Download Cosmological Constraints from the Virial Mass

Cosmological Constraints from the Virial Mass
Function of Nearby Galaxy Groups and Clusters
by
James Colin Hill
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Bachelor of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2008
@ James Colin Hill, MMVIII. All rights reserved.
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Author.
Department of Physics
May 16, 2008
Certified by.
.
(I
Professor Claude R. Canizares
Department of Physics
Thesis Supervisor
Certified by..
Dr. Kenneth J. Rines
Harvard-Smithsonian Center for Astrophysics
Thesis Supervisor
Accepted by ........
MASSACHUSETTS INSTiTUTE
OF TECHNOLOGY
JUN 132008
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Professor David E. Pritchard
Senior Thesis Coordinator, Department of Physics
Cosmological Constraints from the Virial Mass Function of
Nearby Galaxy Groups and Clusters
by
James Colin Hill
Submitted to the Department of Physics
on May 16, 2008, in partial fulfillment of the
requirements for the degree of
Bachelor of Science
Abstract
In this thesis, I present a new determination of the cluster mass function in a volume
107 h -3 Mpc3 using the ROSAT-2MASS-FAST Group Survey (R2FGS). R2FGS is
an X-ray-selected sample of systems from the ROSAT All-Sky Survey in the region
6 > 0' and 0.01 < z < 0.06, with target galaxies for each system compiled from
the Two Micron All Sky Survey (2MASS). The sample is designed to focus on lowmass groups and clusters, so as to break a degeneracy between the cosmological
parameters m, and os. In addition, R2FGS covers a very large area of sky (, 4.13
ster.), which is necessary given the low redshift limit of the survey. I acquire optical
redshifts for the target galaxies in R2FGS from the literature and from new data
collected with the FAST spectrograph on Mt. Hopkins. After removing foreground
and background galaxies (interlopers) using a dynamical maximum-velocity criterion,
I estimate the group and cluster masses using the full virial theorem, and subsequently
verify the results using the projected mass estimator. I briefly investigate the up - Lx
and M - Lx scaling relations, as well as the halo occupation function. Due to
interloper issues with some of the systems, I apply a luminosity-dependent correction
to the virial masses, and subsequently use these masses to compute the virial mass
function of the sample. By comparing this mass function to predictions from various
cosmological models, I constrain the parameters •,m and 8s. I find Qm = 0.26 +0.07
and U8 = 1.02 +3; the R2FGS value for Qm agrees very well with the recent FiveYear WMAP (WMAP5) result, although the R2FGS value for as is somewhat larger
than that found by WMAP5. Future work will include an expansion of the survey to
6> -200 and z < 0.04, which will greatly increase its degeneracy-breaking power.
Thesis Supervisor: Claude R. Canizares
Title: Professor of Physics
Thesis Supervisor: Dr. Kenneth J. Rines
Title: Astronomer, Harvard-Smithsonian Center for Astrophysics
Acknowledgments
First, I would like to thank Ken Rines for his extensive help and advice over the past
year, as well as for giving me the opportunity to work on such an interesting project.
I would also like to thank Prof. Claude Canizares for his helpful advice and support
as my co-supervisor. In addition, I am grateful to Prof. Max Tegmark and Molly
Swanson for their guidance and collaboration with me on my first research project
in cosmology, and to Prof. Scott Hughes for introducing me to astrophysics research
three years ago. Without the support of so many great advisors, I would never have
successfully completed such a breadth of research during my undergraduate years.
Second, I am eternally grateful to my friends, without whom I surely would not
have survived MIT; I have learned more about physics, math, and tomfoolery from
them than any class could ever teach. Most important, I owe great thanks to my
family, particularly my parents, who have always encouraged and supported me in
everything I do, especially in the form of banana bread and coffee. I could not have
made it this far without them.
Contents
1 Introduction
13
8)
..
........................
Degeneracy .
1.1
The (Qm,
1.2
The Cluster Mass Function .
......................
1.2.1
Mass Estimation Techniques .
1.2.2
Recent X-ray Studies .
1.2.3
The CIRS Mass Function
1.2.4
The R2FGS Mass Function
.................
......................
.
.
15
.
15
..
..................
.
14
.................
16
.
17
.
18
2 The R2FGS Sample
21
21
....................
2.1
X-ray Cluster Surveys
......
2.2
Group and Cluster Selection .
.....................
.
25
3 The R2FGS Observations
3.1
Target Galaxy Selection
3.2
FAST Data ........
.
.
.......................
.........................
......
5
27
4.1
Interloper Removal ..
4.2
Virial Masses
4.3
Projected Masses ..
4.4
The Reduced R2FGS Sample
..........................
.............................
............................
.
.
27
.
30
..
31
.....................
32
35
Cluster Scaling Relations
5.1
25
26
4 Estimating Cluster Masses
..
22
The Velocity Dispersion-Luminosity Relation . . . . .
7
. . .....
.
.
35
6
5.2
The Mass-Luminosity Relation . . . . . . . . . . . . . . . . . . . . .
36
5.3
The Velocity Dispersion-Mass Relation . . . . . . . . . . . . . . . .
38
5.4
The Halo Occupation Function
40
...
. . . . . . . .
. . . . . ....
.
The R2FGS Mass Function
45
6.1
Uniform Correction ..................
6.2
Luminosity-Dependent Correction . . . . . . . . . . . . . . . . . . . .
47
6.3
Comparison to Previous Mass Functions . . . . . . . . . . . . . . . .
49
..........
47
7 Cosmological Constraints
55
8
59
Conclusions
A Redshift-Radius Plots
61
B Tables
69
List of Figures
2-1
z versus Lx .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
4-1
M200
oo versus MpM . .............................
33
5-1
o 200 versus Lx.
5-2
M2oo
00 versus Lx ...............................
5-3
00versus
2oo
. ..
M2oo.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
37
39
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
5-4
oo versus M200
oo .
N 200
6-1
Vmx versus Lx...............................
.
46
6-2
R2FGS mass function: uniform correction . . . . . . . . . . . . . . .
48
6-3
R2FGS mass function: luminosity-dependent correction . . . . . . .
50
7-1
Constraints on Qm and
58
8s . . . . . . . . . . . . . . . . . . . . . . . . .
A-1 Acz versus Rp for the first ten R2FGS systems (ordered by increasing
cluster redshift).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
A-2 Acz versus Rp for the next ten R2FGS systems . . . . . . . . . . . .
63
A-3 Acz versus Rp for the next ten R2FGS systems . . . . . . . . . . . .
64
A-4 Acz versus Rp for the next eight R2FGS systems . . . . . . . . . . .
65
A-5 Acz versus Rp for the next eight R2FGS systems . . . . . . . . . . .
66
A-6 Acz versus Rp for the next eight R2FGS systems . . . . . . . . . . .
67
A-7 Acz versus Rp for the final eight R2FGS systems . . . . . . . . . . .
68
List of Tables
B.1 R2FGS Groups and Clusters .
B.2 R2FGS Systems: M 20 0 , MpM, MLx, r200 ,
......................
2 00
. .
. . . . . . . . . . . . .
.
69
Chapter 1
Introduction
The observed abundance of massive systems as a function of mass, that is, the mass
function, is a basic prediction of any viable cosmological model. The hierarchical
theory of cosmic structure formation asserts that groups and clusters of galaxies arise
from rare high peaks of the initial density fluctuation field, and currently represent
the most massive virialized systems in the universe. These systems are a powerful and
fairly clean tool for cosmology because their growth is primarily governed by linear
gravitational processes, as first described analytically by Press and Schechter [1];
their theory implies that the abundances of groups and clusters depend strongly on
the amplitude of the density fluctuations on the cluster mass scale. In particular,
these abundances are highly sensitive to Qm, the matter density of the universe,
and os, which is both the rms amplitude of the density fluctuations on an 8 h Mpc scale and the normalization of the linear power spectrum [2, 3].
1
Moreover,
the evolution of the mass function is a measure of the growth of structure, and can
therefore be used to constrain the (possibly redshift-dependent) properties of dark
energy [4, 5, 6, 7, 8, 9, 10]. In order to obtain robust constraints on this evolution,
though, an accurate measurement of the mass function in the nearby universe is
needed.
1.1
The (Qm,
9s)
Degeneracy
Although the cluster mass function is a sensitive probe of Qm and us, there is a large
degeneracy in this parameter space for high-mass systems: for example, a small value
of as8 (for a given power spectrum) implies that massive clusters are very rare peaks in
the initial density field, and thus predicts a low abundance of such clusters; however,
a similar effect would follow from a small value of Qm, as a smaller amount of matter
in the universe would lead to a smaller abundance of massive clusters. Similarly, a
large value of as8 implies that large fluctuations in the density field (which then form
massive clusters) are more common; again, though, this effect could also occur due
to a larger value of Qm, as a larger amount of matter in the universe would yield a
larger abundance of massive clusters. However, two models that predict the same
number of high-mass clusters predict different abundances of low-mass clusters: the
model with the smaller value of as8 predicts fewer low-mass clusters than the model
with the larger value of us. Therefore, it is important to accurately determine the
cluster mass function at group-scale masses, where the degeneracy between
Qm
and
as8 can be broken by probing not just the amplitude of the mass function, but also its
shape.
Recent estimates of as8 from the local cluster mass function include values in
the range as8
0.6 - 0.8 for Qm = 0.3 [11, 12, 13, 14, 15, 16, 17, 18, 9, 19, 20].
Somewhat larger estimates of as8
0.8- 1.0 have also been obtained recently through
the measurement of cosmic shear [21, 22], which is the distortion of light emitted
by distant background galaxies while traveling through the universe, producing a
small effect on the distribution of galaxy ellipticities. The recently-released Five-Year
WMAP results (WMAP5) found a8 = 0.796+0.036 [23]; recent modeling work has also
converged on values in the range as
8
0.7 - 0.9 [24].
However, until the release of WMAP5, there was significant tension between the
WMAP3+SDSS results [25] and CDM simulations [24]: the WMAP3+SDSS cosmology required significant velocity segregation in clusters and excess specific energy in
the intracluster medium (ICM) in order to explain discrepancies in the value of the
parameter Ss
=
9s(Qm/0.3) 0 .35 . These requirements conflicted strongly with expecta-
tions from numerical simulations. However, the revised constraints from WMAP5 [23]
are sufficient to require only modest velocity segregation and small specific energy in
the ICM, while those from WMAP5+SN+BAO [26] actually make both of these
requirements unnecessary [27].
In particular, the revisions between WMAP3 and
WMAP5 bring their results into much better agreement with the recent cluster mass
function of Rines et al [28].
In short, observational estimates of the cluster mass function provide important
cosmological constraints. In this thesis, I attempt to improve these constraints by
sampling a larger area of sky and probing smaller cluster masses than previous studies.
1.2
The Cluster Mass Function
1.2.1
Mass Estimation Techniques
The largest obstacle to accurately estimating the cluster mass function is computing sufficiently accurate mass estimates. There are three well-known techniques for
obtaining such estimates:
1. Applying the virial theorem to observations of the dynamics of cluster galaxies [29, 30];
2. Observing the properties of the hot ICM, whose distribution and temperature
depend strongly on the cluster's gravitational potential [31];
3. Measuring the gravitational lensing of background clusters by foreground objects, which produces very accurate mass estimates in the centers of clusters (first suggested by Zwicky [30]).
Each of these techniques is subject to possibly large systematic uncertainties. Technique (1) may be inaccurate due to the presence of substructure, as well as velocity
bias between cluster galaxies and dark matter particles, although the exact magnitude
and direction of this bias is still a matter of debate [32, 24]. Method (2) is subject
to uncertainties resulting from the complex physical properties of the ICM and its
interaction with active galactic nuclei (AGN), as demonstrated in observations from
Chandra and XMM-Newton [33], although this is only a significant problem in the
cores of clusters. Finally, errors may arise in method (3), especially at large radii,
due to lensing by other objects (e.g., filaments) along the line-of-sight to the cluster [34, 35]; however, such errors may be overcome by combining observations of both
strong and weak lensing [36, 37]. As a result of these potential uncertainties, some
groups have instead determined cluster properties in an inverse fashion by matching the mass function predicted by a cosmological model to the measured luminosity
function or richness function [38, 39].
1.2.2
Recent X-ray Studies
The cluster mass function has been studied extensively in recent years, primarily via
X-ray data [12, 40, 9] or optical richness data, i.e., the number of galaxies found in
a cluster via optical observations, subject to some magnitude criterion [18]. Measurements of the mass function using X-ray data are subject to several sources of
uncertainty. First, the most significant uncertainty is due to the normalization of
scaling relations, namely, that of M - Tx or M - Lx [41], for which a range of values
has been determined in hydrodynamical simulations [42, 15, 17]. Additionally, these
scaling relations may be subject to Malmquist bias [39]. Some investigators avoid
such scaling relations altogether by estimating the mass of the ICM and measuring
the baryonic mass function [9, 43], although this technique requires additional assumptions regarding the relative contribution of stars and gas to the total baryon
mass, the ratio of the baryon fraction in clusters to the global value of the baryon
fraction, and the mass dependence of this ratio.
