6 research outputs found
Searching for the scale of homogeneity
We introduce a statistical quantity, known as the 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,
, as the local slope of the log--log plot of the 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 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..