Nuclear Physics (Italy) Los Alamos National Laboratories Large Electron-Positron storage ring Left hand side Ministry of Science and Technology (Spain) 107

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MEXT MS NIST NuTeV PDG PV RFBR RHS RMS SK SLAC SNO SSE1 SSE2 SSE3 SSE4 SSE5 SSEx TRISTAN US NSF USDoE ZC ZPF

Ministry of Science (Japan) Mean Square National Institute of Standards & Technology Neutrino's at the Tevatron Particle Data Group: an international research effort of leading scientists Polarisable Vacuum Russian Foundation for Basic Research Right hand side Root Mean Square Super-Kamiokande Collaboration Stanford Linear Accelerator Centre Sudbury Neutrino Observatory Spectral Similarity Equation One Spectral Similarity Equation Two Spectral Similarity Equation Three Spectral Similarity Equation Four Spectral Similarity Equation Five Generalised reference to SSE1, SSE2, SSE3, SSE4 or SSE5 Particle collider at the Japanese High Energy Physics Laboratory (KEK) United States National Science Foundation United States Department of Energy ZEUS Collaboration: an international research effort of leading scientists utilising facilities at HERA Zero-Point-Field NOTES

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3.1.2 Symbols in alphabetical order Symbol A a a1 a2 APP ax(t) a∞

Description 1st Harmonic term Magnitude of acceleration vector Acceleration with respect to General Modelling Equation One Acceleration with respect to General Modelling Equation Two Parallel plate area of a Classical Casimir Experiment Arbitrary acceleration in the time domain Mean magnitude of acceleration over the fundamental period in a FS representation in EGM Magnitude of Magnetic field vector B Magnitude of Magnetic field vector (at infinity) in the PV model of gravity: Ch. 3.2 Amplitude of applied Magnetic field: Ch. 3.6 B0 Magnitude of Magnetic field vector (locally) in the PV model of gravity Magnitude of applied Magnetic field vector BA Critical Magnetic field strength BC Magnitude of PV Magnetic field vector BPV Bottom Quark: elementary particle in the SM bq Root Mean Square of BA Brms Velocity of light in a vacuum c Velocity of light in a vacuum (at infinity) in the PV model of gravity: Ch. 3.1 Velocity of light (locally) in the PV model of gravity c0 Amplitude of fundamental frequency of PV (nPV = 1) CPV(1,r,M) CPV(nPV,r,M) Amplitude spectrum of PV Charm Quark: elementary particle in the SM cq Common difference D Experimental configuration factor: a specific value relating all design criteria; this includes, but not limited to, field harmonics, field orientation, physical dimensions, wave vector, spectral frequency mode and instrumentation or measurement accuracy Offset function DC Down Quark: elementary particle in the SM dq Energy: Ch. 3.3 E Magnitude of Electric field vector Magnitude of Electric field vector (at infinity) in the PV model of gravity: Ch. 3.2 Electronic energy level Charge e, e Electron: subatomic / elementary particle in the SM Exponential function: mathematics Amplitude of applied Electric field: Ch. 3.6 E0 Energy (locally) in the PV model of gravity Magnitude of Electric field vector (locally) in the PV model of gravity Magnitude of applied Electric field vector EA Critical Electric field strength EC Magnitude of PV Electric field vector EPV Root Mean Square of EA Erms F(k,n,t) Complex FS representation of EGM 109

Units m/s2 m2 m/s2

T

T m/s m/s2

% J V/m

J C

V/m J V/m

m/s2

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Magnitude of the ambient gravitational acceleration represented in the time m/s2 domain Amplitude spectrum / distribution of F(k,n,t) F0(k) The Casimir Force by classical representation N FPP The Casimir Force by EGM FPV Gluon: theoretical elementary particle in the SM g Magnitude of gravitational acceleration vector m/s2 Universal Gravitation Constant m3kg-1s-2 G Tensor element g00 Tensor element g11 Tensor element g22 Tensor element g33 Height: Ch. 3.4 m h Higgs Boson: theoretical elementary particle in the SM H Hydrogen Magnetic field strength Oe Planck’s Constant (plain h form) Js h h-bar Planck’s Constant (2π form) HSE4A R Time average form of HSE4 R HSE5A R Time average form of HSE5 R Generalised reference to the reduced form of HSEx HSEx R Complex number i Initial condition Macroscopic intensity of Photons within a test volume W/m2 In,P Vector current density A/m2 J Wave vector 1/m k K0(r,X) ERF by displacement domain precipitation Generalised ERF K0(X) K0(ω ω,r,E,B,X) ERF by wavefunction precipitation K0(ω ω,X) ERF by frequency domain precipitation K0(ω ωPV,r,EPV,BPV,X) ERF equivalent to K0(ω,r,E,B,X) ERF formed by re-interpretation of the primary precipitant (V/m)2 K1 ERF formed by re-interpretation of the primary precipitant T-2 K2 Harmonic wave vector of applied field 1/m kA Critical Factor KC PaΩ Engineered Refractive Index KEGM Harmonic form of KEGM KEGM H Experimentally implicit Planck Mass scaling factor Km The intensity of the background PV field at specific frequency modes W/m2 Kn,P A refinement of a constant in FPP KP Harmonic wave vector of PV 1/m kPV Refractive Index of PV KPV Harmonic form of KPV KPV H Critical Ratio KR Critical harmonic operator (based upon the unit amplitude spectrum) KR H Neutron MS charge radius by EGM m2 KS Neutron MS charge radius (determined experimentally) in the SM KX Experimentally implicit Planck Length scaling factor Kλ Experimentally implicit Planck Frequency scaling factor Kω Length m L f(t)

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L0 L2 L3 L5 M m0 M0 mAMC mbq mcq mdq me ME men mgg mH mh MJ mL(2) mL(3) mL(5) MM mn mp mQB(5) mQB(6) MS msq mtq muq mW mx mZ mε mγ mγg mγγ mµ mµn mτ mτn n n, N nA nB NC nE nPV nq NT

Length (locally) in the PV model of gravity by EGM Theoretical elementary particle (Lepton) by EGM Theoretical elementary particle (Lepton) by EGM Theoretical elementary particle (Lepton) by EGM Mass Mass (locally) in the PV model of gravity by EGM Zero mass (energy) condition of free space Atomic Mass Constant Bottom Quark rest mass (energy) by EGM Charm Quark rest mass (energy) by EGM Down Quark rest mass (energy) by EGM Electron rest mass (energy) according to NIST Mass of the Earth Electron Neutrino rest mass (energy) according to PDG Graviton rest mass (energy) by EGM Higgs Boson rest mass (energy) according to PDG Planck Mass Mass of Jupiter Rest mass (energy) of the L2 particle by EGM Rest mass (energy) of the L3 particle by EGM Rest mass (energy) of the L5 particle by EGM Mass of the Moon Neutron rest mass (energy) according to NIST Proton rest mass (energy) according to NIST Rest mass (energy) of the QB5 particle by EGM Rest mass (energy) of the QB6 particle by EGM Mass of the Sun Strange Quark rest mass (energy) by EGM Top Quark rest mass (energy) according (energy) to PDG Up Quark rest mass (energy) by EGM W Boson rest mass according (energy) to PDG Imaginary particle mass Z Boson rest mass according (energy) to PDG Electron rest mass (energy) in high energy scattering experiments Photon rest mass (energy) threshold according to PDG Graviton rest mass (energy) threshold according to PDG Photon rest mass (energy) by EGM Muon rest mass (energy) according to NIST Muon Neutrino rest mass (energy) according to PDG Tau rest mass (energy) according to NIST Tau Neutrino rest mass (energy) according to PDG Neutron: subatomic particle in the SM Field harmonic (harmonic frequency mode) Harmonic frequency modes of applied field Harmonic mode number of the ZPF with respect to BA Critical mode Harmonic mode number of the ZPF with respect to EA Harmonic frequency modes of PV Quantum number Number of terms 111

m

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NTR NX N∆r nΩ nΩ ZPF nβ P p Q, Qe QB5 QB6 r

r0 rBohr rBoson rbq rc rcq rdq re RE ren RError rgg rH RJ rL RM rp rQB RS rsq rtq ru ruq rW rx rxq rZ rε rγγ rµ rµn rν

The ratio of the number of terms Harmonic inflection mode Permissible mode bandwidth of applied experimental fields Harmonic cut-off mode of PV ZPF beat cut-off mode Mode Number (Critical Boundary Mode) of ωβ Polarisation vector Proton: subatomic particle in the SM Magnitude of Electric charge Theoretical elementary particle (Quark or Boson) by EGM Theoretical elementary particle (Quark or Boson) by EGM Arbitrary radius with homogeneous mass (energy) distribution Generalised notation for length (e.g. r → λ/2π): Ch. 3.1 Generalised notation for length (locally) in the PV model of gravity: Ch. 3.1 Magnitude of position vector from centre of spherical object with homogeneous mass (energy) distribution Reciprocal of the wave number: Ch. 3.1 Length (locally) in the PV model of gravity Classical Bohr radius Generalised RMS charge radius of a Boson by EGM RMS charge radius of the Bottom Quark by EGM Transformed value of generalised length (locally) in the PV model of gravity RMS charge radius of the Charm Quark by EGM RMS charge radius of the Down Quark by EGM Classical Electron radius in the SM Mean radius of the Earth RMS charge radius of the Electron Neutrino by EGM Representation Error RMS charge radius of the Graviton by EGM RMS charge radius of the Higgs Boson utilising ru Mean radius of Jupiter Average RMS charge radii of the rε, rµ and rτ particles Mean radius of the Moon Classical RMS charge radius of the Proton in the SM Average RMS charge radius of the QB5 / QB6 particles by EGM utilising ru Mean radius of the Sun RMS charge radius of the Strange Quark by EGM RMS charge radius of the Top Quark by EGM Heisenberg uncertainty range RMS charge radius of the Up Quark by EGM RMS charge radius of the W Boson utilising ru Bohr radius by EGM Generalised RMS charge radius of all Quarks as determined by the ZC within the SM RMS charge radius of the Z Boson by utilising ru RMS charge radius of the Electron by EGM RMS charge radius of the Photon by EGM RMS charge radius of the Muon by EGM RMS charge radius of the Muon Neutrino by EGM Neutron RMS charge radius (by analogy to KS) 112

C/m2 C

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RMS charge radius of the ν2 particle by EGM RMS charge radius of the ν3 particle by EGM RMS charge radius of the ν5 particle by EGM Neutron Magnetic radius by EGM Generalised reference to rν2, rν3 and rν5 RMS charge radius of the Proton by EGM Proton Electric radius by EGM Proton Magnetic radius by EGM RMS charge radius of the Tau by EGM RMS charge radius of the Tau Neutrino by EGM Rydberg Constant Poynting Vector Strange Quark: elementary particle in the SM nth Harmonic term Range factor 1st Sense check 3rd Sense check 4th Sense check 2nd Sense check 5th Sense check 6th Sense check A positive integer value representing the harmonic cut-off frequency ratio between two proportionally similar mass (energy) systems Poynting Vector of PV Sω Time t Top Quark: elementary particle in the SM tq Initial state GPE per unit mass described by any appropriate method Ug Harmonic form of Ug Ug H Rest mass-energy density Um Up Quark: elementary particle in the SM uq Field energy density of PV Uω Local value of the velocity of light in a vacuum vc W Boson: elementary particle in the SM W All variables within the experimental environment that influence results and X behaviour including parameters that might otherwise be neglected due to practical calculation limitations, in theoretical analysis Impedance function Z Z Boson: elementary particle in the SM Change in electronic energy level ∆Ε Change in the magnitude of the local PV acceleration vector ∆aPV Change in magnitude of the local gravitational acceleration vector ∆g ∆GME1 Change in GME1 ∆GME2 Change in GME2 ∆GMEx Generalised reference to changes in GME1 and GME2 Harmonic form of ∆K0 ∆K0 H ∆K0(ω ω,X) Engineered Relationship Function by EGM Change in K1 by EGM ∆K1 Change in K2 by EGM ∆K2 Change in Critical Factor by EGM ∆ KC rν2 rν3 rν5 rνM rνx rπ rπE rπM rτ rτn R∞ S sq StN Stα Stβ Stδ Stε Stγ Stη Stθ Stω

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J W/m2

PaΩ

W/m2 s (m/s)2 Pa Pa m/s

Ω J m/s2

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∆nS ∆r ∆t ∆t0 ∆ Ug ∆UPV ∆vΩ ∆vδr ∆ΛPV ∆λΩ ∆λδr ∆ωPV ∆ωR ∆ωS ∆ωZPF ∆ωΩ ∆ωδr Π ΣH ΣHR Ω α α1 αx β β1 βx ε0 φ φC φgg φγγ γ γg λ λΑ λΒ λCe λCN λCP λh

Change in the number of ZPF modes Plate separation of a Classical Casimir Experiment Practical changes in benchtop displacement values Change in time (at infinity) in the PV model of gravity by EGM Change in time (locally) in the PV model of gravity by EGM Change in Gravitational Potential Energy (GPE) per unit mass induced by any suitable source Change in energy density of gravitational field Change in rest mass-energy density Terminating group velocity of PV Group velocity of PV Change in the local value of the Cosmological Constant by EGM Change in harmonic cut-off wavelength of PV Change in harmonic wavelength of PV Frequency bandwidth of PV Bandwidth ratio Similarity bandwidth ZPF beat bandwidth Beat bandwidth of PV Beat frequency of PV Dimensional grouping derived by application of BPT The sum of terms The ratio of the sum of terms Harmonic cut-off function of PV An inversely proportional description of how energy density may result in an acceleration: Ch. 3.2 Fine Structure Constant The subset formed, as “N → ∞”, by the method of incorporation Generalised reference to α1 and α2 A directly proportional description of how energy density may result in an acceleration The subset formed, as “N → ∞”, by the method of incorporation Generalised reference to β1 and β2 Permittivity of a vacuum Relative phase variance between EA and BA Critical phase variance RMS charge diameter of the Graviton by EGM RMS charge diameter of the Photon by EGM Mathematical Constant: Euler-Mascheroni (Euler’s) Constant Photon: elementary particle in the SM Graviton: theoretical elementary particle in the SM Wavelength 1st term of the Balmer Series by EGM Classical Balmer Series wavelength Electron Compton Wavelength Neutron Compton Wavelength Proton Compton Wavelength Planck Length 114

m s (m/s)2 Pa m/s Hz2 m Hz Hz

m/s2 m/s2

F/m θc m

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λPV µ, µµ µ0 ν2 ν3 ν5 νe νµ ντ ρ ρ0 τ, τω

Wavelength of PV Muon: elementary particle in the SM Reduced mass of Hydrogen Permeability of a vacuum Theoretical elementary Neutrino of the L2 particle by EGM Theoretical elementary Neutrino of the L3 particle by EGM Theoretical elementary Neutrino of the L5 particle by EGM Electron Neutrino: elementary particle in the SM Muon Neutrino: elementary particle in the SM Tau Neutrino: elementary particle in the SM Charge density Spectral energy density Tau: elementary particle in the SM Field frequency Field frequency (at infinity) in the PV model of gravity: Ch. 3.2 Field frequency (locally) in the PV model of gravity ω0 Field frequency (locally) in the PV model of gravity by EGM Harmonic frequency of the ZPF with respect to BA ωB Critical frequency ωC Harmonic frequency of the ZPF with respect to EA ωE Electron Compton Frequency ωCe Neutron Compton Frequency ωCN Proton Compton Frequency ωCP Planck Frequency ωh Generalised reference to ωPV(nPV,r,M) ωPV Fundamental frequency of PV (nPV = 1) ωPV(1,r,M) Frequency spectrum of PV ωPV(nPV,r,M) Harmonic inflection frequency ωX Harmonic cut-off frequency of PV ωΩ ZPF beat cut-off frequency ωΩ ZPF Critical boundary ωβ 〈 mQuark〉 Average mass (energy) of all Quarks according to PDG Average mass (energy) of all Quarks by EGM Average RMS charge radius of all Bosons in the SM utilising ru 〈rBoson〉 Average RMS charge radius of all Quarks by EGM 〈rQuark〉 Average RMS charge radius of all Quarks and Bosons by EGM utilising ru 〈r〉〉

m kg or eV N/A2

C/m3 Pa/Hz Hz

kg or eV m

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3.2

Quinta Essentia – Part 4

3.2.1 Acronyms •

Refer to “Quinta Essentia – Part 3: Acronyms” where appropriate.

BH CMBR DAT ED EFT EP GA GRP GUT HEP LFT MW NASA QED QFT QM RF SBH SED SM SMBH SPBH ST ToE VP ZP ZPE

Black-Hole Cosmic Microwave Background Radiation Dimensional Analysis Technique Electro-Dynamics Effective Field Theory Experimental Prototype Gravitational Astronomy Galactic Reference Particle Grand Unified Theory High Energy Physics Lattice Field Theory Milky-Way National Aeronautics and Space Administration Quantum Electro Dynamics Quantum Field Theory Quantum Mechanics Radio Frequency Schwarzschild-Black-Hole Spectral Energy Density Standard Model of Particle-Physics or Cosmology Super-Massive-Black-Hole Schwarzschild-Planck-Black-Hole String Theory Theory of Everything Virtual Photon Zero-Point Zero-Point-Energy NOTES

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3.2.2 Symbols by chapter •

Refer to “Quinta Essentia – Part 3: Symbols in alphabetical order” where appropriate.

Symbol aEGM_ωΩ ωΩ CΩ_J CΩ_J1 nΩ_1 Stg StG StJ SωΩ Ω1 ωΩ_1 ωΩ_2 ωΩ_3

Chapter 5: Characterisation of the Gravitational Spectrum Description Gravitational acceleration utilising ωΩ_2 EGM Flux Intensity Non-refractive form of CΩ_J Non-refractive form of nΩ 1st EGM gravitational constant: Stg = 1.828935 x10245 2nd EGM gravitational constant: StG = 8.146982 x10224 3rd EGM gravitational constant: StJ9 = 1.093567 x10-146(kg4m26/s18) Approximated / simplified Poynting Vector Non-refractive form of Ω Non-refractive form of ωΩ Gravitational acceleration form of ωΩ_1 Transformed representation of ωΩ_1

Units m/s2 Jy (Jansky)

m-1s-5 m5kg-2s-9 (kg4m26/s18)(1/9) W/m2 Hz

Chapter 6: Derivation of “Planck-Particle” and “Schwarzschild-Black-Hole” Characteristics Symbol Description Units Propagation energy of a Graviton J Eg Ex Proportional relationship between Eg and Eγ Propagation energy of a Photon J Eγ Hubble constant (present value) Hz H0 Refractive Index of PV by Depp KDepp Planck scale experimental relationship function Kh Generalised mass kg M1 Generalised mass M2 Mass of a SBH MBH 2nd SPBH constant mx nBH Harmonic cut-off mode ratio (nΩ_5 : nΩ_4) ng Average number of Gravitons radiated by a SBH per TΩ_4 period Population of Gravitons within starving matter ngg Transformed representation of nΩ_1 nΩ_2 The form nΩ_2 takes as a function of λx nΩ_3 nΩ_1 at the periphery of a SBH singularity nΩ_4 nΩ_1 at the event horizon of a SBH nΩ_5 Average number of Photons radiated by a SBH per TΩ_4 period nγ Population of Photons within starving matter nγγ Hubble size (present value) m r0 Generalised radial displacement r1 Generalised radial displacement r2 Radius of the event horizon of a SBH RBH Range variable for SBH’s Rbh Singularity radius rS ZPF equilibrium radius rZPF Distance from the centre of mass of a celestial object to the Earth rω Average emission period per Graviton s Tg 117

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TL TΩ_3 TΩ_4 V ∆KPV ∆r λVL λx λX-RAY ρm ρS ωBH ωg ωPV_1 ωX-RAY ωΩ_4 ωΩ_5 ωΩ_6 ωΩ_7

Minimum gravitational lifetime of matter 1 / ωΩ_3 1 / ωΩ_4 Volume Change in Refractive Index of PV Change in displacement within the event horizon of a SBH Wavelength of visible light 1st SPBH constant Wavelength of X-Rays Mass density Mass density of a SPBH Harmonic cut-off frequency ratio (ωΩ_5 : ωΩ_4) Average Graviton emission frequency (1 / Tg) Fundamental harmonic frequency ratio (ωΩ_6 : ωΩ_7) Frequency of X-Rays ωΩ_3 at the event horizon of a SBH ωΩ_3 at the periphery of a SBH singularity ωPV(1,r,MBH) at the periphery of a SBH singularity: r = rS(MBH) ωPV(1,r,MBH) at the event horizon of a SBH: r = RBH(MBH)

s

m3 m

m kg/m3

Hz Hz

Chapter 7: Fundamental Cosmology Symbol AU H HU HU2 Hα KT KU KW M3 Mf MG mg1 mg2 mg3 mg4 mg5 Mi MU r3 rf ri Ro RU rx1 rx2 rx3 rx4

Description EGM Cosmological age (present value) Generalised reference to the Hubble constant EGM Hubble constant Transformed representation of HU Primordial Hubble constant Expansive scaling factor rf / ri Wien displacement constant: 2.8977685 x10-3 [35] Generalised mass or mxmh Total Cosmological mass (present value) Total mass of the Milky-Way Galaxy Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor Total Cosmological mass (initial value) Total EGM Cosmological mass Generalised radial displacement or λxλh Cosmological size (present value) Cosmological size (initial value) Distance from the Sun to the Galactic centre EGM Cosmological size (present value) Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor 118

Units s Hz

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rx5 T0 TU TU2 TW UZPF ∆ Ro ∆T0 Ω ΩEGM Ωm ΩPDG ΩZPF ΩΛ Ωγ Ων λy λΩ_3 ρ ρc ρU ρU2

Computational pre-factor CMBR temperature (present value) CMBR temperature by the EGM method Transformed representation of TU Thermodynamic scaling factor ZPF energy density threshold Experimental tolerance of Ro Experimental tolerance of T0 Community reference to the net Cosmological density parameter Net Cosmological density parameter as defined by the EGM method Visible mass contribution to the net Cosmological density parameter Net Cosmological density parameter as defined by the PDG ZPE contribution to the net Cosmological density parameter Dark energy contribution to the net Cosmological density parameter Photon contribution to the net Cosmological density parameter Neutrino contribution to the net Cosmological density parameter Generalised representation of λx c / ωΩ_3 Community reference to Cosmological mass-density Critical Cosmological mass-density EGM Cosmological mass-density Transformed representation of ρU

K

Pa pc K

m kg/m3

Chapter 8: Advanced Cosmology Symbol CΩ_Jωω dH2dt2 dHdt dT2dt2 dT3dt3 dTdt Hβ Hβ2 Hγ MEGM ML REGM rL t1 t2 t3 t4 t5 t7 tEGM

Description Equal to CΩ_J 2nd time derivative of H 1st time derivative of H 2nd time derivative of TU4 3rd time derivative of TU4 1st time derivative of TU4 Dimensionless range variable Computational pre-factor Dimensionless range variable Convenient form of MU EGM Cosmological mass limit Convenient form of RU EGM Cosmological size limit • Temporal ordinate (local maxima) of the CMBR temperature • The instant of maximum Cosmological temperature Temporal ordinate (local minima) of the 1st time derivative of the CMBR temperature Temporal ordinate (local maxima) of the 2nd time derivative of the CMBR temperature • Temporal ordinate (local maxima) of the 1st time derivative of H • The instant of maximum physical EGM Hubble constant Temporal ordinate (local minima) of the 2nd time derivative of H Equal to t1 Convenient form of AU 119

Units Jy Hz3 Hz2 K/s2 K/s3 K/s

kg m s

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tL TU3 TU4 η µ

EGM Cosmological age limit Transformed representation of TU2 Transformed representation of TU3 Computed index Indicial constant (µ = 1 / 3)

s K

Symbol ag aPV DE2M gav r4 r5

Chapter 9: Gravitational Cosmology Description High frequency harmonic acceleration Gravitational acceleration harmonic Mean distance from the Earth to the Moon Average high frequency harmonic acceleration Distance from the centre of mass of the Earth to the buoyancy point Distance from the centre of mass of the Moon to the buoyancy point

Units m/s2 m m/s2 m

Chapter 10: Particle Cosmology Symbol mgg2 mγγ2 γγ Nγ Qγ Qγ_PDG Qγγ Qγγ2 γγ rgg2 rγγ2 γγ

Description Graviton mass-energy lower limit Photon mass-energy lower limit Photon population at Qγ Photon RMS charge threshold by EGM Photon RMS charge threshold by PDG Photon RMS charge upper limit by EGM Photon RMS charge lower limit by EGM Graviton RMS charge radius lower limit Photon RMS charge radius lower limit

Units eV

C

m

Appendix 4.A Symbol TBH Th TSPBH Φ κ σ

Description BH temperature Planck temperature SPBH temperature Energy flux emitted from a “Black-Body” at temperature “T” Boltzmann’s constant: 1.3806505 x10-23 [35] Stefan-Boltzmann constant: 5.670400 x10-8 [35]

Units K W/m2 J/K Wm-2K-4

NOTES

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3.2.3 Symbols in alphabetical order •

Refer to “Quinta Essentia – Part 3: Symbols in alphabetical order” where appropriate.

Symbol aEGM_ωΩ ωΩ ag aPV AU CΩ_J CΩ_J1 CΩ_Jωω DE2M dH2dt2 dHdt dT2dt2 dT3dt3 dTdt Eg Ex Eγ gav H H0 HU HU2 Hα Hβ Hβ2 Hγ KDepp Kh KT KU KW M1 M2 M3 MBH MEGM Mf MG mg1 mg2 mg3 mg4 mg5 mgg2 Mi ML

Description Gravitational acceleration utilising ωΩ_2 High frequency harmonic acceleration Gravitational acceleration harmonic EGM Cosmological age (present value) EGM Flux Intensity Non-refractive form of CΩ_J Equal to CΩ_J Mean distance from the Earth to the Moon 2nd time derivative of H 1st time derivative of H 2nd time derivative of TU4 3rd time derivative of TU4 1st time derivative of TU4 Propagation energy of a Graviton Proportional relationship between Eg and Eγ Propagation energy of a Photon Average high frequency harmonic acceleration Generalised reference to the Hubble constant Hubble constant (present value) EGM Hubble constant Transformed representation of HU Primordial Hubble constant Dimensionless range variable Computational pre-factor Dimensionless range variable Refractive Index of PV by Depp Planck scale experimental relationship function Expansive scaling factor rf / ri Wien displacement constant: 2.8977685 x10-3 [35] Generalised mass Generalised mass Generalised mass or mxmh Mass of a SBH Convenient form of MU Total Cosmological mass (present value) Total mass of the Milky-Way Galaxy Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor Graviton mass-energy lower limit Total Cosmological mass (initial value) EGM Cosmological mass limit 121

Units m/s2

s Jy (Jansky)

m Hz3 Hz2 K/s2 K/s3 K/s J J m/s2 Hz

mK kg

eV kg www.deltagroupengineering.com

MU mx mγγ2 γγ nBH ng ngg nΩ_1 nΩ_2 nΩ_3 nΩ_4 nΩ_5 nγ Nγ nγγ Qγ Qγ_PDG Qγγ Qγγ2 γγ r0 r1 r2 r3 r4 r5 RBH Rbh REGM rf rgg2 ri rL Ro rS RU rx1 rx2 rx3 rx4 rx5 rZPF rγγ2 γγ rω Stg StG StJ SωΩ T0 t1

Total EGM Cosmological mass 2nd SPBH constant Photon mass-energy lower limit Harmonic cut-off mode ratio (nΩ_5 : nΩ_4) Average number of Gravitons radiated by a SBH per TΩ_4 period Population of Gravitons within starving matter Non-refractive form of nΩ Transformed representation of nΩ_1 The form nΩ_2 takes as a function of λx nΩ_1 at the periphery of a SBH singularity nΩ_1 at the event horizon of a SBH Average number of Photons radiated by a SBH per TΩ_4 period Photon population at Qγ Population of Photons within starving matter Photon RMS charge threshold by EGM Photon RMS charge threshold by PDG Photon RMS charge upper limit by EGM Photon RMS charge lower limit by EGM Hubble size (present value) Generalised radial displacement Generalised radial displacement Generalised radial displacement or λxλh Distance from the centre of mass of the Earth to the buoyancy point Distance from the centre of mass of the Moon to the buoyancy point Radius of the event horizon of a SBH Range variable for SBH’s Convenient form of RU Cosmological size (present value) Graviton RMS charge radius lower limit Cosmological size (initial value) EGM Cosmological size limit Distance from the Sun to the Galactic centre Singularity radius EGM Cosmological size (present value) Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor Computational pre-factor ZPF equilibrium radius Photon RMS charge radius lower limit Distance from the centre of mass of a celestial object to the Earth 1st EGM gravitational constant: Stg = 1.828935 x10245 2nd EGM gravitational constant: StG = 8.146982 x10224 3rd EGM gravitational constant: StJ9 = 1.093567 x10-146(kg4m26/s18) Approximated / simplified Poynting Vector CMBR temperature (present value) • Temporal ordinate (local maxima) of the CMBR temperature • The instant of maximum Cosmological temperature 122

kg eV

C

m

pc (parsec) m

m

m-1s-5 m5kg-2s-9 (kg4m26/s18)(1/9) W/m2 K s

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t2 t3 t4 t5 t7 TBH tEGM Tg Th TL tL TSPBH TU TU2 TU3 TU4 TW TΩ_3 TΩ_4 UZPF V ∆KPV ∆r ∆ Ro ∆T0 Φ Ω Ω1 ΩEGM Ωm ΩPDG ΩZPF ΩΛ Ωγ Ων η κ λVL λx λX-RAY λy λΩ_3 µ ρ ρc ρm

Temporal ordinate (local minima) of the 1st time derivative of the CMBR temperature Temporal ordinate (local maxima) of the 2nd time derivative of the CMBR temperature • Temporal ordinate (local maxima) of the 1st time derivative of H • The instant of maximum physical EGM Hubble constant Temporal ordinate (local minima) of the 2nd time derivative of H Equal to t1 BH temperature Convenient form of AU Average emission period per Graviton Planck temperature Minimum gravitational lifetime of matter EGM Cosmological age limit SPBH temperature CMBR temperature by the EGM method Transformed representation of TU Transformed representation of TU2 Transformed representation of TU3 Thermodynamic scaling factor 1 / ωΩ_3 1 / ωΩ_4 ZPF energy density threshold Volume Change in Refractive Index of PV Change in displacement within the event horizon of a SBH Experimental tolerance of Ro Experimental tolerance of T0 Energy flux emitted from a “Black-Body” at temperature “T” Community reference to the net Cosmological density parameter Non-refractive form of Ω Net Cosmological density parameter as defined by the EGM method Visible mass contribution to the net Cosmological density parameter Net Cosmological density parameter as defined by the PDG ZPE contribution to the net Cosmological density parameter Dark energy contribution to the net Cosmological density parameter Photon contribution to the net Cosmological density parameter Neutrino contribution to the net Cosmological density parameter Computed index Boltzmann’s constant: 1.3806505 x10-23 [35] Wavelength of visible light 1st SPBH constant Wavelength of X-Rays Generalised representation of λx c / ωΩ_3 Indicial constant (µ = 1 / 3) Community reference to Cosmological mass-density Critical Cosmological mass-density Mass density 123

s

K s K s K

s Pa m3 m pc K W/m2

J/K m m m kg/m3

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ρS ρU ρU2 σ ωBH ωg ωPV_1 ωX-RAY ωΩ_1 ωΩ_2 ωΩ_3 ωΩ_4 ωΩ_5 ωΩ_6 ωΩ_7

Mass density of a SPBH EGM Cosmological mass-density Transformed representation of ρU Stefan-Boltzmann constant: 5.670400 x10-8 [35] Harmonic cut-off frequency ratio (ωΩ_5 : ωΩ_4) Average Graviton emission frequency (1 / Tg) Fundamental harmonic frequency ratio (ωΩ_6 : ωΩ_7) Frequency of X-Rays Non-refractive form of ωΩ Gravitational acceleration form of ωΩ_1 Transformed representation of ωΩ_1 ωΩ_3 at the event horizon of a SBH ωΩ_3 at the periphery of a SBH singularity ωPV(1,r,MBH) at the periphery of a SBH singularity: r = rS(MBH) ωPV(1,r,MBH) at the event horizon of a SBH: r = RBH(MBH)

kg/m3

Wm-2K-4 Hz Hz

NOTES

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4 Derivation Processes 4.1

Main sequence

4.1.1 Characterisation of the gravitational spectrum The Electro-Gravi-Magnetic (EGM) equations, utilised to describe fundamental particles in harmonic terms “Stω”, are simplified for values of Refractive Index approaching unity “KPV → 1”. This facilitates the representation of gravitational acceleration “g” utilising the Polarisable Vacuum (PV) harmonic cut-off frequency “ωΩ”, leading to the formulation of a generalised cubic frequency expression “g → ωPV3”. It is demonstrated that the PV spectrum is dominated by “ωΩ” such that the magnitude of the associated gravitational Poynting Vector “SωΩ” is usefully approximated by the total energy density “SωΩ → c⋅Um”, resulting in an expression for EGM Flux Intensity “CΩ_J”. The derivation sequence proceeds as follows, 1. Simplification of the EGM equations. 2. Derivation of gravitational acceleration in terms of “ωΩ”. 3. Formulation of a generalised cubic frequency expression in terms of “g”: “g → ωPV3”. 4. Determination of the gravitationally dominant EGM frequency: “SωΩ → c⋅Um”. 5. Derivation of EGM Flux Intensity “CΩ_J”. 4.1.2 Derivation of “Planck-Particle” and “Schwarzschild-Black-Hole” characteristics The minimum physical dimensions of “Schwarzschild-Planck-Particle” mass and radius is derived, leading to the determination of the value of “KPV” at the event horizon of a “Schwarzschild-Planck-Black-Hole” (SPBH). Consequently, the magnitude of the harmonic cut-off frequency “ωΩ” at the event horizon “RBH” of a “Schwarzschild-Black-Hole” (SBH) is presented, yielding the singularity radius “rS” and harmonic cut-off profiles (“nΩ” and “ωΩ” as “r → RBH”). The minimum gravitational lifetime of matter “TL” is also advanced such that the value of generalised average emission frequency per Graviton “ωg” may be calculated. These determinations assist in the supplemental EGM interpretation with respect to the visibility of “Black-Holes” (BH’s). The derivation sequence proceeds as follows, 6. Derivation of the minimum physical “Schwarzschild-Planck-Particle” mass and radius. 7. Derivation of the value of the “KPV” at the event horizon of a “Schwarzschild-PlanckBlack-Hole” (SPBH). 8. Derivation of “ωΩ” at the event horizon of a SPBH. 9. Derivation of “ωΩ” at the event horizon of a SBH. 10. Derivation of “rS”. 11. “nΩ” and “ωΩ” profiles (as “r → RBH”) of SBH’s. 12. Derivation of the minimum gravitational lifetime of matter “TL”. 13. Derivation of the average emission frequency per Graviton “ωg”. 14. Why can't we observe BH’s? 4.1.3 Fundamental Cosmology The primordial and present values of the Hubble constant are derived (“Hα” and “HU” respectively), leading to the determination of the Cosmic Microwave Background Radiation (CMBR) temperature “TU”. This facilitates the determination of the impact of “Dark Matter / Energy” on “HU” and “TU” such that a generalised expression for “TU” in terms of “HU” is formulated. An experimentally implicit derivation of the ZPF energy density threshold “UZPF” is also presented. The derivation sequence proceeds as follows, 125

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15. Derivation of the primordial and present Hubble constants “Hα, HU”. 16. Derivation of the Cosmic Microwave Background Radiation (CMBR) temperature “TU”. 17. Numerical solutions for “Hα, AU, RU, ρU, MU, HU” and “TU”. 18. Determination of the impact of “Dark Matter / Energy” on “HU” and “TU”. 19. “TU” as a function of a generalised Hubble constant “TU → TU2”. 20. Derivation of “Ro”, “MG”, “HU2” and “ρU2” from “TU2”. 21. Experimentally implicit derivation of the ZPF energy density threshold “UZPF”. 4.1.4 Advanced Cosmology A time dependent derivation of “TU” is performed, including its rate of change and relationship to “HU”. This facilitates the articulation of the Cosmological evolution process into four distinct periods dealing with the inflationary and early expansive phases. Subsequently, the history of the Universe33 is developed and compared to the Standard Model (SM) of Cosmology. This assists in determining the Cosmological limitations of the EGM construct. The question of the practicality of utilising conventional radio telescopes for gravitational astronomy is also addressed. The derivation sequence proceeds as follows, 22. Time dependent CMBR temperature “TU2 → TU3”. 23. Rates of change of CMBR temperature “TU3 → TU4 → d1,2,3TU4/dt1,2,3”. 24. Rates of change of the Hubble constant “d1,2H/dt1,2”. 25. Cosmological evolution process. 26. History of the Universe. 27. EGM Cosmological construct limitations. 28. Are conventional radio telescopes, practical tools for gravitational astronomy? 4.1.5 Gravitational Cosmology An engineering model is developed to explain how gravitational effects are transmitted through space-time in terms of EGM wavefunction propagation and interference. The derivation sequence proceeds as follows, 29. Gravitational propagation: the mechanism for interaction. 30. Gravitational interference: the mechanism of interaction. 4.1.6 Particle Cosmology The following characteristics are derived utilising EGM principles: 31. The Photon and Graviton mass-energies lower limit. 32. The Photon and Graviton Root-Mean-Square (RMS) charge radii lower limit. 33. The Photon charge threshold. 34. The Photon charge upper limit. 35. The Photon charge lower limit.

33

As defined by the EGM construct. 126

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4.2

The Hubble sequence

Note: refer to “Glossary of Terms” where required. 4.2.1 Preconditions i. Derivation of EGM Flux Intensity “CΩ_J”. ii. Derivation of the minimum physical “Schwarzschild-Planck-Particle” mass and radius. iii. Derivation of the minimum gravitational lifetime of matter “TL”. 4.2.2 Assumptions i. At an instant prior to the “Big-Bang”, the Universe (termed the “Primordial Universe”) was analogous to a homogeneous Planck scale particle at the maximum permissible energy density limit (i.e. with the physical proportions defined by precondition “ii”). ii. The “Milky-Way” is analogous to a Planck scale particle of equivalent total Galactic mass-energy (i.e. visible + dark) such that dynamic, kinematic and geometric similarity exists between the “Primordial Universe” and the “Milky-Way” Planck scale particles34. Hence, for Earth based observations, the average EGM Flux Intensity of the “Milky-Way Galaxy” is proportional to the average value of the present Universe and the peak value of the “Primordial Universe” at the instant of the “Big-Bang”. iii. Matter is dispersed throughout the Universe in accordance with the EGM harmonic representation of fundamental particles, i.e. Eq. (3.230) applies to Cosmology. 4.2.3 Simplified sequence The Hubble constant by the EGM method “HU”, may be derived from first principles in agreement with physical measurement, by the following sequence, i. Express the EGM harmonic representation of fundamental particles in Cosmological form. ii. Formulate a relationship between the size of the Planck scale particles and EGM Flux Intensity. iii. Formulate a generalised relationship between the dimensions of a “Schwarzschild-PlanckParticle” and harmonic frequency mode. iv. Formulate a generalised expression for “HU” utilising the output from the three previous steps and evaluate appropriately, NOTES

34

The analogous Planck scale particle for the “Milky-Way” performs the function of a “Galactic Reference Particle” (GRP) in relation to the “Primordial Universe”. 127

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4.3

The CMBR temperature sequence

Note: refer to “Glossary of Terms” where required. 4.3.1 Preconditions i. “HU” has been derived correctly. ii. The total mass-energy of the present Universe is dynamically, kinematically and geometrically similar to a particle at the Planck scale limit, consistent with the formulation of “HU”. 4.3.2 Assumptions i. EGM considers the “Primordial Universe” as a single mode wavefunction, therefore any temperature calculation must be scaled accordingly for application to black-body radiation (i.e. black-bodies emit a spectrum of thermal frequencies, not just one). ii. The CMBR temperature is proportional to the average number of Gravitons radiated per harmonic period by the “Primordial Universe” at the instant prior to the “Big-Bang”. 4.3.3 Simplified sequence The Cosmic Microwave Background Radiation (CMBR) temperature by the EGM method “TU”, may be derived from first principles in precise agreement with physical measurement, by the following sequence, i. Formulate an expansive scaling factor relating the average number of Gravitons radiated per harmonic period by the “Primordial Universe” at the instant prior to the “Big-Bang”, the value of the Hubble constant at the same instant (termed the primordial Hubble constant “Hα”) and “HU”. ii. Formulate a thermodynamic scaling factor relating Wien’s Displacement Law and EGM wavelength. iii. Multiply the expansive and thermodynamic scaling factors to compute “TU”. NOTES

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5 Characterisation of the Gravitational Spectrum

Abstract The Electro-Gravi-Magnetic (EGM) equations, utilised to describe fundamental particles in harmonic terms “Stω”, are simplified for values of Refractive Index approaching unity “KPV → 1”. This facilitates the representation of gravitational acceleration “g” utilising the Polarisable Vacuum (PV) harmonic cut-off frequency “ωΩ”, leading to the formulation of a generalised cubic frequency expression “g → ωPV3”. It is demonstrated that the PV spectrum is dominated by “ωΩ” such that the magnitude of the associated gravitational Poynting Vector “SωΩ” is usefully approximated by the total energy density “SωΩ → c⋅Um”, resulting in an expression for EGM Flux Intensity “CΩ_J”.

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5.1

Simplification of the EGM equations

5.1.1 “Ω → Ω1”, “nΩ → nΩ_1” and “ωΩ → ωΩ_1” Considering celestial objects as point masses radiating a spectrum of gravitational wavefunctions, dominated by “ωΩ” in accordance with [5], the EGM Particle-Physics equations applied to Cosmology produce signature characteristics of the gravitational spectrum resulting in highly precise simplified representations (with negligible error) such that: i. Ω(r,M) → Ω1(r,M) ii. nΩ(r,M) → nΩ_1(r,M) iii. ωΩ(r,M) → ωΩ_1(r,M) Utilising Eq. (3.69, 3.70) yields, 3 .M .c U m( r , M ) U ω( r , M )

For solutions where “ 81.

U m( r , M )

3 4 .π .r

3 .M .c

5

4 3 h . 4 ω PV( 1 , r , M ) 2 .π .r .h .ω PV( 1 , r , M ) 3 2 .c

(4.7)

2

U ω( r , M ) 3

Ω 1( r , M )

2

108.

>> 768”: Eq. (3.72) → Eq. (4.8) [i.e. “Ω → Ω1”] as follows, U m( r , M )

108.

U ω( r , M )

3

U m( r , M )

216.

U ω( r , M )

U m( r , M ) U ω( r , M )

(4.8)

Substituting Eq. (4.7) into Eq. (4.8) yields, Ω 1( r , M )

3

6 .c r .ω PV( 1 , r , M )

.

3 .M .c

2

2 .π .h .ω PV( 1 , r , M )

(4.9)

For solutions where “Ω(r,M) >> 1”: Eq. (3.71) → Eq. (4.10) by substitution of Eq. (4.9) [i.e. “nΩ → nΩ_1”] as follows, 3

n Ω_1( r , M )

3

2 Ω 1( r , M ) 1 U m( r , M ) c 3 .M .c . . 12 2 U ω( r , M ) 2 .r .ω PV( 1 , r , M ) 2 .π .h .ω PV( 1 , r , M )

(4.10)

Hence: Eq. (3.73) → Eq. (4.11) [i.e. “ωΩ → ωΩ_1”] as follows, 3

ω Ω_1( r , M ) n Ω_1( r , M ) .ω PV( 1 , r , M )

2 c . 3 .M .c 2 .r 2 .π .h .ω PV( 1 , r , M )

(4.11)

5.1.2 Computing errors ω Ω_1 R M , M M

ω Ω_1 R E, M E

ω Ω R M, M M

ω Ω R E, M E

ω Ω_1 R J , M J

ω Ω_1 R S , M S

ω Ω R J, M J

ω Ω R S, M S

1=

ω Ω_1 R NS , M NS ω Ω R NS , M NS

130

. 6.66133810

14

. 2.22044610

14

. 6.66133810

14

. 8.88178410

14

(%)

(4.12) 1 = 0 (%)

(4.13)

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5.2

Derivation of gravitational acceleration in terms of “ωΩ”

5.2.1 Transformation: “ωΩ_1 → ωΩ_2” 5.2.1.1 Simplification The cubic form of Eq. (4.11) yields, ω Ω_1( r , M )

3 2 c . 3 .M .c 2 .r 2 .π .h .ω PV( 1 , r , M )

3

(4.14)

For solutions where “KPV(r,M) = 1”, “ωPV(1,r,M)” may be usefully approximated and utilised such that “ωΩ_19 → ωΩ_29” according to, 3

ω Ω_2( r , M )

9

9 2 1 c . 3 .M .c . . . . 3 2r 2πh ω PV( 1 , r , M )

3

9 2 c . 3 .M .c 2 .r 2 .π .h

.

3 14 2 3 .c .M

1 . 1 . 2 c .G.M 3 π .r r

13 5 2 3 2 .r .π .h .G

(4.15)

Recognising, th

G.h c

λh

5

ωh

λh

(4.19)

2

c G.h

mh

2

h .c G

(4.17)

(4.18)

3

1

3

5

1 th

(4.16)

G.h c

2

c G.h

G λh

(4.20)

c 2

3

h

(4.21)

Yields, 3 14 2 3 .c .M

4 3 2 2 3 .G .M .ω h .c

13 5 2 3 2 .r .π .h .G

2 13 5 2 2 .r .π .λ h

5 3 3 .ω h . . GM 13 5 π 2 .λ h .r

2

5 3 3 .ω h .G.M G.M . 2 3 13 2 .λ h .π .r

2

r

6 3 3 .ω h . . GM 2 13 2 2 .π .r .c r

2

(4.22)

Let, 6 3 3 .ω h

St g

13 2 2 .π .c

(4.23)

G.M

g( r , M )

2

r

(4.24)

Hence, ω Ω_2( r , M )

9

St g

.g ( r , M ) 2

r

(4.25)

Let “aEGM_ωΩ(r,M)” denote the magnitude of the gravitational acceleration vector such that: a EGM_ωΩ( r , M )

131

r . 9 ω Ω_2( r , M )

St g

(4.26)

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5.2.1.2 Computing errors 5.2.1.2.1 “ωΩ_1 → ωΩ_2” The error associated with Eq. (4.25) in relation to the refractive inclusive simplified form of harmonic cut-off frequency (i.e. “ωΩ_1”) may be articulated as follows, ω Ω_2 R M , M M

ω Ω_2 R E, M E

ω Ω_1 R M , M M

ω Ω_1 R E, M E

ω Ω_2 R J , M J

ω Ω_2 R S , M S

ω Ω_1 R J , M J

ω Ω_1 R S , M S

ω Ω_2 R NS , M NS ω Ω_1 R NS , M NS

1=

. 1.04678510

9

. 2.32001510

8

. 6.57443310

7

. 7.07196310

5

(%)

(4.27) 1 = 2.491576 ( % )

(4.28)

5.2.1.2.2 “g” The error associated with Eq. (4.26) in relation to “g” may be articulated as follows, a EGM_ωΩ R M , M M

a EGM_ωΩ R E, M E

g R M,M M

g R E, M E

a EGM_ωΩ R J , M J

a EGM_ωΩ R S , M S

g R J, M J

g R S, M S

a EGM_ωΩ R NS, M NS g R NS, M NS

1=

. 1.49880110 . 1.5432110

12 12

. 1.49880110

12

. 1.57651710

12

(%)

(4.29) . 1 = 1.65423210

12

( %)

(4.30)

5.2.1.3 Error analysis The difference in error between Eq. (4.28, 4.30) indicates that the historical weak field representation of “KPV” is limited in applicability to celestial objects with mass approximately less than Neutron Stars. However, Eq. (4.30) demonstrates that the simplified EGM representation is not effected by this constraint. This is due to the manner in which “KPV” is defined. It has two definitions according to “Puthoff et. Al.”: these are (i), it has a value of unity at infinity [i.e. Eq. (3.55)] and (ii), it has a value of “KPV = c / vc” where, “c” is the velocity of light in a vacuum and “vc” denotes the velocity of light in a vacuum affected by a gravitational field as perceived by an observer at infinity. Since “c” is a definition and “vc” is measured, the application of the PV model of gravity in terms of “KPV” is exact when the measurement is made from infinity: however, “g” is a local value. Consequently, when comparing “g” to “aEGM_ωΩ”, the contribution of “KPV” must be either (iii), removed from “aEGM_ωΩ” or (iv), merged with the classical representation of “g” for strong gravitational fields when utilising the weak field approximation specified by Eq. (3.55). Subsequently, two subtle yet important characteristics must be observed when comparing the PV model of gravity to its classical equivalent: these are (v), “KPV” increases from unity at infinity to the local observer and (vi) “g” decreases to zero at infinity from the local observer.

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5.2.2 Transformation: “ωΩ_1 → ωΩ_3” 5.2.2.1 Simplification Utilising Eq. (4.10, 4.11), an alternative expression for harmonic cut-off frequency may be formulated as follows, 3

ω Ω_1( r , M )

3

1 1 . U m( r , M ) . ω PV( 1 , r , M ) . 2 2 U ω( r , M )

U m( r , M ) h . 4 ω PV( 1 , r , M ) 3 . 2c

.ω ( 1 , r , M ) PV

(4.31)

Taking the cubic form yields, ω Ω_1( r , M )

3

3 1 . 2 .c . U m( r , M ) 8 h ω PV( 1 , r , M )

(4.32)

For solutions where “KPV(r,M) = 1”, “ωPV(1,r,M)” may be usefully approximated and utilised such that “ωΩ_19 → ωΩ_39” according to, 3 .M .c ω Ω_3( r , M )

9

3 c . U m( r , M ) 4 .h ω PV( 1 , r , M )

3

3

3

c 4 .h

2 3

. .3 . 4πr 2 .c .G.M

14

2

5

27 . c . M 8192 h 3 π2 .r5 .G

9

2

27 . c . c . M 8192 G.h h 2 π2 .r5

3

9

2

3 . 2.c . M ωh 13 2 2 2 h π .r5

π .r

4

(4.33)

Hence, 9

ω Ω_3( r , M )

c.

3.

2

3 .ω h 4 .π .h

2

. M

2

5

r

(4.34)

Let, St G

3.

3 .ω h

2

. c 2

4 .π .h

9

(4.35)

Therefore, 9

ω Ω_3( r , M )

2

M St G. 5 r

(4.36)

where, G

St G St g

(4.37)

5.2.2.2 Computing errors ω Ω_3 R M , M M

ω Ω_3 R E, M E

ω Ω_2 R M , M M

ω Ω_2 R E, M E

ω Ω_3 R J , M J

ω Ω_3 R S , M S

ω Ω_2 R J , M J

ω Ω_2 R S , M S

ω Ω_3 R NS , M NS ω Ω_2 R NS , M NS 1 . St G G

1=

. 1.11022310

13

. 13 1.11022310

. 1.11022310

13

(%)

(4.38) . 1 = 6.66133810

. 1 = 3.33066910

St g

. 14 8.88178410

14

(%)

(4.39) 14

( %)

(4.40) 133

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5.3

Formulation of a generalised cubic frequency expression in terms of “g”: “g → ωPV3”

Applying the methodology utilised to derive “ωΩ_2” and “ωΩ_3”, a generalised cubic frequency expression in terms of gravitational acceleration at an arbitrary value of “nPV” may be formulated directly from Eq. (3.67, 4.24). For solutions where “KPV(r,M) = 1”, “ωPV(nPV,r,M)3” may be usefully approximated as follows, 2 .c .n PV

3

3

ω PV n PV, r , M

5.4

π .r

.g ( r , M )

2

(4.41)

Determination of the gravitationally dominant EGM frequency: “SωΩ → c⋅Um”

Storti et. Al. demonstrated in [5] that “>> 99.99(%)” of gravitational energy exists well above the “THz” range at the surface of the Earth. Moreover, it was also shown in [5] that, in accordance with DAT’s and BPT, the PV model of gravity may be usefully approximated by a unique wavefunction at a specific frequency. This section advances these conclusions by demonstrating that the gravitational spectrum of PV frequencies may be usefully approximated by a single valued Poynting Vector wavefunction “SωΩ”. Considering the harmonic element on the Right-Hand-Side (RHS) of Eq. (3.68) and simplifying yields, n PV

2

4

4

n PV

(RHS: 3.68)

Let “nPV = nΩ - 2”, “nPV > 0”: nΩ

2

2

4

nΩ

2

substitute , n Ω

4

2

nΩ

2

8 .n Ω

simplify

24.n Ω

3

2

32.n Ω

16

(4.42)

For solutions where “nΩ >> 1”, 8 .n Ω

3

24.n Ω

8 .n Ω . n Ω

2

2

32.n Ω factor

3 .n Ω factor

8 .n Ω . n Ω

2 8 .n Ω . n Ω

2

3

3 .n Ω

4

(4.43)

→ 8 .n Ω 3

(4.44)

Hence, “Sω(r,M)”, in terms of “nΩ(r,M)”, is usefully approximated as “SωΩ(r,M)” according to: S ωΩ ( r , M )

h . 4 ω PV( 1 , r , M ) . 2 2 .c

n Ω ( r, M )

2

4

n Ω ( r, M )

4

h . 4 3 ω PV( 1 , r , M ) .8 .n Ω ( r , M ) 2 2 .c

(4.45)

Assuming “nΩ → nΩ_1” yields, 3

2 4 .h . 4 4 c 3 .M .c 3 4 .h . . S ωΩ ( r , M ) ω PV( 1 , r , M ) .n Ω ( r , M ) ω PV( 1 , r , M ) . 2 2 2 .r .ω PV( 1 , r , M ) 2 .π .h .ω PV( 1 , r , M ) c c

3

(4.46)

4

3 2 3 4 .h .c . ω PV( 1 , r , M ) . 3 .M .c 3 .M .c S ωΩ ( r , M ) 2 3 3 . . . 3 8 .c .r ω PV( 1 , r , M ) 2 π h ω PV( 1 , r , M ) 4 .π .r

Therefore,

S ωΩ ( r , M ) c .U m( r , M )

(4.47) (4.48)

This result demonstrates that “Sω(r,M)” for any solid spherical object with homogeneous mass-energy distribution may be characterised by a single wavefunction at “ωΩ(r,M)”. In other words, all other frequencies [i.e. for “ωPV(nPV,r,M) < ωΩ(r,M)”] within the PV spectrum of the gravitational field generated by the object, may be usefully neglected. 134

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5.5

Derivation of EGM Flux Intensity “CΩ_J”

5.5.1 Simplification By considering each celestial object as a point mass / source radiating a high frequency EGM wavefunction (i.e. in accordance with the conclusion stated in the preceding section), the intensity of gravitational energy (EGM Flux expressed in Jansky's) may be derived as follows: Let, 2 d λ Ω ( r , M ) . U m( r , M ) dr

C Ω_J ( r , M )

(4.49)

Note: “CΩ_J” contains “KPV” because “λΩ → c / ωΩ”. Assuming “λΩ → c / ωΩ_3” yields “CΩ_J → CΩ_J1” (i.e. the refractive exclusive form) as follows, 2

C Ω_J1( r , M )

2

c 9

2

M St G. 5 r

. . . d 3M c d r 4 .π .r3

2

c

2

5

9

. r

9

M

2

St G

2

1 2

5 . . 2 . 4 . 9 M c 9 c .St 9 . M G 4 26 4 .π 4 .π .r r

9

(4.50)

Let, 9 .c . St G 4 .π 4

St J

2 9

(4.51)

Hence, C Ω_J1( r , M )

St J 2

r

9

. M

5

8

r

(4.52)

5.5.2 Computing errors C Ω_J1 100.( km) , M M

C Ω_J1 R S , M M

C Ω_J 100.( km) , M M C Ω_J1 100.( km) , M E C Ω_J 100.( km) , M E C Ω_J1 100.( km) , M J C Ω_J 100.( km) , M J C Ω_J1 100.( km) , M S C Ω_J 100.( km) , M S C Ω_J1 100.( km) , M NS C Ω_J 100.( km) , M NS

C Ω_J R S , M M

1=

. 3.63875410

8

. 2.95903310

6

. 9.40034410

4

C Ω_J1 R S , M E C Ω_J R S , M E C Ω_J1 R S , M J

( %)

C Ω_J R S , M J

0.979587

C Ω_J1 R S , M S

0.979587

C Ω_J R S , M S

1=

. 3.86357610

12

. 4.23450210

10

. 1.3506210

7

. 1.41439110

4

. 1.41439110

4

( %)

C Ω_J1 R S , M NS C Ω_J R S , M NS

(4.53)

5.5.3 Error analysis The preceding equation set indicates that for practical astronomical applications [i.e. “r >> 100(km)”], i. “CΩ_J = CΩ_J1”. ii. The historical definition of “KPV” [i.e. the weak field approximation shown in Eq. (3.55)] does not significantly modify the value of “CΩ_J”. 135

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NOTES

136

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6 Derivation of “Planck-Particle” and “SchwarzschildBlack-Hole” Characteristics

Abstract The minimum physical dimensions of “Schwarzschild-Planck-Particle” mass and radius is derived, leading to the determination of the value of “KPV” at the event horizon of a “Schwarzschild-Planck-Black-Hole” (SPBH). Consequently, the magnitude of the harmonic cut-off frequency “ωΩ” at the event horizon “RBH” of a “Schwarzschild-Black-Hole” (SBH) is presented, yielding the singularity radius “rS” and harmonic cut-off profiles (“nΩ” and “ωΩ” as “r → RBH”). The minimum gravitational lifetime of matter “TL” is also advanced such that the value of generalised average emission frequency per Graviton “ωg” may be calculated. These determinations assist in the supplemental EGM interpretation with respect to the visibility of “Black-Holes” (BH’s).

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6.1

Derivation of the minimum physical “Schwarzschild-Planck-Particle” mass and radius

6.1.1 What does “physical” mean? 6.1.1.1 Conceptualisation The definition and conceptualisation of the term “physical” has been disputed and discussed for millennia. It remains one of the greatest questions in philosophy, dating back to ancient Greek civilization. A standard dictionary definition is “of or pertaining to that which is material”. To many people, this is an adequate and useful definition. The next obvious task is to define what “material” means. Of course, this process can be repeated indefinitely such that no universally acceptable definition is ever reached. Everyday life and experience is extremely forgiving in a “communication” sense. More often than not, one is not required to articulate in unambiguous terms, the key elements in a chance discussion. It is often sufficient, in many casual settings, to communicate thoughts and ideas in the most generalised terms available as brief human encounters can be quite dynamic. However, this luxury does not exist in the realm of Physics where precise definitions are vital for clear communication and understanding. Fortunately, the EGM construct is able to provide a useful mathematical definition facilitating the computation of common “Planck-Particle” and “Schwarzschild-Black-Hole” (SBH) characteristics. Storti. et. Al. demonstrated in [5] that by considering a solid spherical object with homogeneous mass-energy distribution to be a point source spectral wavefunction radiator with ZPF equilibrium radius “rZPF’, the following properties are mathematically observed, i. As “|Um(rZPF,M)|” increases at any arbitrarily fixed value of radial displacement35 “r”: i. “|nΩ(r,M)|” decreases. ii. “|ωΩ(r,M)|” increases. iii. “|ωPV(1,r,M)|” increases. iv. “|ωΩ(r,M)| - |ωPV(1,r,M)|” decreases. v. The PV spectrum is converging. ii. For any arbitrarily fixed value of |Um(rZPF,M)|, as “r” increases: i. “|nΩ(r,M)|” increases. ii. “|ωΩ(r,M)|” decreases. iii. “|ωPV(1,r,M)|” decreases. iv. “|ωΩ(r,M)| - |ωPV(1,r,M)|” increases. v. The PV spectrum is diverging. 6.1.1.2 Assumptions A Cosmological physical limit exists for maximum permissible energy density such that “|Um(rZPF,M)| → the physical limit”. Hence, a consistent interpretation of the results above within the framework of contemporary Quantum Mechanical (QM) expectation provokes the following hypotheses when “r > rZPF” such that “r → rZPF”: i. “|nΩ(r,M)| → 1”. ii. “|ωΩ(r,M)| → |ωPV(1,r,M)|”. iii. “[|ωΩ(r,M)| - |ωPV(1,r,M)|] → 0”. iv. The PV spectrum is convergent.

35

Where, “r > rZPF”. 138

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6.1.1.3 Definitions 6.1.1.3.1 Matter Based upon the preceding hypotheses, we are able to state an EGM definition of a specific class of “physical matter” (i.e. spherical), in addition to contemporary interpretations. Matter with spherical geometry and homogeneous mass-energy distribution is said to be “physical” if: i. The magnitude of the energy density of the object at rest, bounded by the ZPF equilibrium radius, is less than or equal to the Cosmological physical limit for maximum permissible energy density (i.e. “|Um(rZPF,M)| ≤ the physical limit”). ii. The number of harmonic frequency modes is greater than or equal to unity at the ZPF equilibrium radius (i.e. “|nΩ(rZPF,M)| ≥ 1”). 6.1.1.3.2 Energy density As specified in the proceeding construct, the limit for maximum permissible “physical energy density” is defined as “|Um(λxλh,mxmh)|”. Note: this condition implies a state of maximum permissible space-time manifold curvature. 6.1.1.3.3 Planck scale properties The historical derivation and classical definitions of Planck Frequency “ωh”, Length “λh” and Mass “mh” were not performed in accordance with formalised DAT’s. Alternatively, one could argue that the classical definitions were derived correctly, but the experimental relationship function “Kh”, associated with the formalised DAT derivation process of Planck scale properties, was assumed to be unity [i.e. “Kh(ωh,λh,mh) = 1”]. Storti et. Al. derive experimentally implicit values of “ωh”, “λh” and “mh” in [13], based upon the determination of three Experimental Relationship Functions (ERF’s) [i.e. “Kh(ωh) = Kω = (2/π)1/3” and “Kh(λh) = Kλ = Kh(mh) = Km = (π/2)1/3”]. Hence, the experimentally implicit (also termed “EGM adjusted”) values of Planck Frequency, Length and Mass are given by “Kω ωh”, “Kλλh” and “Kmmh” respectively, such that when “h = 6.626069 x10-34(Js)”: i. “Kωωh = 6.365769 x1042(Hz)”. ii. “Kλλh = 4.709446 x10-35(m)”. iii. “Kmmh = 6.341792 x10-8(kg)”. A derivation of the Root-Mean-Square (RMS) charge diameter (i.e. the ZPF equilibrium diameter) of a Photon “φγγ” (i.e. twice the RMS charge radius “rγγ”) is performed in [10] demonstrating that “φγγ = 2rγγ ≈ λh” [to within 15.3(%)]. Subsequently, utilising the EGM adjusted Planck Length, it may be demonstrated that “Kλλh = φγγ” to within “0.83(%)”. [13] Since “φγγ” is extremely close to QM expectation and all EGM adjusted Planck properties were derived in the same manner, it follows that the individual (not combined) “physical Planck scale properties” of frequency “ω”, length “λ” and mass “m” may be defined as: iv. “ω ≤ Kωωh”. v. “λ ≥ Kλλh”. vi. “m ≥ Kmmh”. Note: in the proceeding construct, vii. “λx > Kλ”. viii. “mx > Km”. 139

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6.1.2 What is a “Schwarzschild-Planck-Particle”? 6.1.2.1 Conceptualisation Contemporary Physics believes the maximum permissible energy density of a “Black-Hole” (BH) is governed by classical Planck scale dimensions. Hence, a proportional representation of this belief by an EGM based method is a natural evolutionary step. Subsequently, one would expect that a SBH with “Planck-Particle” energy density represents the natural physical energy density limit [i.e. “Um(λxλh,mxmh)”] as the number of harmonic frequency modes approaches unity [i.e. “nΩ(λxλh,mxmh) = 1”]. For a SBH or “Planck-Particle” in a vacuum, one would intuitively expect that “KPV → Undefined” at (or beyond) the event horizon, relative to a non-local observer without. However, if we consider ourselves to be local observers within the event horizon, the space-time manifold is maximally curved and no variation exists with respect to the PV model of gravity in terms of “KPV”, from point to point. Therefore, by logical induction, we shall assign a value of “KPV = 1” within the event horizon of a SBH due to the dimensional consistency of the space-time manifold within being analogous (only) to a totally flat space-time manifold without [i.e. “∆KPV(∆r) = 0”]. In other words, one cannot geometrically curve the space-time manifold, within the event horizon, more than “completely” (i.e. beyond the maximum permissible curvature limit). Note: conceptualising a local observer within the event horizon negates potential modelling difficulties of a non-local observer beyond the event horizon36. 6.1.2.2 Assumptions In addition to the assumptions specified in the preceding section (i.e. serving as a derivational base), we shall also apply the following, i. Complete dynamic, kinematic and geometric similarity exists, in accordance with DAT’s and BPT principles, between a SBH and a mass-energy equivalent particle on the Planck scale (i.e. a “Planck-Particle”). ii. The relationship between a SBH and “Planck-Particle” may be described by the EGM harmonic representation of fundamental particles [i.e. Eq. (3.230ii)]. iii. The energy density limit also denotes a condition of maximum permissible space-time manifold curvature such that “KPV = 1” is a valid assignment within the even horizon of a SBH. 6.1.2.3 Definition A “Schwarzschild-Planck-Particle” is a SBH with “Planck-Particle” energy density satisfying the condition “Stω(λxλh,mxmh) = 1” and “nΩ(λxλh,mxmh) = 1” such that the event horizon “RBH” and ZPF equilibrium radii “rZPF” coincide at “λxλh”: • i.e. “RBH = rZPF = λxλh”. Note: a SBH with “Planck-Particle” energy density is also termed a “Schwarzschild-Planck-BlackHole” (SPBH).

