A. Higher order correlation functions
The two-point correlation function is not a unique descriptor of clustering, it is merely the first of an infinite hierarchy of such descriptors describing the galaxy distribution of galaxies taken N at a time. Two quite different distributions can have the same two-point correlation function. In particular, the fact that a point distribution generated by any random walk (e.g., as a Lévy flight as proposed by Mandlebrot (1975) has the correct two-point correlation function does not mean much unless other statistical measures of clustering are tested.
The present day galaxy distribution is manifestly not a Gaussian random process: there is, for example, no symmetry about the mean density. This fact alone tells us that there is more to galaxy clustering than the two-point correlation function.
So what kind of descriptors should we look for? Generalizations of the two-point functions to 3-, 4- and higher order functions are certainly possible, but they are difficult to calculate and not particularly edifying. However, they do the job of providing some of the needed extra information and through such constructs as the BBGKY hierarchy 9 they do relate to the underlying physics of the clustering process. We shall describe the observed scaling of the 3-point correlation function below.
One alternative is to go for different clustering models: anything but correlation functions. These may have the virtue of providing immediate gratification in terms of visualization of the process, but they are often difficult to relate to any kind of dynamical process.
If we knew all higher order correlation functions we would have a complete description of the galaxy clustering process. However, calculating an estimate of a two point function from a sample of N galaxies requires taking all pairs from the sample of N, while calculating a three point functions requires taking all triples from N. The amount of computation escalates rapidly and restrictions have to be imposed on what is actually being calculated.
Nevertheless, calculating restricted N-point functions may be useful: these functions may be related to one another and have interesting scale dependence. Gaztañaga (1992) has calculated restricted N-point functions and showed that these have power law behavior over the range of scales where they can be determined.
9 The BBGKY hierarchy, (after Bogolyubov, Born, Green, Kirkwood and Yvon), is an infinite chain of equations adapted from plasma physics (Ichimaru, 1992) to describe self-gravitating non-linear clustering (see for example Fall and Severne (1976), Peebles (1980), and Saslaw (2000).) Back.