The correlation function can be generalized to higher order [1, 52, 53]: The N-point correlation functions. This allows to statistically characterize the galaxy distribution with a hierarchy of quantities which progressively provide us with more and more information about the clustering of the point process. These measures, however, had been difficult to derive with reliability from the galaxy catalogs. The new generation of surveys will surely overcome this problem.
There are, nevertheless, other clustering measures which provide complementary information to the second-order quantities describe above. For example, the topology of the large-scale structure measured by the genus statistic [54] provides information about the phase correlations of the density fluctuations in k-space. To obtain this quantity, first the point process has to be smoothed by means of a kernel function with a given bandwidth. The topological genus of a surface is the number of holes minus the number of isolated regions plus 1. This quantity is calculated for the isodensity surfaces of the smoothed data corresponding to a given density threshold. The genus has an analytically calculable expression for a Gaussian field [55].
Minkowski functionals are very effective clustering measures commonly
used in stochastic geometry
[20].
These quantities are adequate to study the shape and connectivity
of a union of convex bodies. They can easily be adapted to
point processes
[56]
by considering the covering of the point field formed by sets
Ar =
i=1N
Br(xi) where
r is the diagnostic parameter,
{xi}i = 1N
represents the galaxy positions, and
Br(xi) is a ball
of radius r centered at point xi. Minkowski
functionals
[57]
are applied to sets Ar as r
increases. In IR3 there are four functionals: the volume
V, the surface area A, the integral mean curvature
H, and the Euler-Poincaré characteristic
, related with the genus
of the boundary of Ar.
Several quantities based on distances to nearest neighbors have been used in the cosmological literature. The empty space function F(r) is the distribution function of the distance between a given random test particle in IR3 and its nearest galaxy. It is related with the void probability function [58] P0(r) - the probability that a ball of radius r randomly placed contains no galaxies - by F(r) = 1 - P0(r). G(r) is the distribution function of the distance r of a given galaxy to its nearest neighbor. The quotient J(r) = [1 - G(r) ]/ [1 - F(r)] has been successfully applied to describe the spatial pattern interaction in the galaxy distribution [59]. Related with the nearest neighbor distances, the minimal spanning tree is a structure descriptor that has shown powerful capabilities to reveal the clustering properties of different point patterns [60, 46, 61]. The minimal spanning tree (MST) is the unique network connecting the N points of the process with a route formed by N - 1 edges, without closed loops and having minimal total length (the total length is the sum of the lengths of the edges). The frequency histograms of the MST edge lengths can be used to analyze the galaxy distribution and to compare it with the simulated models [61].
The use of wavelets and related integral transforms is an extremely promising tool in the clustering analysis of 3-D catalogs. Some of these techniques are introduced in other contributions published in this volume [62, 63].
Acknowledgments
We are grateful to our collaborators Martin Snethlage and Dietrich Stoyan for common results on the correlation function of shifted Cox processes. We thank Valerie de Lapparent, Luigi Guzzo, Jon Loveday, and John Peacock for kindly providing us with some illustrations. This work was supported by the Spanish MCyT project AYA2000-2045 and by the Estonian Science Foundation under grant 2882.