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

We show that it may be possible to determine both the cosmic density parameter Omega(0) and the amplitude of cosmic mass fluctuations sigma(8) by using large redshift surveys of galaxies and galaxy clusters. If a duster sample can be considered complete above some mass threshold, then hierarchical 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 that 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 this possibility 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 h(-1) Mpc for Abell clusters would require sigma(8) greater than or similar to 1, while r(0) < 25 h(-1) Mpc would require sigma(8) less than or similar to 0.5. With conventional estimates of the relevant mass threshold, these imply Omega(0) less than or similar to 0.3 and Omega(0) greater than or similar to 1 respectively.