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 h2 0.33, b 0.02, m 0.3, where h = H0/100 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 deq 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 , 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 .
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 1014 to 1015 h-1 M for the mass located within the central 1.5h-1 Mpc. region of a cluster . The resulting median mass-to-light ratio for rich clusters is M/LB 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 h3/2 / m = 0.07 ± 0.03. As a result the baryonic mass fraction greatly exceeds nucleosynthesis constraints b h2 = 0.015 ± 0.005 if m = 1, leading to a `baryon catastrophe'. However no catastrophe occurs if m h1/2 = 0.21 ± 0.12 since the value of b is now small enough to be acceptable by nucleosynthesis constraints . 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 kn which can be shown to lie in the range |1 - n| 0.2 
Figure 11. The observed and expected cluster abundance is shown as a function of redshift for massive clusters with Mcl 8 × 1014 M located within the Abell radius of 1.5 h-1 Mpc. The curves show the expected cluster abundances in CDM models with different m. Figure courtesy of Neta Bahcall (1999).
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 ~ 1015 M are expected to have formed by z ~ 0.5 in a low density universe, only a small fraction (< 10%) of the present day 1015 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 . 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  P(k) k for k 0 and P(k) k-3 log2k for k . The maximum value of P(k) occurs near k ~ deq-1. Back.