36

Due to the historical definition of “KPV” being a weak field approximation only. 140

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6.1.3 Construct If we equate the energy density of a “Planck-Particle” of mass “mxmh” and radius “λxλh” to a SBH, we may determine the minimum radius of a SBH with “Planck-Particle” energy density. This is termed a “Schwarzschild-Planck-Black-Hole” and its minimum radius and mass may be determined. Taking the cubic form of Eq. (4.10) yields, n Ω_1( r , M )

1 . U m( r , M ) 1 . 3 .M .c . 2 .c 8 U ω( r , M ) 8 4 .π .r3 h .ω ( 1 , r , M ) 4 PV 2

3

3

(4.54)

For solutions where “KPV(r,M) = 1”, “ωPV(1,r,M)” may be usefully approximated according to, 1 . 3 .M .c . 8 4 .π .r3 h .ω

2 .c

2

1 . 3 .M .c . 8 4 .π .r3

2 .c

2 .c

2

PV( 1 , r , M )

2

1 . 3 .M .c . 8 4 .π .r3

3 4

3 1 2 .c .G.M h. . r π .r

1 . 3 .M .c . 8 4 .π .r3

3

3

2 .c

2

4

3 1 2 .c .G.M h. . r π .r

4

(4.55)

3

3 1 2 .c .G.M . 2 .c .G.M h. . 4 π .r π .r r

(4.56)

3

2 1 . 3 .M .c .

8 4 .π .r3

2 .c

3. c . . 2 cr 1 . 4 h .G

3

3 1 2 .c .G.M . 2 .c .G.M 8 h. . 4 π .r π .r r

3

2 .c .G.M π .r

(4.57)

Substituting Eq. (4.20) into Eq. (4.57) produces, 3 .c . r n Ω_1( r , M )

3

Therefore, “nΩ_1 →

nΩ_19”

9

3

2 .c .G.M π .r

(4.58)

2 3

λh

1. 4 8

8

λh

according to, 3 .c . r

n Ω_1( r , M )

1. 4

3

2

3

r . π .c . 3 . 16 2 2 GM λh 7

2 .c .G.M π .r

2

3

7 r . π .m h . 3 16 2 2 M λh λh

3

(4.59)

Hence, let: 9

n Ω_2( r , M )

. 3 3 .π m h . r 16 M λh 2

7

(4.60)

such that “nΩ_2 → nΩ_3” as “r → λxλh” and “M → mxmh” according to, 9

n Ω_3 λ x.λ h , m x.m h

λ x.λ h 16

2

7

9

m . π . h. 3 m x.m h λ h λ 2 h

141

3

7

9

3 3 3 .π . λ x 1 . 3 .π . λ x 16 m 2 x 2 mx 2

7

(4.61)

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Assuming “nΩ_3(λxλh,mxmh) = 1” (i.e. for the maximum energy density condition), the value of “mx” satisfying this limit may stated as, 3 3 .π . 7 λx 16 2

mx

(4.62)

The value of “λx” may be derived by application of the EGM harmonic representation of fundamental particles [i.e. relating Eq. (4.62) to Eq. (3.230ii)]. Within the boundaries of contemporary Physics, the dimensions of the smallest permissible SBH exist on the Planck scale. Subsequently, if “r1 = RBH”, “M1 = MBH”, “r2 = λxλh” and “M2 = mxmh” then Eq. (3.230ii) may be applied as follows, λ x.λ h

5

m x.m h

2

2

M BH

.

5

R BH

2

2 c .R BH 2 .G

λ x.λ h

5

m x.m h

2

.

5

R BH

(4.63)

where, the radius to the event horizon “RBH” is related to mass “MBH” by, 2 .G . M BH 2 c

R BH

(4.64)

Substituting Eq. (4.64) into Eq. (4.63) yields, 2

2 c .R BH 2 .G

λ x.λ h

5

m x.m h

2

.

5

R BH 5

2 1. λ x . c 4 R 3 G BH

2

λh

.

5

5

1.

m x.m h 5

1.

. c 3 G

4 λ .λ x h

2 λ x.λ h .c

λh

.

λh

.

5

m x.m h

. c 3 G

5

m x.m h

2

2 2

λx

4 λ .λ x h

2

2 2

λx

5

2 1. λ x . c 4 R 3 G BH

λh

.

(4.65) 5

m x.m h

2 λ x.λ h .c 2

2

2 .G.m x.m h

2

(4.66)

2

(4.67)

2

St ω λ x.λ h , m x.m h

2 .G.m x.m h

9

(4.68)

Assuming ideal similarity between a SBH and a “Planck-Particle” satisfying “Stω(λxλh,mxmh) = 1” at “nΩ_3(λxλh,mxmh) = 1”, the value of “λx” and “mx” may be determined as follows, Recognising that Eq. (4.69) may be substituted into Eq. (4.68) yields, c

2

mh

G λh

(4.69)

2 λ x.λ h .c

1.λ x

2 .G.m x.m h 2 m x

1

(4.70)

such that:

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mx

λx 2

(4.71)

Substituting Eq. (4.71) into Eq. (4.62) yields, λx

4 . 2 π 3

6

(4.72)

Evaluating produces, λx

=

mx

2.698709 1.349354

(4.73)

λ x.λ h = 1.093333 10

10 .

. m x.m h = 7.36147410

8

ym

(4.74)

( kg )

(4.75) where, “ym = yoctometre = 10-24(m)”. Mass and energy density characteristics of a SPBH may also be easily derived based upon the preceding results according to, V( r )

4. . 3 πr 3

(4.76)

where, “V” and “ρm” denote volume and mass-density respectively. Hence, ρ m( r , M )

M V( r )

(4.77i)

. 94 kg ρ m λ x.λ h , m x.m h = 1.34467810 3 m

(4.78)

The ratio of the mass-density of a SPBH to the Sun is given by, ρ m λ x.λ h , m x.m h ρ m R S, M S

. = 9.55041510

90

(4.79)

Substituting Eq. (4.74, 4.75) into Eq. (3.70) produces the SPBH energy-density result, . 87 ( YPa) U m λ x.λ h , m x.m h = 1.20853710

(4.80)

24

where, “YPa = yottaPascals = 10 (Pa)”. 6.1.4 Computing errors The “1st” of two simple checks to ensure no algebraic errors were made in relation to the baseline assumption that “Stω(λxλh,mxmh) = 1” at “nΩ_3(λxλh,mxmh) = 1”, is to substitute Eq. (4.71) into Eq. (4.61) producing the result, 3 9

n Ω_3 λ x

π. 2

3.

λx

2

2

(4.81)

Comparing “nΩ_2” to “nΩ_3” confirms a lack of algebraic errors, n Ω_2 λ x.λ h , m x.m h n Ω_3 λ x

143

. 1 = 2.22044610

14

(%)

(4.82) www.deltagroupengineering.com

The “2nd” check substitutes “mx = λx / 2” into Eq. (4.68). Upon simplification one concludes that “Stω(λxλh,mxmh) = 1”, confirming a lack of algebraic errors. 6.1.5 Convergent and divergent spectra Substituting various values into Eq. (4.81) yields a sequence demonstrating the preservation of EGM characteristics and exhibiting PV spectral convergence / divergence behaviour as follows, n Ω_3 n Ω_3

1 3

0.248017

1

0.324994

2

n Ω_3( 1 )

=

0.515897 0.818935

n Ω_3( 2 )

1

n Ω_3 λ x

1.073108

n Ω_3( 3 )

(4.83)

Therefore, for solutions where “KPV = 1” such that “nΩ → nΩ_3”, the following statements may be articulated: i. Values of “nΩ(r,M) < 1” indicate a divergent PV spectrum where “ωΩ(r,M) < ωPV(1,r,M)” and are non-physical. ii. Values of “nΩ(r,M) ≥ 1” indicate a convergent PV spectrum where “ωΩ(r,M) ≥ ωPV(1,r,M)” and are physical. 6.1.6 Honourable mention It should not escape attention that “mx” is very close to the square of the experimentally implicit definition of EGM adjusted Planck mass ERF derived in [13] (see below). However, no specific conclusion may be inferred from this result. mx

1 = 0.14278 ( % )

2

Km

(4.84)

6.1.7 Concluding remarks 6.1.7.1 Characteristics of a physical SPBH i. ii. iii. iv.

It has coinciding singularity and event horizon radii. “r ≥ λxλh”, “M ≥ mxmh” and “mx = λx / 2”. “ρm(r,M) ≤ ρm(λxλh,mxmh)”. A value of “KPV = 1” is assigned within the event horizon (i.e. “r < λxλh”) due to maximum permissible space-time manifold curvature [i.e. “∆KPV(∆r) = 0”]. v. For solutions where “KPV = 1”, only one harmonic mode exists [i.e. “nΩ(λxλh,mxmh) = 1”] such that “ωPV(1,λxλh,mxmh) = ωΩ(λxλh,mxmh)” and the PV spectrum is convergent.

6.1.7.2 Characteristics of a non-physical “Planck-Particle” i. ii. iii. iv.

It is not a SPBH. “r < λxλh” such that “nΩ(r<λxλh) < 1”. “ωΩ(r,M) < ωPV(1,r,M)”. The PV spectrum is described as divergent. 144

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6.1.7.3 Physicality of the EGM adjusted Planck Length A derivation of the Root-Mean-Square (RMS) charge diameter is performed in [10] demonstrating that “φγγ = 2rγγ ≈ λh”. Subsequently, utilising the EGM adjusted Planck Length, it may be demonstrated that “Kλλh = φγγ” to within “0.83(%)” [13] according to, K λ .λ h 2 .r γγ

1 = 0.82832 ( % )

(4.85)

Since “φγγ” is extremely close to QM expectation and all EGM adjusted Planck properties were derived in the same manner, it follows that the individual (not combined) “physical Planck scale properties” of frequency “ω”, length “λ” and mass “m” may be defined as: i. “ω ≤ Kωωh”. ii. “λ ≥ Kλλh”. iii. “m ≥ Kmmh”. NOTES

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6.2

Derivation of the value of the “KPV” at the event horizon of a SPBH

6.2.1 Synopsis It was stated previously that, for a SPBH in a vacuum, one would intuitively expect that “KPV → Undefined” at (or beyond) the event horizon, relative to a non-local observer without. This section demonstrates two independent methods for mathematically verifying this contention utilising “λxλh” and “mxmh”. The “1st” method applies EGM principles; the “2nd” advances the work of Depp et. Al as derived in [47]. 6.2.2 Construct 6.2.2.1 1st Formulation Recognising, U ω λ x.λ h , m x.m h

h . ω PV 1 , λ x.λ h , m x.m h 3 . 2c

4

h . ω Ω λ x.λ h , m x.m h 3 . 2c

4

(4.86)

and utilising Eq. (4.69), it follows that: ω PV 1 , λ x.λ h , m x.m h

3 . . . . 1 . 2 c G mx mh . K PV λ x.λ h , m x.m h λ x.λ h π .λ x.λ h

(4.87)

3

3

1 . λ x.λ h

2 .c .G.m x.m h π .λ x.λ h

3

2 .c .G. 1 . λ x.λ h

λx 2 .c .G. .m h 1 . 2 . K . . PV λ x λ h , m x m h . . λxλh π λ x.λ h

. K . . PV λ x λ h , m x m h

(4.88)

λx

.m h 2 . K . . PV λ x λ h , m x m h π .λ x.λ h 3

c . 1. K PV λ x.λ h , m x.m h λ x.λ h π

3

c . 1. K PV λ x.λ h , m x.m h λ x.λ h π ωh

3

. 1. K . . PV λ x λ h , m x m h λx π

ωh

(4.90)

3

. 1. K . . PV λ x λ h , m x m h λx π

ω PV 1 , λ x.λ h , m x.m h

(4.89)

(4.91)

Performing the appropriate substitutions yields, U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

U m λ x.λ h , m x.m h h . ω PV 1 , λ x.λ h , m x.m h 3 . 2c

U m λ x.λ h , m x.m h h . ω PV 1 , λ x.λ h , m x.m h 3 2 .c

4

(4.92)

U m λ x.λ h , m x.m h 4

3 h . ωh. 1. K PV λ x.λ h , m x.m h 3 2 .c λ x π

146

4

(4.93)

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U m λ x.λ h , m x.m h

U m λ x.λ h , m x.m h

3 h . ωh. 1. K PV λ x.λ h , m x.m h 3 λ π . 2c x

U m λ x.λ h , m x.m h h .ω h

4

3 4 3 2 .π . π .c .λ x

.K . . PV λ x λ h , m x m h

2

4

h .ω h 3

4 4

3.

2 .π . π .c λ x

.K . . PV λ x λ h , m x m h

2

(4.94)

3 4 3 2 .π . π .c .λ x U m λ x.λ h , m x.m h . 4 . h ωh K PV λ x.λ h , m x.m h

2

(4.95)

Partially evaluating Eq. (4.95) numerically within a computational environment yields, 3 4 3 2 .π . π .c .λ x .U λ .λ , m .m = 8 m x h x h 4 . h ωh

Hence,

U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

(4.96)

8 K PV λ x.λ h , m x.m h

2

(4.97)

such that an expression for “KPV” may be formed according to, K PV λ x.λ h , m x.m h

2. 2.

U ω λ x.λ h , m x.m h U m λ x.λ h , m x.m h

(4.98)

Recognising that the EGM spectrum converges to a single mode for a SPBH and utilising Eq. (3.71) yields, n Ω λ x.λ h , m x.m h

Ω λ x.λ h , m x.m h 12

4 Ω λ x.λ h , m x.m h

1 1

(4.99)

where, Ω λ x.λ h , m x.m h

4. 3

(4.100)

3

Ω λ x.λ h , m x.m h

108.

U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

12. 768 81.

U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

2

4. 3

(4.101)

By inspection, the only solution satisfying this equation is: U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

0

(4.102)

Verifying numerically yields, 3

108.0

2 12. 768 81.0 = 6.928203

4 . 3 = 6.928203

(4.103) (4.104)

Therefore, K PV λ x.λ h , m x.m h

147

2. 2 0

Undefined

(4.105)

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6.2.2.2 2nd Formulation Let the expression for “KPV” (shown as “KDepp” below) derived by Depp et. Al. in [47], satisfying the SBH condition be assigned the form, 1

K Depp ( r , M )

2 .G.M

1

r .c

2

(4.106)

such that: K Depp ( r , M )

1 2 .G.M

2

1

r .c

r .c r .c

2

2

2 .G.M

2

(4.107)

Performing a sample calculation at the surface of the Earth and comparing results utilising Eq. (3.55, 4.107) produces, K PV R E, M E = 1.00000000069601

(4.108)

K Depp R E, M E = 1.00000000069601

(4.109)

Hence, the relationship between the “Depp” and weak field exponential form is, K PV( r , M )

K Depp ( r , M )

2

K Depp ( r , M )

K PV( r , M )

(4.110)

To ensure the preceding results were not coincidental due to weak field application, we shall test Eq. (4.110) in the strong field as follows, K PV( r , M )

1

2 .G.M r .c

r .c

1

r .c

2

2

2

2 .G.M

(4.111)

Substituting Eq. (4.64) into Eq. (4.110) validates the relationship for a strong gravitational field and yields the expected result, K PV R BH, M BH

1

2 .G.M BH 2 .G . 2 M BH.c 2 c

1

1

Undefined

0

(4.112)

Substituting Eq. (4.69, 4.71) into Eq. (4.107) produces the “Depp” value for a SPBH as follows, K Depp λ x.λ h , m x.m h

2

1

1 2 .G.m x.m h 2 λ x.λ h .c

1 1

2 .G.m x c2 . 1 2 G . λxc

1

1

2 .m x λx

2. 1

1 λx 0

Undefined

2 λx

(4.113)

6.2.3 Concluding remarks From the results obtained above, we have shown by derivation of identical values of Refractive Index, that a “Planck-Particle” is dynamically, kinematically and geometrically equivalent to a SBH. This may be mathematically stated as follows, KPV(λxλh,mxmh) = KPV(RBH,MBH) = KDepp(λxλh,mxmh) = KDepp(RBH,MBH) 148

(4.114)

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6.3

Derivation of “ωΩ” at the event horizon of a SPBH

6.3.1 Synopsis It was stated previously - for a SPBH in a vacuum, one would intuitively expect that “KPV → Undefined” (not infinity - an important distinction) at (or beyond) the event horizon, relative to a non-local observer without. This section derives the value of “ωΩ” for a SPBH [i.e. “ωΩ_3(λxλh,mxmh)”] relative to a local observer within the event horizon. 6.3.2 Assumptions The value of “ωΩ_3(λxλh,mxmh)” may be determined by assuming the following (remaining consistent with preceding sections), i. The space-time manifold is maximally curved and cannot be geometrically modified beyond the maximum permissible limit. ii. A value of “KPV = 1” is assigned within the event horizon of a SPBH due to the dimensional consistency of the space-time manifold within being analogous (only) to a totally flat spacetime manifold without [i.e. “∆KPV(∆r) = 0”]. 6.3.3 Construct Commencing with the utilisation of the “9th” power form of Eq. (4.36), we substitute “r = λxλh” and “M = mxmh” producing, 2

M St G. 5 r

St G.

m x.m h

2

λ x.λ h

5

λx

2

2

St G.

St G.

2

λ x.λ h

5

(4.115) λx 2

St G.

.m h

λ x.λ h

m x.m h

2

.m h

λ x.λ h

5

(4.116)

2

St G m h . 3 5 4 .λ x λ h

5

(4.117)

Substituting Eq. (4.35) into Eq. (4.117) yields, 2

St G m h . 3 5 . 4λx λh 3 . 4λ

3 . 4λ

3

.

x

3

x

.

ωh

2

π .h

3

.

x

2

π .h

3 . 4λ

5 4 c .ω h . 9

2

2

ωh

2

9 m . c . h 2 λ 5 h

ω h .m h π .h

3 . 4λ

3

x

.

2

9 m . c . h 2 λ 5 h

ω h .m h

3

2

.

π .h

2 c .m h . h λx

3

(4.118) 5 4 c .ω h 9

2 2

.

ωh

(4.119)

7

15 2 2 .π

(4.120)

Substituting Eq. (4.121) into Eq. (4.120) yields Eq. (4.122) as follows, 149

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m h .c

2

ωh

h St G.

m x.m h

2

λ x.λ h

5

(4.121) 3

3

ωh

.

λx

9

15 2 2 .π

(4.122)

Utilising Eq. (4.36, 4.122) and simplifying yields, 9

St G.

9 3

3 λx 9

1. 2

.

m x.m h

2

λ x.λ h

5

ωh

9 3

3

9

λx 9

1.

15 2 2 .π 2

ωh

.

9

15 2 2 .π

(4.123)

3

3

. 1 .ω h λ x 26 .π2 9

3

(4.124)

3

1. 1 . 3 . 1 .ω .ω h h 2 6 2 π2 4 .λ x λ x 2 .π 3

9

3

1. 1 . 3 .ω h 2 π2 4 .λ x

3

1 9

. 2

2. π

(4.125)

3 . ωh

4 .λ x

(4.126)

Substituting Eq. (4.72) into the Left-Hand-Side (LHS) of Eq. (4.126) yields, 9

1. 1 . 2 π2

3

3 6

4. 4.

3

2

9

4

1. 3 . 6 4 25 π3

3 3 .π

(4.127)

Hence, 9

ω Ω_3 λ x.λ h , m x.m h

4

1. 3 . 6 . ωh 4 25 π3

(4.128)

Displaying the numerical result for “ωΩ” at the event horizon of a SPBH yields, . 18 ( YHz) ω Ω_3 λ x.λ h , m x.m h = 1.87219710

(4.129)

6.3.4 “ωPV(1,λxλh,mxmh)” Substituting Eq. (4.81) into the basic EGM relational form “ωΩ(r,M) = nΩ(r,M)ωPV(1,r,M)”, facilitates the derivation of “ωPV(1,λxλh,mxmh)” as follows, ω PV 1 , λ x.λ h , m x.m h

ω Ω_3 λ x.λ h , m x.m h n Ω_3 λ x

(4.130)

ω Ω_3 λ x.λ h , m x.m h

(4.131)

Recognising that “nΩ_3(λx) = 1” yields, ω PV 1 , λ x.λ h , m x.m h

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6.3.5 Honourable mention By inspection, “ωΩ_3(λxλh,mxmh) ≈ ¼ωh” utilising a definition of “h = 6.626069 x10-34(Js)”. However, no specific conclusion may be inferred from the following result, 1. . 18 ( YHz) ω h = 1.84996810 4

(4.132)

Evaluating the approximation with respect to Eq. (4.129) demonstrates a small error according to, ωh

1.

4 ω Ω_3 λ x.λ h , m x.m h

1 = 1.187319 ( % )

(4.133)

Additionally, it is noteworthy that “λx” may be represented in terms of the exponential function “e” and the Fine Structure constant “α” (to high precision). However, no specific conclusion may be inferred from the following results, λx

e 1

e 1

α

α

(4.134)

= 2.698589

(4.135)

Checking the simplification error yields, 1 . e λx 1 α

. 3 (%) 1 = 4.43474910

(4.136)

6.3.6 Concluding remarks The preceding construct demonstrates that the value of “ωΩ” for a SPBH is “≈ ¼ωh”. This “appears” consistent with the QM assertion of a physical frequency limit being “≈ ωh”. However, in the proceeding construct, the QM expectation shall be challenged and shown to be ill-founded. NOTES

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6.4

Derivation of “ωΩ” at the event horizon of a SBH

6.4.1 Synopsis The actual value of “KPV” is unimportant within the event horizon of a SBH. That is, an observer without cannot see past the event horizon, so any value of “KPV” within, has no physical meaning to an observer without. Therefore, we may assign any convenient value of “KPV” inside the event horizon, provided it is done consistently. 6.4.2 Assumptions At the event horizon of a SBH, “KPV(RBH,MBH) = Undefined” (not infinity - an important distinction) relative to a non-local observer without. Subsequently, for an observer within the event horizon of a SBH, we shall assume: i. A physical singularity of radius “rS” exists at the centre of a SBH. ii. “KPV” is constant [i.e. “∆KPV(∆r) = 0”] from the singularity radius “rS” to the event horizon “RBH” [i.e. “KPV(rS≤r≤RBH,MBH) = 1”]. This feature is beyond the determination of an observer without. 6.4.3 Construct Utilising Eq. (4.36, 4.64), we may derive the value of “ωΩ” at the event horizon of a SBH [i.e. “ωΩ_4(MBH)” at “RBH”] as follows, 9

ω Ω_4 M BH

2

St G.

2 .G.M BH c

c .St G

9

M BH

c. 5

5 3 ( 2 .G) .M BH

2

(4.137)

Let, 9

St BH

c.

c .St G 5 ( 2 .G)

(4.138)

Hence, 3

ω Ω_4 M BH

St BH.

1 M BH

(4.139)

6.4.4 Sample calculations Performing sample calculations for a SPBH (i.e. “MBH = mxmh”), a SBH at one solar mass, “105” and “1010” solar masses (i.e. “MBH = MS, 105MS, 1010MS” respectively) yields, ω Ω_4 m x.m h

. 18 1.87219710

ω Ω_4 M S 5 ω Ω_4 10 .M S

=

. 5 6.23977510

( YHz)

. 4 1.34431910

10 ω Ω_4 10 .M S

289.624693

(4.140)

6.4.5 Concluding remarks Since “ωΩ_4(mxmh) = ωΩ_3(λxλh,mxmh)”, no errors have been generated in the formulation of “ωΩ_4(MBH)”. A clear mathematical pattern is articulated demonstrating that “ωΩ_4” increases with energy density (i.e. to “RBH”) and decreases with “MBH”. 152

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Note: the decrease in “ωΩ_4” arises from the increase in “RBH” because: i. The physical geometry of a SBH utilised to determine energy density is defined by “RBH”. The larger it becomes due to an “MBH” increase, the lower the value of the energy density and “ωΩ_4”. ii. Matter is considered to be a point source wavefunction radiator under the EGM construct. As the wavefunction propagates, its frequency decays. Since the event horizon is farther away from the singularity (i.e. the point source) for increasing “MBH”, “ωΩ_4” decreases. NOTES

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6.5

Derivation of “rS”

6.5.1 Synopsis A physical “Planck-Particle” is considered to represent the condition of maximum permissible energy density in the Universe. If a SBH is a real manifestation of a physical “PlanckParticle”, then it follows that the maximum permissible mass-density of the singularity at the centre of a BH is equal to the mass-density of a physical “Planck-Particle” (i.e. a SPBH). 6.5.2 Assumptions Utilising key features defined in preceding sections, we shall assume that, for an observer within the event horizon of a SBH: i. A physical singularity of mass-density “ρm(rS,MBH)” exists at the centre of a SBH. ii. The physical singularity at the centre is a SPBH [i.e. “ρm(rS,MBH) = ρm(λxλh,mxmh)”]. iii. “KPV” is constant [i.e. “∆KPV(∆r) = 0”] from the singularity radius “rS” to the event horizon “RBH” [i.e. “KPV(rS≤r≤RBH,MBH) = 1”]. This feature is beyond the determination of an observer without. 6.5.3 Construct 6.5.3.1 1st Formulation “rS” may be derived utilising Eq. (4.77i) according to, ρ m( r , M )

3 .M 3 4 .π .r

(4.77ii)

Substituting “Planck-Particle” characteristics yields,

ρ m λ x.λ h , m x.m h

3.

3 .m x.m h 4 .π . λ x.λ h

λx 2

3

.m h

3 .m h 3

4 .π . λ x.λ h

2 3 8 .π .λ x .λ h

(4.141)

Let the mass-density of the singularity at the core of a SBH be defined by, 3 .M BH

ρ m r S , M BH

4 .π .r S

3

(4.142)

If “ρm(rS,MBH) = ρm(λxλh,mxmh)” then, M BH 3

rS

mh 2 3 2 .λ x .λ h

(4.143)

Substituting “ωh2 = (c/λh)2” and “c2/G = mh/λh” into the above yields, M BH 3

rS

ωh

2 2

2 .G.λ x

(4.144)

such that:

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3

r S M BH

2 2 .G.λ x

ωh

2

3

. . 2. .M BH λ h 2 λ x

M BH mh

(4.145)

Recognising that “RBH(MBH) = 2GMBH/c2” yields, 3

r S R BH

2 λ x.λ h .R BH

(4.146)

Noting that, 3

r S λ x.λ h

2 λ x.λ h . λ x.λ h

λ x.λ h

(4.147)

6.5.3.2 2nd Formulation Since the singularity mass-density “ρS” is constant, we may express the construct in an alternative form by specifying “ρS = ρm(λxλh,mxmh)” as follows, 3

r S M BH

3 .M BH 4 .π .ρ S

(4.148)

Hence, 3.

2 c .R BH 2 .G

3 4 .π .r S

ρS

3

r S R BH

2 3 .c .R BH 3 8 .π .G.r S

(4.149)

2 3 .c .R BH 8 .π .G.ρ S

(4.150)

6.5.3.3 3rd Formulation An expression for “MBH” as a function of “rS” may be formulated utilising “V(r) = (4/3)πr3” as follows, 4. . 3 π ρ S .r S 3

M BH r S

(4.151)

6.5.4 Sample calculations Performing sample calculations of “rS” expressing “MBH” in terms of proportional solar mass yields, rS MS 5.

r S 10 M S

. 3.28046310 =

10 r S 10 .M S

0.015227

4

( am)

0.706754

(4.152)

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Similarly, performing sample calculations of the BH to solar mass ratio “MBH/MS” as a function of “rS” such that it resides on the fundamental particle scale (see: [12,17]) yields, M BH r ε

M BH r π

M BH r ν

M BH r µ

. 13 1.62379510 . 19 1.60185510 . 19 1.57097210 . 13 4.66247210

M BH r τ 1 . M BH r uq MS M BH r bq

M BH r en

M BH r µn

M BH r τn

. 13 2.45782610 . 7 5.19529810

M BH r dq

M BH r sq

M BH r cq

= 1.28408510 . 10 2.95005410 . 10 1.9828610 . 10 3.68186410 . 10

M BH r tq

M BH r W

. 10 2.27391910 . 10 5.99684310 . 10 3.39015710 . 10 3.47948910

M BH r H

M BH r γγ

M BH r gg

M BH r Z 1.( kg )

. 10 2.3560510

. 9 2.12850410 . 11 7.96867110

0

0

0

(4.153)

Note: i. Disregard “1(kg)” from the above: this was required and included merely to define the matrix for evaluation in the “MathCad” computational environment. ii. “rS” of a “Super-Massive-Black-Hole” (SMBH) [i.e. “1010” solar masses] approaches the dimensions of a Quark or Boson. Calculating the total mass of the Universe has been attempted many times by the Physics community, with no definitive success. We shall perform some qualitative comparisons utilising “rS”. The following results represent various speculative total mass values of the Universe if it were condensed to an “rS” value equal to some well known particles. M BH r ε

. 43 9.27104510

M BH r π

. 49 3.22881910

M BH r e

=

. 51 1.26038310

( kg )

. 63 8.34661610

M BH r Bohr

(4.154)

6.5.5 Honourable mention It should not escape attention that when “rS ≈ rε”, “RBH” approximately equals the size of the observable Universe according to, RBH(MBH(rε)) ≈ 14.56(GLyr)

(4.155)

6.5.6 Concluding remarks The derivation of “rS” has lead to the development of some very useful relationships and characteristics. Six significant results may be emphasised from this section, these are: i. The SPBH and SBH singularity energy densities are equal: U m λ x.λ h , m x.m h

U m r S M BH , M BH

(4.156)

ii. The singularity at the centre of every SBH exists at the energy density limit. iii. A key difference between a SPBH and a SBH singularity is the location of the event horizon. For a SPBH, “rS” and “RBH” coincide: for a SBH, they do not. iv. By observational inference regarding the mass limit of SMBH’s (i.e. approximately “1010” solar masses), the preceding results suggest that the physical dimensions of a “Quark or Boson” might be some sort of natural singularity size limit. v. If the preceding point is correct, it may be possible to discount “Bosons” by recognising them to be force carriers. Subsequently, it may (?) be reasonable to conjecture that the dimensions of a SMBH singularity is generalised by the physical “Quark” range according to “1.28 x1010MS < MBH(rS) < 3.69 x1010MS”. vi. A philosophical question arises: “is a Quark actually a SMBH from another Universe”? 156

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6.6

“nΩ” and “ωΩ” profiles (as “r → RBH”) of SBH’s

6.6.1 “nΩ” 6.6.1.1 Synopsis This section derives the “nΩ” profile (i.e. as “r → RBH”) of any SBH. It is numerically demonstrated (explicitly in terms of “MBH”) and graphically illustrated (in terms of radial displacement) that the profile remains consistent with conclusions defined in [5] such that: i. “nΩ_2” increases as “r → RBH” [see: Eq. (4.157, 4.158), Fig. (4.18)]. ii. “nΩ_4,5” increases with rising “MBH” [see: Eq. (4.160)]. 6.6.1.2 Assumptions Utilising key features defined in preceding sections, we shall assume that, for an observer within the event horizon of a SBH: i. The singularity of mass-density “ρm(rS,MBH)” at the centre of a SBH exists at the physical limit such that: “ρm(rS,MBH) = ρm(λxλh,mxmh)”. ii. “KPV” is constant [i.e. “∆KPV(∆r) = 0”] from the singularity radius “rS” to the event horizon “RBH” [i.e. “KPV(rS≤r≤RBH,MBH) = 1”]. This feature is beyond the determination of an observer without. iii. “nΩ(rS≠λxλh,MBH≠mxmh) > 1” where “nΩ → nΩ_2”. 6.6.1.3 Construct The “nΩ” profile for SBH’s (i.e. as “r → RBH”) may be determined trivially in a computational environment utilising “nΩ_2”. However, for subsequent use within this text, it is more convenient to define a new form explicitly in terms of “MBH” as follows, Let “nΩ” at the periphery of a SBH singularity be given by “nΩ_4” according to, n Ω_2 r S M BH , M BH

n Ω_4 M BH

(4.157)

Let “nΩ” at the event horizon of a SBH be given by “nΩ_5” according to, n Ω_2 R BH M BH , M BH

n Ω_5 M BH

(4.158)

Let the event horizon to singularity cut-off mode ratio be given by “nBH” according to, n BH M BH

n Ω_5 M BH n Ω_4 M BH

(4.159)

6.6.1.4 Sample calculations Evaluating Eq. (4.157 – 4.159) utilising arbitrary values produces the following illustrational results explicitly in terms of “MBH”, n Ω_4 m x.m h

n Ω_5 m x.m h

n BH m x.m h

n Ω_4 M S

n Ω_5 M S

n BH M S

5 n Ω_4 10 .M S

5 n Ω_5 10 .M S

5 n BH 10 .M S

10 n Ω_4 10 .M S

10 n Ω_5 10 .M S

10 n BH 10 .M S

157

1 =

1

1

. 5 9.00254210 . 24 2.56419310 . 19 3.51086810 . 6 1.93953910 . 28 1.0035610 . 22 1.93265910 . 7 4.1786110 . 31 3.92767810 . 24 1.06388810

(4.160)

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6.6.1.5 Sample plots (log vs. log) The increase in “nΩ” (as a function of radial displacement “Rbh” and mass “MBH”) over the range “rS(MS) ≤ Rbh ≤ RBH(MS)” may be graphically illustrated according to, Harmonic Cut-Off Mode vs Radial Disp.

Harmonic Cut-Off Mode

rS MS

R BH M S

n Ω _2 R bh , M S 5 n Ω _2 R bh , 10 .M S n Ω _2 R bh , 10

10 . MS n Ω _4 M S

R bh Radial Displacement

Schwarzschild-Black-Hole (1 Solar Mass) Schwarzschild-Black-Hole (10^5 Solar Masses) Schwarzschild-Black-Hole (10^10 Solar Masses)

Figure 4.18 (log vs. log), Note: “nΩ_2(RBH,mxmh)” has been omitted due to plotting limitations. 6.6.2 “ωΩ” 6.6.2.1 Synopsis This section derives the “ωΩ” profile (i.e. as “r → RBH”) of any SBH. It is numerically demonstrated (explicitly in terms of “MBH”) and graphically illustrated (in terms of radial displacement) that the profile remains consistent with conclusions defined in [5] such that: i. “ωΩ_3” decreases as “r → RBH” [see: Fig. (4.19)]. ii. “ωΩ_4” decreases with rising “MBH” [see: Eq. (4.165)]. iii. “ωΩ_5” increases with rising “MBH” [see: Eq. (4.163, 4.165)]. 6.6.2.2 Assumptions Utilising key features defined in preceding sections, we shall assume that, for an observer within the event horizon of a SBH: i. The singularity of mass-density “ρm(rS,MBH)” at the centre of a SBH exists at the physical limit such that: “ρm(rS,MBH) = ρm(λxλh,mxmh)”. ii. “KPV” is constant [i.e. “∆KPV(∆r) = 0”] from the singularity radius “rS” to the event horizon “RBH” [i.e. “KPV(rS≤r≤RBH,MBH) = 1”]. This feature is beyond the determination of an observer without. iii. “ωh” is not a physical limit. 158

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6.6.2.3 Construct The “ωΩ” profile for SBH’s (i.e. as “r → RBH”) may be determined trivially in a computational environment utilising “ωΩ_3”. However, for subsequent use within this text, it is more convenient to define a new form explicitly in terms of “MBH” as follows, Let “ωΩ” at the periphery of a SBH singularity be given by “ωΩ_5” according to, ω Ω_5 M BH

ω Ω_3 r S M BH , M BH

(4.161)

Let the singularity to event horizon cut-off frequency ratio be given by “ωBH” according to, ω BH M BH

ω Ω_5 M BH ω Ω_4 M BH

(4.162)

6.6.2.4 Sample calculations Evaluating Eq. (4.161, 4.162) utilising arbitrary values produces the following results explicitly in terms of “MBH”, ω Ω_5 m x.m h

. 18 1.87219710

ω Ω_5 M S

. 19 4.55727410

=

5 ω Ω_5 10 .M S

. 19 6.9805610 . 20 1.06924110

10 ω Ω_5 10 .M S

ω BH m x.m h

(4.163)

1

ω BH M S 5 ω BH 10 .M S

( YHz)

=

. 13 7.30358710 . 15 5.19263810 . 17 3.69181510

10 ω BH 10 .M S

(4.164)

Hence, the proportional relationship between “ωh” and “ωΩ_4,5” may be trivially approximated as follows, ω Ω_5 m x.m h

ω Ω_4 m x.m h

ω Ω_5 M S 1 . 5 ω h ω Ω_5 10 .M S

5 ω Ω_4 10 .M S

10 ω Ω_5 10 .M S

10 ω Ω_4 10 .M S

0.253004

ω Ω_4 M S =

0.253004

. 6.158585 8.43227510

14

. 9.433354 1.81667910

15

14.44945

0

(4.165)

The preceding results may be indicative of a natural physical frequency boundary based upon an observational mass limit of SMBH’s. If we conjecture that “ωBH” has harmonic foundations, in accordance with the broader EGM construct for the harmonic representation of fundamental particles, then a set of simultaneous equations may be formulated such that a precise observational mass limit for SMBH’s may be predicted. For example, if we assume that “ωBH = 15” (i.e. an integer value), a precise prediction for the SMBH limit may be calculated. If this result matches the observational limit, then a natural physical frequency limit is implied such that a harmonic relationship exists between “rS” and “RBH”. Of course, this is pure conjecture and no emphatic conclusion may be inferred. 159

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6.6.2.5 Sample plots (log vs. log) The decrease in “ωΩ” (as a function of radial displacement “Rbh” and mass “MBH”) over the range “rS(MS) ≤ Rbh ≤ RBH(MS)” may be graphically illustrated according to, Harmonic Cut-Off Freq. vs Radial Disp.

Harmonic Cut-Off Frequency

rS MS

R BH M S

ω Ω _3 R bh , m x .m h ω Ω _3 R bh , M S 5 ω Ω _3 R bh , 10 .M S 10 ω Ω _3 R bh , 10 .M S

R bh Radial Displacement

Schwarzschild-Planck-Black-Hole Schwarzschild-Black-Hole (1 Solar Mass) Schwarzschild-Black-Hole (10^5 Solar Masses) Schwarzschild-Black-Hole (10^10 Solar Masses)

Figure 4.19 (log vs. log), Note: Eq. (4.140, 4.163) indicate that “ωΩ(rS,mxmh) = ωΩ(RBH,mxmh)”. However, the preceding graph illustrates that, over the radial displacement range specified, “ωΩ” is not constant for a SPBH. This is due to “Rbh > [rS(mxmh) = RBH(mxmh) = λxλh]” (i.e. “ωΩ” decreases beyond the event horizon). 6.6.3 “ωPV(1,r,MBH)” 6.6.3.1 Synopsis This section derives the value of “ωPV(1,r,MBH)” at “rS” and “RBH” of any SBH (i.e. “ωΩ_6,7”). It is numerically demonstrated (explicitly in terms of “MBH”) that results remain consistent with conclusions defined in [5] such that: i. “ωPV(1,r,MBH)” decreases as “r → RBH” [see: Fig. (4.20)]. ii. “ωPV[1,r(MBH),MBH]” decreases with rising “MBH” [see: Eq. (4.166 - 4.169)]. 6.6.3.2 Assumptions Refer to preceding sections.

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6.6.3.3 Construct The value of “ωPV(1,r,MBH)” at “rS” and “RBH” for SBH’s may be determined trivially in a computational environment utilising “ωΩ_4,5” and “nΩ_4,5”. However, for subsequent use within this text, it is more convenient to define new forms explicitly in terms of “MBH” as follows, Note: the following symbols for the fundamental harmonic frequencies of a SBH have been adopted to emphasise that “ωPV(1,r,M) = ωΩ(r,M)” for a SPBH (i.e. when “r = λxλh”, “M = mxmh”). Let “ωPV[1,rS(MBH),MBH]” at the periphery of a SBH singularity be given by “ωΩ_6” according to, ω Ω_6 M BH

ω Ω_5 M BH n Ω_4 M BH

(4.166)

Let “ωPV[1,RBH(MBH),MBH]” at the event horizon of a SBH be given by “ωΩ_7” according to, ω Ω_7 M BH

ω Ω_4 M BH n Ω_5 M BH

(4.167)

Let the singularity to event horizon fundamental frequency ratio be given by “ωPV_1” according to, ω Ω_6 M BH

ω PV_1 M BH

ω Ω_7 M BH

(4.168)

6.6.3.4 Sample calculations Evaluating Eq. (4.166 - 4.168) utilising arbitrary values produces the following approximated results explicitly in terms of “MBH”, ω Ω_6 m x.m h

ω Ω_7 m x.m h

ω Ω_6 M S

ω Ω_7 M S

5 ω Ω_6 10 .M S

5 ω Ω_7 10 .M S

10 ω Ω_6 10 .M S

10 ω Ω_7 10 .M S

. 42 1.87219710 . 42 1.87219710 =

. 38 6.93112610 . 4 1.29804810 . 37 3.61189510

. 37 6.93112610 . 1.00503110

ω PV_1 m x.m h

( Hz)

0.693113 6

(4.169)

1

ω PV_1 M S 5 ω PV_1 10 .M S

=

10 ω PV_1 10 .M S

. 33 1.8727810 . 37 5.21112310 . 42 1.45002610

(4.170)

The preceding results indicate that the PV spectral bandwidth expands as the radial displacement from the singularity increases (i.e. “r → ∞”). 6.6.3.5 Sample plots (log vs. log) The decrease in “ωPV(1,r,MBH)” (as a function of radial displacement “Rbh” and mass “MBH”)37 over the range “rS(MS) ≤ Rbh ≤ RBH(MS)” may be graphically illustrated according to,

37

Shown in the proceeding graph as the ratio “ωΩ_3(Rbh,MBH) / nΩ_2(Rbh,MBH)”. 161

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Fundamental Freq. vs Radial Disp. rS MS

R BH M S

ω Ω _3 R bh , m x .m h n Ω _2 R bh , m x .m h

Fundamental Frequency

ω Ω _3 R bh , M S n Ω _2 R bh , M S 5 ω Ω _3 R bh , 10 .M S 5 n Ω _2 R bh , 10 .M S 10 ω Ω _3 R bh , 10 .M S n Ω _2 R bh , 10

10 . MS

R bh Radial Displacement

Schwarzschild-Planck-Black-Hole Schwarzschild-Black-Hole (1 Solar Mass) Schwarzschild-Black-Hole (10^5 Solar Masses) Schwarzschild-Black-Hole (10^10 Solar Masses)

Figure 4.20, 6.6.3.6 Honourable mention It should not escape attention that the dimensionless ratio shown below approaches an integer value. However, no specific conclusion may be inferred from the following result, 1

.

ωh

10 ω PV_1 10 .M S ( Hz)

= 5.103269

(4.171)

6.6.4 Concluding remarks The key determinations are: i. “nΩ_2” increases as “r → RBH”. ii. “nΩ_4,5” increases with rising “MBH”. iv. “ωΩ_3” decreases as “r → RBH”. v. “ωΩ_4” decreases with rising “MBH”. vi. “ωΩ_5” increases with rising “MBH”. vii. “ωPV(1,r,MBH)” decreases as “r → RBH”. viii. “ωPV[1,r(MBH),MBH]” decreases with rising “MBH”. ix. “ωPV[1,r(Rbh),MBH]” increases with rising “MBH” [Rbh ≠ f(MBH)]. x. The PV spectral bandwidth expands as the radial displacement from the singularity increases (i.e. “r → ∞”). xi. It is conjectured that a natural physical frequency limit may exist such that it influences the SMBH observational limit.

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6.7

Derivation of the minimum gravitational lifetime of matter “TL”

6.7.1 Synopsis 6.7.1.1 Fundamentals EGM assumes that the spectral energy of the polarized ZPF surrounding an object (i.e. the PV spectrum / gravitational field) is equal to the mass-energy of the object itself, which may be characterised by a population of coherent conjugate wavefunction Photon pairs at “ωΩ”. The massenergy equivalence relationship is given by Einstein’s famous equation, E( M )

M .c

2

(4.172)

To derive the minimum gravitational lifetime of matter “TL”, we require a reference particle (i.e. a starting point) for the derivation process. For simplicity and clarity, we shall utilise the SPBH as our reference particle from which to build a construct. The primary reason for this selection is because “rS” coincides with “RBH” which negates any potential “singularity or event horizon” arguments the investigator (i.e. us / the reader) might have. The propagation energy of a single Photon is given by “Eγ(ω)”, E γ( ω )

h .ω

(4.173)

Consequently, the energy of a coupled Photon pair (i.e. a Graviton as defined by EGM) should equal “2Eγ(ω)”. A simple mathematical proof of this may be demonstrated as follows: let the relationship between the propagation energy of a Graviton (i.e. a conjugate Photon pair) be, E g ( ω ) E x.E γ ( ω )

(4.174)

where, “Eg(ω)” denotes the Propagation energy of a Graviton and “Ex” represents the proportional relationship to the propagation energy of a Photon. Moreover, let the population of Gravitons (i.e. a population of coherent conjugate Photon pairs) being radiated per period be given by “TΩ_4” according to, T Ω_4 M BH

1 ω Ω_4 M BH

(4.175)

Hence, the average number of Photons radiated by a SBH, each with propagation energy “Eγ(ω)”, is given by “nγ” according to, n γ ω , M BH

E M BH E γ( ω )

(4.176)

Subsequently, the number of Gravitons is given by “ng” according to, n g ω , M BH

E M BH E g( ω )

(4.177)

However, recognising that a single Graviton is a conjugate Photon pair (i.e. by the EGM definition given) according to, n g ω , M BH

1. n γ ω , M BH 2

(4.178)

It follows that, 1. n γ ω , M BH 2

163

E M BH E g( ω )

(4.179) www.deltagroupengineering.com

Substituting “Eg(ω) = ExEγ(ω)” yields, 1. n γ ω , M BH 2

Hence, Ex

E M BH E x.E γ ( ω )

(4.180)

2 .E M BH 2 n γ ω , M BH .E γ ( ω )

E g( ω )

(4.181)

2 .E γ ( ω )

(4.182)

6.7.1.2 Assumptions To apply the preceding equations, we are required to specifically assign a mechanism facilitating the existence of gravitational fields. EGM considers all matter to be wavefunction radiators of populations of coherent conjugate Photon pairs such that each pair constitutes a Graviton38. Hence, to evaluate the preceding equations we shall assume the following key mathematical modelling criteria for an object at rest, i. Gravitons reside within, until they are spontaneously ejected and the supply has been exhausted. Hence, existing gravitational (PV) field strengths are sustained in this manner. ii. Only whole Gravitons are ejected. iii. The physical reality of the mathematical modelling processes utilised, or theories of Graviton absorption by the object itself from external sources - at this stage, are irrelevant. The true measure will be the complete and accurate derivation of the Hubble constant and Cosmic Microwave Background Radiation (CMBR) temperature later in this text. 6.7.1.3 Sample calculations Considering a SPBH (i.e. “MBH = mxmh”) yields a value of mass-energy equivalence and Photon-Graviton “emission / absorption” period as follows, E m x.m h = 6.616163 ( GJ)

(4.183)

T Ω_4 m x.m h = 5.341319 10

43 .

s

(4.184)

The energy radiated per “TΩ_4” (i.e. per Photon or Graviton) is given by “Eγ(ω)” and “Eg(ω)” respectively according to, E γ ω Ω_4 m x.m h

= 1.240531 ( GJ)

E g ω Ω_4 m x.m h

= 2.481061 ( GJ)

(4.185) (4.186)

Hence, the average number of Photons and Gravitons radiated per “TΩ_4” is evaluated to be, n γ ω Ω_4 m x.m h , m x.m h n g ω Ω_4 m x.m h , m x.m h

=

5.333333 2.666667

(4.187)

38

In a manner of speaking, the typical PV spectrum contains many different Graviton massenergies dependent upon one’s definition of how many Photons constitute “a specific kind / type” of Graviton. However, the EGM definition is applied throughout this construct. 164

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6.7.2 Construct 6.7.2.1 Reconciliation 6.7.2.1.1 Dilemma •

How does one reconcile Eq. (4.187) against the “2nd” assumption?

6.7.2.1.2 Resolution 6.7.2.1.2.1 Uncertainty The preceding results suggest that the Graviton burst per period varies slightly around the average value. Subsequently, over “3” periods, the total number of Gravitons radiated equals “8”. This may be explained a numbers of ways. For example, in the case of a SPBH (conserving coherent population characteristics), Graviton emission profiles could (?) appear as follows, i. “3” Gravitons in the “1st” period + “3” Gravitons in the “2nd” period + “2” Gravitons in the “3rd” period = 3 + 3 + 2 = 8, with an average being = “8/3” = “2.6667” = “ng(ωΩ_4(mxmh),mxmh)”. ii. “2” Gravitons in the “1st” period + “2” Gravitons in the “2nd” period + “4” Gravitons in the “3rd” period = 2 + 2 + 4 = 8 etc. iii. “1” Graviton in the “1st” period + “1” Graviton in the “2nd” period + “6” Gravitons in the “3rd” period = 1 + 1 + 6 = 8 etc. iv. “0” Gravitons in the “1st” period + “0” Gravitons in the “2nd” period + “8” Gravitons in the “3rd” period = 0 + 0 + 8 = 8 etc. Note: the potential emission profiles above re-enforce the uncertainty principle. 6.7.2.1.2.2 Quasi-Uncertainty An alternative possibility is that the sum of the Gravitons radiated over any “3” consecutive periods equals “8” (commencing the count from an arbitrary period position in the emission train), in which case an emission profile could (?) be “3 + 3 + 2 + 3 + 3 + 2 + 3 + 3 + 2 + ….”. Hence, moving from left to right across the emission train yields, i. 3 + 3 + 2 = 8 ii. 3 + 2 + 3 = 8 iii. 2 + 3 + 3 = 8 etc. Therefore, our initial assumption regarding coherent integer Graviton population ejections has been reconciled against “ng” by the existence of a number of different possible emission trains and temporal profiles. 6.7.2.2 “TL” 6.7.2.2.1 Fundamentals The generalised form has been avoided thus far because it did not adequately expose the dilemma surrounding “emission / absorption” trains. For example, if one applies a value of “MBH” other than “mxmh”, one obtains relatively large values of Graviton “emission / absorption” numbers. Hence, the presence of uncertainty principles in the EGM construct may not have been readily apparent. 165

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However, because the possibility of “emission / absorption” trains has been introduced, we may now determine the minimum gravitational lifetime of starving matter in generalised form by considering the mass-energies of a Photon “mγγ” and Graviton “mgg” at rest by the EGM method defined in [10], m γγ m gg

3.195095

=

10

6.39019

45 .

eV

(4.188)

Subsequently, the population of Gravitons and Photons contained within starving matter (“ngg” and “nγγ” respectively) may be stated as, E( M )

n gg ( M )

m gg

(4.189)

2 .n gg ( M )

n γγ( M )

(4.190)

Hence, the minimum gravitational lifetime of starving matter is given by “TL” according to, T L( r , M )

n gg ( M ) .T Ω_3( r , M ) n g ω Ω_3( r , M ) , M

(4.191)

where, 1

T Ω_3( r , M )

ω Ω_3( r , M )

(4.192)

6.7.2.2.2 Sample calculations Evaluating the preceding equations for arbitrary values of SBH mass at “rS” yields, T L r S λ x.λ h , m x.m h

. 13 4.10173110

T L r S M S ,M S =

5 5 T L r S 10 .M S , 10 .M S

. 13 4.10173110

9 10 .yr

. 13 4.10173110 . 13 4.10173110

10 10 T L r S 10 .M S , 10 .M S

(4.193)

Evaluation at “RBH” [recalling that: “rS(λxλh) = RBH(λxλh) = λxλh”] produces, T L R BH λ x.λ h , m x.m h

. 13 4.10173110

T L R BH M S , M S 5 5 T L R BH 10 .M S , 10 .M S 10 10 T L R BH 10 .M S , 10 .M S

=

. 13 4.10173110 . 4.10173110

13

9 10 .yr

. 13 4.10173110

(4.194)

Notably, evaluation at the charge radius (i.e. ZPF equilibrium radius) of several fundamental particles produces, T L r uq , m uq T L r ε, m e T L r π, m p T L r ν,mn

. 13 4.10173110 =

. 13 4.10173110 . 13 4.10173110

9 10 .yr

. 13 4.10173110

(4.195)

Therefore, these results indicate that all starving matter has identical gravitational lifetimes. 166

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6.7.2.2.3 Simplifications By inspection, “TL” may be simplified according to, h

TL

m γγ

(4.196)

Evaluating yields, 1 m γγ

h.

. 13 4.10173110

=

2

. 4.10173110

13

m gg

9 10 .yr

(4.197)

6.7.3 Concluding remarks Utilising the Hubble constant “H0” defined by the Particle Data Group (PDG) in [22], we may determine the minimum gravitational lifetime of the Universe expressed as a scalar multiple of “H0” according to, H0

71.

km s .Mpc

(4.198)

Hence, . T L.H 0 = 2.97830810

12

(4.199)

Therefore, the minimum gravitational lifetime of starving matter is approximately “3” Trillion times the Hubble age of the Universe. NOTES

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6.8

Derivation of the average emission frequency per Graviton “ωg”

6.8.1 Synopsis BH’s are the most extreme gravitational objects in nature and at least “1” Graviton should be emitted within a few periods at “ωΩ”. However, all but a SPBH are described by a spectrum of EGM frequencies. Subsequently, at “ωΩ” for example, not every cycle emits a Graviton with certainty. Hence, the average emission frequency per Graviton shall be determined. 6.8.2 Assumptions i. The “uncertainty” resolution in the preceding section is an adequate representation for mathematical modelling purposes and is appropriate for the objective defined above. ii. The average number of Gravitons radiated by matter at “ωPV” is given by “ng(ωPV,M)” [i.e. the generalised form of Eq.(4.177)]. iii. The maximum number of cycles at “ωPV” it may take to emit the appropriate coherent Graviton population is given by “ng(ωPV,M)-1”. 6.8.3 Construct The average emission period per Graviton at “ωPV” is given by “Tg” according to, T g n PV, r , M

T PV n PV, r , M n g ω PV n PV, r , M , M

(4.200)

Hence, the average emission frequency per Graviton is given by “ωg” according to, ω g n PV, r , M

n g ω PV n PV, r , M , M T PV n PV, r , M

(4.201)

Substituting “TPV = 1 / ωPV” yields, n g ω PV n PV, r , M , M T PV n PV, r , M

ω PV n PV, r , M .n g ω PV n PV, r , M , M

(4.202)

Substituting “ng = ½nγ” yields, 1 ω PV n PV, r , M . .n γ ω PV n PV, r , M , M 2

ω PV n PV, r , M .n g ω PV n PV, r , M , M

(4.203)

Subsequently, ω g n PV, r , M

1 E( M ) ω PV n PV, r , M . . 2 E γ ω PV n PV, r , M

(4.204)

Substituting “Eγ = hωPV” yields, 1 E( M ) ω PV n PV, r , M . . 2 E γ ω PV n PV, r , M

1 E( M ) ω PV n PV, r , M . . 2 h .ω PV n PV, r , M

(4.205)

Simplifying produces, 1 E( M ) ω PV n PV, r , M . . . 2 h ω PV n PV, r , M

E( M ) 2 .h

(4.206)

Therefore, ω g( M )

M .c 2 .h

168

2

(4.207) www.deltagroupengineering.com

Hence, it follows that the total population of Gravitons residing within matter is equal to the average emission frequency per Graviton39 multiplied by the minimum gravitational lifetime of starving matter, given by “ngg(M)” according to, n gg ( M ) T L.ω g ( M )

(4.208)

Note: “ngg(M) = E(M) / mgg = Mc2 / mgg”. 6.8.4 Sample calculations Evaluating the preceding equations for various arbitrary values of SBH mass yields, ω g m x.m h

. 18 4.99252510

ωg MS 5 ω g 10 .M S

=

. 56 1.34855310 . 61 1.34855310

( YHz)

. 66 1.34855310

10 ω g 10 .M S

n gg m x.m h

(4.209)

. 72 6.46222510

n gg M S =

5 n gg 10 .M S 10 n gg 10 .M S

. 110 1.7455410 . 115 1.7455410 . 120 1.7455410

E m x.m h

(4.210)

. 72 6.46222510

E MS 1 . 5 m gg E 10 .M S

=

10 E 10 .M S

. 110 1.7455410 . 115 1.7455410 . 120 1.7455410

(4.211)

6.8.5 Concluding remarks •

39

On average, Graviton emission occurs once every “ωg-1” seconds.

“ωg” is non-physical: it is a mathematical contrivance. 169

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6.9

Why can't we observe BH’s?

6.9.1 Synopsis The “invisibility” of BH’s has been historically attributed to the curvature of the space-time manifold induced by their mass. However, the PV model of gravity attributes this behaviour to the value of “KPV” of the space-time manifold induced by the radial gradient of the energy density of the gravitational field at the event horizon. EGM advances to the next logical step by combining the static PV gravitational field with the harmonic nature of the ZPF resulting in a spectrum of frequencies describing the gravitational field such that the spectral bandwidth converges to a single mode in the case of a SPBH. Moreover, it demonstrates that the PV spectrum may be characterised by a single wavefunction due to the magnitude of “SωΩ” where, “>> 99.99(%)” of gravitational energy exists at “ωΩ” (i.e. all other frequencies may be usefully neglected). Therefore, EGM implies that, for a SBH, a wavefunction radiating from “RBH” with a frequency of “ωΩ”, should degrade into the Visible Light (VL) frequency range if a hypothetical “EGM wavefunction detector” was sufficiently distant from it. However, three principle reasons exist as to why SBH’s will never be detectable. The “1st” reason is discussed in this section, i.e. the Universe is insufficiently large to permit an “EGM wavefunction detector” to detect SBH’s, even if the incoming conjugate pair EGM signal could be appropriately isolated, amplified and filtered “somehow(?)”40. Even if the Universe is much older and larger than current estimates, we show that it remains too small for a device to detect SBH’s in the VL range. Subsequently, it is implied that the EGM wavefunction of a SBH will enter the optical wavelength range in the far distant future – long after our species has probably disappeared from existence. As an alternative to VL detection and confirmation of the EGM construct, we explore the possibility of detecting BH’s within the X-Ray range. It is shown that, whilst VL prospects are doubtful, the X-Ray range may be a potential theoretical direction for future community research, if (and only if), the technical problems of signal isolation, amplification and filtration, emphasised in “Are conventional radio telescopes, practical tools for gravitational astronomy?” and “Gravitational Cosmology” respectively, are overcome. To facilitate this derivation, we must firstly specify “a” size and age of the Universe for subsequent use. Since these values are not precisely known and are “hotly debated” within the scientific community, we shall approximate them directly from the Hubble constant. These are termed the “Hubble” size and age of the Universe (i.e. “r0” and “H0-1” respectively). 6.9.2 Assumptions i. All physical BH’s, for the purpose of this derivation, are usefully represented by approximation to SBH’s (note: SPBH’s are probably non-physical). ii. The present size and age of the observable Universe is adequately approximated utilising “H0” according to “r0 = c / H0” and “H0-1” respectively. iii. “r0 >> RBH” such that “KPV(r0,M) → 1” and “ωΩ(r0,M) → ωΩ_3(r0,M)”. iv. It will “somehow(?)” be technologically possible in the distant future to appropriately isolate, amplify and filter the incoming conjugate wavefunction paired EGM signal to verify the derivation.

40

The “2nd” and “3rd” reasons are discussed in: “Are conventional radio telescopes, practical tools for gravitational astronomy?” and “Gravitational Cosmology” respectively. 170

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6.9.3 Construct The VL spectrum for human beings is approximately bounded by wavelengths in the “nanometre” (nm) range according to “400≤λVL(nm)≤750”. The X-Ray spectrum has an approximate wavelength range of “0.3≤λX-RAY(nm)≤300”. Both these wavelength ranges can be converted to the frequency domain according to the classical relationship “ω = c / λ”. Utilising the expression for “ωΩ_3”, such that “r = rω” represents the distance from the centre of mass of a celestial object to the Earth, we may determine the “visibility” of SBH’s in the VL and X-Ray ranges by transposing for “rω” as follows, 5

St G.

r ω ω Ω_3 , M

M

2

ω Ω_3

9

(4.212)

Therefore, “rω” denotes the distance from the Earth that a celestial object (i.e. an EGM wavefunction radiation source) must be located such that its EGM wavefunction frequency decays to the VL or X-Ray ranges. 6.9.4 Sample calculations 6.9.4.1 SBH’s The value of “ωΩ_3” at the edge of the presently observable Universe for various arbitrary SBH mass configurations (i.e. expressed as solar multiples) is approximated by, ω Ω_3 r 0 , M S 5.

ω Ω_3 r 0 , 10 M S 10 ω Ω_3 r 0 , 10 .M S

0.163994 = 2.118067 ( EHz) 27.355887

(4.213)

where, “EHz = 1018(Hz)”. However, computation of the EGM Flux Intensity at an Earth based detector implies current impossibility of technical achievement according to, C Ω_J1 r 0 , M S 5.

C Ω_J1 r 0 , 10 M S 10 C Ω_J1 r 0 , 10 .M S

. 1.48429110

5

= 8.89809310 .

3

10

20 .

yJy

5.334267

(4.214)

-24

where, “yJy” denotes “yocto-Jansky” [i.e. “10 (Jansky)”]. Ignoring technical feasibility: for SBH’s at the edge of “an” observable Universe, the following results demonstrate (i.e. being significantly greater than unity) that “a” Universe is required to be substantially larger than “r0” for the value of “ωΩ_3” to be within the VL range according to, r ω ω VL( 400 ( nm ) ) , M S 1 . r ω ( 400 ( nm ) ) , 105 .M ω VL S

r0

10 r ω ω VL( 400 ( nm ) ) , 10 .M S

r ω ω VL( 750 ( nm ) ) , M S

. 4 5.05271110 . 4 1.62975410

5 r ω ω VL( 750 ( nm) ) , 10 .M S

= 1.62975410 . 6 5.05271110 . 6

10 r ω ω VL( 750 ( nm ) ) , 10 .M S

. 8 5.05271110 . 8 1.62975410

(4.215)

Repeating the procedure in the X-Ray frequency range (i.e. “30(PHz)≤ωX-RAY≤30(EHz)”) produces favourable detection results according to,

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r ω 30.( PHz) , M S 5.

r ω 30.( EHz) , M S

r ω 30.( PHz) , 10 M S

r ω 30.( EHz) , 10 M S

10 r ω 30.( PHz) , 10 .M S

10 r ω 30.( EHz) , 10 .M S

. 5 2.93002110

5.

= 2.93002110 .

7

1.166462 116.646228

. 9 1.16646210 . 4 2.93002110

6 10 .Lyr

(4.216)

15

where, “PHz = 10 (Hz)” and “Lyr = light-year”. The preceding results indicate that the detection of SBH’s in the X-Ray range “may(?)” be possible within observational distances of approximately “1.2 → 117” million light years from Earth. However, computation of the EGM Flux Intensity for an Earth based detector implies current impossibility of technical achievement according to, C Ω_J1 r ω 30.( PHz) , M S , M S

2.164916

C Ω_J1 r ω 30.( PHz) , 10 M S , 10 M S 5.

5.

. = 2.16491610

3

. 2.16491610

6

10 10 C Ω_J1 r ω 30.( PHz) , 10 .M S , 10 .M S

C Ω_J1 r ω 30.( EHz) , M S , M S 5.

10

29 .

yJy

(4.217)

8.618686

5.

