The cluster abundance in flat and open cosmologies

We use the galaxy cluster X-ray temperature distribution function to constrain the amplitude of the power spectrum of density inhomogeneities on the scale corresponding to clusters. We carry out the analysis for critical density universes, for low density universes with a cosmological constant included to restore spatial flatness and for genuinely open universes. That clusters with the same present temperature but different formation times have different virial masses is included. We give careful consideration to the uncertainties involved, carrying out a Monte Carlo analysis to determine the cumulative errors. For critical density our result agrees with previous papers, but we believe the result carries a larger uncertainty. For low density universes, either flat or open, the required amplitude of the power spectrum increases as the density is decreased. If all the dark matter is taken to be cold, then the cluster abundance constraint remains compatible with both galaxy correlation data and the {\it COBE} measurement of microwave background anisotropies for any reasonable density