The cluster abundance in flat and open cosmologies

We use the galaxy cluster X-ray temperature distribution function toconstrain the amplitude of the power spectrum of density inhomogeneities on thescale corresponding to clusters. We carry out the analysis for critical densityuniverses, for low density universes with a cosmological constant included torestore spatial flatness and for genuinely open universes. That clusters withthe same present temperature but different formation times have differentvirial masses is included. We model cluster mergers using two completelydifferent approaches, and show that the final results from each are extremelysimilar. We give careful consideration to the uncertainties involved, carryingout a Monte Carlo analysis to determine the cumulative errors. For criticaldensity our result agrees with previous papers, but we believe the resultcarries a larger uncertainty. For low density universes, either flat or open,the required amplitude of the power spectrum increases as the density isdecreased. If all the dark matter is taken to be cold, then the clusterabundance constraint remains compatible with both galaxy correlation data andthe {\it COBE} measurement of microwave background anisotropies for anyreasonable density.Comment: Uuencoded package containing LaTeX file (uses mn.sty) plus 7 postscript figures incorporated using epsf. Total length 10 pages. Final version, to appear MNRAS. COBE comparison changed to 4yr data. No change to results or conclusion