The Natural Philosophy Of The Cosmos (c)

The atural Philosophy of the Cosmos (C)

Riccardo C. Storti1

Abstract The principles of mass-energy distribution and similitude by Zero-Point-Field (ZPF) equilibria are utilised to derive the values of the present Hubble constant “H0” and Cosmic Microwave Background Radiation (CMBR) temperature “T0”. It is demonstrated that a mathematical relationship exists between the Hubble constant and CMBR temperature such that “T0” is derived from “H0”. The values derived are “67.0843(km/s/Mpc)” and “2.7248(K)” respectively. The derivations are possible by 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 wavefunction. Simultaneously, we represent the “Milky-Way” (MW) as a Planck scale object of equivalent total Galactic mass “MG”, acting as a Galactic Reference Particle (GRP) characterised by a large number of wavefunctions with respect to the solar distance from the Galactic centre “Ro”. This facilitates a comparative analysis between the Primordial and Galactic particle representations by application of a harmonic relationship, yielding “H0” in terms of “Ro” and “MG”. Consequently, utilising the experimental value of “T0”, we derive improved estimates for “Ro” and “MG” as being “8.1072(kpc)” and “6.3142 x1011(solar-masses)” respectively. The construct herein 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)” [i.e. “< -0.252(mJ/km3)”]. Moreover, it is graphically illustrated that the gradient of the Hubble constant in the time domain is presently positive (i.e. “dH/dt > 0”).

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Electro-Gravi-Magnetics (EGM) Dilemma: All objects and natural systems seek states of lowest energy, so why wouldn’t a solitary Electron at rest in “some other” Universe, diffuse itself into non-existence? A point at infinity denotes a co-ordinate of lower energy, so why wouldn’t the mass-energy of an internally structure-less particle seek to “spread-out” over the entire Universe if mass and energy are truly equivalent? If Quantum behaviour is somehow “the answer”, then what may be responsible for all (or any) forms of Quantum existence? Where might the energy, or the required conditions, to sustain Quantum existence originate?

This manuscript seeks to resolve this dilemma and, in doing so, derives critical Cosmological information such as the present values of the Hubble constant “H0” and CosmicMicrowave-Background-Radiation (CMBR) temperature “T0”. The most important concept developed herein is that “Dark Matter / Energy” is not required to mathematically articulate and precisely numerically determine “H0” and “T0”. The accelerated Cosmological expansion phenomenon (i.e. “dH/dt > 0”) is derived organically from Particle-Physics, in favourable agreement with experimental evidence. Therefore, it is proposed that the observational suggestion for the existence of “Dark Matter” may be explained by halos of ejected Gravitons existing as conjugate wavefunction pairs of non-zero mass Photons. [12, 23, 34] Moreover, it is demonstrated in [4] that “Dark Energy” is analogous to the Zero-Point-Field (ZPF) energy associated with the Casimir Effect, acting on a Cosmological scale. i. Quantum Vacuum (QV) is a generalised theoretical Quantum Mechanical reference to the space-time manifold of General Relativity (GR). ii. ZPF refers to the ground-state of the QV. iii. Polarisable Vacuum (PV) refers to a polarised5 form of the ZPF / QV. Table 1: Applied Definitions, The initial premise in the development of the Electro-Gravi-Magnetics (EGM) method is the assumption that gravity and ElectroMagnetism may be unified via Quantum Mechanics (QM) in terms of the QV, utilising Buckingham “Π” Theory (BPT) and Dimensional Analysis Techniques (DAT’s). In order to compare a mathematical model to a physical system, it must possess Dynamic, Kinematic or Geometric similarity to the real-world (any or all of these if applicable). “Dynamic similarity” relates forces, “Kinematic similarity” relates motion and “Geometric similarity” relates shape6. EGM is an engineering approach, not a theory, it is a tool for mathematically simulating real-world systems in order to model physical problems in GR and QM utilising standard engineering techniques. Gravity is the result of an interaction between matter and the space-time manifold; leading to the following precepts, i. An object at rest polarises the QV surrounding it. ii. An object at rest is in equilibrium with the QV surrounding it. iii. The QVE7 surrounding an object at rest is equivalent to “E = mc2”. iv. The frequency distribution of the spectral energy density of the QV surrounding an object at rest is cubic. 2

Available for download as Ch. 3.8 in [5]. Available for download as Ch. 3.10 in [5]. 4 Available for download as Ch. 4 in [6]. 5 i.e. not in its ground-state. 6 i.e. the topology of space-time curvature within the context of GR. 7 i.e. gravitational field energy. 3

