336 Inflation II: origin of the primordial inhomogeneities
and theδT 00 component is
δT 00 =δε=ε,XδX+ε,φδφ=ε,X(
δX−X′ 0δφ
φ 0 ′)
− 3 H(ε+p)δφ
φ′ 0=
ε+p
c^2 s((
δφ
φ 0 ′)′
+H
δφ
φ 0 ′−
)
− 3 H(ε+p)
δφ
φ 0 ′. (8.49)
We have used here the second equality in (8.45) to expressε,φin terms ofε,X,ε
andp, and introduced the “speed of sound”
c^2 s≡p,X
ε,X=
ε+p
2 Xε,X. (8.50)
For a canonical scalar field the “speed of sound” is always equal to the speed of
light,cs=1. The componentsδTi^0 are readily calculated and the result is
δTi^0 =(ε+p)u^0 δui=(ε+p)g^00
φ 0 ′
√
2 X 0δφ,i
√
2 X 0=(ε+p)(
δφ
φ 0 ′)
,i. (8.51)
Replacingδφbyδφ, defined in (8.16), and substituting (8.49) and (8.51) into (7.38)
and (7.39), one obtains the for the gauge-invariant variables, andδφ:
− 3 H(
′+H
)
= 4 πa^2 (ε+p)[
1
c^2 s((
δφ
φ 0 ′)′
+H
δφ
φ 0 ′−
)
− 3 H
δφ
φ′ 0]
,
(8.52)
(
′+H
)
= 4 πa^2 (ε+p)(
δφ
φ 0 ′)
. (8.53)
SinceδTki=0 fori=k,wehave= ; the two equations above are sufficient
to determine the gravitational potential and the perturbation of the scalar field. It is
useful, however, to recast them in a slightly different, more convenient form. Using
(8.53) to express in terms of′andδφand substituting the result into (8.52),
we obtain
=
4 πa^2 (ε+p)
c^2 sH(
H
δφ
φ 0 ′+
)′
, (8.54)
where the background equations (8.44) and (8.46) have also been used. Because
=, (8.53) can be rewritten as
(
a^2
H
)′
=
4 πa^4 (ε+p)
H^2(
H
δφ
φ 0 ′