Physical Foundations of Cosmology

7.3 Hydrodynamical perturbations 299

Vector perturbations The equations for the vector perturbations take the forms

Vi= 16 πGa^2 δT

0

i(V), (7.41)

(

Vi,j+Vj,i

)′

+ 2 H

(

Vi,j+Vj,i

)

=− 16 πGa^2 δT

i

j(V), (7.42)

whereViis defined in (7.24) andδT
α
β(V)is the vector part of the energy–momentum
tensor.

Tensor perturbations For the gravitational waves we obtain
(
h′′ij+ 2 Hh′ij−hij

)

= 16 πGa^2 δT

i

j(T), (7.43)

whereδT
i
j(T)is that part of the energy–momentum tensor which has the same
structural form ashij.

Problem 7.7Derive (7.41), (7.42) and (7.43).

7.3 Hydrodynamical perturbations

Let us consider a perfect fluid with energy–momentum tensor

Tβα=(ε+p)uαuβ−pδβα. (7.44)

One can easily verify that its gauge-invariant perturbations, defined in (7.35), can
be written as

δT 00 =δε, δTi^0 =

1

a

(ε 0 +p 0 )

(

δu
i+δu⊥i

)

, δTij=−δpδij, (7.45)

whereδε,δu
iandδpare defined in (7.20), (7.21). The only term, which contributes
to the vector perturbations is proportional toδu⊥i; all other terms have the same
structural form as the scalar metric perturbations.

7.3.1 Scalar perturbations

SinceδTji=0 fori=j,(7.40) reduces to

( −),ij= 0 (i=j). (7.46)

The only solution consistent with andbeing perturbations is=.Then
substituting (7.45) into (7.38)–(7.40) we arrive at the following set of equations for
the scalar perturbations:

− 3 H

(

′+H

)

= 4 πGa^2 δε, (7.47)

(a )′,i= 4 πGa^2 (ε 0 +p 0 )δu
i, (7.48)

′′+ 3 H ′+

(

2 H′+H^2

)

= 4 πGa^2 δp. (7.49)