The correlation function of clusters of galaxies and the amplitude of mass fluctuations in the Universe
Abstract
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 cluster 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_8Omega^0.6_0. Thus cluster data should allow both sigma_8 and Omega_0 to be determined observationally. We explore this possibility using Nbody simulations together with a simple but accurate analytical model based on extensions of PressSchechter 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>~1, while r_0<15 h^1 Mpc would require sigma_8<~0.5. With conventional estimates of the relevant mass threshold, these imply Omega_0<~0.3 and Omega_0>~1 respectively.
 Publication:

Monthly Notices of the Royal Astronomical Society
 Pub Date:
 October 1996
 DOI:
 10.1093/mnras/282.3.1096
 arXiv:
 arXiv:astroph/9602052
 Bibcode:
 1996MNRAS.282.1096M
 Keywords:

 GALAXIES: CLUSTERS: GENERAL;
 COSMOLOGY: THEORY;
 DARK MATTER;
 Astrophysics
 EPrint:
 Latex, 25 pages (including 8 PS figures). The PSfile of the paper is also available via anonymous ftp at: ftp://ibm3.mpagarching.mpg.de/pub/jing/xicc.ps . Submitted to MNRAS. In the replaced version, a typo in Eq.(1a) is fixed