Cluster evolution as a diagnostic for Ω

The population of rich galaxy clusters evolves much more rapidly in auniverse with critical density than one with low density, thus offering thepossibility of determining the cosmological density parameter, Omega_0. Wequantify this evolution using the Press-Schechter formalism which we extend toflat models with a cosmological constant. Using new large N-body simulations,we verify that this formalism accurately predicts the abundance of richclusters as a function of redshift in various cosmologies. We normalise themodels by comparing them to the local abundance of clusters as a function oftheir X-ray temperature which we rederive from data compiled by Henry & Arnaud.This gives values of the rms density fluctuation in spheres of radius 8 Mpc/hof sigma_8 = (0.50+/- 0.04) Omega_0^{-0.47+0.10 Omega_0} if Lambda_0=0 andsigma_8 = (0.50 +/- 0.04) Omega_0^{-0.53+0.13 Omega_0} if Lambda_0=1-Omega_0.These values depend very weakly on the shape of the power spectrum. We thenexamine how the distributions of mass, X-ray temperature and Sunyaev-Zel'dovichdecrement evolve as a function of Omega_0. We present the expecteddistributions at z=0.33 and z=0.5 and the predicted number counts of thelargest clusters. We find that even at z=0.33, these distributions depend verystrongly on Omega_0 and only weakly on Lambda_0. For example, at this redshift,we expect 20 times as many clusters per comoving volume with M>3.5 10^{14}Msol/h and 5 times as many clusters with kT>5 keV if Omega_0=0.3 than ifOmega_0=1. The splitting in the integrated counts is enhanced by the largervolume element in low Omega_0 models. There is therefore a real prospect ofestimating Omega_0 from forthcoming surveys of intermediate redshift clustersthat will determine their masses, X-ray temperatures or SZ decrements.Comment: Compressed postscript also available at ftp://star-ftp.dur.ac.uk/pub/preprints/ecf.ps.g