Physical Foundations of Cosmology

8.2 Perturbations on inflation (slow-roll approximation) 325

Thus, in the case of the Gaussian random process we need to know onlyσk^2
or, equivalently,δf(k). For small perturbations Fourier modes evolve indepen-
dently. Therefore, the spatial distribution of inhomogeneities remains Gaussian
and only their spectrum changes with time. When the perturbations enter the non-
linear regime, different Fourier modes start to “interact.” As a result the statistical
analysis of nonlinear structure becomes very complicated.
In this chapter we consider only small inhomogeneities and our main task is
to derive the initial perturbation spectrum generated in inflation. This spectrum

will be characterized by the variance of the gravitational potentialσk^2 ≡| (^) k|^2 or,
equivalently, by the dimensionless variance
δ^2 (k)≡
| (^) k|^2 k^3
2 π^2

. (8.11)

In the following we refer toδ 2 (k)as the power spectrum. Givenδ^2 the corresponding
power spectrum for the energy density fluctuations can easily be calculated.

8.2 Perturbations on inflation (slow-roll approximation)

To aid our intuition we begin with anonrigorousderivation of the inflationary
spectrum in a simple model with a scalar field. Let us consider the universe filled
by a scalar fieldφwith potentialV(φ), and see how small inhomogeneitiesδφ(x,η),
superimposed on a homogeneous componentφ 0 (η), evolve during the inflationary
stage. In curved spacetime the scalar field satisfies the Klein–Gordon equation,

1

√

−g

∂

∂xα

(

√

−ggαβ

∂φ

∂xβ

)

+

∂V

∂φ

= 0 , (8.12)

which follows immediately from the action

S=

∫(

1

2 g

γδφ

,γφ,δ−V

)√

−gd^4 x. (8.13)

A small perturbationδφ(x,η)induces scalar metric perturbations and as a result
the metric takes the form (7.17). Substituting

φ=φ 0 (η)+δφ(x,η)

together with (7.17) into (8.12), we find that the Klein–Gordon equation for the
homogeneous component reduces to

φ 0 ′′+ 2 Hφ′ 0 +a^2 V,φ= 0 (8.14)