Cluster evolution as a diagnostic for Omega

The population of rich galaxy clusters evolves much more rapidly in a universe with critical density than in a universe with low density. Thus, counts of clusters at intermediate redshift offer the possibility of determining the cosmological density parameter, Omega(0), with a minimum of assumptions. We quantify this evolution using the Press-Schechter formalism which we extend to flat cosmological models with a cosmological constant, Lambda(0) = 1 - Omega(0). Using new large N-body simulations, we verify that this formalism accurately predicts the abundance of rich clusters as a function of redshift in various cosmologies. We normalize the models by comparing them with the local abundance of clusters as a function of their X-ray temperature which we rederive from data compiled by Henry & Arnaud. The resulting values of the rms density fluctuation in spheres of radius 8h(-1) Mpc are sigma(8) = (0.52 +/- 0.04)Omega(0)(0.46+0.10 Omega 0) if Lambda(0) = 0 and sigma(8) = (0.52 +/- 0.04)Omega(0)(-0.52+0.13 Omega 0) if Lambda(0) = 1 - Omega(0). These values depend only weakly, and almost not at all if Omega(0) = 1, on the shape of the power spectrum. We then examine how the distributions of mass, X-ray temperature and Sunyaev-Zel'dovich decrement evolve as a function of Omega(0). We present the expected distributions at z = 0.33 and z = 0.5 and the predicted number counts of the largest clusters, both in space and in projection on the sky. We find that, even at z = 0.33,these distributions depend very strongly on Omega(0) and only weakly on Lambda(0). For example, at this redshift, we expect 15 times as many clusters per comoving volume with M > 3.5 x 10(14)h(-1) M(.) and 5 times as many clusters with kT > 5 keV if Omega(0) = 0.3 than if Omega(0) = 1. The splitting in the integrated counts is enhanced by the larger volume element in low-Omega(0) models. There is therefore a real prospect of estimating Omega(0) from forthcoming surveys of intermediate-redshift clusters that will determine their masses, X-ray temperatures or Sunyaev-Zel'dovich decrements.