Properties of galaxy clusters: mass and correlation functions

We analyse parallel N-body simulations of three cold dark matter (CDM) universes to study the abundance and clustering of galaxy clusters. The simulation boxes are 500h^-1Mpc on a side and cover a volume comparable to that of the forthcoming Sloan Digital Sky Survey. The use of a treecode algorithm and 47 million particles allows us at the same time to achieve high mass and force resolution. We are thus able to make robust measurements of cluster properties with good number statistics up to a redshift larger than unity. We extract haloes using two independent, public domain group finders designed to identify virialized objects - `Friends-of-Friends' and `HOP' - and find consistent results. The correlation function of clusters as a function of mass in the simulations is in very good agreement with a simple analytic prescription based upon a Lagrangian biasing scheme developed by Mo & White and the Press-Schechter (PS) formalism for the mass function. The correlation length of clusters as a function of their number density, the R_0-D_c relation, is in good agreement with the APM Cluster Survey in our open CDM model. The critical-density CDM model (SCDM) shows much smaller correlation lengths than are observed. We also find that the correlation length does not grow as rapidly with cluster separation in any of the simulations as suggested by the analysis of very rich Abell clusters. Our SCDM simulation shows a robust deviation in the shape and evolution of the mass function when compared with that predicted by the PS formalism. Critical models with a low sigma_8 normalization or small shape parameter Gamma have an excess of massive clusters compared with the PS prediction. When cluster-normalized, the SCDM universe at z=1 contains 10 times more clusters with temperatures greater than 7keV, compared with the PS prediction. The agreement between the analytic and N-body mass functions can be improved, for clusters hotter than 3keV in the critical-density SCDM model, if the value of delta_c (the extrapolated linear theory threshold for collapse) is revised to be delta_c(z)=1.686[(0.7/sigma_8)(1+z)]^-0.125 (sigma_8 is the rms density fluctuation in spheres of radius 8h^-1Mpc). Our best estimate for the amplitude of fluctuations inferred from the local cluster abundance for the SCDM model is sigma_8=0.5+/-0.04. However, the discrepancy between the temperature function predicted in a critical-density universe and that observed at z=0.33 (Henry et al.) is reduced by a modest amount using the modified PS scheme. The discrepancy is still large enough to rule out Omega_0=1, unless there are significant differences in the relation between mass and temperature for clusters at high and low redshift.