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The EGM method commences by mathematically representing mass as an equivalent localised density of wavefunction energy, contained by the QV surrounding it. Properties of Fourier harmonics are utilised to mathematically decompile the mass-energy into a spectrum of EM frequencies. The QV is predicted and required by QM and Quantum-Electro-Dynamics (QED), both dictating that virtual energy must exist within the fabric of space-time. The EGM construct represents matter as a precisely defined spectrum of EM energy utilising Fourier techniques and models its interaction with the QV as a dynamic system. Subsequently, the unique spectral “signatures” of matter are superimposed upon the QV demonstrating that a change in Poynting Vector (∆P) results in a gravitational effect. All natural systems seek to find equilibrium; this implies that the energy condensed as matter exists in a state of equilibrium within the Universe surrounding it. Consequently, EGM asserts that mass is relativistic because it equilibrates to the ambient energy conditions of its local environment. The methodology articulated in [3] specifies the mass-energy equilibrium point between an object and the space-time manifold such that the metaphysical conception of “curvature” is re-interpreted as being a local polarisation of the QV, explicable by the superposition of EM fields, yielding a change in the Poynting Vector “∆P”. The gradient in “P” is analogous to variations in the Refractive Index “KPV” of the space-time manifold in an optical model of gravity.8 The EGM construct models vacuum polarisation by the superposition of mass-energy and QV spectra. A key difference distinguishing mass-energy from Quantum-Vacuum-Energy (QVE) is that the energy contained within matter is highly localised, whereas QVE is distributed homogeneously throughout the vast regions of free-space. Haisch, Rueda and Puthoff (HRP) determined that the QV spectrum obeys a cubic frequency distribution9. However, this presents a rather formidable dilemma. This type of distribution implies that the energy density of empty space is staggering. Calculating the total energy represented by the HRP interpretation suggests that every cubic centimetre of empty space is so packed full of energy that it should cause the Universe to collapse in upon itself. Because of this theoretical result, many Physicists discount the existence of the QV in cubic frequency form, believing that something must be fundamentally wrong with its formulation, despite the fact that it is derived utilising standard QM. However, the EGM construct does not suffer from this ailment and emphatically rejects the assertion that an infinite quantity of energy is contained within the vanishing volume associated with the QM derivation of the QV. The physical justification for this emphatic rejection spawns from the derivation of “H0” and “T0” utilising the harmonic representation of fundamental particles. The derivation of “H0” and “T0” within the EGM construct yields experimentally impressive results, substantially beyond the abilities of the Standard Model (SM) of Cosmology (SMoC), without the “vanishing volume” implications of QM; hence, the “emphatic rejection” asserted herein is substantiated. Applying EGM to the energy dynamics of Hubble expansion spectrally, “H0” is derived by modelling the QV spectrum of the “Primordial-Universe”10 as a single high-frequency wavefunction representing the energy of the entire Universe. Instantaneously after the “Big-Bang”, the single wavefunction rapidly decomposed into a broad spectrum of lower-frequency wavefunctions, forming localised gradients through the condensation of mass. Summing the energy associated with all lower-frequency wavefunctions in the present QV yields the total energy of the Universe, equalling the total energy at an instant prior to the “Big-Bang”; hence, energy is conserved and the cubic frequency distribution of the QV 8

An optical interpretation of gravity was first suggested approximately three hundred years ago by Sir Isaac Newton in his manuscript entitled Opticks. Newton theorised that the aether should be most dense far away from an object like the Earth, and conversely, more subtle and rarefied nearby or within it. Newton wrote that a gradual change in the density of the aether curves paths of light. Regional changes in “KPV” result in the refraction of light as though passing through a lens. 9 i.e. the energy density of QV spectral modes increases to the cube of the frequency. 10 i.e. instantaneously prior to the “Big-Bang”. 3

