**4.6. Clusters of galaxies and the Large Scale Structure of the Universe**

Observations of large scale structure
indicate that the model which comes closest to
explaining most observational features of galaxy clustering is
CDM, a
model containing a cosmological constant in addition to baryons and cold
dark matter
[114,
144].
Parameters of this model which agree well with observations
are _{}
*h*^{2}
0.33, _{b}
0.02,
_{m}
0.3,
where h = *H*_{0}/100 is the Hubble parameter in units of
100 km/sec/Mpc. (Setting *h* = 0.7 gives
_{} = 0.68.)

There are several reasons as to why the presence of
improves the
performance of the standard cold dark matter model.
The first is related to the fact that in a spatially flat universe
linearized density perturbations grow at a slower rate in the presence of
than in its absence.
(The growth rate is however faster
than that in an open universe for identical values of
1 - _{m}.)
This changes the initial normalization of the density field since
the linearized gravitational potential now becomes time-dependent,
which affects the Sachs-Wolfe integral
discussed in section 4.4.
The slow down in the rate of growth also affects the abundance of very
massive objects (clusters and superclusters) some of which may have
formed only relatively recently and would therefore feel the presence of
long wavelength modes still in the linear regime. A small value of
_{m}
(alternatively, a large value of
_{} = 1 -
_{m})
also affects the matter power spectrum in
CDM models which
is strongly influenced by the epoch of matter radiation equality.
This effect is incorporated in the *shape parameter*
^{(9)}
=
_{m} *h*:
a small value of
_{m} leads to a larger value of the horizon at matter-radiation equality
*d*_{eq}
16 / ( *h*) Mpc
and hence to more long wavelength power in the fluctuation spectrum
*P*(*k*) =
<|_{k}|^{2}>.
Both open models and
CDM models show
better agreement with galaxy clustering data on large scales
[54],
the `best fit'
value of being
0.25.

An independent estimate of
_{m} is
provided by the peculiar velocities of galaxies in our neighborhood (on
scales ~ 10 - 100 h^{-1} Mpc).
The results of a joint estimate from velocity flows and supernovae gives the
most likely values
_{m}
0.5 and
_{}
0.8, thereby
favouring an approximately flat universe
[45].

A low value of
_{m} is also
indicated by studies of clusters of galaxies.
Clusters of galaxies have traditionally been powerful probes of cosmological
structure formation scenario's.
The masses of rich clusters can be estimated using three independent
methods: the velocity dispersion
of member galaxies, the cluster X-ray temperature due to hot intracluster
gas and strong gravitational lensing of background galaxies by the cluster.
All three methods provide an estimate of the cluster mass which ranges from
10^{14} to 10^{15} h^{-1}
M_{} for the mass
located within the central 1.5h^{-1} Mpc. region of a cluster
[6].
The resulting median mass-to-light ratio for rich clusters is
*M*/*L*_{B}
300 ± 100h
*M*_{} /
*L*_{}, which
when integrated over the full range of luminous matter in the universe
gives an estimate for the density parameter
_{m} = 0.2
± 0.1.

A low value of
_{m} is also
indicated by a study of baryonic matter within clusters.
In a detailed study of the composition of the Coma cluster which included
estimates of the baryonic mass fraction provided by X-ray emitting gas
and virial measurements of its total mass,
White et al (1993)
showed that the ratio of baryonic matter to total mass
_{b}
*h*^{3/2} /
_{m} = 0.07
± 0.03. As a result the baryonic mass fraction
greatly exceeds nucleosynthesis constraints
_{b}
*h*^{2} = 0.015 ± 0.005
if _{m} = 1,
leading to a `baryon catastrophe'. However no catastrophe occurs if
_{m}
*h*^{1/2} = 0.21 ± 0.12 since the value of
_{b} is now
small enough to be acceptable by nucleosynthesis constraints
[144].
This result therefore is strongly
supportive of either an open universe or one that is
dominated and flat,
so that
_{m} = 1 -
_{} << 1.

