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We show that if a sample of galaxy clusters is complete above some mass threshold, then hierarchical clustering theories for structure formation predict its autocorrelation function to be determined purely by the cluster abundance and by the spectrum of linear density fluctuations. Thus if the shape of the initial fluctuation spectrum is known, its amplitude \sigma_8 can be estimated directly from the correlation length of a cluster sample in a way which is independent of the value of \Omega_0. If the cluster mass corresponding to the sample threshold is also known, it provides an independent estimate of the quantity \sigma_8\Omega_0^{0.6}. Thus cluster data should allow both \sigma_8 and \Omega_0 to be determined observationally. We explore these questions using N-body simulations together with a simple but accurate analytical model based on extensions of Press-Schechter theory. Applying our results to currently available data we find that if the linear fluctuation spectrum has a shape similar to that suggested by the APM galaxy survey, then a correlation length r_0 in excess of 20\mpch for Abell clusters would require \sigma_8>1, while r_0<15\mpch would require \sigma_8<0.5. With conventional estimates of the relevant mass threshold these imply \Omega_0\la 0.3 and \Omega_0\ga 1 respectively

The correlation function of clusters of galaxies and the amplitude of mass fluctuations in the Universe