300 Gravitational instability in General Relativity
In a nonexpanding universeH= 0 ,and the first equation exactly coincides with the
usual Poisson equation for the gravitational potential. In an expanding universe, the
second and third terms on the left hand side in (7.47) are suppressed on sub-Hubble
scales by a factor∼λ/H−^1 , and hence can be neglected.Thus (7.47) generalizes
the Poisson equation and supports the interpretation of as the relativistic gen-
eralization of the Newtonian gravitational potential. Note that, on scales smaller
than the Hubble radius, (7.47) can be applied even to nonlinear inhomogeneities,
because it requires only| |1 but not necessarily|δε/ε 0 | 1 .From (7.48) it
follows that the time derivative of(a )′serves as the velocity potential.
Givenp(ε,S),the pressure fluctuationsδpcan be expressed in terms of the
energy density and entropy perturbations,
δp=cs^2 δε+τδS, (7.50)wherecs^2 ≡(∂p/∂ε)Sis the square of the speed of sound andτ≡(∂p/∂S)ε.Taking
this into account and combining (7.47) and (7.49), we obtain the closed form
equation for the gravitational potential
′′+ 3(
1 +c^2 s)
H ′−c^2 s +(
2 H′+
(
1 + 3 c^2 s)
H^2
)
= 4 πGa^2 τδS. (7.51)Below we begin by finding the exact solutions of this equation in two particular
cases: (a) for nonrelativistic matter with zero pressure, and (b) for relativistic fluid
with constant equation of statep=wε.Then we analyze the behavior of adiabatic
perturbations(δS= 0 )for a general equation of statep(ε),and finally consider
entropy perturbations.
Nonrelativistic matter(p= 0 )In a flat matter-dominated universea∝η^2 andH=
2 /η.In this case (7.51) simplifies to
′′+6
η′= 0 , (7.52)
and has the solution
=C 1 (x)+C 2 (x)
η^5, (7.53)
whereC 1 (x)andC 2 (x)are arbitrary functions of the comoving spatial coordinates.
From (7.47) we find the gauge-invariant density perturbations
δε
ε 0=
1
6
[
(
C 1 η^2 − 12 C 1)
+
(
C 2 η^2 + 18 C 2) 1
η^5]
. (7.54)
The nondecaying mode of the gravitational potential,C 1 ,remains constant regard-
less of the relative size of the lengthscale of the inhomogeneity relative to the