C Ω_J1 r ω 30.( EHz) , 10 M S , 10 M S

. = 8.61868610

3

. 8.61868610

6

10 10 C Ω_J1 r ω 30.( EHz) , 10 .M S , 10 .M S

10

14 .

yJy

(4.218)

6.9.4.2 SPBH’s The existence of SPBH’s is considered to be a theoretical possibility predicted by the dimensional manipulation of Planck properties. Physicality of such phenomena cannot be completely discounted due to a lack of observational evidence. Hence, EGM predicts a VL range according to, r ω ω VL( 400 ( nm ) ) , m x.m h

=

r ω ω VL( 750 ( nm ) ) , m x.m h

0.239057 0.741144

( Lyr)

(4.219)

However, computation of the EGM Flux Intensity for an Earth based detector implies current impossibility of technical achievement according to, C Ω_J1 r ω ω VL( 400 ( nm ) ) , m x.m h , m x.m h C Ω_J1 r ω ω VL( 750 ( nm ) ) , m x.m h , m x.m h

=

28.979765 1.102778

10

16 .

yJy

(4.220)

A SPBH existing in the X-Ray frequency range would be “visible” within our solar system according to, r ω 30.( PHz) , m x.m h r ω 30.( EHz) , m x.m h

=

. 6 2.95234410

3 10 .km

11.753495

(4.221)

However, computation of the EGM Flux Intensity for an Earth based detector implies current impossibility of technical achievement according to, 7 C Ω_J1 r ω 30.( PHz) , m x.m h , m x.m h = 6.228302 10 .yJy

(4.222)

C Ω_J1 r ω 30.( EHz) , m x.m h , m x.m h = 2.479532 ( fJy )

(4.223)

where, “fJy” denotes “femto-Jansky” [i.e. “10-15(Jansky)”].

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6.9.5 Concluding remarks Assuming it is technically possible to appropriately isolate, amplify and filter an incoming EGM wavefunction signal, the preceding construct implies the following, i. Eventually, as the observable Universe continues to expand to thousands of times its present size, the radiant EGM wavefunctions of BH’s at the edge of the Universe will enter the VL range. This does not mean that they will become “visible” to the naked eye. This only means that their EGM wavefunction will enter the “VL” part of the EM spectrum. ii. It “may(?)” be theoretically possible to detect BH’s utilising the X-Ray range within observational distances of approximately “1.2 → 117” million light years from Earth. However, significant theoretical and technical challenges would be required to be overcome, emphasised in “Are conventional radio telescopes, practical tools for gravitational astronomy?” and “Gravitational Cosmology” respectively. NOTES

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NOTES

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7 Fundamental Cosmology

Abstract The primordial and present values of the Hubble constant are derived (“Hα” and “HU” respectively), leading to the determination of the Cosmic Microwave Background Radiation (CMBR) temperature “TU”. This facilitates the determination of the impact of “Dark Matter / Energy” on “HU” and “TU” such that a generalised expression for “TU” in terms of “HU” is formulated. An experimentally implicit derivation of the ZPF energy density threshold “UZPF” is also presented.

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7.1

Derivation of the primordial and present Hubble constants “Hα, HU”

7.1.1 Synopsis The derivation of the primordial “Hα” and present “HU” Hubble constants by the EGM method is possible by postulating an initial size, shape and mass of the Universe, momentarily prior to the “Big-Bang”: we shall term this state the “Primordial Universe”. Once a description of the “Primordial Universe” has been mathematically articulated in generalised terms, it may be compared to a dimensionally equivalent object in accordance with BPT and similarity principles. The objective herein is to derive a system of generalised equations, withholding numerical evaluation. In a subsequent section, the generalised expressions will be numerically evaluated demonstrating a calculation of “HU” in favourable agreement with expert opinion and physical measurement of “H0”. Moreover, a value of “Hα” is presented demonstrating that the EGM method suggests exciting new avenues for Cosmological research. 7.1.2 Assumptions i. Dynamic, kinematic and geometric similarity: The “Primordial Universe” was analogous to a spherical particle on the Planck scale with radius “r1” and homogeneous mass distribution “M1”, described by a single wavefunction whereas the presently observable Universe is described by a spectrum of wavefunctions. The maximum EGM Flux Intensity measured by an observer at the edge of the “Primordial Universe” is given by “CΩ_J1(r1,M1)”. Matter radiates Gravitons41 at a spectrum of frequencies such that the Cosmological majority of it exists in Photonic form, resulting in an approximately homogeneous massenergy distribution throughout the Universe whereby any Galactic formation is dynamically, kinematically and geometrically equivalent to a spherical particle of homogeneous mass distribution and may be represented as a Planck scale object to be utilised as a Galactic Reference Particle (GRP). The associated EGM Flux Intensity of the GRP is given by “CΩ_J1(r2,M2)” where, “r2” denotes the mean “H0” measurement distance42 to the Galactic centre and “M2” represents total Galactic mass43. The definition of “r2” comes from the scientific requirement to compare calculation or prediction to measurement. Subsequently, one should also utilise parameters within the same frame of reference as the measurement, against which the construct is being tested. It is not known by physical validation that “H0” is measured as being the same from all locations in the Universe. It is believed to be the case by contemporary theory; however it is not factually known to be true. To verify it physically, one would be required to perform the “H0” measurement from a significantly different location in space. Thus, to minimise potential modelling errors, we shall confine “r2” to the same frame of reference44 as the measurement of “H0”. ii. The ratio of the presently observable Cosmological size “rf”, to the initial size “ri” of the “Primordial Universe” instantaneously prior to the “Big-Bang”, is proportional to the corresponding EGM Flux Intensity {i.e. “(rf / ri) ∝ [CΩ_J1(rf) / CΩ_J1(ri)]”}. iii. A relationship exists between the physical proportions of a particle at the Planck scale limit governed by “λx” and “nΩ_2” such that it may be stated as “λx = λy(r1,r2,M1,M2)”. 41

Coherent populations of conjugate Photon pairs for a minimum period of “TL”. i.e. the distance relative to the Galactic centre from where a physical measurement of “H0” is performed. 43 Visible + dark. 44 Our solar system. 42

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Upon consideration of the preceding assumptions, it follows that Eq. (3.230ii) represents the basic form of mass-energy distribution throughout the Cosmos. 7.1.3 Construct 7.1.3.1 “AU, RU, HU” The EGM harmonic representation of fundamental particles may be applied to facilitate the derivation of “HU” by considering the initial “i” and present “f” mass and observable size of the Universe. Hence, utilising Eq. (3.230ii) yields, Mi

2

.

Mf

rf

5

St ω

ri

9

(4.224)

where, “Mi = Mf” due to the conservation of mass. For simplicity, let the “rf” to “ri” ratio be defined according to, rf

KU

ri

(4.225)

“Stω9” in Eq. (3.230ii) represents the harmonic relationship between the values of “ωΩ” of two dimensionally similar particles. Hence, recognising that the value of “Stω” is presently unknown in a Cosmological context, and that the frequency and time domains are interchangeable, let “Stω9” equal the ratio of “TL” to the present “Hubble age” of the Universe “AU” according to, St ω

9

TL AU

(4.226)

Hence, 5

KU

TL AU

(4.227)

Considering the preceding assumptions and equations, one expects that a relationship should exist between “ri,f” and “CΩ_J1(r1,2,M1,2)”; however, their precise values are not yet known. Subsequently, we shall deduce a relationship to be tested against physical observation utilising the following logical statements and deductions, i. If the order of magnitude of “rf” is approximately known by physical measurement45 and “ri” approached the Planck scale limit, then “rf >> ri” such that “e(rf / ri) → ∞”. ii. Without empirical evidence, one’s expectation is that “CΩ_J1(r1,M1) >> CΩ_J1(r2,M2)”, such that “[CΩ_J1(r1,M1) / CΩ_J1(r2,M2)] → ∞”. iii. Hence, it follows that46: “e(rf / ri) → [CΩ_J1(r1,M1) / CΩ_J1(r2,M2)]” according to, rf ri

ln

C Ω_J1 r 1 , M 1 C Ω_J1 r 2 , M 2

(4.228)

It was demonstrated earlier that the appropriate proportions of a particle at the Planck scale limit satisfying the EGM construct are: “r1 = λxλh” and “M1 = mxmh = λxmh / 2”. Although the precise value of “λx” was calculated and shown to be small, we shall remove this constraint and advance the derivation in a more generalised manner.

45 46

i.e. approximately “< 15” billion light-years. “e” denotes the “exponential function”. 177

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Temporarily ignoring the previously computed value of “λx” facilitates the creation of a substantially more robust construct such that the generalisation may be tested against physical observation utilising the following logical statements and deductions, iv. Let “λx = λy(r1,r2,M1,M2)”. v. If “λy(r1,r2,M1,M2) → 0” then “e[1 / λy(r1,r2,M1,M2)] → ∞”. vi. Without empirical evidence to the contrary, one’s expectation is that “nΩ_2(r2,M2) >> nΩ_2(r1,M1)”, such that “[nΩ_2(r2,M2) / nΩ_2(r1,M1)] → ∞”. vii. By definition: “nΩ_2(r1,M1) = 1” at the Planck scale limit for the wavefunction of the particle to remain consistent with the EGM construct47. Subsequently, “nΩ_2(r2,M2) → ∞”. viii. Hence, it follows that “e[1 / λy(r1,r2,M1,M2)] → nΩ_2(r2,M2)” according to, 1

λ y r 2, M 2

ln n Ω_2 r 2 , M 2

(4.229)

Subsequently, “r1” and “M1” may be written in the following block form, λ y r 2 , M 2 .r 3

r1

λ y r 2, M 2 .M 3 2

M1

(4.230)

Thus, “KU” may be written in functional form according to,

K U r 2, r 3, M 2, M 3

ln

λ y r 2, M 2 .M C Ω_J1 λ y r 2 , M 2 .r 3 , 3 2 C Ω_J1 r 2 , M 2

(4.231)

Performing the appropriate substitutions, one obtains the reduced functional form as follows, 5

5

K U r 2, r 3, M 2, M 3

ln

1 2

9

7

.ln n Ω_2 r 2 , M 2

3.

M3

26

9

.

M2

r2 r3

9

(4.232)

Hence, the EGM age of the Universe “AU” is given by, A U r 2, r 3, M 2, M 3

TL K U r 2, r 3, M 2, M 3

5

(4.233)

Consequently, the EGM size of the Universe “RU” may be stated as follows, R U r 2, r 3 , M 2, M 3

c .A U r 2 , r 3 , M 2 , M 3

H U r 2, r 3, M 2, M 3

1

(4.234)

Therefore, for a “flat” Universe, A U r 2, r 3, M 2, M 3

(4.235)

7.1.3.2 “Hα” The energy density of the Universe changes with time and, by mathematical definition, so must the Hubble constant. Assuming the “Primordial Universe” was analogous to a particle at the Planck scale limit, it is possible to predict a value for the Hubble constant at the instant of the “BigBang” (i.e. the primordial Hubble constant “Hα”) by equating it to the mass-density “ρm”. 47

i.e. “rS(r1,M1) = RBH(r1,M1)”. 178

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Utilising the contemporary density relationship according to, 3 .H ρm 8 .π .G 2

(4.236)

such that: “H → Hα(r3,M3)” and “ρm → ρm(r3,M3)” yields, H α r 3, M 3

2.

2. . . π G ρ m r 3, M 3 3

(4.237)

7.1.3.3 “ρU” Utilising the contemporary density relationship, the EGM mass-density of the Universe “ρU” may be determined as follows, ρ U r 2, r 3, M 2, M 3

3 .H U r 2 , r 3 , M 2 , M 3 8 .π .G

2

(4.238)

7.1.3.4 “MU” Approximating the observable Universe to a spherical volume [i.e. “V(r) = 4πr3 / 3”], the total EGM mass of the Universe “MU” (i.e. visible + dark) when “r → RU(r2,M2,r3,M3)” is given by, M U r 2, r 3, M 2, M 3

V R U r 2 , r 3 , M 2 , M 3 .ρ U r 2 , r 3 , M 2 , M 3

(4.239)

7.1.4 Concluding remarks A robust generalised construct for “Hα”, “HU”, “ρU” and “MU” has been formulated which may be tested against physical observation. Non-refractive forms were utilised throughout this derivation (i.e. “CΩ_J1” and “nΩ_2”) because: i. “KPV(r,M)” evaporates when “[CΩ(r1,M1) / CΩ(r2,M2)] → [CΩ_J1(r1,M1) / CΩ_J1(r2,M2)]”. ii. “r2 >> 1” such that “KPV(r2,M2) → 1”. NOTES

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7.2

Derivation of the CMBR temperature “TU”

7.2.1 Synopsis The Cosmic Microwave Background Radiation (CMBR) temperature “TU” may be calculated utilising the EGM method by considering the total mass-energy of the Universe to be dynamically, kinematically and geometrically similar to a particle at the Planck scale limit, consistent with the formulation of “Hα” and “HU” in the preceding section. By generalising the result: “ng(ωΩ_4(mxmh),mxmh) → ng(ωΩ_3(r3,M3),M3)” {see Eq. (4.187)}, we may formulate a relationship between the primordial and Galactic reference average numbers of Gravitons radiated by similar particles. For a “Primordial Universe” particle model at the Planck scale limit, the relationship yields “TU” by the application of proportional similarity principles, wavefunction frequency degradation and the “Wien” Displacement Constant “KW”. The following quotation is taken verbatim from [http://hyperphysics.phy-astr.gsu.edu/hphys.html]. “When the temperature of a blackbody radiator increases, the overall radiated energy increases and the peak of the radiation curve moves to shorter wavelengths. When the maximum is evaluated from the Planck radiation formula, the product of the peak wavelength and the temperature is found to be a constant.

Figure 4.21, This relationship is called “Wien's Displacement Law” and is useful for determining the temperature of hot radiant objects such as stars, and indeed for a determination of the temperature of any radiant object whose temperature is far above that of its surroundings. It should be noted that the peak of the radiation curve in the Wien relationship is the peak only because the intensity is plotted as a function of wavelength. If frequency or some other variable is used on the horizontal axis, the peak will be at a different wavelength.” End of verbatim quotation. 7.2.2 Assumptions i. The primordial average number of Gravitons radiated per “TΩ_3” period, instantaneously after the “Big-Bang”, is given by “ng(ωΩ_3(r3,M3),M3)”. ii. The Galactic reference average number of Gravitons “KT” (also termed the “expansive scaling factor”), radiated per wavefunction period, may be defined as a proportion of the primordial average given by “KT(r2,M2,r3,M3) ∝ ng(ωΩ_3(r3,M3),M3)”. iii. Specific information about “KT’s” wavefunction period is irrelevant due to the assignment of proportional similarity characteristics between the primordial (i.e. “Primordial Universe”) and Galactic reference averages described above. 180

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7.2.3 Construct Generalising the result “ng(ωΩ_4(mxmh),mxmh) → ng(ωΩ_3(r3,M3),M3)” facilitates the creation of a substantially more robust construct such that it may be tested against physical observation utilising the following logical statements and deductions, i. If “ng(ωΩ_3(r3,M3),M3) → 0” then “e[KT(r2,M2,r3,M3) / ng(ωΩ_3(r3,M3),M3)] → ∞”. ii. Without empirical evidence to the contrary, one’s expectation is that “Hα(r3,M3) >> HU(r2,M2,r3,M3)”, such that “[Hα(r3,M3) / HU(r2,M2,r3,M3)] → ∞”. iii. Hence, it follows that “e[KT(r2,M2,r3,M3) / ng(ωΩ_3(r3,M3),M3)] → [Hα(r3,M3) / HU(r2,M2,r3,M3)]”, yielding the expansive scaling factor according to, K T r 2, r 3, M 2, M 3

n g ω Ω_3 r 3 , M 3 , M 3 .ln

H α r 3, M 3 H U r 2, r 3 , M 2, M 3

(4.240)

“Wien's Displacement Law” provides the frequency (or wavelength) at which “Planck’s Law” has maximum specific intensity. [46] Consequently, the hotter an object is, the shorter the wavelength at which it will emit most of its radiation and the frequency for maximal (i.e. peak) radiation power is found by dividing “KW” by the temperature. [24] If the present size of the Universe were held static (i.e. spatial expansion was miraculously halted) and its total mass-energy (i.e. visible + dark) were compressed48 such that it was dynamically, kinematically and geometrically analogous to a particle at the Planck scale limit such that “nΩ_2(r3,M3) = 1” (i.e. only one wavefunction describes the “Primordial Universe”), then a mass-less observer at the periphery of the presently observable Universe, given by “RU(r2,M2,r3,M3)”, would measure its EGM wavefunction frequency to be “ωΩ_3(r,M) → ωΩ_3(RU(r2,M2,r3,M3),M3)”. Recognising that “λΩ_3(RU(r2,M2,r3,M3),M3) = c / ωΩ_3(RU(r2,M2,r3,M3),M3)” yields the expansive independent average temperature of the observable Universe “TW” (also termed the “thermodynamic scaling factor”) according to, T W r 2, r 3, M 2 , M 3

KW λ Ω_3 R U r 2 , r 3 , M 2 , M 3 , M 3

(4.241)

Hence, applying “Wien's Displacement Law” for blackbody radiation, scaled by “KT for application to the EGM domain by preservation of dynamic, kinematic and geometric similarity, yields the CMBR temperature (i.e. the expansive dependent average) as follows, T U r 2, r 3, M 2, M 3

K T r 2 , r 3 , M 2 , M 3 .T W r 2 , r 3 , M 2 , M 3

(4.242) Therefore, i. “TW” denotes the Cosmological expansive independent average temperature because the expression does not contain “HU”. ii. “TU” denotes the Cosmological expansive dependent average temperature because the expression contains “HU”. 7.2.4 Concluding remarks It is clear from the preceding construct that the CMBR temperature is a function of the Hubble constant.

48

Mimicking the “Primordial Universe” and excluding space-time manifold expansion from consideration. 181

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7.3

Numerical solutions for “Hα, AU, RU, ρU, MU, HU” and “TU”

7.3.1 “r2, r3, M2, M3” Thus far, we have determined mathematical relationships for “Hα, AU, RU, ρU, MU, HU” and “TU”. However, to numerically evaluate these expressions, we require precise definitions of “r2, r3, M2” and “M3”. If the “Primordial Universe” was analogous to a particle at the Planck scale limit, then “RBH” must have coincided with “rS” (i.e. the “Primordial Universe” was analogous to a SPBH) because nothing (i.e. including a space-time manifold) existed beyond “RBH”. This assertion is reinforced by the contemporary scientific belief that the “Big-Bang” was not an explosion in the space-time manifold, but was an explosion of the space-time manifold. Moreover, it was previously shown, by the calculation of “rS”, that a Planck scale particle configuration of “r = λh” and “M = mh” is inconsistent with the EGM construct and “non-physical”. The argument for this conclusion is easily demonstrated according to, r S mh 1 . r m .m S x h λh λx

1=

144.219703 . 4.21884710

13

(%)

(4.243)

r S mh

1 = 22.109851 ( % )

R BH m h

(4.244)

These results indicate that a Planck scale particle of radius “λh” and mass “mh” is nonphysical because “rS(mh) > λh” {i.e. “[rS(mh) / λh] > 1”}. Moreover, they also demonstrate that “RBH” is smaller than “rS” {i.e. “[rS(mh) / RBH(mh)] > 1”}. This means that the event horizon is inside the singularity, not outside as expected and required. Notably, “rS” of a particle with radius “λxλh” and mass “mxmh” is equal to the radius of the particle [i.e. “rS(mxmh) = λxλh”] hence, it is physical. Thus, “r3” and “M3” may be given according to, r3 = λxλh

(4.245)

M3 = mxmh = λxmh / 2

(4.246)

At the commencement of the “Hα” and “HU” derivation process, the following assertion was articulated: “any Galactic formation is dynamically, kinematically and geometrically equivalent to a spherical particle of homogeneous mass distribution and may be represented as a Planck scale mass to be utilised as a GRP”. Moreover, it was also stated that - for the Galactic formation to be related by proportional similarity to the “Primordial Universe”, it should be the Galactic formation from which the Hubble constant and the CMBR temperature were measured. This constraint ensures that no currently unknown phenomena influence the calculation. The GRP is formulated by the compression of all matter (i.e. visible + dark), within the Galactic formation, to the Planck scale. Hence, it follows that the GRP’s dimensions must be transformed by the EGM adjusted Planck characteristics of Length “Kλ” and Mass “Km”, as derived by Storti. et. Al. in [13], such that “r2 → r2(r)” and “M2 → M2(M)”. Therefore, for consistent and complete generalised dynamic, kinematic and geometric similarity of any GRP to a SPBH in terms of radius and mass, “r2” and “M2” may be defined according to, r2(r) = Kλ⋅r (4.247) M2(M) = Km⋅M = Kλ⋅M

(4.248)

where, “Kλ = Km = [π / 2](1 / 3) ≈ 1.162447”. 182

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7.3.2 Computational results Utilising the expressions for “r2, r3, M2” and “M3” defined above such that the GRP is formed from the “Milky-Way” (MW), solutions for “Hα, AU, RU, ρU, MU, HU” and “TU” may be given according to, i. “r = Ro” denotes the mean distance from the Sun to the MW Galactic centre. ii. “Ro = 8(kpc)” as defined by the PDG. [20] iii. “M = MG” denotes the total mass (i.e. visible + dark) of the MW Galaxy. iv. “MG ≈ 6 x1011” solar masses as defined by [21]. v. “H0 = 71(km/s/Mpc)” as defined by the PDG. [22] vi. “T0 = 2.725(K)” as defined by the PDG. [20] H α λ x.λ h , m x.m h

ωh λx

(4.249)

9 A U K λ .R o , λ x.λ h , K m.M G, m x.m h = 14.575885 10 .yr

(4.250)

9 R U K λ .R o , λ x.λ h , K m.M G, m x.m h = 14.575885 10 .Lyr

(4.251)

33 kg ρ U K λ .R o , λ x.λ h , K m.M G, m x.m h = 8.453235 10 . 3 cm

(4.252)

. 52 ( kg ) M U K λ .R o , λ x.λ h , K m.M G, m x.m h = 9.28458610

(4.253)

H U K λ .R o , λ x.λ h , K m.M G, m x.m h = 67.084304

km . s Mpc

(4.254)

T U K λ .R o , λ x.λ h , K m.M G, m x.m h = 2.724752 ( K )

(4.255)

The EGM construct error associated with “HU” and “TU” with respect to expert opinion and physical measurement is given by, 1 . H U K λ .R o , λ x.λ h , K m.M G, m x.m h H0

1 = 5.515064 ( % )

1 . T U K λ .R o , λ x.λ h , K m.M G, m x.m h T0

. 1 = 9.08391310

(4.256) 3

( %)

(4.257)

7.3.3 Honourable mention It should not escape attention that the absence of “Kλ”, “λx”, “Km” and “mx” from “r2, r3, M2” and “M3” respectively, continues to produce impressive results, re-affirming the validity of the EGM construct as follows, H U R o , λ x.λ h , M G, m x.m h H U R o , λ h , M G, m h H U K λ .R o , λ h , K m.M G, m h T U R o , λ x.λ h , M G, m x.m h T U R o , λ h , M G, m h T U K λ .R o , λ h , K m.M G, m h

183

66.700842 = 69.672169 70.06923

km s .Mpc

(4.258)

2.716201 = 1.199134 ( K ) 1.202877

(4.259) www.deltagroupengineering.com

The EGM construct error associated with “HU” and “TU” for the various functional deviations with respect to physical measurement is given by, H U R o , λ x.λ h , M G, m x.m h 1 . H0

H U R o , λ h , M G, m h

6.055152 1=

H U K λ .R o , λ h , K m.M G, m h

1.310944

T U R o , λ x.λ h , M G, m x.m h 1 . T0

T U R o , λ h , M G, m h T U K λ .R o , λ h , K m.M G, m h

1.870184 ( % )

(4.260)

0.322893 1=

55.995089 ( % ) 55.857737

(4.261)

7.3.4 Concluding remarks The EGM construct produces highly precise numerical approximations, in agreement with physical measurement as reported by the PDG. The correlation of “HU” and “TU” (including the functional deviations presented) to the experimental evidence, demonstrates a clear relationship between “H0” and “T0”, suggesting exciting new avenues of theoretical Cosmological research. An important question arises as to why the relationship between “Kλ” and “Km” is different to the relationship existing between “λx” and “mx”. The reason for this is because “λx” and “mx” apply specifically to a SPBH – a theoretically physical object with a singularity radius, an event horizon and a value of “nΩ” equal to unity. The GRP does not physically exist, it is a mathematical contrivance formulated by dimensional similarity principles approaching the Planck scale. It does not have a singularity radius or event horizon associated with it and relative to the position of the Earth, has a value of “nΩ” much greater than unity. Therefore, the reasons for the difference in relationship between “λx,mx” and “Kλ,Km” may be summarised as follows: i. The relationship between “λx” and “mx” is governed by “nΩ(λxλh,mxmh) = 1”. ii. The relationship between “Kλ” and “Km” is governed by the Planck scale such that “nΩ(r2(r),M2(M)) >> 1” hence, “M2(M) ≠ (Kλ / 2)⋅M”. NOTES

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7.4

Determination of the impact of “Dark Matter / Energy” on “HU” and “TU”

7.4.1 Synopsis The question of the impact of “Dark Matter / Energy” on “H0” has long thought to be certain. It has been assumed that the “driving” component of the accelerating expansion of the Universe is the presence of “Dark Matter / Energy”. EGM disagrees with this assertion because it (i.e. EGM) maintains that Photon's have mass. Therefore, a significant contribution to the “missing mass” relating to “Dark Matter / Energy” theories, is in-fact - Photonic mass. Note: the impact of “Dark Matter / Energy” on “T0” has never been (to date) seriously considered, supported by meaningful and accurate calculations, by mainstream Physicists. 7.4.2 Assumptions i. The EGM construct is valid. ii. The values of “HU” and “TU” calculated in the preceding section are correct. iii. The “visible mass” of the MW Galaxy is “MG / 3”, as defined by [21]. 7.4.3 Construct It is possible to calculate the value of “HU” and “TU” based upon the “visible mass only” of the MW Galaxy by a simple substitution of values (i.e. “M2 / 3 = KmMG / 3”) as follows, 1 km H U K λ .R o , λ x.λ h , .K m.M G, m x.m h = 67.753267 . 3 s Mpc

(4.262)

1 T U K λ .R o , λ x.λ h , .K m.M G, m x.m h = 2.739618 ( K ) 3

(4.263)

Hence, the magnitude of the impact of “Dark Matter / Energy” on the value of “HU” and “TU” is demonstrated to be “< 1(%)” when compared to the previously derived value according to, H U K λ .R o , λ x.λ h , K m.M G, m x.m h

1 = 0.987352 ( % )

1 H U K λ .R o , λ x.λ h , .K m.M G, m x.m h 3 T U K λ .R o , λ x.λ h , K m.M G, m x.m h 1 T U K λ .R o , λ x.λ h , .K m.M G, m x.m h 3

(4.264) 1 = 0.542607 ( % )

(4.265)

7.4.4 Concluding remarks The preceding results demonstrate that the impact of “Dark Matter / Energy” on “HU” and “TU” is very small. This implies that the constitution of the Universe under the EGM construct is quite different from current thinking. The contemporary view asserted in [23] is that the constitution of the Universe is, i. “72(%) Dark Energy”. ii. “23(%) Dark Matter”. iii. “4.6(%) Atoms”. However, the EGM construct generalises the constitution of the Universe as being, iv. “> 94.4(%) Photons”. v. “< 1(%) Dark Matter / Energy”. vi. “4.6(%) Atoms”. 185

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7.5

“TU” as a function of a generalised Hubble constant “TU → TU2”

Utilising “ωΩ_3”, “TU” may be expressed in terms of a generalised Hubble constant “TU2(H)” according to, 9

λ Ω_3( r , M )

c

c

ω Ω_3( r , M )

9

5

1 . r St G M 2

c. 2

M St G. 5 r

(4.266)

If “r → (c / H)” and “M → M3” then, 9

c c λ x. λ Ω_3 , mh H 2

c.

1 . St G

5 9

H λx 2

c. 2

.m h

1 . 2 St G λ x.m h

2

. c H

5

(4.267)

Recognising that “ng(ωΩ_3(r3,M3),M3) = (8 / 3)” yields “KT(H)” as follows, K T( H )

8 . H α r 3, M 3 ln 3 H

(4.268)

Hence, KW

T W( H) λ Ω_3

c λ x. , mh H 2

(4.269)

T U2( H ) K T( H ) .T W ( H )

(4.270)

Recognising that “Hα(r3,M3) = ωh / λx” yields, K T( H ) .T W ( H )

ωh 8. . ln 3 λ x.H

KW λ Ω_3

c λ x. , mh H 2

(4.271)

Performing the appropriate substitutions produces, 9

ωh λ .m 8 KW. . St . x h T U2( H ) . ln G 3 c 2 λ x.H

2

. H c

5

(4.272)

Let, 9

. 8 . St G . λ x m h St T 5 3 .c 2 c

2

(4.273)

Injecting “StG” and simplifying yields, 9

4. 3. 1 . λ x 3 4 c5 π .λ 2 h 3

St T

2

(4.274)

Therefore, T U2( H )

K W .St T .ln

186

ωh λ x.H

9

. H5

(4.275)

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7.6

Derivation of “Ro”, “MG”, “HU2” and “ρU2” from “TU2”

7.6.1 Synopsis The value of “Ro” has been substantially improved in recent years and is stated by the PDG as being “Ro = 8(kpc)” with an experimental uncertainty given as “∆Ro = 0.5(kpc)”. The value of MW total Galactic mass, expressed in solar masses as being “MG / MS ≈ 6 x1011”, is quite rough. In-fact, one has difficulty finding an “MG / MS” uncertainty value anywhere in the scientific literature [note: “kpc” = kilo-parsec]. The principle reason for “MG” being so generalised is due to the lack of current knowledge around “Dark Matter / Energy”. However, utilising the relationship between “TU” and “HU” articulated in “TU2”, we are able to significantly improve upon the estimates for “Ro” and “MG” by determining a convergent numerical solution bound by the experimental uncertainty associated with “Ro” (i.e. “∆Ro”). Before commencing the derivation process, we shall generalise “HU” such that “Ro → r”, “MG → M” and “HU → HU2(r,M)” according to, H U2( r , M )

H U K λ .r , λ x.λ h , K m.M , m x.m h

(4.276)

A simple test verifying “TU2” is demonstrated below. Since, the computed value of “TU2(HU2(Ro,MG/3))” based upon visible MW Galactic mass “MG/3” is exactly compliant with “TU” (i.e. “TU = TU2”), no technical error exists. Moreover, the result “TU2(H0) ≈ T0” agrees precisely with historical expectation (i.e. prior to measurement by satellite) of “T0”. 1 T U2 H U2 R o , .M G 3

=

T U2 H 0

2.739618

( K)

2.810842

(4.277)

Computing values of “TU2” associated with “∆Ro”, yields violation of “T0” experimental boundaries [i.e. “∆T0 ± 0.001(K)”]. In other words, the “TU2” result returned when “r = (Ro ± ∆Ro)” is beyond “T0 ± ∆T0” when “M = MG” according to, T U2 H U2 R o

∆R o , M G

T U2 H U2 R o

∆R o , M G

=

2.720213 2.729021

( K)

(4.278)

Repeating the calculation based upon visible MW Galactic mass (i.e. “M = MG/3”) yields, T U2 H U2 R o T U2 H U2 R o

1 ∆R o , .K m.M G 3 1 ∆R o , .K m.M G 3

=

2.733025 2.741859

( K)

(4.279)

The preceding results infer numerical avenues for the accurate determination of “Ro” and “MG” based upon precise measurement of “T0”. It is likely that the experimental measurement of “T0” will advance at a substantially greater pace than “Ro” or “MG”. In the proceeding construct, we shall establish a method to accurately determine the values of “Ro” and “MG”, which may be observationally tested in the future when the experimental capability of “∆T0 → 0(K)” is achieved. 7.6.2 Assumptions i. The EGM Cosmological construct thus far is correct. ii. The values of “Ro”, “MG” and “MG/3” are approximately correct. iii. The values of “T0”, “∆T0” and “∆Ro” are precisely correct. 187

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7.6.3 Construct 7.6.3.1 “Ro” or “MG” Compliant mutually exclusive boundary values for “Ro” and “MG” may be determined numerically within the “MathCad 8 Professional” environment utilising the “Given” and “Find” commands as follows, Let “rx1”, “rx2”, “mg1” and “mg2” denote the algorithm pre-factors required by the computational environment with initialisation string: “rx1 = rx2 = mg1 = mg2 =1”. Given T U2 H U2 r x1.R o , M G T U2 H U2 R o , m g1 .M G T U2 H U2 r x2.R o , M G T U2 H U2 R o , m g2 .M G

T0

∆T 0

(4.280) T0

∆T 0

(4.281)

r x1 r x2 m g1

Find r x1, r x2, m g1 , m g2

m g2

(4.282)

Hence, r x1 r x2 m g1

0.989364 =

1.017883 1.057292 0.911791

m g2

(4.283)

Substituting “rx1”, “rx2”, “mg1” and “mg2” into “TU2” produces “T0 ± ∆T0”, confirming that the algorithm executed correctly as follows, T U2 H U2 r x1.R o , M G T U2 H U2 r x2.R o , M G T U2 H U2 R o , m g1 .M G T U2 H U2 R o , m g2 .M G

2.724 =

2.726 2.724

( K)

2.726

(4.284)

Thus, the mutually exclusive boundary values satisfying the condition “TU2 = T0 ± ∆T0” become, R o.

r x1

=

r x2

7.914908 8.143063

( kpc )

(4.285)

. 11 M G m g1 6.34375310 . = M S m g2 . 11 5.47074910 r x1 m g1 r x2 m g2

1=

(4.286)

1.063645 5.729219 1.788292

188

8.820858

(%)

(4.287)

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Therefore, “TU2 = T0 ± ∆T0” is satisfied when: • “0.9894Ro < Ro < 1.0179Ro” or “0.9118MG < MG < 1.0573MG”. 7.6.3.2 “Ro” and “MG” Compliant simultaneous boundary values for “Ro” and “MG” (i.e. to “6” decimal places) satisfying the condition “TU2 = T0 ± ∆T0” may be determined numerically within the “MathCad 8 Professional” environment utilising the “Given” and “Find” commands as follows, Given T U2 H U2 r x1.R o , m g1 .M G T U2 H U2 r x1.R o , m g2 .M G

T0

T U2 H U2 r x2.R o , m g1 .M G T U2 H U2 r x2.R o , m g2 .M G

T0

∆T 0

(4.288) ∆T 0

(4.289)

Let, r x3 r x4 m g3

Find r x1, r x2, m g1 , m g2

m g4

(4.290)

Hence, r x3 r x4 m g3

0.984956 =

1.013348 0.977007 0.977007

m g4

(4.291)

Substituting “rx3”, “rx4”, “mg3” and “mg4” into “TU2” produces “T0 ± ∆T0”, confirming that the algorithm executed correctly as follows, T U2 H U2 r x3.R o , m g3 .M G T U2 H U2 r x4.R o , m g4 .M G

=

2.724 2.726

( K)

(4.292)

Thus, the simultaneous boundary values satisfying the condition “TU2 = T0 ± ∆T0” become, R o.

r x3

=

r x4

7.879647 8.106786

( kpc )

(4.293)

. 11 M G m g3 5.8620410 . = M S m g4 . 11 5.8620410 r x3 m g3 r x4 m g4

1=

1.50441 2.29934 1.334822 2.29934

(4.294) (%)

(4.295)

Therefore, “TU2 = T0 ± ∆T0” is satisfied when: • “0.9850Ro < Ro < 1.0133Ro” and “MG / MS = 5.8620 x1011”. 189

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7.6.3.3 “Ro”, “MG”, “HU2” and “ρU2” 7.6.3.3.1 “Ro” and “MG” Compliant simultaneous values for “Ro” and “MG” satisfying the condition “TU2 = T0” may be determined numerically within the “MathCad 8 Professional” environment utilising the “Given” and “Find” commands as follows, Given T U2 H U2 r x1.R o , m g1 .M G

T0

(4.296)

Let, r x5 m g5

Find r x1, m g1

(4.297)

Hence, r x5 m g5

1.013403

=

1.052361

(4.298)

Substituting “rx5” and “mg5” into “TU2” produces “T0”, confirming that the algorithm executed correctly as follows, T U2 H U2 r x5.R o , m g5 .M G

= 2.725 ( K )

(4.299)

Thus, the simultaneous values satisfying the condition “TU2 = T0” become, r x5.R o = 8.107221 ( kpc ) m g5 .

MG

(4.300)

. = 6.31416710

11

MS

r x5 m g5

(4.301) 1=

1.340256 5.236123

(%)

(4.302)

Therefore, “TU2 = T0” is satisfied when: • “Ro = 8.1072(kpc)” and “MG / MS = 6.3142 x1011”. 7.6.3.3.2 “HU2” and “ρU2” If “T0” is exactly correct, then “HU2” and “ρU2” may be determined utilising the derived values for “rx5” and “mg5” according to, H U2 r x5.R o , m g5 .M G = 67.095419

km s .Mpc

(4.303)

Hence, ρ U2( r , M )

3 .H U2( r , M )

2

8 .π .G

(4.304)

33 kg ρ U2 r x5.R o , m g5 .M G = 8.456036 10 . 3 cm

190

(4.305)

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7.6.4 Concluding remarks The preceding construct demonstrates a method by which it is possible to determine the values of “Ro” and “MG” for the MW Galaxy from an exact measurement of “T0”. NOTES

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7.7

Experimentally implicit derivation of the ZPF energy density threshold “UZPF”

7.7.1 Synopsis The ZPF energy density threshold “UZPF” is very important to Cosmology as it is believed to be the reason for the “flat expansion phenomenon” as determined by the “Wilkinson Microwave Anisotropy Probe” (WMAP). The EGM method may be applied to derive “UZPF” by considering the average EGM mass-density of the Cosmos, given by the form “ρm(r,M)” – according to, ρ m R U K λ .R o , λ x.λ h , K m.M G, m x.m h , M U K λ .R o , λ x.λ h , K m.M G, m x.m h

= 8.453235 10

33 .

kg 3

cm

(4.306)

This result may also be expressed in “ρU2” notation as, ρ U2 R o , M G = 8.453235 10

33 .

kg 3

cm

(4.307)

Hence, if we compare “ρU2(rx5Ro,mg5MG)” to “ρU2(Ro,MG)”, the ratio produces the EGM density parameter “ΩEGM”, leading to the threshold value (i.e. upper limiting estimate) of “UZPF”. 7.7.2 Assumptions i. The experimental value of “T0” is exactly correct. ii. “ρU2(rx5Ro,mg5MG)” being based upon the experimentally measured value of “T0”, differs from the idealised EGM result “ρU2(Ro,MG)” due to the “flat expansion phenomenon”. iii. The ZPF energy density value, responsible for the “flat expansion phenomenon”, is a negative quantity. 7.7.3 Construct The EGM total density parameter “ΩEGM” may be written according to, Ω EGM

ρ U2 r x5.R o , m g5 .M G ρ U2 R o , M G

(4.308)

Evaluating produces, Ω EGM = 1.000331

(4.309)

Subsequently, considering the contemporary representation of Cosmological density parameter “Ω” such that “Ω → ΩEGM”, the critical EGM total density may be identified as “ρU2(Ro,MG)” from, Ω

ρ ρc

(4.310)

where, “ρc” denotes critical Cosmological total density. The PDG state in [20] that the total density parameter is “ΩPDG = 1.003” such that its constitution may be decomposed according to, Ω PDG Ω m Ω γ .. Ω ν

ΩΛ

(4.311)

where, each term on the Right-Hand-Side (RHS) of the equation denotes a physical contribution such as visible matter “Ωm”, Photon’s “Ωγ”, Neutrinos “Ων” and Dark Energy “ΩΛ” etc. However, under the EGM construct all matter radiates populations of high frequency conjugate Photon pairs (possessing non-zero mass). Subsequently, all the typical density terms may be “clumped together” such that “ΩEGM ≈ ΩPDG” according to, 192

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Ω EGM Ω PDG

= 0.997339

(4.312)

The geometry of the Cosmological space-time manifold has been measured by WMAP to be nearly flat, hence; the Friedman equation written in ZPF considerate form is “ΩEGM + ΩZPF = 1” where, “ΩZPF” denotes the “ZPF” density parameter. Approximated evaluation yields, Ω ZPF

Ω EGM

1

. Ω ZPF = 3.31400710

(4.313) 4

(4.314)

Therefore, the Cosmological average ZPF energy density may be approximated according to, 3 .c . Ω ZPF . H U2 R o , M G 8 .π .G 2

U ZPF

U ZPF = 2.51778 10

2

(4.315)

13 .

Pa

(4.316)

7.7.4 Concluding remarks The utilisation of “T0” (i.e. a physical measurement) leads to an experimentally implicit derivation of the ZPF energy density threshold “UZPF” characterised by the following boundary values: i. ΩZPF < -3.32 x10-4. ii. UZPF < -2.52 x10-13(Pa). On a human scale, this translates to levels of ZPF energy according to, iii. “< -252(yJ/mm3)”. On an astronomical scale, this becomes, iv. “< -0.252(mJ/km3)”. v. “< -7.4 x1012(YJ/pc3)”. On a Cosmological scale, this becomes, vi. “< -6.6 x1041(YJ/RU3)”. The deceleration parameter, vii. “ΩEGM” may be utilised to obtain non-zero deceleration parameter solutions. Note: although on the human scale the quantities of ZPF energy are extremely small, on the astronomical or Cosmological scales, they become extremely large when approaching the dimensions of the visible Universe according to “RU → RU(KλRo,λxλh,KmMG,mxmh)”.

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NOTES

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8 Advanced Cosmology

Abstract A time dependent derivation of “TU” is performed, including its rate of change and relationship to “HU”. This facilitates the articulation of the Cosmological evolution process into four distinct periods dealing with the inflationary and early expansive phases. Subsequently, the history of the Universe49 is developed and compared to the Standard Model (SM) of Cosmology. This assists in determining the Cosmological limitations of the EGM construct. The question of the practicality of utilising conventional radio telescopes for gravitational astronomy is also addressed.

49

As defined by the EGM construct. 195

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Time dependent CMBR temperature “TU2 → TU3”

8.1

8.1.1 Synopsis A Hubble constant dependent expression is formulated and graphed for the CMBR temperature in the time domain. This may be further developed into a generalised time dependent representation of the average CMBR temperature; laying foundations such that the relationship to the primordial Hubble constant is emphasised and thermodynamic rates of change may be subsequently articulated in the proceeding section. 8.1.2 Assumptions i. The Universe is “flat” (i.e. as indicated by WMAP). ii. “t = 1 / H”. 8.1.3 Construct Recalling that “Hα(λxλh,mxmh) = ωh / λx” facilitates the derivation of a time dependent expression for CMBR temperature. Simplifying notation such that “ωh / λx = Hα” and substituting into “TU2”, yields a primordially dependent form where “TU2 → TU3” according to, T U3( H ) K W .St T .ln

Hα

9

. H5

H

(4.317)

Let: “µ = 1 / 3” and “H = HβHα” where, “1 ≥ Hβ > 0” such that it denotes a dimensionless range variable. Hence, T U3 H β

K W .St T .ln

1 Hβ

. H .H β α

5 .µ

2

(4.318)

Determining local maxima in the conventional manner (i.e. “dTU3/dHβ = 0”) yields, 1 . d K W .St T .ln H β .H α Hβ dH β

5 .µ

2

0

(4.319)

1

Hβ e

2 5 .µ

(4.320)

If the freezing temperature of water [i.e. “0°(C) = 273(K)”] represents “some sort” of Cosmological milestone, we may determine the value of the Hubble constant and the age of the Universe satisfying this temperature condition numerically utilising the “Given” and “Find” commands within the “MathCad 8 Professional” environment according to the following algorithm, Let “Hβ2” denote the algorithm pre-factor required by the computational environment with an appropriate initialisation value such that the error vector converges to zero. Given T U2 10

H β2

H β2

.H α

273.( K )

(4.321)

Find H β2

(4.322)

Hence, H β2 = 56.450309

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8.1.4 Sample calculations •

The primordial Hubble constant (i.e. the value at the instant of the “Big-Bang”) was, . 61 H α = 8.46094110

•

km s .Mpc

(4.324)

The maximum average Cosmological temperature since the “Big-Bang” was, 1

T U3 e

•

2 5 .µ

. 31 ( K ) = 3.19551810

(4.325)

The value of the Hubble constant at the maximum average Cosmological temperature was, 1

e

•

2 5 .µ .

. 61 H α = 1.39858410

km s .Mpc

The present Cosmological value of “ H β ” is, H U2 R o , M G

= 7.928705 10

61

Hα

•

T U3( 1 )

=

0 0

( K)

(4.328)

The value of the Hubble constant coinciding with an average Cosmological temperature of “273(K)” was, 10

•

(4.327)

The average Cosmological temperature at the moment of the “Big-Bang” was, T U2 H α

•

(4.326)

H β2

.H = 2.99992310 . 5 α

km . s Mpc

(4.329)

The Cosmological age coinciding with an average Cosmological temperature of “273(K)” was, 10

H β2

.H α

1

6 = 3.259461 10 .yr

(4.330)

8.1.5 Sample plots •

See overleaf.

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8.1.5.1 “TU3 vs. Hβ”: Figure 4.22 Av. Cosmological Temperature

1

31 3.5 .10 e

5 .µ

2

3 .1031

Av. Cosmological Temperature (K)

2.5 .1031

31 2 .10

T U3 H β 1 T U3 e

5 .µ

2 1.5 .1031

31 1 .10

5 .1030

1

0.1

0.01

1 .10 3 Hβ Dimensionless Range Variable

1 .10 4

1 .10 5

1 .10 6

Average Cosmological Temperature Maximum Av. Cosmological Temperature

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8.1.5.2 “TU3 vs. t = (HβHα)-1” (i): Figure 4.23 Av. Cosmological Temperature vs. Age

1

31 3.5 .10

2

1 Hα

e

5 .µ . 1 Hα 3 .1031

Av. Cosmological Temperature (K)

2.5 .1031

31 2 .10

T U3 H β 1 T U3 e

5 .µ

2 1.5 .1031

31 1 .10

5 .1030

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

H β .H α Cosmological Age (s)

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8.1.6 Honourable mention It should not escape attention that the preceding graphs clearly exhibit “Planck-Black-Body” radiation characteristics. 8.1.7 Concluding remarks The key determinations are: i. The preceding graphs imply that the “Primordial Universe” prior to the “Big-Bang” was non-physical and at the moment of the “Big-Bang”, it became physical. This suggests that the space-time geometry of the “Primordial Universe” prior to the “BigBang” was “inverted”50 in relation to its present form51. ii. Prior to the “Big-Bang”52, “T0 → -∞(K)”. iii. At the instant of the “Big-Bang”53, “T0 = 0(K)”. iv. Since the “Big-Bang”54, the maximum value of “T0” was “≈ 3.2 x1031(K)”. v. The present value of “T0” is “2.724752(K)”. NOTES

50

i.e. it was analogous to a non-physical “Planck-Particle” such that “RBH < rS”. i.e. analogous to a SBH where “RBH > rS”. 52 i.e. at “t = 0”. 53 i.e. at “t = 1 / Hα”. 54 i.e. at “t = t1”: refer to proceeding section. 51

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Rates of change of CMBR temperature “TU3 → TU4 → d1,2,3TU4/dt1,2,3”

8.2

8.2.1 Synopsis This section develops expressions and graphical representations of the rates of change of CMBR temperature within the first few moments of the “Big-Bang”, based upon the preceding construct. 8.2.2 Assumptions •

No new assumptions are asserted.

8.2.3 Construct If “t = (HβHα)-1” then “TU3 → TU4” according to, 1 T U4( t ) K W .St T .ln H α .t . t

5 .µ

2

(4.331)

Determining the local maxima of CMBR temperature in the time domain utilising standard techniques produces “t1” according to, d T U4( t ) dt

K W .St T .

µ

1 . 1 t t5

1 . 1 K W .St T . t t5

µ

2

. 5 .ln H .t .µ 2 α

1

(4.332)

2

. 5 .ln H .t .µ 2 α

1

0

(4.333)

Subsequently: if “t → t1” then, 1

t1

e

2 5 .µ .

1 Hα

(4.334)

Hence, let the expression for the “1st” derivative of the CMBR temperature be given by, K W .St T .

dT dt ( t )

2 5 .ln H α .t .µ

t

5 .µ

2

1

(4.335)

.t

The local minima of the “1st” CMBR temperature derivative “t2” is determined according to, 2

1 . 1 T U4( t ) K W .St T . 2 5 2 dt t t d

1 . 1 K W .St T . 2 5 t t

µ

µ

2

. 5 .µ 2 . ln H .t . 5 .µ 2 α

1

2

1

(4.336)

2

. 5 .µ 2 . ln H .t . 5 .µ 2 α

1

2

1

0

(4.337)

Subsequently: if “t → t2” then, 10 .µ

t2

e

2

2 2 5 .µ . 5 .µ

201

1 1

. 1 Hα

(4.338) www.deltagroupengineering.com

Hence, let the expression for the “2nd” derivative of the CMBR temperature be given by, K W .St T .

dT2 dt2 ( t )

2 2 5 .µ . ln H α .t . 5 .µ

t

2

5 .µ

1

2

1

.t2

(4.339)

The local maxima of the “2nd” CMBR temperature derivative “t3” is determined according to, d

3

d t3

T U4( t )

1 . 1 K W .St T . 3 5 t t

K W .St T .

1 . 1 t

3

t

5

µ

µ

2

. 5 .µ 2 .ln H .t . 5 .µ 2 . 5 .µ 2 α

3

2

2 2 15.µ . 5 .µ

2

2

(4.340)

2

. 5 .µ 2 .ln H .t . 5 .µ 2 . 5 .µ 2 α

3

2

2 2 15.µ . 5 .µ

2

2

0

(4.341)

Subsequently: if “t → t3” then, 2 2 15 .µ . 5 .µ

t3

e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2 3

. 1 Hα

2

(4.342)

Hence, let the expression for the “3rd” derivative of the CMBR temperature be given by, dT3 dt3 ( t )

K W .St T .

2 2 2 5 .µ .ln H α .t . 5 .µ . 5 .µ

t

3

5 .µ

2

2 .t3

2 2 15.µ . 5 .µ

2

2

(4.343)

8.2.4 Sample calculations Evaluating expressions numerically yields, dT dt t 1 = 0

K

dT2 dt2 t 2 = 0

s

(4.344) K s

dT3 dt3 t 3 = 0

2

(4.345)

K s

3

(4.346)

Expressing values in perspective produces, 1 Hα t1 t2

0.364697 =

2.206287 4.196153

10

42 .

s

6.205726

t3

(4.347)

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dT dt

1 . 74 1.32321810

Hα

dT dt t 1

=

dT dt t 2

s

(4.348) 1 . 116 7.65967810

Hα

dT2 dt2 t 1

=

dT2 dt2 t 2

. 114 2.02615310

K

0

s

2

. 112 8.77595210

dT2 dt2 t 3 dT3 dt3

K

. 72 1.05719310 . 71 9.25283810

dT dt t 3 dT2 dt2

0

(4.349)

1 . 159 6.22716710

Hα

dT3 dt3 t 1

=

. 156 3.77545710 155

. 1.45285710

dT3 dt3 t 2

K s

3

0

dT3 dt3 t 3

(4.350)

T U2 H α T U2 T U2 T U2

1

0

t1 1 t2 1

. 31 3.19551810 =

. 31 3.03432210

( K)

. 31 2.83254210

t3

(4.351)

An example of the CMBR temperature prior to the “Big-Bang” is given by, 4 . 34 ( K ) T U2 10 .H α = 7.41414610

(4.352)

8.2.5 Sample plots •

See overleaf.

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8.2.5.1 “TU3 vs. t = (HβHα)-1” (ii): Figure 4.24 Av. Cosmological Temperature vs. Age 1 Hα

31 3.5 .10

t1

3 .1031

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

T U3 e

T U3 e

5 .µ

2 31 2 .10

2 10 .µ 1 2 2 5 .µ . 5 .µ 1 1.5 .1031

2 2 15 .µ . 5 .µ 2 2 2 2 2 . . . . . 2 5µ 5µ 5µ 3

31 1 .10

5 .1030

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

H β .H α Cosmological Age (s)

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8.2.5.2 “TU3 vs. t = (HβHα)-1” (iii): Figure 4.25 Av. Cosmological Temperature vs. Age t2

31 3.5 .10

t3

3 .1031

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

T U3 e

T U3 e

5 .µ

2 31 2 .10

2 10 .µ 1 2 2 5 .µ . 5 .µ 1 1.5 .1031

2 2 15 .µ . 5 .µ 2 2 2 2 2 . . . . . 2 5µ 5µ 5µ 3

31 1 .10

5 .1030

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

H β .H α Cosmological Age (s)

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8.2.5.3 “TU3 vs. H = (HβHα)” (i): Figure 4.26 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 t1

Hα

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 2 2 1.5 .10 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

H β .H α Hubble Constant (Hz)

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8.2.5.4 “TU3 vs. H = (HβHα)” (ii): Figure 4.27 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 1 t2 t3

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 2 2 1.5 .10 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

H β .H α Hubble Constant (Hz)

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8.2.5.5 “TU3 vs. r = (HβHα)-1c” (i): Figure 4.28 Av. Cosmological Temperature vs. Size c Hα

31 3.5 .10

t 1 .c

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 1.5 .10

2 2 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

1 .10 29

1 .10 28

1 .10 27

1. c

H β .H α EGM Cosmological Size (m)

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8.2.5.6 “TU3 vs. r = (HβHα)-1c” (ii): Figure 4.29 Av. Cosmological Temperature vs. Size

31 3.5 .10

t 2 .c t 3 .c

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 1.5 .10

2 2 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

1 .10 29

1 .10 28

1 .10 27

1. c

H β .H α EGM Cosmological Size (m)

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8.2.5.7 “dTU4/dt vs. t = (HβHα)-1” (i): Figure 4.30 1st Derivative of Av. Cosmological Temp.

1 .1072 t1

t2

71 8 .10

6 .1071

4 .1071

71 2 .10 dT dt

H β .H α

1 0

(K/s)

dT dt t 1 dT dt t 2 dT dt t 3

2 .1071

4 .1071

71 6 .10

71 8 .10

1 .1072

72 1.2 .10 1 .10 42

1 .10 41

1 .10 40

1 .10 39

1

H β .H α Cosmological Age (s)

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8.2.5.8 “dTU4/dt vs. t = (HβHα)-1” (ii): Figure 4.31 1st Derivative of Av. Cosmological Temp.

1 .1072 t2

t3

71 8 .10

6 .1071

4 .1071

71 2 .10 dT dt

H β .H α

1 0

(K/s)

dT dt t 1 dT dt t 2 dT dt t 3

2 .1071

4 .1071

71 6 .10

71 8 .10

1 .1072

72 1.2 .10 1 .10 42

1 .10 41

1 .10 40

1 .10 39

1

H β .H α Cosmological Age (s)

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8.2.5.9 “d2TU4/dt2 vs. t = (HβHα)-1” (i): Figure 4.32 2nd Derivative of Av. Cosmological Temp.

113 5 .10 t1

t2

0

113 5 .10

(K/s^2)

dT2 dt2

H β .H α

1 1 .10114

dT2 dt2 t 1 dT2 dt2 t 2 dT2 dt2 t 3

114 1.5 .10

2 .10114

114 2.5 .10

114 3 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42 1 .10 41 1 H β .H α Cosmological Age (s)

212

1.1 .10 41 1.2 .10 41 1.3 .10 41 1.4 .10 41 1.5 .10 41

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8.2.5.10 “d2TU4/dt2 vs. t = (HβHα)-1” (ii): Figure 4.33 2nd Derivative of Av. Cosmological Temp.

113 5 .10 t2

t3

0

113 5 .10

(K/s^2)

dT2 dt2

H β .H α

1 1 .10114

dT2 dt2 t 1 dT2 dt2 t 2 dT2 dt2 t 3

114 1.5 .10

2 .10114

114 2.5 .10

114 3 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42 1 .10 41 1 H β .H α Cosmological Age (s)

213

1.1 .10 41 1.2 .10 41 1.3 .10 41 1.4 .10 41 1.5 .10 41

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8.2.5.11 “|d3TU4/dt3| vs. t = (HβHα)-1” (i): Figure 4.34 3rd Derivative of Av. Cosmological Temp.

157 1 .10 t1

t2

156 1 .10

(K/s^3)

dT3 dt3

H β .H α

1 .10155 1

dT3 dt3 t 1 dT3 dt3 t 2 1 .10154

153 1 .10

152 1 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42

1 .10 41 1

1.1 .10 41

1.2 .10 41

1.3 .10 41

1.4 .10 41

1.5 .10 41

H β .H α Cosmological Age (s)

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8.2.5.12 “|d3TU4/dt3| vs. t = (HβHα)-1” (ii): Figure 4.35 3rd Derivative of Av. Cosmological Temp.

157 1 .10 t2

t3

156 1 .10

(K/s^3)

dT3 dt3

H β .H α

1 .10155 1

dT3 dt3 t 1 dT3 dt3 t 2 1 .10154

153 1 .10

152 1 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42

1 .10 41 1

1.1 .10 41

1.2 .10 41

1.3 .10 41

1.4 .10 41

1.5 .10 41

H β .H α Cosmological Age (s)

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8.2.6 Concluding remarks The preceding graphs exhibit interesting properties of the CMBR temperature in relation to the Hubble constant. In particular, an inference is presented to precisely differentiate and articulate the differences between the inflationary period of the “Primordial Universe” (i.e. “t < t1”) and the expansion period of the present Universe (i.e. “t > t1”). NOTES

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Rates of change of the Hubble constant “d1,2H/dt1,2”

8.3

8.3.1 Synopsis The rates of change of the Hubble constant in the time domain “d1,2H/dt1,2” are useful relationships confirming the assertion that the Cosmos can never end with a “Big-Crunch”. This shall be comprehensively discussed in the next section, but for the moment, we shall develop the tools (i.e. expressions and graphs) we require to conduct the analysis. This section achieves, by differentially combining the CMBR temperature in the Hubble and time domains – via numerical approximation methods, verification that the assigned temporal property of “t = 1 / H” produces the appropriate “d1,2H/dt1,2” curves resulting in, i. Mathematical expressions for “dH/dt”, “d2H/dt2” and “|H|” in the time domain. ii. Graphical representations of “dH/dt vs. (HβηHα)-1”, “d2H/dt2 vs. (HβηHα)-1” and “|H| vs. (HβηHα)-1” – qualitatively and quantitatively tested against “TU2,3 vs. |H|”. Note: neither an expression nor graphical representation of “d3H/dt3 vs. t = (HβHα)-1” has been included - for reasons of brevity. 8.3.2 Assumptions •

No new assumptions are asserted.

8.3.3 Construct Substituting “µ = 1 / 3” into “TU3”, the generalised “1st” derivative of the CMBR temperature with respect to the Hubble constant “dTU3/dH”: is given by, Hα . 2 d d .H5 µ T U3( H ) K W .St T .ln dH dH H 5 .µ Hα . 2 d .H5 µ K .St . H K W .St T .ln W T dH H H

(4.353)

2

. 5 .ln

Hα

.µ 2

1

H

(4.354)

Recognising that “TU3 → TU4”, the “1st” derivative of the Hubble constant with respect to time “dH/dt” may be determined according to, 5 .µ

H K W .St T .

d d T U3( H ) . t dH d T U4( t )

5 .µ

H K W .St T .

2

. 5 .ln H

1 . 1 K W .St T . t t5

Hα

. 5 .ln

.µ

µ

. 5 .ln H .t .µ 2 α

2

. 5 .ln

2

. 5 .ln H .t .µ α

217

1

1 . 1 t t5

1

(4.355)

H 2

1

2

H

1

.µ 2

H

5 .µ

2

Hα

H

1 . 1 K W .St T . t t5

H µ

2

Hα

.µ 2

1

H µ

2

. 5 .ln H .t .µ 2 α

1

(4.356)

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5 .µ

2

H

. 5 .ln

Hα

H

.µ 2

t . . 5 .µ 2 . . H α . 2 (H t) 5 ln µ H H

1

H

1 . 1 t t5

µ

2

. 5 .ln H .t .µ 2 α

2 5 .ln H α .t .µ

1

1

1

(4.357)

Hence, H

d H dt

. . 2

5µ . ( H .t ) t

2 5 .ln H α .t .µ

5 .ln

Hα

.µ 2

1 1

H

(4.358)

Let: “H → Hα” such that “dH/dt → dHdt” and “Hγ ∝ Hβη” according to, t

1 H γ .H α 5 .ln H α .

Hα

d H dt H α.

1 H γ .H α

(4.359)

.

2 5 .µ

.

5 .ln

1 H γ .H α

1 .µ 2 . Hγ Hα

Hα

.µ 2

1

1

Hα

(4.360)

Therefore, dH dt H γ

2 H α .H γ . 5 .ln 1 .µ 2 2 Hγ 5 .µ Hγ

1

(4.361)

The temporal ordinate of the local maxima “t4” may be determined in the typical manner, 2

H α .H γ d . 5 .ln 1 .µ 2 2 dH γ Hγ 5 .µ Hγ

d dH dt H γ dH γ 2 H α .H γ d . 5 .ln 1 .µ 2 2 Hγ dH γ 5 .µ Hγ

Hα Hγ

Hα

(4.362)

2

. 5 .ln 1 .µ 2 . 1 2 Hγ 5 .µ Hγ

1

1

5 .µ

2

1

(4.363)

2

2 5 .µ

. 5 .ln 1 .µ 2 . 1 Hγ

5 .µ

2

1

0

(4.364) 1

Hγ e

2 2 5 .µ . 5 .µ

1

(4.365)

Hence, 1

t4

e

2 2 5 .µ . 5 .µ

1

. 1 Hα

(4.366)

The “2nd” time derivative “dH2dt2” may be derived similarly as follows, Let,

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d

2

d t2

H

d t2 H

d dt

H

2

d

dH2 dt2

( H .t )

(4.367) .

5 .µ

2

.t

2 5 .ln H α .t .µ

Hα

5 .ln

1

.µ 2

1

H

(4.368)

Subsequently, H

d dt

. . 2

5µ . ( H .t ) t

2 5 .ln H α .t .µ

5 .ln

Hα

1

H

.

2 2 5 .µ . ln H α .t . 5 .µ

. 2

.µ 2

5µ .2 ( H .t ) t

1

5 .ln

H

d

2 H H α.

5 .µ

1 H γ .H α

.µ 2

2

1

1

H 2 5 .µ . ln H α .

Hα

d t2

Hα

1

. 2

.

2

1 H γ .H α

(4.369)

1 . 5 .µ 2 . Hγ Hα Hα

5 .ln

.µ 2

1

2

1

1

Hα

(4.370)

Therefore, 3 2 H α .H γ . 5 .µ 2 . ln 1 . 5 .µ 2 2 Hγ 5 .µ Hγ

dH2 dt2 H γ

1

2

1

(4.371)

The temporal ordinate of the local minima “t5” is determined as follows, 3

2

H α .H γ d . 5 .µ 2 . ln 1 . 5 .µ 2 2 Hγ dH γ 5 .µ Hγ

1

2

1

3 H α .H γ . 5 .µ 2 . 5 .µ 2 . 5 .µ 2 .ln

Hγ

5 .µ

1

ln

Hγ

2

3 H α .H γ . 5 .µ 2 . 5 .µ 2 . 5 .µ 2 .ln 1 2 Hγ 5 .µ Hγ

ln

1 Hγ

1

2 .ln

1

1 Hγ

4

2

4

2

(4.372)

0

Hγ

(4.373) 2 2 5 .µ . 5 .µ

Hγ e

2 .ln

1

1 Hγ

4

2 2 2 5 .µ . 5 .µ . 5 .µ

2 1

2

(4.374)

Hence, 2 2 5 .µ . 5 .µ

t5

e

4

2 2 2 5 .µ . 5 .µ . 5 .µ

2 1

2

. 1 Hα

(4.375)

The magnitude of the Hubble constant “|H|” in the time domain may be derived by numerical approximation utilising “dHdt” as follows55, Let, Hγ Hβ

η

(4.376)

For solutions where “H = 1 / t” (i.e. the deceleration parameter is zero), 55

This terminology is an abbreviated reference to “the square-root of the magnitude of the rate of change of the Hubble constant in the time domain”, as indicated by the proceeding equations. 219

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d d 1 H dt dt t

1 t

2

H

2

(4.377)

56

Hence , d H dt

H

(4.378)

Therefore, “η” may be numerically approximated utilising the “Given” and “Find” commands within the “MathCad 8 Professional” computational environment, subject to the constraint that “dHdt” as a function of the present value of “Hβ” [i.e. “≈ HU2(Ro,MG) / Hα”] raised to an indicial power, is equal to the square of the present Hubble constant as determined by the EGM construct “HU2(Ro,MG)2” according to the following algorithm, Given dH dt

H U2 R o , M G

η

Hα

1

H U2 R o , M G

η

(4.379)

Find( η )

(4.380)

Note: the utilisation of “rx5Ro” and “mg5MG” instead of “Ro” and “MG” does not significantly, nor adversely, influence the otherwise computed value of “η”. 8.3.4 Sample calculations Executing the algorithm to determine the value of “η” yields, η = 4.595349

(4.381)

Evaluating “dHdt” and “dH2dt2” for various temporal ordinates produces the results, 1

dH dt t 1 .H α

1

dH dt e

5 .µ

10 .µ

dH dt t 2 .H α

1

dH dt e

2

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

dH dt t 3 .H α

1

dH dt e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2 3

2

1

dH dt t 4 .H α

1

dH dt e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

dH dt t 5 .H α

1

dH dt e

1

4

2 2 2 5 .µ . 5 .µ . 5 .µ

. 68 7.50137510

. 68 7.50137510

. 83 9.06689310

. 83 9.06689310

= 1.22575310 . 84

. 84 1.22575310

. 84 1.55351810

. 84 1.55351810

. 84 1.38436210

. 84 1.38436210

2

Hz

2 1

2

(4.382)

56

Noting that the terminology utilised is an abbreviated reference to “the square-root of the magnitude of the rate of change of the Hubble constant in the time domain”, as indicated by the equation. 220

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1

dH2 dt2 t 1 .H α

1

2 5 .µ

dH2 dt2 e

10 .µ

dH2 dt2 t 2 .H α

1

dH2 dt2 e

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

dH2 dt2 t 3 .H α

1

dH2 dt2 e

2

1

2 2 2 5 .µ . 5 .µ . 5 .µ

dH2 dt2 e

=

2

3

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

dH2 dt2 t 5 .H α

1

dH2 dt2 e

. 125 8.50679910

0

0

2

1

dH2 dt2 t 4 .H α

. 125 8.50679910

1

4

125

. 1.16257810

. 125 1.16257810

. 124 8.2461110

. 124 8.2461110

. 125 1.33162810

. 125 1.33162810

3

Hz

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2

1

(4.383)

where, 1

e

t1

2 5 .µ .

