Cluster abundance and large-scale structure

We use the presently observed number density of large X-ray clusters and linear mass power spectra to constrain the shape parameter ($\Gamma$), the spectral index ($n$), the amplitude of matter density perturbations on the scale of $8 h^{-1}$Mpc ($\sigma_8$), and the redshift distortion parameter ($\beta$). The non-spherical-collapse model as an improvement to the Press-Schechter formula is used and yields significantly lower $\sigma_8$ and $\beta$. An analytical formalism for the formation redshift of halos is also derived.