Annu. Rev. Astron. Astrophys. 1988. 26: 245-294
Copyright © 1988 by . All rights reserved

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To study voids by means of statistical probabilities, a statistic must first be identified that provides a well-defined signature for the presence of voids. Then the probability-density or distribution function of that parameter is calculated from the locational coordinates of the galaxies in the sample. Finally, a comparison is made between this observed probability function and the predicted probability functions calculated from mathematical and astrophysical models with the same number of galaxies and the same observational selection functions as in the observational sample to identify that model which provides the best fit to the observed probability function.

Early statistical studies of large-scale structure applied primarily the spatial two-point and occasionally n-point correlation functions (cf. 142, 142a, pp. 138-256; 198). Correlation functions are insensitive to void structure (cf. 77, 84, 114a), so the extensive work on determining correlation functions from observational data, while contributing to our general knowledge of large-scale structure, contributed not at all to the discovery of voids. One can think of the observed two-point correlation function, the structure indicator SI of Rij, as a construction from the totality of the N(N - 1) / 2 separation vectors of length Rij (where the N galaxies in the homogeneous sample are numbered from i, j = 1 to N). The two-point correlation function provides information on the observed large-scale structure relative to a (structureless) model with Poisson-distributed locational coordinates, and it is defined as SI = [N(Rij) / N(Rij)Poisson] - 1, where N(Rij) is the number of observed galaxy pairs with Rij to Rij + DeltaRij. Here DeltaRij is a standard interval chosen to be sufficiently large so that SI is statistically well determined at the smallest Rij of interest, but small compared with the effective radius of the cosmic volume containing the sample of N galaxies. N(Rij)Poisson is the corresponding number of pairs with Rij to Rij + DeltaRij averaged over NS samples (where NS rightarrow infty), each sample with N galaxies constructed from an initially Poisson distribution of locational coordinates to which the observationally derived selection functions have been applied. The n-point correlation function is obtained by generalizing these concepts to encompass the joint distribution of the n separations: Rij, Rik, etc. The observed two-point correlation function is observationally well described by a power law, i.e. SI = ARijx, where A is the correlation amplitude and x is the "slope." The correlation scale length RL is the value of Rij for which SI = 1, i.e. RL = A-1/x. Analysis of data on locational coordinates of galaxies and galaxy clusters indicates that x appeq - 1.8 for both types of cosmic object, but RL appeq 10 Mpc for galaxies and RL appeq 50 Mpc for galaxy clusters (cf. Section 2.2.1). Additional information on large-scale structure is contained within the values of corresponding parameters for the hierarchy of n-point correlation functions; for example, within this framework, Fry (78a) studied statistics to quantify the visual impression of filaments in galaxy maps.

White (211c) derived and displayed relations that can be used to express any quantitative measures of clustering in terms of the hierarchy of correlation functions. He found that on scales less than the expected nearest neighbor distance most measures are influenced only by the lowest order correlation functions, and on all larger scales the measures depend significantly on higher order correlations only. White then suggested a particularly appealing statistical probability function that is influenced by correlation of all orders: the probability density function p(r), or, alternatively, the distribution function, g(r) = integrinfty p(r)dr, of the distance r from a randomly chosen point to its nearest galaxy neighbor; p(r) and g(r) are very sensitive to the presence of voids (cf. 3). For a Poisson reference model of galaxies with randomly distributed locational coordinates [i.e. the local number densities n(x, y, z) equal the global number density n to within Poisson uncertainties], we have g(r) = e-nr. The probability functions p(r) and g(r), and other closely related probability functions, seem to encompass the entire set of probability density and distribution functions that have been used to study the nature of voids as indicated by the locational properties of homogeneous samples of galaxies. Void-sensitive probability functions of this type have been derived for many mathematical and astrophysical models (cf. 3, 78, 84, 171); as demonstrated by Fry (78), for example, the main advantage of this type of analysis is that evolution and other effects are reduced to a single statistic whose distribution is model sensitive. Some of these mathematical and astrophysical models have been subjected to the selection functions derived from observational data of homogeneous samples of cosmic objects, and the resulting predicted probability functions have been compared with the corresponding empirical probability function derived directly from the observational data (cf. 31, 140, 167, 186, 208, 209). Probability functions derived from observational samples of galaxies with different ranges of absolute luminosity are consistent with the assumption that the less luminous galaxies do not fill in the voids (167, 186; see also Section 2.1.2). Finally, I note that the above statistical procedures to study voids, which have been applied extensively to samples of galaxies, seem not to have been applied to samples of Abell clusters of galaxies. This may be a consequence of a reluctance to accept the homogeneity of the statistical sample of Abell clusters (which was identified visually). Abell (4) documents the care taken to insure that the sample is statistically homogeneous so that it can be used for statistical studies [with the proviso that care must be exercised to allow for effects of the random and systematic uncertainties of the data, as has been done in the work by, e.g., N. Bahcall et al. (Section 2.2.1)]. Concrete studies to settle this issue are in progress.

Void-sensitive probability functions such as p(r) and g(r) derived from observational galaxy data have been related to the amplitudes of the n-point correlation functions by Sharp (179) [from data in the Zwicky catalogue (220)] and by Bouchet & Lachieze-Rey (31) [from data in the CfA catalog (93a)]. Masson (114a) points out that correlation function analysis cannot distinguish between overdensity and underdensity structure. Fry (77) and Hamilton (84) show that the qualitative and to some degree quantitative aspects of n(x, y, z) << n, the spatial density function at the small local densities appropriate to voids that is recovered from the amplitudes of the n-point correlation functions, is largely independent of the exact sequence of amplitudes [i.e. the correlation amplitudes provide a poor characterization of n(x, y, z) within voids].

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