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

3.2. Theoretical Studies

3.2.1. DYNAMICAL EVOLUTION OF VOIDS     A typical void has a radius R_v ~ 25 Mpc and a sharp contiguous shell of superclusters with a non-Hubble velocity (assumed outward relative to the center of the void) V_vs ~ 600-1400 km s^-1. (The range is defined by the velocity of the Local Group relative to the cosmic microwave background and the relative velocity of Abell clusters in superclusters.) This velocity includes a component V_in ~ 300 km s^-1 from infall of galaxies toward the center of a supercluster. To this reviewer, these observed parameter values seem nicely consistent with the results of the basic theoretical considerations now described.

We adopt a model consisting of a galaxy located in a group (or cluster), within a supercluster, constituting part of the contiguous shell of a void in a universal homogeneous background of matter. The velocity V_g and radius vector R_g of the galaxy are referred to an origin at the center of the void. The velocity of the galaxy is specified by the vector relation V_g = V_vir + V_H + V_in + V_out + V_ex, where V_vir is the virial velocity relative to the barycenter of the group, V_H is the Hubble (cosmological) velocity corresponding to the universal background, V_in is the supercluster infall velocity [caused by the overdensity of the supercluster relative to the mass density of the universal background], V_out is the void outflow velocity (caused by the underdensity of the void relative to the mass density of the universal background), and V_ex is the vector sum of all extra components [the only extra component suggested so far is that caused by a hypothesized explosive origin of a void (98, 133)]. The components V_vir and V_in are discussed in detail elsewhere (cf. 54, 69), and V_H, V_out, and V_ex are discussed below; a previous review with different perspectives is provided by Ostriker (130).

First consider a locally Newtonian, Euclidean, homogeneous, isotropic universe of mass density _u 5 × 10^-30 g cm^-3 consisting of non - interacting unit point masses (which are henceforth simply called "particles"). The relative velocity V_H of any two particles separated by a distance R_ij << c / H₀ (where c is the vacuum speed of light and H₀ is the local Hubble constant) obeys the Hubble relation V_H = H₀R_ij. Now consider a similar universe - the same model except for the presence of a single ellipsoidal void. For simplicity, the mass density of the void, _v, is assumed constant, so that its density deficiency is _uv = _u - _v. As pointed out, for example, by Icke (97), according to this model the void can be considered as a region of negative density in a uniform background. Therefore, a particle located on the perimeter of a void feels an outward force that decreases with increasing distance from the void center (cf. 148c, pp. 161-84); consequently, as time passes, asphericities will tend to disappear. In this way, Icke (97) explained analytically an effect that had been observed by Centrella & Melott (36) from a three-dimensional computer simulation of large-scale structure in the Universe. Icke's result was obtained independently by Fujimoto (79), whose extensive model calculations in the framework of an expanding universe are also applied to estimate time scales for achieving sphericity (which he finds are typically < H₀^-1) and to demonstrate that the neglect of the effect of outfall velocities leads to an overestimate of the radial length of the Boötes void (by a factor of ~ 1.4 according to the model calculations).

Consider a spherical void of radius R_v ~ 25 Mpc with a contiguous shell of particles moving outward at the velocity (relative to the Hubble flow and with virial motions removed) given by V_vs = V_in + V_out + V_ex. We adopt V_in ~ 300 km s^-1 and determine V_out and V_ex as follows. The acceleration of a particle on the perimeter of the void equals the unbalanced force, dV_out / dt = GM_uv / R_v² = 4 G _uv R_v / 3 directed outward from the void, where G is the Newtonian gravitational constant. A crude integration then gives V_out (GM_uv / R_v²)H₀^-1. If _uv = _u, then V_out ~ 600 km s^-1. The contiguous shell tends toward a sharp structure because points that lie at distances r > R_v experience a Keplerian falloff in outfall velocity (V_out ~ r^-1/2), so that particles nearer the void tend to overtake more distant particles.

An estimate of V_ex similar to that by Ostriker (130) is now made. An explosive origin of a void evacuates the mass M_uv. The median distance of this material from the center of the void is 0.8 R_v. Hence V_ex 0.2 R_vH₀^-1. From this relation, for R_v ~ 25 Mpc, we obtain V_ex ~ 250 km s^-1. For M_uv = M_u, the corresponding kinetic energy of the ejected material, E_ex ~ M_uvV_ex² / 2, is E_ex ~ 2 × 10⁶³ ergs (the energy equivalent of 10⁹ solar masses). The source of this large quantity of energy and other problems noted by Oort (128, p. 425) make difficult the creation of voids with a characteristic length of ~ 50 Mpc by this mechanism. Nevertheless, physical processes within blast waves generated in a cosmic explosion may be significant for the formation of galaxies and smaller voids, and details of models within this framework have been worked out (e.g. 34, 98, 99, 130, 174, 210).

From the above discussion, it should be clear that the observed structure and kinematics of voids are consistent with basic theoretical considerations to within the large uncertainties of both the observational data and the theory. The crude analysis presented above is intended both to illustrate some basic aspects of the dynamical evolution of a void and to underline the theoretical need for forthcoming observational results on detailed structural and kinematic parameters of homogeneous samples of voids and their contiguous shells. More sophisticated analyses of the dynamical evolution of voids are given in (26a, 71, 85, 91, 92, 112, 122, 124 - 126, 139, 143, 144b).