Annu. Rev. Astron. Astrophys. 1988. 26:
245-294
Copyright © 1988 by . 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* / *H*_{0} (where
*c* is the vacuum speed of
light and *H*_{0} is the local Hubble constant) obeys the
Hubble relation
*V*_{H} = *H*_{0}*R*_{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
< *H*_{0}^{-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*_{v}^{2} =
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*_{v}^{2})*H*_{0}^{-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*_{v}*H*_{0}^{-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*_{uv}*V*_{ex}^{2} / 2, is
*E*_{ex} ~ 2 × 10^{63} ergs (the energy
equivalent of 10^{9} 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).