Mass To Luminosity Ratios Of Galaxy Groups

Mass to Luminosity ratios of Galaxy Groups Timothy Green, Queens' college. October 13, 2008

Abstract

G The diering results for the minima in M LG vs Luminosity found in the two papers G. Mountrichas 2007

[7] and V. Eke 2004 [4] are investigated by means of examining the dark matter environment of the groups in the simulation data used to generate the 2PIGG mock catalogues. Although the V. Eke mock mass estimation results broadly agree with the discovered group masses, the results still disagree with the halo simulation, raising questions about the degree to which, for the purposes of comparison, the haloes found in the simulation G accurately represent the groups formed. Questions are also raised about the proportionality of to M LG .

1 The problem

As the simulation box was smaller than the observed distance of the 2dFGRS survey (point 3.(ii) in V. Eke 2004 [3]), the simulation box was repeated in a cylic fashion a number of times. The observer then viewed the original galaxies out along the direction of observation, applying appropriate cuts and distortions for viewing to produce mock catalogues for 2dF North Galactic Plane (NGP) and South Galactic Plane (SGP). This was done for both z = 0:127 and z = 0. As well as typical observations, the catalogues include the original galaxy that the observed galaxy derives from.

The two papers investigated, V. Eke et al (2007) [4] and G. Mountrichas & Shanks (2007) [7], both attempt to predict the mass to light ratio of various classes of galaxy groups. V. Eke discoveres a minimum in the MLGG vs L plot, and G. Mountrichas discovers a similar minimum in vs L. The minimum found in V. Eke is at LG  1010 M , however the minimum found in G. Mountrichas is found at LG  1011 M . The essential problem investigated is how this dierences comes about, and whether the two papers are actually discussing the same quantity. 1.1

The 2PIGG (2dFGRS Percolation-Inferred Galaxy Group) group then found groups within the real and mock catalogues as described in V. Eke [3]. A cylindrical linking volume with the major axis aligned along the line of sight was used in spherical redshift space was used to link together nearby galaxies into groups. A cylindrical linking volume rather than a spherical one was used to account for redshift distortion from the perculiar velocity spread of galaxies within groups.

The simulation

As part of the two papers investigated, a number of mock catalogues were made. These catalogues were processed to match the characteristics of the 2dF Galaxy Redshift Survey (2dFGRS) survey. The simulation used for producing mock 2dFGRS catalogues was performed using CDM cosmology with

m = 0:3 and  = 0:7 within a 141.3 MPch 1 cyclic boundary condition box [6]. Spawning 224 particles, each with mass of 1:4  1010 M , the simulation was advanced to a redshift of z = 0:127 (approx 1:6  109 years ago) and z = 0. Haloes of dark matter were then found by using a friends of friends algorithm to group together dark matter particles [3].

1.2

Masses in V. Eke (2007)

The paper attempts to calculate the mass to luminosity ratios of galaxy groups found by the 2PIGG team from both the real observations and mock simulations. Galaxy group masses were estimated by the virial theorem [5]:

Galaxies were then 'painted' on top of the haloes using a semi-analytic method described in Cole 2000 [2] to produce the original galaxies. An observer was then placed into the simulation box at a random translation and rotation by generating three random orthogonal basis vectors and random translations in each direction.

2T +  = 0 Where T is the mean total energy of the system, and

 is the number of bodies in the system times the mean total potential energy of the system. By calcu1

1.4

2

Questions

lating the velocity dispersion () of the galaxies within the group - estimated from the spread of deviations of redshift from the central galaxy of the group, and the root mean squared spatial distribution of galaxies (eective 'radius' scale of the group), we can calculate the mass M of the group ([4] eqn 2.1): M

mass in the universe, L the auto-correlation function of luminosity to luminosity in the universe. It is assumed that these are simply proportional to each other by the factor b (the linear bias) on all scales. m represents the mean mass density, l represents the mean density of luminosity, and g represents the mean density of galaxies. We can show that is related to the mass to light ratio of a group.