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spectrum predicted by HRP is preserved. Setting the QV spectrum temporarily aside, we shall now define and describe the energy spectrum associated with matter; termed “the EGM spectrum”. This is a wavefunction representation of mass-energy obeying a Fourier distribution such that the number of modes decreases as energy density increases, implying that the energy density of free-space approaches zero, avoiding the “infinite energy in a vanishing volume” problem. This is because each mode is representative of the possibility of the existence of virtual Photons only, not that virtual Photons must exist. In other words, free space can accommodate the existence of high energy Photons; however, the probability of their existence in the absence of mass approaches zero. Similarly, the probability of low frequency virtual Photons existing in the QV of free space approaches unity. Consider the action of adding a point mass to an empty Universe. This action superimposes the EGM spectrum of the point mass onto the QV spectrum of the Universe; doing so forms the PV spectrum11 surrounding the point mass, inducing a mode population gradient in space-time between the point mass and the edge of the Universe. The mode population gradient modifies the “KPV” value of the vacuum such that it changes at the same rate as gravitational acceleration “g” from the point mass. Thus, the gradient is “curved” in an analogous manner to space-time within GR. A mass-object pushes the vacuum around it “uphill”, against the natural flux of expansion. Mass may be modelled as doing work on the surrounding vacuum by “curving” it. This occurs because the nature of the Universe is to expand and upon encountering resistance to its normal flux from high to low energy, the Universe “pushes back” as it strives to find balance (i.e. equilibrium). Thus, the matter-Universe interaction is a dynamic mass-energy-vacuum exchange system rather than material inertly suspended in a vast expanse of nothingness. EGM mathematically represents matter as radiating a spectrum of conjugate EM frequencies. However, if we consider matter to radiate a spectrum of “Gravitons”, the EGM construct may be represented in quasi-physical form12 in equilibrium with its environment as a system, such that Gravitons emerge as a vehicle for the feedback of information between the EGM spectrum of matter and the QV spectrum of the local space-time manifold. EGM considers the spectral energy of a gravitational field to be equivalent to the massenergy of the object generating the field, expressible in terms of a PV spectrum analogous to spacetime curvature within GR. It models each of the conjugate EM frequencies as two populations of “conjugate Photon pairs”, i.e., each population is “180°” out of phase with its conjugate, consistent with a Fourier harmonics representation of a constant function in complex form. A conjugate Photon pair constitutes the definition of a Graviton within the EGM construct13. The density of Gravitons surrounding a mass-object is maximal nearby, gradually decreasing with radial distance; thus, the greater the population density of Gravitons, the stronger the gravitational field. These factors are consistent with the manner in which the PV spectrum is defined via Fourier harmonics, resulting in a spectrum which increases in mode number with radial distance from the mass-object14. The tendency of the space-time manifold is to expand; however, the presence of matter interrupts this movement, polarising the QV. Energy is required to alter its state to fewer modes of higher frequency, counteracting the thermodynamic tendency of any system to move towards a state 11

i.e. a quantised representation of the gravitational field. Science has yet to detect or rigorously define Gravitons; consequently, sufficient latitude exists to interpret the Graviton in a manner suitable to the EGM construct. 13 The mathematical summation of conjugate wavefunction pairs (typically represented as oscillating about “zero”) produces a constant function, analogous to the manifold stress tensor within GR. The summation of opposing sinusoids of equal amplitude does not result in the nonexistence of energy; otherwise the law of conservation of energy would be violated. That is, a mathematical “zero” point is an arbitrary assignment and should not be mistaken for the absence of manifold stress energy. 14 i.e. QV mode number decreases with “Graviton” density. 12

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of lowest energy and greatest stability. Subsequently, an observer held fixed within a QV gradient senses that the mode energy is asymmetrical15 and based upon the Quantum-Vacuum-InertiaHypothesis (QVIH), vacuum asymmetry results in an apparent acceleration force on the observer, perceived as gravity. Rather than a geometric curvature of nothingness, the manifestation of “g” is better represented as back-pressure from the vacuum as mass-energy exerts its influence upon it. EGM represents this process as the superposition of two spectra, resulting in a mathematical description of “g”, utilising Fourier harmonics16. Thus, it may be stated that the EGM construct yields a quantised description of gravity17 as articulated in [3]. 2