1 Hα

10 .µ

t2

e

2

1

2 2 5 .µ . 5 .µ

2 2 15 .µ . 5 .µ

t3 e

. 1 Hα

1

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2.206287 2.206287 4.196153 4.196153

2 3

. 1 Hα

2

= 6.205726 6.205726

e

t5 e

s

8.385263 8.385263

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

42 .

20.932666 20.932666

1

t4

10

1

4

2 2 2 5 .µ . 5 .µ . 5 .µ

. 1 Hα 2

1

. 1 Hα

2

(4.384)

The present values of “dHdt” and “dH2dt2” are given by the following approximations, dH dt

dH2 dt2

H U2 R o , M G

η

= 4.726505 10

Hα

36 .

2

Hz

(4.385)

H U2 R o , M G

η 3

= 0 Hz

Hα

(4.386)

The “η” calculation algorithm may be verified against the following two determinations, confirming the validity of the numerical approximation as follows, Hα

=1 η

dH dt 1

221

(4.387) www.deltagroupengineering.com

η

dH dt 1

H U2 R o , M G

dH dt

=

η

. 61 8.46094110

km

67.084257

s .Mpc

Hα

(4.388)

8.3.5 Construct errors 8.3.5.1 How can they be determined? The computational environment utilised for the numerical determination of “η” has limitations. Subsequently, one must consider its effect and the error associated with the introduction of any numerical method into a construct. The error relating to the approximation of “η” and “|H|” are resolved by graphical, analytical and numerical techniques indicating that, i. The expression for “dH/dt” is correctly derived. ii. The approximated value of “η” satisfies boundary conditions such that “TU2 = TU3” and is precisely representative of the present value of CMBR temperature – hence, it is sufficiently accurate for qualitative applications. iii. Graphical comparison of various intermediate thermal ordinates indicates that the approximated value of “η” is sufficiently accurate for qualitative applications. iv. Calculation of various intermediate thermal ordinates demonstrates that “TU2 ≈ TU3” – hence, the approximated value of “η” is sufficiently accurate for qualitative applications. 8.3.5.2 Analytical Separating variables in the “dH/dt” expression and integrating both sides of the equation produces “H = 1 / t”, confirming that the rate of change was correctly derived as follows, 5 .µ

H

2

. 5 .ln

Hα

H

.µ

2

2 5 .ln H α .t .µ

1 dH

H

t

5 .µ

2

1 dt

.t

(4.389) 5 .µ

H

2

Hα

. 5 .ln H

.µ 2

5 .µ

1 dH H

H

2

.ln

Hα H

(4.390) 2 5 .ln H α .t .µ

t

5 .µ

2

ln H α .t

1 dt

.t

t

5 .µ

2

(4.391) 5 .µ

H

2

.ln

Hα H

ln H α .t t

5 .µ

2

(4.392)

Solving for “H” confirms that the expression for “dH/dt” is correct and no error exists. H

1 t

(4.393)

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8.3.5.3 Graphical Graphical representations of the “TU2,3 vs. |H|” curves57 suggests common characteristics. Thus, illustrating that “TU2 ≈ TU3” and the approximated value of “η” is sufficiently accurate for qualitative applications. 8.3.5.4 Numerical 8.3.5.4.1 General case A numerical comparison of results demonstrates that the approximated value of “η” satisfies boundary conditions such that “TU2 = TU3” and is exactly representative of the present value of CMBR temperature. Moreover, determination of various intermediate thermal ordinates demonstrates that “TU2 ≈ TU3” such that the difference between them at “t1” is “≈ 7(%)”. This indicates that the approximated value of “η” is sufficiently accurate for qualitative applications. η

T U2

dH dt 1

T U3( 1 ) η

1

T U2

dH dt e

2 10 .µ

T U2

dH dt e

T U3 e η

2 2 5 .µ . 5 .µ

dH dt e

T U2

dH dt

2 2 2 5 .µ . 5 .µ . 5 .µ

1

H U2 R o , M G

0

2

. 31 3.19551810

= 3.18632310 . 31 3.03432210 . 31 ( K )

1

T U3

3

2.724751

2.724752

2

H U2 R o , M G Hα

1

dH dt

2

2

2 2 2 5 .µ . 5 .µ . 5 .µ

η

T U2

. 2.97174510

. 31 2.83254210 . 31 3.18071410

T U3 e

Hα

0 31

1

2 2 15 .µ . 5 .µ

2 3

2

2 2 5 .µ . 5 .µ

T U3 e η

2

5 .µ

2 10 .µ

1

2 2 15 .µ . 5 .µ

T U2

1

2 5 .µ

(4.394)

η

2 5 .µ

e

1 = 7.002696 ( % ) 1

T U3 e

5 .µ

2

(4.395)

8.3.5.4.2 Specific case For the specific case of “dHdt(Hγ) = 0”, the exact value of “η” may be determined by solving for “Hγ” as follows, 2 H α .H γ . 5 .ln 1 .µ 2 2 Hγ 5 .µ Hγ

1

0

(4.396)

1

Hγ e

5 .µ

2

(4.397)

57

i.e. the “TU2” curve (solid line) is superimposed upon the “TU3” curve (dotted line) – refer to graphs. 223

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Let “Hγ → t7” in the typical manner, hence: 1

t7

2 5 .µ .

e

1 Hα

(4.398)

Comparing “t7” to “t1” (i.e. the instant of maximum Cosmological temperature) yields, t7

=1

t1

(4.399)

Therefore, utilising the relationship “Hγ = Hβη”, the value of “η” at “t7 = t1” is given by, η

ln H γ

ln t 7 .H α

1

ln H β

ln t 1 .H α

1

1

(4.400)

However, when “t7 = t1” in terms of “Hγ” is substituted into “dHdt(Hγ)”, a result consistent with Eq. (4.394) is produced (see below), not the correct result such that “dHdt[(t1Hα)-1] = 0” as required by the preceding derivation. 1 2 H α .e

1

. 5 .ln 2 5 .µ

1

e

2 5 .µ

.µ 2

. 68 Hz2 1 = 7.50137510

1 5 .µ

e

2 5 .µ

2

(4.401)

The obvious question arises as to why this occurs when the “t7 = t1” result is analytically exact. Localising the anomaly is possible by systematically simplifying the expression for “dHdt”. A “1st” level investigation may be conducted by recognising that “dHdt(Hγ)” may be written as, dH dt H γ

2 1 . 2 2 4 .µ . . H α .H γ 5 ln µ Hγ

1

(4.402)

Evaluating at “Hγ → t7 = t1” continues to produce the anomalous result as given by, 4 2.

Hα e

5.

1

5 .ln

.µ 2

. 68 Hz2 1 = 7.50137510

1

e

5 .µ

2

(4.403i)

A “2nd” level investigation produces the correct result by further simplification according to, 1

4 2.

Hα e

5.

5 .ln e

5 .µ

2

.µ 2

2

1 = 0 Hz

(4.403ii)

The cause of the anomalous result becomes apparent when replacing “µ” with “1 / 3” as follows, 1

5 .ln

. 1 3

1 5.

e

1

2

1 =0

2

3

(4.404)

8.3.6 Sample plots •

See overleaf. 224

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8.3.6.1 “dH/dt vs. (HβηHα)-1” (i): Figure 4.36 1st Derivative of the Hubble Constant 1.6 .1084

t1

t4

84 1.4 .10

84 1.2 .10 dH dt H β

(Hz^2)

dH dt e

dH dt e

dH dt e

η

5 .µ

1 2

1 .1084 1

1 2 2 5 .µ . 5 .µ

1

8 .1083

2 2 5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

83 6 .10

83 4 .10

2 .1083

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

225

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8.3.6.2 “dH/dt vs. (HβηHα)-1” (ii): Figure 4.37 1st Derivative of the Hubble Constant 2 .1084 1 Hα

t1

84 1 .10

0 0

dH dt H β

(Hz^2)

dH dt e

η

5 .µ

1 2

1 .1084

1

84 2 .10

1 dH dt e

2 2 5 .µ . 5 .µ

3 .1084 2

dH dt e

1 2

5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

84 4 .10

5 .1084

6 .1084

84 7 .10

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

226

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8.3.6.3 “dH/dt vs. (HβηHα)-1” (iii): Figure 4.38 1st Derivative of the Hubble Constant 2 .1084 t2

t3

84 1 .10

0 0

dH dt H β

(Hz^2)

dH dt e

η

5 .µ

1 2

1 .1084

1

84 2 .10

1 dH dt e

2 2 5 .µ . 5 .µ

3 .1084 2

dH dt e

1 2

5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

84 4 .10

5 .1084

6 .1084

84 7 .10

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

227

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8.3.6.4 “dH/dt vs. (HβηHα)-1” (iv): Figure 4.39 1st Derivative of the Hubble Constant 2 .1084 t5

t4

84 1 .10

0 0

dH dt H β

(Hz^2)

dH dt e

η

5 .µ

1 2

1 .1084

1

84 2 .10

1 dH dt e

2 2 5 .µ . 5 .µ

3 .1084 2

dH dt e

1 2

5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

84 4 .10

5 .1084

6 .1084

84 7 .10

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

228

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8.3.6.5 “d2H/dt2 vs. (HβηHα)-1” (i): Figure 4.40 2nd Derivative of the Hubble Constant 1 Hα

4 .10127

t1

127 3.5 .10

3 .10127

(Hz^3)

2.5 .10127

dH2 dt2 H β

η 127 2 .10

127 1.5 .10

1 .10127

5 .10126

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1

η H β .H α Cosmological Age (s)

229

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8.3.6.6 “d2H/dt2 vs. (HβηHα)-1” (ii): Figure 4.41 2nd Derivative of the Hubble Constant t2

4 .10127

t3

127 3.5 .10

3 .10127

(Hz^3)

2.5 .10127

dH2 dt2 H β

η 127 2 .10

127 1.5 .10

1 .10127

5 .10126

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1

η H β .H α Cosmological Age (s)

230

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8.3.6.7 “d2H/dt2 vs. (HβηHα)-1” (iii): Figure 4.42 2nd Derivative of the Hubble Constant t5

4 .10127

t4

127 3.5 .10

3 .10127

(Hz^3)

2.5 .10127

dH2 dt2 H β

η 127 2 .10

127 1.5 .10

1 .10127

5 .10126

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1

η H β .H α Cosmological Age (s)

231

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8.3.6.8 “d2H/dt2 vs. (HβηHα)-1” (iv): Figure 4.43 2nd Derivative of the Hubble Constant 125 8 .10

t1

t2

7 .10125

125 6 .10

dH2 dt2 H β

(Hz^3)

dH2 dt2 e

dH2 dt2 e

dH2 dt2 e

η

5 .µ

1 2

125 5 .10

1

1 2 2 5 .µ . 5 .µ

4 .10125

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 . . . . . 2 5µ 5µ 5µ 1

125 3 .10

2 .10125

1 .10125 0 0

1 .10125

1 .10 42

1 .10 41 1 η H β .H α Cosmological Age (s)

232

1 .10 40

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8.3.6.9 “d2H/dt2 vs. (HβηHα)-1” (v): Figure 4.44 2nd Derivative of the Hubble Constant 125 8 .10

t3

t4

7 .10125

125 6 .10

dH2 dt2 H β

(Hz^3)

dH2 dt2 e

dH2 dt2 e

dH2 dt2 e

η

5 .µ

1 2

125 5 .10

1

1 2 2 5 .µ . 5 .µ

4 .10125

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 . . . . . 2 5µ 5µ 5µ 1

125 3 .10

2 .10125

1 .10125 0 0

1 .10125

1 .10 42

1 .10 41 1 η H β .H α Cosmological Age (s)

233

1 .10 40

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8.3.6.10 “d2H/dt2 vs. (HβηHα)-1” (vi): Figure 4.45 2nd Derivative of the Hubble Constant 125 8 .10

t5

7 .10125

125 6 .10

dH2 dt2 H β

(Hz^3)

dH2 dt2 e

dH2 dt2 e

dH2 dt2 e

η

5 .µ

1 2

125 5 .10

1

1 2 2 5 .µ . 5 .µ

4 .10125

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 . . . . . 2 5µ 5µ 5µ 1

125 3 .10

2 .10125

1 .10125 0 0

1 .10125

1 .10 42

1 .10 41 1 η H β .H α Cosmological Age (s)

234

1 .10 40

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8.3.6.11 “|H| vs. (HβηHα)-1” (i): Figure 4.46 Mag. of Hubble Cons. vs. Cosm. Age

2.5 .1042

1 Hα

t1

42 2 .10 dH dt H β

dH dt e

5 .µ

η

1 2

1 1.5 .1042

(Hz)

1 dH dt e

dH dt e

2 2 5 .µ . 5 .µ

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 1

42 1 .10

5 .1041

0 1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36 1

1 .10 35

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

η H β .H α Cosmological Age (s)

235

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8.3.6.12 “|H| vs. (HβηHα)-1” (ii): Figure 4.47 Mag. of Hubble Cons. vs. Cosm. Age

2.5 .1042

1 Hα

t4

42 2 .10 dH dt H β

dH dt e

5 .µ

η

1 2

1 1.5 .1042

(Hz)

1 dH dt e

dH dt e

2 2 5 .µ . 5 .µ

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 1

42 1 .10

5 .1041

0 1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36 1

1 .10 35

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

η H β .H α Cosmological Age (s)

236

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8.3.6.13 “TU2,3 vs. |H|”: Figure 4.48 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 t1

Hα 31 3 .10

Av. Cosmological Temperature (K)

2.5 .1031

T U2

dH dt H β

η 2 .1031

T U3 H β 1 T U3 e

5 .µ

2 31 1.5 .10

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

η dH dt H β , H β .H α Hubble Constant (Hz)

237

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8.3.6.14 “TU2 vs. |H|”: Figure 4.49 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 t1

Hα 31 3 .10

Av. Cosmological Temperature (K)

2.5 .1031

T U2

dH dt H β

η 2 .1031

1 T U3 e

5 .µ

2 31 1.5 .10

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

η dH dt H β Hubble Constant (Hz)

238

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8.3.7 Concluding remarks For most computational environment application users, it is naturally assumed that the environment will always produce numerically valid results. However, one should be cautious in this regard because numerical output is a direct function of the computational structure of the environment. For example, an anomalous result became apparent when replacing “µ” with “1 / 3” in a specific calculation. Importantly, the same computational anomaly encountered when “dHdt = 0” does not occur when “dH2dt2 = 0”. This suggests that the anomalous effect is a localised environmental defect – an assertion supported by the following scaled representation of the “1st” derivative of the Hubble constant versus Cosmological age. The axis scaling technique nullifies potential “large number” error effects, clearly demonstrating that at maximal resolution, the interpolated line between the final “2 of 5000” data points – spanning an identical domain to that utilised in preceding curves, passes precisely through the origin (i.e. at “dHdt = 0”) – in agreement with the exact analytical result. Moreover, it is known that within the “MathCad 8 Professional” environment, the graphical generation engine is sufficiently distinct from the user facing “work pad calculation engine”, such that graphical results may sometimes be considered more reliable. Hence, in our case at the very least, the graphical representations are sufficiently accurate for qualitative applications, to disregard the computed result of “dHdt[(t1Hα)-1] ≠ 0” in favour of the agreement between the analytical proof and the graphical evidence. 1st Derivative of the Hubble Constant

10 42 t 1 .10 8

Scaled Derivative (Hz^2)

6

4 dH dt H β 10

η

79 2

0 0

2

4

2.20624

2.20625

2.20626

2.20627

2.20628

2.20629

2.2063

2.20631

2.20632

2.20633

2.20634

2.20635

2.20636

2.20637

2.20638

1 η .1042 H β .H α Scaled Cosmological Age (s)

Figure 4.50, Therefore, the construct error relating to the approximation of “η” and “|H|” is resolved by graphical, analytical and numerical techniques indicating that, i. The expression for “dH/dt” is correctly derived. ii. Numerical comparisons of results demonstrate that the approximated value of “η” satisfies boundary conditions such that “TU2 = TU3” and is exactly representative of the present value of CMBR temperature – hence, it is sufficiently accurate for qualitative applications. iii. Determination of various intermediate thermal ordinates demonstrates that “TU2 ≈ TU3” 239

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such that the difference between them at “t1” is “≈ 7(%)”. This indicates that the approximated value of “η” is sufficiently accurate for qualitative applications. iv. Graphical representations of the “TU2,3 vs. |H|” curves [i.e. the “TU2” curve (solid line) is superimposed upon the “TU3” curve (dotted line) – refer to graphs] suggests common characteristics. Thus, illustrating that “TU2 ≈ TU3” and the approximated value of “η” is sufficiently accurate for qualitative applications. v. The graphical representations of “dH/dt, d2H/dt2 and |H|” demonstrate that the rate of the change of the Hubble constant in the time domain is presently positive – indicating that the Universe is “flatly” expanding. NOTES

240

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8.4

Cosmological evolution process

The Cosmological evolution process may be categorised into two regimes, comprised of four distinct periods (i.e. three inflationary and one expansive) as follows, Time Temperature Hubble Constant -1 0 < t < Hα -∞ < TU2 < 0 +∞ > |H| > Hα 0 → Hα-1 -∞ → 0 +∞ → Hα -1 -1 Hα ≤ t < t1 0 ≤ TU2 < TU2(t1 ) Hα ≥ |H| > 0 Hα-1 → t1 0 → TU2(t1-1) Hα → 0 -1 -1 t1 ≤ t < t4 TU2(t1 ) ≥ TU2 > TU2(t4 ) 0 ≤ |H| < √|dHdt[(t4Hα)-1]| t1 → t4 TU2(t1-1) → TU2(t4-1) 0 → √|dHdt[(t4Hα)-1]| -1 t4 ≤ t < AU TU2(t4 ) ≥ TU2 ≥ TU2(HU2) √|dHdt[(t4Hα)-1]| ≥ |H| ≥ HU2 t4 → AU TU2(t4-1) → TU2(HU2) √|dHdt[(t4Hα)-1]| → HU2 Description Primordial Inflation (prior to the “Big-Bang”): the Universe may be described as “inverted and non-physical” such that the interior of the Cosmos existed outside the exterior boundary “RBH” in accordance with the “Primordial Universe” model described previously such that: 1. “TU2” increases from negative infinity to zero. 2. “dHdt” increases from negative infinity to “-Hα2”. 3. “|H|” decreases from positive infinity to “Hα”. Thermal Inflation: the period from the instant of the “Big-Bang” to the instant of maximum Cosmological temperature such that: 4. “TU2” increases from zero to its maximum value “TU2(t1-1)”. 5. “dHdt” increases from “-Hα2” to zero. 6. “|H|” decreases from “Hα” to zero. Hubble Inflation: the period from the instant of maximum Cosmological temperature to the instant of maximum post-primordial “|H|” such that: 7. “TU2” decreases from its maximum value to “TU2(t4-1)”. 8. “dHdt” increases from zero to its maximum physical value “dHdt[(t4Hα)-1]”. 9. “|H|” increases from zero to its maximum physical value “√|dHdt[(t4Hα)-1]|”. Hubble Expansion: the period from the maximum post-primordial “|H|” to the present day such that: 10. “TU2” decreases from “TU2(t4-1)” to “TU2(HU2)”. 11. “dHdt” decreases from its maximum physical value to “HU22”. 12. “|H|” decreases from its maximum physical value to “HU2”. Symbol Definition / Value The EGM Hubble constant at the instant of the “Big-Bang”: Hα ≈ 2.742004 x1042(Hz) ≈ 8.460941 x1061(km/s/Mpc) -Hα2 ≈ -7.518587 x1084(Hz2) ≈ -7.158752 x10123(km/s/Mpc)2 HU2 The present value of the EGM Hubble constant: = HU2(Ro,MG) ≈ 67.084304(km/s/Mpc) HU22 ≈ 4.500304 x103(km/s/Mpc)2 H0 The PDG Hubble constant: ≈ 71(km/s/Mpc) H02 ≈ 5.041 x103(km/s/Mpc)2 Hα-1 The instant of the “Big-Bang”: ≈ 3.646967 x10-43(s) t1 The instant of maximum Cosmological temperature: ≈ 2.206287 x10-42(s) t4 The instant of maximum physical “|H|”: ≈ 2.093267 x10-41(s)

Physical @ {RBH ≥ rS}

Non-Physical @ {RBH < rS}

Period Primordial Inflation Thermal Inflation Hubble Inflation Hubble Expansion Regime

241

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AU TU2(Hα) TU2(t1-1) TU2(t4-1) TU2(HU2) T0 dHdt[(t4Hα)-1] √|dHdt[(t4Hα)-1]| RU 2 Hα ⋅(dHdt[(t4Hα)-1])-1 Hα⋅(√|dHdt[(t4Hα)-1]|)-1

The EGM Cosmological age: = HU2-1 ≈ 14.575885 x109(yr) The EGM Cosmological temperature at the instant of the “Big-Bang”: = 0(K) The Maximum EGM Cosmological temperature: ≈ 3.195518 x1031(K) The EGM Cosmological temperature at the instant of maximum physical “|H|”: ≈ 2.059945 x1031(K) The present EGM Cosmological temperature: = TU3(HU2Hα-1) ≈ 2.724752(K) The present experimentally measured CMBR temperature: ≈ 2.725(K) The approximated maximum rate of change of the physical EGM Hubble constant: ≈ 1.553518 x1084(Hz2) ≈ 1.479167 x10123(km/s/Mpc)2 The approximated maximum physical “|H|”: ≈ 1.246402 x1042(Hz) ≈ 3.845994 x1061(km/s/Mpc) The EGM Cosmological size: = c⋅HU2-1 ≈ 14.575885 x109(Lyr) ≈ 4.839718 ≈ 2.199936 Table 4.10,

Time 0 Hα-1 ≈ 3.646967 x10-43(s) t1 ≈ 2.206287 x10-42(s) t4 ≈ 2.093267 x10-41(s) AU ≈ 14.575885 x109(yr)

TU2 (K) dHdt (km/s/Mpc)2 -∞ -∞ 0 ≈ -7.158752 x10123 0 ≈ 3.195518 x1031 31 ≈ 2.059945 x10 ≈ 1.479167 x10123 ≈ 2.724752 ≈ 4.500304 x103 Table 4.11,

|H|| (km/s/Mpc) +∞ ≈ 8.460941 x1061 0 ≈ 3.845994 x1061 ≈ 67.084304

NOTES

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8.5

History of the Universe

8.5.1 According to the Standard Model (SM) 8.5.1.1 Graphical representation (i)

Figure 4.51, 243

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8.5.1.2 Graphical representation (ii)

Figure 4.52,

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8.5.1.3 Graphical representation (iii)

Figure 4.53,

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8.5.1.4 Graphical representation (iv)

Figure 4.54,

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8.5.2 According to EGM Epoch or Event

Time Domain t

Primordial epoch

Boundary Temperature Value

1

T U2 H α = 0 ( K )

Hα

1

Grand unification epoch

Hα

< t 10

34 .

10

10-34 < t(s) ≤ 10-10

Electroweak / Quark Epoch

1

T U2

(s)

10-10 < t(s) ≤ 102

28

T U2

( K)

(s)

1

T U2 10

Lepton Epoch

. = 1.92400510

34 .

. 15 ( K ) = 3.43308810

10 .

(s)

1

. 9 ( K) = 1.01325410

2.

10 ( s )

Photon Epoch

Universe becomes transparent

102 < t(s) ≤ 1013 1013(s) ≈ 3 x105(yr)

T U2

3 x105 < t(yr) ≤ 109

T U2

1 13 .

= 978.724031( K )

10 ( s )

1 9.

= 11.838588 ( K )

10 ( yr )

109 < t(yr) ≤ 5 x109

First Supernovae

Present Epoch

T U2

5 x109 < t(yr) ≤ 14.58 x109

1 9.

5 .10 ( yr )

= 4.898955 ( K )

T U2 H U2 R o , M G

= 2.724752 ( K )

Table 4.12, T U2

1

T U2 T U2

1 .( day ) 1 31.( day )

T U2 T U2

1 1 .( s )

1 1 .( yr ) 1 2.

T U2

1 3 10 .( yr )

1 4 10 .( yr )

5.

10 ( yr ) 1

T U2

6.

10 ( yr ) T U2 T U2

1

T U2 T U2

. 10 1.84076810 . 3 1.2497710 . 7 2.52413210

521.528169

. 6 3.86401510

147.71262

= 1.00307810 . 6

41.823796

. 4 8.07751510

11.838588

10 ( yr )

. 4 2.29089210

3.35005

1

. 6.49496110

0.947724

7 10 .( yr )

1 8 10 .( yr )

1

T U2

10 ( yr ) T U2

1

T U2

9.

3

( K)

10 10 .( yr )

1 11 10 .( yr )

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8.6

EGM Cosmological construct limitations

8.6.1 Synopsis Any complete physical model requires construct limitations to be clearly defined so as to acknowledge the boundaries of applicability. This section determines the maximum permissible values of Cosmological mass, size and age (i.e. “ML”, “rL” and “tL” respectively) at which the EGM construct remains valid – expressed as, i. MU ≤ M < ML. ii. RU ≤ r < rL. iii. AU ≤ t < tL. 8.6.2 Assumptions •

No new assumptions are asserted.

8.6.3 Construct 8.6.3.1 The mass limit “ML” Utilising “CΩ_J1(r,M)”, we may formulate an estimation for the maximum permissible Cosmological mass “ML” for which the EGM construct remains valid. This is facilitated by considering “ML” to be concentrated at the geometric centre of a spherical Universe, with an observer at its periphery. Hence, CΩ_J1(r1,M1) = CΩ_J1(r2,M2) (4.406) 5

M1

5

M2

26

26

r1 M 2 M 1.

r2

r2 5 5

.

r1

(4.407) r2 r1

(4.408)

Let, “r1 = KλRo”, “r2 = REGM”, “M1 = KmMG” and “M2 = ML” such that the maximum permissible Cosmological mass “ML” is given by, ML

K m.M G.

where, R EGM

R EGM K λ .R o

5 5

.

R EGM K λ .R o

(4.409)

R U K λ .R o , λ x.λ h , K m.M G, m x.m h

(4.410)

8.6.3.2 The size limit “rL” For a SBH of mass “ML”, the maximum permissible Cosmological size “rL” is given by, rL

R BH M L

248

(4.411)

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8.6.3.3 The age limit “tL” The maximum permissible EGM age limit of the Universe “tL” may be determined as follows, tL

rL c

(4.412)

8.6.4 Boundary ratio EGM construct boundary relationships for “ML”, “rL” and “tL” may be expressed in ratio form as follows, Let: “MEGM = MU” and “tEGM = AU” given by, M EGM

t EGM

M U K λ .R o , λ x.λ h , K m.M G, m x.m h

A U K λ .R o , λ x.λ h , K m.M G, m x.m h

(4.413) (4.414)

Hence, M L M EGM rL

tL

R EGM t EGM

(4.415)

8.6.5 Sample calculations Evaluating “ML”, “rL”, “tL” and boundary ratio’s yields, . 71 ( kg ) M L = 4.86482110

(4.416)

. r L = 7.6372910

(4.417)

9 10 .Lyr

19

. 19 109 .yr t L = 7.6372910

(4.418)

ML M EGM rL R EGM

. 18 5.23967510 = 5.23967510 . 18

tL

. 18 5.23967510

t EGM

(4.419)

Notably, tL

. = 1.86196810

6

TL

(4.420)

8.6.6 Concluding remarks The boundaries of Cosmological mass, size and age at which the EGM construct remains valid are given by, i. MU ≤ M < ML. ii. RU ≤ r < rL. iii. AU ≤ t < tL. 249

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8.7

Are conventional radio telescopes, practical tools for gravitational astronomy?

8.7.1 Synopsis A very important question to address is the possibility of utilising conventional Radio Telescopes (RT’s) for application to Gravitational Astronomy (GA). The practicality of this may be determined by consideration of “CΩ_J1” expressed in terms of “ωΩ_3” such that “CΩ_J1 → CΩ_Jω”. Subsequently, if “ωΩ_3” represents the observational Radio Frequency (RF) limit, the required RF Flux Intensity for direct gravitational observation may be determined. 8.7.2 Assumptions •

No new assumptions are asserted.

8.7.3 Construct Substituting the expression for “ωΩ_3” into “CΩ_J1” produces “CΩ_Jω” as follows, 9

9

5

M St J . St J . 26 r

M

5 26

M

St G.

5

2 9

ω Ω_3

(4.421)

9 9

M

St J .

5

M

St G.

2

5

St G

5

.

ω Ω_3

26

5

M

5

2

9

ω Ω_3

(4.422)

9

St J .

M

St J .

26

26 9

26

M

5

ω Ω_3

.

26

St G

M

5

St J .ω Ω_3

9

St J .ω Ω_3

2

26 9 5 .

M

M 52

M

5

26

5 26

.St G

5

52

M

26 9 5 .

26

5

St J .ω Ω_3

5

5

26

.St 5 G

5

9

1

.

27 5

M

26

.St 5 G

1

9

1

5 St J .ω Ω_3 .

27

M

5

26

(4.423)

4

5 4 9 .c .ω Ω 5. 5. .ω Ω_3 St G M 4 .π

.St 5 G

(4.424) 3 5

(4.425)

Recognising that, 26

St J .St G

45

2

9 .c . 9 St G .St G . 4π 4

250

26 45

9 .c . St G 4 .π 4

4 5

(4.426)

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Therefore, 5.2

C Ω_Jω ω Ω_3 , M

4 9 .c . ω Ω_3 4 .π St 0.8 .M 0.6 G

(4.427)

8.7.4 Sample calculations For illustrational purposes only, let “ωΩ_3” equal the value of “ωΩ” at the event horizon of a SMBH, such that “ωΩ_3 = ωΩ_4(1010MS)”. Although the illustrational value of “ωΩ_3” is substantially above the RF range, it clearly demonstrates that the required RF flux intensity for direct gravitational observation far exceeds present capability as follows, 10 10 C Ω_Jω ω Ω_4 10 .M S , 10 .M S = 180.283336( nJy )

(4.428)

where, “nJy” denotes nano-Jansky’s [i.e. “10-9(Jansky)”]. 8.7.5 Concluding remarks It is exceedingly obvious from the illustrational result that conventional RF telescopes are not practical tools for GA. To compound the problem, the signal strength required to be detected from an Earth base telescope, due to amplitude decay by the time it reaches the detector, makes the problem even more difficult (i.e. “CΩ_Jω” will be many orders of magnitude less than computed above). In addition to signal strength detection issues, frequency decay poses another substantial issue to be resolved. By the time the signal has reached the detector, it will be hidden amongst background radiation as noise. Subsequently, the signal would require the appropriate filtration (also considering the conjugate wavefunction pair nature of the signal in accordance with the EGM construct) such that it might “somehow” be identified. Note: the propagation characteristics of gravitational signals are discussed in the proceeding chapter. NOTES

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NOTES

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9 Gravitational Cosmology

Abstract An engineering model is developed to explain how gravitational effects are transmitted through space-time in terms of EGM wavefunction propagation and interference.

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9.1

Gravitational propagation: the mechanism for interaction

9.1.1 Synopsis EGM is a method, not a theory, producing experimentally verified results from the fundamental to the Cosmological scale. This chapter develops an engineering model to explain how gravitational effects are transmitted through space-time. The intention herein is not to advance an explanation (theory) on how gravitational transmission occurs, only to present an engineering approach leading to observationally validated physical results as derived in the preceding chapters. To achieve this, we are compelled to distinguish between physical theory and modelling technique. As stated in the very first sentence, EGM is a method, a tool by which we may attempt to conceptualise space-time manifold effects. The tool has been repeatedly tested against physical observation in order to ensure that, at the very least; the method itself is robust and consistent. One of the most fundamental questions in Physics is the phenomenon of gravity and its propagation. The standard view of this was advanced by Einstein involving the geometry of spacetime curvature. His approach has been highly successful in describing and predicting many astronomical situations and has been rigorously tested by the scientific community. Commonly in engineering solutions, one is not required to understand the physical nature of a specific phenomenon in great detail. Very often, the observed behaviour of a system is modelled in a non-physical way, permitting and facilitating the manipulation and prediction of desired effects for commercial gain. For example, the technique of “discontinuity functions” is often applied to beam loading configurations to avoid the loss of life or property through structural failure. This particular approach to stress analysis is such that physical loads (uniformly distributed and point alike) are represented as discontinuous functions along the beam. The reason for this is because no single equation can model deflections along a beam continuously (other than the simplest situations). Each time the loading situation changes, so does the mathematical equation describing the deflection of the beam. Subsequently, the analysis is “broken-up” into a set of manageable stress sections. From this, one obtains shear force and bending moment diagrams and is able to determine permissible loading boundaries and beam deflections. The significance of the above is that it emphasises the fact that the structural member (the beam in the example given) is modelled and analysed in a manner which is vastly distant from what is physically real. In this case, the engineer is seeking to predict an effect and quantify safe working loads, not necessarily model the Physics of what is happening within the beam in great detail. For a far more detailed analysis, an engineer requires finite difference or element methods. The point of the beam example is to help the reader understand that only the result of a mathematical modelling method is required to agree with physical observation. The technique applied to derive the physically verified result is not necessarily important. Only if the logically derived result disagrees with observation does the mathematical modelled utilised become questionable. If it agrees with physical observation, particularly on a broad scale of application, it defies logic to disregard it in favour of that which cannot achieve comparable results. A second example of a mathematical description being potentially dissimilar to physical reality is an EM wave. The true nature of an EM wave is unknown to contemporary Physics, yet it is considered by many, to be exactly as it appears mathematically (i.e. a sinusoid). However, one should be very careful to draw the distinction between “what physically is” and the tool utilised to describe observed effects and behaviour. Hence, we shall demonstrate that basic engineering Control Theory is a useful tool by which to develop a gravitational propagation model, consistent with the Cosmological results obtained in the preceding chapters. Note: two propagation models are presented herein (i.e. broadband and narrowband), with the key characteristics of the broadband model preserved in the narrowband approximation.

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9.1.2 Construct 9.1.2.1 Broadband The propagation of EGM waves may be represented by two distinct models (i.e. broadband and narrowband). The broadband propagation model refers to the entire PV spectrum surrounding an object, whilst the narrowband propagation model refers to the same spectrum, but usefully approximated to a single wavefunction at a frequency of “ωΩ”. Since the narrowband model is an analogous representation of the broadband model, key characteristics of the broadband model are required to be preserved in the narrowband analogy. In was demonstrated in [5] that, in the case of broadband propagation, the group velocity of a large number of superimposed wavefunctions is zero. However, if one could filter-out all EGM wavefunctions except a specific frequency, an EM signal would be detected. The next important issue to reconcile against the “zero group velocity” behaviour of broadband propagation is how narrowband propagation might work such that it remains consistent with broadband characteristics. A single wavefunction representation of a broadband PV spectrum implies that the evidence of propagation of high frequency gravitational waves from celestial bodies should be clear and obvious – contradicting physical observation. Hence, it shall be demonstrated in the proceeding section that broadband characteristics are preserved in the analogous narrowband approximation by: i. The utilisation of Control System principles to describe EGM wavefunction propagation and space-time curvature in terms of a control loop. ii. Simplifying the constitution of a Graviton to be: a Photon coupled to its ZPF (space-time manifold) response, representing the conjugate EGM wavefunction associated with a Photon pair. Subsequently, the narrowband approximation propagates with characteristics preserving the broadband group velocity condition. iii. Recognising that the conjugate EGM wavefunction associated with a Photon pair (i.e. the ZPF response) may be considered, for practical solution purposes, to be a “Virtual Photon” (VP) popping into existence as a result of the ZPF response to EGM stress. 9.1.2.2 Narrowband EGM considers all masses to be radiators of conjugate wavefunction pairs. That is to say, all mass radiates a spectrum of wavefunctions at frequencies according to “ωPV(1,r,M) ≤ ω ≤ ωΩ(r,M)”: both being dependent upon the objects mass-energy distribution over space-time. At each frequency in the spectrum, wavefunctions are being radiated with positive and negative amplitudes of equal magnitude. This is at the heart of a Fourier representation of any constant function in complex form. Hence, each positive amplitude wavefunction is coupled to its negative amplitude counterpart. If we assume that Electricity, Magnetism and Gravity are unified, then EGM propagation may be graphically represented as follows,

Figure 3.14, 255

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where, the Electric Field Wave is at right angles (orthogonal58) to the Magnetic Field Wave, and the Electro-Gravitic Coupling Wave is at right angles (orthogonal) to the Magneto-Gravitic Coupling Wave. The arrow denotes the orientation of the associated Poynting Vector (i.e. the propagation of energy). A very big and obvious question is “what is the nature of the Gravitic coupling waves?” Simply put, the Gravitic coupling waves are the responses of the space-time manifold to the work being done to it by the Electric and Magnetic Field Waves, consistent with “Newton’s 1st Law of motion” (i.e. for every action, there is an equal and opposite reaction). Explaining EGM propagation may be reduced in complexity by initially considering only the contribution of the Electric Field Wave. That is, we may consider the Electric and Magnetic Field Waves as being independent of each other. As will be shown graphically, the Magnetic Field Wave may then be considered and the “explanation process” repeated. To facilitate this, let “M” denote the mass of an object radiating an Electric Field Wave in accordance with the EGM construct (i.e. not generated in the classical EM wave production manner). This characteristic is conceptualised diagrammatically by the function “G(s)”, existing in the Laplace Domain (i.e. the classical form of representation in Control System Engineering59). Subsequently, “H(s)” denotes the response of the space-time manifold to the Electric Field Wave (i.e. the Electro-Gravitic Coupling Wave).

Figure 4.55: control system representing EGM propagation (illustrational only), Mathematically, this means that “G(s) = -H(s)”, or in other words, the response of the spacetime manifold is equal and opposite of the work being done to it by the Electric Field Wave. Diagrammatically, “G(s) = H(s)” but note that it feeds into the summing junction illustrated as a negative input, producing “G(s) = -H(s)” mathematically. Graphing forcing function and space-time manifold response [i.e. “G(s)” and “H(s)” respectively], for each EM component separately, yields the following two illustrations,

Electric Field

EGM Propagation (Electric Component)

Time

Electric Field Wave (Forcing Function) Space-Time Manifold Electric Response

Figure 4.56: Electric forcing function “G(s)” and its space-time manifold response “H(s)”, 58

http://en.wikipedia.org/wiki/Orthogonal http://en.wikipedia.org/wiki/Control_systems; http://en.wikipedia.org/wiki/Control_engineering; http://en.wikipedia.org/wiki/Transfer_Function 59

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Magnetic Field

EGM Propagation (Magnetic Component)

Time

Magnetic Field Wave (Forcing Function) Space-Time Manifold Magnetic Response

Figure 4.57: Magnetic forcing function “G(s)” and its space-time manifold response “H(s)”, The function “E(s)”, in both cases, represents signal degradation (in the Laplace domain) over distance (in the direction of propagation). A complete and thorough control systems engineering analysis60 utilising Transfer Functions, Characteristic Equations, Root Locus, Nyquist Stability, Bode Plots and Proportional-Integral-Derivative (PID) Controllers etc., is beyond the scope of this text and has been omitted for brevity. 9.1.3 Testing 9.1.3.1 Newtonian It is extremely important to test the analogous narrowband approximation against well established classical and contemporary gravitational models. The testing to be conducted is not intended to replace any widely accepted model, but rather to ensure that key aspects of mainstream Physics are qualitatively contained within the narrowband approximation. Immediately, one can see that the Electric and Magnetic Field Waves mathematically “cancel-out” with respect to their conjugate space-time manifold responses, producing a constant mathematical result of zero force (i.e. action equals reaction), at right angles to the direction of the Poynting Vector61. In terms of EM propagation, the Poynting Vector travels the path of least resistance through the space-time manifold. Hence, key Newtonian aspects are qualitatively (in principle) satisfied by the action-reaction force pairing in the analogous narrowband approximation. 9.1.3.2 Relativistic From a General Relativity (GR) perspective, energy has been deposited into the region by the EM Field Wave and the space-time manifold reacts via the physical manifestation of “spacetime curvature”. The analogous narrowband approximation regards the ZPF as an “infinite store” of available reactive space-time manifold “bending” stress as an EM wave propagates through it. Hence, key aspects of GR are qualitatively (in principle) satisfied by the deposition of energy manifesting as space-time curvature (i.e. “bending” stress) in the analogous narrowband approximation.

60

Suggest reading: Linear Control System Analysis and Design, John J. D’Azzo and Constantine Houpis, Third Edition, 1988, McGraw-Hill. 61 The direction of energy flow: http://en.wikipedia.org/wiki/Poynting_Vector. 257

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9.1.3.3 PV Gravitational acceleration is proportional to the cubic frequency of the PV spectrum62, of the specific mass being considered. As the EM signal degrades, the Poynting Vector diminishes. EGM considers Gravity to be a direct result of the gradient of the Energy Density (realised by the Poynting Vector). The EM field deposits energy into a region, but without a gradient over distance, there would be no change in Energy Density, no change in Poynting Vector and “no Gravity”. Since the Energy Density of the gravitational field surrounding an object is less, farther away from the objects centre of mass than closer to it, the change in Energy Density always acts toward the centre of mass of the object. That is why Gravity always acts downward because the change in Energy Density is always negative. The Electro-Gravitic and Magneto-Gravitic Coupling Waves may each be described as populations of Virtual Photons (VP’s), “popping” into existence from the Zero-Point Vacuum as it seeks a lower state of potential energy in response to work being done to it. That is, in response to it being “bent” by the input of energy, in the case of EGM, a propagating EM wave. In other words, one may consider the ZPF as being an energy sink which is always full. It seeks equilibrium with the applied EM forcing function by “curving” the space-time manifold, thereby producing an EGM wave. Hence, key aspects of the PV model are satisfied by the analogous narrowband approximation. 9.1.4 Concluding remarks The analogous narrowband approximation may be summarised as follows: i. The constitution of a Graviton is simplified to be: a Photon coupled to its ZPF (space-time manifold) response, representing the conjugate EGM wavefunction associated with a Photon pair. Subsequently, the narrowband approximation propagates with characteristics preserving the broadband group velocity condition such that an EGM Wave may be described as an EM Wave coupled to its ZPF (space-time manifold) response. ii. Gravity propagates as EGM Waves with EM characteristics, but remains undetectable unless the ZPF response can be appropriately filtered-out by “a” detector. iii. The conjugate EGM wavefunction associated with a Photon pair (i.e. the ZPF response) may be considered, for practical solution purposes, to be a “Virtual Photon” (VP) popping into existence as a result of the ZPF response to EGM stress, such that the ZPF acts as an infinite store of available reactive space-time manifold “bending” stress as an EM wave propagates through it. iv. The response of the ZPF to an applied forcing function is reactionary (consistent with Newtons “1st” Law of Motion) and equivalent (in principle) to “space-time manifold curvature”. v. The mechanism of the ZPF response may be usefully described by VP’s, propagating “180°-out-of-phase” with respect to an EM forcing function. vi. The EGM Wave may be categorised into two key couplings. That is, the Electric Field Wave couples to its Electro-Gravitic conjugate, whilst the Magnetic Field Wave couples to its Magneto-Gravitic conjugate. vii. The gravitational effect arises from the degradation of the EM Wave Poynting Vector over distance (change in Energy Density) associated with EGM propagation. Note: the wavefunction describing each population of Photon pairs (i.e. a population of Photons and their ZPF response) may be considered to be representative of either side of a Fourier distribution in Complex form, symmetrical about the “0th” mode. 62

A bandwidth of the EGM spectrum. 258

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9.2

Gravitational interference: the mechanism of interaction

9.2.1 Synopsis The gravitational interaction between two bodies may be represented by the EGM construct as (i), a broadband interference pattern or (ii), a narrowband interference pattern such that the entire PV spectrum surrounding each mass is usefully approximated by a single wavefunction at “ωΩ”. We shall illustrate both of these situations by graphical example. Consider the location of zero net acceleration [0(m/s2) - termed the buoyancy point] between the Earth and the Moon with the lunar orbit usefully approximated as being circular. Let, “DE2M”, “r4” and “r5” denote the mean distance from the Earth to the Moon, the mean distance from the centre of mass of the Earth to the buoyancy point and the mean distance from the centre of mass of the Moon to the buoyancy point respectively such that: r4

r 5 D E2M

r5

(4.429)

D E2M

r4

(4.430)

Hence, when accelerations are balanced: G.M E G.M M 2

2

r4

r5

(4.431)

Solving for “r4” yields, r4

D E2M. M M .M E MM

M M .M E

(4.432)

Evaluating produces, r4 r5

=

. 5 3.46028110 . 4 3.83719110

( km)

(4.433)

Calculating “g” at “r4” and “r5” yields, g r 4, M E g r 5, M M

=

. 3 3.33165310

m

. 3 3.33165310

s

2

(4.434)

where, the resultant acceleration is given by, g r 4, M E

g r 5, M M = 0

m s

2

(4.435)

9.2.2 Construct 9.2.2.1 Broadband The broadband interference pattern of the buoyancy point between the Earth and the Moon may be formulated by graphing the harmonics of gravitational acceleration “aPV”. Summing the first “21” modes only (i.e. “nPV = 21”), an approximation of the resultant interference pattern may be represented (illustrational only) utilising the EGM construct as follows, a PV( r , M , t )

i .

C PV n PV, r , M .e

n PV

π .n PV .ω PV( 1 , r , M ) .t .i

(4.436)

where, “CPV” and “nPVωPV(1,r,M)” represent the gravitational amplitude and frequency spectra respectively – mindful that in physical reality, “|nPV| → ∞”. 259

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Harmonic Acc. & Grav. Interference T PV 1 , r 5 , M M

Acceleration

a PV r 4 , M E , t a PV r 5 , M M , t a PV r 4 , M E , t

a PV r 5 , M M , t

t Time

Gravitational Acceleration due to The Earth Gravitational Acceleration due to The Moon Resultant Acceleration (Interference)

Figure 4.58, The graph above only includes the first “21” harmonics. A complete representation would involve summing all modes (i.e. “> 1029” wavefunctions for each celestial object). Subsequently, complete graphical representation at the zero “g” position [“aPV(r4,ME,t)” and “aPV(r5,MM,t)” respectively: producing a resultant acceleration of zero “aPV(r4,ME,t) - aPV(r5,MM,t)”], will appear as flat lines. The resultant magnitude of acceleration curve “aPV(r4,ME,t) - aPV(r5,MM,t)” will run along the x-axis with a value of zero. 9.2.2.2 Narrowband The narrowband representation is formed by usefully approximating the PV spectrum of the gravitational field, as a single wavefunction at “ωΩ”. Firstly, we shall validate that the EGM method produces the correct result with negligible error as follows, a EGM_ωΩ r 4 , M E a EGM_ωΩ r 5 , M M

=

. 3.33165310

3

. 3.33165310

3

m s

2

(4.437)

where, the resultant acceleration is given by, a EGM_ωΩ r 4 , M E

a EGM_ωΩ r 5 , M M = 0

m s

2

(4.438)

Graphing the high frequency harmonic accelerations “ag” of the gravitational interaction of the Earth and the Moon at the buoyancy point (illustrational only), based solely upon “ωΩ” [i.e. “ωΩ(r4,ME)” and “ωΩ(r5,MM)”], produces a visualisation of beats utilising a generalised form according to, a g ( r , M , φ, t )

π g ( r , M ) . .sin 2 .π .ω Ω ( r , M ) .t 2

φ

(4.439)

At “φ = 0”, the average acceleration is given by “gav”,

g av ( r , M )

2 T Ω ( r, M )

1. T Ω ( r, M ) 2 . 0 .( s )

260

a g( r, M , 0, t ) d t

(4.440) www.deltagroupengineering.com

Evaluating yields the correct result as demonstrated by, g av R E, M E = 9.809009

m s

2

(4.441)

such that: ω Ω r 4 , M E = 56.499573 ( YHz)

(4.442)

ω Ω r 5 , M M = 72.138509( YHz)

(4.443)

and, Hence, the conjugate wavefunction acceleration pairs may be illustrated as follows, Conjugate WaveFunction Acc. Pairs

Acceleration

a g r 4, M E, 0, t a g r 4, M E, 0, t a g r 5, M M, π , t a g r 5, M M, π , t

t Time

+ve WaveFunction From The Earth -ve WaveFunction From The Earth +ve WaveFunction From The Moon -ve WaveFunction From The Moon

Figure 4.59, The Moon’s EGM narrowband wavefunction approximation contribution is phase-shifted “180°” (polarized) relative to the Earth because it approaches the zero “g” position from the opposite direction. This is inconsequential because “aPV” is equal to the time averaged magnitude of the curves above.

Acceleration

Conjugate WaveFunction Acc. Beats

a g r 4, M E, φ , t a g r 4, M E, φ , t

a g r 5, M M, π , t a g r 5, M M, π , t

t Time

+ve WaveFunction Interference Beat -ve WaveFunction Interference Beat (Conjugate)

Figure 4.60,

261

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Note: for narrowband representations of non-zero acceleration, one may apply either of two techniques to the approximation, i. Disregard the conjugate wavefunction or, ii. Apply an appropriately adjusted magnitude constraint. 9.2.3 Concluding remarks 9.2.3.1 Broadband EGM interference patterns form when two or more gravitational fields interact. EGM considers all masses to be radiators of conjugate wavefunction pairs. That is to say, all mass radiates a spectrum of wavefunctions at frequencies according to “ωPV(1,r,M) ≤ ω ≤ ωΩ(r,M)”. Each positive amplitude wavefunction is coupled to its negative amplitude counterpart. The total gravitational influence of the wavefunction pair is characterised by the sum of the magnitudes. Gravitational interaction between two bodies may be written mathematically in the time domain as an interference pattern such that “nPV” has the odd number harmonic distribution from “-nΩ(r,M)” to “+nΩ(r,M)” as follows, n PV

n Ω ( r, M ) , 2

n Ω ( r , M ) .. n Ω ( r , M )

(4.444)

The magnitude of “nΩ(r,M)” is extremely important in EGM as it defines the breadth of the double sided EGM spectrum. Without a sufficiently large magnitude of “nΩ(r,M)”, a measurably constant function is not possible. In other words, if “nΩ(r,M)” is too low, then EGM would imply that gravity varies with time (noticeably) at the surface of the Earth and we would all be able to feel this behaviour on our bodies. Therefore, “nΩ(r,M)” is required to be sufficiently large such that “aPV” produces a “flatlined” graph, consistent with human experience of “g” at the surface of the Earth. Fortunately, the EGM method produces extremely large values of “nΩ(r,M)” for all masses, even at the fundamental particle level. Typically, real world values of “1014 to +∞”, depending on the mass being considered. It should be noted that “acceptably constant” graphical behaviour can be observed with values of “nΩ(r,M)” as low as several hundred. The value of “nΩ(r,M)”, as determined by the EGM method, is intimately tied to the massenergy distribution of the object under consideration. For example, the value of “nΩ(r,M)” for free space (zero gravity) is “+∞”. This decreases as the energy density of the space-time manifold increases. In other words, as mass is added to the space-time manifold, the value of “nΩ(r,M)” decreases acting to compresses the local PV spectrum. Note: gravitational interference patterns form due to broadband propagation of EGM wavefunctions at a group velocity of zero. 9.2.3.2 Narrowband A narrowband interference pattern may be produced by approximating the entire PV spectrum surrounding each mass as a single wavefunction, existing as a population of conjugate Photon pairs. Where, one population of Photons propagate with positive amplitude, coupled to a population of Virtual Photons propagating with negative amplitude, such that each population of pairs is polarized “180°” apart (i.e. a pair existing as equal and opposite forms)63. Note: the broadband group velocity condition is preserved in the narrowband approximation.

63

Refer to previous section for an expanded description. 262

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10 Particle Cosmology

Abstract The following characteristics are derived utilising EGM principles: i. The Photon and Graviton mass-energies lower limit. ii. The Photon and Graviton Root-Mean-Square (RMS) charge radii lower limit. iii. The Photon charge threshold. iv. The Photon charge upper limit. v. The Photon charge lower limit.

263

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10.1 Derivation of the Photon and Graviton mass-energies lower limit The lower limit of the Photon and Graviton mass-energies (“mγγ2” and “mgg2” respectively) may be determined trivially by equating “tL” to “TL” as follows, Let, “TL → tL” such that: tL

h m γγ

(4.445)

Therefore: if “mγγ → mγγ2” then, m γγ2

h tL

(4.446)

2 .m γγ2

(4.447)

m gg2

Evaluating yields, m γγ2

1.715978

=

m gg2

10

3.431956

51 .

eV

(4.448)

Hence, m γγ2 < m m γγ

(4.449)

m gg2 < m m gg

(4.450)

10.2 Derivation of the Photon and Graviton RMS charge radii lower limit The lower limit of the Photon and Graviton Root-Mean-Square (RMS) charge radii (“rγγ2” and “rgg2” respectively) may be determined utilising “mγγ2” and the radii relationships derived in [10] as follows, 5

r γγ2

r e.

m γγ2 m e .c

2

2

(4.451)

where, “re” and “me” denote the classical Electron radius and rest mass respectively. Thus, r gg2

5

4 .r γγ2

(4.452)

Evaluating yields, r γγ2 r gg2

=

7.250508 9.567103

10

38 .

m

(4.453)

Hence, r γγ2 < r r γγ

(4.454)

r gg2 < r r gg

(4.455)

10.3 Derivation of the Photon charge threshold The Root-Mean-Square (RMS) charge threshold of a free Photon “Qγ” may be derived in a similar manner to the mass-energy threshold in [8]. Utilising a generalised form for the energy of a Photon “EΩ” [10] propagating at “ωΩ”, “Qγ” may be derived in highly favourable agreement with the PDG estimate [49] as follows, E Ω ( r, M )

h .ω Ω ( r , M )

264

(4.456) www.deltagroupengineering.com

Subsequently, the Photon population at the charge threshold is given by “Nγ”, E Ω ( r, M )

N γ( r, M )

mγ

(4.457)

where, “mγ” denotes the mass-energy threshold. Hence, “Qγ” is given by, Qe

Q γ( r, M )

N γ( r, M )

(4.458) where, “Qe” denotes the Electric charge. Let: “r → rε” and “m → me” such that “Qγ(rε,me) → Qγ” where “rε” denotes the RMS charge radius of the Electron as determined in [9]. Hence, “Qγ” may be evaluated according to, Qγ

= 2.655018 10

30

Qe

(4.459)

30 Q γ < 2.7.10 .Q e

(4.460)

Therefore,

Comparing “Qγ” to the PDG value (i.e. “Qγ_PDG = 5 x10-30Qe”) produces a highly favourable result as follows, Q γ_PDG

= 1.883226

Qγ

(4.461)

10.4 Derivation of the Photon charge upper limit The upper limit of the RMS charge of a free Photon “Qγγ” may be derived trivially by considering the value of the charge threshold in proportion to the Photon population as follows, Q γ( r, M )

Q γγ( r , M )

N γ( r, M )

(4.462)

Subsequently it follows that, Q γ( r, M )

Q γγ ( r , M )

2

Qe

(4.463)

Let: “r → rε” and “m → me” such that “Qγ(rε,me) → Qγ” according to, Q γγ

Qγ

2

Qe

(4.464)

Evaluating yields, Q γγ = 1.129394 10 Q γγ

= 7.049122 10

78 .

C

(4.465)

60

Qe

(4.466)

Therefore, Q γγ 7.05. 10

265

60

.Q e

(4.467)

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10.5 Derivation of the Photon charge lower limit The lower limit of the RMS charge of a free Photon “Qγγ2” may be derived trivially by recognising that the Cosmological age, Photon mass-energy and Photon RMS charge limit ratios are equal as follows, tL

m γγ

Q γγ

T L m γγ2 Q γγ2 m γγ

(4.468)

. = 1.86196810

6

m γγ2

(4.469)

Hence, Q γγ

Q γγ2

m γγ

.m γγ2

(4.470)

Evaluating yields, Q γγ2 = 6.065593 10 Q γγ2

85 .

= 3.785846 10

C

(4.471)

66

Qe

Therefore,

(4.472)

Q γγ2 .Q e < Q Q γγ .Q e

(4.473)

10.6 Other useful relationships i. Relationship: mγ

2

E Ω r ε,me

ω Ω r e, m e .m γγ ω Ω r ε, m e

(4.474)

Verification, mγ

2

E Ω r ε, m e ω Ω r e, m e .m γγ ω Ω r ε, m e

=

1.525768 1.525768

10

46 .

eV

(4.475)

ii. Relationship: E Ω r e, m e

mγ

2

m γγ

(4.476)

Verification, E Ω r e, m e mγ

2

=

0.165603 0.165603

m γγ

( µJ )

(4.477)

iii. Relationship: ω Ω r e, m e

266

mγ

2

h .m γγ

(4.478)

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Verification, ω Ω r e,m e mγ

2

=

h .m γγ

249.926816 249.926816

( YHz)

(4.479)

iv. Relationship: Qe me 2.

=

c Q γγ

. 11 1.7588210

C

198.286288

kg

m γγ

(4.480)

NOTES

267

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NOTES

268

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11 Equation Summary The following is an itemised account of the key relationships derived herein: 11.1 Gravitation 11.1.1 “Stg” 6 3 3 .ω h

St g

13 2 2 .π .c

(4.23)

11.1.2 “ωΩ_2” ω Ω_2( r , M )

9

St g r

.g ( r , M ) 2

(4.25)

11.1.3 “aEGM_ωΩ” r . 9 ω Ω_2( r , M ) St g

a EGM_ωΩ( r , M )

(4.26)

11.1.4 “StG” 3.

St G

2

3 .ω h

9

. c 2

4 .π .h

(4.35)

11.1.5 “ωΩ_3” 9

2

M St G. 5 r

ω Ω_3( r , M )

(4.36)

11.1.6 “λΩ_3” λΩ_3 = c / ωΩ_3 11.1.7 “G” St G

G

St g

(4.37)

11.1.8 “ωPV(nPV,r,M)3” 2 .c .n PV

3

ω PV n PV, r , M

3

π .r

.g ( r , M )

2

(4.41)

11.1.9 “StJ” 9 .c . St G 4 .π 4

St J

2 9

(4.51)

11.1.10 “CΩ_J1, CΩ_Jω” C Ω_J1( r , M )

St J 2

r

9

. M

5

8

r

(4.52) 5.2

C Ω_Jω ω Ω_3 , M

269

4 9 .c . ω Ω_3 4 .π St 0.8 .M 0.6 G

(4.427)

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11.1.11 “nΩ_2” . 3 3 .π m h . r 16 M λh 2

9

n Ω_2( r , M )

7

(4.60)

11.1.12 “KDepp” 1

K Depp ( r , M )

2 .G.M

1

11.1.13 “KPV”

K PV( r , M )

r .c

2

(4.106)

K Depp ( r , M )

2

K Depp ( r , M )

K PV( r , M )

(4.110)

11.1.14 “TL” TL

h m γγ

(4.196)

11.1.15 “ωg” M .c 2 .h

ω g( M )

11.1.16 “ngg”

2

(4.207)

n gg ( M ) T L.ω g ( M )

(4.208)

11.1.17 “rω” 5

St G.

r ω ω Ω_3 , M

M

2

ω Ω_3

9

(4.212)

11.1.18 “aPV” a PV( r , M , t )

i .

C PV n PV, r , M .e

π .n PV .ω PV( 1 , r , M ) .t .i

n PV

(4.436)

11.1.19 “ag” a g ( r , M , φ, t )

π g ( r , M ) . .sin 2 .π .ω Ω ( r , M ) .t 2

φ

(4.439)

11.1.20 “gav” g av ( r , M )

2

1. T Ω ( r, M ) 2 .

T Ω ( r, M )

0 .( s )

a g( r, M , 0, t ) d t

(4.440)

11.2 Planck-Particles 11.2.1 “mx” mx

λx 2

(4.71)

11.2.2 “λx” λx

4 . 2 π 3

6

270

(4.72)

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11.2.3 “ρm, ρS” . 94 kg ρ m λ x.λ h , m x.m h = 1.34467810 3 m

(4.78)

ρS = ρm(λxλh,mxmh) 11.2.4 “r3, M3” r3 = λxλh

(4.245)

M3 = mxmh = λxmh / 2

(4.246)

11.3 SBH’s 11.3.1 “StBH” c .St G

9

St BH

c.

( 2 .G)

5

(4.138)

11.3.2 “ωΩ_4” 3

St BH.

ω Ω_4 M BH

1 M BH

(4.139)

11.3.3 “rS” 3

2 λ x.λ h .R BH

r S R BH

3

r S M BH 3

r S R BH

(4.146)

3 .M BH 4 .π .ρ S

(4.148)

2 3 .c .R BH 8 .π .G.ρ S

(4.150)

11.3.4 “nΩ_4” n Ω_4 M BH

n Ω_2 r S M BH , M BH

(4.157)

n Ω_5 M BH

n Ω_2 R BH M BH , M BH

(4.158)

11.3.5 “nΩ_5” 11.3.6 “nBH” n BH M BH

n Ω_5 M BH n Ω_4 M BH

(4.159)

11.3.7 “ωΩ_5” ω Ω_5 M BH

ω Ω_3 r S M BH , M BH

(4.161)

11.3.8 “ωBH” ω BH M BH

ω Ω_5 M BH ω Ω_4 M BH

(4.162)

11.3.9 “ωΩ_6” ω Ω_6 M BH

ω Ω_5 M BH n Ω_4 M BH

(4.166)

11.3.10 “ωΩ_7” ω Ω_7 M BH

271

ω Ω_4 M BH n Ω_5 M BH

(4.167)

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11.3.11 “ωPV_1” ω Ω_6 M BH

ω PV_1 M BH

ω Ω_7 M BH

(4.168)

11.3.12 “ng” n g ω , M BH

E M BH E g( ω )

(4.177)

11.4 Cosmology 11.4.1 “r2, M2” r2(r) = Kλ⋅r

(4.247)

M2(M) = Km⋅M = Kλ⋅M

(4.248)

11.4.2 “λy” 1

λ y r 2, M 2

ln n Ω_2 r 2 , M 2

(4.229)

11.4.3 “KU” K U r 2, r 3, M 2, M 3

ln

λ y r 2, M 2 .M C Ω_J1 λ y r 2 , M 2 .r 3 , 3 2 C Ω_J1 r 2 , M 2 5

5

K U r 2, r 3, M 2, M 3

1

ln

2

9

(4.231)

7

.ln n Ω_2 r 2 , M 2

3.

M3

26

9

.

M2

r2

9

r3

(4.232)

11.4.4 “AU” A U r 2, r 3, M 2, M 3

11.4.5 “RU”

TL K U r 2, r 3, M 2, M 3

5

(4.233)

R U r 2, r 3 , M 2, M 3

c .A U r 2 , r 3 , M 2 , M 3

H U r 2, r 3, M 2, M 3

1

(4.234)

11.4.6 “HU, HU2, HU5, |H|”

H U5( r , M )

1 . ln TL

( 3 .π )

µ

(4.235)

H U K λ .r , λ x.λ h , K m.M , m x.m h

H U2( r , M )

7 .µ .

A U r 2, r 3 , M 2, M 3

2

32

256

. µ m .ln ( 3 π ) . h 4 M

H

µ

(4.276)

7 .µ

2

. r λh

d H dt

2 7 .µ

5

.

mh M

5 .µ

2

. r λh

2 26 .µ

(4.529)

(4.378)

11.4.7 “Hα” H α r 3, M 3

2.

