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

Abstract

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 $σ_8$ can be estimated directly from the correlation length of a cluster sample in a way which is independent of the value of $Ω_0$. If the cluster mass corresponding to the sample threshold is also known, it provides an independent estimate of the quantity $σ_8Ω_0^{0.6}$. Thus cluster data should allow both $σ_8$ and $Ω_0$ 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 $σ_8>1$, while $r_0<15\mpch$ would require $σ_8<0.5$. With conventional estimates of the relevant mass threshold these imply $Ω_0\la 0.3$ and $Ω_0\ga 1$ respectively.