Cluster abundance and large scale structure

We use the presently observed number density of large X-ray clusters and the linear mass power spectra to constrain $\sigma_8$ and the redshift distortion parameter $\beta$, in both OCDM and $\Lambda$CDM models. The best fit to the observed mass power spectra gives $n=0.84\pm 0.67$ and $\Gamma=0.27^{+0.42}_{-0.16}$, with the theoretically expected degeneracy $\Gamma'=0.247\Gamma\exp(1.4n)=0.220^{+0.036}_{-0.031}$ (all at 95% confidence level). Based on this, we then calculate the cluster-abundance-normalized $\sigma_8$, using different models of mass function: Press & Schechter (1974), Sheth & Tormen (1999), and Lee & Shandarin (1999). The $\sigma_8$ based on the non-spherical-collapse models (ST & LS) are significantly lower, mainly due to the larger mass function within the scale range of our interest. In particular, we found $\sigma_{8{\rm (ST+LS)}}=0.477\Omega_{\rm m0}^\alpha$, where $\alpha=-0.3-0.17\Omega_{\rm m0}^{0.34}-0.13\Omega_{\Lambda 0}$. We also derive the probability distribution function of cluster formation redshift using the Lacey-Cole formalism (1993), but with modifications to incorporate non-spherical collapse. The uncertainties in our $\sigma_8$ are mainly contributed from the normalization in the virial mass-temperature relation. We also obtain for the IRAS galaxies $\sigma_{8\rm (I)}=0.78\pm 0.06$ (at 95% confidence level), and found $\beta_{\rm I(ST+LS)}=0.613\Omega_{\rm m0}^{0.24-0.16(\Omega_{\rm m0}+\Omega_{\Lambda 0})}$. This is more consistent with the recent observations than the result based on the PS formalism.