Physical Foundations of Cosmology

316 Gravitational instability in General Relativity

perturbations.For example, the dark matter might initially be distributed inho-
mogeneously on a homogeneous radiation background. It is clear that, at early
times when radiation dominates,δγand both vanish and the constant of integra-
tion on the right hand side of (7.118) is equal to zero. After equality, the dark matter
inhomogeneities induce the gravitational potential. Then, it follows from (7.118)
that the fluctuations in the radiation component on supercurvature scales are equal
toδγ= 4 ,where is mainly due to the cold dark matter fluctuations (see also
(7.90)). The differences between this case with entropic perturbations and the adi-
abatic case give rise to distinctive cosmic microwave background anisotropies.

Short-wavelength perturbations (kηr> 1 )We next consider the subcurvature
modes which enter the horizon before recombination. These perturbations are es-
pecially interesting since they are responsible for the acoustic peaks in the cosmic
microwave background spectrum.
To simplify the consideration, we neglect the contribution of baryons to the grav-
itational potential. This approximation is valid in realistic models where the baryons
constitute only a small fraction of the total matter density. Although we neglect the
contribution of the baryons to the gravitational potential, they cannot be completely
ignored since they substantially affect the speed of sound after equality.
In general, there exist four independent instability modes in the two-component
medium. The set of equations for the perturbations is rather complicated and they
cannot be solved analytically without making further approximations. Let us con-
sider the evolution of perturbations after equality, atη>ηeq.In this case the prob-
lem is greatly simplified if one notes that atη>ηeqthe gravitational potential
is mainly due to the perturbations in the cold dark matter and is therefore time-
independent for both the long-wavelength and the short-wavelength perturbations.
Thus, atη>ηeq,the potential can be considered as an external source in (7.113)
and the general solution of this equation can be written as a sum of the general
solution of the homogeneous equation (with =0) and a particular solution for
δγ. Using the variablexdefined bydx=c^2 sdηand taking into account that the time
derivatives of the potential on the right hand side of (7.113) are zero ( =const),
(7.113) becomes

d^2 δγ

dx^2

−

4 τγ

5 a



dδγ

dx

−

1

cs^2

δγ=

4

3 cs^4

, (7.122)

where the second term is due to the viscosity. If the speed of sound is slowly varying,
(7.122) has an obvious approximate particular solution:

δγ−

4

3 c^2 s

. (7.123)