Secondly, all of these X-ray studies are subject to a potentially large systematic
error (-
20 - 30%) resulting from a difference between the temperature Tspec mea-
sured by X-ray satellites and the emission-weighted temperature Tew computed in
the aforementioned simulations [44]. This effect arises due to an ICM with a variety
of temperatures, leading to an excess contribution of line emission from cooler gas,
which causes Tspec to underestimate Tew. However, this effect should not pose a problem if observations of the cores of clusters are omitted. Vikhlinin et al. [45] found
a similar systematic effect, namely, a higher normalization of the mass-temperature
relation using Chandra observations of the temperature profiles of relaxed clusters,
which would lead to an increase in both the estimated cluster masses as well as the
inferred constraints on Qm and u8s.
Finally, some investigators have combined X-ray data with weak lensing measurements to compute the mass function [14, 20], and have obtained results generally
consistent with those of other X-ray techniques.
1.2.3
The CIRS Mass Function
The most recent observation-based cluster virial mass function is that computed from
the Cluster Infall Regions in SDSS (CIRS) sample [28]. Virial mass estimates have
improved dramatically in recent years as a result of the size and uniformity of largescale redshift surveys, such as the Sloan Digital Sky Survey [46] (SDSS). Furthermore,
virial mass estimates provide several advantages over X-ray studies: they are sensitive
to larger scales (r 2o0o rather than r5 00oo); they can be compared with theoretical mass
functions with much less extrapolation [47]; they are less sensitive to the complicated
physics in the centers of clusters; and virial masses can be estimated for poor clusters and rich groups, while X-ray mass estimates for these systems are complicated
by nongravitational physics, such as energy from AGN [48]. Importantly, the mass
estimates for these low-mass systems allow one to eliminate any uncertainty resulting
from the possible scale dependence of the estimate of as;
8 in other words, one directly
constrains fluctuations on the scale 8 h - 1 Mpc, as opposed to the
scales probed by
-
1015 h-
1
-,
14 h - 1 Mpc
M® clusters [11]. Recently, Eke et al. [19] estimated
the group mass function from an optically-selected group catalog in the 2dFGRS [49]
using a simplified version of the virial theorem. However, estimates of the group
mass function based on measurements of virial masses can be hindered by systematic
uncertainties in the group selection function, mass estimation techniques, and cosmic
variance [50, 51, 52].
Rines et al. [28] improve on this measurement by overcoming many of these difficulties. First, they utilize X-ray selection rather than optical selection of clusters,
which both reduces the influence of projection effects and allows the selection function to be computed directly. Second, they compute virial masses using the full virial
theorem, including corrections for the surface pressure term [53, 54, 55]. Third, the
CIRS survey [56] includes much better sampling of individual systems than the 2dFGRS catalog. Finally, they remove interlopers in a much more conservative manner,
thus greatly reducing scatter in the mass estimates. With the CIRS mass function
alone, they find 8 = 0.84 + 0.03 when holding
1.2.4
•m
= 0.3 fixed.
The R2FGS Mass Function
This thesis is a complementary study to that of Rines et al [28]. In particular, it
is difficult to measure the abundance of X-ray groups and clusters of - 1014 MD
from SDSS data, since available X-ray surveys detect these systems only in the very
nearby universe, and SDSS does not cover the whole sky (although it does cover a
large volume of space). Hence, R2FGS is designed to cover a much larger area of sky
to a shallower depth; in fact, the area of the R2FGS region is nearly twice the area
of the SDSS DR6 spectroscopic footprint. To construct the R2FGS dataset, I utilize
an X-ray selected sample from the ROSAT All-Sky Survey (RASS) group and cluster
catalogs [57]. I then select targets in these systems from 2MASS [58]. I combine these
measurements with optical redshift data found in the literature, as well as new data
collected with the FAST spectrograph on Mt. Hopkins in Amado, AZ.
After carefully removing interlopers, I estimate the virial masses of these X-rayselected systems and compute the mass function. Note that this study avoids significant problems with Malmquist bias (despite using an X-ray selected sample) by
using virial mass estimates that do not depend on X-ray data, as well as by using
the Vmax weighting technique in my calculation of the mass function. I also use the
virial mass data to probe the scaling relations between X-ray luminosity and virial
masses over a wide mass range. This result is important for future measurements of
the evolution of cluster abundances, which will be used to constrain the properties of
dark energy. In addition, I use the virial mass data to investigate the halo occupation
function, which is an important link between numerical simulations and observables,
especially in studies of galaxy formation [59]. Finally, by measuring the slope of the
mass function at low mass, I break the aforementioned degeneracy between Qm and
os, and obtain constraints on these cosmological parameters.
The remainder of this thesis is organized as follows. In §2, I describe the data and
the cluster sample. In §3, I discuss observations taken for this project. I estimate
the group and cluster virial masses in §4. I examine the up - Lx, M - Lx, and
N - M scaling relations in §5. In §6, I compute the mass function. I then constrain
the aforementioned cosmological parameters in §7. Finally, I discuss the results and
conclude in §8. I assume H0 = 70h 70 km s- 1 , and a flat ACDM cosmology (QA =
1 - Qm) throughout. Where not stated otherwise, I assume Qm = 0.3 and h 70 = 1.0
for initial calculations when necessary.
Chapter 2
The R2FGS Sample
2.1
X-ray Cluster Surveys
This study is primarily focused on low-mass systems. As these systems are rather
faint in X-ray emission, RASS only finds them in the very nearby universe (z < 0.06).
Although RASS is a shallow survey, it covers essentially the entire sky and is the
most complete X-ray survey for nearby groups and clusters. Conveniently, 2MASS
provides photometry over the entire sky to a depth corresponding to M* + 1 for
these systems, where M* is the magnitude of the characteristic knee in the Schechter
function describing each system's luminosity [60]. CIRS showed that this depth is
sufficient to obtain large samples of cluster galaxies needed for accurate dynamical
mass estimates [28].
Moreover, R2FGS is unique in that it covers a much larger area of sky than
previous mass function surveys (or even the SDSS). In particular, the area of the
R2FGS footprint is r 4.13 ster., nearly 3 times larger than the CIRS mass function
area of - 1.46 ster. For comparison, the area of the SDSS DR6 spectroscopic footprint
is - 2.26 ster. Thus, although R2FGS is not a particularly deep survey, it still includes
a significant volume of space due to the large size of the survey region.
The exact criteria for group and cluster selection are described below. I search
several published cluster catalogs derived from RASS, including: the X-ray Brightest
Abell Cluster Survey (XBACS, [61]); the Bright Cluster Survey and its extension
(BCS/eBCS, [62, 63]); the NOrthern ROSAT All-Sky galaxy cluster survey (NORAS,
[64]); and the ROSAT-ESO Flux-Limited X-ray galaxy cluster survey (REFLEX,
[65]). For detailed descriptions of the construction of the catalogs, please consult the
original catalog papers (see Rines and Diaferio [56] for a summary).
None of these catalogs is complete; the worst, NORAS, is - 50% complete,
while the best, REFLEX, is - 90% complete.
Regardless, I obtain a fairly com-
plete composite catalog after combining them (more complete than any individual catalog).
In particular, as mentioned above, the composite catalog covers es-
sentially the entire northern sky at high Galactic latitude (Ibl
limit of fx
5 x 10-
12
erg cm- 2
S- 1
(ROSAT 0.5-
> 200) to a flux
2.0 keV band corrected for
Galactic absorption). Since the catalogs are nominally complete to a flux limit of
3 x 10- 12 erg cm - 2 s- 1 , my imposed flux limit ensures that the sample is essen-
fx
tially fully complete. For the purposes of this study, I disregard any modest possible
incompleteness, as these clusters are an unbiased sample selected purely based on Xray flux. I confirm this claim with a V/Vmax test [66]: I find (V/Vmax) = 0.492 + 0.038
compared to an expected value of 0.5 for a complete, uniform sample.
When the composite catalog contains multiple X-ray fluxes for a given cluster,
I choose the most recently published value. Hence, the order of preferences is: REFLEX, NORAS, BCS/eBCS, XBACS. Note that the various surveys determine fluxes
in somewhat different ways: REFLEX and NORAS measure fluxes with Growth
Curve Analysis (GCA), while BCS/eBCS utilizes Voronoi Tessellation and Percolation (VTP). Nevertheless, the measured fluxes are in fairly good agreement; see
Figure 21 of B6hringer et al. [64] for a direct comparison of NORAS and BCS/eBCS.
Note that since I use the most recently published flux value, my first preference is
GCA, while my second is VTP.
2.2
Group and Cluster Selection
The overall observational goal for this study is to obtain complete spectroscopic samples to roughly the 2MASS magnitude limit [58] for all groups and clusters in the
ROSAT catalogs, restricted to the region 6 > 00 and 0.01 < z < 0.06, in addition
to the flux limit described above. Note that the aforementioned differences in the
flux determination techniques for the various surveys may slightly alter the exact flux
limit. After imposing these restrictions, the sample consists of 62 groups and clusters,
which I will hereafter refer to as the R2FGS systems. Figure 2-1 shows the redshift
versus X-ray luminosity of each R2FGS system, as well as the sample X-ray luminosity and redshift limits. A complete list of the R2FGS systems and their properties is
given in Table 1 of Appendix B.
'IO
t.4
CQ
It4.
J-4
0.1
.V I
0
0.02
0.04
0.06
Figure 2-1: Redshift versus X-ray luminosity (0.5 - 2.0 keV) for X-ray clusters from
XBACS, BCS/eBCS, NORAS, and REFLEX contained in the R2FGS region. The
X-ray cluster catalogs are nominally complete to fx
3 x 10- 12 erg cm - 2 S-1; thus,
2
1
the imposed R2FGS limit of fx > 5 x 10-12 erg cm
s
(denoted by the solid curved
line) ensures that the R2FGS sample is essentially fully complete. The solid vertical
line shows the R2FGS redshift limit.
Chapter 3
The R2FGS Observations
3.1
Target Galaxy Selection
For a given cluster, I select as targets all galaxies in 2MASS within a projected distance Rp = 2.14 Mpc/h
70
of the cluster's X-ray center, such that either the galaxy's
Kron magnitude K, < 13.0 or its absolute magnitude MK, < -22.2 + 5 log h70
(- Mk + 1), where I assume the target galaxy is at the distance of the cluster. I
supplement the 2MASS redshift catalogs with literature data from the NASA/IPAC
Extragalactic Database (NED) and from the SDSS for galaxies fainter than the magnitude limits stated above.
Approximately half of the targets in the sample have already been surveyed with
the FAST spectrograph for the 2MASS Abell Cluster Survey [67], and about half of
the remaining systems are included in SDSS Data Release 6 (DR6) [68]. The process
of obtaining spectra with the FAST spectrograph for galaxies in the • 25 remaining
systems is nearly complete; A2665, A2626, and A2271 are the only remaining systems
with significant incompleteness. Overall, I have targeted 1048 galaxies in groups and
clusters across the northern sky, 964 of which have been observed through January
2008.
Combined with existing redshift data, the entire survey includes - 14800
galaxies, for an average of
-
(after removing interlopers).
240 galaxies per field and
-
120 members per system
3.2
FAST Data
The new redshifts were obtained with the FAST spectrograph [69] on the 1.5-m Tillinghast telescope at the Fred Lawrence Whipple Observatory (FLWO) in Amado,
AZ. FAST is a high-throughput, long-slit spectrograph with a thinned, backsideilluminated, antireflection-coated CCD detector. The length of the slit is 180"; the
R2FGS observations used a slit width of 3" and a 300 lines mm
-
grating. This setup
provides spectral resolution of 6 - 8A and spans the wavelength range 3600 - 7200A.
Redshifts are computed by cross-correlation with spectral templates of emissiondominated and absorption-dominated galaxy spectra created from FAST observations [70]. The uncertainty in the redshifts is usually < 30 km s 1 .
One significant difference between the FAST spectra collected for this project
and those collected for other redshift surveys [49, 46] is that the completeness of the
R2FGS sample is not hindered by fiber placement constraints. Another difference
is that the long-slit FAST spectra sample light from larger fractions of galaxy areas
than do fiber spectra. Hence, the effects of aperture bias on spectral classification are
significantly lessened [71, 72].
Chapter 4
Estimating Cluster Masses
4.1
Interloper Removal
The removal of interlopers is a challenging problem in any galaxy cluster study. Recently, Wojtak et al. used cosmological N-body simulations to perform a detailed
analysis of many widely-used interloper removal schemes [73]. Of all the direct methods, they recommend a dynamical maximum velocity criterion first proposed by Den
Hartog and Katgert [74]. This method removes the largest fraction of interlopers
(73%) and avoids many of the difficulties of indirect methods, such as the need for
large kinematic samples that can only be obtained by stacking data from many objects.
Based on these results, I utilize this dynamical maximum velocity criterion in this
thesis. In this method, I select as an interloper any particle at a given projected
radius R whose velocity exceeds a maximum attainable velocity for halo particles at
this radius. For the maximum velocity profiles, I consider two characteristic velocities:
the circular velocity Vcjr and the infall velocity Vinf, given respectively by
vir =
v/GM(r)/r
Vinf = vfVcir.
(4.1)
(4.2)
The infall velocity is an upper limit to the particles' velocities for which the virial
theorem is violated; it can be thought of as an escape velocity from the mass interior
to the radius r [73].
The following formula then gives the maximum velocity profile:
Vmax = maxR {vinf
cosO, Vcir sin O},
(4.3)
where 0 is the angle between the position vector of the object with respect to the
cluster center and the line of sight. This formula assumes a particular kinematic
model that allows objects to fall onto the cluster center with velocity Vinf or to move
tangentially with circular velocity vir. This is a fairly restrictive maximum velocity
criterion, which gives accurate limits at large R
-
rvir
[73].