Cosmology18

2.1

Introduction

EGM represents a single paradigm which may be applied to precisely derive Cosmological measurements such as “H0” and19 “T0”. The EGM harmonic representation of fundamental particles serves to validate and substantiate the evolutionary epochs of our Universe, as science has come to understand them, since the time of the “Big-Bang”. EGM models mass-objects as being in equilibrium with the QV such that the energy state of matter describes the energy state of the vacuum. Hence, “H0” and “T0” represent observational evidence of Cosmological mass-energy equilibration. Invoking principles of similitude, “H0” is derived by relating the PV spectrum of a “PlanckParticle”20 to the present-day utilising the “Milky-Way” Galaxy as a basis for comparison. Within the EGM construct, a “Planck-Particle” denotes the condition of maximum permissible energy density, representing the Universe compacted to a point. As mass-energy density increases, the PV modal bandwidth compresses such that for a particle approaching the Planck Scale, the PV spectrum converges into a single mode approaching the Planck Frequency. Galaxies are homogeneously distributed throughout the Universe and are “approximately” in the same stage of evolution. Hence, it follows that we may utilise our own “Milky-Way” Galaxy as a universal reference to yield an average value of Cosmological gravitational intensity. Utilising astronomical estimates of total Galactic mass and radius, we may represent the “Milky-Way” as a “particle” at the centre of the galaxy, termed the “Galactic Reference Particle” (GRP). The radiant gravitational intensity of the GRP may be calculated from its PV spectral limit. The GRP is representative of the total mass-energy density and vacuum equilibrium state of the Universe at the present time; as viewed by instrumentation within our solar system. Thus, “H0” is derived by comparing the “Planck-Particle Universe” at the instant of creation to the GRP; facilitated by utilisation of the harmonic representation of fundamental particles. Relating the Cosmological expansion of the primordial “Planck-Particle Universe” to the GRP yields an expansive scaling factor “KT”. Subsequently, Wien’s displacement law is applied to determine a thermodynamic scaling factor “TW” quantifying the manner in which Photons radiated at the instant of the “Big-Bang” have red-shifted to the microwave range after Hubble time. The microwave frequency is converted to temperature by relating “KT” and “TW”, producing a value of “T0” precisely matching physical measurement. 15

i.e. higher in the direction of the centre of mass of an object and lower out in space. Enhanced by the PV representation of GR; developed by “Puthoff et. Al.”. 17 EGM derives the Casimir Force from first principles, demonstrating that it differs depending on the gravitational field strength of where it is measured. For example, EGM asserts that the strength of the Casimir Force on Jupiter will be smaller than on the surface of the Moon. (Ch. 3.7 in [5]). 18 Refer to Appendices when required. 19 It is demonstrated that “T0” may be derived from “H0”. 20 Representing the Universe at the instant of the “Big-Bang”. 16

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The resulting history of the CMBR temperature corroborates with all epochs of cosmic evolution as predicted by the SMoC. The theory of early “cosmic inflation” is reinforced and the recently measured “accelerated expansion” is derived21. Even though the cosmic inflation epoch is a contrivance introduced to fit a theory, EGM substantiates its existence because it emerges as a natural consequence of the derivation of “H0” and “T0”. However, the existence of dark energy / matter must be questioned due to the fact that the EGM method predicts “H0”, “T0” and Cosmological inflation / accelerated expansion, without invoking dark matter or energy; producing results substantially more precise than the SMoC22. The EGM construct produces “H0” and “T0” formulations of approximately “67.0843(km/s/Mpc)” and “2.7248(K)” respectively23. The derivation of “H0” and “T0” is possible assuming that, instantaneously prior to the “Big-Bang”, the “Primordial Universe” was analogous to a homogeneous Planck scale particle of maximum permissible energy density, characterised by a single EGM wavefunction. Simultaneously, the “Milky-Way” is represented as a Planck scale object of equivalent total Galactic mass “MG”, acting as a 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 yielding “H0” in terms of “Ro” and “MG”. Moreover, the analysis is extended by determining the theoretical frequency shift of a fictitious Graviton radiated from the Primordial particle, yielding “T0” in terms of “H0”. Consequently, by utilising the experimental value of “T0”, improved estimates for “Ro” and “MG” are derived 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 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(%) Gravitons”. • “< 1(%) Dark Matter / Energy”. • “4.6(%) Atoms”.

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The Cosmological inflation and accelerated expansion phenomena emerge naturally within the EGM construct and are not presumed “a priori” as part of the modelling process. The EGM construct generates the inflationary epoch from first principles, derived from Particle-Physics. 22 Within the EGM construct, the contribution of dark matter / energy to the Cosmological model is shown to be “< 1(%)”. 23 “T0” experimental tolerance is presently “2.725 ± 0.001(K)”. 6

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Derivation of the primordial and present Hubble constants “Hα, HU” [4]24

2.2 2.2.1

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. Generalised expressions derived 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. 2.2.2

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)” (Jansky’s). Matter radiates Gravitons25 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 distance26 to the Galactic centre and “M2” represents total Galactic mass27. 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 reference28 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)”.

