1
Cosmology with clusters of galaxies
Neta A. Bahcall
Princeton University Observatory, Princeton, NJ 08544, USA
ABSTRACT
Rich clusters of galaxies, the largest virialized systems known, place some of the most powerful constraints on
cosmology. I discuss below the use of clusters of galaxies in determining two fundamental properties of the
universe: the mass-density of the universe, and the distribution of dark matter. I show that several independent
methods utilizing clusters of galaxies -- cluster dynamics and the mass-to-light function, baryon fraction in
clusters, and cluster abundance and its evolution -- all indicate the same result: a universe with a low massdensity, ~20% of the critical density needed to halt the universal expansion, and a mass distribution that
approximately traces light on large scales.
1. INTRODUCTION
2. MASS-TO-LIGHT RATIO
Rich clusters of galaxies -- families of hundreds of
galaxies held together by the gravitational potential
of the cluster -- are the most massive bound systems
known. As such, they serve as powerful tools in
constraining the cosmological model of the universe.
Cluster masses can be directly determined using
three independent methods:
The simplest argument for a low-density universe
involves the mass-to-light method. The masses of
rich clusters of galaxies range from ~10 14 to 1015 h-1
Mo within 1.5 h-1 Mpc radius of the cluster center
(where h = Ho/100 km/s/Mpc denotes Hubble's
constant, representing the expansion rate of the
universe). When normalized by the cluster
luminosity, a median mass-to-light ratio of M/LB ≈
300 +- 100h in solar units (Mo/Lo) is observed for
rich clusters, nearly independent of the cluster
luminosity, velocity dispersion, or other parameters
[3,11]. (LB is the total luminosity of the cluster in the
blue band, corrected for internal and galactic
absorption.) When integrated over the entire
observed luminosity density of the universe, this
mass-to-light ratio yields a mass density of 0.4x10-29
h2 g/cm3, or only ~20% of the critical density needed
to halt the universal expansion. The inferred density
assumes that all galaxies exhibit the same high M/L
ratio as clusters, and that mass follows light on large
scales. Thus, even if all galaxies have as much mass
per unit luminosity as do massive clusters, the total
mass of the universe is only ~20% of the critical
density. For a critical density universe, most of the
1) the motion (velocity dispersion) of galaxies within
clusters reflects the dynamical cluster mass, within
a given radius, assuming the clusters are in
hydrostatic equilibrium [1-3];
2) the temperature of the hot intracluster gas, like the
galaxy motion, traces the cluster mass [4-6];
3) gravitational lensing distortions of background
galaxies provide a direct measure of the intervening
cluster mass that causes the distortions [7-10].
All three independent methods yield consistent
cluster masses (typically within radii of ~1 Mpc),
indicating that we can reliably determine cluster
masses (within the observed scatter of ~30%).
2
Figure 1. The dependence of mass-to-light ratio,
M/LB, on scale, R, for average spiral galaxies (spiral
symbols), elliptical galaxies (ellipses), and groups
and clusters (filled circles). (From Bahcall, Lubin
and Dorman 1995 [11]). A flattening of the mass-tolight function is suggested at Ωm ~ 0.2 (see also
[14]).
mass of the universe has to be unassociated with
galaxies (i.e., with light); on large scales, the mass
has to reside in 'voids', where there is no light. This
would imply a large bias in the distribution of mass
versus light, with mass distributed considerably more
diffusely than light. Is there a strong bias in the
universe, with most of the dark matter residing on
large scales, well beyond galaxies and clusters of
galaxies? Analysis of the mass-to-light ratio of
galaxies, groups, and clusters by Bahcall, Lubin and
Dorman [11] shows that there is not a large bias.
They show that the M/L ratio of galaxies
increases with scale up to radii of ~ 0.2h-1 Mpc,
due to very large dark halos around galaxies (see also
[12,13]). The M/L ratio, however, flattens and
remain approximately constant for groups and rich
clusters on larger scales, to at least 1.5h-1 Mpc and
even beyond (Fig.1). The flattening occurs at M/LB ≈
250h, corresponding to a universal mass-density of
~20%. (An M/LB ≈ 1350h is needed for a critical
density universe.) This observation contradicts the
classical belief that the relative amount of dark
matter increases continuously with scale, possibly
reaching the critical density on large scales. The data
suggest that most of the dark matter is associated
with very large dark halos of galaxies and that
clusters do not contain a substantial amount of
additional dark matter, other than that associated
with (or torn-off from) the galaxy halos, plus the hot
intracluster gas. This flattening of M/L with scale,
recently confirmed by new larger-scale observations
using gravitational lensing, suggests that the relative
amount of dark matter does not increase significantly
with scale on large scales. This directly indicates the
'end' of the increasing dark matter contribution, and
yields a low-density universe, with only ~20% of the
critical density, and with no significant bias -- i.e.,
mass approximately follows light on large scales.
Comparison with cosmological simulations reveals a
nice agreement with these conclusions [14].
