Physical Foundations of Cosmology

8.3 Quantum cosmological perturbations 339

stage is over we must use (8.67) and (8.68) directly. Inflation is usually followed
by an oscillatory stage where the scale factor grows as some power of time,a∝tp,
withpdepending on the scalar field potential. We have found that for the quadratic
potentialp= 2 /3 and for the quartic potentialp= 1 /2. Neglecting the decaying
mode we obtain from (8.67) and (8.68)



A

p+ 1

, δφ

Atφ ̇ 0

p+ 1

, (8.71)

that is, the amplitude of the gravitational potential freezes out after inflation.
The scalar field finally converts its energy into ultra-relativistic matter corre-
sponding top= 1 /2. This influences the perturbations only via the change of the
effective equation of state and the resulting amplitude is

^23 A. (8.72)

Using (8.70), we can expressAin terms ofδφ, ̇φ 0 andHat the moment of sound
horizon crossing, whencsk∼Ha. For those perturbations which leave the horizon
during inflation the final result is



2

3

(

H

δφ

φ ̇ 0

)

csk∼Ha

. (8.73)

Given initial quantum fluctuations, this is consistent with the estimate in ( 8.35) and
we infer that the amplitude of the perturbation in the radiation-dominated epoch
differs from its amplitude at the end of inflation only by a numerical factor of order
unity. Note that (8.73) can also be applied to calculate the perturbations in theories
with a non-minimal kinetic term.

Problem 8.7Using the integral representation of (8.59) (see (7.75)), calculate the
k^2 -corrections to the long-wavelength solution (8.61). Verify that the “conserved”
quantityζ∝θ^2 (u/θ)′(see also (7.72)) blows up during an oscillatory stage. Hence,
contrary to the claims often made in the literature, it cannot be used to trace the
evolution of perturbation through this stage.

Problem 8.8Synchronous coordinate system.(a) Verify that a scalar field pertur-
bation in the synchronous coordinate system can be expressed through the gravita-
tional potential as

δφs=δφ−φ ̇ 0

∫

dt=Fsφ ̇ 0 −φ ̇ 0

∫

cs^2

Ha ̇^2 dt, (8.74)

whereFsis a constant of integration corresponding to the fictitious mode. The
relation above is exact. In the long-wavelength limit the physical mode ofδφsis of
orderk^2. Considering a long-wavelength perturbation and using (8.70), show that