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Ch. 7.1. Coherent populations of conjugate Photon pairs for a minimum period of “TL”. 26 i.e. the distance relative to the Galactic centre from where a physical measurement of “H0” is performed. 27 Visible + dark. 28 Our solar system. 25

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2.2.3

Construct

2.2.3.1 “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 the harmonic representation of fundamental particles articulated in [3], 2

M1

.

M2

r2

5

St ω

r1

9

(1)

and applying it Cosmologically yields, 2

Mi

.

Mf

rf ri

5

St ω

9

(2)

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

KU

ri

(3)

“Stω9”

in Eq. (1) represents the relationship between the values of harmonic cut-off frequency “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 the minimum gravitational lifetime of starving matter “TL” [4]29 to the present “Hubble age” of the Universe “AU” according to, St ω

TL

9

AU

(4)

Hence, TL

5

KU

AU

(5)

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 measurement30 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 that31: “e(rf / ri) → [CΩ_J1(r1,M1) / CΩ_J1(r2,M2)]” according to, rf

ln

ri

C Ω_J1 r 1 , M 1 C Ω_J1 r 2 , M 2

(6)

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Ch. 6.7. i.e. approximately “< 15” billion light-years. 31 “e” denotes the “exponential function”. 30

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It is demonstrated in [4]32 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. 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 construct33. 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

(7)

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

(8)

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

(9)

Hence, the Hubble 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

(10)

Consequently, the Hubble 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

(11)

Therefore, for a “flat” Universe; the Hubble constant may be written as, H U r 2, r 3, M 2, M 3

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

(12)

2.2.3.2 “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”. 32 33

Ch. 6.1.3. i.e. “rS(r1,M1) = RBH(r1,M1)”. 9

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

(13)

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

(14)

2.2.3.3 “ρU” Utilising the contemporary density relationship, the present Cosmological 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

(15)

2.2.3.4 “MU” Approximating the observable Universe to a spherical volume [i.e. “V(r) = 4πr3 / 3”], the total 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

(16)

Derivation of the CMBR temperature “TU” [4]34

2.3 2.3.1

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)” derived in [4]35, 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”. 2.3.2

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. 34 35

Ch. 7.2. Ch. 6.7.1. 10

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2.3.3

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

(17)

“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. If the present size of the Universe were held static and its total mass-energy (i.e. visible + dark) were compressed36 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

(18)

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

(19) 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”. 2.4 2.4.1

Numerical solutions for “Hα, AU, RU, ρU, MU, HU” and “TU” [4]37 “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, 36

Mimicking the “Primordial Universe” and excluding space-time manifold expansion from consideration. 37 Ch. 7.3. 11

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M2” and “M3”. If the “Primordial Universe” was analogous to a particle at the Planck scale limit, it is demonstrated in [4]38 that “nΩ_2(r3,M3) = 1” occurs when, r3 = λxλh

(20)

M3 = mxmh = λxmh / 2

(21)

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 in [5]39, 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, (22) r2(r) = Kλ⋅r M2(M) = Km⋅M = Kλ⋅M

(23)

where, “Kλ = Km = [π / 2](1 / 3) ≈ 1.162447”. 2.4.2

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. [7] iii. “M = MG” denotes the total mass (i.e. visible + dark) of the MW Galaxy. iv. “MG ≈ 6 x1011” solar masses as defined by [8]. v. “H0 = 71(km/s/Mpc)” as defined by the PDG. [9] vi. “T0 = 2.725(K)” as defined by the PDG. [7] H α λ x.λ h , m x.m h

ωh λx

(24)

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

(25)

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

(26)

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

(27)

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

(28)

km s .Mpc

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

39

3

cm

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

38

kg

33 .

(29) (30)

Ch. 6.1.3. Ch. 3.13. 12

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Determination of the impact of “Dark Matter / Energy” on “HU” and “TU” [4]40

2.5 2.5.1

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”41. 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. 2.5.2

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 [8]. 2.5.3

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

(31)

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

(32)

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

(33) 1 = 0.542607 ( % )

(34)

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 [10] 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(%) Gravitons”. v. “< 1(%) Dark Matter / Energy”. vi. “4.6(%) Atoms”. 40

Ch. 7.4. “Dark Matter” is associated with Galactic rotation; “Dark Energy” is associated with Cosmological expansion. However, since matter is equivalent to energy, we have phrased the expression “Dark Matter / Energy”. 41

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2.6

“TU” as a function of a generalised Hubble constant “TU → TU2” [4]42

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

(35)

If “r → (c / H)” and “M → M3” then, 9

c c λ x. λ Ω_3 , mh H 2

c.

