6.4 Beyond linear approximation 283factor of
εm
ε(R→∞)=
9 π^2
165. 55 , (6.101)
the matter detaches from the Hubble flow and begins to collapse.
Formally the energy density becomes infinite att= 2 tm; in reality, however,
this does not happen because there always exist deviations from exact spherical
symmetry. As a result a spherical cloud of particles virializes and forms a stationary
spherical object.
Problem 6.9Consider a homogeneous spherical cloud of particles at rest and,
using the virial theorem, verify that after virialization its size is halved. Assuming
that virialization is completed att= 2 tm, compare the density inside the cloud with
the average density in the universe at this time. (HintThe virial theorem states
that at equilibrium,U=− 2 K, whereUandKare the total potential and kinetic
energies respectively.)
Problem 6.10Assuming thatη1 and expanding the expressions in (6.94) in
powers ofη, derive the following expansion for the energy density in powers of
(t/tm)^2 /^3 1:
ε=1
6 πt^2[
1 +
3
20
(
6 πt
tm) 2 / 3
+O
((
t
tm) 4 / 3 )]
, (6.102)
wheretmis defined in (6.99). The second term inside the square brackets is obviously
the amplitude of the linear perturbationδ.Thus, when the actual density exceeds
the averaged density by a factor of 5.5, according to the linearized theory
δ(tm)= 3 ( 6 π)^2 /^3 / 20 1. 06.Later, att= 2 tm, the Tolman solution formally givesε→∞, while the linear
perturbation theory predictsδ( 2 tm) 1. 69.
6.4.2 Zel’dovich solution
The geometrical shapes of realistic inhomogeneities are typically far from spher-
ical and their collapse is strongly anisotropic. To build intuition about the main
features of anisotropic collapse we consider the Zel’dovich solution. This solution
describes the nonlinear behavior of aone-dimensionalperturbation, superimposed
on three-dimensional Hubble flow. In this case the relation between the Eulerian
and Lagrangian coordinates can be written as
xi=a(t)(
qi−fi(
qj,t