The power spectrum of clusters of galaxies and the Press-Schechter approximation

We examine the power spectrum of clusters in the Press-Schechter (PS) theory and in N-body simulations to see how the power spectrum of clusters is related to the power spectrum of matter density fluctuations in the universe. An analytic model for the power spectrum of clusters for their given number density is presented, for both real space and redshift space. We test this model against results from N-body simulations and find that the agreement between the analytic theory and the numerical results is good for wavelengths lambda > 60 h(-1) Mpc. On smaller scales nonlinear processes that are not considered in the linear PS approximation influence the result. We also use our analytic model to study the redshift space power spectrum of clusters in cold dark matter models with a cosmological constant and with a scale-invariant Harrison-Zeldovich initial spectrum of density fluctuations. We find that the power spectra of clusters in these models are not consistent with the observed power spectra of the APM and Abell-ACO clusters. One possible explanation for the observed power spectra of clusters is an inflationary scenario with a scalar held with the potential that has a localized steplike feature. We use the PS theory to examine the power spectrum of clusters in this model.