  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, Omega_0. 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 sigma_8 = (0.50+/- 0.04) Omega_0^{-0.47+0.10 Omega_0} if Lambda_0=0 and
sigma_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 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. 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 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 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 SZ decrements.
