Physical Foundations of Cosmology

330 Inflation II: origin of the primordial inhomogeneities

Since 3H^2  8 πVduring inflation, we obtain the equation

d(yV)

dt

= 0 , (8.32)

which is readily integrated to give

y=A/V, (8.33)

whereAis a constant of integration. The final result for the nondecaying mode is

δφk=Ak

V,φ

V

,k= 4 πAk

φ ̇ 0

H

V,φ

V

=−

1

2

Ak

(

V,φ

V

) 2

. (8.34)

The behavior ofδφk(a)is shown in Figure 8.1. Fora<ak∼k/Hthe perturbation
is still inside the horizon and its amplitude decreases in inverse proportion to the
scale factor. After horizon crossing, fora>ak, the perturbation amplitude slowly
increases sinceV,φ/Vgrows towards the end of inflation. In particular, for a power-
law potential,V∝φn,wehaveδφk∝φ−^1. The integration constantAkin (8.34)
can be fixed by requiring thatδφkhas the minimal vacuum amplitude at the moment
of horizon crossing. Comparing (8.34) to (8.25), we find

Ak∼

k−^1 /^2

ak

(

V

V,φ

)

k∼Ha

,

where the indexk∼Hameans that the corresponding quantity is estimated at the
moment of horizon crossing. At the end of inflation

(

t∼tf

)

, the slow-roll condition
is violated andV,φ/Vbecomes of order unity. Therefore, it follows from (8.34)
that at this time the typical amplitude of the metric fluctuations on supercurvature

∼a−^1 V,

φ

V

ak af a

∼

δφk

ka−^1 /^2

k

Fig. 8.1.