306 Gravitational instability in General Relativity
where[X]±≡X+−X−denotes the jump of a variableXon#.The integrand in
(7.78) is singular and to perform the integration we note that
c^2 sθ^2 =(
p′ 0
ε′ 0)(
1
a^2ε 0
ε 0 +p 0)
=
ε 0
3 a^2 H(
1
ε 0 +p 0)′
−
ε 0
a^2 (ε 0 +p 0 ).
The second term in the latter equality does not contribute to the integral,while the
first term gives
#∫+ 0#− 0c^2 sθ^2 (u
θ)
dη=[
ε 0
3 a^2 H(
1
ε 0 +p 0)
(u
θ)]
±, (7.79)
taking into account thata,εandu/θare continuous. Substituting (7.79) into (7.78)
and expressinguin terms of the gravitational potential, we finally arrive at the
following matching conditions:
[ ]±= 0 ,[
ζ−2
9 H^2
1 +w]
±= 0 , (7.80)
whereζis defined in (7.72), (7.73). For long-wavelength perturbations the term
proportional to can be neglected. Hence on superhorizon scales the matching
conditions reduce to the continuity of and to the conservation law forζ.
Problem 7.12Assuming a sharp transition from the radiation- to the matter-dom-
inated epoch, determine the amplitude of metric perturbations after the transition
for both short- and long-wavelength perturbations.
Problem 7.13Write down the matching conditions explicitly in terms of the metric
perturbations in the synchronous coordinate system.
Entropy perturbationsUntil now we have been considering adiabatic perturba-
tions in an isentropic fluid where the pressure depends only on the energy density.
In a multi-component media both adiabatic and entropy perturbations can arise.
Generally speaking, the analysis of perturbations in this case is rather complicated
because of extra instability modes due to the relative motion of the components. We
will consider the problem of cold matter mixed with a baryon–radiation plasma in
Section 7.4. Here we content ourselves with studying a fluid of cold baryons tightly
coupled to radiation. The baryons do not move with respect to the radiation and this
simplifies our task enormously. In particular, we can still use the one-component
perfect fluid approximation. However, in this case the pressure depends not only on
the energy density, but also on the baryon-to-radiation distribution, characterized
by the entropy per baryon,S∼Tγ^3 /nb,wherenbis the number density of baryons.
Consequently, entropy perturbations can arise. We can study the evolution of these