The final component needed for the maximum velocity profiles is the mass profile.
In accordance with the rest of this study, I utilize the mass estimator MVT derived
from the virial theorem [75]:
MvT(r = Rmax)-
)2
37N Ei (v
2G -i.j1/ Rij
(4.4)
where N is the number of galaxies enclosed on the sky by a circle with radius Rmax,
vi is the line-of-sight velocity of the ith galaxy, and Ri,j is the projected distance
between the
ith
and jth galaxies. Note that this formula is valid for spherical systems
with arbitrary anisotropy. I then approximate the mass profile as M(r) _ MVT(Ri <
r < Ri+I), where Ri is the sequence of projected radii of galaxies in increasing order.
However, the virial theorem applies to an entire system; thus, since I am applying it to
a subset of the cluster members, a surface pressure term is required: 2T + U = 3PV,
instead of the usual 2T + U = 0 [55]. However, as I am concerned here with interloper
removal and not accurate mass estimation, I neglect the surface term in this analysis,
although it is included in the final mass estimates (see §4.2).
In order to determine cluster membership, I initially include all galaxies within
S2.5 h- 1 Mpc of a given cluster X-ray center (larger radii are used for some clusters
that clearly have a large virial radius, such as Coma and A2147). I then apply the
Vmax criterion described in the preceding paragraphs, so as to discard any galaxies
with vi >
Vmax.
Figures A-1-A-7 in Appendix A display the infall patterns for all
62 groups and clusters in the R2FGS sample, starting with NGC1550 (z = 0.0131),
the nearest cluster in the sample, and increasing in redshift up to A2457 (z = 0.0594),
the most distant cluster in the sample. The plots show both interlopers and member
galaxies, as well as the maximum velocity profiles calculated using Eq. (4.3). Most
of the systems have a sufficient amount of data to accurately estimate their virial
mass, although FAST data is still being collected for a few (e.g., A2271). Interestingly, several of the systems in the R2FGS sample have very few existing redshifts
in the literature, despite their proximity. For instance, 2A0335, which has been observed extensively with Chandra [76], had only two published redshifts before this
new survey.
Lastly, note that there appear to be interlopers remaining in some systems (e.g.,
UGC04052, NGC4325, A1142, and A2626) after the removal procedure described
above. This is partly an unfortunate consequence of the fact that the fraction of
removed interlopers is limited in principle to values less than
75% using the Vmax
technique, because - 1/4 of the unbound particles within the observation cylinder
are within the envelope of bound velocities and hence inaccessible to direct removal
techniques [73]. Further analysis of particularly outlying objects, such as those in
A2626, may be needed before a truly robust virial mass estimate for such systems
can be obtained. Perhaps the most difficult system to analyze in the entire sample is
A2147, a member of the Hercules supercluster (with A2151 and A2152). Due to the
extreme proximity of the two other clusters, it is very difficult to accurately determine
cluster membership for A2147; rather sophisticated interloper removal techniques are
required, such as the KMM mixture-modeling algorithm [77]. In general, though, the
Vmax
technique successfully removes most of the evident interlopers from the R2FGS
systems.
4.2
Virial Masses
After removing interlopers from each system, I use Eq. (4.4) to compute the virial
masses. To start with, it is necessary to define a radius of virialization within which
the member galaxies are relaxed. I use r200 , where rA refers to the radius at which
the mean enclosed density is Ape, where Pc is the critical density:
PC=
3H2
(9.21
8-7rG
x 10-27
kg m-3)h270×
0 .
(4.5)
If the system does not lie completely within r200 , the surface pressure term 3PV
in the virial theorem must be included, as described in the previous section. The
virial mass is then an overestimate of M 200 , so that a correction C must be applied.
Assuming that mass follows the galaxy distribution, the correction is given by
47rr 3
200
f
where
[ _rr 2 2)
p(r200oo)
47rr 2pdr [o(< r 2 o00) '
(4.6)
r,(r 20oo) is the radial velocity dispersion at r200
oo and u(< r 200oo)is the inte-
grated velocity dispersion within r200 [78]. Considering the limiting cases of circular,
isotropic, and radial orbits, the maximum values of the term involving the velocity
dispersion are 0, 1/3, and 1, respectively.
After obtaining an initial estimate of r200 using the mass profiles calculated from
the interloper-cleaned systems, I apply a correction factor of 8% to account for
the aforementioned surface term. This factor is calculated from Eq. (4.6) assuming isotropic orbits of galaxies and an NFW mass profile [79] with a concentration
parameter c200
=
r200/rs = 5, where r, is a scale radius in the NFW profile. Note
that the assumption of isotropic orbits is supported by many observations [32], as is
the value c20 0 = 5 [80]. After this correction, the enclosed density within r 20 0 is
17 8 pc;
thus, I reduce the corresponding estimate of M 200 by 3.3%.
Lastly, I estimate the uncertainties on the virial masses by using the limiting
fractional uncertainty 7r-V 2lnNN - 1/ 2 [81]. Note that these uncertainties do not
include systematic uncertainties due to errors in interloper identification. Table 2 in
Appendix B lists the M 200 and r 200 estimates.
Unfortunately, I am not able to obtain mass estimates for four of the systems in the
R2FGS sample using the techniques described above, because the density enclosed by
any projected radius R never drops below
2 0 0 Pc.
I instead estimate the virial masses
of these systems by letting M 200 be the total mass enclosed by the projected radius
of the most distant galaxy (from the cluster center) left after interloper removal. I
then estimate r 200 using the relation
7200 =
(
3M2oo00
1/3
00r p
8007F Pc
(4.7)
The groups and clusters for which I use this method are A2147, A0576, MKW3s, and
A2256. The reasons behind the failures of these systems to converge to Pend <
2 0 0 pc
appear to be mostly related to interlopers. The difficulties of analyzing A2147 were
already discussed in the previous section, but its failure to converge demonstrates
the seriousness of the aforementioned membership assignment problems. For A0576,
it appears that interlopers within the bound velocity envelope are responsible. For
MKW3s, the maximum velocity scheme evidently fails to remove several galaxies far
outside the infall region, which are very likely interlopers. Essentially, the interlopers
remaining in these three systems lead to a significant overestimate of the enclosed
mass at a given radius, so that the enclosed mean density is never < 200pc (at least
not within radii of < 3h-1 Mpc). A2256, on the other hand, simply seems to be
an extremely massive cluster, and it is not particularly surprising that its enclosed
density does not converge in this scheme.
4.3
Projected Masses
As a "sanity check" on my calculation of the virial masses, I confirm these results
using the projected mass estimator MpM [81]:
MPM(r = R200)
PM
G(N - a)
(Vi - V)2 i,
(4.8)
where N is the number of galaxies lying within a distance R 200 of the cluster center,
vi is the line-of-sight velocity of the ith galaxy, and Ri is the projected radius of the
ith galaxy with respect to the cluster center. The parameter a is intended to account
for the difference between measuring velocities and radii relative to the center of mass
of the system and measuring these quantities relative to the centroid of the tracers.
Following Heisler et al., I set a = 1.5, which is appropriate for radial or isotropic
orbits [81]. The parameter fPM is equal to 64/w for radial orbits and 32/7 for isotropic
orbits [81]; I use the value 32/7, since I have no specific information regarding the
distribution of orbit eccentricities. Finally, I approximate the fractional uncertainty
on MpM as 1.4/vN.
The projected mass estimator avoids some of the difficulties encountered by the
virial mass estimator. For example, it is less sensitive to galaxies which are accidentally projected very close to each other. Thus, it provides a good method with which
to confirm the virial mass results above. I calculate MPM for each system using all
of the member galaxies within r 200 ; the results are listed in Table 2. In addition, to
investigate the correlation of these two mass estimators, I plot MPM versus M200 in
Figure 4-1. In an idealized situation, I would expect the best-fit line to have a slope
of 1 (the dashed line in the figure). Computing the bisector of the two weighted leastsquares fits, I find a slope of 1.053 + 0.017, which confirms that the mass estimates
computed in the previous section are reasonably robust.
Finally, although the projected mass estimator is a useful tool for comparison,
the virial mass estimator is still generally preferred for galaxy cluster studies, since it
does not require any model-dependent parameters which must be specified by hand.
Hence, I utilize only the M 200 estimates throughout the rest of this thesis.
4.4
The Reduced R2FGS Sample
Although the Vmx,,
scheme is fairly effective when applied to the R2FGS systems,
there are still many groups and clusters that appear to contain interlopers even after
100
10
0 IUU
1
0
U.I
0.1
10
1
100
M2oo (1014 h-o Me)
Figure 4-1: M 200 versus MpM. The solid line displays the bisector of the weighted
+ (0.078 ± 0.016). For comleast-squares fits: logloMpM = (1.053 ± 0.017)logIoM 200oo
parison, the dashed line has slope equal to 1.
applying the removal algorithm. In some cases, this is due to a sparseness of data,
but in most, it is due to the presence of galaxies which hinder the effectiveness of a
generalized algorithm. One possible solution to this problem is to apply a different
interloper removal technique for each system (e.g., the "shifting gapper" technique [82]
or the M 2oo/MPM
ratio test technique [73]). However, this would come at the cost
00
of generality in the analysis; it is a delicate procedure to combine data sets analyzed
using very different techniques.
Nevertheless, in order to calculate a robust mass function and accurately constrain
Qm
and Us, the masses utilized in the analysis must be very accurate. For many of
the R2FGS systems, the redshift-radius plots given in Appendix A demonstrate that
this is clearly not the case. Therefore, based on a visual inspection of the phasespace plots, I define a "reduced" R2FGS sample which consists only of those systems
that appear to be accurately analyzed by the Vmax interloper removal technique (e.g.,
Coma or A2052). Since this is a difficult issue to define precisely, I err on the side of
caution and attempt to leave in any system that does not have clear problems in its
redshift-radius plot.
As a characteristic example of a problematic system, consider A2063. Upon visual
inspection, this system appears to have a well-defined caustic profile; however, the
Vmax criterion (denoted by the solid black curve) is significantly weakened by the
presence of a handful of outlying objects at large projected radii. An interloper
removal scheme based on deviations from the mean, such as the shifting gapper
technique [82], would likely identify these particles as interlopers. As such, I do not
include this system in the reduced R2FGS sample. Overall, after similarly analyzing
all 62 groups and clusters, I include 45 in the reduced sample. The 17 clusters which
do not pass the cut are marked with an asterisk (*) in Table 2 of Appendix B.
Finally, this cut introduces an incompleteness in the sample which must be taken
into account when calculating the mass function. I consider two methods of accounting for this incompleteness in §6.1 and §6.2; the second method relies upon using the
reduced R2FGS sample to calibrate scaling relations between Lx and M200 and U200 ,
which is the main focus of the following chapter.
Chapter 5
Cluster Scaling Relations
Scaling relations between simple cluster observables (e.g., temperature or luminosity) and cluster masses probe the nature of cluster assembly, as well as the detailed
properties of cluster components. It is extremely important that these relations are
well-established for clusters in the local universe, as future studies of distant clusters
that aim to constrain dark energy will rely on these results.
5.1
The Velocity Dispersion-Luminosity Relation
I utilize the results of Danese et al. to calculate the mean redshift c2 and the projected
velocity dispersion
up
of each cluster using the galaxies remaining after interloper
removal [83]. For a system of n galaxies, up is given by
2PZ
n
=
v2
2
(5.1)
n - 1 (I+ VP/c)2
where vp, = (Vp1 - Vp)/(1 + Vp/c) is the line-of-sight component of the velocity of the
ith galaxy with respect to the cluster center of mass, Vp, = czi is the radial velocity
of the ith galaxy uncorrected for the motion of the local observer, Vp = cf , and 6
is the uncertainty in the measured values of cz. For this study, I am particularly
interested in the velocity dispersion at r 200 , denoted by
UO2
00 ,
which I calculate for
each system using only the galaxies projected within r 200 . The values of o 200 for the
R2FGS groups and clusters are given in Table 2 of Appendix B.
Perhaps the simplest cluster observable is X-ray luminosity. The X-ray luminosities for the systems in the R2FGS sample are in the ROSAT band (0.5 - 2.0 keV) and
corrected for Galactic absorption. Figure 5-1 shows U200 versus Lx, using the velocity dispersions given in Table 2 of Appendix B. The solid dots in the figure represent
systems in the reduced R2FGS sample, while the open squares represent systems that
did not pass the. cut. The up - Lx relation of the RASS-SDSS [84] is also displayed
in Figure 5-1. A weighted least-squares fit to the reduced sample of R2FGS systems
yields:
logo10
20 0
= (0.205 ± 0.020) logo10 Lx + (2.878 ± 0.011).
(5.2)
Although the scatter is moderate, the R2FGS systems follow roughly the same relation
as the RASS-SDSS sample.
5.2
The Mass-Luminosity Relation
Figure 5-2 displays the M200 - Lx relation, using the virial masses computed in
§4.2 (and given in Table 2 of Appendix B). As in Figure 5-1, the solid dots represent systems in the reduced R2FGS sample, while the open squares represent the
other systems in the original sample.
I also plot the M 200 - Lx relations of the
RASS-SDSS [841, computed using both optical masses derived from the virial theorem (dashed line) and masses estimated from X-ray temperature data (dotted line).