2

= AG r

MG LG

A value of A = 5:0 is given, with a suggested error on the order of 'a few tens of percent', the most signi cant error in the virial method. The method for calculation luminosity for each group is as described in section 2.1 of V. Eke 2004 [4]. Given this data, the paper nds that there is a possible minimum in the MLGG versus group membership at NG  20 and in group luminosity at LG  1010 M . This indicates a 'peak' eciency of converting mass to luminosity. The minimum more or less matches for both real and mock observations. Model results were also generated, using the haloes used to generate the galaxies to estimate the group masses. 1.3

  MG LG

1.4

G

m g



(3)



bG m g g m G m

0:6 g

(7)

(8) (9)

Questions

2 Methods 2.1

(2)

Mass estimation

First, we can consider what the appropriate integration radius is for a galaxy group. Although the G. Mountrichas method doesn't take galaxy radius into account, the V. Eke virial method considers mass internal to a group. To get an idea of group size, we can look at gure 1, showing the root mean squared distance of member galaxies to the centre of their group (r ) and the distance to the nearest galaxy group.

(4)

Where M is the auto-correlation function of mass to Tim Green

(6)

A number of questions arise from the two papers. Firstly, do the assumptions that MLGG / hold in the Mountrichas paper? Does the virial mass re ect the same mass that the in fall gravitational eect might, i.e. could the infall method be measuring mass on a larger scale than the virial method? Also, is the halo model an appropriate point of comparison?

The redshift space correlation function is distorted by the velocity dispersion of galaxies within the groups and by infall of other galaxies outside the group towards the group. By tting a model for these distortions, we can attempt to estimate rather than MLGG , which is de ned as, for MG and LG being the group mass and group luminosity, and Mg and Lg being the galaxy luminosity. (1)

 0G:6 m L G M g  0:6 L

(5)

This demonstrates that MLGG / , and so that the minima of the two papers are comparable. Taking the assumptions, then we nd that the minimum in the G. Mountrichas MLGG vs LG plot is approximately an order of magnitude higher in luminosity than in V. Eke. If any of the assumptions fail, there might be another eect that is shifting the minimum in .

This paper attempts to calculate the group mass to luminosity ratios by a statistical test. It uses only the centres and luminosities of the 2PIGG groups. The cross-correlation function is then calculated for group centres to galaxies for galaxy bins by membership and luminosity. The correlation function in this case is effectively a measure of the excess probability of nding a galaxy at a certain displacement from a group centre compared to a uniform distribution of galaxies.

= gg mm 0:6



m L m g g L G m g

0:6 g L

g

Masses in G. Mountrichas (2007)

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Figure 1: On the left the distance in MPc h 1 to the nearest other galaxy group versus the number of members in the group, and on the right the root mean squared radius of the galaxy group versus the membership summed over a particular set of centres; while for another it was tried to a constant multiple of r to capture the characteristic radius of a galaxy group. You can observe the cross-correlation functions shown in gure 2 the sort of scale that r is over. Performing the counting requires that we are able to take the 2PIGG mock groups and transform them back into the coordinate system used for the dark matter particles. From now on, the mass estimated by this method is termed the 'integrated mass', while the mass estimated by the code in V. Eke 2004 is termed the 'virial' mass.

As the dark matter data is only available for the simulations, we will only be able to compare the results of the mock catalogues. Since the two papers show large internal agreement between mock and real observations, this shouldn't be an issue. At the time of writing, the only appropriate dark matter data available was for the z = 0:127 simulation, not the z = 0 simulation used in the two papers - though the dierence between the two should hopefully just be small statistical variation. Using this dark matter particle data, we can then just count the number of dark matter particles around each group centre out to a prede ned radius minus the expected number of mass particles assuming uniform uncorrelated distribution. This is eectively integrating the correlation function multiplied by density: (r ) 1 = GM GR(r) GR(r) (r) = GM (r) GR(r)  (r)

2.2

The simulation data available included the dark matter particle positions; the dark matter haloes; the galaxies seeded on top of the dark matter, in simulation coordinates; and the galaxies observed in observation coordinates to match 2dFGRS, with a record of which original galaxy the observed galaxy is from.

