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 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 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 halos using two independent , public domain group finders designed to identify virialised objects – ‘ Friends-of-Friends ’ ( Davis et al . 1985 ) and ‘ HOP ’ ( Eisenstein & Hut 1998 ) – 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 ( 1996 ) 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 7 keV , compared with the Press & Schechter prediction .
The agreement between the analytic and N-body mass functions can be improved , for clusters hotter than 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 ) = 1.685 \left [ ( 0.7 / \sigma _ { 8 } ) ( 1 + z ) \right ] ^ { -0.125 } ( \sigma _ { 8 } is the rms density fluctuation in spheres of radius 8 h ^ { -1 } Mpc ) .
Our best estimate for the amplitude of fluctuations inferred from the local cluster abundance for the SCDM model is \sigma _ { 8 } = 0.5 \pm 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 . 1998 ) is reduced by a modest amount using the modified Press-Schechter 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 .