Observations of cluster abundances can be used to
provide good estimates of
_{8} -
the average root-mean-square mass fluctuation in a sphere of
radius 8h^{-1} Mpc. The best-fit value of
_{8} consistent with
present day cluster abundances is
_{8}
0.5
_{m}^{-0.5}.
This value gives a measure of the clustering amplitude on small scales
and therefore
can be used to normalize the power spectrum of density perturbations.
A complementary method of normalization is provided by large angle
CMB anisotropies measured by COBE. Taken together the
_{8} normalization
on small scales and the COBE normalization on large scales (~ 1000 Mpc.)
provide very useful constraints on the cosmological parameters
_{m},
_{},
_{B}, on the
biasing parameter
*b* = _{lum}
/ _{dark} and on
the `primordial tilt' in the power spectrum
|_{k}|^{2}
*k*^{n} which can be shown to lie in the range |1 - *n*|
0.2
[144]

A potentially powerful method for discriminating between different
cosmological models is provided by the abundance of rich clusters of
galaxies measured at high redshifts.
The presence of large amounts of X-ray emitting gas in many rich clusters
provides us with a useful observational tool with which
to probe cluster mass. Observations of galaxy clusters are then
matched against theoretical models which model cluster formation and
evolution
using either Press-Schechter techniques or N-body/Hydro-simulations
[145,
88,
61,
19,
41,
7,
58,
194].
As discussed earlier the growth of long wavelength
perturbations which are still in the linear regime, is significantly
slower in
low density models (both with and without a cosmological constant) than in a
critical density
_{m} = 1
universe. This leads to dramatic differences in
the redshift dependence of the rich cluster abundance in cosmological
models: rich clusters are much rarer at high redshifts in an
_{m} = 1
universe than they are in a low density
universe (see figure 11). For instance,
whereas almost all massive clusters with *M* ~ 10^{15}
*M*_{}
are expected to have formed by *z* ~ 0.5 in a low density universe,
only a small fraction (< 10%) of the present day 10^{15}
*M*_{}
clusters would have been in place by *z* ~ 0.5 in an
_{m} = 1
universe
[78,
194].
The existence of three massive clusters in the
redshift range *z* ~ 0.5 - 0.9
has therefore been viewed as a difficulty for
the standard cold dark matter model with
_{m} = 1
for which 10^{-3} rich clusters are expected at *z* > 0.5
[78,
7,
6].
It must be noted however that large uncertainties in
both the observational data (only a few very massive clusters have been
reliably observed at high *z*) and in our theoretical understanding
of rich clusters, makes it difficult at present to place unambiguous
constraints on the values
of _{m} and
_{}
[195].
It is hoped that better quality data from satellite
launches planned for the immediate future (XMM) and more accurate
modelling of large scale structure will improve the situation
significantly in the near future.

Constraints on the abundance of rich clusters also come from
*arcs* caused by the strong gravitational lensing of extended
background sources (galaxies, radio sources) by foreground clusters.
Since clusters act as gravitational lenses for background sources, the
larger number of clusters at early epochs in (i) open,
low _{m}
models, and (ii) flat, high
_{} models, relative to
(iii) flat
_{m} = 1
models leads to a greater abundance of arcs in
both (i) and (ii) relative to (iii).
An estimate by Bartelmann et al. (1998) based on numerical
simulations of large scale structure, has shown that
an order of magnitude more arcs are predicted in flat models with
_{m}
0.3,
_{}
0.7
(_{arcs} ~
280) than in the flat
_{m} = 1
model (_{arcs}
~ 36). In open models this effect is even more dramatic
(_{arcs} ~
2400 _{m}
0.3,
_{} = 0).
However, impressive as these results are,
the absence of a comprehensive data base for arcs
and uncertainties in the modelling of galaxy clusters
makes it difficult to attempt to constrain theoretical models
on the basis of observations at present. (Bartelmann et al. (1998)
however make a case for
a low density universe by arguing that the observed number of arcs in the
EMSS arc survey extrapolated to the full sky is 1500 - 2300, which is
close to
what one observes for low density models in their numerical simulations.)
Both observational data sets and the theoretical modelling of clusters
are likely to improve significantly in the near future
giving this method potentially great importance in the ongoing
`quest for '.

Finally, the Lyman-
forest which populates the spectra of quasars
provides a potentially powerful means of discriminating between rival models
of structure formation and in probing the presence of a cosmological
-term
at intermediate redshifts
0 *z*
5
[100,
206].

^{9} The
shape parameter is so named because it affects the shape of the Power spectrum
*P*(*k*), which interpolates between the asymptotic regimes
Einstein is quoted as saying
[170]
*P*(*k*)
*k* for
*k* 0 and
*P*(*k*)
*k*^{-3} log^{2}*k* for
*k*
. The maximum value
of *P*(*k*) occurs near
*k* ~ *d*_{eq}^{-1}.
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