326 Inflation II: origin of the primordial inhomogeneities
(compare with (5.24)). To linear order in metric perturbations andδφit becomes
δφ′′+ 2 Hδφ′−(
δφ−φ′ 0(
B−E′
))
+a^2 V,φφδφ
−φ 0 ′( 3 ψ+φ)′+ 2 a^2 V,φφ= 0. (8.15)This equation is valid in any coordinate system. Using the background equation
(8.14), we can easily recast it in terms of the gauge-invariant variables and,
defined in (7.19), and the gauge-invariant scalar field perturbation
δφ≡δφ−φ′ 0(
B−E′
)
. (8.16)
The result is
δφ
′′
+ 2 Hδφ
′
−δφ+a^2 V,φφδφ−φ 0 ′( 3 + )′+ 2 a^2 V,φ = 0. (8.17)Problem 8.3Derive (8.15) and (8.17). (HintAs previously noted, the quickest
way to derive gauge-invariant equations is to use the longitudinal gauge. After the
equations are obtained in this gauge, we simply have to replace the perturbation
variables by the corresponding gauge-invariant quantities:φl→ ,ψl→and
δφl→δφ. Then using the explicit expressions for , andδφwe can write the
equations in an arbitrary coordinate system.)
Equation (8.17) contains three unknown variables,δφ, and, and should be
supplemented by the Einstein equations. For these we need the energy–momentum
tensor for the scalar field, which follows from the action in (8.13) upon variation
with respect to the metricgαβ:
Tβα=gαγφ,γφ,β−(
gγδφ,γφ,δ−V(φ))
δαβ. (8.18)It is convenient to use (7.39). For the energy–momentum tensor (8.18), the perturbed
gauge-invariant componentsδT
0
i, defined in (7.35), are
δT
0
i=1
a^2φ′ 0 δφ,i−1
a^2φ′ 02(
B−E′
)
,i=1
a^2(
φ′ 0 δφ)
,i. (8.19)Equation (7.39) becomes
′+H = 4 πφ′ 0 δφ, (8.20)where we have setG=1. Finally, we note that the nondiagonal spatial components
of the energy–momentum tensor are equal to zero, that is,δTki=0 fori=k, and
hence=.
We will solve (8.17) and (8.20) in two limiting cases: for perturbations with a
physical wavelengthλphmuch smaller than the curvature scaleH−^1 and for long-
wavelength perturbations withλphH−^1. The curvature scale does not change
very much during inflation. On the other hand the physical scale of a perturbation,