312 Gravitational instability in General Relativity
is the projection operator. We will analyze small perturbations using the conformal
Newtonian coordinate system in which the metric takes the form
ds^2 =a^2 (η)[
( 1 + 2 )dη^2 −( 1 + 2 )δijdxidxj]
. (7.102)
The potentials andare equal only if the nondiagonal spatial components of
the energy–momentum tensor are equal to zero. This is obviously not true for an
imperfect fluid. However, one can easily check that even when the main contribution
to the gravitational potential comes from the imperfect fluid, the difference −
is suppressed compared to ,at least by the ratio of the photon mean free path
to the perturbation scale. After equality, the gravitational potential is mainly due
to the cold dark matter and the contribution of the nondiagonal components of the
energy–momentum tensor can be completely neglected. Therefore we set= .
In this case the Christoffel symbols are
^000 =H+ ′; ^00 i=i 00 = (^) ,i; ij^0 =
(
( 1 − 4 )H− ′
)
δij;
ij 0 =(
H− ′
)
δij; ikj = (^) ,jδik− (^) ,iδjk− (^) ,kδij,
(7.103)
and the zero component of the 4-velocity to first order in perturbations is equal to
u^0 =1
a
( 1 − ), u 0 =a( 1 + ). (7.104)Given these relations, it is easy to show that, to zeroth order in the perturbations,
the relationT0;αα=0 reduces to the homogeneous energy–momentum conservation
law andTiα;α=0 is trivially satisfied. To the next order, the equationT0;αα=0 leads
to
δε′+ 3 H(δε+δp)− 3 (ε+p) ′+a(ε+p)ui,i= 0. (7.105)Note that the shear viscosity does not appear in this relation. As for the remaining
equations,Tiα;α=0, if we are only interested in scalar perturbations, it suffices to
take the spatial divergence. We find
1
a^4(
a^5 (ε+p)ui,i)′
−
4
3
ηui,i+δp+(ε+p) = 0. (7.106)As already noted, the two equations above are separately valid for the dark matter
and the baryon–radiation plasma components.
Problem 7.16Derive (7.105) and (7.106).
Dark matterFor dark matter, the pressurepand the shear viscosityηare both
equal to zero. Taking into account thatεda^3 =const, we obtain from (7.105) that
the fractional perturbation in the energy of dark matter component,δd≡δεd/εd,