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We present a new determination of the cluster mass function and velocity dispersion function in a volume ~107 h3 Mpc-3 using data from the Fourth Data Release of the Sloan Digital Sky Survey (SDSS) to determine virial masses. We use the caustic technique to remove foreground and background galaxies. The cluster virial mass function agrees well with recent estimates from both X-ray observations and cluster richnesses. Our determination of the mass function lies between those predicted by the 1 and 3 year WMAP data. We constrain the cosmological parameters Om and s8 and find good agreement with WMAP and constraints from other techniques. With the CIRS mass function alone, we estimate Om = 0.24img1.gif and s8 = 0.92img2.gif, or s8 = 0.84 +-0.03 when holding Om = 0.3 fixed. We also use the WMAP parameters as priors and constrain velocity segregation in clusters. Using the 1 and 3 year results, we infer velocity segregation of sgxy/sDM [?] 0.94 +- 0.05 or 1.28 +- 0.06, respectively. The good agreement of various estimates of the cluster mass function shows that it is a useful independent constraint on estimates of cosmological parameters. We compare the velocity dispersion function of clusters to that of early-type galaxies and conclude that clusters comprise the high-velocity end of the velocity dispersion function of dark matter halos. Future studies of galaxy groups are needed to study the transition between dark matter halos containing individual galaxies and those containing systems of galaxies. The evolution of cluster abundances provides constraints on dark energy models; the mass function presented here offers an important low-redshift calibration benchmark.

In hierarchical clustering, galaxy clusters accrete mass through the aggregation of smaller systems. Thus, the velocity field of the infall regions of clusters contains significant random motion superposed on radial infall. Because the purely spherical infall model does not predict the amplitude of the velocity field correctly, methods estimating the cosmological density parameter O0 based on this model yield unreliable biased results. In fact, the amplitude of the velocity field depends on local dynamics and only very weakly on the global properties of the universe. We use N-body simulations of flat and open universes to show that the amplitude of the velocity field of the infall regions of dark matter halos is a direct measure of the escape velocity within these regions. We can use this amplitude to estimate the mass of dark matter halos within a few megaparsecs from the halo center. In this region dynamical equilibrium assumptions do not hold. The method yields a mass estimate with better than 30% accuracy. If galaxies trace the velocity field of the infall regions of clusters reliably, this method provides a straightforward way to estimate the amount of mass surrounding rich galaxy clusters from redshift data alone.

Using high‐resolution cosmological N‐body simulations, we investigate the survival of dark matter satellites falling into larger haloes. Satellites preserve their identity for some time after merging. We compute their loss of mass, energy and angular momentum as they are dissolved by dynamical friction, tidal forces and collisions with other satellites. We also analyse the evolution of their internal structure. Satellites with less than a few per cent of the mass of the main halo may survive for several billion years, whereas larger satellites rapidly sink into the centre of the main halo potential well and lose their identity. Penetrating encounters between satellites are frequent and may lead to significant mass loss and disruption. Only a minor fraction of cluster mass (10–15 per cent on average) is bound to substructure at most redshifts of interest. We discuss the application of these results to the survival and extent of dark matter haloes associated with galaxies in clusters, and to their interactions. We find that a minor fraction of galaxy‐size dark matter haloes are disrupted by redshift z= 0. The fraction of satellites undergoing close encounters is similar to the observed fraction of interacting or merging galaxies in clusters at moderate redshift.