(10) (11) (12)

Unfortunately the observer position and rotation were not kept from an intermediate stage in the production of the catalogue. In order to discover the dark matter environment around each galaxy group, this will need to be determined.

Where GM (r) is the actual number of mass particles at radius r from a galaxy group centre, and GR(r) = 4r2 m is the expected number of random particles at radius r. As GR(r) (r) is the excess mass at a particular radius r, we can simply integrate this to get the full mass excess of a galaxy group: MG

=

MG

=

MG

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Z

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GR() ()d

(13)

 ()42 m d

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GM ()d

4 r3  m 3

Since each galaxy had a record of which original galaxy they belonged to, it is possible to produce a set of child galaxies for each original galaxy. Given the cyclic coordinate system of the observations, it is expected that each child galaxies for a particular original galaxy will be separated from one another by an integral vector with orthogonal bases along the axes of the dark matter cube. This artifact will be preserved after the rotation and transformations are applied. Thus, plotting the separation of each child galaxy from the other child galaxy in its original galaxy set we nd a pattern of basis vectors - indicating the orientation of the simulation basis with respect to the observation

(15)

For one set of results this was calculated to a xed radius r, as the correlation function is considered Tim Green

Recovering transformations

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4 basis. This can be roughly measured o the plot to give a rough rotation matrix.

tion with dierent distances scales and methods for a given amount of computing time.

This rough matrix, however, could have its axes interchanged or inverted with respect to the true matrix used. Discovering this relation was a case of generating the galaxy catalogue in wrapped simulation coordinates (i.e. all points are roughly rotated into place and then wrapped to 0:::141:3 MPc h 1 plus an oset to the cube's unwrapped position. The axes can then be swizzled and inverted as needed until a consistant match between the catalogue galaxies and their original galaxy is found.

Fully balanced KD-trees were generated for the dark matter particles, galaxies and galaxy groups in order to facilitate any needed calculations. In order to take the cyclic coordinate system of the simulation into account, the spherical counting volume was tested at 26 locations:

0 p~0 = @

The transpose of this rough rotation matrix can then be compared to the matrices generated by the original catalogue generating code to discover the original seed used, and thus the precise rotation and translation. A seed range of s = 0 ! 109 can then be scanned for matching rotations, along with their translations.

1 A

If the sphere centred at these locations would also intersect the simulation cube (a rapid box-sphere intersection test will determine this [1]), it will be tested in addition to the default position p~. So taking each galaxy group centres in simulation coodinates, a number of wrapped points were generated, and then from each one the number of dark matter particles within the spherical volume were counted. Taking this count the mean background count was subtracted to give the corrected integrated mass. This mass was then binned by the group's luminosity and the group's membership, in addition to binning the number of galaxies within rg of the group centre in catalogue space and the luminosity as calculated in V. Eke 2004.

Taking the group catalogue in observation spherical redshift coordinates and transforming into cosmological cartesian coordinates, we can then apply the matrix and translation to give the catalogue in simulation coordinates. Perculiar velocity redshift distortion effects wont be present in the group positions as the group centres will average out between the galaxies that are redshifted and those that are blueshifted. If galaxies are transformed similarly, it may be necessary to correct for perculiar velocity distortions using the recorded velocities of the original galaxies. It is also worth noting that as a result of original galaxies spawning multiple child galaxies the statistical signi cance of a given group in its environment may be reduced - as it is might be a copy of another galaxy group, but with a slightly dierent number of galaxies due to observation limits and random selection when generating the catalogue. 2.3

p~x [141:3] p~y [141:3] p~z [141:3]

3 Results In recovering the transformation, after a fair amount of processing, a single accurate rotation was discovered. To three signi cant gures, the rotation matrix and associated translation:

KD-trees

0 0:979 R = @ 0:120

In order to count the number of points within a certain distance of a galaxy group centre, for calculating the correlation function or the integrated correlation function, you would naively have to compare every galaxy group centre (n  104 points) to every dark matter particle (m = 224 points) to nd if it is within the appropriate radius. This would result in a number of total comparisons on the order of O(nm) / 1:7  1011 . Using a KD-tree this time can be signi cantly reduced. KD-trees are a form of data structure that allows rapid spatial lookups of points, allowing you to discover the nearest neighbouring point or number of points within a certain radius in O(log m) time. This gives the total number of comparison required as O(n log m) / 2:4  105 a signi cant reducation, allowing more experimentaTim Green