3. BARYON FRACTION IN CLUSTERS
Clusters contain many baryons, observed as gas
and stars. Within 1.5h-1 Mpc of a rich cluster of
galaxies, the X-ray emitting gas contributes ~7h-1.5%
of the cluster virial mass [15,16]. Stars contribute
another ~3%. The baryon fraction observed in
clusters is thus:
Ωb / Ωm ≥ 0.07h-1.5 + 0.03,
(1)
representing the ratio of cluster mass in baryons to
the total cluster mass. The Ωb and Ωm parameters
reflect their universal mass-density values (in units
of the critical density), since no significant
segregation between baryons and dark matter is
expected on these large scales. Standard big bang
nucleosynthesis and the cosmic microwave
background anisotropy spectrum limit the baryon
density of the universe to [17,18]:
3
Ωb ≈ 0.02h-2.
(2)
observed at these redshifts than expected for Ω m=1.
The results yield Ωm=0.3+- 0.1.
These facts suggest that the baryon fraction observed
in rich clusters (eq.1) exceeds that of a critical
density universe (Ωm = 1) by a factor of ~4 (for h ~
0.7). Since detailed hydrodynamic simulations [15]
show that baryons do not segregate into rich clusters,
the above results imply that the mean density of the
universe is lower than the critical density by a factor
of four. Thus, the observed high baryonic mass
fraction in clusters (eq.1), combined with the
nucleosynthesis limit (eq.2), suggest (for h = 0.7+0.1)
Ωm ≤ 0.27 +- 0.05.
4.
CLUSTER
EVOLUTION
(3)
ABUNDANCE
AND
ITS
Bahcall, Fan and Cen [19,20] show that the
evolution of the number density of clusters as a
function of cosmic time (redshift) provides a
powerful constraint on the mass density of the
universe [19-22]. The growth of high-mass clusters
from initial Gaussian fluctuations depends strongly
on the cosmological parameters Ωm and σ8 (where σ8
is the root-mean-square mass fluctuation on 8h-1 Mpc
scale [23-27]). In low-density models, density
fluctuations evolve and freeze out at early times, thus
producing only relatively little evolution at recent
times (z < 1). In a critical density universe, the
fluctuations start growing more recently thereby
producing strong evolution in recent times; a large
increase in the abundance of massive clusters is
expected from z ~ 1 to z ~ 0. The evolution is so
strong in high-density models that finding even a few
Coma-like clusters at z > 0.5 over ~103 deg2 of sky
contradicts an Ωm=1 model where only ~10-2 such
clusters would be expected (when normalized to the
observed present-day cluster abundance). The
evolution of the number density of Coma-like
clusters was determined from observations and
compared with cosmological simulations [19-21].
The data show only a slow evolution of the cluster
abundance to z ~ 0.5, with ~100 times more clusters
Figure 2. Observed versus expected evolution of the
number density of massive clusters as a function of
redshift (for clusters with mass ≥ 8x1014 h-1 Mo
within a comoving radius of 1.5 h-1 Mpc). From
Bahcall and Fan 1998 [29]. The expected evolution
is presented for different mass-density parameters by
the different curves. The data points show only a
slow evolution in the cluster abundance, consistent
with Ωm ≈ 0.2 (+-0.1). Higher density models predict
considerably fewer clusters than observed at z ~ 0.51 (see also [33]).
The evolutionary effects increase with cluster mass
and with redshift. The existence of several very
massive clusters, observed so far at z=0.5-0.9, places
the strongest constraint yet on Ωm and σ8. These
clusters are nearly twice as massive as the Coma
cluster, and they have reliably measured masses
(including
gravitational
lensing
masses,
temperatures, and velocity dispersions [3,9,29-32]).
4
These clusters possess the highest masses, the
highest velocity dispersions (≥ 1200 km/s), and the
highest temperatures (≥ 8 kev) in the z>0.5 EMSS
survey; [28]). The existence of these massive distant
clusters yields a mass-density parameter of Ωm = 0.2
+- 0.1 and σ8 ≈ 1 +- 0.1 (See Bahcall et al [29,33]).
Figure 2 presents a comparison between the
observed versus expected evolution of the number
density of such massive clusters for different values
of the cosmological parameters. The difference
between the cosmological models is dramatic in their
prediction of the evolution of these high-mass
clusters, and it thus provides a powerful constraint on
the cosmological models [29,33].
5. SUMMARY
We have shown that several independent
observations of clusters of galaxies all indicate that
the mass-density of the universe is low: Ωm ≈ 0.2 +0.1. These include:
1. The mass-to-light ratio of clusters of galaxies and
the flattening of the mass-to-light function of
galaxies, clusters and superclusters on large scales
suggest Ωm ~ 0.2.
2. The high baryon fraction observed in clusters of
galaxies suggests Ωm ≤ 0.27 +- 0.05.
3. The weak evolution of the observed cluster
abundance to z ~ 1 indicates Ωm = 0.2 +- 0.1, valid
for any Gaussian models.
4. All the above-described independent measures are
consistent with each other and indicate a low-density
universe with ~20% of the critical density. These
results have been recently supported by observations
of the fluctuation spectrum of the cosmic microwave
background radiation, which yields Ωm = 0.27 +0.07 [18,34] and the distant supernovae observations,
which yield Ωm = 0.28 +- 0.05 for a flat universe
[35].
The independent cluster observations indicate that
we live in a lightweight universe with only ~20% 30% of the critical density. The universe will thus
expand forever.
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