A weighted least-squares fit to the reduced sample of R2FGS systems yields:
logo
10 M 20 0 = (0.568 ± 0.059) loglo Lx + (0.880 ± 0.033).
(5.3)
This fit agrees roughly with the RASS-SDSS scaling relations, although it is somewhat closer to the RASS-SDSS relation computed using X-ray masses than to that
computed using optical masses. Also, note that the obvious (solid dot) outlier at
1000
toC00
0
0.01
0.1
1
10
L, (1044 h- 2 erg s-1)
Figure 5-1: Velocity dispersions at r 200 versus X-ray luminosities. The solid dots
represent systems in the reduced R2FGS sample (see §4.4), while the open squares
represent the other systems in the original R2FGS sample. The solid line displays the
weighted least-squares fit to only the reduced R2FGS systems: log 10 U200 = (0.205 +
0.020) loglo Lx + (2.878 ± 0.011). The dashed line shows the a 200 - Lx relation for
RASS-SDSS [84].
low mass in Figure 5-2 is A2271, which is still significantly undersampled, as seen in
Figure A-7.
As an additional verification that the selection of the reduced R2FGS sample
improves the robustness of the data, I calculate the scatter in the M 20oo0 - Lx relation
using only the reduced sample and using the entire original sample. I compute the
unidirectional scatter:
M-Lx = j(log(M200) - 0log(M
2 00(Lx)))
2
, where the average
is taken over all systems in either the reduced or original sample, and M 200 (Lx) is
the value of M 20 0 calculated from applying the best-fit scaling relation of either the
reduced or original sample to the Lx values. I find UM-Lx = 0.313 for the original
sample and UM-Lx = 0.279 for the reduced sample. Thus, the reduced R2FGS sample
has a smaller scatter in the M - Lx relation, which provides additional support to
the assertion that it is a more robust data set than the original sample.
5.3
The Velocity Dispersion-Mass Relation
Figure 5-3 shows the M 200 - a 2oo
00 relation. Since this scaling relation is not needed
for the mass function analysis in the subsequent chapters, I do not divide the data
into the reduced sample and the other systems (as for the other scaling relations
above); the main reason to compute this relation is to ensure that M 200 and a 200 are
well-correlated, as should be the case if the mass estimates are robust. The bisector
of the weighted least-squares fits (one using the M20 0 uncertainties and one using the
a 200 uncertainties) is given by:
log1 0 a 200
=
(0.357 ± 0.045) loglo M20 0 + (2.559 ± 0.127).
(5.4)
As in Figure 5-2, the obvious outlier in Figure 5-3 is A2271, whose mass is most likely
underestimated significantly due to its incomplete sampling (see Figure A-7). Nevertheless, the low scatter in Figure 5-3 implies that the virial masses are well-correlated
with the velocity dispersion estimates, although this is not necessarily surprising, be-
100
10
0
-o
I>.
000
..
1
U.I
-I
0.01
1
0.1
Lx (104
h-2
erg
10
s-1 )
Figure 5-2: Virial masses at r 200 compared to X-ray luminosities. The solid line
displays the weighted least-squares fit: loglo M 20oo
0 = (0.568+0.059) log1 o Lx +(0.880+
0.033). The solid dots represent systems in the reduced R2FGS sample (see §4.4),
while the open squares represent the other systems in the original R2FGS sample.
The dashed and dotted lines show the M200 - Lx relations for RASS-SDSS [84] for
optical and X-ray masses, respectively.
cause both quantities depend similarly on the galaxy velocity distribution. Overall,
this is a good indicator that the R2FGS virial mass estimates are fairly robust.
5.4
The Halo Occupation Function
The halo occupation function (HOF) is an important link between the physics of
galaxy formation and the clustering of matter, both dark and baryonic [85]. The HOF
assumes that cosmology is the main factor governing the evolution and clustering of
halos, while the physics of galaxy formation determines the particular manner in
which galaxies populate the halos.
As a result, one can calculate various power
spectra of dark matter and galaxies.
The primary components of this model are the mean number of galaxies N(M)
per halo of mass M, the probability distribution P(NIM) that a halo of mass M
contains N galaxies, and the relative distribution in real space and velocity space of
dark matter and baryonic matter within halos [86]. In this study, I use the R2FGS
sample to measure the mean number of galaxies N(M) (brighter than some minimum
mass or luminosity) per halo of mass M, which I will hereafter refer to as the HOF.
The simplest possibility is that N oc M, which would imply that galaxy formation
is equally efficient for all halos with mass greater than some cut-off value. If galaxies
form more efficiently in more massive halos, then the relation might be a power law
(N oc M") with exponent p greater than 1; if galaxies form less efficiently in more
massive halos, then the relation might be a power law with exponent less than 1.
The latter situation could arise due to the heating of gas by the halo potential, which
would prevent the gas from collapsing into galaxies, or due to galaxy disruption
through dynamical friction or tidal stripping.
Semi-analytic models predict p
0.8 - 0.9 [87, 88], while Springel and Hernquist use numerical simulations to show
that gas heating suppresses galaxy formation in the most massive halos [89].
Recent observational results include those of Marinoni and Hudson [90], who find
M = 0.55 ± 0.03 using virial masses and blue luminosities of objects in the Nearby
1000
0
1CM'
100
0.1
1
M
2oo (10o1 h-I Mo)
10
100
Figure 5-3: Virial masses at r200 compared to velocity dispersions within r 200. The
solid line is the bisector of the weighted least squares fits: log 10 a 200 = (0.357 +
0.045) loglo M 20oo
0 + (2.559 ± 0.127).
Optical Catalog, as well as those of Pisani et al. [91], who find p = 0.70 ± 0.04 using
a sample of groups. Of greater interest are the results of Rines et al., who constrain
N(M) using the Cluster and Infall Region Nearby Survey (CAIRNS), a spectroscopic
survey of the infall regions surrounding nine nearby rich clusters [92]. They utilize
the same magnitude limit that I use in this analysis (see next paragraph), and also
consider N200 , the number of galaxies projected within r200 , and M2oo
00 , which are the
same quantities that I investigate below. They find p = 0.70 + 0.09. Also of great
interest are the results of Lin et al., who analyze the HOF using a sample of 93 clusters
with 2MASS photometry and X-ray mass estimates [93]. They find p = 0.84 ± 0.04,
which agrees fairly well with theoretical models.
In this study, I calculate the number of bright galaxies N 200 that lie within r200 of
the center of each group or cluster in the R2FGS sample, where "bright" is defined
by the magnitude criterion MK, <_ Mk, + 1. Here, Mk, is the magnitude of the
characteristic knee in the Schechter function describing each system's luminosity [60];
for the R2FGS systems, M,
= -23.2 + 5 log h70 . I plot N200 against M20 0, the virial
mass of each system, in Figure 5-4. The bisector of the two weighted least-squares
fits is given by:
loglo N 20 0 = (0.83 ± 0.04) logto M20 0 + (1.00 ± 0.06).
In other words, N20 0 C M120 0.
0 4,
(5.5)
4.25a shallower than a linear relation (shown as a
dashed line in Figure 5-4). This result agrees extremely well with that of Lin et al [93].
The R2FGS HOF thus provides an excellent independent verification that A < 1. This
conclusion holds for cluster masses derived from either the virial theorem or from Xray data and associated scaling relations. This nonlinear HOF has important physical
implications which lie beyond this scope of this thesis; for a detailed discussion, please
refer to Lin et al [93].
100
o
z
10
I
0.1
10
1
100
M2oo (1014 h-o1 Mo)
Figure 5-4: Number of bright galaxies within r200 compared to virial masses at r 200. A
"bright" galaxy is defined by the magnitude criterion MKs < Mk + 1. The solid line is
oo= (0.826±0.040) loglo M 200 +
the bisector of the weighted least squares fits: loglo N 200
(1.003 ± 0.060). For comparison,the dashed line has slope equal to 1.
Chapter 6
The R2FGS Mass Function
I estimate the standard cluster mass function dn(M)/dlogo M using the 1/Vmax
estimator [66], where Vm,(Lx) is the maximum comoving volume a cluster with Xray luminosity Lx would lie within in the flux- and redshift-limited R2FGS sample.
In each logarithmic mass bin, I sum the clusters:
dn(M)
1
dloglo M
d loglo M
1
. Vmax(Lx,i)'
(6.1)
where the sum is taken over all clusters within the mass bin. The uncertainty in the
mass function is then given by:
(
S dn (M) - 2(6.2)
dloglo M
[Vmax(Lx,i)2
The major advantage of using Vma(Lx) instead of Vmax(M) is that the slope, normalization, and scatter of the M - Lx scaling relation are not needed in order to
calculate Vm,
[12]. Figure 6-1 shows the maximum volume probed by the R2FGS
sample as a function of X-ray luminosity. I calculate Vm,
assuming a flat •m, = 0.3
cosmology. Note that this assumption should not affect the final results significantly
due to the local nature of the R2FGS sample.
0.8
S0.6
0.2
0 E 0.4
"•
0.2
03
0.01
0.1
1
10
L (1044 h-• erg s-1)
Figure 6-1: The maximum volume sampled by R2FGS as a function of X-ray luminosity.
6.1
Uniform Correction
As mentioned in §4.4, the reduced R2FGS sample introduces an incompleteness that
must be accounted for when computing the mass function. The simplest possibility
is to include a correction factor in Eq. (6.1):
dn(M)
_
1
dloglo M dloglo M
E
1
KVmax(Lx,i)'
(6.3)
where K = Nred/Ntot, that is, the number of systems included in the reduced sample
divided by the total number of systems in the original sample. Based on the analysis
in §4.4, K = 45/62. This correction re-scales the mass function in a uniform manner
by assuming that the incompleteness in the sample is distributed uniformly over all
of the mass bins.
Figure 6-2 presents the R2FGS mass function computed using the virial masses
from §4.2 and this correction; the results are discussed in more detail in §6.3.
6.2
Luminosity-Dependent Correction
Although the uniform correction discussed in the previous section is the simplest
method of accounting for the incompleteness in the reduced R2FGS sample, it is not
necessarily the most robust. Thus, I also consider a luminosity-dependent correction,
which consists of the following steps:
1. Calibrate the M 2 00 - Lx and U200 - Lx relations using the reduced R2FGS
sample, as given in Eqs. (5.3) and (5.2), respectively;
2. Calculate the X-ray luminosity of each system in the entire R2FGS sample from
the X-ray flux according to:
Lx = 47fxd
4rfx (
Ho
0
,
(6.4)
(6.4)
-3
4--4
O
0U
C)
-5
0
~ 6
-7
13.5
14
14.5
15
M20oo (h-1 Mo )
Figure 6-2: The R2FGS mass function (thick solid line), computed using a uniform
correction applied to the virial masses of the reduced sample. The thick dash-dotted
lines show the mass functions computed using the cosmological parameters from the
WMAP1 results (upper) and WMAP3 results (lower), following the results of Jenkins
et al [94]. The dashed (curved) line shows the best-fit mass function for the CIRS
virial mass function; the unfit CIRS virial mass function is displayed by the dashed
(straight) lines [28]. The light dotted line and error bars show the CIRS virial mass
function computed after removing the minimum redshift and including all possible
mergers as separate systems. This demonstrates the importance of cosmic variance
at these low masses. The vertical line indicates the minimum mass I use to constrain
cosmological parameters, although I do not perform that calculation for this mass
function.
where dL is the luminosity distance to the cluster and z- is the mean value of
cz calculated from the redshifts of the member galaxies;
3. Insert these X-ray luminosities into Eqs. (5.3) and (5.2) in order to calculate
M 200 and U200 for each system in the original R2FGS sample.
Note that in Step (2), I do not simply use the X-ray luminosities from Table 1 of
Appendix B because these observational values are not as accurate as those computed from the observed X-ray fluxes according to Eq. (6.4). Also, note that in this
luminosity-dependent correction, I utilize the resulting mass and velocity dispersion
estimates for all 62 of the original R2FGS systems in order to compute the mass
function. The mass estimates are listed in Table 2 of Appendix B under the column
heading "MLx".-
Figure 6-3 presents the R2FGS mass function computed using these MLx and
U200,Lx
estimates; the results are discussed in more detail in §6.3. In addition, I use
this mass function to constrain Qm and as8 in the next chapter.
6.3
Comparison to Previous Mass Functions
The original formalism for computing the mass function was based on Press-Schechter
theory [1]. However, numerical simulations have predicted comparatively more massive systems and fewer less massive systems than the Press-Schechter formalism [95].