2. . . π G ρ m r 3, M 3 3

(4.237)

Hα(r3,M3) = ωh / λx 272

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ωh

H α λ x.λ h , m x.m h

λx

(4.249)

11.4.8 “ρU, ρU2” 3 .H U r 2 , r 3 , M 2 , M 3

ρ U r 2, r 3, M 2, M 3

8 .π .G 3 .H U2( r , M )

ρ U2( r , M )

11.4.9 “MU” M U r 2, r 3, M 2, M 3

2

(4.238)

2

8 .π .G

(4.304)

V R U r 2 , r 3 , M 2 , M 3 .ρ U r 2 , r 3 , M 2 , M 3

(4.239)

11.4.10 “KT” K T r 2, r 3 , M 2, M 3

H α r 3, M 3

n g ω Ω_3 r 3 , M 3 , M 3 .ln

H U r 2, r 3, M 2, M 3

8 . H α r 3, M 3 ln 3 H

K T( H )

(4.240)

(4.268)

11.4.11 “TW” T W r 2, r 3, M 2 , M 3

KW λ Ω_3 R U r 2 , r 3 , M 2 , M 3 , M 3

(4.241)

KW

T W( H) λ Ω_3

c λ x. , mh H 2

(4.269)

11.4.12 “StT” 9

4. 3. 1 . λ x 3 4 c5 π .λ 2 h 3

St T

2

(4.274)

11.4.13 “TU, TU2, TU3, TU4, TU5” T U r 2, r 3, M 2, M 3

K T r 2 , r 3 , M 2 , M 3 .T W r 2 , r 3 , M 2 , M 3 ωh

K W .St T .ln

T U2( H )

T U3 H β

9

λ x.H

K W .St T .ln

1 Hβ

. H5

(4.275) . H .H β α

KW c

µ

.ln

. . 4µ H U5( r , M ) λ h Hα

273

2 .µ

.

5 .µ

2

(4.318)

1 T U4( t ) K W .St T .ln H α .t . t

T U5( r , M )

(4.242)

5 .µ

1 π .H α

2

(4.331) 2 .µ

2

. 2

.H ( r , M ) 5 µ U5

(4.530)

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11.4.14 “dTdt, dT2dt2, dT3dt3” K W .St T .

dT dt ( t )

2 5 .ln H α .t .µ

t

dT2 dt2 ( t )

K W .St T .

dT3 dt3 ( t )

.t

(4.335)

2 2 5 .µ . ln H α .t . 5 .µ

t K W .St T .

2

5 .µ

5 .µ

2

1

2

1

.t2

2 2 2 5 .µ .ln H α .t . 5 .µ . 5 .µ

t

1

(4.339)

3 5 .µ

2

2 2 15.µ . 5 .µ

2 .t3

2

2

(4.343)

11.4.15 “dHdt, dH2dt2” 1

t

H γ .H α

Hγ Hβ

dH dt H γ

dH2 dt2 H γ

(4.359)

η

(4.376)

2 H α .H γ . 5 .ln 1 .µ 2 2 Hγ 5 .µ Hγ

1

(4.361)

3 2 H α .H γ . 5 .µ 2 . ln 1 . 5 .µ 2 2 Hγ 5 .µ Hγ

1

2

1

(4.371)

11.4.16 “t1, t2, t3, t4, t5” 1

t1

e

2 5 .µ .

1 Hα

10 .µ

t2

e

2

(4.334) 1

2 2 5 .µ . 5 .µ

1

. 1 Hα

2 2 15 .µ . 5 .µ

t3

e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

(4.338) 2

3

2

. 1 Hα

(4.342)

1

t4

e

2 2 5 .µ . 5 .µ

1

2 2 5 .µ . 5 .µ

t5

e

. 1 Hα 4

2 2 2 5 .µ . 5 .µ . 5 .µ

274

(4.366)

2 1

2

. 1 Hα

(4.375)

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11.5 ZPF 11.5.1 “ΩEGM” Ω EGM

ρ U2 r x5.R o , m g5 .M G ρ U2 R o , M G r x5

1.013403

=

m g5

(4.308)

1.052361

(4.298)

11.5.2 “ΩZPF” Ω ZPF

Ω EGM

1

(4.313)

11.5.3 “UZPF” 3 .c . H U2 R o , M G Ω ZPF . 8 .π .G 2

U ZPF

2

(4.315)

11.6 EGM Construct limits 11.6.1 “ML” ML

K m.M G.

R EGM

5 5

R EGM

.

K λ .R o

R EGM K λ .R o

(4.409)

R U K λ .R o , λ x.λ h , K m.M G, m x.m h

(4.410)

11.6.2 “rL” rL

R BH M L

(4.411)

11.6.3 “tL” rL

tL

c

(4.412)

11.6.4 “ML / rL = MEGM / REGM = tL / tEGM” M L M EGM rL M EGM

t EGM

tL

R EGM t EGM

(4.415)

M U K λ .R o , λ x.λ h , K m.M G, m x.m h

A U K λ .R o , λ x.λ h , K m.M G, m x.m h

(4.413) (4.414)

11.7 Particle-Physics 11.7.1 “mγγ2” m γγ2

11.7.2 “mgg2” m gg2

h tL

(4.446)

2 .m γγ2

(4.447)

11.7.3 “rγγ2” 5

r γγ2

r e.

m γγ2 m e .c

275

2

2

(4.451) www.deltagroupengineering.com

11.7.4 “rgg2” r gg2

11.7.5 “Nγ”

5

4 .r γγ2

(4.452)

E Ω ( r, M )

N γ( r, M )

mγ

(4.457)

11.7.6 “Qγ” Qe

Q γ( r, M )

N γ( r, M )

(4.458)

11.7.7 “Qγγ” Q γ( r, M )

Q γγ( r , M )

N γ( r, M ) Q γ( r, M )

Q γγ ( r , M )

Q γγ

Qγ

(4.462)

2

Qe

(4.463)

2

Qe

(4.464)

11.7.8 “Qγγ2” Q γγ2

Q γγ m γγ

.m γγ2

(4.470)

11.7.9 “tL / TL = mγγ / mγγ2 = Qγγ / Qγγ2” tL

m γγ

Q γγ

T L m γγ2 Q γγ2

(4.468)

11.8 Other useful relationships mγ

2

E Ω r ε,me

ω Ω r e, m e .m γγ ω Ω r ε, m e

E Ω r e, m e

ω Ω r e, m e

276

mγ

(4.474)

2

m γγ mγ

(4.476) 2

h .m γγ

(4.478)

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11.9 Quick symbol guide Symbol aEGM_ωΩ ωΩ ag aPV AU CΩ_J1 CΩ_Jωω dH2dt2 dHdt dT2dt2 dT3dt3 dTdt gav H H0 HU HU2 HU5 Hα Hβ Hγ KDepp KT KU KW M1 M2 M3 MBH MEGM mg5 mgg2 ML MU mx mγγ2 γγ nBH ng ngg nΩ_2 nΩ_4 nΩ_5 Nγ Qγ Qγγ Qγγ2 γγ r1

Description Gravitational acceleration utilising ωΩ_2 High frequency harmonic acceleration Gravitational acceleration harmonic EGM Cosmological age (present value) Non-refractive form of CΩ_J Equal to CΩ_J 2nd time derivative of H 1st time derivative of H 2nd time derivative of TU4 3rd time derivative of TU4 1st time derivative of TU4 Average high frequency harmonic acceleration Generalised reference to the Hubble constant Hubble constant (present value) EGM Hubble constant Transformed representation of HU Simplest functionally dependent form of HU Primordial Hubble constant Dimensionless range variable Dimensionless range variable Refractive Index of PV by Depp Expansive scaling factor rf / ri Wien displacement constant: 2.8977685 x10-3 [35] Generalised mass Generalised mass Generalised mass or mh Mass of a SBH Convenient form of MU Computational pre-factor Graviton mass-energy lower limit EGM Cosmological mass limit Total EGM Cosmological mass 2nd SPBH constant Photon mass-energy lower limit Harmonic cut-off mode ratio (nΩ_5 : nΩ_4) Average number of Gravitons radiated by a SBH per TΩ_4 period Population of Gravitons within starving matter Transformed representation of nΩ_1 nΩ_1 at the periphery of a SBH singularity nΩ_1 at the event horizon of a SBH Photon population at Qγ Photon RMS charge threshold by EGM Photon RMS charge upper limit by EGM Photon RMS charge lower limit by EGM Generalised radial displacement 277

Units m/s2

s Jy (Jansky) Hz3 Hz2 K/s2 K/s3 K/s m/s2 Hz

mK kg

eV kg

eV

C

m

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r2 r3 RBH REGM rgg2 rL Ro rS RU rx5 rγγ2 γγ rω Stg StG StJ T0 t1 t2 t3 t4 t5 tEGM TL tL TU TU2 TU3 TU4 TU5 TW UZPF ΩEGM ΩZPF η λx λy λΩ_3 µ ρm ρS ρU ρU2 ωBH ωg ωPV_1

Generalised radial displacement m Generalised radial displacement or λh Radius of the event horizon of a SBH Convenient form of RU Graviton RMS charge radius lower limit EGM Cosmological size limit Distance from the Sun to the Galactic centre pc (parsec) Singularity radius m EGM Cosmological size (present value) Computational pre-factor Photon RMS charge radius lower limit m Distance from the centre of mass of a celestial object to the Earth 1st EGM gravitational constant: Stg = 1.828935 x10245 m-1s-5 nd 224 2 EGM gravitational constant: StG = 8.146982 x10 m5kg-2s-9 3rd EGM gravitational constant: StJ9 = 1.093567 x10-146(kg4m26/s18) (kg4m26/s18)(1/9) CMBR temperature (present value) K s • Temporal ordinate (local maxima) of the CMBR temperature • The instant of maximum Cosmological temperature Temporal ordinate (local minima) of the 1st time derivative of the CMBR temperature Temporal ordinate (local maxima) of the 2nd time derivative of the CMBR temperature • Temporal ordinate (local maxima) of the 1st time derivative of H • The instant of maximum physical EGM Hubble constant Temporal ordinate (local minima) of the 2nd time derivative of H Convenient form of AU Minimum gravitational lifetime of matter EGM Cosmological age limit CMBR temperature by the EGM method K Transformed representation of TU Transformed representation of TU2 Transformed representation of TU3 Simplest functionally dependent form of TU Thermodynamic scaling factor ZPF energy density threshold Pa Net Cosmological density parameter as defined by the EGM method ZPE contribution to the net Cosmological density parameter Computed index 1st SPBH constant Generalised representation of λx m c / ωΩ_3 Indicial constant (µ = 1 / 3) Mass density kg/m3 Mass density of a SPBH EGM Cosmological mass-density Transformed representation of ρU Harmonic cut-off frequency ratio (ωΩ_5 : ωΩ_4) Average Graviton emission frequency (1 / Tg) Hz Fundamental harmonic frequency ratio (ωΩ_6 : ωΩ_7) 278

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ωΩ_2 ωΩ_3 ωΩ_4 ωΩ_5 ωΩ_6 ωΩ_7

Gravitational acceleration form of ωΩ_1 Transformed representation of ωΩ_1 ωΩ_1 at the event horizon of a SBH ωΩ_1 at the periphery of a SBH singularity ωPV(1,r,MBH) at the periphery of a SBH singularity: r = rS(MBH) ωPV(1,r,MBH) at the event horizon of a SBH: r = RBH(MBH)

Hz

NOTES

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NOTES

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APPENDIX 4.A Thermodynamic “Π Π” Groupings of BH’s The temperature of a BH is given by (“κ” denotes Boltzmann’s constant [35]), h .c

T BH

3

2 16.π .κ .G.M BH

(4.481)

This may be represented in “Π” form as follows h .c

3

1

2 2 16.π .κ .G.M BH ( 4 .π ) .M BH

2 . . c . hc G κ

(4.482)

Recognising that “c2 / G = mh / λh”, “ωh = c / λh” and64 “Th = mhc2 / κ = hωh / κ” yields, 1 2 ( 4 .π ) .M BH

mh 2 ( 4 .π ) .M BH

mh

2 . . c . hc G κ

. . hc . κ λh .

mh 2 ( 4 .π ) .M BH

mh

.

2 ( 4 .π ) .M BH

h .ω h

T h .m h

κ

2 ( 4 .π ) .M BH

2 ( 4 .π ) .M BH

Hence, T BH

. . hc κ .λ h

(4.483)

h .ω h κ

(4.484)

(4.485)

T h .m h 2 ( 4 .π ) .M BH

(4.486)

Therefore, the thermodynamic “Π” representation of BH temperature is given by, Th

.

T BH

mh

( 4 .π )

2

M BH

(4.487)

Note: it is a personal preference of the author, never to apply the “h-bar” form of Planck’s Constant. Conventional calculation of SPBH temperature “TBH” The EGM construct identifies that the “Primordial Universe” may be modelled as a SPBH of mass “mxmh”. Hence, substituting “mxmh = λxmh / 2” into the “Π” form of BH temperature yields, Th T BH

.

mh m x.m h

Th

. 1 T BH m x

Th T BH

. 2 λx

2 ( 4 .π )

(4.488)

64

Planck temperature = Planck energy divided by “κ”. http://www.planck.com/plancktemperature.htm 281

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Th

2 ( 4 .π ) .λ x

T BH

2

2 8 .π .λ x

(4.489)

Therefore, the conventional representation of the temperature of a SPBH is given by, Th

T BH m x.m h

2 8 .π .λ x

(4.490)

Evaluating yields, Th

. = 1.66667410

30

( K)

2

8 .π .λ x

(4.491)

“TU2 : TBH” The ratio between “TBH” and the maximum value of “TU2” since the “Big-Bang” (i.e. at “t1”) may be determined numerically, leading to a simple relationship between them [i.e. an approximation to within “1.72(%)”] as follows, T U2

1 t1

= 19.173025

T BH m x.m h

(4.492)

Hence, T U2

1

6 .π .T BH m x.m h

t1

(4.493)

Therefore by approximation, T U2

1

3 .T h

t1

4 .π .λ x

(4.494)

Evaluating the difference yields, T U2

1 t1

6 .π .T BH m x.m h

1 = 1.716054 ( % )

(4.495)

Approximations of “TU2(t1-1)” •

“1st” Form

The peak value of Cosmological temperature may be usefully approximated [i.e. to within 0.163(%)] by applying Dimensional Analysis Techniques (DAT’s) to the gravitational Poynting Vector “SωΩ”, yielding the dimensional limit of the Cosmological temperature “TSPBH” (“σ” denotes the Stefan-Boltzmann constant [35]) as follows, 4

T SPBH

S ωΩ λ x.λ h , m x.m h σ

(4.496)

Evaluating yields, . T SPBH = 5.02766910

31

( K)

(4.497)

Comparing the result to the conventional BH form produces,

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T SPBH T BH m x.m h

= 30.165887

(4.498)

Comparing the result to the precise EGM form produces, T SPBH T U2

= 1.57335

1 t1

(4.499)

Therefore by approximation, T U2

1

3 K ω .T SPBH

t1

(4.500)

Evaluating, 3 . 31 ( K ) K ω .T SPBH = 3.20071410

(4.501)

Hence, the difference yields, 3 K ω .T SPBH

T U2

1 = 0.162602 ( % )

1 t1

(4.502)

Since the difference between forms is small [i.e. “< 0.163(%)”], we may usefully approximate “TU2(t-1)” according to, 4 3.

K ω T SPBH K ω

3 3 .m x.m h .c

3.

4 .π .σ . λ x.λ h 4

4 3 Kω .

4 .π .σ . λ x.λ h

4

3. Kω

3.

3.

2

3

3 .m h .c

3.

Kω

3 3 .c . ω h 8 .π .G.σ λ x

Kω

3.

4

2

Kω

4

.m .c3 h

3 .m h .c

3.

3 3 .c . ω h 8 .π .G.σ λ x

3 2 3 .c .H α 8 .π .G.σ

3

(4.504)

3

2 3 8 .π .σ .λ x .λ h 4

3

λx

4 .π .σ . λ x.λ h

4 3.

(4.503)

2

3 Kω .

.m .c3 h

2 3 8 .π .σ .λ x .λ h

4

Kω

λx

3

4 .π .σ . λ x.λ h 4

Kω

3.

3 3 .m x.m h .c

3

(4.505)

2

(4.506)

(4.507)

4

3 2 3 2 3 .c .H α 2 3 .c .H α 3. . Kω 8 .π .G.σ π 8 .π .G.σ 4

(4.508)

4

2 2 . 3. . 3. 2. 3 c Hα 1. 6 c Hα π 8 .π .G.σ π π .G.σ

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(4.509) www.deltagroupengineering.com

4

2 . 3. 1. 6 c Hα π π .G.σ

Hα

4

6 .c π .σ .G 3

.

π

(4.510)

Therefore by approximation, T U2

1

Hα

t1

π

4

.

6 .c π .σ .G 3

(4.511)

Checking the result confirms the simplification, Hα

4

.

π

•

6 .c . 31 ( K ) = 3.20071410 π .σ .G 3

(4.512)

“2nd” Form

“TU2(t-1)” may be approximated explicitly in terms of physical constants as follows (to within “0.249(%)” where, “KW” denotes Wien’s displacement constant [35]), T U2

2 c . KW 5 G.κ

1 t1

(4.513)

Evaluating, 2 c . KW . 31 ( K ) = 3.18758510 5 G.κ

(4.514)

Hence, the approximation error is, T U2

1

1 t1

2 KW .c . 5 G.κ

1 = 0.248248 ( % )

(4.515)

Approximation of “λ λx” in terms of physical constants The value of “λx” may be usefully approximated in terms of physical constants to within “1.45(%)” as follows, 3 .T h

2 c . KW 4 .π .λ x 5 G.κ

(4.516)

Hence by approximation, λx

15.T h

. . Gκ KW 4 .π .c 2

(4.517)

Simplifying yields, m .c . h 15 15.T h G.κ κ . G.κ . 2 2 KW KW 4 .π .c 4 .π .c

(4.518)

m h .c 15. . κ . G.κ 15 m h . G.κ 2 K W 4 .π .κ K W 4 .π .c

(4.519)

2

2

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15.m h

15.m h . G . Gκ . 4 .π .κ K W 4 .π κ .K W

15.m h 4 .π

.

(4.520)

15 . h .c . G κ .K W 4 .π G κ .K W G

(4.521)

15 . h .c . G 15 . h .c 4 .π G κ .K W 4 .π κ .K W

(4.522)

Therefore by approximation, λx

15 . h .c . . 4 π κ KW

(4.523)

Evaluating, 15 . h .c = 2.659782 4 .π κ .K W

(4.524)

1 . 15 . h .c . . λx 4 π κ KW

(4.525)

Hence, the approximation error is, 1 = 1.442436 ( % )

Physical interpretation of “λ λx” A physical interpretation of “λx” is possible utilising the Stefan-Boltzmann Law where, “Φ” denotes the energy flux emitted from a “Black-Body” at temperature “T” according to, Φ σ .T

4

(4.526)

Subsequently, if we equate the Stefan-Boltzmann Law to the peak average Cosmological temperature (correlating to the approximated temperature of a SPBH), a physical interpretation of “λx” is possible as follows, Φ σ.

3 .T h

4

4 .π .λ x

(4.527) th

Therefore by inspection, “λx” is proportional to the “4 power-root” of the energy flux of the Universe at the peak average Cosmological temperature, in accordance with the Stefan-Boltzmann Law shown by, λx ∝

285

4

1 Φ

(4.528)

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Bibliography 4 Note: [1 - 19] refer to: http://stores.lulu.com/dge; “Riccardo C. Storti”, Quinta Essentia: A Practical Guide to Space-Time Engineering, Part 3, Metric Engineering & The Quasi-Unification of ParticlePhysics. [1] Ch. 3.1, Dimensional Analysis. [2] Ch. 3.2, General Modelling and the Critical Factor. [3] Ch. 3.3, The Engineered Metric. [4] Ch. 3.4, Amplitude and Frequency Spectra. [5] Ch. 3.5, General Similarity. [6] Ch. 3.6, Harmonic and Spectral Similarity. [7] Ch. 3.7, The Casimir Effect. [8] Ch. 3.8, Derivation of the Photon Mass-Energy Threshold. [9] Ch. 3.9, Derivation of Fundamental Particle Radii (Electron, Proton and Neutron). [10] Ch. 3.10, Derivation of the Photon and Graviton Mass-Energies and Radii. [11] Ch. 3.11, Derivation of Lepton Radii. [12] Ch. 3.12, Derivation of Quark and Boson Mass-Energies and Radii. [13] Ch. 3.13, The Planck Scale, Photons, Predicting New Particles and Designing an Experiment to Test the Negative Energy Conjecture. [14] App. 3.G, Derivation of ElectroMagnetic Radii. [15] App. 3.H, Calculation of L2, L3 and L5 Associated Neutrino Radii. [16] App. 3.I, Derivation of the Hydrogen Atom Spectrum (Balmer Series) and an Experimentally Implicit Definition of the Bohr Radius. [17] App. 3.K, Numerical Simulations, MathCad 8 Professional, Complete Simulation. [18] App. 3.L, Numerical Simulations, MathCad 8 Professional, Calculation Engine. [19] App. 3.M, Numerical Simulations, MathCad 12, High Precision Calculation Results. [20] http://pdg.lbl.gov/2006/reviews/astrorpp.pdf [21] http://zebu.uoregon.edu/~imamura/123/lecture-2/mass.html [22] http://pdg.lbl.gov/2006/reviews/hubblerpp.pdf (pg. 20 - “WMAP + All”). [23] http://map.gsfc.nasa.gov/m_mm.html [24] http://en.wikipedia.org/wiki/Main_Page [25] “Alfonso Rueda, Bernard Haisch”, Contribution to inertial mass by reaction of the vacuum to accelerated motion, Found.Phys. 28 (1998) 1057-1108: http://arxiv.org/abs/physics/9802030v1 [26] “Alfonso Rueda, Bernard Haisch”, Inertia as reaction of the vacuum to accelerated motion, Phys.Lett. A240 (1998) 115-126: http://arxiv.org/abs/physics/9802031v1 [27] “Bernard Haisch, Alfonso Rueda, Hal Puthoff”, Advances in the proposed ElectroMagnetic zero-point field theory of inertia, presentation at 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, July 13-15, 1998, Cleveland, OH, 10 pages: http://arxiv.org/abs/physics/9807023v2 [28] “Puthoff et. Al.”, Polarizable-Vacuum (PV) approach to general relativity, Found. Phys. 32, 927 - 943 (2002): http://xxx.lanl.gov/abs/gr-qc/9909037v2 [29] Particle Data Group, Photon Mass-Energy Threshold: “S. Eidelman et Al.” Phys. Lett. B 592, 1 (2004): http://pdg.lbl.gov/2006/listings/s000.pdf [30] The SELEX Collaboration, Measurement of the Σ- Charge Radius by Σ- - Electron Elastic Scattering, Phys.Lett. B522 (2001) 233-239: http://arxiv.org/abs/hep-ex/0106053v2 [31] “Karmanov et. Al.”, On Calculation of the Neutron Charge Radius, Contribution to the Third International Conference on Perspectives in Hadronic Physics, Trieste, Italy, 7-11 May 2001, Nucl. Phys. A699 (2002) 148-151: http://arxiv.org/abs/hep-ph/0106349v1 [32] “P. W. Milonni”, The Quantum Vacuum – An Introduction to Quantum Electrodynamics, Academic Press, Inc. 1994. Page 403. [33] Mathworld, http://mathworld.wolfram.com/Euler-MascheroniConstant.html [34] “Hirsch et. Al.”, Bounds on the tau and muon Neutrino vector and axial vector charge radius, Phys. Rev. D67: http://arxiv.org/abs/hep-ph/0210137v2 287

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[35] National Institute of Standards and Technology (NIST): http://physics.nist.gov/cuu/ [36] The D-ZERO Collaboration, A Precision Measurement of the Mass of the Top Quark, Nature 429 (2004) 638-642: http://arxiv.org/abs/hep-ex/0406031v1 [37] Progress in Top Quark Physics (Evelyn Thomson): Conference proceedings for PANIC05, Particles & Nuclei International Conference, Santa Fe, New Mexico (USA), October 24 – 28, 2005. http://arxiv.org/abs/hep-ex/0602024v1 [38] Combination of CDF and D0 Results on the Mass of the Top Quark, Fermilab-TM-2347-E, TEVEWWG/top 2006/01, CDF-8162, D0-5064: http://arxiv.org/abs/hep-ex/0603039v1 [39] “Hammer and Meißner et. Al”., Updated dispersion-theoretical analysis of the nucleon ElectroMagnetic form factors, Eur. Phys.J. A20 (2004) 469-473: http://arxiv.org/abs/hep-ph/0312081v3 [40] “Hammer et. Al”, Nucleon Form Factors in Dispersion Theory, invited talk at the Symposium "20 Years of Physics at the Mainz Microtron MAMI", October 20-22, 2005, Mainz, Germany, HISKP-TH-05/25: http://arxiv.org/abs/hep-ph/0602121v1 [41] Spectrum of the Hydrogen Atom, University of Tel Aviv. http://www.tau.ac.il/~phchlab/experiments/hydrogen/balmer.htm [42] The CDF & D0 Collaborations, W Mass & Properties, FERMILAB-CONF-05-507-E. http://arxiv.org/abs/hep-ex/0511039v1 [43] The L3 Collaboration, Measurement of the Mass and the Width of the W Boson at LEP, Eur. Phys.J. C45 (2006) 569-587: http://arxiv.org/abs/hep-ex/0511049v1 [44] The ALEPH, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak Working Group, the SLD Electroweak & Heavy Flavor Groups, Precision Electroweak Measurements on the Z Resonance, CERN-PH-EP/2005-041, SLAC-R-774: http://arxiv.org/abs/hep-ex/0509008v3 [45] The ZEUS Collaboration, Search for contact interactions, large extra dimensions and finite quark radius in ep collisions at HERA, Phys. Lett. B591 (2004) 23-41: http://arxiv.org/abs/hep-ex/0401009v2 [46] “James William Rohlf”, Modern Physics from α to Z, John Wiley & Sons, Inc. 1994. [47] “Joseph Depp”, Polarizable Vacuum and the Schwarzschild Solution. [48] Scienceworld, http://scienceworld.wolfram.com/ [49] http://pdg.lbl.gov/2006/listings/s000.pdf Other useful references: [50] “J. F. Douglas”, Solving Problems in Fluid Mechanics, Vol. 2, Third Edition, Longman Scientific & Technical, ISBN 0-470-20776-0 (USA only), 1986. [51] Software: MathCad 8 Professional, http://www.mathsoft.com/ [52] University of Illinois, http://archive.ncsa.uiuc.edu/Cyberia/NumRel/mathmine1.html [53] http://en.wikipedia.org/wiki/Dimensional_analysis [54] http://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem [55] University of California, Riverside, http://math.ucr.edu/home/baez/physics/Quantum/casimir.html [56] Georgia State University, http://hyperphysics.phy-astr.gsu.edu/hbase/forces/exchg.html [57] “B.S. Massey”, Mechanics of Fluids sixth edition, Van Nostrand Reinhold (International), 1989, Ch. 9. [58] “Rogers & Mayhew”, Engineering Thermodynamics Work & Heat Transfer third edition, Longman Scientific & Technical, 1980, Part IV, Ch. 22. [59] “Douglas, Gasiorek, Swaffield”, Fluid Mechanics second edition, Longman Scientific & Technical, 1987, Part VII, Ch. 25. [60] “Erwin Kreyszig”, Advanced Engineering Mathematics Seventh Edition, John Wiley & Sons, 1993, Ch. 10. [61] “R.H. Dicke”, Gravitation without a principle of equivalence. Rev. Mod. Phys. 29, 363 – 376, 1957. 288

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[62] “R.H. Dicke”, Mach’s principle and equivalence, in Proc. Of the International School of Physics “Enrico Fermi” Course XX, Evidence for Gravitational Theories, ed. C. Møller, Academic Press, New York, 1961, pp. 1 – 49. [63] “Puthoff et. Al.”, Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight, JBIS, Vol. 55, pp.137, 2002, http://xxx.lanl.gov/abs/astro-ph/0107316v2 [64] “K.A. Stroud”, “Further Engineering Mathematics”, MacMillan Education LTD, Camelot Press LTD, 1986, Programme 17. [65] “Lennart Rade, Bertil Westergren”, “Beta Mathematics Handbook Second Edition”, ChartwellBratt Ltd, 1990, Page 470. [66] Scienceworld, http://scienceworld.wolfram.com/physics/BeatFrequency.html [67] Scienceworld, http://scienceworld.wolfram.com/physics/MaxwellEquationsSteadyState.html [68] Scienceworld, http://scienceworld.wolfram.com/physics/ElectromagneticRadiation.html [69] Georgia State University, http://hyperphysics.phy-astr.gsu.edu/hbase/ems1.html [70] Scienceworld, http://scienceworld.wolfram.com/physics/MaxwellEquations.html [71] http://www.mathcentre.ac.uk/students.php/all_subjects/series [72] Georgia State University, http://hyperphysics.phy-astr.gsu.edu/hbase/astro/whdwar.html [73] Georgia State University, http://hyperphysics.phy-astr.gsu.edu/hbase/astro/redgia.html [74] Georgia State University, http://hyperphysics.phy-astr.gsu.edu/hbase/astro/pulsar.html [75] “Stein, B. P”. Physics Update, Physics Today 48, 9, Oct. 1995. [76] “Simon et Al.”, Nucl. Phys. A333, 381 (1980). [77] Scienceworld, http://scienceworld.wolfram.com/physics/Proton.html [78] “Andrews et Al.”, 1977 J. Phys. G: Nucl. Phys. 3 L91 – L92. [79] “L.N. Hand, D.G. Miller, and R. Wilson”, Rev. Mod. Phys. 35, 335 (1963). [80] Stanford Linear Accelerator, http://www.slac.stanford.edu/ http://www2.slac.stanford.edu/vvc/theory/quarks.html [81] Scienceworld, http://scienceworld.wolfram.com/physics/PlanckLength.html [82] Scienceworld, http://scienceworld.wolfram.com/physics/Photon.html [83] Stanford Linear Accelerator, http://www2.slac.stanford.edu/vvc/theory/fundamental.html [84] “Joshipura et. Al.”, Bounds on the tau Neutrino magnetic moment and charge radius from Super-K and SNO observations, http://arxiv.org/abs/hep-ph/0108018v1 [85] Scienceworld, http://scienceworld.wolfram.com/physics/BohrRadius.html [86] “Jean Baptiste Joseph Fourier”, http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fourier.html [87] http://stores.lulu.com/dge [88] http://www.veoh.com/users/DeltaGroupEngineering [89] http://www.deltagroupengineering.com/Docs/QE3_-_Summary.pdf [90] http://www.deltagroupengineering.com/Docs/QE3_-_Calculation_Engine.pdf [91] http://www.deltagroupengineering.com/Docs/QE3_-_High_Precision_(MCAD12).pdf [92] http://www-cdf.fnal.gov/physics/new/top/top.html#PAIR [93] Cornell University Library: http://www.arxiv.org/

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APPENDIX 4.B Note: “Quinta Essentia – Part 4” is a companion to “Quinta Essentia – Part 3”. Subsequently, the simulation and calculation engine in “Part 4” is a natural extension of “Part 3”, utilising it as a foundational construct. Hence, the calculation engine developed in “Part 3” has been included (verbatim) herein for reference. Please consult “Part 3” if required. Quinta Essentia – Part 3 •

MathCad 8 Professional: calculation engine a. Computational environment

NOTE: KNOWLEDGE OF MATHCAD IS REQUIRED AND ASSUMED • • •

Convergence Tolerance (TOL): 0.001. Constraint Tolerance (CTOL): 0.001. Calculation Display Tolerance: 6 figures – unless otherwise indicated. b. Units of measure (definitions)

Scale 1

10

Scale 2

10

3

3

10 6

10

6

10

9

9

10

10

12

10 12

10 15

10

15

10

18

10

( mm µm nm pm fm am zm ym )

18

10

21

10

10

10

24

24

Scale 1 .( m)

( mHz µHz nHz pHz fHz aHz zHz yHz ) ( mJ µJ nJ pJ fJ aJ zJ yJ )

21

Scale 1 .( Hz)

Scale 1 .( J ) Scale 1 .( W )

( mW µW nW pW fW aW zW yW ) ( mΩ µΩ nΩ pΩ fΩ aΩ zΩ yΩ ) ( mV µV nV pV fV aV zV yV )

Scale 1 .( ohm )

Scale 1 .( V)

( mPa µPa nPa pPa fPa aPa zPa yPa ) ( mT µT nT pT fT aT zT yT )

Scale 1 .( T )

( mNs µNs nNs pNs fNs aNs zNs yNs ) ( mN µN nN pN fN aN zN yN )

Scale 1 .( Pa )

Scale 1 .( Ns )

Scale 1 .( newton )

( mgs µgs ngs pgs fgs ags zgs ygs )

Scale 1 .( gauss )

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Scale 1 .( gm )

( mgm µgm ngm pgm fgm agm zgm ygm ) ( mSt µSt nSt pSt fSt aSt zSt ySt )

Scale 1

( kSt MSt GSt TSt PSt ESt ZSt YSt )

Scale 2

( kHz MHz GHz THz PHz EHz ZHz YHz)

Scale 2 .( Hz)

Scale 2 .( newton )

( kN MN GN TN PN EN ZN YN )

Scale 2 .( J )

( kJ MJ GJ TJ PJ EJ ZJ YJ )

Scale 2 .( W )

( kW MW GW TW PW EW ZW YW )

Scale 2 .( ohm )

( kΩ MΩ GΩ TΩ PΩ EΩ ZΩ YΩ )

Scale 2 .( V)

( kV MV GV TV PV EV ZV YV)

Scale 2 .( Pa )

( kPa MPa GPa TPa PPa EPa ZPa YPa )

Scale 2 .( T )

( kT MT GT TT PT ET ZT YT )

( keV MeV GeV TeV PeV EeV ZeV YeV)

Scale 2 .( eV)

Ns newton .s

c. Constants (definitions)

G

ε0

α

. 6.674210

3

m

11 .

kg .s

. 8.85418781710

2

12 .

F

c

m 299792458. s

h

. 6.626069310

µ0

34 .

( J .s )

7 newton 4 .π .10 . 2 A

. eV 1.6021765310

19 .

( J)

m . 7.29735256810

3

. 1.6021765310

Qe

19 .

( C)

γ

0.5772156649015328

d. Fundamental particle characteristics (definitions or initialisation values) m e m p m n m µ m τ m AMC

. 31 1.6726217110 . 27 1.6749272810 . 27 1.883531410 . 28 3.1677710 . 27 1.6605388610 . 27 .( kg ) 9.109382610

λ Ce λ CP λ CN λ Cµ λ Cτ

ω Ce ω CP ω CN ω Cµ ω Cτ

h. 1 c

1

1

1

1

me mp mn mµ mτ 2 2 .π .c .

h

me mp mn mµ mτ

292

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eV 6 6 3 0.19.10 18.2.10 . 2 c

m en m µn m τn

Note: for the Bottom Quark, the “SLAC” estimate is utilised initially. 4 .10

m uq m dq m sq m cq m bq m tq

( 80.425 91.1876 114.4 ) .

mW mZ mH

GeV 0.13 1.35 4.7 179.4 . 2 c

2

( 2.817940325 0.875 0.85 ) .( fm)

re rp rn 0.85.10

3

GeV c

r xq

8 .10

3

16 .

( cm)

. 0.529177210810

r Bohr

10

.( nm ) 656.469624182052

λB

( m)

e. Planck characteristics (definitions) G.h

λh

c

3

h .c

mh

G.h

th

G

c

1

ωh

5

th

f. Astronomical statistics MM ME MJ MS

5 1738 6377.18 71492 6.96.10 .( km)

RM RE RJ RS 2 c .R E 2 .G

M BH

200.R S

R RG

24 24 24 30 0.0735.10 5.977.10 1898.8.10 1.989.10 .( kg )

R BH

2 .G . M BH 2 c

M NS

1 .M S

R NS

M RG

4 .M S

R WD

4200.( km)

M WD

20.( km)

3 300.10 .M E

g. Other .10 M BH = 4.29379067958471

33

( kg )

mx

mp

rx

r Bohr

h. Arbitrary values for illustrational purposes ω

KR

1 .( Hz)

1

k

1

R max

X

4 10 .( km)

r

1

∆R max

RE

F 0( k )

1

K 0( ω , X )

1

R max 250

293

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i. PV / ZPF equations 2.

K PV( r , M )

e

G .M

3

2 r .c

K 0( r , M )

K PV( r , M ) .e

K EGM_N( r , M )

K PV( r , M )

2

2 . ∆K 0( r , M )

G.M . KR 2 r .c

∆K 0( r , M )

K PV( r , M )

K EGM_E( r , M )

e

G.M .

C PV n PV, r , M

2

r

2

1

T PV n PV, r , M

n PV 3 2 .c .G.M . . K ( r, M ) PV r π .r

ω PV n PV, r , M

π .n PV

c

λ PV n PV, r , M

ω PV n PV, r , M

2 . ∆K 0( r , M )

U m( r , M )

ω PV n PV, r , M

3 .M .c

2

4 .π .r

3

3

U ω( r , M )

n Ω ( r, M )

h . 4 ω PV( 1 , r , M ) 3 2 .c Ω ( r, M )

4

12

Ω ( r, M )

∆ω PV( r , M )

ω Ω ( r, M )

∆ω δr n PV, r , ∆r , M

c.

λ PV n PV, r

ω Ω ( r, M )

n Ω ( r , M ) .ω PV( 1 , r , M )

U ω( r , M )

S m( r , M )

∆r , M ∆r , M

1 ω Ω(r

108.

ω PV( 1 , r , M )

ω PV n PV, r

∆λ δr n PV, r , ∆r , M ∆λ Ω ( r , ∆r , M )

1

U m( r , M )

Ω ( r, M )

12. 768 81.

c .U m( r , M )

∆ω ZPF( r , ∆r , M )

ω β r , ∆r , M , K R

U ω( r , M )

N ∆r( r , M )

ω Ω ( r, M ) .

∆r c

ω PV n PV, r , M λ PV n PV, r , M

1 ∆r , M )

ω Ω ( r, M )

3 .M .c . 4 .π 2

∆K C( r , ∆r , M )

2

∆ω δr n PV, r , ∆r , M .∆λ δr n PV, r , ∆r , M

∆v δr n PV, r , ∆r , M

∆v Ω ( r , ∆r , M )

U m( r , M )

∆v δr n Ω ( r , M ) , r , ∆r , M

∆U PV( r , ∆r , M ) .

4

4

µ0

ω Ω_ZPF( r , ∆r , M )

ε0

ω Ω_ZPF( r , ∆r , M )

ω Ω_ZPF( r , ∆r , M )

∆U PV( r , ∆r , M )

∆ω δr( 1 , r , ∆r , M )

4

(r

1

∆r )

3

2 .c . ∆U PV( r , ∆r , M ) h

3

r

3

n Ω_ZPF( r , ∆r , M )

4 K R . ω Ω_ZPF( r , ∆r , M )

294

1

∆ω δr( 1 , r , ∆r , M )

4

ω Ω_ZPF( r , ∆r , M ) ω PV( 1 , r , M )

∆ω δr( 1 , r , ∆r , M )

4

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ω β r , ∆r , M , K R

n β r , ∆r , M , K R

∆ω Ω ( r , ∆r , M )

ω Ω(r

∆ω S r , ∆r , M , K R

∆r , M )

∆ω Ω ( r , ∆r , M )

St γ ( r , ∆r , M )

ω Ω ( r, M )

ω Ω_ZPF( r , ∆r , M )

∆U PV( r , ∆r , M ) .

St α ( r , ∆r , M )

∆n S r , ∆r , M , K R

ω PV( 1 , r , M )

ω Ce

µ0

n β r , ∆r , M , K R

∆ω ZPF( r , ∆r , M )

∆ω R( r , ∆r , M )

∆ω Ω ( r , ∆r , M )

ω β r , ∆r , M , K R

St β ( r , ∆r , M )

ε0

n Ω_ZPF( r , ∆r , M )

St δ( r , ∆r , M )

n Ω(r

∆ω ZPF( r , ∆r , M ) ω Ce ∆r , M )

n Ω ( r, M )

∆v δr n PV, r , ∆r , M

St ε n PV, r , ∆r , M

∆v Ω ( r , ∆r , M )

j. Casimir equations ω C( ∆r )

c . 2 ∆r

E C( r , ∆r , M )

ω X( r , ∆r , M )

N C( r , ∆r , M )

λ C( ∆r )

π .N X( r , ∆r , M ) N X( r , ∆r , M ) .ω PV( 1 , r , M )

ω C( ∆r ) ω PV( 1 , r , M )

Σ HR( A , D , r , ∆r , M )

∆Λ ( r , ∆r , M )

ω C( ∆r )

c .K PV( r , M ) . St α ( r , ∆r , M )

N TR( A , D , r , ∆r , M )

F PP( r , ∆r )

c

A D St N

N T A , D , N X( r , ∆r , M ) N T A , D , N C( r , ∆r , M )

Σ H A , D , N X( r , ∆r , M ) Σ H A , D , N C( r , ∆r , M )

π .h .c .A PP( r ) 4 480.∆r

F PV( r , ∆r , M )

N X( r , ∆r , M )

B C( r , ∆r , M )

λ X( r , ∆r , M )

(1 1 1 )

Σ H A , D, N T

N R( r , ∆r , M )

n Ω_ZPF( r , ∆r , M )

1

ln 2 .n Ω_ZPF( r , ∆r , M )

γ

E C( r , ∆r , M ) c c ω X( r , ∆r , M ) St N

N T A , D , St N

NT

. 2 .A

2

N C( r , ∆r , M )

A PP( r ) .∆U PV( r , ∆r , M ) .

D

D

D. N T

N X( r , ∆r , M )

A

1

A PP( r )

N C( r , ∆r , M ) N X( r , ∆r , M )

2

.ln

4 .π .r

2

N X( r , ∆r , M )

4

N C( r , ∆r , M )

8 .π .G . ∆U PV( r , ∆r , M ) 2 3 .c

295

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2

4

8 .π .G . F PV( r , ∆r , M ) . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln 2 A PP( r ) N C( r , ∆r , M ) N C( r , ∆r , M ) 3 .c

St ∆Λ ( r , ∆r , M )

∆Λ ( r , ∆r , M )

Λ R( r , ∆r , M )

∆ω δr_Error( r , ∆r , M )

St ∆Λ ( r , ∆r , M )

9 .G.M . ∆ω δr( 1 , r , ∆r , M )

∆Λ EGM( r , ∆r , M )

3

2 U m( r , M ) . 3 . ∆ω δr( 1 , r , ∆r , M ) 2 ∆U PV( r , ∆r , M )

∆Λ Error( r , ∆r , M )

ω PV( 1 , r , M )

2 .r

2

2 3 1 . 16.π .r .h . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln K P 27.c .M .∆r4 N C( r , ∆r , M ) N C( r , ∆r , M )

St PP K P , r , ∆r , M

2

4

2 3 16.π .r .h . N X( r , ∆r , M ) . N X( r , ∆r , M ) ln 4 N ( r , ∆r , M ) N C( r , ∆r , M ) . . . 27 c M ∆r C

K P( r , ∆r , M )

1

ω PV( 1 , r , M )

1

∆Λ ( r , ∆r , M )

1

4

∆Λ EGM( r , ∆r , M ) ω PV( 1 , r , M )

.

∆ω δr( 1 , r , ∆r , M )

ω PV( 1 , r , M )

.

∆ω δr( 1 , r , ∆r , M )

k. Fundamental particle equations

mγ

512.h .G.m e

.

2

c . π .r e

n Ω r e, m e ln 2 .n Ω r e , m e 5

2 .m γγ

m gg

φ γγ

r e.

r γγ

2.

φ gg

r γγ

γ

2

m γγ m e .c

r gg

2

ω Ω ( r, M )

St ζ( r , M )

r gg

h .ω Ω r e , m e

EΩ

ω Ce

5

3

4 .r γγ

Kω

St η ( r , M )

Nγ

2

Kλ

π

ω Ω ( r, M ) ω CP

EΩ

m γγ

mγ

1

Km

Kω

mγ Nγ

Kλ

ω Ω ( r, M )

St θ ( r , M )

ω CN

Note: the highlighted equation is not included as a constraint. This is the most significant difference between the calculation engine and the “complete algorithm” of Appendix 3.K. 5

1 rπ

c .ω Ce

rν

4

5

.

2 4 27.ω h .ω Ce 4 32.π

.

ω CP

3

.

ω CP

5

1

.

rµ rτ

r ε.

1 . mµ 9 4 me

2 5

1 . mτ 9 6 me

1

5

rε

1 . me r π. 9 2 mp

2

ω CN

3

ω CN 5

1

5

2

r en r µn r τn

r ε.

m en me

2

5

r µ.

m µn mµ

2

5

r τ.

m τn

2

mτ

Given 5

r ε r π.

1 . me 9 2 mp

2

296

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rε

α

2

.e

3

rπ rµ

rε

α

.e

rτ

rν rε rπ

π rν

rε rπ rν rµ rτ

1

5

Find r ε , r π , r ν , r µ , r τ , r en , r µn , r τn

3 .r xq. 2

r uq

m dq

5

2

r dq

m uq

r uq

m dq

.

2

m uq

r en r µn r τn 5

2

m sq

9

St sq

St dq

ω Ω r dq , m dq

St dq

floor St dq

St sq

ω Ω r xq, m sq

St sq

floor St sq

St cq St bq

1 ω Ω r uq , m uq

St tq

. ω Ω r xq, m cq

St cq

floor St cq

ω Ω r xq, m bq

St bq

floor St bq

ω Ω r xq, m tq

St tq

floor St tq

5

m cq

r sq r cq r bq

5

r uq .

1 m uq

. 2

St cq

9

m bq

2

5

r tq

St bq 5

m tq

ω Ω r uq , m uq

St uq

floor St uq

St dq

ω Ω r dq , m dq

St dq

floor St dq

ω Ω r sq , m sq

St sq

floor St sq

ω Ω r cq , m cq

St cq

floor St cq

St bq

ω Ω r bq , m bq

St bq

floor St bq

St tq

ω Ω r tq , m tq

St tq

floor St tq

St sq

1

St cq

ω Ω r ε, m e

.

297

9

2

St tq St uq

2

9

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9 5 St uq .r uq

m uq

9 5 St dq .r dq

m dq m sq

me

m cq

rε

m bq

5

9 5 St sq .r sq

.

1 . m tq r uq . 9 10 m uq

r tq

9 5 St cq .r cq

5

2

r u( M )

h . 4 π .c .M

rW

r u mW

rZ

r u mZ

rH

r u mH

9 5 St bq .r bq

m tq

9 5 St tq .r tq

ω Ω r u mW ,mW

St W

round St W , 0

. ω Ω r u mZ ,mZ

St Z

round St Z , 0

ω Ω r u mH ,mH

St H

round St H , 0

St W 1

St Z

ω Ω r uq , m uq

St H

5

1 St W

rW

5

5

1

r uq .

rZ

.

m uq

rH

2

9

.m 2 W

1 . 2 mZ 9 St Z

5

rµ

rL

rε

rτ

3

1 . 2 mH 9

St H

1.

r QB

9

r uq

m QB St ω , r QB

Let:

r dq

r sq

r cq

9 m uq . St ω .

r bq

r QB r uq

r tq

rW

rZ

rH

m L St ω , r L

9 m e . St ω .

rL

5

rε

5

4. . 3 πr 3

V( r )

Q( r )

1 V( r )

Q ch ( r )

Q( r ) 3

r dr

5. rν 3

1

x

2

Given 2

x

ln( x) . 2

x x

KS

1 1 3

Find( x) 2 3 . π .r ν ( 1 x) .x3 . 2 8 1 x x

b1

2 3 .r ν

. 2

KS 2

x

KX

2 0.113. fm

1

298

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6 .b 1 .K X . x

2

rX

3 .b 1 . x

2

1

1

r νM

KS

2. 3

. e

3 5 2 π .r ν . x

rν

r πE

rν

r πM

r dr .

r dr

fm

r x .r

1. e 3 x

1

rν

1

2

r

ρ ch ( r )

rν.

rν

fm

1

K S.

KS

2

fm

2 ν

10.r ν

∞

rν

1

V

volt

Given r dr rν r ν .ρ ch r νM

ρ ch ( r ) d r rν r dr

r ν .ρ ch r πE

ρ ch ( r ) d r rν ∞

r ν .ρ ch r πM

ρ ch ( r ) d r r dr rν

r νM Find r νM , r πE, r πM

r πE r πM

r νM

r νM

r πE

r πE .( fm)

r πM

r πM

5 5

r ν2 r ν3 r ν5

λ A( r, M )

m en 1 . r ε. 2 9 me 2

2

5

r µ.

m µn 9

3

2

rν

5

r τ.

m τn

r ν .( fm)

KS

K S . fm

2

2

9

5

λ PV( 1 , r , M ) 2 .n Ω ( r , M )

Given

λ A K ω .r x, m AMC λB

rx

1

Find r x

299

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l. Particle summary matrix 3.1 2 0.69. fm 0.848.( fm)

rπ r πE

KX KS

=

0.113

2

0.113364

0.857.( fm)

r πM

fm

1.

r πE

2

830.702612 830.662386

= r νM

rν

rp

848.636631

848

850.059022

857

874.696943

875

( am)

826.944318 825.617412 rν

rX

r νM

0.879.( fm)

879.064943

879

2

rε

.e

3

rπ

m tq = 178.440506

GeV c

rε

2

.e

rµ

. 7.29735310

3

rτ

= 7.29735310 .

3

rν

λ A K ω .r Bohr , m p λB

3.141593

=

657.329013 656.469624

( nm )

rε rπ

1 .r ε . e α rπ

rν rµ

2

1 .r ε . e α rν

3

rπ 2 0.69. fm

M Error

1 . 1. r νM rp 2

rν

0.848.( fm)

0.857.( fm) 1

r πE

rν

KS

rX

KX

178.( GeV) . 1.11022310

0

0.034635 . 7.38826910

Error Av

r πM

0.879.( fm)

3

3

0.809916

0.160717

0.321692

0.247475

0.130911

M Error

0, 1

M Error

2, 1

2

λ A K ω .r Bohr , m p λB

13

0.075074

1. M Error 0,0 12 + M Error 2, 0

rν

r πE

m tq .c

. 4.8425510

rε

1. π rπ

r νM

0 M Error =

rτ

(%)

M Error

0, 2

M Error

2, 2

M Error

1, 0

M Error

3,0

M Error

1,1

M Error

3, 1

M Error

1, 2

...

M Error

3, 2

Error Av = 0.149388 ( % )

300

www.deltagroupengineering.com

m. Particle summary matrix 3.2 2

rε

. c .e r e ω Ce

r π_1 r π_2

3

rπ r ν_1

5

r π_av

3

r π_2

∆r π

r π_av

r π_1

2 r ν_1

r ν_2

∆r ν

r ν_av

r ν_1

1 . r π_av

r π_Error

r π_2

r ν_Error

1 . r ν_av r ν_2

∆r π

rX KX

3 .b 1 . x

2

r π_Error

1

r π_1

r π_2

r ν_1

r ν_2

r π_av r ν_av

1

∆r π

∆K X

=

( 0.69 0.02) . fm

2

(%)

830.594743

826.944318

826.941624

830.648674

826.942971 . 1.34683810

( am) 3

2 0.005. fm

=

( 0.69 0.02) . fm

14

830.702606

0.053931

2 0.69. fm

1.

2 4 27.ω h ω Ce . 4 32.π ω CN

0

∆r ν

. 3 ( YHz) ω Ω r π , m p = 2.61722210

π

. 2.22044610

1=

r ν_Error

∆r ν

2 6 .b 1 .K X . x

.

4 .ω CN

r π_1

1.

r ν_av

c .ω Ce

r ν_2

2 4 27.ω h ω Ce . . 3 4 4 .ω CP 32.π ω CP

c .ω Ce

5

rε

830.662386 12.03985

2

( am)

2 r X_av

r X_Error

1.

∆K X

rX KX

2

∆K X

rX KX

rX KX

∆r X_av

∆K X

∆r X_av

r X_av

rX KX

∆K X

1

r X_av

rX KX

∆K X

rX KX

∆K X

843.685579 807.144886

=

r X_av

825.415232

∆r X_av

18.270346

m γ = 5.746734 10

17 .

eV

r X_Error = 0 ( % )

( am)

m γγ m gg

=

3.195095 6.39019

10

301

45 .

eV

www.deltagroupengineering.com

φ 1 . γγ λ h φ gg

=

1.152898 1.521258

φ 1 . γγ K λ .λ h φ gg

=

0.991785 1.308668

n. Particle summary matrix 3.3 The following is accurate to “1 or 2” decimal places (as implied by the results): ω Ω r π, m p

ω Ω r ν ,mn

ω Ω r ν,mn

ω Ω r ε, m e

ω Ω r ε, m e

0.5

ω Ω r en , m en

0.5

2

ω Ω r L, m L 2 , r L

1

4

ω Ω r L, m L 3 , r L

1

6

ω Ω r µn , m µn

ω Ω r µ,mµ

8

3

ω Ω r L, m L 5 , r L

8

ω Ω r µn , m µn

4

10

ω Ω r L, m L 5 , r L

4

ω Ω r τ, m τ

12

ω Ω r τ, m τ

12

ω Ω r τn , m τn

ω Ω r en , m en

1

ω Ω r L, m L 2 , r L

2

ω Ω r L, m L 3 , r L ω Ω r µ,mµ

ω Ω r τn , m τn 1 ω Ω r π, m p

.

ω Ω r uq , m uq

= 14

ω Ω r dq , m dq

14

1 ω Ω r ε, m e

.

ω Ω r uq , m uq

2

5 6 =

6 7 7

28

ω Ω r dq , m dq

42

ω Ω r cq , m cq

ω Ω r sq , m sq

56

ω Ω r bq , m bq

70

ω Ω r cq , m cq

28

84

ω Ω r bq , m bq

35

ω Ω r QB, m QB 5 , r QB

ω Ω r QB, m QB 5 , r QB

42

ω Ω r QB, m QB 6 , r QB

98 112

ω Ω r W,mW

ω Ω r QB, m QB 6 , r QB

126

56

ω Ω r Z, m Z

140

ω Ω r W,mW

63

ω Ω r Z, m Z

70

ω Ω r sq , m sq

ω Ω r H, m H

14 21

49

ω Ω r H, m H

ω Ω r tq , m tq

ω Ω r tq , m tq

302

www.deltagroupengineering.com

ω Ω r π, m p ω Ω r ν ,mn ω Ω r ε,me

0.07

ω Ω r en , m en

0.07

ω Ω r L, m L 2 , r L

0.14

ω Ω r L, m L 3 , r L

0.14

1 14

0.29

1

ω Ω r µ,mµ

0.43

7

ω Ω r µn , m µn

0.57

0.07

1

0.57

0.07

ω Ω r L, m L 5 , r L

7

0.71

2

0.86

7

0.86

3

1

7

1

4

2

7

3

4

ω Ω r cq , m cq

4

7

0.86

ω Ω r bq , m bq

5

5

0.86

ω Ω r QB, m QB 5 , r QB

6

ω Ω r τ,mτ 1

1 14

ω Ω r τn , m τn

.

=

ω Ω r uq , m uq

ω Ω r uq , m uq

ω Ω r dq , m dq ω Ω r sq , m sq

8

ω Ω r W,mW

9

ω Ω r Z, m Z

10

0.14 0.29 = 0.43 0.57 0.57 0.71

7 6

7

ω Ω r QB, m QB 6 , r QB

0.14

7 6 7

ω Ω r H, m H ω Ω r tq , m tq

o. Particle summary matrix 3.4 φ γγ φ gg r Bohr rx

=

4.670757 6.163101

10

35 .

1 = 0.352379 ( % )

m

. r x = 5.27319110

m γγ m gg

=

11

( m)

3.195095 6.39019

10

303

φ 1 . γγ K λ .λ h φ gg

=

0.991785 1.308668

45 .

eV

www.deltagroupengineering.com

rε

me

rπ

11.807027

mp

. 5.10998910

rν

830.702612

mn

0.938272

rµ

826.944318

mµ

0.939565

rτ

8.215954

mτ

r en

12.240673

m en

0.095379

r µn

0.105658 1.776989

0.655235

m µn

1.958664

m τn

r uq =

0.768186

m uq

r dq

1.013628

r τn

( am)

0.887904

r sq

1.091334

3 .10

9

1.9.10

4

0.0182 =

. 3.50490310

3

m dq

. 7.00980510

3

m sq

0.113909 1.182905

r cq

1.070961

m cq

r bq

0.92938

m bq m tq

rW

1.061716

mW

91.1876

mZ

114.4

rH

c

2

4.11826

1.284033

rZ

GeV

178.440506

r tq

0.940438

4

80.425

mH m L 2, r L

. 9.15554710

m L 3, r L

rL

=

r QB

10.754551 1.005287

0.056767

m L 5, r L

( am)

3

=

0.565476

m QB 5 , r QB

9.596205

m QB 6 , r QB

21.797922

GeV c

2

. 3 1.32141 1.319591 11.734441 0.697721 ( fm) λ Ce λ CP λ CN λ Cµ λ Cτ = 2.4263110 . ω Ce ω CP ω CN ω Cµ ω Cτ = 7.76344110

1. 6

1. 6

r uq

m uq

r dq

r sq

m dq

r cq

m sq

r bq

m cq

4

1.425486 1.427451 0.160523 2.699721 ( YHz)

r tq = 0.960232 ( am)

m bq

m tq = 30.644349

GeV c

2

p. Similarity equations SSE 3 E rms , B rms , r , ∆r , M

φ 4C_S( r , ∆r , M )

K PV( r , M ) . St α ( r , ∆r , M ) ln 2 .n Ω_ZPF( r , ∆r , M ) γ . π .E rms .B rms n Ω_ZPF( r , ∆r , M ) 1

Re acos SSE 3 E C( r , ∆r , M ) , B C( r , ∆r , M ) , r , ∆r , M

304

www.deltagroupengineering.com

φ 5C_S( r , ∆r , M )

Re asin SSE 3 E C( r , ∆r , M ) , B C( r , ∆r , M ) , r , ∆r , M 1

SSE 4 φ , DC_E, DC_B, E rms , B rms , r , ∆r , M

.SSE ( 1 3

DC_E) .E rms , ( 1

DC_B) .B rms , r , ∆r , M

.SSE ( 1 3

DC_E) .E rms, ( 1

DC_B) .B rms , r , ∆r , M

cos ( φ )

1

SSE 5 φ, DC_E, DC_B, E rms , B rms, r , ∆r , M

sin ( φ)

q. Calculation results K PV R E, M M K PV R E, 2 .M M

K PV R E, M E K PV R E, 2 .M E

K PV R E, M J K PV R E, 2 .M J

K 0 R E, M M

K 0 R E, M E

K 0 R E, M J

∆K 0 R E, M M

∆K 0 R E, M E

∆K 0 R E, M J

K EGM_N R E, M M

K EGM_N R E, M E

K EGM_N R E, M J

1

1

1.000001

K EGM_E R E, M M

K EGM_E R E, M E

K EGM_E R E, M J

1

1

1

=

1

1

1

1

1

1.000001

1

1

0.999999

. 8.55887110

12

. 6.96005110

K PV R E, M S K PV R E, 2 .M S 3 K PV R E, M E .e

3 K PV R S , M S .e

∆K 0 R E , M E

∆K 0 R S , M S

ω PV 1 , R E, M M ω PV 1 , R E, M E

K 0 R E, M E

= 1.000008

. 8.27226110 =

e

=1

∆K 0 R E , M E

0.035839

e

( Hz)

K 0 R S, M S

2.484128

T PV 1 , R E, M S

λ PV 1 , R E, M M

. 7 3.62406910

λ PV 1 , R E, M E

. 8.36497210

λ PV 1 , R E, M J λ PV 1 , R E, M S

. 6 1.2259310 . 5 1.20683210

1.000927

K EGM_E R E, M S

1

4

120.885935 =

U m R E, M E U m R E, M J U m R E, M S

305

27.902544 4.089263

(s)

0.402556

U m R E, M M ( km)

. 2.31613510

K EGM_N R E, M S

T PV 1 , R E, M E

ω PV 1 , R E, M S

=

0.999305

= 1.000008

T PV 1 , R E, M J

6

=

∆K 0 R S , M S

0.244543

ω PV 1 , R E, M J

7

1.000927

∆K 0 R E, M S

T PV 1 , R E, M M

3

. 2.211110

1.000463

K 0 R E, M S

=1

10

6.080707 494.481475 =

. 5 1.57089110

( EPa)

. 8 1.64551410

www.deltagroupengineering.com

Ω R E, M M Ω R E, M E Ω R E, M J

=

. 29 2.83606210

n Ω R E, M M

. 29 1.73968910

n Ω R E, M E

. 28 9.17216810

n Ω R E, M J

. 28 4.2341410

n Ω R E, M S

Ω R E, M S ω Ω R E, M M

519.573099 =

ω Ω R E, M J

. 3 1.86915710

ω Ω R E, M S

. 3 8.76512110

S m R E, M M

0.182295

S m R E, M E S m R E, M J

( YHz)

14.824182 =

S m R E, M S

. 3 4.70941210

∆ω δr 1 , R E, ∆r , M E

=

∆ω δr 1 , R E, ∆r , M J

. 27 3.5284510 195.505363

∆ω PV R E, M E

519.573099 =

∆ω PV R E, M J

N ∆r R E, M E

YW

N ∆r R E, M J

2

cm

. 14 6.52135710 =

N ∆r R E, M S

1.729554

∆λ δr 1 , R E, ∆r , M M

7.493187

∆λ δr 1 , R E, ∆r , M E

( pHz)

51.128768

∆λ δr 1 , R E, ∆r , M J

. 1.33585910

4

∆v δr 1 , R E, ∆r , M M

∆λ Ω R E, ∆r , M E

. 5.02660110

5

∆v δr 1 , R E, ∆r , M E

. 1.39724710

5

∆λ Ω R E, ∆r , M S

. 2.97920610

6

∆v Ω R E, ∆r , M M

13.105112

∆λ Ω R E, ∆r , M J

∆v Ω R E, ∆r , M E ∆v Ω R E, ∆r , M J

=

∆K C R E, ∆r , M M ∆K C R E, ∆r , M E ∆K C R E, ∆r , M J ∆K C R E, ∆r , M S

( ym )

∆v δr 1 , R E, ∆r , M J

13.105121

pm

∆U PV R E, ∆r , M E

13.105115

s

∆U PV R E, ∆r , M J

87.634109 . 4 2.78399910

. 16 2.9237310

7.577156 =

∆U PV R E, ∆r , M S

1.74894 0.256316

13.105101 =

13.10513

pm

13.105131

s

13.109717

2.860531 232.617621 =

. 7 7.74094810

ω Ω_ZPF R E, ∆r , M E ω Ω_ZPF R E, ∆r , M J

. 7 2.9162510

( GPa)

4 7.3899.10

ω Ω_ZPF R E, ∆r , M M ( MPa .MΩ )

( m)

0.025237

∆U PV R E, ∆r , M M

1.077649 =

. 15 6.23483610

∆v δr 1 , R E, ∆r , M S

13.109693

∆v Ω R E, ∆r , M S

. 15 1.73310910

∆λ δr 1 , R E, ∆r , M S

∆λ Ω R E, ∆r , M M =

( YHz)

. 3 1.86915710 . 3 8.76512110

N ∆r R E, M M

519.469801

∆ω δr 1 , R E, ∆r , M S

. 27 7.64347410

∆ω PV R E, M S

. 6 4.93312710

∆ω δr 1 , R E, ∆r , M M

. 28 1.44974110

=

∆ω PV R E, M M

195.505363

ω Ω R E, M E

. 28 2.36338510

ω Ω_ZPF R E, ∆r , M S

123.501066 370.868276 =

. 3 1.56573710

( PHz)

. 3 8.90753610

KR2 = 99.99999999999999(%)

306

www.deltagroupengineering.com

∆ω ZPF R E, ∆r , M M ∆ω ZPF R E, ∆r , M E

ω β R E, ∆r , M M , K R2

14.793206

ω β R E, ∆r , M E, K R2

=

ω β R E, ∆r , M J , K R2

∆n S R E, ∆r , M M , K R2 ∆n S R E, ∆r , M E, K R2 ∆n S R E, ∆r , M J , K R2 ∆n S R E, ∆r , M S , K R2 ∆ω R R E, ∆r , M M =

∆ω R R E, ∆r , M J ∆ω R R E, ∆r , M S St α R E, ∆r , M M

. 18 6.40202410

∆ω Ω R E, ∆r , M J

. 18 3.58539910

∆ω Ω R E, ∆r , M S

∆ω S R E, ∆r , M E, K R2

9.615565

∆ω S R E, ∆r , M J , K R2

11.66707

∆ω S R E, ∆r , M S , K R2 St β R E, ∆r , M M

. 2.78399910

St β R E, ∆r , M E St β R E, ∆r , M J

. 2.9162510

7

St β R E, ∆r , M S

. 2.19383110

5

St δ R E, ∆r , M M

St γ R E, ∆r , M E

. 5.83032610

5

St δ R E, ∆r , M E

. 2.0974410

St γ R E, ∆r , M S

. 9.83425710

St ε 1 , R E, ∆r , M M St ε 1 , R E, ∆r , M E St ε 1 , R E, ∆r , M J

=

St ε 1 , R E, ∆r , M S 2.

