The supercluster-void network - III. The correlation function as a geometrical statistic

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Abstract
We investigate properties of the correlation function of clusters of galaxies using geometrical models. We show that the correlation function contains useful information on the geometry of the distribution of clusters. On small scales the correlation function depends on the shape and the size of superclusters. On large scales it describes the geometry of the distribution of superclusters. If superclusters are distributed randomly then the correlation function on large scales is featureless. If superclusters have a quasi-regular distribution then this regularity can be detected and measured by the correlation function. Superclusters of galaxies separated by large voids produce a correlation function with a minimum which corresponds to the mean separation between centres of superclusters and voids, followed by a secondary maximum corresponding to the distance between superclusters across voids. If superclusters and voids have a tendency to form a regular lattice then the correlation function on large scales has quasi- regularly spaced maxima and minima of decaying amplitude; i.e. it is oscillating. The period of oscillations is equal to the step size of the grid of the lattice. We also calculate the power spectrum and the void diameter distribution for our models and compare the geometrical information of the correlation function with other statistics. We find that geometric properties (the regularity of the distribution of clusters on large scales) are better quantified by the correlation function. We also analyse errors in the correlation function and the power spectrum by generating random realizations of models and finding the scatter of these realizations.
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