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 member galaxies assigned to a cluster. Such a survey can strongly constrain $σ_8$ ($Δσ_8\approx 0.05$), the power law index $α$ where $<N_{gal}|m>= 1+(m/M_1)^α$ ($Δα\approx0.03$), and perhaps even the Hubble parameter ($Δh\approx 0.07$). We find cluster abundances and bias not well suited for constraining $Ω_m$ or the amplitude $M_1$. We also find that without bias information $σ_8$ and $α$ 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.

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