Jenkins et al. calculated fitting formulae for a universal mass function that can be
evaluated for a variety of cosmological models [94]. In fact, their mass function accurately replicates the mass function of dark matter halos in the Hubble Volume
simulation. In recent years, many investigators have concluded that the following
equation from Jenkins et al. [94] provides a nearly universal mass function, so that it
can be used to constrain cosmological parameters [96]:
f(M) = 0.301 exp [-I In o - + 0.6413.82] ,
(6.5)
-3
O
O
0
-4
5
C)
0
6
N~-6
13.5
14
14.5
15
M200 (h-IM®)
Figure 6-3: The R2FGS mass function (thick solid line), computed from masses found
using the M - Lx relation calibrated with the reduced sample. The thick dash-dotted
lines show the mass functions computed using the cosmological parameters from the
WMAP1 results (upper) and WMAP3 results (lower), following the results of Jenkins
et al [94]. The dashed (curved) line shows the best-fit mass function for the CIRS
virial mass function; the unfit CIRS virial mass function is displayed by the dashed
(straight) lines [28]. The light dotted line and error bars show the CIRS virial mass
function computed after removing the minimum redshift and including all possible
mergers as separate systems. This demonstrates the importance of cosmic variance
at these low masses. The vertical line indicates the minimum mass I use to constrain
cosmological parameters in §7.
for the range -0.5
< In a 1 < 1.0 where f(M) is the mass function as defined in
Eq. (6.7) below. In this formula, u 2 (M, z) is the variance of the linear density field,
extrapolated to the redshift z at which halos are identified, after smoothing with a
spherical top-hat filter which encloses a mean mass M. This variance can be expressed
in terms of the power spectrum P(k) of the linear density field extrapolated to z = 0
as:
a2 (M, z) =
2(Z )
k 2 P(k)W 2 (k; M)dk,
(6.6)
where b(z) is the growth factor of linear perturbations normalized so that b = 1
at z = 0, and W(k; M) is the k-space representation of a real-space top-hat filter
enclosing mass M at the mean density of the universe. Note that Jenkins et al. define
the mass function as [94]:
M dn(M, z)
n u- 1,
f (, z)- Po dIn
(6.7)
where n(M, z) is the abundance of halos with mass less than M at redshift z, and
po(z) is the mean density of the universe at that redshift.
In order to use Eq. (6.5), halos are placed at the most bound galaxies, then the
halo radius is increased until the enclosed spherical overdensity is
18 0 pb,
where Pb
is the background (not critical) density (see discussion in White [47]). This mass,
denoted as M180b, must then be converted to M200 , for which I assume an NFW
profile with c = 5 [96]. For comparison, note that X-ray mass estimates often require
conversion to Msoo
0 , which is more of an extrapolation than the conversion to M 200
and more sensitive to the assumed value of c.
Figures 6-2 and 6-3 show the mass functions for the best-fit cosmological parameters from WMAP1 [97] and WMAP3 [98] at z = 0.037 (the mean redshift of the
R2FGS sample). Note that the mass function for the cosmological parameters from
WMAP5 [26] lies between the WMAP1 and WMAP3 mass functions. The WMAP
mass functions in Figures 6-2 and 6-3 are convolved with an assumed mass uncertainty alogM = 0.056 for appropriate comparison with the observed mass functions.
Note that some of the largest differences between the WMAP1 results and those of
WMAP3 were in Qm and a8 , the two parameters which most strongly influence the
mass function. In fact, the WMAP5 value for as
8 is again significantly different from
those of WMAP1 and WMAP3, although it lies between them. The R2FGS mass
functions lie closest to the WMAP1 prediction (or slightly above it). Hence, R2FGS
u8 WMAP1 found as8 = 0.84±0.04
favors a larger value of as;
Figure 6-2 and 6-3 also show the CIRS mass function [28], which lies between
the WMAP1 and WMAP3 results (and hence very close to the WMAP5 results).
According to Reiprich [99], the HIFLUGCS mass function computed using the X-ray
properties of clusters lies closer to the WMAP3 results [12], although other mass
functions derived using X-ray data find values closer to the WMAP1 results [15, 17].
Lastly, Figures 6-2 and 6-3 also show the CIRS mass function estimated without
imposing a minimum redshift limit on the CIRS sample [28]. This plot shows that
the systematic uncertainty in the mass function due to cosmic variance becomes
quite large at M 200
<
1014 h0 1 M®, decreasing confidence in the data points from
both CIRS and R2FGS in that mass range (although both R2FGS mass functions
contain essentially no systems in that region). In general, the R2FGS virial mass
functions agree roughly with the datapoints from the CIRS virial mass function.
However, R2FGS has comparatively more clusters in the mass range M200
1014.9
,
1014.5-
h-1 M®.
As a result of the high-mass bins containing so many clusters, the low-mass bins
are essentially devoid of systems (entirely so in Figure 6-3). In the uniformly-corrected
mass function, one might suspect that many low-mass systems were simply trimmed
out of the sample in §4.4. However, a cursory inspection of the mass estimates in Table
2 of Appendix B demonstrates that this is not actually the case; most of the discarded
systems (marked with an asterisk in the table) have virial masses in the range which
already appears overpopulated in Figure 6-2, namely, M 2 00
-
1014.5 - 1014.9 h - 1 M®.
In the Lx-corrected mass function, the shortage of low-mass systems is even more
pronounced. Referring to Figure 5-2, this is likely a consequence of the R2FGS M 200 Lx relation (computed using the reduced sample) possessing a shallower slope than
the RASS-SDSS relations, which leads to larger mass estimates for low-luminosity
systems in the Lx-dependent correction of §6.2. Again, though, it does not appear
that I have discarded only low-mass systems in constructing the reduced sample;
inspection of Figure 5-2 shows that the discarded systems (open squares in the figure)
are distributed throughout the range of virial masses.
Overall, then, since both mass functions display this characteristic, it seems rather
likely that the dearth of low-mass systems is an actual property of the R2FGS sample.
This is extremely surprising given that the R2FGS sample was designed to investigate
low-mass groups and clusters in the nearby universe by selecting many systems with
fairly low X-ray luminosities. This result may be due to a relative overabundance
of massive clusters in the nearby universe (z < 0.06), or it may indicate a failure in
the general understanding of the M - Lx relation for low-luminosity systems. For
example, it is possible that the low-luminosity groups and clusters selected for the
R2FGS sample are not actually low-mass systems, but are instead moderate- or highmass systems that are faint in X-ray emissions for unknown reasons (e.g., they could
lack the necessary hot X-ray gas, or perhaps only a fraction of their X-ray emissions
reach our telescopes).
Nevertheless, before seriously considering any such explanations, this result requires further investigation and more thorough verification. A clear first step would
be to obtain X-ray temperature data for the R2FGS systems, which could then be
used to estimate their masses independently using a given M - Tx scaling relation;
alternatively, the R2FGS virial masses could be used to constrain this scaling relation.
More importantly, there are a multitude of systematic effects which could affect the
R2FGS results, such as:
1. Peculiar velocities of nearby groups and clusters (i.e., one needs very accurate
estimates of the actual distances to the R2FGS systems);
2. Velocity segregation between galaxies and dark matter in the R2FGS systems,
which could be significant in the centers of clusters [100, 101];
3. Differences in the spatial distribution of red and blue galaxies in clusters, which
could lead to overestimates for virial masses calculated using all member galaxies
instead of only red galaxies [102];
4. A possible absence of low surface-brightness X-ray systems in the R2FGS parent
catalogs (this hypothesis will be tested by the 2MASS Abell Cluster Survey [67],
which will analyze all nearby Abell clusters and reanalyze the RASS X-ray data);
5. Superposition of systems along the line of sight: because X-ray emission from
groups and clusters has a larger angular size, it is possible for emission from
multiple systems to be projected along a single line of sight;
6. Errors in the redshift identification of one or more systems due to X-ray emission
from a higher-redshift system being mistaken for emission from a galaxy group
with a faint intragroup medium;
7. Potentially, an excessively conservative method for interloper removal (the Vmax
scheme), which could lead to the inclusion of non-members in the mass estimate
calculations, and hence a systematic bias toward higher masses.
Unfortunately, a thorough treatment of the possible systematic errors in the R2FGS
mass function is beyond the scope of this thesis. For a discussion of potential systematic effects in the mass function of a similar survey, see Rines et al [28].
Chapter 7
Cosmological Constraints
I use the virial mass function computed with a luminosity-dependent correction in
§6.2 to constrain Qm and as. This choice is primarily influenced by the much smaller
uncertainties (except for the lowest-mass data point in Figure 6-3) and more consistent
shape of the Lx-corrected mass function compared to those of the uniformly-corrected
mass function. I minimize X2 for the Lx-corrected mass function in the mass range
loglo0 (M 2oo00)
= [13.9, 15.3] by calculating the Jenkins et al. mass function [94] for given
values of Qm and as8 and then shifting the mass scale from M180b to
M 200 .
Following
Sugiyama [103], I fix F (the shape parameter of the linear matter power spectrum)
according to:
F(Qm, h) = Qmh
2.K) exp
(-Qb
-
2
h-b
(7.1)
with To = 2.726 K, h = 0.7, and Qb =0.0223h 2 [98].
I also convolve the mass function with an assumed mass uncertainty alogo M
0.056 according to
dh(M)
1
dM
Vmax(M)
00 dn(M')Vma(M)
JOO
d
max'
(2U2ogM) - 1 / 2 exp
-(log M'- logA
M)2 dlog M, (7.2)
log M
where Vmax(M) is calculated using the Rines and Diaferio scaling relation [56]:
log(M 2ooh/M®) = 0.763log(Lx, 44 ) + 14.62,
(7.3)
where Lx, 44 is the X-ray luminosity in units of 1044 h - 2 ergs - 1 . Although this relation
has a different normalization than that found by Popesso et al. [84], it is consistent
with the relation of Reiprich and B6hringer [12]. The weighting by Vma., is necessary
because the R2FGS mass function is flux-limited for z < 0.06. Note that the volume
probed at smaller masses is quite small (Figure 6-1), so the mass function estimate in
that range is subject to large uncertainties due to cosmic variance [96]. In addition,
the number of systems in the low-mass bins is quite clearly insufficient to probe
the scatter in Lx. Thus, for the purpose of constraining cosmological parameters, I
neglect the mass bins below the vertical line in Figure 6-3.
Figure 7-1 shows the 68%, 95%, and 99.7% confidence levels for Qm and ua8 inferred
and o 8
from the R2FGS virial mass function. I find Qm = 0.26+07
-- u-0.08
=
-
+0.30
1.02
.015"
0..
-
To compute the constraints in Figure 7-1, I assumed h7 o = 1.0 and calculated F
from Eq. (7.1).
The two sets of solid contours in Figure 7-1 show the 68% and
95% confidence levels from WMAP3, while the larger dashed contours show these
levels for WMAP1. The dash-dotted contours are the same confidence levels from
CFHTLS [104]. Lastly, the smaller dashed contours are the 68%, 95%, and 99.7%
confidence levels from the CIRS virial mass function [28]. Note that the contours for
WMAP1 are adapted from Figure 1 of Spergel et al. [98], which displays the contours
in the Qmh 2 - a 8 plane. I thus increase the uncertainties in Qm to account for the
uncertainty in h.
The 68% confidence level of R2FGS overlaps the 68% confidence level of all four
of the other studies except for WMAP3, whose 95% confidence level just intersects
the 68% confidence level of R2FGS. However, note that the WMAP5 constraints (not
shown on the plot) lie significantly closer to the R2FGS constraints than do those
of WMAP3. To be specific, I find excellent agreement between the R2FGS value
of Qm = 0.26+0.07 and the WMAP5 value of Qm = 0.2708+0.15 and rough agreement
between the R2FGS value of as8 = 1.0201
and the WMAP5 value of
8=
0.796
.036
The large R2FGS value of as8 is due to the comparatively large number of systems
seen in the high-mass bins, which was discussed in the previous chapter. Further
investigation of possible systematic effects is necessary in order to more thoroughly
verify the R2FGS constraints.
I
*
I
'I
'
'
I'
1.5
1.0
I
=1---
```-----.,
-
0.5
0.0
0.0
I
I
I
I
0.2
I
I
0.4
I
0.6
I
0.8
1.0
Figure 7-1: Cosmological constraints from the Lx-corrected R2FGS virial mass function compared to other results. The shaded contours show 1 - 2 - 3a confidence
. The (larger) dashed and solid contours display the 68% and
levels for Q,, and 8su
95% confidence levels from WMAP1 and WMAP3, respectively. The dash-dotted
contours show the 68% and 95% confidence levels from the CFHTLS Wide survey;
the (smaller) dashed contours display the 1- 2- 3a confidence levels from the CIRS
virial mass function. Partially adapted from Figure 6 of Rines et al [28].
Chapter 8
Conclusions
I use an X-ray-selected sample of systems (R2FGS) from the ROSAT All-Sky Survey to measure the virial mass function of galaxy groups and clusters in the nearby
universe (z < 0.06). X-ray selection has the distinct advantage over optical selection
of yielding a well-defined selection function. I remove interlopers from the systems
using a dynamical maximum velocity criterion. I then estimate the group and cluster
masses from the dynamics of their member galaxies using the virial theorem. I verify
these masses using the projected mass estimator, and also define a "reduced" R2FGS
sample consisting of the systems for which the redshift-radius plots are reasonably
robust. Using this data, I investigate the op - Lx and M - Lx scaling relations,
for which I find results roughly consistent with those of the RASS-SDSS [84]. In
addition, I find the halo occupation function to be N 200
oC
M203+±0.0 4 , which agrees
extremely well with the results of Lin et al. [93], who measure the HOF in a similar
manner using 2MASS data, but with X-ray mass estimates.
Finally, I compute the group and cluster virial mass function using a uniform
correction and a luminosity-dependent correction. I compare the latter to predictions
of various cosmological models to constrain the parameters Qm and oUs. With the Lxcorrected R2FGS virial mass function (and assuming h70o= 1.0), I find Q,
and oa
1.0201
.
0.26
.07
While these results are intriguing, further work remains to be
done in order to prove that they are not altered by various possible systematic effects.
