Physical Foundations of Cosmology

8.3 Quantum cosmological perturbations 337

Finally, in terms of the new variables

u≡



4 π(ε+p)^1 /^2

,v≡

√

ε,Xa

(

δφ+

φ′ 0

H



)

, (8.56)

(8.54) and (8.55) become

csu=z

(v

z

)′

, csv=θ

(u

θ

)′

(8.57)

where

z≡

a^2 (ε+p)^1 /^2

csH

,θ≡

1

csz

=

√

8 π

3

1

a

(

1 +

p

ε

)− 1 / 2

. (8.58)

8.3.2 Classical solutions

Substitutingvfrom the second equation in (8.57) into the first gives a closed form,
second order differential equation foru:

u′′−c^2 su−

θ′′

θ

u= 0. (8.59)

The variablesuandθcoincide (up to irrelevant numerical factors) with the cor-
responding quantities defined in (7.63) and (7.66) for the hydrodynamical fluid.
However, now they describe the perturbations in the homogeneous scalar conden-
sate.
The solutions of (8.59) were discussed in the previous chapter. Considering a
short-wavelengthplane wave perturbation with a wavenumberk(c^2 sk^2 

∣

∣θ′′/θ

∣

∣),

we obtain in the WKB approximation

u

C

√

cs

exp

(

±ik

∫

csdη

)

, (8.60)

where∣ Cis a constant of integration. Thelong-wavelengthsolution, valid forcs^2 k^2 
∣θ′′/θ∣∣,is

u=C 1 θ+C 2 θ

∫

η 0

dη

θ^2

+O

(

(kη)^2

)

. (8.61)

Givenu, the gravitational potential can be inferred from the definition in (8.56):

== 4 π(ε+p)^1 /^2 u (8.62)

and a perturbation of the scalar field is calculated using (8.53):

δφ=φ′ 0

(a )′

4 πa^3 (ε+p)

=φ ̇ 0

(

̇+H

)

4 π(ε+p)

. (8.63)