The Zero-Point of the Cluster-Cluster Correlation Function: A Key Test of Cosmological Power Spectra

We propose the zero-point of the cluster-cluster correlation function r_0_ as a sensitive test for the shape of the power spectrum of initial fluctuations. With a wealth of new available redshifts for rich clusters the correlation function is measured to higher accuracy than has previously been done. It is now possible to go beyond the power law description to measure the point at which the correlation function becomes zero. A number of measurements of the zero-point of the rich galaxy cluster correlation function indicate that the zero-point should be in the range (40-60) h^-1^ Mpc. We discuss different effects, which could affect the zero-point and conclude that it is reasonably stable. The large value of r_0_ at which the zero-point occurs rules out conventional CDM models independently of the assumed amplitude. The most severe constraints are imposed on the CDM models with the cosmological constant. Models with {OMEGA} < 0.25 should he rejected because they predict too large r_0_. If the age of the universe is assumed to be larger than 15 Gyr, models with either {OMEGA} < 0.5 or h > 0.55 are rejected. We present the results of numerical simulations of clusters in the mixed model (the cold-plus-hot-dark-matter or CHDM). The correlation function of clusters in the model has a zero-point, r_0_ = 55 h^-1^ Mpc that matches the zero-point of the observed function and is very close to the zero-point predicted by the linear theory. The shape of the function on all available scales (up to 100 h^-1^ Mpc) is reproduced for both the Abell and the APM clusters. The CHDM model predicts the mass function for the clusters, which is higher than that given by Bahcall & Cen (1993) (masses are larger by a factor 1.5), but is compatible with the results of Biviano et al. (1993).