7.3 Hydrodynamical perturbations 303For a plane wave perturbation,u∝exp(ikx),the solutions of (7.65) can easily be
found in two limiting cases: on scales much larger and scales much smaller than
the Jeans length.
Long-wavelength perturbationsWhencskη 1 ,we can omit the spatial deriva-
tive term in (7.65) and thenu∝θis obviously the solution of this equation. The
second solution is derived using the Wronskian,
uC 1 θ+C 2 θ∫
η 0dη
θ^2=C 2 θ∫
η ̄ 0dη
θ^2, (7.67)
where the latter equality is obtained by changing the lower limit of integration
fromη 0 to ̄η 0 and, thus, absorbing theC 1 mode.Using the definition in (7.66) and
integrating by parts, we reduce the integral in (7.67) to
∫
dη
θ^2
=
2
3
∫
a^2[
1 +
(
1
H
)′]
dη=2
3
(
a^2
H−
∫
a^2 dη)
. (7.68)
With this result the gravitational potential becomes
=(ε 0 +p 0 )^1 /^2 u=A(
1 −
H
a^2∫
a^2 dη)
=A
d
dt(
1
a∫
adt)
, (7.69)
wheret=
∫
adη.
Let us apply this result to find the behavior of long-wavelength, adiabatic per-
turbations(δS= 0 )in a universe with a mixture of radiation and cold baryons. In
this case the scale factor increases as
a(η)=aeq(
ξ^2 + 2 ξ)
, (7.70)
whereξ≡η/η(see (1.81)) Substituting (7.70) into (7.69), we obtain
=ξ+ 1
(ξ+ 2 )^3[
A
(
3
5
ξ^2 + 3 ξ+1
ξ+ 1+
13
3
)
+B
1
ξ^3]
, (7.71)
whereAandBare constants of integration multiplying the nondecaying and de-
caying modes respectively.
Problem 7.10Calculate the energy density fluctuations.
The behavior of the gravitational potential and the energy density perturbation
for an inhomogeneity which enters the Hubble horizon after equality, is shown in
Figure 7.2, where we have neglected the decaying mode. We see that andδε/ε 0
are constant at times both early and late compared toηeq∼η.After the transition
from the radiation- to the matter-dominated epoch, the amplitude of both and
δε/ε 0 decreases by a factor of 9/ 10 .During the period of matter domination,
remains constant whileδε/ε 0 begins to increase after the perturbation enters horizon