Physical Foundations of Cosmology

338 Inflation II: origin of the primordial inhomogeneities

Taking into account that

ε+p= 2 Xp,X=

1

a^2

φ 0 ′^2 p,X (8.64)

and substituting (8.60) into (8.62) and (8.63 ), we have

 4 πCφ ̇ 0

√

p,X

cs

exp

(

±ik

∫

cs

a

dt

)

, (8.65)

δφC

√

1

csp,X

(

±ics

k

a

+H+···

)

exp

(

±ik

∫

cs

a

dt

)

, (8.66)

for ashort-wavelengthperturbation.
In thelong-wavelengthlimit the calculation is identical to that done in deriving
(7.69), and the result is

A

d

dt

(

1

a

∫

adt

)

=A

(

1 −

H

a

∫

adt

)

, (8.67)

δφAφ ̇ 0

(

1

a

∫

adt

)

, (8.68)

whereAis a constant of integration. (A second constant of integration corresponding
to the decaying mode can always be shifted to the lower limit of integration.)
Let us first find out how a perturbation behaves during inflation. It follows from
(8.65) and (8.66) that in the short-wavelength regime both metric and scalar field
perturbations oscillate. The amplitude of the metric perturbation is proportional to
φ ̇ 0 and it grows only slightly towards the end of inflation, while the amplitude of
scalar field perturbation decays in inverse proportion to the scale factor. After a
perturbation enters the long-wavelength regime it is described by (8.67) and (8.68).
These formulae are simplified during slow-roll. Integrating by parts, we obtain the
following asymptotic expansion:

1

a

∫

adt=

1

a

∫

da

H

=H−^1 −

1

a

∫

da

H

(

H−^1

)•

=H−^1

[

1 −

(

H−^1

)•

+

(

H−^1

(

H−^1

)•)•

−···

]

+

B

a

, (8.69)

whereBis a constant of integration corresponding to the decaying mode. Neglecting
this mode we find that to leading order

A

(

H−^1

)•

=−A

H ̇

H^2

, δφA

φ ̇ 0

H

. (8.70)

It is easy to see that for standard slow-roll inflation these formulae are in agreement
with (8.34). Result (8.70) is applicable only during inflation. After the slow-roll