Evolution of the cluster correlation function

We study the evolution of the cluster correlation function and its richness dependence from z = 0 to z = 3 using large-scale cosmological simulations. A standard flat LCDM model with Ω_m = 0.3 and, for comparison, a tilted Ω_m = 1 model (TSCDM) are used. The evolutionary predictions are presented in a format suitable for direct comparisons with observations. We find that the cluster correlation strength increases with redshift: high-redshift clusters are clustered more strongly (on a comoving scale) than low-redshift clusters of the same mass. The increased correlation with redshift, in spite of the decreasing mass correlation strength, is caused by the strong increase in cluster bias with redshift: clusters represent higher density peaks of the mass distribution as the redshift increases. The richness-dependent cluster correlation function, presented as the correlation scale versus cluster mean separation relation, R₀-d, is found to be, remarkably, independent of redshift to z ≲ 2 for LCDM and z ≲ 1 for TSCDM for a fixed correlation function slope and a cluster mass within a fixed comoving radius. The nonevolving R₀-d relation implies that both the comoving clustering scale and the cluster mean separation increase with redshift for the same mass clusters, so that the R₀-d relation remains essentially unchanged. For LCDM, this relation is R₀(Z) ≃ 2.6[d(z}]^1/2 for z ≲ 2 in comoving h^-1 Mpc scales. The TSCDM model has smaller correlation scales, as expected. Evolution in the relation is seen at z ≳ 2 for LCDM and z ≳ 1 for TSCDM, where the amplitude of the relations declines. The evolution of the R₀-d relation from z ∼ 0 to z ∼ 3 provides an important new tool in cosmology; it can be used to break degeneracies that exist at z ∼ 0 and provide precise determination of cosmological parameters.