0:163

0:172 6:66  10 0:983

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0 1 102:0 ~t = @ 68:8 A 25:8

Taking this transformation and generating the appropriate catalogues, the data could then be processed to examine the nature of the group masses. A number of plots of the data generated using the above methods are shown over the next few pages. 4

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Figure 2: The correlation functions using group centres, against dark matter particles, galaxy centres, and other group centres respectively

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Mean integrated mass for varying group radius 1e+15 r = 0.1 MPc/h r = 0.5 MPc/h r = 1.0 MPc/h r = 2.0 MPc/h r = 5.0 MPc/h r = 10.0 MPc/h

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Figure 3: Mean of membership bins of excess mass in groups out to a xed radius, rg = k (upper), and to a multiple of r , rg = kr (lower)

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Figure 4: Mean of luminosity bins of excess mass in groups out to a xed radius (upper), rg = k, and to a multiple of r , rg = kr (lower)

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Mass vs. Luminosity for r_g = 4 sigma_r 1e+16 Groups Power law fit f2(x)

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Figure 5: Scatterplots of mass to luminosity for mass estimted using four times the root-mean squared radius as the group radius, a xed integration out to 10 MPch 1 and the virially estimated mass respectively Tim Green

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Figure 6: Scatter plots of virial estimated masses versus integrated estimated masses. Upper plot shows density of points with contours, lower plot shows a bivariate log normal distribution density tted - both showing bivariate major axis

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Figure 7: Residuals from gure 6 1e+019 Groups x x**2 / 1.08056e+016 1e+018

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Figure 8: Plot of proposed corrected integrated mass versus virially estimated mass. Lines show a line of direct proportionality, and a x2 tted line corresponding to a line of direct proportionality without the transformation Tim Green

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Mass/Lum vs. Lum for r_g = 4 sigma_r (transformed masss) 1e+007 Groups

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Figure 9: Plot of the mass to light ratios as found using the mass particle counting method out to rg = 4r with the transformation applied to the mass given the correlation to the virial mass. Shown is the scatter of the individual groups, and the mean mass to light ratio of each luminosity bin

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Figure 10: Plot of the mass to light ratios as found using the mass particle counting method out to rg = 4r and using the virial method. Shown is the scatter of the individual groups, and the mean mass to luminosity ratio of each luminosity bin. Bars show the population variance

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Figure 11: Plot of galaxies found within the counting volume versus the 2PIGG estimated group luminosities

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Comparison of the three estimates of M/L vs. Luminosity

M/L ratio (scaled units)

V. Eke model V. Eke mock Integrated 4rms mean G. Mountrichas mock

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Figure 12: A comparison of the MLGG vs. Luminosity curves estimated in V. Eke (in which the model is generated from the simulation haloes, and the mock is from the mock 2PIGG catalogues), G. Mountrichas and this report. The vertical axis is arbitrary for the purposes of comparing the values.

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4 Discussion

data is actually better described untransformed, possibly indicating that the bivariate log normal model of deviations was insucient to describe the relationship of the virial mass and the integrated estimation. Using the tranformed masses in plotting MLGG vs luminosity in gure 9 gives a strong downwards trend completely unlike the ratios in 10, bearing no similarity to the virial mass to light ratio estimation. Further use of the data in calculating ratios is with the untransformed masses.

