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The observed mass function (MF), n( > M), of clusters of galaxies, which describes the number density of clusters above a threshold mass M, can be used as a critical test of theories of structure formation in the universe. The richest, most massive clusters are thought to form from rare high peaks in the initial mass-density fluctuations; poorer clusters and groups form from smaller, more common fluctuations. Bahcall and Cen (1993) determined the MF of clusters of galaxies using both optical and X-ray observations of clusters. Their MF is presented in Figure 5. The function is well fit by the analytic expression

Equation 50 (50)

with M* = (1.8 ± 0.3) × 1014h-1 Modot, (where the mass M represents the cluster mass within 1.5h-1 Mpc radius).

Figure 5

Figure 5. Cluster mass functions from observations and from CDM dimulations (Bahcall and Cen 1992).

The observed cluster mass function is compared in Figure 5 with expectations from different cold-dark-matter cosmologies using large-scale simulations (Bahcall and Cen 1992). The comparison shows that the cluster MF is indeed a powerful discriminant among models. The standard CDM model (Omegam = 1) cannot reproduce the observed MF for any bias parameter; when normalized to the COBE microwave background fluctuations on large scales, this model produces too many massive clusters, unseen by the observations. A low-density CDM model on the other hand, with Omegam ~ 0.2-0.3 (with or without a cosmological constant), appears to fit well the observed cluster MF (Fig. 5).

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