320 Gravitational instability in General Relativity
noting that, at earlier times when the wavelength of the mode exceeds the curvature
scale, one should match the result for the long-wavelength perturbations:
δd 3 δγ/ 4 − (^3 0) k/ 2. (7.137)
Problem 7.19Determine the constants of integration in (7.136) and show that after
the perturbation enters the Hubble scale, but before equality,
δd− 9
[
C−
1
2
+ln(
kη/√
3
)
+O
(
(kη)−^1)]
(^0) k, (7.138)
whereC= 0. 577 ...is the Euler constant.
It is easy to see from (7.116) that, before equality, the contribution of the dark
matter perturbations to the gravitational potential is suppressed by a factorεd/εγ
compared to the fluctuations in the radiation component. At equality, the dark matter
contribution begins to dominate and the density perturbationδdstarts to grow∝η^2 ,
as shown in Section 7.3.1. The gravitational potential “freezes” at the value
(^) k
(
η>ηeq)
∼−
4 πGa^2 ε
k^2δd∣
∣
∣∣
ηeq∼O( 1 )
ln(
kηeq)
(
kηeq) 2 0 k (7.139)and stays constant until recombination.
More work is required to obtain the exact coefficients in (7.139).
Problem 7.20For short-wavelength perturbations, the time derivatives of the grav-
itational potential in (7.108) and (7.116) can be neglected compared to the spatial
derivatives. From these relations, it follows that
(
aδd′)′
− 4 πGa^3(
εdδd+1
3 c^2 s
εγδγ)
= 0. (7.140)
Show that the second term here induces corrections to (7.138) that become signifi-
cant only near equality. These corrections are mainly due to theεdδdcontribution.
Show that theεγδγterm remains negligible throughout and, hence, can be omitted
in (7.140). Then (7.140) coincides with the equation describing the instability in
the nonrelativistic cold matter component on a homogeneous radiation background
and its solution is given in (6.72). Atx1 this solution should coincide with
(7.138). Considering this limit, show that the integration constants in (6.72) are
C 1 − 9
(
ln(
2 kη∗
√
3)
+C−
7
2
)
(^0) k, C 2 (^9 0) k, (7.141)