In gure 2 you can see the cross-correlation  (r) of the group centres and dark matter particles, group centres and galaxies, and the autocorrelation of group centres, split by group membership. You can see how the larger a group, the greater excess there is, and how they all have a roughly similar shape. Figure 3 shows the mean (stacked) excess mass for galaxy groups binned by membership over a number of scales - proportional to r and to a xed radius. The excess mass found is largely proportional to the cube of the radius. Going out to a multiple of r gives roughly a 30 times increase in mass over the membership bins. Going to a xed radius gives roughly a magnitude increase in mass over the membership bins. Similarly, gure 4 shows the mean excess mass for galaxy groups binned by luminosity over a number of scales. Again, the scales show a roughly similar trend to a factor of proportion. There is also a larger change in mean mass from low luminosity to high luminosity groups - roughly a 300 times increase in mass for scales of multiples of r but only a magnitude change for xed radius. This indicates that the estimated mass is largely proportional over small to large scales.

In order to test the assumed proportionality in the derivation of MLGG / , in gure 11 the group luminosity as calculated from the 2PIGG groups has been plotted against the number of galaxies found in the same counting volume as the dark matter particles. The plot for the count of galaxies within rg = 4r of the group centre shows a rough upwards trend though with large variation. However, in the plot for rg = 10MPc h 1 there is a strong downwards trend, indicating an inverse proportionality between the 2PIGG luminosity and the integral of Gg to a xed radius. This might be explained as bright groups tending to be more tightly packed with other galaxy groups, thus the 2PIGG group luminosity is only counting luminosity from the member galaxies while Gg is looking at the entire environment of the supercluster - so this could indicate an issue with the assumptions of MLGG / , in that the galaxy-group cross correlation function doesn't necessarily translate into the description of the luminosity of a single 2PIGG group. This will need closer examination to con rm as it may be a product of the processing.

Figure 5 shows the mass to luminosity scatterplots of all groups using the two methods, and the two ways of setting the group radius. The xed radius method eectively demonstrates no variation in mass for varying luminosity, while the root mean squared estimation of radius appears to be correlated similarly to virial mass.

A principal question is whether the predicted masses using the virial mock estimate and halo model and integrated estimated give similar estimations of the MG LG ratio or if they show some sort of dierence. If they are similar, the virial mass is consistent with the dark matter environment surrounding and so the simulation haloes are a reasonable model to use for generating the galaxies. If the integrated mass diers from the halo mass model but agrees with the virial estimates, this may indicate a de ciency in the use of haloes. Figure 10 shows the ratios plotted for integrated masses (upper) and virial masses (lower), the vertical axes are a scale factor of about 100 out. Both seem to be consistent with a minimum in MLGG at about LG = 2:5  1010 L , similar to what was reported in V. Eke 2004. The main cause of this minimum appears to be the presence of a population of low mass groups with median luminosity.

In gure 6 the masses for r = 4r have been plotted against the virially estimated masses. A clear bivariate log normal distribution correlation can be observed. The covariance matrix  is

  07 0:397  = 01::397 0:738 The principal axis of the distribution as found by nding the eigenvectors of  is logMvirial = 0:492logMinteg + 9:13 p Mvirial  109:1  M integ

Comparing all the mass to luminosity versus luminosity plots in gure 12, it is possible to see the variation in predictions. The model in V. Eke was generated using the haloes found in the simulation which were then used to generate the galaxies. The virially estimated masses from the V. Eke mock data seems to give a minimum consistent with the masses predicted in this

The overlaid density plots show the measured density and expected density from the bivariate log normal model. Figure 7 shows the residual of the the density minus the expected density. However, gure 8 shows the integrated mass using the correction formula, along with a line of direct proportionality and a ax2 line best t. This seems to indicate that the Tim Green

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6 Acknowledgements

paper, though the shape appears to be signi cantly dierent with the lack of a similar half-magnitude rise in MLGG as seen in the V. Eke mock results - it seems that the integrated method underestimates the mass for higher luminosity groups compared to the virial method, though the plot calculated in this paper for the virial masses seems to have a lack of a similar rise - possibly due to dierent collation or selection. The halo model seems to deviate signi cantly from the integrated masses and the virial estimate, suggesting that it might not be a statistical abberation.

I would like to thank my project supervisor Professor Tom Shanks, along with Vince Eke, Georgios Mountrichas, Carlton Baugh, the 2PIGG team and everyone involved with the 2dFGRS survey.

