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 model cluster mergers using two completely
different approaches, and show that the final results from each are extremely
similar. 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.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
