Properties of galaxy clusters: mass and correlation functions

We analyse parallel N-body simulations of three cold dark matter (CDM) univ erses to study the abundance and clustering of galaxy clusters. The simulat ion boxes are 500 h(-1) Mpc on a side and cover a volume comparable to that of the forthcoming Sloan Digital Sky Survey. The use of a treecode algorit hm 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 clus ter properties with good number statistics up to a redshift larger than uni ty. We extract haloes using two independent, public domain group finders de signed to identify virialized objects - 'Friends-of-Friends' and 'HOP' - an d find consistent results. The correlation function of clusters as a functi on of mass in the simulations is in very good agreement with a simple analy tic prescription based upon a Lagrangian biasing scheme developed by Mo & W hite and the Press-Schechter (PS) formalism for the mass function. The corr elation 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 C DM model. The critical-density CDM model (SCDM) shows much smaller correlat ion lengths than are observed. We also find that the correlation length doe s 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 wh en compared with that predicted by the PS formalism, Critical models with a low as normalization or small shape parameter Gamma have an excess of mass ive clusters compared with the PS prediction. When cluster-normalized, the SCDM universe at z = 1 contains 10 times more clusters with temperatures gr eater than 7 keV, compared with the PS prediction. The agreement between th e analytic and N-body mass functions can be improved, for clusters hotter t han 3 keV 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 fl uctuation in spheres of radius 8 h(-1) Mpc). Our best estimate for the ampl itude of fluctuations inferred from the local cluster abundance for the SCD M model is sigma(8) = 0.5 +/- 0.04. However, the discrepancy between the te mperature function predicted in a critical-density universe and that observ ed at z = 0.33 (Henry et al.) is reduced by a modest amount using the modif ied PS scheme. The discrepancy is still large enough to rule out Omega(0) = 1, unless there are significant differences in the relation between mass a nd temperature for clusters at high and low redshift.