6.4 Beyond linear approximation 285
The decaying mode soon becomes negligible and does not influence the evolution
even in the nonlinear phase. Ignoring this mode we have
ε(q,t)=ε 0 (t)
(1−α(q^1 )δi(t)). (6.110)
In those places whereα
(
q^1)
is positive, the energy density exceeds the averaged
densityε 0 (t)and the relative density contrast grows. However, during the linear
stage, whenαδi1, the energy density itself decays. Only after the perturbation
enters the nonlinear regime (αδi∼1) does the inhomogeneous region drop out of
the Hubble flow and start to collapse. To estimate when this turnaround happens
we have to find when ̇ε(q,t)=0. The time derivative of the expression in (6.110)
vanishes when
ε(q,t)
ε 0 (t)= 1 + 3
H
(lnδi)•. (6.111)
In a flat matter-dominated universe,δi∝aand, according to (6.111), as soon as the
energy density exceeds the averaged density by a factor of 4, the region detaches
from the Hubble flow and begins to collapse (compare to (6.101)). The collapse is
one-dimensional and produces a two-dimensional structure known as a Zel’dovich
“pancake.” According to (6.110), at some moment of time the energy density of
the pancake becomes infinite. However, in contrast with spherical collapse, the
gravitational force and velocities at this moment remain finite. Once the matter
trajectories cross, the solution in (6.110) becomes invalid.
In places whereα
(
q^1)
is negative, the energy density always decreases. Matter
“escapes” from these regions and they eventually become empty.
In reality, the situation is more complicated because a typical inhomogeneity is
neither spherical nor one-dimensional. To describe the evolution of a perturbation
with an arbitrary shape, Zel’dovich suggested generalizing the solution in (6.110)
to
ε(q,t)=ε 0 (t)
(1−αδi(t))(1−βδi(t))(1−γδi(t)), (6.112)
whereα, β, γ characterize the deformation along the three principle axes of the
strain tensor and they now depend on all coordinatesqi. The corresponding strain
tensor
J=aI−aδi⎛
⎝
α 00
0 β 0
00 γ⎞
⎠, (6.113)
whereIis the unit matrix, satisfies (6.89) only to leading order. Therefore, the
approximate solution in (6.112) has a very limited range of applicability. In fact,