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

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Previous reviews of topics discussed in Section 3 include de Lapparent et al. (57), Doroshkevich et al. (63), Fall (69a), Krauss (108), Peebles (144), Rees (148d), Shandarin & Zel'dovich (176), Silk et al. (182), Sunyaev & Zel'dovich (189), Waldrop (210b), White (211e), and Zel'dovich et al. (216).

3.1. Statistical Representations of Void Data

Once a homogeneous sample of superclusters or individual voids is identified, their structural properties can be determined and then studied statistically, just as is done for other cosmic objects (e.g. stars, supernova remnants, galaxies); in 1981, Oort (127) pointed the way. The locational properties of the galaxies or clusters that constitute the homogeneous data bases used to identify voids can also be studied by statistical methods to learn about void properties and their relation to predictions of models; the techniques and results are described below.

3.1.1. POISSON VOIDS (SUBTLETIES)     Individual voids in the space distribution of galaxies with characteristic lengths of ~ 50 Mpc were first recognized definitively by visual inspection (Sections 1.3, 2.1.1); the Coma and Hercules voids are illustrated in Figure 4. Bahcall & Soneira (15) presented a variety of evidence that suggests the physical presence of a void in the space distribution of the 71 northern Abell clusters of galaxies (statistical sample, distance classes D leq 4) corresponding to Nt appeq 52 superclusters distributed according to the selection function f (b) = 100.3(1 - csc b) (where b is Galactic latitude) in solid angle Omegat appeq pi The void was detected visually as an empty region of solid angle Omegav = gv Omegat (where gv appeq 1/7) in the surface distribution of Abell clusters. From an analysis of 100 computer model simulations and an analytical calculation for a model of Nt superclusters with uniformly Poisson-distributed locational coordinates, they estimated that the statistical probability for the chance occurrence of a void of solid angle Omega > Omegav is Pv leq 0.01. Politzer & Preskill [147; see also (135)] proved that a void search-procedure correction must be applied to Bahcall & Soneira's analytical calculation [to allow for the fact that Bahcall & Soneira identified the void by scanning the survey area Omegat to find the largest empty region, Omegav, and not by placing search windows of solid angle Omegav at random (or, alternatively, evenly spaced) locations within Omegat]. Hence, Bahcall & Soneira's probability formula Pv appeq Omegat / Omegav × e-Ntgv became replaced (for circular voids) by Politzer & Preskill's Pv appeq Omegat / Omegav × (Nt gv)2 × e-Ntgv / fc (where fc is a fiducial correction factor to exclude circles that overlap the boundary of the sample), so that Pv ~ 0.2. The value is even larger if the shape of the void is allowed additional degrees of freedom [see (147)]. The factor of > 20 discrepancy between the value of Pv derived from computer model simulations and the value derived analytically is unexplained at this time.

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