7 Appendix Code is available on request or at http://people.pwf.cam.ac.uk/tfgg2/code/ as there is too much to include here. Source code includes code for KD-trees (shared/kdtree.h), counting and binning in C++ (mass lum/mass lum.cpp) and the brute forcing of the random number generator seeds to recover the transformation (random test.f90).

5 Conclusions

References

The correlation in gure 6 indicates that although there are statistical variations, the two methods of integrating the mass particles and virial estimation are partially in agreement. The large spreads might be explained by uncorrelated errors in both of the methods - such as shot noise for counting mass particles, and errors in the calculation of the virial mass - particually the quoted error in A. However, the integrated masses actually seem to correspond better in the mass to light ratios when not transformed - indicating that the distribution is more complex.

[1] J Arvo. [2]

[3]

The plots of mass to luminosity versus luminosity seem to indicate that given the data the integrated masses are largely consistant with the virial mock estimates given in V. Eke 2004. However, the integrated mass estimates using the simulation data do not agree with the estimates using the haloes from the same simulations, though they are closer than the estimates using . This suggests that the 2PIGG groups may not have much to do with the haloes. In addition, the accuracy of using to represent MLGG is questionable, especially for groups with few galaxies around them, giving a large spread as seen in gure 11.

[4]

Further investigation could be done in examining the relationship of the 2PIGG groups to the haloes used to generate the member galaxies. The mass values could also be compared to values found using gravitational lensing, and to further analysis of other surveys and associated mock catalogues. During this investigation a mistake was also found in the fortran code used in G. Mountrichas where an integral had been calculated incorrectly. The reanalysis of the data hasn't been completed, but preliminary results indicate that it doesn't aect the fundamental conclusion of the paper. Tim Green

A simple method for box-sphere inter-

, pages 335{339. Aacdemic Press, Boston, MA, 1990. Shaun Cole, Cedric Lacey, Carlton Baugh, and Carlos Frenk. Hierarchical galaxy formation. astro-ph/0007281, July 2000. Mon.Not.Roy.Astron.Soc. 319 (2000) 168. V. R Eke, C. M Baugh, S. Cole, C. S Frenk, P. Norberg, J. A Peacock, I. K Baldry, J. BlandHawthorn, T. Bridges, R. Cannon, M. Colless, C. Collins, W. Couch, G. Dalton, R. De Propris, S. P Driver, G. Efstathiou, R. S Ellis, K. Glazebrook, C. Jackson, O. Lahav, I. Lewis, S. Lumsden, S. Maddox, D. Madgwick, B. A Peterson, W. Sutherland, and K. Taylor. Galaxy groups in the 2dfgrs: the group- nding algorithm and the 2pigg catalogue. astro-ph/0402567, February 2004. Mon.Not.Roy.Astron.Soc. 348 (2004) 866. V. R Eke, C. S Frenk, C. M Baugh, S. Cole, P. Norberg, J. A Peacock, I. K Baldry, J. BlandHawthorn, T. Bridges, R. Cannon, M. Colless, C. Collins, W. Couch, G. Dalton, R. De Propris, S. P Driver, G. Efstathiou, R. S Ellis, K. Glazebrook, C. Jackson, O. Lahav, I. Lewis, S. Lumsden, S. Maddox, D. Madgwick, B. A Peterson, W. Sutherland, and K. Taylor. Galaxy groups in the 2dfgrs: the luminous content of the groups. astro-ph/0402566, February 2004. Mon.Not.Roy.Astron.Soc. 355 (2004) 769. II George W. Collins. The Virial Theorem in Stellar Astrophysics. Pachart Press, 1978. A Jenkins, C. S Frenk, F. R Pearce, P. A Thomas, J. M Colberg, S. D. M White, H. M. P Couchman, section testing

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References

J. A Peacock, G. Efstathiou, and A. H Nelson. Evolution of structure in cold dark matter universes. astro-ph/9709010, September 1997. Astrophys.J. 499 (1998) 20. [7] Georgios Mountrichas and Tom Shanks. Clustering of 2pigg galaxy groups with 2dfgrs galaxies. 0712.3255, December 2007.

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