G .M M . 1 2 R E .c

1. 2

St δ R E, ∆r , M J

4

St δ R E, ∆r , M S

4

=1

=

( PHz)

. 3 8.90658910

3

0.011474

1 =

1 1 1

St ε n Ω_ZPF R E, ∆r , M J , R E, ∆r , M J

1.000002

St ε n Ω_ZPF R E, ∆r , M S , R E, ∆r , M S

e

. 3 1.56556910

. 2.01680710

1.000001

2

370.826434 =

4

1.000001

1.

( PHz)

123.486273

. 4.77711210

St ε n Ω_ZPF R E, ∆r , M E , R E, ∆r , M E

G .M E . 1 2 R E .c

162.833549

4

St ε n Ω_ZPF R E, ∆r , M M , R E, ∆r , M M

2.

45.263389

. 1.59080310

0.999999

2

. 14 6.84403710

763.476685

8.19356

St γ R E, ∆r , M M

St γ R E, ∆r , M J

=

∆ω S R E, ∆r , M M , K R2

( MPa .MΩ )

. 15 1.16748410

. 14 3.81125810

7.251258

4

=

=

17.031676

∆ω Ω R E, ∆r , M E

. 1.034710

87.634109 =

. 15 1.78829110

∆ω Ω R E, ∆r , M M

19

=

n β R E, ∆r , M M , K R2

n β R E, ∆r , M S , K R2

. 19 1.49277510

. 18 6.40270810 . 18 3.5857810

n β R E, ∆r , M J , K R2

1.077649

St α R E, ∆r , M S

e

167.366022

. 19 1.03481710

n Ω_ZPF R E, ∆r , M S

n β R E, ∆r , M E, K R2

( THz)

946.765196

ω β R E, ∆r , M S , K R2

∆ω R R E, ∆r , M E

41.841506

=

n Ω_ZPF R E, ∆r , M J

. 3 8.90753610

∆ω ZPF R E, ∆r , M S

St α R E, ∆r , M J

( PHz)

. 3 1.56573710

. 19 1.49295410

n Ω_ZPF R E, ∆r , M E

370.868276 =

∆ω ZPF R E, ∆r , M J

St α R E, ∆r , M E

n Ω_ZPF R E, ∆r , M M

123.501066

1.000001 =

1 1.000003 1

2

=1

307

www.deltagroupengineering.com

G .M J . 1 2 R E .c

2.

e

1. 2

2.

2

= 1.000001

e

G .M S . 1 2 R E .c

1. 2

2

= 1.000927

N X R M , ∆r , M M

. 17 2.15162910

E C R M , ∆r , M M

N X R E, ∆r , M E

. 17 2.29685210

E C R E, ∆r , M E

. 3.15778710

E C R J , ∆r , M J

N X R J , ∆r , M J

=

N X R S , ∆r , M S

. 17 3.76223110

B C R M , ∆r , M M B C R E, ∆r , M E B C R J , ∆r , M J

=

λ X R M , ∆r , M M

λ X R J , ∆r , M J

2

=

36.419294 97.406507

=

=

N C R J , ∆r , M J N C R S , ∆r , M S

ln 2 .N X R E, ∆r , M E

γ

ln 2 .N C R E, ∆r , M E

ln 2 .N X R J , ∆r , M J

γ

ln 2 .N C R J , ∆r , M J

ln 2 .N X R S , ∆r , M S

γ

ln 2 .N C R S , ∆r , M S

8.231693 3.077746

( PHz)

. 12 3.20180310

N C R E, ∆r , M E

γ

m

1.791481

N C R M , ∆r , M M

167.343325

volt

23.079214

10.073108

ω X R S , ∆r , M S

( nm )

190.811924 7.220558

ω X R J , ∆r , M J

ln 2 .N C R M , ∆r , M M

1.

2

( mgs )

γ

2

1.

6.364801

ω X R E, ∆r , M E

ln 2 .N X R M , ∆r , M M

1.

2

ω X R M , ∆r , M M

29.761666

λ X R S , ∆r , M S 1.

9.8181 0.76984

=

E C R S , ∆r , M S

0.240852

B C R S , ∆r , M S

λ X R E, ∆r , M E

17

294.339224

. 12 4.18248610 . 13 1.53794510 . 13 3.14792110

1 . N X R M , ∆r , M M ln 2 N C R M , ∆r , M M

γ

γ

γ

1 . N X R E, ∆r , M E ln 2 N C R E, ∆r , M E

5.557718 5.557718 =

1 . N X R J , ∆r , M J ln 2 N C R J , ∆r , M J

5.45678 5.45678 4.964882 4.964882 4.694305 4.694305

1 . N X R S , ∆r , M S ln 2 N C R S , ∆r , M S

N T 1 , 2 , N C R M , ∆r , M M

N T 1 , 2 , N C R J , ∆r , M J

. 12 7.68972610 . 12 1.60090210

N T 1 , 2 , N X R M , ∆r , M M

N T 1 , 2 , N X R J , ∆r , M J

. 17 1.57889410 . 17 1.07581410

N T 1 , 2 , n Ω_ZPF R M , ∆r , M M

N T 1 , 2 , n Ω_ZPF R J , ∆r , M J

N T 1 , 2 , N C R E, ∆r , M E

N T 1 , 2 , N C R S , ∆r , M S

N T 1 , 2 , N X R E, ∆r , M E

N T 1 , 2 , N X R S , ∆r , M S

. 17 1.88111510 . 17 1.14842610

N T 1 , 2 , n Ω_ZPF R E, ∆r , M E

N T 1 , 2 , n Ω_ZPF R S , ∆r , M S

. 18 8.57004510 . 18 5.17408410

N TR 1 , 1 , R M , ∆r , M M N TR 1 , 1 , R E, ∆r , M E N TR 1 , 1 , R J , ∆r , M J N TR 1 , 1 , R S , ∆r , M S

=

=

. 18 7.16489910 . 18 4.83975610 . 12 1.57396110 . 13 2.09124310

. 4 6.72005410

Σ H 1 , 2 , n Ω_ZPF R M , ∆r , M M

. 4 5.49159510

Σ H 1 , 2 , n Ω_ZPF R E, ∆r , M E

. 4 2.05325110

Σ H 1 , 2 , n Ω_ZPF R J , ∆r , M J

. 4 1.19514810

Σ H 1 , 2 , n Ω_ZPF R S , ∆r , M S

308

. 37 9.36929710 =

. 38 1.07084610 . 38 2.05343110 . 38 2.93782710

www.deltagroupengineering.com

F PP R M , ∆r A PP R M

Σ HR 1 , 2 , R M , ∆r , M M

. 9 4.51591310

F PP R E, ∆r

Σ HR 1 , 2 , R E, ∆r , M E

. 9 3.01576110

A PP R E

. 4.21583910

F PP R J , ∆r

. 8 1.42837810

A PP R J

=

Σ HR 1 , 2 , R J , ∆r , M J Σ HR 1 , 2 , R S , ∆r , M S

8

1.300126 =

1.300126

( fPa )

1.300126 1.300126

F PP R S , ∆r A PP R S F PV R M , ∆r , M M

F PP R M , ∆r

A PP R M

F PV R M , ∆r , M M

F PV R E, ∆r , M E

2.349179

F PP R E, ∆r

A PP R E

1.300007

F PV R E, ∆r , M E

=

F PV R J , ∆r , M J

0.074224

( fPa )

F PP R J , ∆r

0.015617

A PP R J

F PV R J , ∆r , M J

F PV R S , ∆r , M S

F PP R S , ∆r

A PP R S

F PV R S , ∆r , M S

∆Λ R M , ∆r , M M ∆Λ R E, ∆r , M E ∆Λ R J , ∆r , M J

=

44.65616

1 = 1

1.447168

St ∆Λ R E, ∆r , M E

0.029107

15 .

2

Hz

. 3 1.65163110

St ∆Λ R J , ∆r , M J

3

(%)

1

St ∆Λ R M , ∆r , M M 10

. 3 9.15864310

. 3 8.22480110

3.225809

. 3.39437710

∆Λ R S , ∆r , M S

1

3.225809 =

St ∆Λ R S , ∆r , M S

1.447168 10

0.029107 . 3.39437710

15 .

2

Hz

3

ω PV 1 , R M , M M ∆ω δr 1 , R M , ∆r , M M Λ R R M , ∆r , M M Λ R R E, ∆r , M E Λ R R J , ∆r , M J Λ R R S , ∆r , M S

=

1

ω PV 1 , R E, M E

1

∆ω δr 1 , R E, ∆r , M E

1

ω PV 1 , R J , M J

1

∆ω δr 1 , R J , ∆r , M J

9 1.3035.10

=

. 9 4.78288210 . 10 5.36192210 . 11 5.22005110

ω PV 1 , R S , M S ∆ω δr 1 , R S , ∆r , M S

309

www.deltagroupengineering.com

2 U m R M, M M 3 . 2 ∆U PV R M , ∆r , M M 2 U m R E, M E 3 . 2 ∆U PV R E, ∆r , M E

9 1.3035.10

=

2 U m R J, M J 3 . 2 ∆U PV R J , ∆r , M J

. 9 4.78288510 . 10 5.361910 . 11 5.21985810

2 U m R S, M S 3 . 2 ∆U PV R S , ∆r , M S

∆ω δr_Error R M , ∆r , M M

∆ω δr_Error R E, ∆r , M E

∆ω δr_Error R J , ∆r , M J

∆ω δr_Error R S , ∆r , M S

∆ω δr_Error R WD , ∆r , M WD

∆ω δr_Error R RG, ∆r , M RG

∆ω δr_Error R NS, ∆r , M NS

∆ω δr_Error R BH, ∆r , M BH

. 2.45448210 =

7

. 4.09314210

4

. 6.56319310

5

. 3.69917510

3

0.023754

0.195216

5.248215

27.272806

∆Λ EGM R M , ∆r , M M

∆Λ EGM R E, ∆r , M E

3.225809

1.447169

∆Λ EGM R J , ∆r , M J

∆Λ EGM R S , ∆r , M S

0.029107

. 3 3.39425210

∆Λ EGM R WD , ∆r , M WD

∆Λ EGM R RG, ∆r , M RG

∆Λ EGM R NS , ∆r , M NS

∆Λ EGM R BH, ∆r , M BH

∆Λ Error R M , ∆r , M M

∆Λ Error R E, ∆r , M E

∆Λ Error R J , ∆r , M J

∆Λ Error R S , ∆r , M S

∆Λ Error R WD , ∆r , M WD

∆Λ Error R RG, ∆r , M RG

∆Λ Error R NS , ∆r , M NS

∆Λ Error R BH, ∆r , M BH

K P R M , ∆r , M M K P R E, ∆r , M E K P R J , ∆r , M J K P R S , ∆r , M S

=

. 6 2.30813410 . 15 5.25385210

=

. 8.47616310

12

(%)

10

15 .

2

Hz

. 9 1.42948610

. 2.45448210

7

. 6.56319310

5

. 4.09314210

4

. 3.69917510

3

0.023754

0.195216

5.248215

27.272806

(%)

265.650431 480.043646 =

. 3 8.40786210 . 4 3.99605210

2 .G.M M ∆U PV R M , ∆r , M M . 3 U m R M,M M RM 2 .G.M E ∆U PV R E, ∆r , M E . 3 U m R E, M E RE 2 .G.M J ∆U PV R J , ∆r , M J . 3 U m R J, M J RJ

3.225809 =

1.447168 10

0.029107 . 3.39437710

15 .

2

Hz

3

2 .G.M S ∆U PV R S , ∆r , M S . 3 U m R S, M S RS

310

www.deltagroupengineering.com

1

2 .G.M M .

1 ∆r

RM

3

RM

1

2 .G.M E.

1 ∆r

RE

3

RJ

1 ∆r

3

∆r

3

RJ

1

2 .G.M S . RS

3.225809 3

RE

1

2 .G.M J .

3

1.447168

=

3

. 3.39437710

15 .

10

0.029107

2

Hz

3

1 3

RS

2 .G.M M ∆U PV R M , ∆r , M M 1 . . 2 .G.M . M 3 U R , M RM m M M R M ∆r 2 .G.M E ∆U PV R E, ∆r , M E 1 . . 2 .G.M . E 3 U m R E, M E RE R E ∆r 2 .G.M J ∆U PV R J , ∆r , M J 1 . . 2 .G.M . J 3 U R , M RJ m J J R J ∆r

3

3

3

RJ

3

1

1

∆Λ EGM R E, ∆r , M E . 2 .G.M E.

3

1

∆Λ EGM R J , ∆r , M J . 2 .G.M J .

∆r

RJ

3

1

∆Λ EGM R S , ∆r , M S . 2 .G.M S . RS

3

0

1

1

1 1

1 = 1

. 2.45448210

7

. 6.56319710

5

. 4.09312510

4

. 3.69903810

3

(%)

1

1

1

1 3

1 3

RE

=

1

1 3

RJ

. 2.45448210

7

. 6.56319710

5

. 4.09312510

4

. 3.69903810

3

(%)

1

1 ∆r

(%)

3

1

1

0

1

1

RM 1

∆r

RE

1 3

0

RS

1 3

0 =

1

1

2 .G.M S ∆U PV R S , ∆r , M S . 3 U m R S, M S RS

∆r

1

1

2 .G.M J ∆U PV R J , ∆r , M J . ∆Λ EGM R J , ∆r , M J . 3 U m R J, M J RJ

RM

1 3

2 .G.M E ∆U PV R E, ∆r , M E . 3 U m R E, M E RE

∆Λ EGM R M , ∆r , M M . 2 .G.M M .

3

RE

2 .G.M M ∆U PV R M , ∆r , M M . ∆Λ EGM R M , ∆r , M M . 3 U m R M,M M RM

∆Λ EGM R S , ∆r , M S .

1

RM 1

2 .G.M S ∆U PV R S , ∆r , M S 1 . . 2 .G.M . S 3 U m R S, M S RS R S ∆r

∆Λ EGM R E, ∆r , M E .

1

1

1 3

RS

311

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5

λ CP c .m e

5

27.m e

.

4

.

K PV r p , m p .m p

3 128.G.π .h

2

8 .π

3

λ CN

5

2 16.π .λ Ce

c .ω Ce

5

λ Ce m p λ Ce m n λ CP m e λ CN m e

r ν λ CN ω CP m p r π λ CP ω CN m n rν

.

830.594743

.

3

2 4 27.ω h ω Ce . 4 32.π ω CN 5

.

2 3 16.c .π .m n

826.941624 = 826.941624 ( am) 826.941624

2 4 27.m h m e . mn 4 .π

= ( 0.315205 0.315205 0.315205) ( % )

St θ r ν , m n

. 5 1.8360210 . 3 1.8386810 . 3 = 3.21927910

ω PV 1 , r e , m e

ω PV 1 , r π , m p

ω PV 1 , r ν , m n

ω Ω r e, m e

ω Ω r π, m p

ω Ω r ν ,mn

ω PV 1 , r π , m p

ω Ω r π, m p

ω PV 1 , r e , m e

ω Ω r e, m e

ω PV 1 , r ν , m p

ω Ω r ν ,mn

ω PV 1 , r e , m e

ω Ω r e, m e

=

ω Ω r ν ,mn

ω PV 1 , r e , m e

ω PV 1 , r ν , m p

ω PV 1 , r ν , m n

2 .π .c .

λ Ce 2

λ CP

0.568793

35.500829

. 2.49926810

. 2.61722210

17

18

35.73252 . 18 2.62462610

( GHz)

62.792864 10.50158

ω Ω r π, m p

2

=

62.414364 10.471952

ω Ω r e,m e

ω Ce

5

h .m e

St η r π , m p

ω CP

2 4 4 .π .λ h λ Ce

4 .ω CN

830.594743

λ . CN

= ( 0.995476 0.998623 0.998623 0.998623)

r π λ CN ω CP m p

ω Ω r π, m p

c .ω Ce

= 830.594743 ( am)

27

. 3 1.83615310 . 3 1.83868410 . 3 1.83868410 . 3 = 1.83615310

λ CP ω CN m n

St ζ r e , m e

5

.

2 16.π .λ Ce

2 4 27.m h m e . mp 4 .π

2 3 16.c .π .m p

( am)

4 2 K PV r n , m n .m n 3

2 4 27.ω h ω Ce . . 3 4 4 .ω CP 32.π ω CP

1

λ CN

.

2 4 4 .π .λ h λ Ce

.

826.941624

λ CN

5

h .m e

830.594743

=

4

λ . CP

27

.

4 2 K PV r p , m p .m p 5

K PV r n , m n .m n λ CP

4

λ CP

.

ω CP.

mp

. 17 7.32784510 . 16 7.34520410 . 16 = 4.39398910

. 3 2.6174110 . 3 2.6174110 . 3 2.6174110 . 3 ( YHz) = 2.61722210

me

312

www.deltagroupengineering.com

2

ω CN

ω Ω r ν,mn

ω Ce

2 .π .c .

λ Ce

mn ω CN. me

2

λ CN

2

ω Ω r ε, m e 2 .ω Ω r π , m p

. 3 2.62463110 . 3 2.62463110 . 3 2.62463110 . 3 ( YHz) = 2.62462610

2

ω . CP ω Ω r π , m p ω Ce

ω . CN ω Ω r ν , m n ω Ce 1

1

= ( 99.985611 100.007215 100.000181) ( % )

m L 1, r ε

m L 2, r L

m L 3, r L

m L 4, r µ

m L 5, r L

m L 6, r τ

m L 7, r L

m L 8, r L

m L 9, r L

m L 10, r L

m L 11, r L

m L 12, r L

m L 13, r L

m L 14, r L

m L 15, r L

m L 16, r L

m L 17, r L

m L 18, r L

m L 19, r L

m L 20, r L

. 5 1.80208610 . 5 2.29847910 . 5 2.89523810 . 5 1.3933810

m L 21, r L

m L 22, r L

m L 23, r L

m L 24, r L

. 5 4.44581510 . 5 5.4303110 . 5 6.57657710 . 5 3.60608710

0.510999

=

9.155547

56.766874

105.677748

. 3 2.5703410 . 3 565.476231 1.77526210

. 3 4.6876410

. 3 1.27952710 . 4 1.96479110 . 4 2.90646410 . 4 7.96417210

MeV

. . . . 4.16672110 5.81601510 7.93341210 1.06069210

c

4

4

4

5

m QB 1 , r dq

m QB 2 , r sq

m QB 3 , r cq

m QB 4 , r bq

. 7.00980510

m QB 5 , r QB

m QB 6 , r QB

m QB 7 , r W

m QB 8 , r Z

9.596205

21.797922

80.425

91.1876

m QB 9 , r H

m QB 10, r tq

m QB 11, r QB

m QB 12, r QB

114.4

178.440506

333.427609

493.23068

m QB 13, r QB

m QB 14, r QB

m QB 15, r QB

m QB 16, r QB

707.097922

986.98519

. 3 1.80000810 . 3 1.3463110

=

3

0.113909

1.182905

4.11826

m QB 17, r QB

m QB 18, r QB

m QB 19, r QB

m QB 20, r QB

. 2.36458310

m QB 21, r QB

m QB 22, r QB

m QB 23, r QB

m QB 24, r QB

. 3 7.54460610 . 3 9.21530610 . 3 1.11605410 . 4 6.11957610

∆U PV R E, ∆r , M M ∆U PV R E, ∆r , M E ∆U PV R E, ∆r , M J ∆U PV R E, ∆r , M S

3

. . 3.05816410 3.90054810 3

2

3

GeV c

2

. 4.91325710

3

2.860531 232.617621 =

4 7.3899.10

( GPa)

. 7 7.74094810

The following two result sets are accurate to “13” decimal places: 1

.

ω Ω r uq , m uq

1 ω Ω r ε, m e

.

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

ω Ω r dq , m dq

ω Ω r sq , m sq

ω Ω r cq , m cq

ω Ω r bq , m bq

ω Ω r W,mW

ω Ω r Z, m Z

ω Ω r H, m H

ω Ω r tq , m tq

=

=

1 2 3 4 7 8 9 10

7 14 21 28 49 56 63 70

r. Resonant Casimir cavity design specifications (experimental) Given ∆ω R R E, ∆r , M E

∆r

Find( ∆r )

1

∆r = 16.518377( mm)

ω X R E, ∆r , M E = 16.340851 ( PHz)

313

www.deltagroupengineering.com

E C R E, ∆r , M E = 550.422869

V m

B C R E, ∆r , M E = 18.360131 ( mgs )

SSE 4 0 .( deg ) , 0 .( % ) , 0 .( % ) , E C R E, ∆r , M E , B C R E, ∆r , M E , R E, ∆r , M E SSE 5 90.( deg ) , 0 .( % ) , 0 .( % ) , E C R E, ∆r , M E , B C R E, ∆r , M E , R E, ∆r , M E

=

1 1

NOTES

314

www.deltagroupengineering.com

•

MathCad 12: High precision calculation results a. Computational environment

NOTE: KNOWLEDGE OF MATHCAD IS REQUIRED AND ASSUMED The high precision calculation results are obtained via the “MathCad 12” computational environment utilising the calculation engine defined in the preceding section. • • •

Convergence Tolerance (TOL): 10-14. Constraint Tolerance (CTOL): 10-14. Calculation Display Tolerance: 6 figures – unless otherwise indicated. b. Particle summary matrix 3.1

      rπE +    

( 2)

0.69⋅ fm

rπ

rπM 1 2

⋅ ( rνM − rν ) rν rνM

830.647087 830.662386    848.579832 848    0.857⋅ ( fm)   849.993668 857   ( am) =   874.643564 875 rp   826.889045 825.617615  rX   879.016508 879   0.879⋅ ( fm)  0.848⋅ ( fm)

rπE

( )

 KX   −0.113  2   =  fm  KS   −0.113348 2  − rε  ⋅e 3  rπ  rµ  −  rε ⋅ e rτ  rν   rε  rπ − rν

         

mtq = 178.470327

GeV  2   c 

 λA( Kω ⋅ rBohr , mp )   657.329013  =  ( nm) λB    656.469624

 7.297353× 10− 3    = −3 7.297353× 10    3.141593 

rµ   2 − − r r r r 1 ε 1 ε 1 ε  3 τ  ⋅ ⋅e ⋅ ⋅e ⋅   α rπ α rν π rπ − rν   rπ rπE rπM     0.848⋅ ( fm) 0.857⋅ ( fm) 2 0.69⋅ fm M Error :=  −1 1 1  r K ν S  ⋅  ⋅ ( rνM − rν ) + rπE  rX KX   rp  2    2 rνM mtq ⋅ c λA ( Kω ⋅ rBohr , mp )     0.879⋅ ( fm) 178⋅ ( GeV) λB  

( )

315

www.deltagroupengineering.com

− 14 − 13   0 2.220446× 10 1.110223× 10    −1.841834× 10− 3  0.068376 −0.817542 M Error =   ( %) −0.040736 0.153997 0.308232     −3 0.264229 0.130911  1.87806× 10 

1

ErrorAv :=

12

⋅  MError

0, 0

+ M Error

0,1

+ MError

0, 2

+ M Error

1,0

+ MError

+ M Error

1, 1

1,2

...

+ M Error2 , 0 + MError2 , 1 + M Error2 , 2 + MError3 , 0 + M Error3 , 1 + MError3 , 2 

  

ErrorAv = 0.148979(%) c. Particle summary matrix 3.2 2  −  rε c 3  ⋅ ⋅e re ωCe  rπ_1     :=  5 2 4  rπ_2   c⋅ ωCe 27⋅ ωh ωCe  ⋅ ⋅  4⋅ ω 3 32⋅ π4 ωCP  CP

 ∆rπ

 rπ_Error 

 rπ_av − rπ_1

  rν_2  rν_av   ∆rν  rπ_2

rε   rπ − π   rν_1   5   := 2 4  r ν_2 ⋅ 27⋅ ωh ωCe c ω   Ce  ⋅ ⋅  4⋅ ωCN3 32⋅ π4 ωCN 

 1 ⋅ (r  r π_av + ∆rπ )  π_2    :=    1  rν_Error   rν_2 ⋅ ( rν_av + ∆rν )   

    :=    ∆rν   rν_av − rν_1 

 rπ_1   rν_1  rπ_av   ∆rπ

        

 830.647081 830.594743 826.889045 826.941624 = ( am)  830.620912 826.915335  −0.026169 0.02629   

rX ( KX ) :=

       

 rπ_av  1  rπ_1 + rπ_2    := ⋅    rν_av  2  rν_1 + rν_2 

 rπ_Error  0   − 1 =   ( %) 0  rν_Error 

(2 ) 2 3⋅ b 1⋅ ( x − 1)

−6⋅ b 1⋅ KX ⋅ x − 1

( 2)

ωΩ ( rπ , mp ) = 2.617319× 10 ( YHz) 3

∆KX := 0.005⋅ fm

( )

2   0.69⋅ fm    830.662386 = 1   ( am) 2 2    ⋅  ( 0.69 + 0.02) ⋅ fm − ( 0.69 − 0.02) ⋅ fm    12.03985  2 

( )

rX_av :=

1 2

( (

( )

)

(

⋅ rX KX − ∆KX + rX KX + ∆KX

rX_Error :=

(

)

rX KX − ∆KX − ∆rX_av rX_av

−1

))

(

∆rX_av := rX_av − rX KX + ∆KX

)

 rX( KX − ∆KX)   843.685786  rX( KX + ∆KX)   807.145085   ( am)  =   825.415435 rX_av    18.270351      ∆rX_av  

316

www.deltagroupengineering.com

− 14

rX_Error = 2.220446× 10

 φγγ   1.152898 =  λh  φgg   1.521258 1

⋅

(

− 17

mγ = 5.746734 10

( %)

)

⋅ eV

(

)

 mγγ   3.195095 − 45   =  10 ⋅ eV  mgg   6.39019 

 φγγ   0.991785 =  Kλ ⋅ λh  φgg   1.308668 1

⋅

d. Particle summary matrix 3.3 The following is accurate to “1 or 2” decimal places (as implied by the results): ωΩ ( rν , mn)     ωΩ ( rε , me)     ωΩ ( ren , men)    ωΩ ( rL , mL( 2 , rL))     ωΩ ( rL , mL( 3 , rL))    ωΩ ( rµ , mµ )   ωΩ ( rµn , mµn)    ω ( r , m ( 5, r ))  Ω L L L     ωΩ ( rτ , mτ )   ωΩ ( rτn , mτn)   1   ω r , m ⋅ Ω ( uq uq)  ωΩ ( rπ , mp )  ωΩ ( rdq , mdq)     ωΩ ( rsq , msq )     ωΩ ( rcq , mcq)   ωΩ ( rbq , mbq)    ω (r , m (5 , r ) )   Ω QB QB QB   ωΩ (rQB , mQB(6 , rQB) )    ωΩ ( rW , mW)     ωΩ ( rZ , mZ )   ωΩ ( rH , mH )     ωΩ ( rtq , mtq )  

ωΩ ( rπ , mp )     ωΩ ( rν , mn)     ωΩ ( rε , me)     ωΩ ( ren , men)    ωΩ ( rL , mL( 2, rL) )   ω (r , m (3, r ) )  Ω L L L   ωΩ ( rµ , mµ )     ωΩ ( rµn , mµn)    ωΩ ( rL , mL( 5, rL) )    ωΩ ( rτ , mτ )     ωΩ ( rτn , mτn) 1   ⋅  ωΩ ( rε , me)  ωΩ ( ruq , muq)   ωΩ ( rdq , mdq)     ωΩ ( rsq , msq)   ωΩ ( rcq , mcq)     ωΩ ( rbq , mbq)    ωΩ ( rQB , mQB( 5, rQB) )     ωΩ ( rQB , mQB( 6, rQB) )    ωΩ ( rW , mW)     ωΩ ( rZ , mZ )   ωΩ ( rH , mH)     ωΩ ( rtq , mtq )  

 1   2     2   4     6   8   8     10   12   12    =  14   14     28   42   56     70   84   98     112   126     140 

317

 0.5   0.5     1   1   2     3   4     4   5   6    6  =  7   7     14   21     28   35   42     49   56   63     70 

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ωΩ ( rπ , mp )     ωΩ ( rν , mn)     ωΩ ( rε , me)     ωΩ ( ren , men)    ωΩ ( rL , mL( 2, rL) )   ω (r , m ( 3, r ))  Ω L L L    ωΩ ( rµ , mµ )    ωΩ ( rµn , mµn)    ωΩ ( rL , mL( 5, rL) )    ωΩ ( rτ , mτ )     ωΩ ( rτn , mτn) 1   ⋅  ωΩ ( ruq , muq)  ωΩ ( ruq , muq)   ωΩ ( rdq , mdq)     ωΩ ( rsq , msq )     ωΩ ( rcq , mcq)   ωΩ ( rbq , mbq)    ωΩ ( rQB , mQB( 5, rQB) )     ωΩ ( rQB , mQB( 6, rQB) )    ωΩ ( rW , mW)     ωΩ ( rZ , mZ )   ωΩ ( rH , mH)     ωΩ ( rtq , mtq )  

 0.07   0.07     0.14   0.14   0.29     0.43   0.57     0.57   0.71   0.86    0.86   =  1   1     2   3     4   5   6     7   8   9     10 

e. Particle summary matrix 3.4  rε     rπ   11.806238   r   830.647087  ν     rµ   826.889045    8.214055   rτ  12.237844   ren    0.095379     rµn   0.655235    rτn    1.958664     ruq  =  0.768186  ( am)  r   1.013628     dq     rsq   0.887904   r   1.091334   cq   1.070961   rbq       0.92938  r tq    1.283947   rW     1.061645   rZ   0.940375  r   H

 me      5.109989× 10− 4   mp   0.938272  m     n   0.939565   mµ   0.105658      1.776989  mτ     men  −9  3 × 10     mµn   1.9 × 10− 4    mτn     0.0182     GeV  muq  =  −3    m   3.505488× 10   c2    dq    7.010977× 10− 3   msq     m   0.113928  cq    1.183102   mbq       4.118949  m tq    178.470327   mW  80.425       91.1876 m Z     m   114.4   H

318

(

)

 φγγ   4.670757 − 35  =  10 ⋅ m  φgg   6.163101

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− 11

rx = 5.273191× 10

 φγγ   0.991785 =  Kλ ⋅ λh  φgg   1.308668 1

( m)

(

⋅

)

 mγγ   3.195095 − 45  rL   10.752712  =  10 ⋅ eV   =  ( am)  mgg   6.39019   rQB   1.005262 

1 6 1 6

rBohr

− 1 = 0.352379( %)

rx

 mL( 2 , rL)     mL( 3 , rL)   m (5, r )   L L   mQB( 5 , rQB)   m (6, r )   QB QB 

 9.153163× 10− 3   0.056752    GeV =  0.565329   2   9.597226   c   21.800242 

⋅ ( ruq + rdq + rsq + rcq + rbq + rtq ) = 0.960232( am) ⋅ ( muq + mdq + msq + mcq + mbq + mtq ) = 30.649471

GeV  2   c 

The following two result sets are accurate to “13” decimal places: 1 ωΩ ( ruq , muq)

 ωΩ ( rdq , mdq) ωΩ ( rsq , msq) ωΩ ( rcq , mcq) ωΩ ( rbq , mbq)    ωΩ ( rW , mW) ωΩ ( rZ , mZ ) ωΩ ( rH , mH) ωΩ ( rtq , mtq ) 

⋅

1 2 3 4    7 8 9 10 

=

 ωΩ ( rdq , mdq) ωΩ ( rsq , msq) ωΩ ( rcq , mcq) ωΩ ( rbq , mbq)   7 14 21 28  =  ωΩ ( rε , me)  ωΩ ( rW , mW) ωΩ ( rZ , mZ ) ωΩ ( rH , mH) ωΩ ( rtq , mtq )   49 56 63 70  1

⋅

NOTES

319

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NOTES

320

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Quinta Essentia – Part 4 •

MathCad 8 Professional a. Complete simulation i. Computational environment

NOTE: KNOWLEDGE OF MATHCAD IS REQUIRED AND ASSUMED • • •

Convergence Tolerance (TOL): 0.001. Constraint Tolerance (CTOL): 0.001. Calculation Display Tolerance: 6 figures – unless otherwise indicated. ii. Units of measure (definitions)

Jy

10

W

26 .

pc

2.

. 16 .( m) 3.085677580710

m Hz

( mJy µJy nJy pJy fJy aJy zJy yJy )

Scale 1 .( Jy )

( mpc µpc npc ppc fpc apc zpc ypc )

Scale 1 .( pc )

( kJy MJy GJy TJy PJy EJy ZJy YJy )

Scale 2 .( Jy )

( kpc Mpc Gpc Tpc Ppc Epc Zpc Ypc )

Scale 2 .( pc )

iii. Constants (definitions) σ

. 8. 5.67040010

W

κ

2. 4

. 1.380650510

Th

J K

m K

m h .c

23 .

KW

. 3 .( m.K ) 2.897768510

2

κ

iv. Astronomical statistics Lyr

∆T 0

c .yr

D E2M

0.001.( K )

8 3.844.10 .( m)

Ro

8 .( kpc )

AU

∆R o

149597870660.( m) H 0

0.5.( kpc )

MG

71.

km . s Mpc

T0

2.725.( K )

11 6 .10 .M S

v. Characterisation of the gravitational spectrum 1. “Ω → Ω1, nΩ → nΩ_1, ωΩ → ωΩ_1” Note: “the complete simulation” is the computational algorithm developed for this text and is predominantly without comment. 321

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Commencing with the following relationship set, significant simplifications to primary EGM equations may be derived as follows, 4 h . ω PV( 1 , r , M ) 3 2 .c

U ω( r , M )

c .U ω n PV, r , M

S ω n PV, r , M

U ω( r , M ) .

U ω n PV, r , M

U m( r , M )

3 .M .c

n PV

2

4

4

n PV

2

4 .π .r

3

3

Ω ( r, M )

108.

U m( r , M ) U ω( r , M )

2

U m( r , M )

12. 768 81.

n Ω ( r, M )

U ω( r , M )

Ω ( r, M )

4

12

Ω ( r, M )

1

Hence, 3 .M .c U m( r , M )

2

4 .π .r

3 .M .c

3

U ω( r , M )

5

4 3 h . 4 ω PV( 1 , r , M ) 2 .π .r .h .ω PV( 1 , r , M ) 3 2 .c

U m( r , M )

108.

3

Ω 1( r , M ) 6 .

>> 768”, hence simplifying / approximating forms yields,

U ω( r , M )

3

Ω 1( r , M )

2

U m( r , M )

Typically: “ 81.

108.

U ω( r , M )

U m( r , M )

3 .M .c

6.

U ω( r , M )

216.

U ω( r , M )

3

U m( r , M )

3

U m( r , M ) U ω( r , M )

Ω 1( r , M )

4 3 2 .π .r .h .ω PV( 1 , r , M )

3

6 .c

5

r .ω PV( 1 , r , M )

.

3 .M .c

2

2 .π .h .ω PV( 1 , r , M )

Typically: “ Ω ( r , M ) >> 1” hence, 3

n Ω_1( r , M )

Ω 1( r , M ) 1 U m( r , M ) .

C Ω_1( r , M )

12

G.M . 2

r

ω Ω_1( r , M )

3

c

U ω( r , M ) 2 .r .ω PV( 1 , r , M )

2

.

3 .M .c

2

2 .π .h .ω PV( 1 , r , M )

n Ω_1( r , M )

Ω 1( r , M ) 12

3

2

ω Ω_1( r , M ) n Ω_1( r , M ) .ω PV( 1 , r , M )

π .n Ω_1( r , M )

n Ω_1( r , M ) .ω PV( 1 , r , M )

T Ω_1( r , M )

1 ω Ω_1( r , M )

2 c . 3 .M .c 2 .r 2 .π .h .ω PV( 1 , r , M )

λ Ω_1( r , M )

c ω Ω_1( r , M )

Checking errors yields, Ω 1 R M,M M

Ω 1 R E, M E

Ω R M, M M

Ω R E, M E

Ω 1 R J, M J

Ω 1 R S, M S

Ω R J, M J

Ω R S, M S

1=

. 14 4.44089210 . 6.66133810

14

. 14 6.66133810 . 4.44089210

14

322

(%)

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Ω 1 R NS , M NS

1 = 0 (%)

Ω R NS , M NS

n Ω_1 R M , M M

n Ω_1 R E, M E

n Ω R M,M M

n Ω R E, M E

n Ω_1 R J , M J

n Ω_1 R S , M S

n Ω R J, M J

n Ω R S, M S

n Ω_1 R NS , M NS n Ω R NS , M NS

ω Ω_1 R E, M E

ω Ω R M,M M

ω Ω R E, M E

ω Ω_1 R J , M J

ω Ω_1 R S , M S

ω Ω R J, M J

ω Ω R S, M S

ω Ω R NS , M NS

T Ω_1 R E, M E

T Ω R M,M M

T Ω R E, M E

T Ω_1 R J , M J

T Ω_1 R S , M S

T Ω R J, M J

T Ω R S, M S

T Ω R NS, M NS

λ Ω_1 R E, M E

λ Ω R M,M M

λ Ω R E, M E

λ Ω_1 R J , M J

λ Ω_1 R S , M S

λ Ω R J, M J

λ Ω R S, M S

λ Ω R NS , M NS

. 2.22044610

14

. 4.44089210

14

. 8.88178410

14

(%)

1=

. 6.66133810

14

. 2.22044610

14

. 6.66133810

14

. 8.88178410

14

(%)

1=

. 7.77156110

14

. 2.22044610

14

. 5.55111510

14

. 7.77156110

14

. 7.77156110

14

. 2.22044610

14

. 6.66133810

14

. 7.77156110

14

(%)

1 = 0 ( %)

λ Ω_1 R M , M M

λ Ω_1 R NS , M NS

14

1 = 0 (%)

T Ω_1 R M , M M

T Ω_1 R NS, M NS

. 6.66133810

1 = 0 (%)

ω Ω_1 R M , M M

ω Ω_1 R NS , M NS

1=

1=

(%)

1 = 0 (%)

2. “g → ωΩ” i. “ωΩ_1 → ωΩ_2” 3

ω Ω_1( r , M )

2 c . 3 .M .c 2 .r 2 .π .h .ω PV( 1 , r , M )

ω Ω_1( r , M )

323

3

3 2 c . 3 .M .c 2 .r 2 .π .h .ω PV( 1 , r , M )

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ω PV( 1 , r , M )

. 3 . . . 2r . 2πh 3 2 c 3 .M .c ω Ω_1( r , M ) 1

16.π .h . r 5 3 .M .c ω Ω_1( r , M )

3

ω Ω_2( r , M )

th

9

9 2 c . 3 .M .c . 1 . 3 2r 2 .π .h ω PV( 1 , r , M )

G.h c

ωh

5

th

2

c G.h

3 14 2 3 .c .M

4 3 2 2 3 .G .M .ω h .c

13 5 2 3 2 .r .π .h .G

2 13 5 2 2 .r .π .λ h

ω Ω_2( r , M )

g( r , M )

9

6 3 3 .ω h . . GM 2 13 . 2 . . 2 π rc r

G.M

mh

2

St g

9

r

λh

G

5 3 3 .ω h . . GM 13 5 π 2 .λ h .r

ω Ω_2( r , M )

2

h .c

5

1

2

9 2 c . 3 .M .c 2 .r 2 .π .h

2

3

3

G.h c

λh

2

r

6 3 3 .ω h

13 5 2 3 2 .r .π .h .G

3

1

3

5 3 3 .ω h .G.M G.M . 2 3 13 2 .λ h .π .r

3 14 2 3 .c .M

1 . 1 . 2 c .G.M 3 π .r r

.

2

c G.h

G λh

6 3 3 .ω h . . GM 2 13 2 2 .π .r .c r

c 2

3

h

2

245

10

St g = 1.828935

13 2 2 .π .c

1. 2 St g .g ( r , M ) r

ω Ω_2( r , M )

9

5 m.s

St g

.g ( r , M ) 2

r

Checking errors yields, ω Ω_2 R M , M M

ω Ω_2 R E, M E

ω Ω_1 R M , M M

ω Ω_1 R E, M E

ω Ω_2 R J , M J

ω Ω_2 R S , M S

ω Ω_1 R J , M J

ω Ω_1 R S , M S

ω Ω_2 R NS , M NS

1=

. 1.04678510

9

. 2.32001510

8

. 6.57443310

7

. 7.07196310

5

(%)

1 = 2.491576 ( % )

ω Ω_1 R NS , M NS

Therefore, a EGM_ωΩ( r , M )

r . 9 ω Ω_2( r , M )

St g

a EGM_ωΩ R E, M E = 9.809009

m s

2

Checking errors yields, a EGM_ωΩ R M , M M

a EGM_ωΩ R E, M E

g R M,M M

g R E, M E

a EGM_ωΩ R J , M J

a EGM_ωΩ R S , M S

g R J, M J

g R S, M S

1=

. 1.49880110

12

. 1.49880110

12

. 1.5432110

12

. 1.57651710

12

324

(%)

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a EGM_ωΩ R NS , M NS

. 1 = 1.65423210

g R NS, M NS

12

(%)

ii. “ωΩ_1 → ωΩ_3” 3

3

1 U m( r , M ) . 1 ω Ω_1( r , M ) . ω PV( 1 , r , M ) . 2 U ω( r , M ) 2

ω Ω_1( r , M )

U m( r , M ) 4 h . ω PV( 1 , r , M ) 3 2 .c

3 1 . 2 .c . U m( r , M ) 8 h ω PV( 1 , r , M )

3

3 .M .c ω Ω_3( r , M )

.ω ( 1 , r , M ) PV

3 c . U m( r , M ) 4 .h ω PV( 1 , r , M )

9

3

3 3

c 4 .h

3

2

. . . 4πr . 2 c .G.M 3

14

2

27 . c . M 8192 h 3 π2 .r5 .G

5

9

2

27 . c . c . M 8192 G.h h 2 π2 .r5

3

9

2

3 . 2 .c . M ωh 13 2 2 2 h π .r5

π .r

4

9

ω Ω_3( r , M )

c.

3.

2

9

ω Ω_3( r , M )

3 .ω h

2

. M

4 .π .h

2

St G

5

r

3.

3 .ω h

2

. c 2

4 .π .h

M St G. 5 r

ω Ω_3( r , M )

224 .

St G = 8.146982 10

5

m

2 9 kg .s

2

1

2

9

9 M St G .

St G

9

G

St g

5

r

9

Checking errors yields, ω Ω_3 R M , M M

ω Ω_3 R E, M E

ω Ω_2 R M , M M

ω Ω_2 R E, M E

ω Ω_3 R J , M J

ω Ω_3 R S , M S

ω Ω_2 R J , M J

ω Ω_2 R S , M S

ω Ω_3 R NS , M NS ω Ω_2 R NS , M NS

. 1 = 6.66133810

1=

14

(%)

. 14 8.88178410

. 1.11022310

13

. 13 1.11022310

. 1.11022310

13

1 . St G

. 1 = 3.33066910

G

(%)

14

(%)

St g

3. “g → ωPV3” 2 .c .n PV

3

ω PV n PV, r , M

3

2 π .r

.g ( r , M )

325

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4. “SωΩ → c⋅Um” Reducing / simplifying / approximating utilising computational features of the environment yields, nΩ

8 .n Ω

3

2

24.n Ω

8 .n Ω . n Ω

4

2

2

nΩ

2

nΩ

2

8 .n Ω

simplify

32.n Ω factor

3 .n Ω factor

2

substitute , n Ω

4

2

8 .n Ω . n Ω

2 8 .n Ω . n Ω

3 .n Ω

2

3

24.n Ω

2

32.n Ω

16

4

3

Hence, nΩ 8 .n Ω

2 3

4

2

2 24.n Ω

S ωΩ ( r , M )

nΩ

2

4

nΩ

4

32.n Ω 8 .n Ω . n Ω

h . 4 ω PV( 1 , r , M ) . 2 . 2c

nΩ 2

2

8 .n Ω

4

3 .n Ω

n Ω ( r, M )

4

24.n Ω

2

32.n Ω

8 .n Ω . n Ω

2

3 .n Ω

4

2

3

n Ω ( r, M )

4

16 2 8 .n Ω . n Ω

3

8 .n Ω

3

h . 4 3 ω PV( 1 , r , M ) .8 .n Ω ( r , M ) 2 . 2c 3

2 c 4 .h . 4 4 3 .M .c 3 4 .h . . S ωΩ ( r , M ) ω PV( 1 , r , M ) .n Ω ( r , M ) ω PV( 1 , r , M ) . 2 2 2 .r .ω PV( 1 , r , M ) 2 .π .h .ω PV( 1 , r , M ) c c

3

4

3 2 3 4 .h .c . ω PV( 1 , r , M ) . 3 .M .c 3 .M .c S ωΩ ( r , M ) 2 3 3 . . . 3 8 .c .r ω PV( 1 , r , M ) 2 π h ω PV( 1 , r , M ) 4 .π .r

Hence, S ωΩ ( r , M ) c .U m( r , M )

5. “CΩ_J” C Ω_J ( r , M )

2 d λ Ω ( r , M ) . U m( r , M ) dr

2

C Ω_J1( r , M )

2

c 9

2

M St G. 5 r 9 .c . St G 4 .π 4

St J

. . . d 3Mc d r 4 .π .r3

2 9

C Ω_J1( r , M )

2

c

2

. r

9

2

St G

St J 2

r

9

5

9

. M

M

2

1

. . .9 M c 4 4 .π .r

2

2

5 9

9 .c . 9 M St G . 26 4 .π r 4

5

8

r

326

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Checking errors yields, C Ω_J1 100.( km) , M M

C Ω_J1 R S , M M

C Ω_J 100.( km) , M M C Ω_J1 100.( km) , M E C Ω_J 100.( km) , M E C Ω_J1 100.( km) , M J C Ω_J 100.( km) , M J C Ω_J1 100.( km) , M S C Ω_J 100.( km) , M S C Ω_J1 100.( km) , M NS C Ω_J 100.( km) , M NS

C Ω_J R S , M M

1=

. 3.63875410

8

. 2.95903310

6

. 9.40034410

4

C Ω_J1 R S , M E C Ω_J R S , M E C Ω_J1 R S , M J

(%)

1=

C Ω_J R S , M J

0.979587

C Ω_J1 R S , M S

0.979587

C Ω_J R S , M S

. 3.86357610

12

. 4.23450210

10

. 1.3506210

7

. 1.41439110

4

. 1.41439110

4

(%)

C Ω_J1 R S , M NS C Ω_J R S , M NS

vi. Derivation of “Planck-Particle” and SBH characteristics 1. “λx, mx” n Ω_1( r , M )

2 .c 1 . U m( r , M ) 1 . 3 .M .c . 8 U ω( r , M ) 8 4 .π .r3 h .ω ( 1 , r , M ) 4 PV 2

3

2 1 . 3 .M .c .

2 .c

2 1 . 3 .M .c .

3

8 4 .π .r3 h .ω ( 1 , r , M ) 4 8 4 .π .r3 PV

1 . 3 .M .c . 8 4 .π .r3

2 .c

2

3

2 .c

3 1 2 .c .G.M h. . r π .r

1 . 3 .M .c . 8 4 .π .r3

3

3

2 .c

2

3 1 2 .c .G.M h. . r π .r

4

4

3

3 1 2 .c .G.M . 2 .c .G.M h. . 4 π .r π .r r

3 .c . r

3

2 1 . 3 .M .c .

8 4 .π .r3

2 .c

3. c . . 2 cr 1 . 4 h .G

3

3 1 2 .c .G.M . 2 .c .G.M 8 h. . 4 π .r π .r r

3 .c . r n Ω_1( r , M )

1. 4

9

8

3

3

n Ω_1( r , M )

2 .c .G.M π .r

3

n Ω_2( r , M )

3

λh

r . π .c . 3 . 16 2 2 GM λh 7

2 .c .G.M π .r

3 . 16

2

π .m h

8

3

λh

2 .c .G.M π .r

2 3 2

3

7 r . π .m h . 3 16 2 2 M λh λh

3

1

1

9

1. 4

2

. r M λh

3 9

7

n Ω_2( r , M )

1. 3 2

327

7

2

.

π .m h M

7

9

. r λh

9

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9

λ x.λ h

n Ω_3 λ x.λ h , m x.m h

7

16

2

mx

3 3 .π . 7 λx 16 2

2 c .R BH 2 .G

R BH

9

m . π . h. 3 m x.m h λ h λ 2 h

3

2

2 .G . M BH 2 c

9

7

3 3 3 .π . λ x 1 . 3 .π . λ x 16 m 2 x 2 mx 2

M BH

.

5

R BH

7

2

2 c .R BH 2 .G

λ x.λ h

5

m x.m h

2

λ x.λ h

5

m x.m h

2

.

5

R BH

2

.

5

R BH 5

2 1. λ x . c 4 R 3 G BH

2

2 λ x.λ h .c

2

λ x.λ h

5

m x.m h

2

λh

.

5

2 1. λ x . c 4 R 3 G BH

5

5

1.

m x.m h

St ω λ x.λ h , m x.m h

2 .G.m x.m h

6

λ x 33 .π .λ 7 x 2 216

9

3

2

λ x 4.

3 3 .π

λh

.

λx

5

m x.m h 2 2

λx

4 λ .λ x h

2

2

. c 3 G

.

λh

2

5

m x.m h

c

2

5

1.

. c 3 G

4 λ .λ x h

2

2 λ x.λ h .c

mh

λx

mx

2

λh

.

1.λ x

1

2

mx

2 .G.m x.m h

λx 2

9

n Ω_3 λ x

2

2 λ x.λ h .c

5

m x.m h

2 .G.m x.m h 2 m x

G λh

4 . 2 6 π 3

2 2

λx

3 1 . 3 .π . λ x 2 λx 2

9

7

1. 2

3.

3 π.

λx

6

2

2 3 9

n Ω_3 λ x

π.

3.

2

λx

2

2

λ x.λ h = 1.093333 10

10 .

ym

n Ω_3 λ x

λx

1 = 0 ( %)

. m x.m h = 7.36147410

8

mx

( kg )

mx

=

2.698709 1.349354

1 = 0.14278 ( % )

2

Km n Ω_3 n Ω_3

1 3

0.248017

1

0.324994

2

n Ω_3( 1 )

=

0.515897 0.818935

n Ω_3( 2 )

1

n Ω_3 λ x

1.073108

n Ω_3( 3 )

328

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2. “ρm(λxλh,mxmh), Um(λxλh,mxmh)” V( r )

4. . 3 πr 3

ρ m( r , M )

. 94 kg ρ m λ x.λ h , m x.m h = 1.34467810 3 m

M V( r )

ρ m λ x.λ h , m x.m h

. 87 ( YPa) U m λ x.λ h , m x.m h = 1.20853710

ρ m R S, M S

. = 9.55041510

90

3. Physicality of “Kλ” . 42 ( Hz) K ω .ω h = 6.36576910 K λ .λ h

K λ .λ h = 4.709446 10

35 .

m

. K m.m h = 6.34179210

8

( kg )

1 = 0.82832 ( % )

2 .r γγ

4. “KPV @ λxλh” i. “KPV = Undefined” Recognising, U ω λ x.λ h , m x.m h

h . ω PV 1 , λ x.λ h , m x.m h 3 . 2c

4

h . ω Ω λ x.λ h , m x.m h 3 . 2c

4

m h c2 λh

G

It follows that, ω PV 1 , λ x.λ h , m x.m h

3 . . . . 1 . 2 c G mx mh . K PV λ x.λ h , m x.m h λ x.λ h π .λ x.λ h

3

3

1 . λ x.λ h

2 .c .G.m x.m h π .λ x.λ h

3

2 .c .G. 1 . λ x.λ h

. K . . PV λ x λ h , m x m h

λx 2 .c .G. .m h 1 . 2 . K . . PV λ x λ h , m x m h . . λxλh π λ x.λ h

λx

.m h 2 . K . . PV λ x λ h , m x m h π .λ x.λ h

3

c . 1. K PV λ x.λ h , m x.m h λ x.λ h π ω PV 1 , λ x.λ h , m x.m h

ωh

ωh

3

c . 1. K PV λ x.λ h , m x.m h λ x.λ h π

3

. 1. K . . PV λ x λ h , m x m h λx π

3

. 1. K . . PV λ x λ h , m x m h λx π

329

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Performing substitutions yields, U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

U m λ x.λ h , m x.m h h . ω PV 1 , λ x.λ h , m x.m h 3 . 2c

U m λ x.λ h , m x.m h h . ω PV 1 , λ x.λ h , m x.m h 3 . 2c

U m λ x.λ h , m x.m h 4

U m λ x.λ h , m x.m h 4

3 h . ωh. 1. K PV λ x.λ h , m x.m h 3 2 .c λ x π

4

3.

4

2 .π . π .c λ x

.K . . PV λ x λ h , m x m h

3 4 3 2 .π . π .c .λ x U m λ x.λ h , m x.m h . 4 . h ωh K PV λ x.λ h , m x.m h

4

.K . . PV λ x λ h , m x m h

h .ω h 3

U m λ x.λ h , m x.m h

3 4 3 2 .π . π .c .λ x

4

3 h . ωh. 1. K PV λ x.λ h , m x.m h 3 λ π . 2c x

U m λ x.λ h , m x.m h

h .ω h

4

2

2

2

Checking errors yields, 3 h . ωh. 1 3 2 .c λ x π

h .ω h

4

. 1 = 6.66133810

14

(%)

4

3 4 3 2 .π . π .c .λ x ω 3 . h . h. 1 4 3 2 .c λ x π h .ω h

4

. 1 = 6.66133810

14

(%)

3 4 3 2 .π . π .c .λ x

Evaluating, 3 4 3 2 .π . π .c .λ x .U λ .λ , m .m = 8 m x h x h 4 . h ωh

Checking errors yields, 3 4 3 2 .π . π .c .λ x .U λ .λ , m .m m x h x h 4 . h ωh

. 8 = 8.88178410

13

(%)

3 4 3 2 .π . π .c .λ x

h .ω h

4

= 6.619576

10

87

YPa

. 87 ( YPa) U m λ x.λ h , m x.m h = 1.20853710

330

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Simplifying, U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

8 K PV λ x.λ h , m x.m h

2

K PV λ x.λ h , m x.m h

2. 2.

U ω λ x.λ h , m x.m h U m λ x.λ h , m x.m h

Recognising that the EGM spectrum converges to a single mode for a SPBH yields, Ω λ x.λ h , m x.m h

n Ω λ x.λ h , m x.m h

12

Ω λ x.λ h , m x.m h

4. 3

4 . Ω λ x λ h , m x.m h

1 1

4 . 3 = 6.928203

3

Ω λ x.λ h , m x.m h

108.

U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

12. 768 81.

U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

2

4. 3

By inspection, the only solution which satisfies this equation is, U m λ x.λ h , m x.m h U ω λ x.λ h , m x.m h

0

Checking yields, 3

108.0

2 12. 768 81.0 = 6.928203

Therefore, 2. 2

K PV λ x.λ h , m x.m h

K PV R BH, M BH

Undefined

0

K PV λ x.λ h , m x.m h

ii. “KDepp = KPV”

K Depp ( r , M )

1 2 .G.M

2

1

r .c

1

r .c

2

2

K PV( r , M )

2 .G.M

2 .G.M r .c

K Depp ( r , M )

2

K Depp ( r , M )

K PV( r , M )

2

1

K Depp ( r , M )

r .c

K Depp R E, M E = 1.00000000069601

2

K PV R E, M E = 1.00000000069601

K PV( r , M )

1

331

2 .G.M r .c

2

r .c

1

r .c

2

2

2 .G.M

R BH

2 .G . M BH 2 c

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K PV R BH, M BH

2 .G.M BH

1

1

1

2 .G . 2 M BH.c 2 c

1

K Depp λ x.λ h , m x.m h

λh

0

1 . 2 G.m x.m h

2

K Depp λ x.λ h , m x.m h

m h c2

Undefined

1 . 2 G.m x c2 . 1 2 G . λxc

1

2 λ x.λ h .c

K Depp R BH, M BH

λx

mx

G

1

1

2 .m x λx

2

1

2.

λx 0

Undefined

2

1

λx

Undefined

5. “ωΩ_3”

2

M St G. 5 r

St G.

m x.m h

2

λ x.λ h

5

2

St G m h . 3 5 . 4λx λh

3 4 .λ

3

.

x

3 4 .λ

3

π .h

x 2

ω h .m h π .h

2

ωh

.

St G.

9 St G .

m x.m h

9

9

15 . 2

2

π

9

ω Ω_3 λ x.λ h , m x.m h

9

3 λx

3

.

ωh

9

St G.

9

1.

15 2 2 .π 2

3

2

.

ωh

2

λ x.λ h 3

ωh

.

m h .c

7

2

π .h

x

St G.

5

2

3 4 .λ

2

St G.

ωh

2

.m h

λ x.λ h

2

9 m . c . h 2 λ 5 h

h

15 2 2 .π

λx

.m h

3

.

2

5

ω h .m h

x

π .h

m x.m h

2

λ x.λ h

5

St G m h . 3 5 4 .λ x λ h 2

.

5 4 c .ω h 9

2

3

3

λx

.

ωh

9

15 2 2 .π

1

9

1

.

λx

5

λ x.λ h

3

3 4 .λ

3

1

3

5

2 c .m h . h λx

3

2

.

λ x.λ h

2

St G.

2

5 4 c .ω h .

9

2

9 m . c . h 2 λ 5 h

2 1

m x.m h

λx

.ω h

. 1 = 1.11022310

m x.m h

2

λ x.λ h

5

9

St G.

9

3

. 1 .ω h λ x 26 .π2

1. 2

13

(%)

m x.m h

2

λ x.λ h

5

3 λx

3

.

ωh

9

15 2 2 .π

9

3

3

9

3

1. 1 . 3 . 1 .ω .ω h h 2 π2 4 .λ x λ x 26 .π2 2

9

3

1. 1 . 3 .ω h 2 π2 4 .λ x

3

1 9

. 2

2. π

1

3 . ωh . 4λx

9 St G .

m x.m h

9

. 5

λ x.λ h

9

332

3

1 9

2. π

. 2

1

3 . ωh . 4λx

. 1 = 1.11022310

13

(%)

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9

1. 1 . 2 π2

6

4. 4.

9

3

3

9

4

1. 3 . 6 4 25 π3

3

2

m x.m h 9 St G . λ x.λ h

5 9

1

9

9

4

. 1 . 3 . 6 .ω h 4 25 π3

ω Ω_3 λ x.λ h , m x.m h 9

4

1. 3 . 6 . ωh 4 25 π3

3 3 .π 2

1

ω Ω_3 λ x.λ h , m x.m h

. 1 = 1.11022310

. 1 = 1.11022310

13

13

(%)

. 18 ( YHz) ω Ω_3 λ x.λ h , m x.m h = 1.87219710

(%)

4

1. 3 . 6 . ωh 4 25 π3

1. . 18 ( YHz) ω h = 1.84996810 4

λx

e

e 1

α

1

α

ωh

1.

1 . e λx 1 α

= 2.698589

. 1 = 4.43474910

ω Ω_3 λ x.λ h , m x.m h

ω PV 1 , λ x.λ h , m x.m h

ω PV 1 , λ x.λ h , m x.m h

3

(%)

n Ω_2 λ x.λ h , m x.m h = 1

n Ω_3 λ x

n Ω_3 λ x = 1

1 = 1.18731904721517( % )

4 ω Ω_3 λ x.λ h , m x.m h

ω Ω_3 λ x.λ h , m x.m h

6. “ωΩ_4” 9

ω Ω_4 M BH

2

9

M BH

St G.

2 .G.M BH c

c. 5

c .St G

9

5 3 ( 2 .G) .M BH

ω Ω_4 M BH

St BH.

c .St G 5 ( 2 .G)

2

ω Ω_4 m x.m h 3

St BH

c.

1 M BH

ω Ω_4 M S 10 ω Ω_4 10 .M S

333

. 18 1.87219710 = 6.23977510 . 5

( YHz)

289.624693

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7. “rS” i. “rS(λxλh)”

ρ m( r , M )

3 .M

ρ m λ x.λ h , m x.m h

4 .π .r

3

ρ m λ x.λ h , m x.m h 3 .m h

. 1 = 2.22044610

14

3.

3 .m x.m h

λx 2

4 .π . λ x.λ h

3

.m h

3 .m h

4 .π . λ x.λ h

3 .M BH

ρ m r S , M BH

(%)

2 3 8 .π .λ x .λ h

3

4 .π .r S

3

2 3 8 .π .λ x .λ h

M BH

ρ m λ x.λ h , m x.m h

ρ m r S , M BH

3

ωh

2

R BH( M )

2 .G.λ x

3

2

rS

3

r S M BH

λ x.λ h

2

ωh

ωh

2

2 .G.λ x

2

3

.M . . 2. BH λ h 2 λ x

= 1.195378 10

32 .

G

2

λh

2

ωh

2

kg

. = 5.63257510

94

2

3

m

M BH

3

r S R BH

mh

r S λ x.λ h

2

am

λh

c

2

2 .G.M c

2 2 .G.λ x

m h c2

2 3 2 .λ x .λ h

rS

M BH

mh

3

2 λ x.λ h .R BH

2 λ x.λ h . λ x.λ h

λ x.λ h

ii. “rS(ΜΒΗ), rS(RΒΗ)” 3

ρS

ρ m λ x.λ h , m x.m h

r S M BH

r S R BH

3

2 3 .c .R BH

8 .π .G.ρ S

2 λ x.λ h .R BH M S 3

2.

3 .c R BH M S 8 .π .G.ρ S

1 rS MS

3 .M BH 4 .π .ρ S

3 3

3.

.

4 .π .r S

3

2 λ x.λ h .R BH M S 3

2.

3 .c R BH M S 8 .π .G.ρ S

rS MS 1 = 0 (%)

2 c .R BH 2 .G

1=

. 3.28046310

5.

r S 10 M S

=

10 r S 10 .M S

334

0.015227

ρS

2 3 .c .R BH

8 .π .G.r S

3

. 1.11022310

14

. 1.11022310

14

(%)

4

( am)

0.706754

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ρ m r S m x.m h , m x.m h ρ m r S M S ,M S 1 . 5 5 ρ S ρ m r S 10 .M S , 10 .M S

1=

. 1.29896110

12

. 8.32667310

13

. 7.66053910

13

. 6.7723610

10 10 ρ m r S 10 .M S , 10 .M S

(%)

13

U m r S M S ,M S

. 8.10462810

1 . U m r S 105 .M S , 105 .M S . . U m λ x λ h,m x mh 10 10 U m r S 10 .M S , 10 .M S ρ m r S m x.m h , m x.m h

. 7.2164510

13

. 6.7723610

13

(%)

U m λ x.λ h , m x.m h

ρ m r S M BH , M BH 10 r S 10 .M S

r uq = 0.768186 ( am)

1=

13

U m r S M BH , M BH

1 = 7.996993 ( % )

r uq

iii. “MBH(rS)” 4. . 3 π ρ S .r S 3

M BH r S

M BH r tq

. 10 = 2.27391910

. 40 ( kg ) M BH r tq = 4.52155110 M BH r uq

MS

. 10 = 1.28408510

MS

M BH r ε

M BH r π

M BH r ν

M BH r µ

. 13 1.62379510 . 19 1.60185510 . 19 1.57097210 . 13 4.66247210

M BH r τ

M BH r en

M BH r µn

M BH r τn

. 13 2.45782610 . 7 5.19529810

1 . M BH r uq

M BH r dq

M BH r sq

M BH r cq

= 1.28408510 . 10 2.95005410 . 10 1.9828610 . 10 3.68186410 . 10

M BH r bq

M BH r tq

M BH r W

. 10 2.27391910 . 10 5.99684310 . 10 3.39015710 . 10 3.47948910

M BH r H

M BH r γγ

M BH r gg

M BH r Z 1 .( kg )

MS

. 10 2.3560510

. 9 2.12850410 . 11 7.96867110

0

0

14.554628

. 6 5.06892810

R BH M BH r τn

16.217926

. 7.67248410

R BH M BH r sq

R BH M BH r cq

= 4.00847210 .

