Physical Foundations of Cosmology

6.4 Beyond linear approximation 279

to the dark matter. They are tightly coupled to radiation before recombination and
perturbations in the baryon component can start to grow only after recombination,
when the dark matter must already have begun to cluster. Furthermore, a high
baryon density would spoil nucleosynthesis.

6.4 Beyond linear approximation

The Hubble flow stretcheslinearinhomogeneities; their spatial size is proportional
to the scale factor. The relative amplitude of the linear perturbation grows, while
its energy density, equal to

ε=ε 0

(

1 +δ+O

(

δ^2

))

,

decays only slightly more slowly than the background energy densityε 0 .It is
obvious that when the perturbation amplitude reaches unity(δ∼ 1 ), the neglected
nonlinear terms∼δ^2 etc., become important. At this time the gravitational field
created by the perturbation leads to a contraction which overwhelms the Hubble
expansion. As a result the inhomogeneity drops out of the Hubble flow, reaches its
maximal size and recollapses to form a stable nonlinear structure.
Even for pressureless matter, exact solutions describing nonlinear evolution can
be obtained only in a few particular cases where the spatial shape of the inho-
mogeneity possesses a special symmetry. To build intuition about the nonlinear
behavior of perturbations we will derive exact solutions in two special cases: for
a spherically symmetric perturbation and for an anisotropic one-dimensional in-
homogeneity. The behavior of realistic nonsymmetric perturbations can then be
qualitatively understood on the basis of these two limiting cases.
Let us first recast the hydrodynamical equations (6.3), (6.8), (6.10) in a slightly
different form, which is more convenient for finding their nonlinear solutions. The
continuity equation (6.3) can be written as
(
∂
∂t

∣∣

∣∣

x

+Vi∇i

)

ε+ε∇iVi= 0 , (6.75)

where∇i≡∂/∂xi.Taking the divergence of the Euler equations (6.8) and using
the Poisson equation (6.10), we obtain
(
∂
∂t

∣

∣∣

∣

x

+Vi∇i

)(

∇jVj

)

+

(

∇jVi

)(

∇iVj

)

+ 4 πGε= 0 , (6.76)

where we have assumed that the pressure is equal to zero.
In the next step we replace the Eulerian coordinatesxwith the comoving La-
grangian coordinatesq,enumerating thematter elements:

x=x(q,t). (6.77)