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

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*
4) corresponding to
*N*_{t} 52
superclusters distributed according to the selection function
*f* (*b*) = 10^{0.3(1 - csc b)} (where *b* is
Galactic latitude) in solid angle
_{t}
The void
was detected visually as an empty region of solid angle
_{v} =
*g*_{v}
_{t}
(where *g*_{v}
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 *N*_{t} superclusters with uniformly
Poisson-distributed locational coordinates, they estimated that the
statistical probability for the chance occurrence of a void of solid
angle
>
_{v} is
*P*_{v}
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
_{t}
to find the largest empty region,
_{v}, and
not by placing search windows of solid angle
_{v} at
random (or, alternatively, evenly spaced) locations within
_{t}]. Hence, Bahcall & Soneira's probability
formula
*P*_{v}
_{t} /
_{v} ×
*e*^{-Ntgv} became replaced
(for circular voids) by Politzer & Preskill's
*P*_{v}
_{t}
/ _{v}
× (*N*_{t} *g*_{v})^{2} ×
*e*^{-Ntgv} / *f*_{c}
(where *f*_{c} is a fiducial correction factor to
exclude circles that overlap the boundary of the sample), so that
*P*_{v} ~
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 *P*_{v} derived from
computer model simulations and the value derived analytically is
unexplained at this time.