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3.2.3. Cluster Abundance Versus the COBE Normalisation

There are a number of ways to infer sigma8 from galaxy clustering and peculiar velocity fields. The problem with the information from galaxy clustering is that it involves an unknown biasing factor, which hinders us from determining an accurate sigma8. The velocity data are susceptible to noise from the distance indicators. Therefore, the cluster abundance discussed above seems to give us a unique method to derive an accurate estimate of sigma8 for a low z universe. Another place we can extract an accurate sigma8 is the fluctuation power imprinted on cosmic microwave background radiation (CBR) anisotropies. Currently only the COBE observation (Bennett et al. 1996) gives sufficiently accurate sigma8 = sigma8(H0, Omega, lambda, OmegaB, ...). Assuming the model transformation function, the matching of COBE sigma8 with that from the cluster abundance gives a significant constraint on cosmological parameters Omega = Omega(H0, lambda) (Efstathiou et al. 1992; Eke et al. 1996). Figure 4 shows allowed regions for two cases, open and flat universes, assuming a flat perturbation spectrum n = 1 and ignoring possible tensor perturbations.

The transfer function is modified if n neq 1. The possible presence of the tensor perturbations in CBR anisotropies causes another uncertainty. The COBE data alone say n being between 0.9 and 1.5 (Bennett et al. 1996), but the allowed range is narrowed to n = 0.9-1.2 if supplemented by smaller angular-scale CBR anisotropy data (Hancock et al. 1998; Lineweaver 1998; Efstathiou et al. 1999; Tegmark 1999). The presence of the tensor mode would make the range of n more uncertain as well as it reduces the value of sigma8. The limit of n when the tensor mode is maximally allowed is about < 1.3 (4). Notwithstanding these uncertainties, Omega > 0.5 is difficult to reconcile with the matching condition. On the other hand, a too small Omega (ltapprox 0.15) is not consistent with the cluster abundance.

Figure 4

Figure 4. Parameter regions allowed by matching the rms fluctuations from COBE with those from the cluster abundance. A flat spectrum (n = 1) is assumed and the tensor perturbations are neglected. The lower band is for a flat universe, and the upper one for a universe with Lambda = 0.

4 In Tegmark's analysis n < 1.5 is quoted as an upper bound, but this is obtained by making OmegaB (and H0) a free parameter. If one would fix the baryon abundance, the allowed range is narrower, n ltapprox 3. Back.

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