5 9

H

1 . St G

λx 2

c. 2

.m h

1 . 2 St G λ x.m h

2

. c H

5

(36)

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

(37)

Hence, KW

T W ( H) λ Ω_3

c λ x. , mh H 2

(38)

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

(39)

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

(40)

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

(41)

Let, 9

. 8 . St G . λ x m h St T 5 2 3 .c c

2

(42)

Injecting “StG” and simplifying yields, 9

4. 3. 1 . λ x 3 4 c5 π .λ 2 h 3

St T

2

(43)

Therefore, T U2( H )

42

K W .St T .ln

ωh λ x.H

9

. H5

(44)

Ch. 7.5. 14

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2.7 2.7.1

Derivation of “Ro”, “MG”, “HU2” and “ρU2” from “TU2” [4]43 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. 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

(45)

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

(46)

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)

(47)

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)

(48)

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.

43

Ch. 7.6. 15

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2.7.2

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. 2.7.3

Construct

2.7.3.1 “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

T0

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

T0

∆T 0

(49) ∆T 0

(50)

r x1 r x2 m g1

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

m g2

(51)

Hence, r x1 r x2 m g1

0.989364 =

1.017883 1.057292 0.911791

m g2

(52)

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

(53)

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

r x1 r x2

16

=

7.914908 8.143063

( kpc )

(54)

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. 11 M G m g1 6.34375310 . = M S m g2 . 11 5.47074910

r x1 m g1

1=

r x2 m g2

(55)

1.063645 5.729219 1.788292

8.820858

( %)

(56)

Therefore, “TU2 = T0 ± ∆T0” is satisfied when: • “0.9894Ro < Ro < 1.0179Ro” or “0.9118MG < MG < 1.0573MG”. 2.7.3.2 “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

(57) ∆T 0

(58)

Let, r x3 r x4 m g3

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

m g4

(59)

Hence, r x3 r x4 m g3

0.984956 =

1.013348 0.977007 0.977007

m g4

(60)

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)

(61)

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

r x3 r x4

=

7.879647 8.106786

( kpc )

(62)

. 11 M G m g3 5.8620410 . = M S m g4 . 11 5.8620410

17

(63)

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r x3 m g3

1.50441 2.29934

1=

r x4 m g4

1.334822 2.29934

( %)

(64)

Therefore, “TU2 = T0 ± ∆T0” is satisfied when: • “0.9850Ro < Ro < 1.0133Ro” and “MG / MS = 5.8620 x1011”. 2.7.3.3 “Ro”, “MG”, “HU2” and “ρU2” 2.7.3.3.1 “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

(65)

Let, r x5 m g5

Find r x1, m g1

(66)

Hence, r x5 m g5

1.013403

=

1.052361

(67)

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 )

(68)

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

MG

(69)

. 11 = 6.31416710

MS

r x5

(70) 1=

m g5

1.340256 5.236123

( %)

(71)

Therefore, “TU2 = T0” is satisfied when: • “Ro = 8.1072(kpc)” and “MG / MS = 6.3142 x1011”. 2.7.3.3.2 “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

18

km . s Mpc

(72)

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Hence, 3 .H U2( r , M )

ρ U2( r , M )

2

8 .π .G

(73)

ρ U2 r x5.R o , m g5 .M G = 8.456036 10

33 .

kg 3

cm

(74)

Experimentally implicit derivation of the ZPF energy density threshold “UZPF” [4]44

2.8 2.8.1

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

(75)

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

33 .

kg 3

cm

(76)

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”. 2.8.2

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. 2.8.3

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

(77)

Evaluating produces, Ω EGM = 1.000331

(78)

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

(79)

where, “ρc” denotes critical Cosmological total density. 44

Ch. 7.7. 19

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The PDG state in [7] 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

(80)

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, Ω EGM Ω PDG

= 0.997339

(81)

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

(82) 4

(83)

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

13 .

Pa

(84) (85)

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. &ote: 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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2.9 2.9.1

Discussion Conceptualisation

2.9.1.1 “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 [4]45. 2.9.1.2 “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” [4]46. 2.9.1.3 “CΩ_J” The initial step in conceptualising 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 frequency47 “ωΩ” [4]48. Subsequently, all gravitational objects may be usefully represented by approximation as wavefunction radiators of a single population of conjugate Photon pairs [4]49. 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 spectrum50) 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.

