The Correlation Function of Clusters of Galaxies and the Amplitude of Mass Fluctuations in the Universe

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 σ8 can be estimated directly from the correlation length of a cluster sample in a way which is independent of the value of Ω0. If the cluster mass corresponding to the sample threshold is also known, it provides an independent estimate of the quantity σ8Ω 0.6 0 . Thus cluster data should allow both σ8 and Ω0 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 r0 in excess of 20 hMpc for Abell clusters would require σ8 > 1, while r0 < 15 h Mpc would require σ8 < 0.5. With conventional estimates of the relevant mass threshold these imply Ω0 < ∼ 0.3 and Ω0 > ∼ 1 respectively.