Furthermore, plans are currently underway to expand the R2FGS sample to 6 >
-20'
and z < 0.04. More data in this redshift range will greatly increase R2FGS's
ability to break the degeneracy between Qm and o-V-Vmax will increase by 50% for
the range of Lx corresponding to - 0.5 - 3 x 1014 M®. This new low-luminosity
data will also allow the survey to more accurately probe the scatter in the M - Lx
scaling relation, which is generally assumed to obey a log-normal distribution, but
may follow a different distribution.
Additionally, existing redshift data from the
6dF Galaxy Survey [105] will significantly decrease the number of targets needed to
be observed with the FAST spectrograph (compared to the targets in the northern
hemisphere). The extended survey should be completed by the end of the second
trimester of 2008.
Overall, the R2FGS sample demonstrates that clusters can probe cosmology, as
shown previously in studies such as CIRS [28]. I have collected a significant amount of
redshift data in order to measure the virial masses of the R2FGS systems, and these
results have yielded interesting constraints on fundamental cosmological parameters.
Furthermore, I expect these constraints to improve significantly in the near future.
Appendix A
Redshift-Radius Plots
0 0.5 11.5 2 00.5 11.5 2 00.5 11.5 2 00.5 11.5 2 0 0.5 11.5 2
0 0.5 11.5 2 00.5 11.5 2 00.5 11.5 2 00.5 11.5 2 0
RP(h-
1
Mpc)
Figure A-1: Redshift versus projected radius for galaxies in the first ten groups and
clusters in the R2FGS sample (ordered from left to right by increasing cluster redshift
starting with NGC1550). Interlopers are denoted by hollow triangles, while cluster
members are denoted by solid dots. The solid curved lines show the maximum velocity
profiles calculated for each system using Eq. (4.3). Figures A-2-A-7 show similar
plots for the rest of the sample.
0 0.5 11.5 2 00.5 11.5 2 00.5 11.5 2 00.5 11.5 2 00.5 11.5 2
0 0.5 11.5 2 0 0.5 11.5 2 0 0.5 11.5 2 0 0.5 11.5 2 00.5 11.5 2
RP(h-o Mpc)
Figure A-2: See Figure A-1.
0 0.5 11.5 2 00.5 11.5 2 00.5 11.5 2 00.5 11.5 2 00.5 11.5 2
0 0.5 1 1.5 2 00.5 1 1.5 2 00.5 1 1.5 2 0
R,(h-
1
Mpc)
Figure A-3: See Figure A-1.
64
2
0
1
2
3
0 0.5
1 1.5
2 0 0.5
1 1.5
2
0 0.5
1 1.5
2 0 0.5
1 1.5
2
0 0.5
1 1.5
2 0 0.5
1 1.5
2 0 0.5
1 1.5
2 0 0.5
1 1.5
2
Re(hA- Mpc)
Figure A-4: See Figure A-1.
0 0.5
1 1.5
2 0 0.5
1 1.5
2 0 0.5
1 1.5
2 0 0.5
1 1.5
2
0 0.5
1 1.5
2
1 1.5
2 0 0.5
1 1.5
2 0 0.5
1 1.5
2
m
C)
I
.4
0 0.5
R,(h- Mpc)
Figure A-5: See Figure A-1.
0 0.5 1 1.5
2 2(5 0.5
1 1.5
2 0 0.5
1 1.5
2 0 0.5
1 1.5
2
0 0.5
2 0 0.5
1 1.5
2 0 0.5
1 1.5
2 0 0.5
1 1.5
2
1 1.5
R (h-o Mpc)
Figure A-6: See Figure A-1.
0 0.5 1 1.5 2 2(B 0.5
1 1.5
2 0 0.5
1 1.5
2
0 0.5
1 1.5
2
0 0.5 1 1.5 2 2.5D 0.5
1 1.5
2 0 0.5
1 1.5
2 0 0.5
1 1.5
2
rn
0
C)
.9
RP(h-I Mpc)
Figure A-7: See Figure A-1.
68
Appendix B
Tables
Table B.1: R2FGS Groups and Clusters
System
X-ray Coordinates
ze
fx
erg cm-2 S- 1 )
Lx
(1044 erg s- 1)
RA (J2000)
DEC (J2000)
NGC1550
64.90751
2.41390
0.0131
40.10
0.35
A0262
28.19480
36.15130
0.0163
71.13
0.81
RXJ0123
20.92100
33.26100
0.0164
17.10
0.20
IV-Zw-038
16.86800
32.46200
0.0170
9.90
0.12
A189
21.35300
1.74100
0.0181
6.10
0.09
MKW4
181.10501
1.90060
0.0199
17.19
0.34
A1185
167.63571
28.71640
0.0220
5.26
0.11
MKW11
202.38361
11.78920
0.0220
5.70
0.12
UGC04052
117.84370
50.21250
0.0228
6.22
0.14
Coma
194.92940
27.93860
0.0231
305.89
7.01
A400
44.41200
6.00600
0.0238
16.20
0.40
NGC4325
185.77721
10.62400
0.0258
7.01
0.20
MKW8
220.15921
3.47650
0.0263
11.84
0.35
RXJ2214
333.69901
13.83400
0.0263
5.10
0.15
A1367
176.17902
19.81880
0.0276
45.83
1.51
(10-12
(continued on next page)
Table B.1 - continued from previous page
System
X-ray Coordinates
zO
fx
Lx
(10-12 ergcm-2 S- 1)
(10 44 erg s-1
0.0276
14.81
0.49
4.35600
0.0293
9.60
0.36
247.15820
39.54870
0.0299
97.92
3.77
IC1262
263.26071
43.76290
0.0307
11.55
0.47
A2634
354.60709
27.01260
0.0309
21.29
0.88
RXJO341
55.32060
15.40740
0.0311
17.42
0.73
NGC6107
244.36351
34.93670
0.0315
7.78
0.33
AWM4
241.23770
23.92060
0.0326
11.98
0.55
AWM5
254.50320
27.85440
0.0337
7.33
0.36
UGC03957
115.24760
55.43200
0.0340
17.60
0.88
2A0335
54.66990
9.97450
0.0347
80.91
4.21
RXJ0228
37.04010
28.19430
0.0353
5.62
0.30
A1142
165.20370
10.55960
0.0353
5.12
0.28
A2052
229.18341
7.01850
0.0353
47.79
2.58
A2147
240.57800
16.02000
0.0353
53.20
2.84
A2063
230.77240
8.60250
0.0355
35.48
1.94
Hercules
241.14890
17.72440
0.0370
16.46
0.98
A0576
110.34261
55.78640
0.0381
22.54
1.41
RXJ1205
181.30211
39.34930
0.0381
10.04
0.63
A76
10.00200
6.81800
0.0395
11.00
0.74
UGC12890
0.02950
8.27440
0.0396
5.39
0.37
A2657
356.23761
9.19800
0.0400
26.27
1.81
A2572
349.63232
18.70540
0.0403
8.21
0.58
A2107
234.90999
21.78900
0.0411
15.20
1.10
A2589
350.97299
16.80900
0.0416
25.30
1.88
Pegasus-II
347.60989
7.57690
0.0424
10.37
0.81
RA (J2000)
DEC (J2000)
NGC6338
258.84140
57.40740
Z1665
125.79800
A2199
(continued on next page)
Table B.1 - continued from previous page
System
X-ray Coordinates
zO
fx
Lx
RA (J2000)
DEC (J2000)
RXJ1740
265.13623
35.64750
0.0428
6.24
0.49
A2593
351.08398
14.65100
0.0428
14.80
1.17
MKW3s
230.45830
7.70880
0.0442
32.15
2.70
A1983
223.24489
16.69980
0.0444
5.64
0.48
A0168
18.80000
0.33000
0.0450
10.43
0.94
A407
45.45600
35.83500
0.0464
5.40
0.50
SHK352f
170.42799
2.81840
0.0468
5.60
0.53
Z8338
272.71002
49.92000
0.0473
8.20
0.79
II-Zw-108
318.52600
2.56900
0.0483
21.90
2.19
A0376
41.51080
36.88790
0.0488
13.20
1.36
A193
21.26600
8.68800
0.0491
15.40
1.59
IC1365
318.47449
2.55550
0.0494
13.49
1.42
Zw0959
150.66090
32.69950
0.0499
5.01
0.54
RXJ1811
272.75031
49.91100
0.0501
9.45
1.03
A0671
127.16209
30.42800
0.0503
7.08
0.78
A2665
357.71091
6.16110
0.0562
13.24
1.81
A2626
354.12610
21.14240
0.0565
11.22
1.55
A2256
255.95320
78.64429
0.0581
48.19
7.01
A2271
259.54242
78.02200
0.0584
5.07
0.75
A1991
223.63091
18.64200
0.0586
9.79
1.46
A2457
338.91922
1.47170
0.0594
11.14
1.78
(10-12 erg cm
-2
s- 1)
(1044 ergs -1)
Table B.2: R2FGS Systems: M 20 0 , MPM, MLx, 7 200,
System
M200
(1014
h-1 MO)
MPM
(1014 h01 M)
MLx
0 20 0
'r200
(1014 h; M®) (h1 Mpc)
0200
(km s- 1 )
NGC1550 (*)
6.09 ± 0.86
12.55 ± 2.93
2.72 ± 0.36
1.74 + 0.74
681+97
-68
A0262
2.71 ± 0.25
3.54 + 0.48
4.56 + 0.43
1.35 + 0.38
517 +-3239
RXJ0123
2.87 ± 0.30
3.93 + 0.63
2.09 ± 0.32
1.38 + 0.44
533+48
IV-Zw-038
2.37 ± 0.25
3.05 - 0.48
1.58 ± 0.28
1.27 ± 0.40
4
468+-33.
