Physical Foundations of Cosmology

8.2 Perturbations on inflation (slow-roll approximation) 331

scales is

δ

(

k,tf

)

∼Akk^3 /^2 ∼

(

H

V

V,φ

)

k∼Ha

∼

(

V^3 /^2

V,φ

)

k∼Ha

. (8.35)

In particular, for the power-law potentialV=λφn/nwe obtain

δ (k,tf)∼λ^1 /^2

(

φ^2 k∼Ha

)n+ 42

∼λ^1 /^2 (lnλphHk)

n+ 42

, (8.36)

where (5.53) has been used ink∼aH≡akHkto expressφk^2 ∼Hain terms of the
physical wavelengthλph∼a

(

tf

)

k−^1. The spectrum (8.36) is shown in Figure 8.2.
The effect ofHkin the logarithm in (8.36) is not very significant; we make only a
slight error by takingHk∼H

(

tf

)

.

For a massive scalar fieldV=m^2 φ^2 /2 and the amplitude of the metric fluctua-
tions is

δ ∼mln

(

λphHk

)

. (8.37)

We will show in the next section that perturbations present at the end of inflation
survive the subsequent reheating phase practically unchanged. Since galactic scales
correspond to ln (λphHk)∼50 and we require the amplitude of the gravitational
potential to beO( 1 )× 10 −^5 , the mass of the scalar field should be about 10−^6
in Planck units orm∼ 1013 GeV. This determines the energy scale at the end of
inflation to beε∼m^2 ∼ 10 −^12 εPl. In the absence of a fundamental particle the-
ory we cannot predict the amplitude of the perturbations; it is a free parameter
of the theory. However, the shape of the spectrum is predicted: it has logarith-
mic deviations from a flat spectrum with the amplitude growing slightly towards
larger scales. We will see that this is a rather generic and robust prediction of
inflation.

H−^1 H− 1 af

ai

λph

∼λ

∼

1/2

δΦ

(ln (λphH))

n+ 42

Fig. 8.2.