Physical Foundations of Cosmology

8.3 Quantum cosmological perturbations 341

whereOˆ ≡Oˆ()is a time-independent operator to be determined. Using the first
equation in (8.57) to expressuin terms of(v/z)′, we obtain

S=

1

2

∫ [

z^2

(v

z

)′Oˆ



(v

z

)′

+cs^2 vOˆv

]

dηd^3 x. (8.78)

Problem 8.9Write down the action for a massless scalar field in a flat de Sitter
universe. Comparing action (8.78) in the limiting case ̇φ 0 /H→0 to the action for
a free scalar field in the de Sitter universe, verify thatOˆ=.

With the result of the above problem, (8.78) becomes

S≡

∫

Ldηd^3 x=

1

2

∫(

v′^2 +c^2 svv+

z′′

z

v^2

)

dηd^3 x, (8.79)

after we drop the total derivative terms. Varying this action with respect tovwe
obtain

v′′−c^2 sv−

z′′

z

v= 0. (8.80)

Note that this equation also follows from the second equation in (8.57) after sub-
stitutinguin terms ofv.

Problem 8.10The long-wavelength solution of (8.80) can be written in a similar
manner to (8.61):

v=C( 1 v)z+C( 2 v)z

∫

η 0

dη

z^2

+O

(

(kη)^2

)

, (8.81)

whereC( 1 v)andC( 2 v)are constants of integration. Becauseuandvsatisfy a system
of two first order differential equations, there are only two independent constants
of integration. ThereforeC( 1 v)andC( 2 v)can be expressed in terms of theC 1 andC 2
in (8.61). Verify thatC( 1 v)=C 2 andC( 2 v)=−k^2 C 1.

QuantizationThe quantization of cosmological perturbations with action (8.79) is
thus formally equivalent to the quantization of a “free scalar field”vwith time-
dependent “mass”m^2 =−z′′/zin Minkowski space. The time dependence of the
“mass” is due to the interaction of the perturbations with the homogeneous expand-
ing background. The energy of the perturbations is not conserved and they can be
excited by borrowing energy from the Hubble expansion.
The canonical quantization variable

v=

√

ε,Xa

(

δφ+

φ 0 ′

H



)

=

√

ε,Xa

(

δφ+

φ 0 ′

H

ψ

)

(8.82)

is a gauge-invariant combination of the scalar field and metric perturbations.