A189
1.74 ± 0.23
2.71 + 0.60
1.29 + 0.26
1.08 ± 0.44
446 +--42
MKW4
2.41 ± 0.23
2.50 - 0.35
2.61 + 0.35
1.31 ± 0.37
510+-3239
A1185 (*)
7.91 ± 0.68
11.81 - 1.42
2.32 + 0.34
1.92 ± 0.49
756 +-42
MKW11
1.26 ± 0.15
1.89 ± 0.34
1.62 + 0.29
1.04 ± 0.37
403+43
-33
UGCO4052
1.37 ± 0.18
1.86 ± 0.40
1.61 + 0.29
1.05 + 0.41
4245439
Coma
16.05 . 0.68
18.75 ± 0.97
15.86 + 1.72
2.45 ± 0.31
A400
4.44 . 0.44
7.43 + 1.09
983+27
983-25
63+53
3.09 + 0.37
1.50 ± 0.45
633_42
1.14
0.17
1.38 + 0.34
2.05 + 0.32
1.00 ± 0.44
361 +-37
MKW8
2.32 . 0.21
2.87 + 0.37
2.95 + 0.37
1.27 ± 0.34
461-27
RXJ2214
1.39 + 0.16
1.63 + 0.29
1.75 + 0.30
1.04 + 0.36
3974132
A1367
7.93 + 0.63
8.29 + 0.91
5.04 + 0.44
1.87 ± 0.45
79541
NGC6338
3.33 - 0.33
3.82 ± 0.56
3.68 + 0.39
1.42 ± 0.42
551+46
-37
Z1665 (*)
1.42 + 0.18
1.45 + 0.30
2.94 + 0.37
0.90 ± 0.35
447+56
NGC4325 (*)
-38
1
59
50
54
41+33
-32
75+48
4-41
72+30
732_2o
A2199
9.93 + 0.60
12.18 ± 0.95
11.20 + 0.97
2.05 + 0.37
IC1262
1.92 + 0.27
1.76 + 0.42
3.58 + 0.39
1.18 + 0.50
A2634
8.31 + 0.68
12.25 ± 1.42
4.78 + 0.43
1.97 ± 0.48
519752
782+49
RXJO341
5.72 + 0.69
10.32 ± 1.95
4.05 + 0.41
1.72 ± 0.62
688 +-57
NGC6107
4.39 * 0.40
5.82 ± 0.76
2.74 + 0.36
1.59 ± 0.43
573+42
AWM4 (*)
3.64 + 0.40
4.95 + 0.81
3.66 + 0.39
1.50 ± 0.49
618 +--545
AWM5
3.46 + 0.37
4.62 - 0.77
3.00 * 0.37
1.45 ± 0.47
51+48
UGC03957 (*)
7.06 + 0.80
11.84 ± 2.07
4.81 * 0.43
1.83 ± 0.62
739 -57
-27
-42
76
-34
8
-738
+ 75
(continued on next page)
Table B.2 - continued from previous page
System
M200
MPM
ML
7200
x
U 200
(1014 h~1 MO)
(1014 h-1 MO)
(1014 h1 Me)
(h1 Mpc)
2A0335
8.20 ± 1.05
11.61 ± 2.40
11.57 A 1.02
1.94 ± 0.75
733+89
RXJ0228
1.46 ± 0.21
1.67 - 0.41
2.59 A 0.35
1.02 ± 0.44
419+63
-43
A1142 (*)
6.23 + 0.66
8.55 ± 1.35
2.50 A 0.35
1.74 + 0.55
A2052
3.80 + 0.39
3.98 + 0.61
8.68 A 0.68
1.36 ± 0.42
625_43
-43
A2147 (*)
28.86 + 1.64
48.81
3.56
10.00 ± 0.82
2.99 ± 0.51
1129_34
A2063 (*)
9.64 ± 0.90
16.51 + 2.23
7.34 A 0.57
2.01 ± 0.56
898_15
Hercules (*)
9.43 ± 0.74
12.22 ± 1.34
4.94 ± 0.44
2.05 ± 0.48
789+47
78940
A0576
12.79 ± 0.93
14.99 ± 1.48
6.26 A 0.50
2.28 ± 0.50
850-39
RXJ1205
0.79 ± 0.13
0.78 + 0.23
3.88 ± 0.40
0.86 ± 0.42
308+53
-37
A76
2.72 ± 0.39
4.88 ± 1.15
4.38 A 0.42
1.35 + 0.57
494+72
(km s-
1)
7 65
63+63
693_49
-49
65+55
8
+46
-50
22+113
UGC12890 (*)
0.28 ± 0.07
0.24 ± 0.14
2.93 ± 0.37
0.58 ± 0.41
228_45
A2657
4.41 ± 0.55
6.88 + 1.38
7.19 ± 0.56
1.57 ± 0.59
A2572 (*)
4.52 ± 0.50
7.37 ± 1.25
3.65 ± 0.39
1.56 ± 0.52
721+85
7-163
549+53
A2107
3.16 ± 0.34
3.64 ± 0.59
5.50 A 0.46
1.41 ± 0.46
59 1+5
-43
A2589
6.74 ± 0.68
7.19 ± 1.07
7.33 ± 0.57
1.70 ± 0.51
8567258
Pegasus-II
6.52 + 0.73
9.63 ± 1.63
4.37 ± 0.42
1.80 ± 0.60
782+77
"-59
RXJ1740
3.83 + 0.53
4.33 ± 1.00
3.60 ± 0.39
1.52 + 0.64
560_55
A2593
5.14 ± 0.44
6.74 ± 0.81
5.45 A 0.46
1.60 + 0.41
656+43
MKW3s (*)
17.87 ± 1.93
44.15 A 7.14
8.88 A 0.70
2.55 + 0.83
-36
5
1126810
A1983
2.50 ± 0.27
3.32 A 0.56
3.44 A 0.39
1.31 ± 0.43
477+46
A0168
4.09 + 0.37
5.44 A 0.72
4.89 A 0.44
1.54 + 0.42
558+41
-34
A407
5.49 ± 0.58
9.28 A 1.48
3.47 A 0.39
1.71 ± 0.54
6666047
SHK352f (*)
4.79 + 0.51
7.08 A 1.12
3.80 A 0.40
1.60 ± 0.51
639+58
639-46
Z8338
5.54 ± 0.65
7.27 A 1.34
4.77 ± 0.43
1.66 ± 0.59
6386851
II-Zw-108
3.92 ± 0.47
4.97 A 0.96
8.14 A 0.63
1.48 ± 0.54
571+63
-748
A0376
7.96 ± 0.78
9.60 A 1.37
6.07 A 0.49
1.92 ± 0.56
769+62
-50
-41
50+79
82
-35
(continued on next page)
Table B.2 - continued from previous page
System
M200
(1014 h-1 Me)
MPM
(1014 h-o1
M®)
ML
x
(1014 h71 Me®)
r200
-1
(h Mpc)
"200
(km s- 1)
63+67
683_i2
A193
3.70 ± 0.41
5.87 + 1.00
6.66 + 0.52
1.46 + 0.49
IC1365 (*)
4.24 ± 0.50
6.28 + 1.16
6.18 ± 0.49
1.54 ± 0.55
572+6
Zw0959.6
1.63 ± 0.20
2.08 ± 0.39
3.67 ± 0.39
1.15 + 0.42
469+52
4-939
RXJ1811
5.88 + 0.69
7.39 + 1.36
5.17 + 0.45
1.71 + 0.60
A0671
6.71 ± 0.64
7.40 + 1.02
4.39 ± 0.42
1.80 + 0.51
638_51
793±61
-52
68+68
-50
72+1 13
A2665
11.24 + 1.67
13.69 ± 3.44
7.19 ± 0.56
2.06 + 0.92
A2626 (*)
7.92 + 0.96
10.20 ± 1.94
6.49 + 0.51
1.87 + 0.68
A2256
24.94 + 1.58
29.86 ± 2.47
15.63 + 1.68
2.85 ± 0.54
1208+544
A2271
0.23 ± 0.05
0.26 + 0.14
4.23 ± 0.41
0.13 + 0.09
382+167
u•72
A1991 (*)
5.06 ± 0.54
6.23 + 1.01
6.32 + 0.50
1.65 ± 0.53
61444
A2457
4.84 ± 0.53
6.24 + 1.04
6.82 ± 0.53
1.64 + 0.54
584+64
726_77
779_8
779±+87
-65
58443
Bibliography
[1] W. H. Press and P. Schechter. Formation of Galaxies and Clusters of Galaxies
by Self-Similar Gravitational Condensation. ApJ, 187:425-438, February 1974.
[2] J. P. Henry and K. A. Arnaud. A measurement of the mass fluctuation spectrum
from the cluster X-ray temperature function. ApJ, 372:410-418, May 1991.
[3] N. A. Bahcall and R. Cen. The mass function of clusters of galaxies. ApJL,
407:L49-L52, April 1993.
[4] Z. Haiman, J. J. Mohr, and G. P. Holder. Constraints on Cosmological Parameters from Future Galaxy Cluster Surveys. ApJ, 553:545-561, June 2001.
[5] W. Hu. Self-consistency and calibration of cluster number count surveys for
dark energy. Phys. Rev. D, 67(8):081304-+, April 2003.
[6] A. Vikhlinin, A. Voevodkin, C. R. Mullis, L. VanSpeybroeck, H. Quintana,
B. R. McNamara, I. Gioia, A. Hornstrup, J. P. Henry, W. R. Forman, and
C. Jones. Cosmological Constraints from the Evolution of the Cluster Baryon
Mass Function at z - 0.5. ApJ, 590:15-25, June 2003.
[7] J. P. Henry. X-Ray Temperatures for the Extended Medium-Sensitivity Survey
High-Redshift Cluster Sample: Constraints on Cosmology and the Dark Energy
Equation of State. ApJ, 609:603-616, July 2004.
[8] S. Majumdar and J. J. Mohr. Self-Calibration in Cluster Studies of Dark Energy: Combining the Cluster Redshift Distribution, the Power Spectrum, and
Mass Measurements. ApJ, 613:41-50, September 2004.
[9] A. Voevodkin and A. Vikhlinin. Constraining Amplitude and Slope of the Mass
Fluctuation Spectrum Using a Cluster Baryon Mass Function. ApJ, 601:610620, February 2004.
[10] A. Mantz, S. W. Allen, H. Ebeling, and D. Rapetti. New constraints on dark
energy from the observed growth of the most X-ray luminous galaxy clusters.
ArXiv e-prints, 709, September 2007.
[11] E. Pierpaoli, D. Scott, and M. White. Power-spectrum normalization from the
local abundance of rich clusters of galaxies. MNRAS, 325:77-88, July 2001.
[12] T. H. Reiprich and H. Bohringer. The Mass Function of an X-Ray Flux-limited
Sample of Galaxy Clusters. ApJ, 567:716-740, March 2002.
[13] U. Seljak. Cluster number density normalization from the observed masstemperature relation. MNRAS, 337:769-773, December 2002.
[14] P. T. P. Viana, R. C. Nichol, and A. R. Liddle. Constraining the Matter Power
Spectrum Normalization Using the Sloan Digital Sky Survey/ROSAT All-Sky
Survey and REFLEX Cluster Surveys. ApJL, 569:L75-L78, April 2002.
[15] E. Pierpaoli, S. Borgani, D. Scott, and M. White. On determining the cluster
abundance normalization. MNRAS, 342:163-175, June 2003.
[16] P. Schuecker, H. Bohringer, C. A. Collins, and L. Guzzo. The REFLEX galaxy
cluster survey. VII. m, and as from cluster abundance and large-scale clustering.
A&A, 398:867-877, February 2003.
[17] P. T. P. Viana, S. T. Kay, A. R. Liddle, O. Muanwong, and P. A. Thomas. The
power spectrum amplitude from clusters revisited: as using simulations with
pre-heating and cooling. MNRAS, 346:319-326, November 2003.
[18] N. A. Bahcall, F. Dong, P. Bode, R. Kim, J. Annis, T. A. McKay, S. Hansen,
J. Schroeder, J. Gunn, J. P. Ostriker, M. Postman, R. C. Nichol, C. Miller,
T. Goto, J. Brinkmann, G. R. Knapp, D. O. Lamb, D. P. Schneider, M. S.
Vogeley, and D. G. York. The Cluster Mass Function from Early Sloan Digital
Sky Survey Data: Cosmological Implications. ApJ, 585:182-190, March 2003.
[19] V. R. Eke, C. M. Baugh, S. Cole, C. S. Frenk, and J. F. Navarro. Galaxy groups
in the 2dF Galaxy Redshift Survey: the number density of groups. MNRAS,
370:1147-1158, August 2006.
[20] H. Dahle. The Cluster Mass Function from Weak Gravitational Lensing. ApJ,
653:954-962, December 2006.
[21] R. Massey, A. Refregier, D. J. Bacon, R. Ellis, and M. L. Brown. An enlarged
cosmic shear survey with the William Herschel Telescope. MNRAS, 359:12771286, June 2005.
[22] E. Semboloni, Y. Mellier, L. van Waerbeke, H. Hoekstra, I. Tereno, K. Benabed,
S. D. J. Gwyn, L. Fu, M. J. Hudson, R. Maoli, and L. C. Parker. Cosmic shear
analysis with CFHTLS deep data. A&A, 452:51-61, June 2006.
[23] J. Dunkley, E. Komatsu, M. R. Nolta, D. N. Spergel, D. Larson, G. Hinshaw,
L. Page, C. L. Bennett, B. Gold, N. Jarosik, J. L. Weiland, M. Halpern, R. S.
Hill, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack, and E. L.
Wright. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Likelihoods and Parameters from the WMAP data. ArXiv e-prints, 803,
March 2008.
[24] A. E. Evrard, J. Bialek, M. Busha, M. White, S. Habib, K. Heitmann, M. Warren, E. Rasia, G. Tormen, L. Moscardini, C. Power, A. R. Jenkins, L. Gao, C. S.
Frenk, V. Springel, S. D. M. White, and J. Diemand. Virial Scaling of Massive
Dark Matter Halos: Why Clusters Prefer a High Normalization Cosmology.
ApJ, 672:122-137, January 2008.
[25] M. Tegmark et al. Cosmological constraints from the SDSS luminous red galaxies. Phys. Rev. D, 74(12):123507-+, December 2006.
[26] E. Komatsu, J. Dunkley, M. R. Nolta, C. L. Bennett, B. Gold, G. Hinshaw,
N. Jarosik, D. Larson, M. Limon, L. Page, D. N. Spergel, M. Halpern, R. S.
Hill, A. Kogut, S. S. Meyer, G. S. Tucker, J. L. Weiland, E. Wollack, and
E. L. Wright. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP)
Observations: Cosmological Interpretation. ArXiv e-prints, 803, March 2008.
[27] K. Rines, A. Diaferio, and P. Natarajan. WMAP5 and the Cluster Mass Function. ApJL, 679:L1-L4, May 2008.
[28] K. Rines, A. Diaferio, and P. Natarajan. The Virial Mass Function of Nearby
SDSS Galaxy Clusters. ApJ, 657:183-196, March 2007.
[29] F. Zwicky. Die Rotverschiebung von extragalaktischen Nebeln. Helvetica Physica Acta, 6:110-127, 1933.
[30] F. Zwicky. On the Masses of Nebulae and of Clusters of Nebulae.
86(3):217-246, 1937.
ApJ,
[31] D. Fabricant, M. Lecar, and P. Gorenstein. X-ray measurements of the mass of
M87. ApJ, 241:552-560, October 1980.
[32] L. Benatov, K. Rines, P. Natarajan, A. Kravtsov, and D. Nagai. Galaxy orbits
and the intracluster gas temperature in clusters. MNRAS, 370:427-434, July
2006.
[33] A. C. Fabian, J. S. Sanders, G. B. Taylor, S. W. Allen, C. S. Crawford, R. M.
Johnstone, and K. Iwasawa. A very deep Chandra observation of the Perseus
cluster: shocks, ripples and conduction. MNRAS, 366:417-428, February 2006.
[34] C. A. Metzler, M. White, and C. Loken. The Effect of the Cosmic Web on
Cluster Weak Lensing Mass Estimates. ApJ, 547:560-573, February 2001.
[35] H. Hoekstra. A comparison of weak-lensing masses and X-ray properties of
galaxy clusters. MNRAS, 379:317-330, July 2007.
[36] P. Natarajan, A. Loeb, J.-P. Kneib, and I. Smail. Constraints on the Collisional
Nature of the Dark Matter from Gravitational Lensing in the Cluster A2218.
ApJL, 580:L17-L20, November 2002.
[37] M. Bradaý, M. Lombardi, and P. Schneider. Mass-sheet degeneracy: Fundamental limit on the cluster mass reconstruction from statistical (weak) lensing.
AeiA, 424:13-22, September 2004.
[38] A. A. Berlind, E. Kazin, M. R. Blanton, S. Pueblas, R. Scoccimarro, and D. W.
Hogg. The Clustering of Galaxy Groups: Dependence on Mass and Other
Properties. ArXiv Astrophysics e-prints, October 2006.