45

App. 4.A. Ch. 6.7.2.2, 6.8. 47 i.e. the high-end terminal spectral frequency. 48 Ch. 5.4. 49 Ch. 9.2.2.2, 9.2.3.2. 50 The PV spectrum is a bandwidth of the EGM spectrum. 46

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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 unity51, it is demonstrated that “ωΩ → ωΩ_3” [4]52, consequently “CΩ_J” may be simplified to “CΩ_J1” [4]53 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. 2.9.1.4 “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 particles54, 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” [4]55.

51

The typical representation of “KPV” is an isomorphic weak field approximation to General Relativity (GR). 52 Ch. 5.1, 5.2. 53 Ch. 5.5.1. 54 One of them being an arbitrarily selected reference particle from which to compare all others. 55 Ch. 6.7.2.2. 22

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2.9.2

Dynamic, kinematic and geometric similarity

2.9.2.1 “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)”. Matter radiates Gravitons56 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 distance57 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 reference58 as the measurement of “H0” [4]59. 2.9.2.2 “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” [4]60. 2.10 Conclusion 1. The CBMR temperature is a function of the Hubble constant. 2. The present Hubble constant and average CMBR temperature is calculated to be: • HU2(Ro,MG) ≈ 67.084304(km/s/Mpc), TU2[HU2(Ro,MG)] ≈ 2.724752(K). 3. Improved estimates of “Milky-Way” Galactic radius and mass are derived to be: • Ro ≈ 8.1072(kpc), MG ≈ 6.3142 x1011(solar-masses).

56

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. 58 The solar system. 59 Ch. 7.1. 60 Ch. 7.2. 57

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4. The Universe is composed of: • “> 94.4(%) Gravitons”61, “< 1(%) Dark Matter / Energy”62, “4.6(%) Atoms”. 5. The magnitude of the impact of “Dark Matter / Energy” on the value of the Hubble constant and CMBR temperature is “< 1(%)”. 6. The EGM construct exhibits characteristics satisfying the observed phenomena of “accelerated Cosmological expansion” due to: • The ZPF energy density threshold value63 “UZPF < -2.52 x10-13(Pa)”. • The gradient of the Hubble constant in the time domain is presently positive64. 7. The Polarisable Vacuum model of gravity65, in concert with the Electro-Gravi-Magnetic construct, may be a superior formulation to General Relativity. Acknowledgements: many thanks to Geoffrey S. Diemer for assistance with the preparation of this manuscript.

61

Existing as populations of conjugate wavefunction pairs of Photons. Pertaining to observations which cannot be explained by the EGM construct. 63 i.e. “< -0.252(mJ/km3)”. 64 i.e. “dH/dt > 0”. 65 Introduced by Puthoff et. Al. 62

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3

Appendix A

Figure66 A1, 66

Available for download: http://www.lulu.com/content/2588584 25

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Figure67 A2,

67

Available for download: http://www.lulu.com/content/2486994 26

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Figure68 A3,

68

Available for download: http://www.lulu.com/content/2526284 27

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4

Appendix B

4.1

Derivation process (concise)

Chapter references: [4].

4.1.1

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

(86)

where, St J

9 .c . St G 4 .π

St G

3.

4

• • •

3 .ω h 4 .π .h

2 9

(87) 2

. c 2

9

(88)

“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

(89)

where, • “mγγ” denotes the mass-energy of a Photon defined in [5]. • “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

(7)

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

(9)

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

5

(10)

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

(12)

where, “nΩ_2” denotes the non-refractive form of “nΩ” defined in [5]. 28

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4.1.2

CMBR temperature “TU”

iv. Derive an expression for the average number of Gravitons “ng” radiated by a Schwarzschild-Black-Hole (SBH) at frequency “ω”: (see: Ch. 6.7.1.1), Output: E M BH

n g ω , M BH

E g( ω )

(90)

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

(14)

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

(17)

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

ω Ω_3( r , M )

2

M St G. 5 r

(91)

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

(18)

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

(11)

“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

29

(19)

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4.1.3

“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

6

(92)

4 . 2 π 3

(93)

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

(22)

M2(M) = Km⋅M = Kλ⋅M

(23)

where, “Kλ = Km = [π / 2](1 / 3) ≈ 1.162447” as defined in [5]. 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

(24)

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

30

(45)

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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

(44)

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

(43)

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

(94)

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. 4.1.4

“r = Ro” denotes the mean distance from the Sun to the MW Galactic centre. “Ro = 8(kpc)” as defined by the PDG (“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 [8]. “H0 = 71(km/s/Mpc)” as defined by the PDG (“Mpc” = Mega-parsec). “T0 = 2.725(K)” as defined by the PDG.

