We use the presently observed number density of large X-ray clusters and the linear mass power spectra from galaxy surveys to constrain the amplitude of matter density perturbations on the scale of 8 h ^ { -1 } Mpc ( \sigma _ { 8 } ) , and the redshift distortion parameter ( \beta ) , in both open cosmologies and flat models with a non-zero cosmological constant .
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 ^ { \prime } = 0.247 \Gamma \exp ( 1.4 n ) = 0.220 ^ { +0.036 } _ { -0.031 } ( all at 95 per cent confidence level ) .
These are consistent with the recent CMB results .
Based on this , we then calculate the cluster-abundance-normalized \sigma _ { 8 } , using different models of mass function .
The models considered are by Press & Schechter ( PS ; 1974 ) , by Sheth & Tormen ( ST ; 1999 ) , and by Lee & Shandarin ( LS ; 1999 ) .
The last two incorporate non-spherical gravitational collapse , and the \sigma _ { 8 } based on these two models are significantly lower .
This lower normalization results from the larger mass function within the scale range of our interest .
In particular , we combine the results of these two models to yield \sigma _ { 8 { ( ST + LS ) } } = 0.477 \Omega _ { m 0 } ^ { \alpha } , where \alpha = -0.3 - 0.17 \Omega _ { m 0 } ^ { 0.34 } -0.13 \Omega _ { \Lambda 0 } .
In our analysis , 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 origins of uncertainties in our \sigma _ { 8 } results are also investigated separately , with the main contributer being the normalization in the virial mass-temperature relation .
From the PSCz power spectrum alone and using \Gamma ^ { \prime } = 0.220 ^ { +0.036 } _ { -0.031 } as the prior , we also obtain for the IRAS galaxies \sigma _ { 8 ( I ) } = 0.78 \pm 0.06 ( at 95 per cent confidence level ) .
By combining this with the \sigma _ { 8 } result , we are able to constrain the redshift distortion parameter \beta _ { I } , which is in turn lower in the non-spherical-collapse models .
We found \beta _ { I ( ST + LS ) } = 0.613 \Omega _ { m 0 } ^ { 0.24 - 0.16 ( \Omega _ { m 0 } + \Omega _ { % \Lambda 0 } ) } .
This is more consistent with the recent observations than the result based on the PS formalism .