Searching for the scale of homogeneity

We introduce a statistical quantity, known as the $K$ function, related to the integral of the two--point correlation function. It gives us straightforward information about the scale where clustering dominates and the scale at which homogeneity is reached. We evaluate the correlation dimension, $D_2$, as the local slope of the log--log plot of the $K$ function. We apply this statistic to several stochastic point fields, to three numerical simulations describing the distribution of clusters and finally to real galaxy redshift surveys. Four different galaxy catalogues have been analysed using this technique: the Center for Astrophysics I, the Perseus--Pisces redshift surveys (these two lying in our local neighbourhood), the Stromlo--APM and the 1.2 Jy {\it IRAS} redshift surveys (these two encompassing a larger volume). In all cases, this cumulant quantity shows the fingerprint of the transition to homogeneity. The reliability of the estimates is clearly demonstrated by the results from controllable point sets, such as the segment Cox processes. In the cluster distribution models, as well as in the real galaxy catalogues, we never see long plateaus when plotting $D_2$ as a function of the scale, leaving no hope for unbounded fractal distributions.Comment: 9 pages, 11 figures, MNRAS, in press; minor revision and added reference

A global descriptor of spatial pattern interaction in the galaxy distribution

We present the function J as a morphological descriptor for point patterns formed by the distribution of galaxies in the Universe. This function was recently introduced in the field of spatial statistics, and is based on the nearest neighbor distribution and the void probability function. The J descriptor allows to distinguish clustered (i.e. correlated) from ``regular'' (i.e. anti-correlated) point distributions. We outline the theoretical foundations of the method, perform tests with a Matern cluster process as an idealised model of galaxy clustering, and apply the descriptor to galaxies and loose groups in the Perseus-Pisces Survey. A comparison with mock-samples extracted from a mixed dark matter simulation shows that the J descriptor can be profitably used to constrain (in this case reject) viable models of cosmic structure formation.Comment: Significantly enhanced version, 14 pages, LaTeX using epsf, aaspp4, 7 eps-figures, accepted for publication in the Astrophysical Journa

Multiscaling

. We introduce the unbiased way statisticians look at the 2-- point correlation function and study its relation to multifractal analysis. We apply this method to a simulation of the distribution of galaxy clusters in order to check the dependence of the correlation dimension on the cluster richness. 1. Introduction The statistical description of the galaxy clustering is usually based on the twopoint correlation function ¸(r). This function is, following the terminology used by statisticians working in point field statistics, a second-order characteristic of the point process (Diggle 1993; Stoyan & Stoyan 1994). The first-order characteristic is just the intensity measure (~r) (Mart'inez et al. 1993). Assuming the Cosmological principle, we accept that galaxies in large volumes represent a stationary and isotropic point process, having therefore constant intensity equal to the number density of galaxies per unit volume, denoted by n. 2. The K-function and the correlation dimension Amo..