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

(95)

&ote: “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 domain69: (see: Ch. 8.3.3), Output:

69

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. 31

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d H dt

H

(96)

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 H γ .H α

t

Hγ Hβ

(97)

η

(98)

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

U ZPF

2

(84)

where, Ω ZPF Ω EGM

4.2

1

Ω EGM

(82)

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

(77)

Sample results

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

(99)

Find( η )

(100)

Hence, “η = 4.595349”.

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5 •

Appendix C Maximum permissible mass-density . 94 kg ρS = ρm(r3,M3) = ρ m λ x.λ h , m x.m h = 1.34467810 3

(101)

m

•

Schwarzschild-Black-Hole Singularity Radius (RBH ≥ “λxλh”) 3

r S R BH

•

r S M BH

(102)

K W .St T .

5 .µ

(104)

5 .µ

2

K W .St T .

5 .µ

2

K W .St T .

2

1

(106)

3 5 .µ

2

2 2 15.µ . 5 .µ

2

2

2

.t3

(107)

2nd time derivative of “H” (Hz3) dH2 dt2 H γ

Symbol Hβ Hγ t1

t5

1

.t2

2 2 2 5 .µ .ln H α .t . 5 .µ . 5 .µ

t

t4

(105)

3rd time derivative of “TU4” (K/s3) dT3 dt3 ( t )

t3

.t

2 2 5 .µ . ln H α .t . 5 .µ

t

t2

1

2nd time derivative of “TU4” (K/s2) dT2 dt2 ( t )

•

2

2 5 .ln H α .t .µ

t

•

(103)

1st time derivative of “TU4” (K/s) dT dt ( t )

•

4 .π .ρ S

“TU → TU4” 1 T U4( t ) K W .St T .ln H α .t . t

•

3 .M BH

3

2 λ x.λ h .R BH

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

1

2

1

Description Dimensionless range variable: {1, (1 - 10-100) …10-100} Dimensionless range variable: {1, (1 - 10-100) …10-100} • 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 Table C.1,

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(108) Units

s

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6

Bibliography

[1] Derivation of the photon mass-energy threshold; Riccardo C. Storti and Todd J. Desiato, Proc. SPIE 5866, 207 (2005), DOI:10.1117/12.614634. [2] Derivation of the photon and graviton mass-energies and radii; Riccardo C. Storti and Todd J. Desiato, Proc. SPIE 5866, 214 (2005), DOI:10.1117/12.633511. [3]70 The natural philosophy of fundamental particles; Riccardo C. Storti, Proc. SPIE 6664, 66640J (2007), DOI:10.1117/12.725545. [4]71 Quinta Essentia: A Practical Guide to Space-Time Engineering – Part 4; Riccardo C. Storti, ISBN-13: 978-1847533548, LuLu Press. [5]72 Quinta Essentia: A Practical Guide to Space-Time Engineering – Part 3; Riccardo C. Storti, ISBN-13: 978-1847539427, LuLu Press. [6]73 Quinta Essentia: A Practical Guide to Space-Time Engineering – Part 2; Riccardo C. Storti, & G. S. Diemer, ISBN-13: 978-1847993618, LuLu Press. [7] http://pdg.lbl.gov/2006/reviews/astrorpp.pdf [8] http://zebu.uoregon.edu/~imamura/123/lecture-2/mass.html [9] http://pdg.lbl.gov/2006/reviews/hubblerpp.pdf (pg. 20 - “WMAP + All”). [10] http://map.gsfc.nasa.gov/m_mm.html Suggested material [11]74 Quinta Essentia: A Practical Guide to Space-Time Engineering – Part 1; G. S. Diemer, ISBN13: 978-1847993601, LuLu Press. [12] http://www.veoh.com/users/DeltaGroupEngineering

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Available for download as Ch. 4 in [6]. Available for download: http://www.lulu.com/content/795547 72 Available for download: http://www.lulu.com/content/471178 73 Available for download: http://www.lulu.com/content/1540406 74 Available for download: http://www.lulu.com/content/2671468 71

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