278 Gravitational instability in Newtonian theory
Problem 6.6Verify that ifw=−1orw=− 1 /3, thenδ∝H is the solution
of (6.46) in the universe with arbitrary curvature.Using (6.55), find the general
solutions in these cases and analyze their behavior in open and closed universes.
Radiation backgroundThe Jeans length for radiation is comparable to the horizon
size because the speed of sound in the radiation component is of order the speed of
light (c^2 s= 1 /3). Therefore cold dark matter, which interacts only gravitationally
with radiation, does not induce significant perturbations in radiation and to study
the growth of inhomogeneities in cold matter alone we can still use (6.64), setting
w= 1 /3. In this case,
δ 1 (x)= 1 +3
2
x (6.71)satisfies (6.64). The other independent solution can be found by substituting (6.71)
into (6.66). The integral can be calculated explicitly and the general solution of
(6.64) is
δ(x)=C 1(
1 +
3
2
x)
+C 2
[(
1 +
3
2
x)
ln√
1 +x+ 1
√
1 +x− 1− 3
√
1 +x]
. (6.72)
At early times, during the radiation-dominated stage (x1), the amplitude of
perturbations grows as
δ(x)=(C 1 − 3 C 2 )−C 2 ln(x/ 4 )+O(x), (6.73)that is, logarithmically at most. Thus, by influencing the rate of expansion, the
radiation suppresses the growth of inhomogeneities in the cold component. After
matter–radiation equality, matter overtakes radiation and atx1, the amplitude
of perturbation is
δ(x)=C 1(
1 +
3
2
x)
+
4
15
C 2 x−^3 /^2 +O(
x−^5 /^2)
, (6.74)
that is, it grows proportionally to the scale factor. Since the perturbations cannot
be amplified significantly during the radiation epoch, small initial perturbations
can produce nonlinear structure only if the cold matter starts to dominate early
enough. This imposes a lower bound on the amount of cold matter. In particular,
if the amplitude of the initial inhomogeneities is of order 10−^4 , as favored by the
observed CMB fluctuations, we can explain the nonlinear structure seen only if the
initial perturbations started to grow before recombination. This is possible if there
exists a colddarkmatter component ofnon-baryonicorigin, which interacts only
gravitationally with radiation. We can reconcile the small initial perturbations with
the observed large-scale structure only if this dark matter constitutes about 30% of
the present critical density. It is clear that baryons cannot substantially contribute