R BH M BH r tq

R BH M BH r W

R BH M BH r Z

0.010862

R BH M BH r γγ

R BH M BH r gg

R BH( 1.( kg ) )

. 7.35477510

R BH M BH r ε

R BH M BH r π

R BH M BH r ν

R BH M BH r µ

R BH M BH r τ

R BH M BH r en

R BH M BH r µn

R BH M BH r uq

R BH M BH r dq

R BH M BH r bq R BH M BH r H

ω Ω_4 m x.m h

5 ω Ω_4 10 .M S 10 ω Ω_4 10 .M S

=

. 5 6.23977510

. 9.2090510

3

. 7.0983910

3

3

0

U m λ x.λ h , m x.m h

. 18 1.87219710

ω Ω_4 M S

3

. 4 1.34431910

5 5 U m R BH 10 .M S , 10 .M S

289.624693

10 10 U m R BH 10 .M S , 10 .M S

335

. 6 5.0004410

4.904034

. 2.48754410

3

0.066445

. 6.18980410

3

0.011494 ( Lyr)

0.01872

0.010583

0

0

. 87 1.20853710

U m R BH M S , M S ( YHz)

6

0

=

. 12 1.65639710

( YPa)

165.639685 . 1.65639710

8

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r S mh 1 . r m .m S x h λh λx

r S m x.m h R BH m x.m h

144.219703

1=

. 4.21884710

(%)

13

. 1 = 4.44089210

M BH r π

1 = 22.109851 ( % )

=

M BH r e

R BH m h

(%)

. 43 9.27104510

M BH r ε r S mh

13

M BH r Bohr

. 49 3.22881910 . 51 1.26038310

( kg )

. 63 8.34661610

8. “r → RBH” i. “nΩ → nΩ_4, nΩ_5, nBH” n Ω_2 r S M BH , M BH

n Ω_4 M BH

n BH M BH

n Ω_5 M BH

n Ω_5 M BH n Ω_4 M BH

n Ω_4 m x.m h

n Ω_5 m x.m h

n BH m x.m h

n Ω_4 M S

n Ω_5 M S

n BH M S

5 n Ω_4 10 .M S

5 n Ω_5 10 .M S

5 n BH 10 .M S

10 n Ω_4 10 .M S

10 n Ω_5 10 .M S

10 n BH 10 .M S

R BH M S

∆R bh

n Ω_2 R BH M BH , M BH

rS MS

200

R bh

1 =

1

1

. 5 9.00254210 . 24 2.56419310 . 19 3.51086810 . 6 1.93953910 . 28 1.0035610 . 22 1.93265910 . 7 4.1786110 . 31 3.92767810 . 24 1.06388810

r S M S , ∆R bh .. R BH M S Harmonic Cut-Off Mode vs Radial Disp.

Harmonic Cut-Off Mode

rS MS

R BH M S

n Ω _2 R bh , M S 5 n Ω _2 R bh , 10 .M S 10 n Ω _2 R bh , 10 .M S n Ω _4 M S

R bh Radial Displacement

Schwarzschild-Black-Hole (1 Solar Mass) Schwarzschild-Black-Hole (10^5 Solar Masses) Schwarzschild-Black-Hole (10^10 Solar Masses)

336

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ii. “ωΩ → ωΩ_5, ωBH” ω Ω_5 M BH

ω Ω_3 r S M BH , M BH

ω Ω_4 m x.m h

. 5 6.23977510

=

5 ω Ω_4 10 .M S 10 ω Ω_4 10 .M S

. 18 1.87219710

ω Ω_5 M S ( YHz)

5 ω Ω_5 10 .M S

289.624693

10 ω Ω_5 10 .M S

1

ω BH M S

. 13 7.30358710

10 ω BH 10 .M S

ω Ω_4 M BH

. 4 1.34431910

ω BH m x.m h =

ω Ω_5 M BH

ω Ω_5 m x.m h

. 18 1.87219710

ω Ω_4 M S

5 ω BH 10 .M S

ω BH M BH

=

. 19 4.55727410 . 19 6.9805610

( YHz)

. 20 1.06924110

. 15 5.19263810 . 17 3.69181510

ω Ω_5 m x.m h ω Ω_5 M S 1 . 5 ω h ω Ω_5 10 .M S 10 ω Ω_5 10 .M S

ω Ω_4 m x.m h

0.253004

ω Ω_4 M S 5 ω Ω_4 10 .M S 10 ω Ω_4 10 .M S

=

0.253004

. 6.158585 8.43227510

14

. 9.433354 1.81667910

15

14.44945

0

Harmonic Cut-Off Freq. vs Radial Disp.

Harmonic Cut-Off Frequency

rS MS

R BH M S

ω Ω _3 R bh , m x .m h ω Ω _3 R bh , M S 5 ω Ω _3 R bh , 10 .M S 10 ω Ω _3 R bh , 10 .M S

R bh Radial Displacement

Schwarzschild-Planck-Black-Hole Schwarzschild-Black-Hole (1 Solar Mass) Schwarzschild-Black-Hole (10^5 Solar Masses) Schwarzschild-Black-Hole (10^10 Solar Masses)

337

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iii. “ωΩ_6, ωΩ_7, ωPV_1” ω Ω_6 M BH

ω Ω_5 M BH n Ω_4 M BH

ω PV_1 M BH

ω Ω_4 M BH

ω Ω_7 M BH

n Ω_5 M BH

ω Ω_6 M BH ω Ω_7 M BH

ω Ω_6 m x.m h

ω Ω_7 m x.m h

. 42 1.87219710 . 42 1.87219710

ω Ω_6 M S

ω Ω_7 M S

. 38 6.93112610 . 4 1.29804810

5 ω Ω_6 10 .M S

5 ω Ω_7 10 .M S

10 ω Ω_6 10 .M S

10 ω Ω_7 10 .M S

ω PV_1 m x.m h

=

. 37 3.61189510

( Hz)

0.693113

. 37 6.93112610 . 1.00503110

6

1

ω PV_1 M S =

5.

ω PV_1 10 M S

. 33 1.8727810 . 5.21112310

37

1

.

ωh

10 ω PV_1 10 .M S ( Hz)

= 5.103269

. 42 1.45002610

10 ω PV_1 10 .M S

Fundamental Freq. vs Radial Disp. rS MS

R BH M S

Fundamental Frequency

ω Ω _3 R bh , m x .m h n Ω _2 R bh , m x .m h ω Ω _3 R bh , M S n Ω _2 R bh , M S 5 ω Ω _3 R bh , 10 .M S 5 n Ω _2 R bh , 10 .M S ω Ω _3 R bh , 10 n Ω _2 R bh , 10

10 . MS

10 . MS

R bh Radial Displacement

Schwarzschild-Planck-Black-Hole Schwarzschild-Black-Hole (1 Solar Mass) Schwarzschild-Black-Hole (10^5 Solar Masses) Schwarzschild-Black-Hole (10^10 Solar Masses)

338

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9. “TL” M .c

E( M )

2

n γ ω , M BH

1. n γ ω , M BH 2 E g( ω )

h .ω

E γ( ω )

E g ( ω ) E x.E γ ( ω )

E M BH

E M BH

n g ω , M BH

E γ( ω ) E M BH

E M BH E x.E γ ( ω )

E m x.m h = 6.616163 ( GJ)

2 .E γ ( ω )

n γ ω Ω_4 m x.m h , m x.m h n g ω Ω_4 m x.m h , m x.m h

=

Ex

= 6.616163 ( GJ)

P g ω Ω_4 m x.m h

n γγ( M )

E g ω Ω_4 m x.m h

= 1.240531 ( GJ)

P γ ω Ω_4 m x.m h

c

= 8.275929 ( Ns )

2 .n gg ( M )

T Ω _3( r , M )

T L r S λ x.λ h , m x.m h

10 10 T L r S 10 .M S , 10 .M S

=

m γγ m gg

3.195095

=

6.39019

1

P g( ω )

10

=2

E g( ω ) c

45 .

T L( r , M )

ω Ω _3( r , M )

n gg ( M )

eV

E( M ) m gg

n gg ( M ) .T Ω _3( r , M ) n g ω Ω _3( r , M ) , M

9 10 .yr

. 13 4.10173110 . 13 4.10173110

T L r uq , m uq

. 13 4.10173110

T L R BH M S , M S

10 10 T L R BH 10 .M S , 10 .M S

s

E g ω Ω_4 m x.m h E γ ω Ω_4 m x.m h

= 2.481061 ( GJ)

= 4.137964 ( Ns )

. 13 4.10173110

T L R BH λ x.λ h , m x.m h

5 5 T L R BH 10 .M S , 10 .M S

43 .

. 13 4.10173110

T L r S M S ,M S 5 5 T L r S 10 .M S , 10 .M S

2 .E M BH 2 n γ ω , M BH .E γ ( ω )

2.666667

n g ω Ω_4 m x.m h , m x.m h .E g ω Ω_4 m x.m h

E γ( ω )

1. n γ ω , M BH 2

5.333333

= 6.616163 ( GJ)

P γ( ω )

ω Ω_4 M BH

T Ω_4 m x.m h = 5.341319 10

n γ ω Ω_4 m x.m h , m x.m h .E γ ω Ω_4 m x.m h

E γ ω Ω_4 m x.m h

1

n g ω , M BH

E g( ω )

1. n γ ω , M BH 2

E g( ω )

T Ω_4 M BH

=

. 13 4.10173110 . 4.10173110

13

9 10 .yr

T L r ε, m e T L r π, m p T L r ν,mn

. 13 4.10173110

339

. 13 4.10173110 =

. 13 4.10173110 . 4.10173110

13

9 10 .yr

. 13 4.10173110

www.deltagroupengineering.com

1 m γγ

h.

=

2

. 13 4.10173110 . 4.10173110

13

m γγ T L λ x.λ h , m x.m h . m gg h 2

9.

10 yr

m gg H0

71.

km . s Mpc

=

1

TL

1

h m γγ

. T L.H 0 = 2.97830810

12

10. “ωg, ngg” T PV n PV, r , M

T g n PV, r , M

n g ω PV n PV, r , M , M

n g ω PV n PV, r , M , M T PV n PV, r , M

T PV n PV, r , M

ω PV n PV, r , M .n g ω PV n PV, r , M , M

ω PV n PV, r , M .n g ω PV n PV, r , M , M

ω g n PV, r , M

n g ω PV n PV, r , M , M

ω g n PV, r , M

1 ω PV n PV, r , M . .n γ ω PV n PV, r , M , M 2

E( M ) 1 ω PV n PV, r , M . . 2 E γ ω PV n PV, r , M

1 E( M ) ω PV n PV, r , M . . 2 E γ ω PV n PV, r , M 1 E( M ) ω PV n PV, r , M . . 2 h .ω PV n PV, r , M ω g m x.m h

E( M ) 2 .h

=

10 ω g 10 .M S

. 56 1.34855310 . 61 1.34855310

E MS 1 . 5 m gg E 10 .M S 10 E 10 .M S

M .c 2 .h

2

n gg ( M )

5 n gg 10 .M S 10 n gg 10 .M S

T L.ω g ( M )

. 72 6.46222510

n gg M S

( YHz)

. 66 1.34855310

E m x.m h

ω g( M )

n gg m x.m h

. 18 4.99252510

ωg MS 5 ω g 10 .M S

1 E( M ) ω PV n PV, r , M . . . 2 h ω PV n PV, r , M

=

. 110 1.7455410 . 115 1.7455410 . 120 1.7455410

. 72 6.46222510 =

. 110 1.7455410 . 115 1.7455410 . 120 1.7455410

340

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11. BH’s r0

c

1

9 r 0 = 13.772016 10 .Lyr

H0

5

St G.

r ω ω Ω_3 , M

M

ω VL λ VL

c

ω VL( 750 ( nm ) )

9

λ VL

ω VL( 400 ( nm ) )

ω Ω_3 r 0 , M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

5 ω Ω_3 r 0 , 10 .M S

5 ω Ω_3 r 0 , 10 .M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

10 .

218.810356

410.269418

. 4 6.84370610 . 4 3.64997710

ω Ω_3 r 0 , 10 M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

4 ω Ω_3 1.63.10 .r 0 , M S

4 ω Ω_3 5.052.10 .r 0 , M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

6 5 ω Ω_3 1.63.10 .r 0 , 10 .M S

6 5 ω Ω_3 5.052.10 .r 0 , 10 .M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

ω Ω_3 1.63.10 r 0 , 10 M S

10 .

8.

ω Ω_3 5.052.10 r 0 , 10

ω VL( 400 ( nm ) )

10 C Ω_J1 r 0 , 10 .M S

( THz)

. 3 5.29883310 . 3 = 2.82604410

10 .

MS

5 C Ω_J1 r 0 , 10 .M S

749.481145

27.355887

ω Ω_3 r 0 , M S

C Ω_J1 r 0 , M S

399.723277

= 2.118067 ( EHz)

10 ω Ω_3 r 0 , 10 .M S

8.

=

0.163994

5 ω Ω_3 r 0 , 10 .M S

ω Ω_3 r 0 , 10

H0

2

ω Ω_3

ω Ω_3 r 0 , M S

9 = 13.772016 10 .yr

0.999916 1.000078 = 0.999916 1.000078 0.999916 1.000078

10 .

MS

ω VL( 750 ( nm ) ) . 1.48429110

5

= 8.89809310 .

3

10

20 .

yJy

5.334267

1 10 C Ω_J1 r 0 , 10 .M S .

C Ω_J1 r 0 , M S 1

=

. 5 3.59381410 599.48425

5 C Ω_J1 r 0 , 10 .M S

341

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r ω ω VL( 400 ( nm ) ) , M S

r ω ω VL( 750 ( nm ) ) , M S

5 r ω ω VL( 400 ( nm ) ) , 10 .M S

5 r ω ω VL( 750 ( nm ) ) , 10 .M S

10 r ω ω VL( 400 ( nm ) ) , 10 .M S

10 r ω ω VL( 750 ( nm ) ) , 10 .M S

r ω ω VL( 400 ( nm ) ) , M S

5 r ω 30.( EHz) , 10 .M S

10 r ω 30.( PHz) , 10 .M S

10 r ω 30.( EHz) , 10 .M S

C Ω_J1 r ω 30.( PHz) , 10 M S , 10 M S 10 10 C Ω_J1 r ω 30.( PHz) , 10 .M S , 10 .M S

C Ω_J1 r ω 30.( EHz) , 10 M S , 10 M S 10 10 C Ω_J1 r ω 30.( EHz) , 10 .M S , 10 .M S

=

. 6 2.95234410

0.741144

( Lyr)

16 .

yJy

= 2.12751776034345 .103 8.46980075872643 .10

3

.105 2.12751776034345

= 2.93002110 .

7

0.846980075872643

1.166462 116.646228

6 10 .Lyr

. 9 1.16646210 . 4 2.93002110

2.164916 . = 2.16491610

3

. 2.16491610

6

C Ω_J1 r ω 30.( EHz) , M S , M S 5.

0.239057

5

. 5 2.93002110

C Ω_J1 r ω 30.( PHz) , M S , M S 5.

10

=

.10 21.2751776034345 8.46980075872643

10 r ω 30.( EHz) , 10 .M S

5 r ω 30.( PHz) , 10 .M S

r ω 30.( EHz) , m x.m h

1.102778

5 r ω 30.( EHz) , 10 .M S

r ω 30.( EHz) , M S

r ω 30.( PHz) , m x.m h

28.979765

=

r ω 30.( PHz) , M S

5.

. 8 5.05271110 . 8 1.62975410

r ω ω VL( 750 ( nm ) ) , m x.m h

r ω 30.( EHz) , M S

5.

= 1.62975410 . 6 5.05271110 . 6

r ω ω VL( 400 ( nm ) ) , m x.m h

C Ω_J1 r ω ω VL( 750 ( nm ) ) , m x.m h , m x.m h

1 . r 30.( PHz) , 105 .M ω S r0 10 r ω 30.( PHz) , 10 .M S

. 4 5.05271110 . 4 1.62975410

10 r ω ω VL( 750 ( nm ) ) , 10 .M S

C Ω_J1 r ω ω VL( 400 ( nm ) ) , m x.m h , m x.m h

9 10 .Lyr

9 . 9 2.2445.10 6.95860210

5 r ω ω VL( 750 ( nm ) ) , 10 .M S

1. 10 ( Lyr) , 10 .M S = 1.031709 10

r ω 30.( PHz) , M S

= 2.2445.107 6.95860210 . 7

r ω ω VL( 750 ( nm ) ) , M S

1 . r ω ( 400 ( nm ) ) , 105 .M ω VL S r0 10 r ω ω VL( 400 ( nm) ) , 10 .M S

K PV

5 . 5 2.2445.10 6.95860210

10

29 .

10

14 .

yJy

8.618686 . = 8.61868610

3

. 8.61868610

6

yJy

3 10 .km

11.753495

7 C Ω_J1 r ω 30.( PHz) , m x.m h , m x.m h = 6.228302 10 .yJy

C Ω_J1 r ω 30.( EHz) , m x.m h , m x.m h = 2.479532 ( fJy )

342

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vii. Fundamental Cosmology 1. “Hα, HU” i. “AU, RU, HU” 5

C Ω_J1 r 1 , M 1

9

C Ω_J1( r , M )

M St J . 26 r

ln

C Ω_J1 r 1 , M 1 C Ω_J1 r 2 , M 2

λ y r 2, M 2

ln

9

5

M1

ln

.

M2

r2

r1 M1

r1

K U r 2, r 3, M 2, M 3

9

ln

ln

A U r 2, r 3, M 2, M 3

H U r 2, r 3, M 2, M 3

ln

.

r2

26

5

ln

ri

C Ω_J1 r 1 , M 1 C Ω_J1 r 2 , M 2

λ y r 2 , M 2 .r 3 λ y r 2, M 2 .M 3 2 5

λy M3 r . . 1 . 2 2 M2 λy r3

C Ω_J1 r 2 , M 2

λy M3 r . . 1 . 2 2 M2 λy r3

rf

r1

C Ω_J1 r 3 , M 3

26

26

λ y r 2, M 2 .M C Ω_J1 λ y r 2 , M 2 .r 3 , 3 2 C Ω_J1 r 2 , M 2 5

5

K U r 2, r 3, M 2, M 3

5

M2

26

ln n Ω_2 r 2 , M 2

C Ω_J1 r 2 , M 2

M1

C Ω_J1 r 2 , M 2

9

1

C Ω_J1 r 3 , M 3

9

1 2

9

7

.ln n Ω_2 r 2 , M 2

TL K U r 2, r 3, M 2, M 3

5

1

3.

M3

26

9

.

M2

r2

9

r3

R U r 2, r 3 , M 2, M 3

c .A U r 2 , r 3 , M 2 , M 3

9 A U K λ .R o , λ x.λ h , K m.M G, m x.m h = 14.575885 10 .yr

A U r 2, r 3 , M 2, M 3

9 R U K λ .R o , λ x.λ h , K m.M G, m x.m h = 14.575885 10 .Lyr

H U K λ .R o , λ x.λ h , K m.M G, m x.m h = 67.084304 1 . H U K λ .R o , λ x.λ h , K m.M G, m x.m h H0

km s .Mpc

1 = 5.515064 ( % )

343

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H U R o , λ x.λ h , M G, m x.m h H U R o , λ h , M G, m h

66.700842

H U K λ .R o , λ h , K m.M G, m h

70.06923

H U R o , λ x.λ h , M G, m x.m h 1 . H0

km . s Mpc

= 69.672169

6.055152

H U R o , λ h , M G, m h

1=

1.870184 ( % )

H U K λ .R o , λ h , K m.M G, m h

1.310944

1 km H U K λ .R o , λ x.λ h , .K m.M G, m x.m h = 67.753267 . 3 s Mpc

H U K λ .R o , λ h , K m.M G, m h 1 H U K λ .R o , λ h , .K m.M G, m h 3

1=

H U K λ .R o , λ x.λ h , K m.M G, m x.m h

0.978843 0.987352

(%)

1 H U K λ .R o , λ x.λ h , .K m.M G, m x.m h 3

ii. “Hα” 3 .H ρm 8 .π .G 2

H α r 3, M 3

2.

. 61 H α λ x.λ h , m x.m h = 8.46094110

λx H α λ x.λ h , m x.m h . ωh

2. . . π G ρ m r 3, M 3 3

H α λ x.λ h , m x.m h

ωh

km s .Mpc

. 1 = 4.44089210

λx

14

. = 8.46094110

61

ωh λx km s .Mpc

(%)

iii. “ρU” 3 .H U r 2 , r 3 , M 2 , M 3 8 .π .G

ρ U r 2, r 3, M 2, M 3

2

ρ U K λ .R o , λ x.λ h , K m.M G, m x.m h ρ U K λ .R o , λ h , K m.M G, m h 3 .H 0

2

8.453235 = 9.222226

10

33 .

kg 3

cm

9.468862

8 .π .G

Hence, 8.45 ρ U . 10

33 .

kg

9.23

3

cm

344

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iv. “MU” M U r 2, r 3, M 2, M 3

V R U r 2 , r 3 , M 2 , M 3 .ρ U r 2 , r 3 , M 2 , M 3

. 52 ( kg ) M U K λ .R o , λ x.λ h , K m.M G, m x.m h = 9.28458610

2. “TU” K T r 2, r 3 , M 2, M 3

λ Ω_3 r 3 , M 3

n g ω Ω_3 r 3 , M 3 , M 3 .ln

c

T0

ω Ω_3 r 3 , M 3

T W r 2, r 3, M 2 , M 3

T U r 2, r 3, M 2, M 3

H α r 3, M 3 H U r 2, r 3, M 2, M 3

2.725.( K )

. K W = 2.89776910

3

( m.K )

KW λ Ω_3 R U r 2 , r 3 , M 2 , M 3 , M 3 K T r 2 , r 3 , M 2 , M 3 .T W r 2 , r 3 , M 2 , M 3

T U K λ .R o , λ x.λ h , K m.M G, m x.m h = 2.724752 ( K )

1 . T U K λ .R o , λ x.λ h , K m.M G, m x.m h

. 1 = 9.08391310

3

(%)

T0

T U R o , λ x.λ h , M G, m x.m h T U R o , λ h , M G, m h

2.716201 = 1.199134 ( K )

T U K λ .R o , λ h , K m.M G, m h

1.202877

T U R o , λ x.λ h , M G, m x.m h 1 . T0

T U R o , λ h , M G, m h T U K λ .R o , λ h , K m.M G, m h

0.322893 1=

55.995089 ( % ) 55.857737

1 T U K λ .R o , λ x.λ h , .K m.M G, m x.m h = 2.739618 ( K ) 3

T U K λ .R o , λ x.λ h , K m.M G, m x.m h

1 = 0.542607 ( % )

1 T U K λ .R o , λ x.λ h , .K m.M G, m x.m h 3

345

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3. “TU → TU2” 9

c

c

λ Ω_3( r , M )

ω Ω_3( r , M )

9

2

M St G. 5 r

9

5

c c λ x. λ Ω_3 , mh H 2

1 . St G

c.

λx

8 . H α r 3, M 3 ln 3 H

K T( H ) .T W ( H )

ωh 8. . ln 3 λ x.H

ωh 8. . ln 3 λ x.H

2

.m h

1 . 2 St G λ x.m h

2

. c H

λ Ω_3

KW

T U2( H ) K T( H ) .T W ( H )

c λ x. mh λ Ω_3 , H 2

c λ x. , mh H 2

ωh 8. . ln 3 λ x.H

c λ x. , mh H 2

KW 9

c.

1 . 2 St G λ x.m h

2

. c H

9

c.

1 . 2 St G λ x.m h

2

. c H

5

ωh λ .m 8 .K W . . St . x h ln G 3 c 2 λ x.H

. H α = 8.46094110

km . s Mpc

. H c

5

ωh λ .m 8 KW. . St . x h T U2( H ) . ln G 3 c 2 λ x.H 9

. 8 . St G . λ x m h St T 5 3 .c 2 c

8 . 3 .c

2

9

61

9

3.

5

9

KW

λx

9

5

KW

KW

ωh

Hα

c.

T W( H)

ωh 8. . ln 3 λ x.H

λ Ω_3

9

H

2

K T( H )

5

1 . r St G M 2

c.

3 .ω h 4 .π .h c

5

2

. c 2

9

2

3.

2

. 8 . St G . λ x m h 5 3 .c 2 c

8 . 3 .c

3 .ω h 4 .π .h c

5

2

. c 2

2

. H c

5

9

.

λ x.m h

2

2

9

.

λ x.m h 2

2

9

. 8 .c . 3 . 3 ω h 3 .c 2 c5 4 .π .h

2

.

λ x.m h

2

2

346

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9

. 8 .c . 3 . 3 ω h 3 .c 2 c5 4 .π .h 9

4. 3

3

3 4 .c

.

9

3

λ x.m h

9

2

2

2

4.

3

3

6 5 2 .c

3

3 4 .c

3

3

4.

9

2

m λ . h. x π .h λ h

4. 3. . λ x c 3 4 π .h .G

T U2( H )

2

2 λ . x .c π .h G

9

3 λx 4. 3. . c 2 3 3 4 π .c .λ h

K W .St T .ln

ωh λ x.H

.

λ x.m h .ω h π .h

3

4.

3

3

6 5 2 .c

.

4. 3

3 4 .c

4.

π .h

3

9

4. 3. 1 . λ x 3 4 c5 π .λ 2 h

9

2

4. 3

3

.

mh λ x . π .h λ h

3 3 . λx c. 4 π .h .G

2

2

2 9 . 95 St T = 6.35557910

s

5 9

m

T U2 H U K λ .R o , λ x.λ h , K m.M G, m x.m h

. H5

3 4 .c

3

2 λ . x .c π .h G

3

St T

9

2

λ x.m h .ω h

9

2

2

9

9

2

= 2.72475246336977( K )

4. “TU2 → Ro, MG, HU2, ρU2” ∆R o

0.5.( kpc )

T U2 H U2 R o , M G

T U2 H 0

T U2 H U2 R o

1 . T U2 H U2 R o , M G T0

= 2.724752 ( K )

1 T U2 H U2 R o , .M G 3

T U2 H U2 R o

H U K λ .r , λ x.λ h , K m.M , m x.m h

H U2( r , M )

=

2.739618

( K)

2.810842

1 ∆R o , .K m.M G 3 1 ∆R o , .K m.M G 3

=

2.733025 2.741859

. 1 = 9.08391310

T U2 H U2 R o

∆R o , M G

T U2 H U2 R o

∆R o , M G

∆T 0

( K)

Computational environment initialisation values →

=

2.720213 2.729021

3

(%)

( K)

0.001.( K )

r x1

1

m g1

1

r x2

1

m g2

1

Given T U2 H U2 r x1.R o , M G T U2 H U2 R o , m g1 .M G

T0

∆T 0

T U2 H U2 r x2.R o , M G T U2 H U2 R o , m g2 .M G

r x1

r x1

r x2

r x2

m g1 m g2

Find r x1, r x2, m g1 , m g2

m g1

T0

∆T 0

0.989364 =

m g2

347

1.017883 1.057292 0.911791

www.deltagroupengineering.com

T U2 H U2 r x1.R o , M G T U2 H U2 r x2.R o , M G T U2 H U2 R o , m g1 .M G T U2 H U2 R o , m g2 .M G

2.724 =

2.726

R o.

( K)

2.724

r x1

=

r x2

7.914908 8.143063

( kpc )

2.726

. 11 M G m g1 6.34375310 . = M S m g2 . 11 5.47074910

r x1 m g1

1.063645 5.729219

1=

r x2 m g2

1.788292

8.820858

(%)

Given T U2 H U2 r x1.R o , m g1 .M G T U2 H U2 r x1.R o , m g2 .M G

∆T 0

T0

T U2 H U2 r x2.R o , m g1 .M G T U2 H U2 r x2.R o , m g2 .M G

r x3

r x3

r x4

r x4

m g3

Find r x1, r x2, m g1 , m g2

=

2.724

=

2.726

1.013348 0.977007 0.977007

m g4

T U2 H U2 r x3.R o , m g3 .M G T U2 H U2 r x4.R o , m g4 .M G

∆T 0

0.984956

m g3

m g4

T0

R o.

( K)

. 11 M G m g3 5.8620410 . = M S m g4 . 11 5.8620410

r x3

=

r x4

7.879647 8.106786

r x3 m g3 r x4 m g4

1=

( kpc )

1.50441 2.29934 1.334822 2.29934

(%)

Hence, if “T0” is exactly correct (i.e. zero experimental uncertainty); “Ro”, “MG” and “HU2” may be approximated as follows, Given T U2 H U2 r x1.R o , m g1 .M G r x5 m g5

Find r x1, m g1

r x5.R o = 8.107221 ( kpc )

T0 r x5 m g5

m g5 .

H U2 r x5.R o , m g5 .M G = 67.095419

=

MG

1.013403

T U2 H U2 r x5.R o , m g5 .M G

1.052361

. = 6.31416710

11

MS

r x5 m g5

1=

1.340256 5.236123

= 2.725 ( K )

( %)

km . s Mpc

ρ m R U K λ .R o , λ x.λ h , K m.M G, m x.m h , M U K λ .R o , λ x.λ h , K m.M G, m x.m h

= 8.453235 10

33 .

kg 3

cm

348

www.deltagroupengineering.com

5. “UZPF”

Ω

ρ ρc

Ω EGM

Ω PDG

Ω EGM Ω PDG

3 .H U2( r , M )

ρ U2( r , M )

ρ U2 R o , M G = 8.453235 10

8 .π .G

Ω EGM = 1.000331

ρ U2 R o , M G

. Ω ZPF = 3.31400710

1

U ZPF

3 .c . H U2 R o , M G Ω ZPF . 8 .π .G

13 .

U ZPF = 2.51778 10

Ω EGM

U ZPF = 251.778016

Pa

fJ

U ZPF = 251.778016

U ZPF = 842.934914

3

kg 3

Ω PDG Ω m Ω γ .. Ω ν

Ω ZPF

2

= 0.997339

33 .

cm

ρ U2 r x5.R o , m g5 .M G

1.003

2

m

2

ΩΛ

4

U ZPF = 251.778016( fPa )

yJ

U ZPF = 0.251778

3

mJ 3

mm

km

EJ

. 12 U ZPF = 7.39723510

AU

3

YJ pc

3

YJ

. 41 U ZPF = 6.60189810

R U K λ .R o , λ x.λ h , K m.M G, m x.m h

3

viii. Advanced Cosmology 1. “nΩ_2 → nΩ_6” 9

n Ω_6( r , M )

. . 3 3 . π mh . K λ r . 16 2 KmM λh

9

7

9 9

9

1 6.

. 3 3 .K λ π m h . r 16 M λh 2 2 . 3 3 . π .π m h . r 16 2 M λh 2

7

7

9

n Ω_6( r , M )

3 .π . m h 4

M

9

6

1

.

π .m h

. r M λh

3 ( 3 .π ) . m h . r 18 M λh 2 1

3

3

3

3 . π 16 2 2

. . 3 3 . π mh . K λ r . 16 2 KmM λh

7

3

3 . π 16 2 2 9

7

3

7 9

6. . 3 3 .K λ π m h . r 16 M λh 2

7

6

.

π .m h

. r M λh

3 ( 3 .π ) . m h . r 18 M λh 2

7

9

1. 4

7

9

2 . 3 3 . π .π m h . r 16 2 M λh 2

3 mh . r ( 3 .π ) . M λh

7

7

7

9

. r λh

9

349

www.deltagroupengineering.com

2. “KU2 → KU3”

K U2( r , M )

ln

Kλ λx

7

5

3

9

.ln n Ω_6( r , M )

.

mh

26 9

. r λh

4 .M

K U3( r , M ) ln ( 3 .π )

7

5

2 . ln n Ω_6( r , M ) 256

3.

3

Kλ

( 3 .π )

λx

7 18 6.

5

2

256

26 9

. r λh

M

.

4

9

mh

9

1

5 7 18 6.

7

5

3. “HU2 → HU3, TU2 → TU3” K U2( r , M )

H U3( r , M )

5

TL 7 5

7 18 5 6 2

ln ( 3 .π ) .

256

7

.ln n Ω_6( r , M )

3.

9

mh M

7 18 5 6 2

9

. r λh

3

1

26

ln ( 3 .π ) .

.ln

3

9

3 .π . m h

256

4

9

. r λh

M

5

7

.

mh M

26

9

. r λh

9

7 7

5

1 . 18 . . 6 . 2 ( 3 π ) ln 256

K U3( r , M )

ln

T U3( r , M )

T U2 H U3( r , M )

1

1

3 ( 3 .π ) . m h

9

4

3

. r λh

M

5

7 9

.

mh M

26

9

9

. r λh

4. “HU3 → HU4, TU3 → TU4” K U3( r , M )

H U4( r , M )

5

T U4( r , M )

TL

T U2 H U4( r , M )

5. “HU4 → HU5, TU4 → TU5”

µ

1

H U5( r , M )

3

9 3

m γγ

( 3 .π )

.ln

h

7 .µ .

µ

2

32

256

µ

m . .ln ( 3 π ) . h 4 M

µ

7 .µ

2

. r λh

9

λx

4 3 1. St T . . 3 4 c5 π .λ 2 h

2

St T

9

9

4 . 3 3 4

3

2 7 .µ

5

.

mh M

5 .µ

2

. r λh

2 26 .µ

9

.1 . c

5

2

λx π .λ h

350

2

9

4 . 3 3 4

3

.1 . c

5

2

λx π .λ h

2

4 .1 . λx 3 c5 π .λ 2 h 6

2

www.deltagroupengineering.com

ωh λx

6

H α r 3, M 3

Hα

2

6 ( 4 .µ ) .

ωh

5

π .H α .λ h

c

St T

1 c

St T

4 .1 . λx 3 c5 π .λ 2 h

3

1 c

µ

2

c

1

2

9

9

.

1 π .H α

. . 4µ λh

6 ( 4 .µ ) .

2 .µ

.

π .H α .λ h

3

. . 4µ λh

5

π .H α .λ h

c

c

π .H α .λ h

1

2

9

9

9

1

2 .µ

3

.

1 π .H α

2

T U5( r , M )

π .H α

. . 4µ λh

KW c

µ

.ln

2

2

1

3

6

c

1

ωh

6

3

2

( 4 .µ ) .

6 ( 4 .µ ) .

2

1

2

1.

3

c

2

1 π .H α

3

6

1

2

9

3

9

1 c

.

1 π .H α

. . 4µ H U5( r , M ) λ h Hα

. . 4µ λh

6

2

. . 4µ λh 2 .µ

.

3

2 .µ

1 π .H α

2

. 2

.H ( r , M ) 5 µ U5

6. “HU3, HU4, HU5, TU3, TU4, TU5” H U3 R o , M G H U4 R o , M G H U5 R o , M G T U3 R o , M G T U4 R o , M G T U5 R o , M G

H U3 r x5.R o , m g5 .M G H U3 r x5.R o , m g5 .M G H U3 r x5.R o , m g5 .M G

= 67.084304 67.095419

T U3 r x5.R o , m g5 .M G T U3 r x5.R o , m g5 .M G T U3 r x5.R o , m g5 .M G

= 2.724752 2.725 ( K )

67.084304 67.095419 67.084304 67.095419

2.724752 2.725 2.724752 2.725

1 . H U4 R o , M G H0 H U5 R o , M G

H U3 r x5.R o , m g5 .M G H U3 r x5.R o , m g5 .M G H U3 r x5.R o , m g5 .M G

T U3 R o , M G

T U3 r x5.R o , m g5 .M G

1 . T U4 R o , M G T0 T U5 R o , M G

T U3 r x5.R o , m g5 .M G T U3 r x5.R o , m g5 .M G

H U3 R o , M G

km . s Mpc

5.515064 5.499409 1=

5.515064 5.499409 ( % ) 5.515064 5.499409

1=

. 9.08391310

3

. 8.37394610

9

. 9.08391310

3

. 8.37394610

9

. 9.08391310

3

. 8.37394610

9

(%)

7. Time dependent characteristics T U3( H ) K W .St T .ln

Hα

9

H H β .H α

. H5

H

T U3 H β

K W .St T .ln

1 Hβ

. H .H β α

5 .µ

2

1

1 . d K W .St T .ln H β .H α dH β Hβ

Hβ

H β_min ,

2 5 .µ

H β_max H β_min 1 .10

5

0

Hβ e

2 5 .µ

.. H β_max

T U3 H β

351

H β_min

10

H β_max

1

K W .St T .ln

1 Hβ

6

. H .H β α

5 .µ

2

www.deltagroupengineering.com

1

T U3 e

1

2 5 .µ

. 31 ( K ) = 3.19551810

2 5 .µ .

. 61 H α = 1.39858410

e

km s .Mpc

1

e

2 5 .µ .

1

= 2.206287 10

H U2 R o , M G

42 .

s

= 7.928705 10

T U2 H α

61

T U3( 1 )

Hα

Hα

=

0

( K)

0

Computational environment initialisation value → H β2 56.4503086205567 Given T U2 10 H β2

10

H β2

273.( K )

.H α

H β2 = 56.450309

Find H β2

H β2

1

.H α

. = 1.02858610

14

10

(s)

10

H β2

H β2

1

.H α

km . s Mpc

.H = 2.99992310 . 5 α

6 = 3.259461 10 .yr

T U2 10

H β2

.H α = 273 ( K )

See Fig. 4.22, 4.23. 1 H β .H α

t

T U3 H β

1 . 1 K W .St T . t t5

d T U4( t ) dt

1

K W .St T .ln

µ

Hβ

t1

e

1

. 5 .ln H .t .µ 2 α

2

1 . 1 K W .St T . 2 5 t t

dT2 dt2 ( t )

µ

µ

3

d t3

T U4( t )

1

2 5 .ln H α .t .µ 5 .µ

2

µ

2

K W .St T .

. 5 .µ 2 . ln H .t . 5 .µ 2 α

ln H α .t . 5 .µ

2

2 2 5 .µ . ln H α .t . 5 .µ

1 . 1 K W .St T . 3 5 t t

. 5 .ln H .t .µ 2 α

1

0

.t

1

2

1

10 .µ

. 5 .µ

2

1

2 2.

5 .µ

2

t

d

1 . 1 K W .St T . 5 t t

t

1 . 1 T U4( t ) K W .St T . 2 5 d t2 t t d

K W .St T .

dT dt ( t )

Hα

1 T U4( t ) K W .St T .ln H α .t . t

2

1 2 5 .µ .

2

5 .µ

. H .H β α

µ

5 .µ

2

1

2

1

1

2

0

t2

e

2

2 2 5 .µ . 5 .µ

1 1

. 1 Hα

1

.t2

2

. 5 .µ 2 .ln H .t . 5 .µ 2 . 5 .µ 2 α

352

3

2

2 2 15.µ . 5 .µ

2

2

www.deltagroupengineering.com

K W .St T .

1 . 1 t

3

t

µ

5

2 2 15 .µ . 5 .µ

t3

e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

. 5 .µ 2 .ln H .t . 5 .µ 2 . 5 .µ 2 α

3

dT2 dt2 t 2 = 0

s

4.196153

10

s

6.205726

1

=

dT2 dt2 t 1

=

dT2 dt2 t 2

. 114 2.02615310

K

0

2

s

. 112 8.77595210

dT2 dt2 t 3

2

K 3

0

K

. 72 1.05719310

s

. 71 9.25283810

dT3 dt3

. 116 7.65967810

2

. 74 1.32321810

Hα

dT dt t 3

Hα

0

1

dT dt t 2

t3

dT2 dt2

2

s

dT dt t 1

42 .

2 2 15.µ . 5 .µ

dT3 dt3 t 3 = 0

2

dT dt

0.364697

t2

K s

2.206287

2

. 1 Hα

2

1

=

2 2 15.µ . 5 .µ

2

2

K

dT dt t 1 = 0

t1

2

2 2 2 5 .µ .ln H α .t . 5 .µ . 5 .µ 3 2 K W .St T . 2 5 .µ . 3 t t

dT3 dt3 ( t )

Hα

3

1 . 159 6.22716710

Hα

dT3 dt3 t 1

=

. 156 3.77545710

K

. 1.45285710

s

155

dT3 dt3 t 2

3

0

dT3 dt3 t 3

T U2 H α T U2 T U2 T U2

1

0

t1 =

1 t2

. 31 3.19551810 ( K)

. 31 3.03432210

4 . 34 ( K ) T U2 10 .H α = 7.41414610

. 31 2.83254210

1 t3

4 10 .H α

1

= 0.364697 10

46 .

s

T U2 H U2 R o , M G

= 2.724752 ( K )

See Fig. 4.24 – 4.35.

353

www.deltagroupengineering.com

Hα

T U3( H ) K W .St T .ln

Hα . 2 d d .H5 µ K W .St T .ln T U3( H ) dH dH H

9

. H5

H

5 .µ Hα . 2 d .H5 µ K .St . H K W .St T .ln W T dH H H 5 .µ

H K W .St T .

d d T U3( H ) . t dH d T U4( t )

5 .µ

5 .µ

2

H

µ

( H .t )

.

2 5 .µ .

t

µ

2

. 5 .ln H .t .µ 2 α

2

H

1

Hα

1

.µ 2

1

Hα

. 5 .ln

.µ 2

1

H µ

2

. 5 .ln H .t .µ 2 α

2 5 .ln H α .t .µ

2 5 .ln H α .t .µ

5 .ln

1 . 1 t t5

1

2

H

1

t . . 5 .µ 2 . . H α . 2 (H t) 5 ln µ H H

1

. 5 .ln H .t .µ 2 α

.µ 2

H

1

. 5 .ln H .t .µ 2 α

.µ 2

1

H

H

1 . 1 t t5

H

2

Hα

. 5 .ln

H

d H dt

µ

Hα

. 5 .ln

5 .µ

.µ 2

.µ 2

H

H

H

1 . 1 t t5

K W .St T .

Hα

. 5 .ln

Hα

. 5 .ln

2

1 . 1 K W .St T . t t5

2

H

K W .St T .

2

d H dt

1

1

1

2

t

H

1

1 H γ .H α

H

5 .ln H α .

Hα

d H dt H α.

dH dt H γ

1 H γ .H α

5 .µ

. 2

.

5 .ln

1 H γ .H α

Hα

.µ 2

1

1

Hα

2 H α .H γ . 5 .ln 1 .µ 2 2 Hγ . 5µ Hγ

2 H α .H γ d . 5 .ln 1 .µ 2 2 dH γ Hγ 5 .µ Hγ

1 .µ 2 . Hγ Hα

1

Hα

1

Hγ

d dH dt H γ dH γ

2 H α .H γ d . 5 .ln 1 .µ 2 2 dH γ Hγ . 5µ Hγ

2

2 5 .µ

. 5 .ln 1 .µ 2 . 1 Hγ

5 .µ

2

1

1

Hα Hγ

2

2 5 .µ

. 5 .ln 1 .µ 2 . 1 Hγ

5 .µ

2

1

0

1

Hγ e

2 2 5 .µ . 5 .µ

354

1 1

t4

e

2 2 5 .µ . 5 .µ

1

. 1 Hα

www.deltagroupengineering.com

dH2 dt2

d

2

d

H

d t2

2

H

d dt

H

d t2

2 5 .ln H α .t .µ

. . 2

5µ . ( H .t ) t

5 .ln

Hα

.µ 2

1 1

H

H

d dt

. . 2

5µ . ( H .t ) t

2 5 .ln H α .t .µ

Hα

5 .ln

1

H

2 2 5 .µ . ln H α .t . 5 .µ

.

. 2

.µ 2

5µ .2 ( H .t ) t

1

Hα

5 .ln

H

d

2 H H α.

1 H γ .H α

2 5 .µ . ln H α .

5 .µ

. 2

.

1 H γ .H α

H α .H γ d . 5.µ 2 . ln 1 . 5 .µ 2 2 dH γ Hγ 5 .µ Hγ 3

2 2 5 .µ . 5 .µ

4

2 2 2 5 .µ . 5 .µ . 5 .µ

Let:

Hγ Hβ

1

1

1

Hα

2

ln

1

1

1

2

2

1

1

1

3 H α .H γ . 5 .µ 2 . 5.µ 2 . 5.µ 2 .ln 1 2 Hγ 5 .µ Hγ

1

Hγ

1

2 .ln

t5

4

2

ln

1 Hγ

1

2 .ln

1

4

2

Hγ

0

Hγ

2 2 5 .µ . 5 .µ 2

.µ 2

1

Hα

2 1

1 . 5 .µ 2 . Hγ Hα

5 .ln

2

3 H α .H γ . 5 .µ 2 . 5 .µ 2 . 5 .µ 2 .ln 1 2 Hγ 5 .µ Hγ

Hγ e

2

3 2 H α .H γ . 5 .µ 2 . ln 1 . 5 .µ 2 2 Hγ . 5µ Hγ

dH2 dt2 H γ

.µ 2

2

H

Hα

d t2

1

e

4

2 2 2 5 .µ . 5 .µ . 5 .µ

2 1

2

. 1 Hα

η

Computational environment initialisation value → η

4.595349

Given

dH dt

H U2 R o , M G Hα

H U2 R o , M G

η

η

1

Find( η )

355

www.deltagroupengineering.com

1

t1

e

2 5 .µ .

Hα 10 .µ

t2

1

e

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

t3 e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

. 1 Hα

2.206287 2.206287 4.196153 4.196153

2 2

3

= 6.205726 6.205726

. 1 Hα

e

t5 e

s

8.385263 8.385263

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

42 .

20.932666 20.932666

1

t4

10

1

4

2 2 2 5 .µ . 5 .µ . 5 .µ

. 1 Hα 2

1

2

. 1 Hα 1

dH dt t 1 .H α

1

dH dt e

5 .µ

10 .µ

dH dt t 2 .H α

1

dH dt e

2

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

dH dt t 3 .H α

1

dH dt e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2 3

2

1

dH dt t 4 .H α

1

dH dt e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

dH dt t 5 .H α

η = 4.595349

1

dH dt e

1

4

2 2 2 5 .µ . 5 .µ . 5 .µ

dH dt

. 68 7.50137510

. 68 7.50137510

. 83 9.06689310

. 83 9.06689310

= 1.22575310 . 84

. 84 1.22575310

. 84 1.55351810

. 84 1.55351810

. 84 1.38436210

. 84 1.38436210

2

Hz

2 1

H U2 R o , M G

2

η

= 4.726505 10

36 .

2

Hz

Hα

356

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1

dH2 dt2 t 1 .H α

1

dH2 dt2 e

2 5 .µ

10 .µ

dH2 dt2 t 2 .H α

1

2

1

2 2 5 .µ . 5 .µ

dH2 dt2 e

1

2 2 15 .µ . 5 .µ

dH2 dt2 t 3 .H α

1

dH2 dt2 e

. 125 8.50679910

0

0

2

2

2 2 2 5 .µ . 5 .µ . 5 .µ

. 125 8.50679910

=

2

3

1

dH2 dt2 t 4 .H α

1

dH2 dt2 e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

dH2 dt2 t 5 .H α

dH2 dt2

1

dH2 dt2 e

1

. 125 1.16257810

. 124 8.2461110

. 124 8.2461110

. 125 1.33162810

. 125 1.33162810

3

Hz

2

4

2 2 2 5 .µ . 5 .µ . 5 .µ

. 1.16257810

125

2

1

η

H U2 R o , M G

3

= 0 Hz

Hα

See Fig. 4.36 – 4.45. H

d d 1 H dt dt t

1 t

1 t

2

H

2

Hα

d H dt

H

=1 η

dH dt 1

η

dH dt 1

η

H U2 R o , M G

dH dt

=

. 61 8.46094110 67.084257

km . s Mpc

Hα

See Fig. 4.46, 4.47. Checking errors yields,

H

d H dt

( H .t )

5 .µ

. 2

2 5 .ln H α .t .µ

.t

5 .ln

Hα

.µ

2

5 .µ

1

H

2

. 5 .ln

H

Hα

.µ 2

2 5 .ln H α .t .µ

1 dH

H

t

1

5 .µ

2

1 dt

.t

H 5 .µ

H

2

. 5 .ln

H

5 .µ

H

2

.ln

Hα

.µ 2

H

Hα H

5 .µ

1 dH H

2

.ln

2 5 .ln H α .t .µ

Hα H

ln H α .t t

5 .µ

2

t

has the solution:

H

5 .µ

2

.t

1 dt

ln H α .t t

5 .µ

2

1 t

357

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T U2

η

T U2

dH dt 1

dH dt

T U3( 1 )

T U3

dH dt

e

dH dt e

dH dt

T U2

dH dt

dH dt

1

T U3 e η

2

2 2 5 .µ . 5 .µ

1

T U3 e

0 2

2 2 5 .µ . 5 .µ

0

. 31 3.19551810 . 31 2.97174510

1

2 2 15 .µ . 5 .µ

2 3

2

= 3.18632310 . 31 3.03432210 . 31 ( K )

1

. 31 2.83254210 . 31 3.18071410

η 2

5 .µ

10 .µ

1

2

T U3 e

T U3

Hα

2

2 2 2 5 .µ . 5 .µ . 5 .µ

η

H U2 R o , M G

1

T U2

Hα

η

2 2 2 5 .µ . 5 .µ . 5 .µ

e

( K)

T U3( 1 )

2 2 15 .µ . 5 .µ

T U2

0 2.724752

H U2 R o , M G

2 5 .µ

10 .µ

0 2.724751

=

dH dt 1

1

T U2

Hα

η

T U2

T U2

η

H U2 R o , M G

2 3

2.724751

2.724752

2

H U2 R o , M G Hα

η

2 5 .µ

e

1 = 7.002696 ( % )

1

T U3 e

5 .µ

2

See Fig. 4.48, 4.49.

dH dt H γ

2 H α .H γ . 5 .ln

0

Hγ t7

t 7 = 2.206287 10

=1

t1

1

.µ 2

1

Hγ e

0

Hγ

2 5 .µ

1

1

42 .

s

Hγ Hβ

η

η

2 5 .µ

t7

e

2 5 .µ .

1 Hα

ln H γ

ln t 7 .H α

1

ln H β

ln t 1 .H α

1

1

1

ln t 7 .H α

1

ln t 1 .H α

1

=1

2 H α .e

e

2

1

. 5 .ln 2 5 .µ

1 5 .µ

5 .µ

2

.µ 2

. 68 Hz2 1 = 7.50137510

1

e

5 .µ

2

358

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4

2.

dH dt H γ

Hα Hγ

4 .µ

2

. 5 .ln 1 .µ 2 Hγ

2.

1

5.

Hα e

1

5 .ln

5 .µ

2

1

4

Hα e

. 68 Hz2 1 = 7.50137510

1

e

2.

.µ 2

5.

5 .ln e

5 .µ

2

.µ 2

1

5 .ln

2

1 = 0 Hz

. 1 3

1 1

5.

2

1 =0

2

3

e

1st Derivative of the Hubble Constant

10 42 t 1 .10 8

Scaled Derivative (Hz^2)

6

4 dH dt H β 10

η

79 2

0 0

2

4

2.20624

2.20625

2.20626

2.20627

2.20628

2.20629

2.2063

2.20631

2.20632

η H β .H α

1

2.20633

2.20634

2.20635

2.20636

2.20637

2.20638

.1042

Scaled Cosmological Age (s)

Hα =

dH dt t 4 .H α

1

Hα

H U2 R o , M G H0

=

1

=

. 123 1.47916710

71

2

km s .Mpc

km s .Mpc

1

2

. 1.55351810

84

Hα dH dt

H U2 R o , M G 2

H0

359

. 3.84599410

61

1

. 84 7.51858710

=

. 61 8.46094110

=

dH dt t 4 .H α

dH dt t 4 .H α

. 123 7.15875210

67.084304

Hα ( Hz)

Hα

1

2

dH dt t 4 .H α

. 1.24640210

42

= 2.199936

dH dt t 4 .H α Hα

. 42 2.74200410

km . s Mpc

2

Hz

2

t 4 .H α 2

=

= 4.839718 1

. 3 4.50030410 3 5.041.10

km s .Mpc

2

www.deltagroupengineering.com

1 Hα t1

3.646967 = 22.062867

10

1

43 .

s

9 = 14.575885 10 .yr

H U2 R o , M G

209.326658

t4

A U K λ .R o , λ x.λ h , K m.M G, m x.m h .H U2 R o , M G = 1 T U2 H α T U2 t 1

1

T U2 t 4

1

0 . 31 3.19551810 . 31 = 2.05994510 2.724752

T U2 H U2 R o , M G T U3 H U2 R o , M G .H α

c.

( K)

t1 t4

=

6.614281

10

62.754553

34 .

m

2.724752

1

2.725

T0

t 16.326238 c . 1 = 154.899031 λh t4

c H U2 R o , M G

9 = 14.575885 10 .Lyr

R U K λ .R o , λ x.λ h , K m.M G, m x.m h .H U2 R o , M G

=1

c

8. History of the Universe T U2 H α = 0 ( K )

1

T U2 10 T U2

10 .

T U2

1

. 15 ( K ) = 3.43308810

31

1 13 .

1 9.

5 .10 ( yr )

( K)

1

T U2 10

T U2

(s)

1

. = 1.92400510

28

34 .

( K)

(s)

. 9 ( K) = 1.01325410

2.

10 ( s ) = 978.724031 ( K )

10 ( s ) T U2

. = 3.19551810

t1

T U2

1 9.

= 11.838588 ( K )

10 ( yr ) = 4.898955 ( K )

T U2 H U2 R o , M G

360

= 2.724752 ( K )

www.deltagroupengineering.com

T U2

1 . 1 (s) 1

1 .( day )

T U2

T U2

1 31.( day )

T U2

T U2

1 .( yr ) 1

T U2

2 10 .( yr )

1

T U2

1

1

. 7 521.528169 2.52413210

41.823796

. 4 8.07751510

11.838588

9 10 .( yr )

. 4 2.29089210

3.35005

1

. 3 6.49496110

0.947724

( K)

10 .

( yr )

1 11 .

10

1

= 1.00307810 .

1

10

=

147.71262

6

8 10 .( yr )

T U2

1 . 116 ( day )

. 6 3.86401510

1

T U2

10 ( yr )

T U2

. 10 1.84076810 . 3 1.2497710

7.

T U2

4.

6 10 .( yr )

10 ( yr )

10 ( yr )

T U2

1

T U2

3.

T U2

5.

10 ( yr )

T U2

1

1

T U2

( yr )

. 6 1.87808710 . 3.98831410

( K)

7

TL

9. “ML, rL, tL, tEGM” 5

C Ω_J1 r 1 , M 1

M1

C Ω_J1 r 2 , M 2

26

r1

R EGM

M 2 M 1.

26

r2

R U K λ .R o , λ x.λ h , K m.M G, m x.m h

. 71 ( kg ) M L = 4.86482110

tL

5

M2

rL

rL

. t L = 7.6372910

19

c

M EGM 2 R EGM.c

=1

t EGM

ML

.

r1

K m.M G.

M EGM

r2 r1

R EGM K λ .R o

5 5

.

R EGM K λ .R o

M U K λ .R o , λ x.λ h , K m.M G, m x.m h

A U K λ .R o , λ x.λ h , K m.M G, m x.m h

2 .G

M EGM

5 5

. 19 109 .Lyr r L = 7.6372910

R BH M L

9 10 .yr

r2

t EGM

=1

R EGM c

2 R EGM.c

2 .G

t EGM

R EGM c

M L M EGM rL

tL

R EGM t EGM

361

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ML M EGM

. 18 5.23967510

rL

tL

= 5.23967510 . 18

R EGM

. 18 5.23967510

tL

. 6 = 1.86196810

TL

t EGM

10. Radio astronomy 9 9

9

5

M St J . St J . 26 r

M

M

9

M

St J .

26

St G.

St J .

9

5

5 26

5

2

St G.

9

ω Ω_3

M

M

St J .

5

2

26 9

5

.

ω Ω_3

26

St G

M

5

5

2

9

ω Ω_3

26

M

5

ω Ω_3

.

26

St G

M

5

9

2

26 9

M M

St J .ω Ω_3

5

St J .St G

45

26

5

52

26

5

.St 5 G

M

1

.

27 5

26

5

26

.St 5 G

9

1

5 St J .ω Ω_3 .

27

M

5

26

.St 5 G 4

5 4 9 .c .ω Ω 5. 5. .ω Ω_3 St G M . 4π

3 5

.St 5 G 2

9 .c . 9 St G .St G . 4π 4

5

52

1

9

M 26

26 9 5 St J .ω Ω_3 .

M

5 St J .ω Ω_3 .

26

5

26 45

9 .c . St G 4 .π 4

4 5

5

9 .c . 4 .π 4

C Ω_Jω ω Ω_3 , M

ω Ω_3 4.

St G M

3

5 .ω Ω_3

5.2

C Ω_Jω ω Ω_3 , M

4 9 .c . ω Ω_3 4 .π St 0.8 .M 0.6 G

10 10 C Ω_Jω ω Ω_4 10 .M S , 10 .M S = 180.283336( nJy )

Checking errors yields,

362

www.deltagroupengineering.com

5

M

St J .

Test 1 ω Ω_3 , M

9

St G.

M

ω Ω_3

27

.M

45

9

Test 2 3 .( EHz) , M S = 5.438023 10

43 .

Jy

Test 3 3 .( EHz) , M S

Test 2 ω Ω_3 , M Test 1 ω Ω_3 , M

C Ω_Jω 3 .( EHz) , M S = 5.438023 10

. 1 = 5.70654610

43 .

12

12

(%)

C Ω_Jω ω Ω_3 , M

. 1 = 3.66373610

10 10 C Ω_Jω ω Ω_4 10 .M S , 10 .M S

(%)

Jy

. 1 = 2.0428110

Test 4 3 .( EHz) , M S

Jy

43 .

Test 2 ω Ω_3 , M

Test 5 ω Ω_3 , M

C Ω_Jω ω Ω_3 , M

Test 5 3 .( EHz) , M S

5

45

2

Test 1 ω Ω_3 , M

Test 4 ω Ω_3 , M

26

45 .

ω Ω_3

Test 1 3 .( EHz) , M S = 5.438023 10

Test 3 ω Ω_3 , M

St J .St G

Test 2 ω Ω_3 , M

26

26

10 10 C Ω_J1 R BH 10 .M S , 10 .M S

12

(%)

= 0.999999999999968

ix. Gravitational Cosmology G.M E G.M M 2

2

r4 r4 r5

r4

r4

r5

=

. 5 3.46028110 . 3.83719110

4

( km)

a EGM_ωΩ r 5 , M M 0 .( s ) ,

=

T PV 1 , r 4 , M E 500

a PV( r , M , t )

D E2M. M M .M E

i .

g r 4, M E g r 5, M M

. 3.33165310

3

. 3.33165310

3

.. T PV 1 , r 4 , M E

C PV n PV, r , M .e

=

. 3 3.33165310 . 3.33165310

m s

r5

M M .M E

MM

a EGM_ωΩ r 4 , M E

t

r 5 D E2M

3

m s

2

g r 4, M E

r4

g r 5, M M = 0

m s

a EGM_ωΩ r 4 , M E

2

D E2M

a EGM_ωΩ r 5 , M M = 0

2

m s

N

21

n PV

N, 2

2

N .. N

π .n PV .ω PV( 1 , r , M ) .t .i

n PV

363

www.deltagroupengineering.com

Harmonic Acc. & Grav. Interference T PV 1 , r 5 , M M

Acceleration

a PV r 4 , M E , t a PV r 5 , M M , t a PV r 4 , M E , t

a PV r 5 , M M , t

t Time

Gravitational Acceleration due to The Earth Gravitational Acceleration due to The Moon Resultant Acceleration (Interference)

0

ξ

9

a g ( r , M , φ, t )

g av ( r , M )

t

0 .( s ) ,

ξ .T Ω r 5 , M M 200

.. ξ .T Ω r 5 , M M

π g ( r , M ) . .sin 2 .π .ω Ω ( r , M ) .t 2

2 T Ω ( r, M )

φ

1. T Ω ( r, M ) 2 . 0 .( s )

g av R E, M E = 9.809009

m s

2

a g( r, M , 0 , t ) d t

ω Ω r 4 , M E = 56.499573 ( YHz) ω Ω r 5 , M M = 72.138509( YHz)

Conjugate WaveFunction Acc. Pairs

a g r 4, M E, 0, t Acceleration

φ

a g r 4, M E, 0, t a g r 5, M M, π , t a g r 5, M M, π , t

t Time

+ve WaveFunction From The Earth -ve WaveFunction From The Earth +ve WaveFunction From The Moon -ve WaveFunction From The Moon

364

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Conjugate WaveFunction Acc. Beats

Acceleration

a g r 4, M E, φ , t

a g r 5, M M, π , t

a g r 4, M E, φ , t

a g r 5, M M, π , t

t Time

+ve WaveFunction Interference Beat -ve WaveFunction Interference Beat (Conjugate)

x. Particle Cosmology h

tL

m γγ 5

m γγ2

r e.

r γγ2

h

m γγ2

m gg2

tL

m γγ2

2 .m γγ2

m gg2

2

r gg2

2 m e .c

5

r γγ2

4 .r γγ2

r gg2

r γγ2

λh

λh

r γγ2

2 .r γγ2

K λ .λ h

K λ .λ h 2 .r γγ2 λh r gg2

2 .r γγ2

0.178967

=

0.357933 0.236148

2 .r γγ2

(%)

λh

7.250508 9.567103

246.127068

2 .r gg2

0.472296

K λ .λ h

211.731798

λh

λh

2 .r gg2

E Ω ( r, M )

Qγ

h .ω Ω ( r , M )

Q γ r ε, m e

N γ( r, M )

Qγ

E Ω ( r, M ) mγ

= 2.655018 10

eV

38 .

m

423.463597

r gg2

2 .r gg2

10

51 .

279.381783

=

0.406294

2 .r gg2

10

3.431956

324.766614

λh K λ .λ h

1.715978

558.763566

λh

0.307913

=

=

30

Qe

365

Q γ( r, M )

Q γ_PDG

Qe N γ( r, M )

5 .10

30 .

Qe

www.deltagroupengineering.com

Q γ_PDG

= 1.883226

Q γγ( r , M )

Qγ

Q γγ

tL

Qγ

Q γ( r, M ) N γ( r, M )

2

Q γγ = 1.129394 10

Qe

m γγ

Q γγ

m γγ

T L m γγ2 Q γγ2

m γγ2

Q γγ2 = 6.065593 10

mγ

85 .

C

Q γγ2

78 .

C

Q γ( r, M )

Q γγ( r , M )

Q γγ

2

Qe

= 7.049122 10

60

Qe

. = 1.86196810

6

= 3.785846 10

Q γγ2

mγ

66

Q γγ m γγ

.m γγ2

2

ω Ω r e, m e .m γγ ω Ω r ε, m e

E Ω r ε,me

Qe

2

E Ω r ε, m e ω Ω r e, m e .m γγ ω Ω r ε, m e

ω Ω r e, m e

mγ

2

h .m γγ

=

1.525768 1.525768

10

46 .

eV

E Ω r e, m e

mγ

2

mγ

h .m γγ

mγ

m γγ

2

=

0.165603 0.165603

( µJ )

m γγ

Qe

ω Ω r e,m e 2

E Ω r e, m e

=

249.926816 249.926816

me ( YHz)

2.

c Q γγ

=

. 11 1.7588210

C

198.286288

kg

m γγ

NOTES

366

www.deltagroupengineering.com

b. Calculation engine i. Computational environment NOTE: KNOWLEDGE OF MATHCAD IS REQUIRED AND ASSUMED • • •

Convergence Tolerance (TOL): 0.001. Constraint Tolerance (CTOL): 0.001. Calculation Display Tolerance: 6 figures – unless otherwise indicated. ii. Standard relationships 1

A0

c

r0

H0

H0

M

ρ m( r , M )

ω VL λ VL

c

9 A 0 = 13.772016 10 .yr

V( r )

2

r

M .c

E( M )

2

G.M

g( r , M )

λ VL

2 .G.M

R BH( M )

V( r )

c

4. . 3 πr 3

E γ ( ω ) h .ω

2

9 r 0 = 13.772016 10 .Lyr

iii. Derived constants 4 . 2 6 π 3

λx

3.

St G

λx

mx

3 .ω h 4 .π .h

2

. c 2

Hα

2

4

St J

10 .µ

t1

e

1

t2

Hα

e

2

t4

e

1

µ

3 1

r3

. 1 Hα

λ x.λ h

= 3.646967 10

2 2 5 .µ . 5 .µ

245

10

5 m.s

t5

M3

43 .

Hα

St g = 1.828935

2

9

9

s

m x.m h

c.

St BH

e

6 3 3 .ω h 13 2 2 .π .c 9

c .St G ( 2 .G)

1

. 1 Hα

t3

4

2 2 2 5 .µ . 5 .µ . 5 .µ

ng

St T

5

1

. 1 Hα

2

mx

3 9 10 .yr

224 .

