Halo model analysis of cluster statistics

We use the halo model formalism to provide expressions for cluster abundances and bias, as well as estimates for the correlation matrix between these observables. Off-diagonal elements due to scatter in the mass-tracer scaling with mass are included, as are observational effects such as biases/scatter in the data, detection rates (completeness), and false detections (purity). We apply the formalism to a hypothetical volume limited optical survey where the cluster mass-tracer is chosen to be the number of satellite galaxies assigned to a cluster. Such a survey can strongly constrain sigma(8) (Deltasigma(8)approximate to0.05), the power law index alpha where =(m/M-1)(alpha) (Deltaalphaapproximate to0.03), and perhaps even the Hubble parameter (Deltahapproximate to0.07). We find cluster abundances and bias are not well suited for constraining Omega(m) or the amplitude M-1. We also find that without bias information sigma(8) and alpha are degenerate, implying constraints on the former are strongly dependent on priors used for the latter and vice-versa. The degeneracy stems from an intrinsic scaling relation of the halo mass function, and hence it should be present regardless of the mass-tracer used in the survey.