We analyze an extended redshift sample of Abell/ACO clusters and compare the results with those coming from numerical simulations of the cluster distribution , based on the truncated Zel ’ dovich approximation ( TZA ) , for a list of eleven dark matter ( DM ) models .
For each model we run several realizations , so that we generate a set of 48 independent mock Abell/ACO cluster samples per model , on which we estimate cosmic variance effects .
Other than the standard CDM model , we consider ( a ) \Omega _ { 0 } = 1 CDM models based on lowering the Hubble parameter and/or on tilting the primordial spectrum ; ( b ) \Omega _ { 0 } = 1 Cold+Hot DM models with 0.1 \leq \Omega _ { \nu } \leq 0.5 ; ( c ) low–density flat \Lambda CDM models with 0.3 \leq \Omega _ { 0 } \leq 0.5 .
We compare real and simulated cluster distributions by analysing correlation statistics , the probability density function , and supercluster properties from percolation analysis .
We introduce a generalized definition of the spectrum shape parameter \Gamma in terms of \sigma _ { 25 } / \sigma _ { 8 } , where \sigma _ { r } is the rms fluctuation amplitude within a sphere of radius r .
As a general result , we find that the distribution of galaxy clusters provides a constraint only on the shape of the power spectrum , but not on its amplitude : a shape parameter 0.18 \raise - 2.0 pt \hbox { \hbox to 0.0 pt { \hbox { $ \sim$ } } \raise 5.0 pt \hbox { $ < $ } } % \Gamma \raise - 2.0 pt \hbox { \hbox to 0.0 pt { \hbox { $ \sim$ } } \raise 5.0 pt \hbox { $ < $ } } % 0.25 and an effective spectral index at the 20 h ^ { -1 } { Mpc } scale -1.1 \raise - 2.0 pt \hbox { \hbox to 0.0 pt { \hbox { $ \sim$ } } \raise 5.0 pt \hbox { $ < $ } } n _ % { eff } \raise - 2.0 pt \hbox { \hbox to 0.0 pt { \hbox { $ \sim$ } } \raise 5.0 pt \hbox { $ < $ } % } -0.9 are required by the Abell/ACO data .
In order to obtain complementary constraints on the spectrum amplitude , we consider the cluster abundance as estimated using the Press–Schechter approach , whose reliability is explicitly tested against N–body simulations .
By combining results from the analysis of the distribution and the abundance of clusters we conclude that , of the cosmological models considered here , the only viable models are either Cold+Hot DM ones with 0.2 \raise - 2.0 pt \hbox { \hbox to 0.0 pt { \hbox { $ \sim$ } } \raise 5.0 pt \hbox { $ < $ } } % \Omega _ { \nu } \raise - 2.0 pt \hbox { \hbox to 0.0 pt { \hbox { $ \sim$ } } \raise 5.0 pt \hbox { $% < $ } } 0.3 , better if shared between two massive \nu species , and \Lambda CDM ones with 0.3 \raise - 2.0 pt \hbox { \hbox to 0.0 pt { \hbox { $ \sim$ } } \raise 5.0 pt \hbox { $ < $ } } % \Omega _ { 0 } \raise - 2.0 pt \hbox { \hbox to 0.0 pt { \hbox { $ \sim$ } } \raise 5.0 pt \hbox { $ < $% } } 0.5 .