St G = 8.146982 10

2

2 3

2

. 1 Hα

1

λx

13

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2

8

. T L = 4.10173110

e

4. 3. 1 . λ x 3 4 c5 π .λ 2 h 3

2 2 15 .µ . 5 .µ

2 2 5 .µ . 5 .µ 1

St g

m γγ

1

1 2 2 5 .µ . 5 .µ

h

TL

λx

9 .c . St G 4 .π

9

1 2 5 .µ .

ωh

5

m

2. 9

kg s

367

=

t7

2.698709 1.349354

e

5 .µ

2

. 1 Hα

. H α = 8.46094110

61

km . s Mpc

. T L.H 0 = 2.97830810

12

1 . St G G

. 1 = 3.33066910

14

(%)

St g

www.deltagroupengineering.com

146 . kg

9

St J = 1.093567 10

4 . 26

s

m

3

18

119

s

10 .

r 3 = 1.093333 10

kg

. St BH = 4.83080210

. M 3 = 7.36147410

ym

8

. St T = 6.35557910 9

3

95

s

5 9

m

( kg )

1

t1

e

5 .µ

10 .µ

t2

e

2

. 1 Hα

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

t3 e

. 1 Hα

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2.206287 2.206287

2 3

2

. 1 Hα

4.196153 4.196153 =

1

t4

e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

t5 e

1

4

. 1 Hα

6.205726 6.205726

10

20.932666 20.932666

t7

42 .

s

=1

t1

8.385263 8.385263 2.206287 2.206287

2

2 2 2 5 .µ . 5 .µ . 5 .µ

1

2

. 1 Hα

1

t7

e

2 5 .µ .

1 Hα

iv. Base approximations / simplifications

Ω 1( r , M )

6 .c r .ω PV( 1 , r , M )

C Ω_1( r , M )

G.M . 2

r

T Ω_1( r , M )

3

.

3 .M .c

2 . π n Ω_1( r , M )

ω Ω_1( r , M )

Ω 1 R M,M M

Ω 1 R E, M E

Ω R M, M M

Ω R E, M E

Ω 1 R J, M J

Ω 1 R S, M S

Ω R J, M J

Ω R S, M S

Ω R NS , M NS

n Ω_1( r , M )

2 .π .h .ω PV( 1 , r , M )

1

Ω 1 R NS , M NS

2

ω Ω_1( r , M )

λ Ω_1( r , M )

1=

Ω 1( r , M ) 12

n Ω_1( r , M ) .ω PV( 1 , r , M )

c ω Ω_1( r , M )

1

K Depp ( r , M ) 1

. 14 4.44089210 . 6.66133810

14

. 14 6.66133810 . 4.44089210

14

2 .G.M r .c

2

(%)

1 = 0 (%)

368

www.deltagroupengineering.com

n Ω_1 R M , M M

n Ω_1 R E, M E

n Ω R M,M M

n Ω R E, M E

n Ω_1 R J , M J

n Ω_1 R S , M S

n Ω R J, M J

n Ω R S, M S

n Ω_1 R NS , M NS n Ω R NS , M NS

ω Ω_1 R E, M E

ω Ω R M,M M

ω Ω R E, M E

ω Ω_1 R J , M J

ω Ω_1 R S , M S

ω Ω R J, M J

ω Ω R S, M S

ω Ω R NS , M NS

T Ω_1 R E, M E

T Ω R M,M M

T Ω R E, M E

T Ω_1 R J , M J

T Ω_1 R S , M S

T Ω R J, M J

T Ω R S, M S

T Ω R NS , M NS

λ Ω_1 R E, M E

λ Ω R M,M M

λ Ω R E, M E

λ Ω_1 R J , M J

λ Ω_1 R S , M S

λ Ω R J, M J

λ Ω R S, M S

λ Ω R NS , M NS

. 2.22044610

14

. 4.44089210

14

. 8.88178410

14

(%)

1=

. 6.66133810

14

. 2.22044610

14

. 6.66133810

14

. 8.88178410

14

(%)

1=

. 7.77156110

14

. 2.22044610

14

. 5.55111510

14

. 7.77156110

14

. 7.77156110

14

. 2.22044610

14

. 6.66133810

14

. 7.77156110

14

(%)

1 = 0 (%)

λ Ω_1 R M , M M

λ Ω_1 R NS , M NS

14

1 = 0 (%)

T Ω_1 R M , M M

T Ω_1 R NS , M NS

. 6.66133810

1 = 0 (%)

ω Ω_1 R M , M M

ω Ω_1 R NS , M NS

1=

1=

K Depp R E, M E

1 = 0 (%)

K PV R E, M E

=

(%)

1.00000000069601 1.00000000069601

v. SBH mass and radius 3

ρS

ρ m λ x.λ h , m x.m h

r S M BH

. 94 kg ρ m λ x.λ h , m x.m h = 1.34467810 3 m

3 .M BH 4 .π .ρ S

M BH r S

4. . 3 π ρ S .r S 3

. 87 ( YPa) U m λ x.λ h , m x.m h = 1.20853710

369

www.deltagroupengineering.com

r S m x.m h

ρ m λ x.λ h , m x.m h

. 90 = 9.55041510

ρ m R S, M S

0

rS MS

. 3.28046310

=

5.

r S 10 M S

1=

144.219703 . 4.21884710

(%)

13

r S mh

( am)

0.015227 0.706754

10 r S 10 .M S

r S mh 1 . r m .m S x h λh λx

4

1 = 22.109851 ( % )

R BH m h

M BH r ε

M BH r π

M BH r ν

M BH r µ

. 43 3.22881910 . 49 3.18519310 . 49 3.12378410 . 43 9.27104510

M BH r τ

M BH r en

M BH r µn

M BH r τn

. 44 4.88723910 . 37 1.58452310 . 40 4.23240210 . 41 1.03305410

M BH r uq

M BH r dq

M BH r sq

M BH r cq

= 2.55332710 . 40 5.86600510 . 40 3.94279810 . 40 7.32116510 . 40

M BH r bq

M BH r tq

M BH r W

M BH r Z

. 40 4.52155110 . 40 1.19243610 . 41 6.74112410 . 40 6.91875410

M BH r H

M BH r e

M BH r Bohr

M BH r gg

. 40 1.26038310 . 51 8.34661610 . 63 1.64821910 . 9 4.68486410

( kg )

M BH r ε

M BH r π

M BH r ν

M BH r µ

. 13 1.62379510 . 19 1.60185510 . 19 1.57097210 . 13 4.66247210

M BH r τ

M BH r en

M BH r µn

M BH r τn

. 13 2.45782610 . 7 5.19529810

1 . M BH r uq

M BH r dq

M BH r sq

M BH r cq

= 1.28408510 . 10 2.95005410 . 10 1.9828610 . 10 3.68186410 . 10

M BH r bq

M BH r tq

M BH r W

M BH r Z

. 10 2.27391910 . 10 5.99684310 . 10 3.39015710 . 10 3.47948910

M BH r H

M BH r e

M BH r Bohr

M BH r gg

. 10 2.3560510

MS

. 20 6.3385510

. 9 2.12850410 . 11 7.96867110

. 33 4.1975710

14.554628

. 6 5.06892810

R BH M BH r τn

16.217926

. 7.67248410

R BH M BH r sq

R BH M BH r cq

= 4.00847210 .

R BH M BH r tq

R BH M BH r W

R BH M BH r Z

0.010862

R BH M BH r e

R BH M BH r Bohr

R BH M BH r gg

. 7.35477510

R BH M BH r ε

R BH M BH r π

R BH M BH r ν

R BH M BH r µ

R BH M BH r τ

R BH M BH r en

R BH M BH r µn

R BH M BH r uq

R BH M BH r dq

R BH M BH r bq R BH M BH r H

3

3

6

. 9.2090510

3

. 7.0983910

3

. 1.97867710

8

0

. 6 5.0004410

4.904034

. 2.48754410

3

0.066445

. 6.18980410

3

0.011494 ( Lyr )

0.01872

0.010583

. 1.31033610

21

0

vi. “nΩ” 1

1 3 9

n Ω_2( r , M )

n Ω_4 M BH

n BH M BH

1. 3 2

7

2

.

π .m h M

7

9

. r λh

n Ω_2 r S M BH , M BH

n Ω_5 M BH n Ω_4 M BH

3 9

9

n Ω_3 λ x

n Ω_5 M BH

π.

3.

2

2

2

n Ω_2 R BH M BH , M BH

n Ω_2 λ x.λ h , m x.m h n Ω_2 r S m x.m h , m x.m h n Ω_2 λ x.λ h , m x.m h n Ω_2 R BH m x.m h , m x.m h

370

λx

1=

. 3.33066910

13

. 14 4.44089210

(%)

www.deltagroupengineering.com

n Ω_2 r S m x.m h , m x.m h

n Ω_2 λ x.λ h , m x.m h

n Ω_2 r S M S , M S

n Ω_2 R BH M S , M S

5 5 n Ω_2 r S 10 .M S , 10 .M S

5 5 n Ω_2 R BH 10 .M S , 10 .M S

10 10 n Ω_2 r S 10 .M S , 10 .M S

10 10 n Ω_2 R BH 10 .M S , 10 .M S

n Ω_3 n Ω_3

1

1

. . 24 9.00254210 3.51086810 5

=

. 6 1.93953910 . 28 1.93265910 . 7 4.1786110 . 31 1.06388810

1 3

0.248017

1

0.324994

2 =

n Ω_3( 1 )

0.515897 0.818935

n Ω_3( 2 )

1

n Ω_3 λ x

1.073108

n Ω_3( 3 ) n Ω_4 m x.m h

n Ω_5 m x.m h

n BH m x.m h

n Ω_4 M S

n Ω_5 M S

n BH M S

5 n Ω_4 10 .M S

5 n Ω_5 10 .M S

5 n BH 10 .M S

10 n Ω_4 10 .M S

10 n Ω_5 10 .M S

10 n BH 10 .M S

1 =

1

1

. 5 9.00254210 . 24 2.56419310 . 19 3.51086810 . 6 1.93953910 . 28 1.0035610 . 22 1.93265910 . 7 4.1786110 . 31 3.92767810 . 24 1.06388810

vii. “ωΩ, TΩ, λΩ” 2 .c . n PV. g( r, M ) 2 π .r 3

ω PV2 n PV, r , M

9 M St G .

ω Ω_7 M BH

ω PV_1 M BH

ω Ω_4 M BH

St BH.

9

5

r

ω Ω_5 M BH

1. 2 St g .g ( r , M ) r

2

1

ω Ω_3( r , M )

9

ω Ω_2( r , M )

3

9

ω Ω_3 r S M BH , M BH

ω Ω_4 M BH n Ω_5 M BH

ω Ω_6 M BH

ω BH M BH

1 M BH

ω Ω_5 M BH n Ω_4 M BH ω Ω_5 M BH ω Ω_4 M BH

ω Ω_6 M BH ω Ω_7 M BH

371

www.deltagroupengineering.com

ω PV2 1 , R M , M M

ω PV2 1 , R E, M E

ω PV 1 , R M , M M

ω PV 1 , R E, M E

ω PV2 1 , R J , M J

ω PV2 1 , R S , M S

ω PV 1 , R J , M J

ω PV 1 , R S , M S

ω PV2 1 , R NS , M NS ω PV 1 , R NS , M NS

1=

. 3.14037710

9

. 6.96004310

8

. 1.9723310

6

. 2.12158610

4

1 = 7.117159 ( % )

ω PV2 n Ω R M , M M , R M , M M

ω PV2 n Ω R E, M E , R E, M E

ω Ω_3 R M , M M

ω Ω_3 R E, M E

ω PV2 n Ω R J , M J , R J , M J

ω PV2 n Ω R S , M S , R S , M S

ω Ω_3 R J , M J

ω Ω_3 R S , M S

ω PV2 n Ω R NS , M NS , R NS , M NS

ω Ω_2 R M , M M

ω Ω_2 R E, M E

ω Ω_1 R M , M M

ω Ω_1 R E, M E

ω Ω_2 R J , M J

ω Ω_2 R S , M S

ω Ω_1 R J , M J

ω Ω_1 R S , M S

ω Ω_1 R NS , M NS

1=

. 4.1871410

9

. 9.2800510

8

. 6 2.62977310

. 2.8287810

4

1=

. 1.04678510

9

. 2.32001510

8

. 6.57443310

7

. 7.07196310

5

(%)

1 = 2.491576 ( % )

ω Ω_3 R M , M M

ω Ω_3 R E, M E

ω Ω_2 R M , M M

ω Ω_2 R E, M E

ω Ω_3 R J , M J

ω Ω_3 R S , M S

ω Ω_2 R J , M J

ω Ω_2 R S , M S

1=

. 14 8.88178410

. 1.11022310

13

. 13 1.11022310

. 1.11022310

13

(%)

ω Ω_4 m x.m h ω Ω_3 R NS , M NS ω Ω_2 R NS , M NS

. 1 = 6.66133810

14

(%)

1 ω Ω_3 λ x.λ h , m x.m h

.

ω Ω_5 m x.m h ω Ω_6 m x.m h ω Ω_7 m x.m h

ω Ω_3 r 0 , m x.m h

. 7.88327910

ω Ω_3 r 0 , M S 5 ω Ω_3 r 0 , 10 .M S 10 ω Ω_3 r 0 , 10 .M S

(%)

1 = 9.375146 ( % )

ω Ω_3 R NS , M NS

ω Ω_2 R NS , M NS

(%)

=

1 =

1 1 1

10

0.163994

( EHz)

2.118067 27.355887

372

www.deltagroupengineering.com

ω Ω_3 r 0 , m x.m h

ω Ω_3 r 0 , m x.m h

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

ω Ω_3 r 0 , M S

ω Ω_3 r 0 , M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

5.

. 1.05183110

ω Ω_3 r 0 , 10 M S

ω Ω_3 r 0 , 10 M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

10 ω Ω_3 r 0 , 10 .M S

10 ω Ω_3 r 0 , 10 .M S

ω VL( 400 ( nm ) )

ω VL( 750 ( nm ) )

. 3 5.29883310

. 4 3.64997710

. 4 6.84370610

ω Ω_6 m x.m h

ω Ω_7 m x.m h

ω Ω_4 M S

ω Ω_5 M S

ω Ω_6 M S

ω Ω_7 M S

5.

410.269418

. 2.82604410

ω Ω_5 m x.m h

5.

. 42 1.87219710 . 42 1.87219710 . 42 1.87219710 . 42 1.87219710 =

5.

ω Ω_4 10 M S

ω Ω_5 10 M S

ω Ω_6 10 M S

ω Ω_7 10 M S

10 ω Ω_4 10 .M S

10 ω Ω_5 10 .M S

10 ω Ω_6 10 .M S

10 ω Ω_7 10 .M S

. 29 4.55727410 . 43 1.29804810 . 38 6.93112610 . 4 6.23977510 . 28 6.9805610 . 43 3.61189510 . 37 1.34431910

ω Ω_5 m x.m h

ω Ω_6 m x.m h

ω Ω_7 m x.m h

ω Ω_4 M S

ω Ω_5 M S

ω Ω_6 M S

ω Ω_7 M S

5 ω Ω_5 10 .M S

5 ω Ω_6 10 .M S

5 ω Ω_7 10 .M S

10 .

10 .

ω Ω_5 10 M S

ω BH m x.m h

ω PV_1 m x.m h

ω BH M S

ω PV_1 M S

5 ω BH 10 .M S

5 ω PV_1 10 .M S

10 ω BH 10 .M S

10 ω PV_1 10 .M S

ω Ω_4 10 M S

10 .

0.253004

14

. 6.158585 1.75414910

5

0

. 1.81667910

15

. 9.433354 4.88102410

6

0

. 14.44945 1.35817410

6

0

ω Ω_7 10 M S

1

0

1

. 7.30358710

. 33 1.8727810

13

=

=

0.253004

6

. 8.43227510

10 .

ω Ω_6 10 M S

0.253004

0.253004

( Hz)

0.693113

. 26 1.06924110 . 44 1.00503110 . 37 6.93112610 . 2.89624710

ω Ω_4 m x.m h 1 . 5 ω h ω Ω_4 10 .M S

6

3

ω Ω_4 m x.m h

5.

. 1.97218410

218.810356

=

5.

6

. 15 5.21112310 . 37 5.19263810 . 17 1.45002610 . 42 3.69181510

viii. Gravitation r . 9 ω Ω_2( r , M ) St g

a EGM_ωΩ( r , M )

a g ( r , M , φ, t )

g av ( r , M )

MM

π g ( r , M ) . .sin 2 .π .ω Ω ( r , M ) .t 2

2 T Ω ( r, M )

D E2M. M M .M E

r4

r5

M M .M E

D E2M r 4

φ

1. T Ω ( r, M ) 2 . 0 .( s )

a g( r, M , 0 , t ) d t

a EGM_ωΩ R M , M M

a EGM_ωΩ R E, M E

g R M,M M

g R E, M E

a EGM_ωΩ R J , M J

a EGM_ωΩ R S , M S

g R J, M J

g R S, M S

1=

. 1.49880110

12

. 1.49880110

12

. 1.5432110

12

. 1.57651710

12

373

(%)

www.deltagroupengineering.com

a EGM_ωΩ R NS, M NS

. 1 = 1.65423210

g R NS, M NS g r 4, M E g r 5, M M

=

. 3 3.33165310

m

. 3 3.33165310

a EGM_ωΩ r 4 , M E

=

a EGM_ωΩ r 5 , M M

g av R E, M E = 9.809009

s

r5

=

. 5 3.46028110 . 4 3.83719110

g r 5, M M = 0

. 3.33165310

3

m s

a EGM_ωΩ r 4 , M E

2

( km)

m s

3

s

r4

( %)

g r 4, M E

2

. 3.33165310

m

12

2

m

a EGM_ωΩ r 5 , M M = 0

s

2

ω Ω r 4 , M E = 56.499573 ( YHz) ω Ω r 5 , M M = 72.138509 ( YHz)

2

ix. Flux intensity 5

r ω ω Ω_3 , M

St G.

M

2

ω Ω_3

C Ω_J ( r , M )

9

2 d λ Ω ( r , M ) . U m( r , M ) dr

5

C Ω_J1( r , M )

St J .

M

5.2

9

C Ω_Jω ω Ω_3 , M

26

r

9

r ω ω VL( 400 ( nm ) ) , m x.m h

=

r ω ω VL( 750 ( nm ) ) , m x.m h

r ω ( 400 ( nm ) ) , m x.m h 1 . ω VL r 0 r ω ω VL( 750 ( nm ) ) , m x.m h

0.239057 0.741144

=

( Lyr)

. 1.73581410

11

. 5.38152510

11

r ω ω VL( 400 ( nm ) ) , M S

r ω ω VL( 750 ( nm ) ) , M S

5 r ω ω VL( 400 ( nm ) ) , 10 .M S

5 r ω ω VL( 750 ( nm ) ) , 10 .M S

10 r ω ω VL( 400 ( nm ) ) , 10 .M S

10 r ω ω VL( 750 ( nm ) ) , 10 .M S

r ω ω VL( 400 ( nm ) ) , M S

1 . r 30.( PHz) , 105 .M ω S r0 10 r ω 30.( PHz) , 10 .M S

5 . 5 2.2445.10 6.95860210

= 2.2445.107 6.95860210 . 7

r ω ω VL( 750 ( nm ) ) , M S

1 . r ω ( 400 ( nm ) ) , 105 .M ω VL S r0 10 r ω ω VL( 400 ( nm) ) , 10 .M S r ω 30.( PHz) , M S

4 9 .c . ω Ω_3 4 .π St 0.8 .M 0.6 G

5 r ω ω VL( 750 ( nm ) ) , 10 .M S 10 r ω ω VL( 750 ( nm ) ) , 10 .M S

r ω 30.( EHz) , M S 5 r ω 30.( EHz) , 10 .M S 10 r ω 30.( EHz) , 10 .M S

9 10 .Lyr

9 . 9 2.2445.10 6.95860210

. 4 5.05271110 . 4 1.62975410 = 1.62975410 . 6 5.05271110 . 6 . 8 5.05271110 . 8 1.62975410

21.2751776034345 8.46980075872643.10

5

= 2.12751776034345 .103 8.46980075872643.10

3

.105 2.12751776034345

374

0.846980075872643

www.deltagroupengineering.com

r ω 30.( PHz) , m x.m h

=

r ω 30.( EHz) , m x.m h

. 6 2.95234410

r 30.( PHz) , m x.m h 1 . ω r 0 r ω 30.( EHz) , m x.m h

3 10 .km

11.753495

r ω 30.( PHz) , M S

r ω 30.( EHz) , M S

5 r ω 30.( PHz) , 10 .M S

5 r ω 30.( EHz) , 10 .M S

10 r ω 30.( PHz) , 10 .M S

10 r ω 30.( EHz) , 10 .M S

r ω 30.( PHz) , M S 1 . r 30.( PHz) , 105 .M ω S r0 10 r ω 30.( PHz) , 10 .M S

. 5 2.93002110 = 2.93002110 .

7

5 r ω 30.( EHz) , 10 .M S 10 r ω 30.( EHz) , 10 .M S

. 21.275178 8.46980110

5

= 2.12751810 . 3 8.46980110 .

3

. 5 2.12751810

0.84698

C Ω_J 100.( km) , M M

C Ω_J R S , M M

C Ω_J 100.( km) , M E C Ω_J1 100.( km) , M J

. 2.95903310

6

. 9.40034410

4

1=

C Ω_J 100.( km) , M J

C Ω_J1 R S , M E C Ω_J R S , M E C Ω_J1 R S , M J

(%)

C Ω_J1 R S , M S

0.979587

C Ω_J 100.( km) , M S

C Ω_J R S , M S

C Ω_J1 100.( km) , M NS

C Ω_J1 R S , M NS

C Ω_J 100.( km) , M NS

C Ω_J R S , M NS

C Ω_Jω ω VL( 400 ( nm) ) , m x.m h C Ω_J1 r ω ω VL( 400 ( nm ) ) , m x.m h , m x.m h C Ω_Jω 30.( PHz) , m x.m h C Ω_J1 r ω 30.( PHz) , m x.m h , m x.m h C Ω_Jω 30.( PHz) , M S C Ω_J1 r ω 30.( PHz) , M S , M S

C Ω_Jω ω VL( 750 ( nm) ) , m x.m h C Ω_J1 r ω ω VL( 750 ( nm ) ) , m x.m h , m x.m h C Ω_Jω 30.( EHz) , m x.m h C Ω_J1 r ω 30.( EHz) , m x.m h , m x.m h C Ω_Jω 30.( EHz) , M S C Ω_J1 r ω 30.( EHz) , M S , M S

5 C Ω_Jω 30.( PHz) , 10 .M S

5 C Ω_Jω 30.( EHz) , 10 .M S

5 5 C Ω_J1 r ω 30.( PHz) , 10 .M S , 10 .M S

5 5 C Ω_J1 r ω 30.( EHz) , 10 .M S , 10 .M S

10 C Ω_Jω 30.( PHz) , 10 .M S

10 C Ω_Jω 30.( EHz) , 10 .M S

10 10 C Ω_J1 r ω 30.( PHz) , 10 .M S , 10 .M S

10 10 C Ω_J1 r ω 30.( EHz) , 10 .M S , 10 .M S

C Ω_J1 r 0 , m x.m h

10 C Ω_J1 r 0 , 10 .M S

1=

. 3.57491810

12

. 4.23150410

10

. 1.35061710

7

. 1.41439110

4

. 1.41439110

4

(%)

. 1.90958410

12

. 1.92068610

12

. 1.9428910

12

. 1.93178810

12

. 1.58761910

12

. 1.59872110

12

. 1.50990310

12

. 1.50990310

12

. 1.48769910

12

. 1.50990310

12

(%)

0

C Ω_J1 r 0 , M S 5 C Ω_J1 r 0 , 10 .M S

1=

C Ω_J R S , M J

0.979587

C Ω_J1 100.( km) , M S

0

6 10 .Lyr

116.646228

C Ω_J1 R S , M M

8

14

1.166462

C Ω_J1 100.( km) , M M

. 3.63872410

. 2.2659710

. 9 1.16646210 . 4 2.93002110

r ω 30.( EHz) , M S

C Ω_J1 100.( km) , M E

=

=

. 1.48429110

5

. 8.89809310

3

10

20 .

yJy

5.334267

375

www.deltagroupengineering.com

1 C Ω_J1 r 0 , m x.m h

. 26 2.24315810

1

10 C Ω_J1 r 0 , 10 .M S .

C Ω_J1 r 0 , M S

= 3.59381410 . 5 599.48425

1 5 C Ω_J1 r 0 , 10 .M S

C Ω_Jω ω VL( 400 ( nm ) ) , m x.m h

=

C Ω_Jω ω VL( 750 ( nm ) ) , m x.m h

28.979765 1.102778

10

16 .

yJy

C Ω_Jω 30.( EHz) , m x.m h = 2.479532 ( fJy )

7 C Ω_Jω 30.( PHz) , m x.m h = 6.228302 10 .yJy

C Ω_Jω 30.( PHz) , M S 5.

2.164916

C Ω_Jω 30.( PHz) , 10 M S

. = 2.16491610

3

10 C Ω_Jω 30.( PHz) , 10 .M S

. 2.16491610

6

C Ω_Jω 30.( EHz) , M S 5.

10

29 .

10

14 .

yJy

8.618686

C Ω_Jω 30.( EHz) , 10 M S

. = 8.61868610

3

10 C Ω_Jω 30.( EHz) , 10 .M S

. 8.61868610

6

10 10 C Ω_Jω ω Ω_4 10 .M S , 10 .M S 10 10 C Ω_J1 R BH 10 .M S , 10 .M S

yJy

= 0.999999999999968

x. Photon and Graviton populations ω g( M )

M .c 2 .h

2

n gg ( M )

ω g m x.m h

. 18 4.99252510

ωg MS

. 56 1.34855310

5 ω g 10 .M S 10 ω g 10 .M S

=

. 61 1.34855310

T L.ω g ( M )

n γγ( M )

2 .n gg ( M )

( YHz)

. 66 1.34855310

376

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ω g m x.m h ω Ω_4 m x.m h

ω g m x.m h ω Ω_5 m x.m h

ωg MS

ωg MS

2.666667

50

ω Ω_5 M S

. 36 2.95912210

. 57 1.0031510

5 ω g 10 .M S

. 63 4.65620810

5 ω Ω_5 10 .M S

2.666667

ω Ω_4 M S

. 2.1612210

=

5 ω g 10 .M S 5 ω Ω_4 10 .M S

=

. 46 1.26122510

10 ω g 10 .M S

10 ω g 10 .M S

10 ω Ω_4 10 .M S

10 ω Ω_5 10 .M S

n gg m x.m h

. 72 6.46222510

n γγ m x.m h

n gg M S

. 110 1.7455410

n γγ M S

. 115 1.7455410

5 n γγ 10 .M S

. 120 1.7455410

10 n γγ 10 .M S

=

5 n gg 10 .M S 10 n gg 10 .M S

. 41 1.93186910

. 73 1.29244510 . 110 3.4910810

=

. 115 3.4910810 . 120 3.4910810

xi. Hubble constant and CMBR temperature r 2( r )

K λ .r

K m.M

M 2( M )

5

5

K U( r , M )

A U( r , M )

1

ln

9

2

7

.ln n Ω_2 r 2( r ) , M 2( M )

TL K U( r , M )

R U( r , M )

5

Hα

K T( r , M )

n g .ln

T U( r , M )

K T( r , M ) .T W ( r , M )

M U( r , M )

H U5( r , M )

T U5( r , M )

KW c

µ

( 3 .π )

7 .µ .

256

µ

32

2

. .ln . 4µ H U5( r , M ) λ h Hα

2 .µ

.

r3

1 π .H α

H U( r , M )

1 A U( r , M )

KW λ Ω_1 R U( r , M ) , M 3 3 .H U( r , M )

2

8 .π .G

K W .St T .ln

T U2( H )

. µ m .ln ( 3 π ) . h 4 M

9

r 2( r )

c .A U( r , M )

ρ U( r , M )

V R U( r , M ) .ρ U( r , M )

.

M 2( M )

T W( r, M )

H U( r , M )

1 . ln TL

M3

3.

26

9

µ

. 2

.H5 µ

H 7 .µ

2

. r λh 2 .µ

Hα

2

2 7 .µ

5

.

mh M

5 .µ

2

. r λh

2 26 .µ

. 2

.H ( r , M ) 5 µ U5

377

www.deltagroupengineering.com

K W .St T .

dT dt ( t )

2 5 .ln H α .t .µ

t

5 .µ

2

1

K W .St T .

dT2 dt2 ( t )

2 2 5 .µ . ln H α .t . 5 .µ

.t

t

dT3 dt3 ( t )

2 2 2 5 .µ .ln H α .t . 5 .µ . 5 .µ 3 2 K W .St T . 2 5 .µ . 3 t t

dH dt H γ

2 H α .H γ . 5 .ln 1 .µ 2 2 Hγ 5 .µ Hγ

2 2 15.µ . 5 .µ

dH2 dt2 H γ

1

5 .µ

2

1

2

1

.t2

2

2

3 2 H α .H γ . 5 .µ 2 . ln 1 . 5 .µ 2 2 Hγ 5 .µ Hγ

1

2

1

Given T U2 H U r x1.R o , M G T U2 H U R o , m g1 .M G

T U2 H U r x2.R o , M G T U2 H U R o , m g2 .M G

∆T 0

T0

T0

∆T 0

r x1 r x2 m g1

Find r x1, r x2, m g1 , m g2

m g2

Given T U2 H U r x1.R o , m g1 .M G T U2 H U r x1.R o , m g2 .M G

∆T 0

T0

T U2 H U r x2.R o , m g1 .M G T U2 H U r x2.R o , m g2 .M G

T0

∆T 0

r x3 r x4 m g3

Find r x1, r x2, m g1 , m g2

m g4

Given T U2 H U r x1.R o , m g1 .M G r x5 m g5

T0

Find r x1, m g1

T U2 H U R o T U2 H U R o

1 ∆R o , .K m.M G 3 1 ∆R o , .K m.M G 3

r x1 =

2.733025 2.741859

r x2

( K)

m g1 m g2

378

0.989364 =

1.017883 1.057292 0.911791

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T U2 H U r x1.R o , M G T U2 H U r x2.R o , M G T U2 H U R o , m g1 .M G T U2 H U R o , m g2 .M G

2.724 =

2.726 2.724

m g3

7.914908 8.143063

( kpc )

1.063645 5.729219 1.788292

=

(%)

8.820858

2.724 2.726

( K)

0.977007

r x3

=

r x4

r x5

7.879647 8.106786

T U2 H U r x5.R o , m g5 .M G

1.052361

MG

. 11 M G m g3 5.8620410 . = M S m g4 . 11 5.8620410

( kpc )

1.013403

=

m g5

m g5 .

=

T U2 H U r x3.R o , m g3 .M G T U2 H U r x4.R o , m g4 .M G

0.977007

m g4

R o.

1=

r x2 m g2

1.013348

=

r x2

r x1 m g1

0.984956

r x4

r x1

2.726

. 11 M G m g1 6.34375310 . = M S m g2 . 11 5.47074910 r x3

R o.

( K)

r x5

. 11 = 6.31416710

m g5

MS

1=

r x3 m g3 r x4 m g4

5.236123

( %)

1.50441 2.29934 1.334822 2.29934

( %)

r x5.R o = 8.107221 ( kpc )

= 2.725 ( K )

1.340256

1=

H U r x5.R o , m g5 .M G = 67.095419

km . s Mpc

Given

dH dt

H U R o,M G

η

Hα

1

H U R o,M G

η

Find( η )

9 A U R o , M G = 14.575885 10 .yr

ρ U R o , M G = 8.453235 10

33 .

9 R U R o , M G = 14.575885 10 .Lyr

kg 3

. M U R o , M G = 9.28458610

52

( kg )

cm H U R o,M G 1 H U R o , .M G 3

=

67.084304 67.753267

km . s Mpc

T U R o,M G 1 T U R o , .M G 3

379

=

2.724752 2.739618

( K)

www.deltagroupengineering.com

H U R o,M G

T U R o,M G

H0

T0

H U R o,M G

T U R o,M G

1 H U R o , .M G 3

1 T U R o , .M G 3

T U2 H U R o

. 5.515064 9.08391310 0.987352

3

(%)

0.542607

∆R o , M G 2.720213

T U2 H U R o , M G T U2 H U R o

1=

2.724752

∆R o , M G

H U5 R o , M G H U5 r x5.R o , m g5 .M G

= 2.729021 ( K )

1 T U2 H U R o , .M G 3

2.739618

=

67.084304

km

67.095419

s .Mpc

2.810842

T U2 H 0

H U5 r x5.R o , m g5 .M G T U5 R o , M G T U5 r x5.R o , m g5 .M G

=

2.724752 2.725

H0

( K)

1=

T U5 r x5.R o , m g5 .M G

5.499409 . 8.3739910

9

(%)

T0

T U2 H α T U2 t 1

dT dt

0

1

. 3.19551810

31

T U2 t 2

1

T U2 t 3

1

T U2 t 4

1

. 31 2.05994510

T U2 t 5

1

. 31 2.65086510

dT2 dt2

. 3.03432210 . 2.83254210

dT2 dt2 t 2

. 114 2.02615310

dT2 dt2 t 3 dT2 dt2 t 4 dT2 dt2 t 5

0 . 8.77595210

112

s

dT3 dt3 t 3

. 112 1.612210

dT3 dt3 t 4

. 112 7.1945910

dT3 dt3 t 5

380

s

1 . 159 6.22716710

Hα

dT3 dt3 t 2

2

. 9.25283810

. 71 7.47950610

. 156 3.77545710

dT3 dt3 t 1 K

K

. 71 3.03728910

dT dt t 5

dT3 dt3

. 72 1.05719310 71

dT dt t 4

. 116 7.65967810

=

=

dT dt t 3

1

dT2 dt2 t 1

0

dT dt t 2

( K)

31

Hα

. 74 1.32321810

dT dt t 1

31

=

1 Hα

=

. 155 1.45285710

K

0

s

3

. 153 1.48902210 . 153 9.53337910

www.deltagroupengineering.com

dH dt ( 1 )

dH dt e

0 1

dH dt t 1 .H α

1

dH dt e

5 .µ

10 .µ

dH dt t 2 .H α

1

dH dt e

2

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

dH dt t 3 .H α

1

dH dt e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

=

2 3

2

1

dH dt t 4 .H α

1

dH dt e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

dH dt t 5 .H α

1

dH dt e

1

dH2 dt2 e

. 123 7.15875210

. 107 7.14236410

. 107 7.14236410

. 122 8.63295710

. 122 8.63295710

. 123 1.16708910

. 123 1.16708910

. 123 1.47916710

. 123 1.47916710

. 123 1.31810810

. 123 1.31810810

km s .Mpc

2

2

4

2 2 2 5 .µ . 5 .µ . 5 .µ

dH2 dt2 ( 1 )

. 123 7.15875210

1

2

0 1

dH2 dt2 t 1 .H α

1

dH2 dt2 e

5 .µ

10 .µ

dH2 dt2 t 2 .H α

1

dH2 dt2 e

2

2

1

2 2 5 .µ . 5 .µ

1

2 2 15 .µ . 5 .µ

dH2 dt2 t 3 .H α

1

dH2 dt2 e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

=

2 3

2

1

dH2 dt2 t 4 .H α

1

dH2 dt2 e

2 2 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

dH2 dt2 t 5 .H α

dH dt

1

H U R o,M G Hα

Hα

dH2 dt2 e

1

4

2 2 2 5 .µ . 5 .µ . 5 .µ

η

. 3 = 4.50029710

km . s Mpc

. 186 1.27869510

. 186 1.27869510

. 184 2.49929710

. 184 2.49929710

0

0

. 3.41565310

. 3.41565310

. 183 2.42270610

. 183 2.42270610

. 183 3.91232210

. 183 3.91232210

183

183

km s .Mpc

3

2 1

2

2

dH2 dt2

H U R o,M G Hα

η

=0

km . s Mpc

3

=1 η

dH dt 1

381

www.deltagroupengineering.com

η

dH dt 1

η

1

dH dt e

5 .µ

10 .µ

2

η

2

1

2 2 5 .µ . 5 .µ

dH dt e

. 61 2.55267410 η

2 2 15 .µ . 5 .µ

dH dt e

2

2 2 2 5 .µ . 5 .µ . 5 .µ

2 2 5 .µ . 5 .µ

km . s Mpc

. 60 4.13447210 . 60 9.11289510

1

67.084257 η 4

2 2 2 5 .µ . 5 .µ . 5 .µ

dH dt

= 1.16926910 . 61

2

3

η

2 2 5 .µ . 5 .µ

dH dt e

. 61 1.59787310

2

1

dH dt e

. 61 8.46094110

1

2 1

H U R o,M G

2

η

Hα

Hα

Hα

. 61 8.46094110

dH dt t 4 .H α

1

. 61 = 3.84599410 67.084304

H U R o,M G

dH dt t 4 .H α

km s .Mpc

H U R o,M G

71

H0

2

2

H0

. 123 7.15875210 1 2

=

. 123 1.47916710 . 3 4.50030410

km s .Mpc

2

3 5.041.10

1 H α.

dH dt t 4 .H α

1

=

Hα dH dt t 4 .H α

2.199936 4.839718

1

382

www.deltagroupengineering.com

η

T U2

dH dt 1

η

1

T U2

5 .µ

dH dt e

10 .µ

T U2

2

η

2

1

2 2 5 .µ . 5 .µ

dH dt e

. 31 2.97174510 η

2 2 15 .µ . 5 .µ

T U2

2

dH dt e

2 2 5 .µ . 5 .µ

T U2

2.724751 η 2

2 3

2

η

H U R o,M G

dH dt

. 31 3.18071410

1

2 2 2 5 .µ . 5 .µ . 5 .µ

dH dt e

. 31 2.72300610

η

2 2 15 .µ . 5 .µ

T U2

= 3.18071410 . 31 ( K )

2

3

1

T U2

. 31 3.18632310

2

2 2 2 5 .µ . 5 .µ . 5 .µ

dH dt e

0

1

Hα η

1

1 T U2 t 1

1

.T U2

dH dt e

5 .µ

10 .µ

1 T U2 t 2

1

.T U2

dH dt e

2

η

2

1

2 2 5 .µ . 5 .µ

η

2 2 15 .µ . 5 .µ

1 T U2 t 3

1

.T U2

dH dt e

1

2

2 2 2 5 .µ . 5 .µ . 5 .µ

3

1 T U2 t 4

η

1

.T U2

dH dt e

2 2 5 .µ . 5 .µ

T U2 t 5

.T U2

1 T U2 H U R o , M G

dH dt e

.T U2

(%)

32.18827 . 3.90264410

5

η 2

2 2 2 5 .µ . 5 .µ . 5 .µ

dH dt

12.291857 19.987768

1

2 2 15 .µ . 5 .µ

1

5.00939

2

1= 1

1

7.002696

2

2 3

2

H U R o,M G

η

Hα

383

www.deltagroupengineering.com

T U2 H α

1

T U2

13 .

10 ( s ) 1

T U2 10

10

10 .

(s)

1

T U2

1

T U2

1 . 31 ( day )

T U2 T U2

1 .( yr ) 1

T U2

2 10 .( yr )

1

T U2

3.

1

. 9 1.01325410

2.724752

( K)

1

1 6.

10 ( yr )

. 10 1.84076810 . 3 1.2497710

1

. 7 521.528169 2.52413210

7.

. 6 3.86401510

147.71262

= 1.00307810 . 6

41.823796

. 4 8.07751510

11.838588

9 10 .( yr )

. 2.29089210

3.35005

1

. 6.49496110

0.947724

1 8 10 .( yr )

1

=

1

3

( yr )

11 .

10

1 116.( day )

4

( K)

1

T U2

4.

T U2

4.898955

5.

10

10 ( yr ) T U2

. 15 3.43308810

10 .

10 ( yr ) T U2

9 5 .10 .( yr )

10 ( yr )

1

T U2

11.838588

10 ( yr )

1 .( day )

T U2

1

T U2

1 .( s ) 1

T U2

=

T U2 H U R o , M G

2 10 .( s )

T U2

T U2

T U2

. 1.92400510

28

9 10 .( yr )

(s)

1

T U2

T U2

34 .

978.724031

0

1

T U2

( yr )

. 6 1.87808710 . 3.98831410

7

( K)

TL

xii. SBH temperature Th

T BH( M )

Th

( 4 .π )

. 2

mh M

. 30 ( K ) = 1.66667410

2 8 .π .λ x

4

T SPBH

. 1. mh c κ h .ω h

2

c. U m λ x.λ h , m x.m h σ

=

. 32 3.55137410 . 32 3.55137410

T BH m x.m h

. 30 ( K ) T BH m x.m h = 1.66667410

h .c

( K)

=1

3

2 16.π .κ .G.m x.m h

10 T BH 10 .M S

h .c

3

T U2 =1

1 t1

T BH m x.m h

T U2 = 19.173025

1 t1

6 .π .T BH m x.m h

1 = 1.716054 ( % )

2 10 16.π .κ .G.10 .M S

384

www.deltagroupengineering.com

T U2

1 t1

3 .T h

T SPBH

. 31 ( K ) T SPBH = 5.02766910

1 = 1.716054 ( % )

T BH m x.m h

= 30.165887

4 .π .λ x

T SPBH T U2

Hα

. 31 ( K ) K ω T SPBH = 3.20071410

= 1.57335

1

T U2

t1 4

.

π

T U2

3 K ω .T SPBH

3.

1

6 .c . 31 ( K ) = 3.20071410 π .σ .G

t1

2 K W .c . 5 G.κ

1 . 15 . h .c λ x 4 .π κ .K W

15 . h .c = 2.659782 4 .π κ .K W

1 = 0.248248 ( % )

1 = 1.442436 ( % )

T BH m e

T BH m p

T BH m n

T BH m µ

T BH m τ

T BH m en

T BH m µn

T BH m τn

T BH m uq

T BH m dq

T BH m sq

T BH m cq

T BH m bq

T BH m tq

T BH m W

T BH m Z

T BH m H T SPBH

=

t1

2 c . KW . 31 ( K ) = 3.18758510 5 G.κ

3

1

1 = 0.162602 ( % )

1

. 47 6.01617410 . 31 5.02766910

. 53 7.33529610 . 49 7.32519910 . 49 6.51392110 . 50 1.34687210 =

. 49 2.29416810 . 58 3.6223710 . 53 3.78159510 . 51 3.87312710 . 52 9.81839510 . 51 6.04208910 . 50 5.81830810 . 49 1.96367910

( K)

. 49 3.8570310 . 47 8.55766610 . 47 7.54763110 . 47 1.67121610

( K)

xiii. ZPF Ω EGM

ρ U r x5.R o , m g5 .M G

3 .c . Ω ZPF . H U R o,M G 8 .π .G 2

U ZPF Ω EGM Ω PDG

Ω PDG

ρ U R o,M G

= 0.997339

U ZPF = 251.778016

U ZPF = 842.934914

yJ 3

2

Ω ZPF

1.003

. Ω ZPF = 3.31400710

4

U ZPF = 251.778016( fPa )

U ZPF = 2.51778 10

13 .

U ZPF = 0.251778

mJ

U ZPF = 251.778016

3

km

EJ

. 12 U ZPF = 7.39723510

AU

Ω EGM

Ω EGM = 1.000331

mm

3

1

fJ 3

m

YJ pc

385

Pa

3

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. 41 U ZPF = 6.60189810

YJ R U R o,M G

3

xiv. Cosmological limits M U R o,M G

M EGM

ML

R EGM

K m.M G.

K λ .R o

R U R o,M G

R EGM 5 5

.

R EGM

rL

K λ .R o

. 71 ( kg ) M L = 4.86482110

t EGM

R BH M L

. 19 109 .Lyr r L = 7.6372910

A U R o,M G

rL

tL

c

. 19 109 .yr t L = 7.6372910

ML M EGM

M EGM 2 R EGM.c

t EGM

=1

2 .G

. 18 5.23967510

rL

=1

R EGM

R EGM

c

tL

tL

= 5.23967510 . 18

. 6 = 1.86196810

TL

. 18 5.23967510

t EGM

xv. Particle Cosmology 5

m γγ2

h tL

E Ω ( r, M )

Qγ

m gg2

h .ω Ω ( r , M )

Q γ r ε, m e

Q γ_PDG

5 .10

2 .m γγ2

r γγ2

N γ( r, M )

Q γγ ( r , M )

30 .

Qe

r e.

m γγ2 m e .c

mγ

m gg2

1.715978 3.431956

4 .r γγ2

5

Q γ( r, M )

Q γγ

N γ( r, M )

=

r gg2

2

E Ω ( r, M )

Q γ( r, M )

m γγ2

2

10

386

51 .

eV

Qγ

Qe N γ( r, M )

2

Qe

Q γγ2

r γγ2 r gg2

Q γγ m γγ

=

.m γγ2

7.250508 9.567103

10

38 .

m

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r γγ2

λh

λh

r γγ2

2 .r γγ2

K λ .λ h

K λ .λ h

2 .r γγ2

0.178967

2 .r γγ2 λh

=

r gg2

0.357933 0.236148

558.763566

λh

0.307913

324.766614

2 .r γγ2

(%)

279.381783

=

λh

423.463597

λh

0.406294

r gg2

246.127068

2 .r gg2

0.472296

K λ .λ h

211.731798

K λ .λ h

2 .r gg2

2 .r gg2

λh

λh

2 .r gg2

Qγ

= 2.655018 10

Q γ_PDG

30

Qe

Q γγ

Q γγ = 1.129394 10

= 1.883226

Qγ

= 7.049122 10

m γγ

60

Qe

. = 1.86196810

6

Q γγ2

Q γγ2 = 6.065593 10

m γγ2 mγ

= 3.785846 10

ω Ω r e, m e .m γγ ω Ω r ε, m e

Qe

E Ω r e, m e mγ

2

C

85 .

C

2

E Ω r ε, m e

66

78 .

=

1.525768 1.525768

10

46 .

eV

ω Ω r e,m e =

0.165603 0.165603

( µJ )

mγ

2

=

h .m γγ

m γγ

249.926816 249.926816

( YHz)

Qe me 2.

c Q γγ

=

. 11 1.7588210

C

198.286288

kg

m γγ

387

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NOTES

388

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•

MathCad 12 c. High precision calculation engine i. Computational environment

NOTE: KNOWLEDGE OF MATHCAD IS REQUIRED AND ASSUMED • • •

Convergence Tolerance (TOL): 10-14. Constraint Tolerance (CTOL): 10-14. Calculation Display Tolerance: 6 figures – unless otherwise indicated. ii. Astronomical statistics

H0 := 71⋅ 

   s ⋅ Mpc 

∆H0 := 2⋅ 

   s ⋅ Mpc 

km

km

∆T0 := 0.001( K)

T0 := 2.725⋅ ( K)

H0 − ∆H0 = 69 

   s ⋅ Mpc  km

T0 + ∆T0 = 2.726( K)

iii. Derived constants

µ :=

λx := 4⋅

1

2⋅ µ µ

π

3

Stt := 2⋅ ωh ⋅  4

  µ  3⋅ π  2

7

⋅ 

2

Hα :=

ωh

TL :=

λx

h mγγ

9

 c

iv. Algorithm 7⋅ µ   2 2 2 2 2  5⋅ µ 7⋅ µ µ µ m µ 7⋅ µ  26⋅ µ  m   ( ) ( ) 1  h  3⋅ π ⋅ 32 ⋅ ln 3⋅ π ⋅  h  ⋅  r    r  HU5( r , M) := ⋅ ln  4  M   λ   ⋅  M  ⋅  λ   TL  256   h   h



  H ( r , M )  U5 

KT ( r , M) := 8⋅ µ ⋅ ln

Hα

TU( r , M ) := KT ( r , M ) ⋅ TW( r , M )

TW( r , M) :=

KW



c



HU5 ( r , M)

λΩ 

,

λx 2



⋅ mh 



 Hα  9 5  ⋅ Stt ⋅ H  H 

TU2( H) := KW⋅ ln

 HU5( Ro , MG)   67.084134  km   =    HU5( Ro , µ ⋅ M G)   67.753095  s ⋅ Mpc 

 TU( Ro , MG)   2.724749  =  ( K)  TU( Ro , µ ⋅ MG)   2.739614

 HU5( Ro , M G)   −2.776618 ( %)  −1=  H0 − ∆H0  HU5( Ro , µ ⋅ MG)   −1.807108 1

5

 TU( Ro , M G)   −0.045904 ( %)  −1=  T0 + ∆T0  TU( Ro , µ ⋅ MG)   0.499413  1

⋅

389

⋅

www.deltagroupengineering.com

 TU2( H0 − ∆H0)    TU2( H0)    ∆H    TU2 H0 + 0   2   

 2.767146 =  2.810842 ( K)    2.832481

 H0 − ∆H0   H0  ∆H  H0 + 0 2 

     

 69  km  =  71      s ⋅ Mpc   72 

d. Various forms of the derived constants

 4 ⋅   6π   4⋅ µ ⋅  

  π   2.698709 2.698709  =  6 2⋅ µ   2.698709 2.698709 4⋅ µ µ  π π  2

4

3

3

⋅

6

µ

6 2 1  ⋅  1  ⋅  4⋅ µ    c3  π⋅ Hα   λh     7 9  4  2   2  2⋅ ωh ⋅  µ  ⋅  c      3⋅ π   5 7   1 ⋅ 4  ⋅ 2    4  c   µ    3⋅ π    λh

 6.355579× 1095    5  s   = 6.355579× 1095     9  6.355579× 1095   m   

NOTES

390

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Periodic Table of the Elements

391

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NOTES

392

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Quick Reference Graphs

8.1.5.1 “TU3 vs. Hβ”: Figure 4.22 Av. Cosmological Temperature

1

31 3.5 .10 e

5 .µ

2

3 .1031

Av. Cosmological Temperature (K)

2.5 .1031

31 2 .10

T U3 H β 1 T U3 e

5 .µ

2 1.5 .1031

31 1 .10

5 .1030

1

0.1

0.01

1 .10 3 Hβ Dimensionless Range Variable

1 .10 4

1 .10 5

1 .10 6

Average Cosmological Temperature Maximum Av. Cosmological Temperature

393

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Quick Reference Graphs

8.1.5.2 “TU3 vs. t = (HβHα)-1” (i): Figure 4.23 Av. Cosmological Temperature vs. Age

1

31 3.5 .10

2

1 Hα

e

5 .µ . 1 Hα 3 .1031

Av. Cosmological Temperature (K)

2.5 .1031

31 2 .10

T U3 H β 1 T U3 e

5 .µ

2 1.5 .1031

31 1 .10

5 .1030

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

H β .H α Cosmological Age (s)

394

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Quick Reference Graphs

8.2.5.1 “TU3 vs. t = (HβHα)-1” (ii): Figure 4.24 Av. Cosmological Temperature vs. Age 1 Hα

31 3.5 .10

t1

3 .1031

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

T U3 e

T U3 e

5 .µ

2 31 2 .10

2 10 .µ 1 2 2 5 .µ . 5 .µ 1 1.5 .1031

2 2 15 .µ . 5 .µ 2 2 2 2 2 . . . . . 2 5µ 5µ 5µ 3

31 1 .10

5 .1030

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

H β .H α Cosmological Age (s)

395

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Quick Reference Graphs

8.2.5.2 “TU3 vs. t = (HβHα)-1” (iii): Figure 4.25 Av. Cosmological Temperature vs. Age t2

31 3.5 .10

t3

3 .1031

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

T U3 e

T U3 e

5 .µ

2 31 2 .10

2 10 .µ 1 2 2 5 .µ . 5 .µ 1 1.5 .1031

2 2 15 .µ . 5 .µ 2 2 2 2 2 . . . . . 2 5µ 5µ 5µ 3

31 1 .10

5 .1030

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

H β .H α Cosmological Age (s)

396

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Quick Reference Graphs

8.2.5.3 “TU3 vs. H = (HβHα)” (i): Figure 4.26 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 t1

Hα

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 2 2 1.5 .10 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

H β .H α Hubble Constant (Hz)

397

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Quick Reference Graphs

8.2.5.4 “TU3 vs. H = (HβHα)” (ii): Figure 4.27 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 1 t2 t3

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 2 2 1.5 .10 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

H β .H α Hubble Constant (Hz)

398

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Quick Reference Graphs

8.2.5.5 “TU3 vs. r = (HβHα)-1c” (i): Figure 4.28 Av. Cosmological Temperature vs. Size c Hα

31 3.5 .10

t 1 .c

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 1.5 .10

2 2 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

1 .10 29

1 .10 28

1 .10 27

1. c

H β .H α EGM Cosmological Size (m)

399

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Quick Reference Graphs

8.2.5.6 “TU3 vs. r = (HβHα)-1c” (ii): Figure 4.29 Av. Cosmological Temperature vs. Size

31 3.5 .10

t 2 .c t 3 .c

31 3 .10

2.5 .1031

Av. Cosmological Temperature (K)

T U3 H β 1 T U3 e

2 5 .µ 2 .1031 2

T U3 e

T U3 e

10 .µ 1 2 2 5 .µ . 5 .µ 1 31 1.5 .10

2 2 2 15 .µ . 5 .µ 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 3

31 1 .10

5 .1030

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

1 .10 29

1 .10 28

1 .10 27

1. c

H β .H α EGM Cosmological Size (m)

400

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Quick Reference Graphs

8.2.5.7 “dTU4/dt vs. t = (HβHα)-1” (i): Figure 4.30 1st Derivative of Av. Cosmological Temp.

1 .1072 t1

t2

71 8 .10

6 .1071

4 .1071

71 2 .10 dT dt

H β .H α

1 0

(K/s)

dT dt t 1 dT dt t 2 dT dt t 3

2 .1071

4 .1071

71 6 .10

71 8 .10

1 .1072

72 1.2 .10 1 .10 42

1 .10 41

1 .10 40

1 .10 39

1

H β .H α Cosmological Age (s)

401

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Quick Reference Graphs

8.2.5.8 “dTU4/dt vs. t = (HβHα)-1” (ii): Figure 4.31 1st Derivative of Av. Cosmological Temp.

1 .1072 t2

t3

71 8 .10

6 .1071

4 .1071

71 2 .10 dT dt

H β .H α

1 0

(K/s)

dT dt t 1 dT dt t 2 dT dt t 3

2 .1071

4 .1071

71 6 .10

71 8 .10

1 .1072

72 1.2 .10 1 .10 42

1 .10 41

1 .10 40

1 .10 39

1

H β .H α Cosmological Age (s)

402

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Quick Reference Graphs

8.2.5.9 “d2TU4/dt2 vs. t = (HβHα)-1” (i): Figure 4.32 2nd Derivative of Av. Cosmological Temp.

113 5 .10 t1

t2

0

113 5 .10

(K/s^2)

dT2 dt2

H β .H α

1 1 .10114

dT2 dt2 t 1 dT2 dt2 t 2 dT2 dt2 t 3

114 1.5 .10

2 .10114

114 2.5 .10

114 3 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42 1 .10 41 1 H β .H α Cosmological Age (s)

403

1.1 .10 41 1.2 .10 41 1.3 .10 41 1.4 .10 41 1.5 .10 41

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Quick Reference Graphs

8.2.5.10 “d2TU4/dt2 vs. t = (HβHα)-1” (ii): Figure 4.33 2nd Derivative of Av. Cosmological Temp.

113 5 .10 t2

t3

0

113 5 .10

(K/s^2)

dT2 dt2

H β .H α

1 1 .10114

dT2 dt2 t 1 dT2 dt2 t 2 dT2 dt2 t 3

114 1.5 .10

2 .10114

114 2.5 .10

114 3 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42 1 .10 41 1 H β .H α Cosmological Age (s)

404

1.1 .10 41 1.2 .10 41 1.3 .10 41 1.4 .10 41 1.5 .10 41

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Quick Reference Graphs

8.2.5.11 “|d3TU4/dt3| vs. t = (HβHα)-1” (i): Figure 4.34 3rd Derivative of Av. Cosmological Temp.

157 1 .10 t1

t2

156 1 .10

(K/s^3)

dT3 dt3

H β .H α

1 .10155 1

dT3 dt3 t 1 dT3 dt3 t 2 1 .10154

153 1 .10

152 1 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42

1 .10 41 1

1.1 .10 41

1.2 .10 41

1.3 .10 41

1.4 .10 41

1.5 .10 41

H β .H α Cosmological Age (s)

405

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Quick Reference Graphs

8.2.5.12 “|d3TU4/dt3| vs. t = (HβHα)-1” (ii): Figure 4.35 3rd Derivative of Av. Cosmological Temp.

157 1 .10 t2

t3

156 1 .10

(K/s^3)

dT3 dt3

H β .H α

1 .10155 1

dT3 dt3 t 1 dT3 dt3 t 2 1 .10154

153 1 .10

152 1 .10 2 .10 42

3 .10 42

4 .10 42

5 .10 42

6 .10 42

7 .10 42

8 .10 42

9 .10 42

1 .10 41 1

1.1 .10 41

1.2 .10 41

1.3 .10 41

1.4 .10 41

1.5 .10 41

H β .H α Cosmological Age (s)

406

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Quick Reference Graphs

8.3.6.1 “dH/dt vs. (HβηHα)-1” (i): Figure 4.36 1st Derivative of the Hubble Constant 1.6 .1084

t1

t4

84 1.4 .10

84 1.2 .10 dH dt H β

(Hz^2)

dH dt e

dH dt e

dH dt e

η

5 .µ

1 2

1 .1084 1

1 2 2 5 .µ . 5 .µ

1

8 .1083

2 2 5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

83 6 .10

83 4 .10

2 .1083

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

407

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Quick Reference Graphs

8.3.6.2 “dH/dt vs. (HβηHα)-1” (ii): Figure 4.37 1st Derivative of the Hubble Constant 2 .1084 1 Hα

t1

84 1 .10

0 0

dH dt H β

(Hz^2)

dH dt e

η

5 .µ

1 2

1 .1084

1

84 2 .10

1 dH dt e

2 2 5 .µ . 5 .µ

3 .1084 2

dH dt e

1 2

5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

84 4 .10

5 .1084

6 .1084

84 7 .10

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

408

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Quick Reference Graphs

8.3.6.3 “dH/dt vs. (HβηHα)-1” (iii): Figure 4.38 1st Derivative of the Hubble Constant 2 .1084 t2

t3

84 1 .10

0 0

dH dt H β

(Hz^2)

dH dt e

η

5 .µ

1 2

1 .1084

1

84 2 .10

1 dH dt e

2 2 5 .µ . 5 .µ

3 .1084 2

dH dt e

1 2

5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

84 4 .10

5 .1084

6 .1084

84 7 .10

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

409

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Quick Reference Graphs

8.3.6.4 “dH/dt vs. (HβηHα)-1” (iv): Figure 4.39 1st Derivative of the Hubble Constant 2 .1084 t5

t4

84 1 .10

0 0

dH dt H β

(Hz^2)

dH dt e

η

5 .µ

1 2

1 .1084

1

84 2 .10

1 dH dt e

2 2 5 .µ . 5 .µ

3 .1084 2

dH dt e

1 2

5 .µ . 5 .µ 4 2 2 2 2 1 5 .µ . 5 .µ . 5 .µ 2

84 4 .10

5 .1084

6 .1084

84 7 .10

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36

1

η H β .H α Cosmological Age (s)

410

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Quick Reference Graphs

8.3.6.5 “d2H/dt2 vs. (HβηHα)-1” (i): Figure 4.40 2nd Derivative of the Hubble Constant 1 Hα

4 .10127

t1

127 3.5 .10

3 .10127

(Hz^3)

2.5 .10127

dH2 dt2 H β

η 127 2 .10

127 1.5 .10

1 .10127

5 .10126

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1

η H β .H α Cosmological Age (s)

411

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Quick Reference Graphs

8.3.6.6 “d2H/dt2 vs. (HβηHα)-1” (ii): Figure 4.41 2nd Derivative of the Hubble Constant t2

4 .10127

t3

127 3.5 .10

3 .10127

(Hz^3)

2.5 .10127

dH2 dt2 H β

η 127 2 .10

127 1.5 .10

1 .10127

5 .10126

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1

η H β .H α Cosmological Age (s)

412

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8.3.6.7 “d2H/dt2 vs. (HβηHα)-1” (iii): Figure 4.42 2nd Derivative of the Hubble Constant t5

4 .10127

t4

127 3.5 .10

3 .10127

(Hz^3)

2.5 .10127

dH2 dt2 H β

η 127 2 .10

127 1.5 .10

1 .10127

5 .10126

0 0

1 .10 43

1 .10 42

1 .10 41

1 .10 40

1

η H β .H α Cosmological Age (s)

413

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8.3.6.8 “d2H/dt2 vs. (HβηHα)-1” (iv): Figure 4.43 2nd Derivative of the Hubble Constant 125 8 .10

t1

t2

7 .10125

125 6 .10

dH2 dt2 H β

(Hz^3)

dH2 dt2 e

dH2 dt2 e

dH2 dt2 e

η

5 .µ

1 2

125 5 .10

1

1 2 2 5 .µ . 5 .µ

4 .10125

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 . . . . . 2 5µ 5µ 5µ 1

125 3 .10

2 .10125

1 .10125 0 0

1 .10125

1 .10 42

1 .10 41 1 η H β .H α Cosmological Age (s)

414

1 .10 40

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8.3.6.9 “d2H/dt2 vs. (HβηHα)-1” (v): Figure 4.44 2nd Derivative of the Hubble Constant 125 8 .10

t3

t4

7 .10125

125 6 .10

dH2 dt2 H β

(Hz^3)

dH2 dt2 e

dH2 dt2 e

dH2 dt2 e

η

5 .µ

1 2

125 5 .10

1

1 2 2 5 .µ . 5 .µ

4 .10125

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 . . . . . 2 5µ 5µ 5µ 1

125 3 .10

2 .10125

1 .10125 0 0

1 .10125

1 .10 42

1 .10 41 1 η H β .H α Cosmological Age (s)

415

1 .10 40

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8.3.6.10 “d2H/dt2 vs. (HβηHα)-1” (vi): Figure 4.45 2nd Derivative of the Hubble Constant 125 8 .10

t5

7 .10125

125 6 .10

dH2 dt2 H β

(Hz^3)

dH2 dt2 e

dH2 dt2 e

dH2 dt2 e

η

5 .µ

1 2

125 5 .10

1

1 2 2 5 .µ . 5 .µ

4 .10125

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 . . . . . 2 5µ 5µ 5µ 1

125 3 .10

2 .10125

1 .10125 0 0

1 .10125

1 .10 42

1 .10 41 1 η H β .H α Cosmological Age (s)

416

1 .10 40

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8.3.6.11 “|H| vs. (HβηHα)-1” (i): Figure 4.46 Mag. of Hubble Cons. vs. Cosm. Age

2.5 .1042

1 Hα

t1

42 2 .10 dH dt H β

dH dt e

5 .µ

η

1 2

1 1.5 .1042

(Hz)

1 dH dt e

dH dt e

2 2 5 .µ . 5 .µ

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 1

42 1 .10

5 .1041

0 1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36 1

1 .10 35

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

η H β .H α Cosmological Age (s)

417

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8.3.6.12 “|H| vs. (HβηHα)-1” (ii): Figure 4.47 Mag. of Hubble Cons. vs. Cosm. Age

2.5 .1042

1 Hα

t4

42 2 .10 dH dt H β

dH dt e

5 .µ

η

1 2

1 1.5 .1042

(Hz)

1 dH dt e

dH dt e

2 2 5 .µ . 5 .µ

1

2 2 5 .µ . 5 .µ 4 2 2 2 2 2 5 .µ . 5 .µ . 5 .µ 1

42 1 .10

5 .1041

0 1 .10 43

1 .10 42

1 .10 41

1 .10 40

1 .10 39

1 .10 38

1 .10 37

1 .10 36 1

1 .10 35

1 .10 34

1 .10 33

1 .10 32

1 .10 31

1 .10 30

η H β .H α Cosmological Age (s)

418

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8.3.6.13 “TU2,3 vs. |H|”: Figure 4.48 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 t1

Hα 31 3 .10

Av. Cosmological Temperature (K)

2.5 .1031

T U2

dH dt H β

η 2 .1031

T U3 H β 1 T U3 e

5 .µ

2 31 1.5 .10

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

η dH dt H β , H β .H α Hubble Constant (Hz)

419

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8.3.6.14 “TU2 vs. |H|”: Figure 4.49 Av. Cosmological Temp. vs. Hubble Cons.

31 3.5 .10 1 t1

Hα 31 3 .10

Av. Cosmological Temperature (K)

2.5 .1031

T U2

dH dt H β

η 2 .1031

1 T U3 e

5 .µ

2 31 1.5 .10

31 1 .10

5 .1030

1 .1043

1 .1042

1 .1041

1 .1040

1 .1039

1 .1038

1 .1037

1 .1036

η dH dt H β Hubble Constant (Hz)

420

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Quinta Essentia: A Practical Guide to Space-Time Engineering - Part 4

ISBN 978-1-84753-403-3