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3.2.2. Evolution of the Rich Cluster Abundance

The cluster abundance at z approx 0 requires the rms mass fluctuation sigma8 = <(deltaM/M)>1/2|r = 8h-1 Mpc to satisfy (White et al. 1993a; Eke et al. 1996; Pen 1998; Viana & Liddle 1999; see also Henry & Arnaud 1991)

Equation 13 (13)

The evolution of the cluster abundence is sensitive to sigma8 in early epochs of growth for a given mass; it is z gtapprox 0.3 for rich clusters. The rich cluster abundance at z ~ 0.3-1, when compared with that at a low z, determines both sigma8 and Omega (Oukbir & Blanchard 1992). Carlberg et al. (1997b) derived Omega = 0.4 ± 0.2, and Bahcall & Fan (1998) obtained Omega = 0.2+0.3-0.1 corresponding to a slow growth of the abundance. On the other hand, Blanchard & Bartlett (1998) obtained Omega appeq 1 from a more rapid growth. A high value is also claimed by Reichart et al. (1999), while Eke et al. (1998) reported Omega = 0.43 ± 0.25 for an open, and Omega = 0.36 ± 0.25 for a flat universe.

The controversy among authors arises from different estimates of the cluster mass at high z. This is a subtle effect, since the mass varies little over the range of relevant redshift, while the cluster number density evolution is sufficiently rapid at fixed mass (Pen 1998). At low z we have an established mass temperature relation, and the cluster mass is securely estimated (Henry & Arnaud 1991). At high z, however, such direct information is not available. Blanchard & Bartlett and Eke et al. used mass temperature relations as a function of z derived from hydrodynamic simulations. Reichart et al. used an extrapolated mass X-ray luminosity relation. Bahcall and Fan used more direct estimates of the cluster mass at higher z for three clusters. A change of a factor of two in the mass estimate would modify the conclusion.