Evolution of the cluster mass and correlation functions in a ACDM cosmology

The evolution of the cluster mass function and the cluster correlation function from z = 0 to 3 is determined using ∼106 clusters obtained from high-resolution simulations of the current best-fit ACDM cosmology (Ω_m = 0.27, σ₈ = 0.84, h = 0.7). The results provide predictions for comparisons with future observations of high-redshift clusters. A comparison of the predicted mass function of low-redshift clusters with observations from early Sloan Digital Sky Survey data and the predicted abundance of massive distant clusters with observational results favor a slightly larger amplitude of mass fluctuations (σ₈ ∼ 0.9) and lower density parameter (Ω_m ∼ 0.2); these values are consistent within 1 σ with the current observational and model uncertainties. The cluster correlation function strength increases with redshift for a given mass limit; the clusters were more strongly correlated in the past because of their increasing bias with redshift - the bias reaches b ∼ 100 at z = 2 for M > 5 × 10¹³ h^-1 M_⊙ clusters. The richness-dependent cluster correlation function, represented by the correlation scale versus cluster mean separation relation, R₀-d, is generally consistent with observations. This relation can be approximated as R₀ = 1.7d^0.6 h^-1 Mpc for d ∼ 20-60 h ^-1 Mpc. The R₀-d relation exhibits surprisingly little evolution with redshift for z < 2; this finding can provide a new test of the current ACDM model when compared with future observations of high-redshift clusters.