[39] R. Stanek, A. E. Evrard, H. Bohringer, P. Schuecker, and B. Nord. The X-Ray
Luminosity-Mass Relation for Local Clusters of Galaxies. ApJ, 648:956-968,
September 2006.
[40] S. W. Allen, R. W. Schmidt, A. C. Fabian, and H. Ebeling. Cosmological
constraints from the local X-ray luminosity function of the most X-ray-luminous
galaxy clusters. MNRAS, 342:287-298, June 2003.
[41] B. Nord, R. Stanek, E. Rasia, and A. E. Evrard. Effects of selection and
covariance on X-ray scaling relations of galaxy clusters. MNRAS, 383:L10-L14,
January 2008.
[42] S. Borgani, P. Rosati, P. Tozzi, S. A. Stanford, P. R. Eisenhardt, C. Lidman,
B. Holden, R. Della Ceca, C. Norman, and G. Squires. Measuring m, with the
ROSAT Deep Cluster Survey. ApJ, 561:13-21, November 2001.
[43] K. Shimasaku. Measuring the Density Fluctuation from the Cluster Gas Mass
Function. ApJ, 489:501-+, November 1997.
[44] E. Rasia, P. Mazzotta, S. Borgani, L. Moscardini, K. Dolag, G. Tormen, A. Diaferio, and G. Murante. Mismatch between X-Ray and Emission-weighted Temperatures in Galaxy Clusters: Cosmological Implications. ApJl, 618:L1-L4,
January 2005.
[45] A. Vikhlinin, A. Kravtsov, W. Forman, C. Jones, M. Markevitch, S. S. Murray,
and L. Van Speybroeck. Chandra Sample of Nearby Relaxed Galaxy Clusters:
Mass, Gas Fraction, and Mass-Temperature Relation. ApJ, 640:691-709, April
2006.
[46] C. Stoughton et al. Sloan Digital Sky Survey: Early Data Release. AJ, 123:485548, January 2002.
[47] M. White. The Mass Function. ApJS, 143:241-255, December 2002.
[48] M. Loewenstein. Heating of Intergalactic Gas and Cluster Scaling Relations.
ApJ, 532:17-27, March 2000.
[49] M. Colless et al. The 2dF Galaxy Redshift Survey: spectra and redshifts.
MNRAS, 328:1039-1063, December 2001.
[50] M. Girardi and G. Giuricin. The Observational Mass Function of Loose Galaxy
Groups. ApJ, 540:45-56, September 2000.
[51] H. J. Martinez, A. Zandivarez, M. E. Merchin, and M. J. L. Dominguez. Galaxy
groups in the 2dF Galaxy Redshift Survey: luminosity and mass statistics.
MNRAS, 337:1441-1449, December 2002.
[52] P. Heinamiaki, J. Einasto, M. Einasto, E. Saar, D. L. Tucker, and V. Miiller. The
mass function of the Las Campanas loose groups of galaxies. A &A, 397:63-70,
January 2003.
[53] J. Binney and S. Tremaine. Galactic dynamics. Princeton, NJ, Princeton University Press, 1987, 747 p., 1987.
[54] R. G. Carlberg, H. K. C. Yee, E. Ellingson, R. Abraham, P. Gravel, S. Morris,
and C. J. Pritchet. Galaxy Cluster Virial Masses and Omega. ApJ, 462:32-+,
May 1996.
[55] L. S. The and S. D. M. White. The mass of the Coma cluster. AJ, 92:1248-1253,
December 1986.
[56] K. Rines and A. Diaferio. CIRS: Cluster Infall Regions in the Sloan Digital
Sky Survey. I. Infall Patterns and Mass Profiles. AJ, 132:1275-1297, September
2006.
[57] W. Voges et al. The ROSAT all-sky survey bright source catalogue. A&A,
349:389-405, September 1999.
[58] M. F. Skrutskie et al. The Two Micron All Sky Survey (2MASS). A J, 131:11631183, February 2006.
[59] A. Cooray and R. Sheth. Halo models of large scale structure. Phys. Rep.,
372:1-129, December 2002.
[60] P. Schechter. An analytic expression for the luminosity function for galaxies.
ApJ, 203:297-306, January 1976.
[61] H. Ebeling, W. Voges, H. Bohringer, A. C. Edge, J. P. Huchra, and U. G. Briel.
Properties of the X-ray-brightest Abell-type clusters of galaxies (XBACs) from
ROSAT All-Sky Survey data - I. The sample. MNRAS, 281:799-829, August
1996.
[62] H. Ebeling, A. C. Edge, H. Bohringer, S. W. Allen, C. S. Crawford, A. C.
Fabian, W. Voges, and J. P. Huchra. The ROSAT Brightest Cluster Sample
- I. The compilation of the sample and the cluster log N-log S distribution.
MNRAS, 301:881-914, December 1998.
[63] H. Ebeling, A. C. Edge, S. W. Allen, C. S. Crawford, A. C. Fabian, and J. P.
Huchra. The ROSAT Brightest Cluster Sample - IV. The extended sample.
MNRAS, 318:333-340, October 2000.
[64] H. B6hringer et al. The Northern ROSAT All-Sky (NORAS) Galaxy Cluster
Survey. I. X-Ray Properties of Clusters Detected as Extended X-Ray Sources.
ApJS, 129:435-474, August 2000.
[65] H. B6hringer et al. The ROSAT-ESO flux limited X-ray (REFLEX) galaxy
cluster survey. I. The construction of the cluster sample. A &A, 369:826-850,
April 2001.
[66] M. Schmidt. Space Distribution and Luminosity Functions of Quasi-Stellar
Radio Sources. ApJ, 151:393-+, February 1968.
[67] M. Ramella et al. . In preparation,2008.
[68] J. K. Adelman-McCarthy et al. The Sixth Data Release of the Sloan Digital
Sky Survey. ApJS, 175:297-313, April 2008.
[69] D. Fabricant, P. Cheimets, N. Caldwell, and J. Geary. The FAST Spectrograph
for the Tillinghast Telescope. PASP, 110:79-85, January 1998.
[70] M. J. Kurtz and D. J. Mink. RVSAO 2.0: Digital Redshifts and Radial Velocities. PASP, 110:934-977, August 1998.
[71] B. J. Carter, D. G. Fabricant, M. J. Geller, M. J. Kurtz, and B. McLean. Star
Formation in a Complete Spectroscopic Survey of Galaxies. ApJ, 559:606-619,
October 2001.
[72] L. J. Kewley, R. A. Jansen, and M. J. Geller. Aperture Effects on Star Formation Rate, Metallicity, and Reddening. PASP, 117:227-244, March 2005.
[73] R. Wojtak, E. L. Lokas, G. A. Mamon, S. Gottl6ber, F. Prada, and M. Moles.
Interloper treatment in dynamical modelling of galaxy clusters. A &A, 466:437449, May 2007.
[74] R. den Hartog and P. Katgert. On the dynamics of the cores of galaxy clusters.
MNRAS, 279:349-388, March 1996.
[75] J. N. Bahcall and S. Tremaine. Methods for determining the masses of spherical
systems. I - Test particles around a point mass. ApJ, 244:805-819, March 1981.
[76] P. Mazzotta, A. C. Edge, and M. Markevitch. A Chandra Study of the Complex
Structure in the Core of 2A 0335+096. ApJ, 596:190-203, October 2003.
[77] P. Barmby and J. P. Huchra. Kinematics of the Hercules supercluster. A J,
115:6-+, January 1998.
[78] M. Girardi, S. Borgani, G. Giuricin, F. Mardirossian, and M. Mezzetti. The Observational Mass Function of Nearby Galaxy Clusters. ApJ, 506:45-52, October
1998.
[79] J. F. Navarro, C. S. Frenk, and S. D. M. White. A Universal Density Profile
from Hierarchical Clustering. ApJ, 490:493-+, December 1997.
[80] M. Markevitch, A. Vikhlinin, W. R. Forman, and C. L. Sarazin. Mass Profiles
of the Typical Relaxed Galaxy Clusters A2199 and A496. ApJ, 527:545-553,
December 1999.
[81] J. Heisler, S. Tremaine, and J. N. Bahcall. Estimating the masses of galaxy
groups - Alternatives to the virial theorem. ApJ, 298:8-17, November 1985.
[82] D. Fadda, M. Girardi, G. Giuricin, F. Mardirossian, and M. Mezzetti. The
Observational Distribution of Internal Velocity Dispersions in Nearby Galaxy
Clusters. ApJ, 473:670-+, December 1996.
[83] L. Danese, G. de Zotti, and G. di Tullio. On velocity dispersions of galaxies in
rich clusters. A&A, 82:322-327, February 1980.
[84] P. Popesso, A. Biviano, H. B6hringer, M. Romaniello, and W. Voges. RASSSDSS galaxy cluster survey. III. Scaling relations of galaxy clusters. ADA,
433:431-445, April 2005.
[85] A. J. Benson, S. Cole, C. S. Frenk, C. M. Baugh, and C. G. Lacey. The nature
of galaxy bias and clustering. MNRAS, 311:793-808, February 2000.
[86] A. A. Berlind and D. H. Weinberg. The Halo Occupation Distribution: Toward
an Empirical Determination of the Relation between Galaxies and Mass. ApJ,
575:587-616, August 2002.
[87] R. K. Sheth and A. Diaferio. Peculiar velocities of galaxies and clusters. MNRAS, 322:901-917, April 2001.
[88] A. A. Berlind, D. H. Weinberg, A. J. Benson, C. M. Baugh, S. Cole, R. Dav6,
C. S. Frenk, A. Jenkins, N. Katz, and C. G. Lacey. The Halo Occupation
Distribution and the Physics of Galaxy Formation. ApJ, 593:1-25, August
2003.
[89] V. Springel and L. Hernquist. The history of star formation in a A cold dark
matter universe. MNRAS, 339:312-334, February 2003.
[90] C. Marinoni and M. J. Hudson. The Mass-to-Light Function of Virialized Systems and the Relationship between Their Optical and X-Ray Properties. ApJ,
569:101-111, April 2002.
[91] A. Pisani, M. Ramella, and M. J. Geller. The Mass Function and Distribution of
Velocity Dispersions for UZC Groups of Galaxies. AJ, 126:1677-1689, October
2003.
[92] K. Rines, M. J. Geller, A. Diaferio, M. J. Kurtz, and T. H. Jarrett. CAIRNS:
The Cluster and Infall Region Nearby Survey. II. Environmental Dependence
of Infrared Mass-to-Light Ratios. AJ, 128:1078-1111, September 2004.
[93] Y.-T. Lin, J. J. Mohr, and S. A. Stanford. K-Band Properties of Galaxy Clusters
and Groups: Luminosity Function, Radial Distribution, and Halo Occupation
Number. ApJ, 610:745-761, August 2004.
[94] A. Jenkins, C. S. Frenk, S. D. M. White, J. M. Colberg, S. Cole, A. E. Evrard,
H. M. P. Couchman, and N. Yoshida. The mass function of dark matter haloes.
MNRAS, 321:372-384, February 2001.
[95] R. K. Sheth and G. Tormen. Large-scale bias and the peak background split.
MNRAS, 308:119-126, September 1999.
[96] W. Hu and A. V. Kravtsov. Sample Variance Considerations for Cluster Surveys. ApJ, 584:702-715, February 2003.
[97] D. N. Spergel et al. First-Year Wilkinson Microwave Anisotropy Probe
(WMAP) Observations: Determination of Cosmological Parameters. ApJS,
148:175-194, September 2003.
[98] D. N. Spergel, R. Bean, O. Dore, M. R. Nolta, C. L. Bennett, J. Dunkley, G. Hinshaw, N. Jarosik, E. Komatsu, L. Page, H. V. Peiris, L. Verde, M. Halpern, R. S.
Hill, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weiland,
E. Wollack, and E. L. Wright. Three-Year Wilkinson Microwave Anisotropy
Probe (WMAP) Observations: Implications for Cosmology. ApJS, 170:377408, June 2007.
[99] T. H. Reiprich. The galaxy cluster X-ray luminosity-gravitational mass relation
in the light of the WMAP 3rd year data. A&A, 453:L39-L42, July 2006.
[100] J. Diemand, B. Moore, and J. Stadel. Velocity and spatial biases in cold dark
matter subhalo distributions. MNRAS, 352:535-546, August 2004.
[101] A. Faltenbacher, A. V. Kravtsov, D. Nagai, and S. Gottlober. Supersonic motions of galaxies in clusters. MNRAS, 358:139-148, March 2005.
[102] A. Biviano, G. Murante, S. Borgani, A. Diaferio, K. Dolag, and M. Girardi.
On the efficiency and reliability of cluster mass estimates based on member
galaxies. A&IA, 456:23-36, September 2006.
[103] N. Sugiyama. Cosmic Background Anisotropies in Cold Dark Matter Cosmology. ApJS, 100:281-+, October 1995.
[104] H. Hoekstra, Y. Mellier, L. van Waerbeke, E. Semboloni, L. Fu, M. J. Hudson,
L. C. Parker, I. Tereno, and K. Benabed. First Cosmic Shear Results from the
Canada-France-Hawaii Telescope Wide Synoptic Legacy Survey. ApJ, 647:116127, August 2006.
[105] D. H. Jones et al. The 6dF Galaxy Survey: samples, observational techniques
and the first data release. MNRAS, 355:747-763, December 2004.