Annu. Rev. Astron. Astrophys. 1998. 36: 599-654
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Cosmological simulations have numerous applications besides testing of structure formation models. This section presents some of the applications that have become active areas of research. Space limitations preclude discussing many other valuable and often ingenious uses of simulations.

5.1. Clusters of Galaxies

Rich clusters of galaxies are the most massive virialized objects in the Universe. At the same time, they are sufficiently young, and the relevant physics sufficiently simple (compared, for example, with galaxy formation), so that present-day simulations are effective in exploiting them as a probe of initial conditions. A great many papers have applied structure-formation simulations to clusters of galaxies because of their power to constrain cosmological parameters, including Omega, Omegab, and sigma8 (the power spectrum normalization).

Numerical simulations of galaxy clusters have been used for the following goals, among others: (a) understanding the general processes of formation and evolution of single clusters (galaxy evolution and dynamics, intracluster medium); (b) testing observational methods of mass estimation (for both dark matter and baryons); (c) using the distribution of X-ray temperature (or luminosity, mass, or cluster velocity dispersion) and its evolution to constrain cosmological parameters; and (d) using substructure, morphology, shape, or radial profile to constrain cosmological parameters.

The first cosmological simulations of cluster formation with gas were performed by Evrard (1990), Thomas & Couchman (1992), both of whom combined P3M for gravity with SPH for gas dynamics. These and later simulations (Frenk et al 1996 and references therein) have explored several issues, including the equilibrium and distribution of the hot intracluster gas, the cluster response to mergers, and, in simulations with radiative cooling (Katz & White 1993, Frenk et al 1996), the survival of dissipatively condensed galaxies within dense cluster cores. Figure 3 shows maps of a simulated X-ray cluster at several epochs.

Figure 3

Figure 3. Evolution of an X-ray cluster in the standard cold dark matter (CDM) model. Columns from left to right show the projected dark matter density, projected baryon density, emission-weighted temperature, and predicted ROSAT X-ray surface brightness. Rows from top to bottom show the cluster at redshifts z = 0.7, 0.3, 0.1, and 0.03, respectively. From Frenk et al (1996).

Observers and theorists have devoted much attention to the ratio of velocity dispersions of galaxies and gas, beta ident sigma2 / (kT / µ mp), where sigma is the one-dimensional velocity dispersion of galaxies in the cluster, T is the gas temperature, µ is the mean molecular weight, and mp is the proton mass. (Note that this is a different use of the symbol beta than in Section 3.4.) Direct measurement of this quantity in high-resolution simulations (e.g. Navarro et al 1995) yields beta approx 1, as expected for gas and galaxies that have fallen through the same potential. Values estimated from fitting the X-ray surface-brightness distribution yield underestimates by a variety of effects (Evrard 1990, Bahcall & Lubin 1994, Navarro et al 1995).

The reliability of cluster mass estimates based on X-ray observations of the hot gas when assuming hydrostatic equilibrium has been examined by several groups, including Tsai et al (1994), Evrard et al (1996), Bartelmann & Steinmetz (1996). Their papers show that the reliability of the deduced masses depends somewhat on how the X-ray surface-brightness profiles are fit but that accuracies of better than 25% are readily achievable. Bartelmann & Steinmetz (1996), Cen (1997a) also argued that cluster projection effects systematically bias the ratio of masses estimated from X-ray data and gravitational lenses below unity (Bartelmann et al 1996 and references therein).

X-ray measurements of intracluster gas naturally provide an estimate of the gas density as well as the total binding mass. Including the relatively small contribution to baryons made by galaxies in luminous X-ray clusters, the baryon fraction of the mass is found to be

Equation 12 (12)

(White et al 1993b;, Evrard 1997 and references therein). Simulations show that the baryon fraction in clusters is expected to vary little from the cosmic mean value (White et al 1993b, Cen & Ostriker 1994a). As White et al noted in the title of their paper ("The baryon content of galaxy clusters - a challenge to cosmological orthodoxy"), Equation 12 represents a challenge to cosmological orthodoxy, which favors Omega = 1, 0.5 < h < 0.9, and Omegab approx 0.015 h-2 from standard Big Bang nucleosynthesis (Copi et al 1995). Although a recent measurement of the deuterium abundance in QSO absorption lines implies the higher value Omegab approx 0.019 h-2 (Tytler et al 1996; S Burles & D Tytler, unpublished manuscript, astro-ph/9712109), even this value from Big Bang nucleosynthesis is incompatible with Equation 12 if Omega = 1. However, the measured baryon fraction is perfectly compatible for an open universe or one with a cosmological constant, lending further support to the LCDM and OCDM models.

Rich clusters are rare objects corresponding to high-density peaks in the initial conditions. Their abundance is therefore highly sensitive to the normalization of the power spectrum. Because the virialized mass of rich clusters roughly equals the mass within a sphere of radius 8 h-1 Mpc at the cosmic mean density, the cluster abundance and its evolution with redshift therefore provide a strong constraint on sigma8 (Evrard 1989, Bahcall & Cen 1992, White et al 1993a). The mean mass density parameter Omega enters the argument directly through the mass within the sphere; it enters indirectly through the scaling of fluctuations from the linear regime (implicit in sigma8) to the nonlinear regime of virialized clusters.

Conceptually, this cluster abundance test compares predicted and measured distributions of cluster masses (although X-ray luminosity or velocity dispersion may be used instead). Eke et al (1996) provided a comprehensive analysis leading to the conclusion

Equation 13 (13)

(Their exponent on Omega actually differs slightly from -1/2, and it depends weakly on Omega and Lambda.) Fan et al (1997) have shown recently that the rate of evolution of the cluster abundance depends on sigma8 but is insensitive to Omega, enabling the degeneracy between these parameters to be broken. They obtained sigma8 = 0.83 ± 0.15 and Omega = 0.3 ± 0.1. These results are exciting, but a prudent observer may wish to wait for confirmation that we really know the cosmological parameters this well.

A different test of Omega was proposed by Richstone et al (1992), who noted that the presence of substructure in clusters argues that they are dynamically young. Because the growth of clustering slows greatly when Omega << 1, if clusters indeed formed recently, this would favor a high Omega. This argument is qualitative, as there is neither a perfect measure of substructure nor a unique relation between substructure and age. Nevertheless, it has inspired much attention from simulators, beginning with Evrard et al (1993), who confirmed the qualitative connection between cluster morphology and Omega. Several groups have tested a range of statistical measures of substructure (with either galaxy counts or X-ray data) using simulations in an attempt to make a quantitative and robust test (e.g. Dutta 1995, Crone et al 1996, Buote & Tsai 1995, Pinkney et al 1996). The latest results appear to favor a low-density universe with Omega approx 0.35 (Buote & Xu 1997). In related work, Wilson et al (1996b) showed that reconstructions of cluster mass distributions by using weak gravitational lensing inversion should provide enough substructure information to allow a test of Omega.

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