Poisson clusters and Poisson voids

Abstract

Expressions are derived for the expected abundance of clusters and voids in a sample of randomly distributed objects.PACS numbers: 02.50.-fs, 05.40. +j The interpretation of an observed distribution of pointlike objects often involves assessing the significance of clusters of voids. For example, one may observe with a neutrino telescope 1 a cluster of several events within a small solid angle, or one may observe a large void in the distribution of rich clusters of galaxies. 2 One then wishes to determine the likelihood of the observed cluster or void arising as a statistical fluctuation, under the assumption that the objects are actually randomly distributed. Unless this probability is low, it is not reasonable to regard the signal detected by the telescope as evidence for an astrophysical point source of neutrinos, nor to regard the rich-cluster void as dramatic evidence for very large scale structure in the universe.One way to test the hypothesis that a set of points is randomly distributed is to check the validity of the Poisson formula,for the probability that a randomly selected region of volume V contains k points, if n is the mean density of the points. However, for clusters so dense or voids so dilute that the corresponding Poisson probability is extremely small, it is far more useful to know the expected number of clusters or voids located anywhere in the entire observed region. In principle, the expected abundance of such dense clusters or dilute voids could be determined by Monte Carlo simulations. However, rare occurrences are not so easily simulated, and at any rate it is evidently desirable to be able to express the answer in an analytic form. Thus, in this Letter, we derive such expressions for the expected abundance of clusters and voids in a sample of randomly distributed objects, in the limit in which the clusters or voids are rare. These formulas can be generalized to the case of correlated distributions. To illustrate their use, we apply our results to the distribution of rich clusters of galaxies in the sky. The Poisson probability P k is independent of the shape of the region in question. In contrast, to define what constitutes a cluster or void requires some specification of an acceptable shape, and the expected abundance of such configurations will, in general, depend on that specification. For example, we may decide that k objects distributed in d dimensions constitute a cluster if there is a cube of volume Kand a priori specified orientation which contains these k objects and no others. Then, as we will show, the expected number of such clusters per unit volume iswhere n is the mean density of the objects, and k is large compared to n V. For k comparable to n K, the mean number of objects in the volume V, there are typically many overlapping clusters, and the abundance of clusters is not of great interest; there is considerable arbitrariness in counting the clusters when they overlap. The OinV/k) correction in Eq.(2) depends on how the overlapping clusters are counted.A void in a d-di...

Statistics and geometry of cosmic voids

“…However, the calculation of the probability of having a void of given size and shape at any place is difficult. Politzer & Preskill's analysis [31] yields the following formula for the probability per unit total volume that there is some region of volume V and given shape that contains k points:…”

On an analytical framework for voids: their abundances, density profiles and local mass functions

“…The general framework for evaluating the mean number densities (i.e. probability densities) of structures like voids (Politzer & Preskill 1986; Otto et al 1986) has been applied to the standard large‐scale structure models (i.e. Gaussian initial density fluctuation growing gravitationally).…”

The statistics of voids as a tool to constrain cosmological parameters: σ8 and Γ

“…Note also that the coefficient 3π 2 /32 shown in is replaced by 0.68 in . The former coefficient was originally introduced by Politzer & Preskill (1986), but we found the latter to be more accurate (Betancort‐Rijo, in preparation). The coefficient 0.68 in has been obtained analytically with an error smaller than 10 −3 , although it has been checked within a 5 per cent.…”