Cluster evolution as a diagnostic for Ω

The population of rich galaxy clusters evolves much more rapidly in a universe with critical density than one with low density, thus offering the possibility of determining the cosmological density parameter, Omega₀. We quantify this evolution using the Press-Schechter formalism which we extend to flat models with a cosmological constant. 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 normalise the models by comparing them to the local abundance of clusters as a function of their 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/h of sigma₈ = (0. 50+/- 0. 04) Omega₀^-0. 47+0. 10 Omega₀ if Lambda₀=0 and sigma₈ = (0. 50 +/- 0. 04) Omega₀^-0. 53+0. 13 Omega₀ if Lambda₀=1-Omega₀. These values depend very weakly 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 Omega₀. We present the expected distributions at z=0. 33 and z=0. 5 and the predicted number counts of the largest clusters. We find that even at z=0. 33, these distributions depend very strongly on Omega₀ and only weakly on Lambda₀. 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 Omega₀=0. 3 than if Omega₀=1. The splitting in the integrated counts is enhanced by the larger volume element in low Omega₀ models. There is therefore a real prospect of estimating Omega₀ from forthcoming surveys of intermediate redshift clusters that will determine their masses, X-ray temperatures or SZ decrements.