340 Inflation II: origin of the primordial inhomogeneities
in an inflationary phase
δφsFsφ ̇ 0 +1
2
A
φ ̇ 0
H(
kcs
Ha) 2
. (8.75)
Skipping the fictitious mode, expressAin terms ofδφs,Hand ̇φ 0 at the moment
of the sound horizon crossing. Compare the result obtained with the expression
previously derived forAin terms ofδφ.(HintTo derive the second equality in
(8.74), use (8.52) to expressδφin terms of and .)
(b) Substituting (8.67) into (7.29), verify that for a long-wavelength perturbation
ψsA+F 1 H, EsA∫
1
a^3(∫t
ad ̃t)
dt+F 1∫
dt
a^2+F 2 , (8.76)
whereF 1 andF 2 are constants of integration corresponding to fictitious modes.
Find the relation betweenFsandF 1 ,F 2. Write down the metric componentsδgik
in the synchronous coordinate system.
Starting with quantum fluctuations, the resulting amplitude of perturbations in
the post-inflationary epoch can be fixed if we knowδφat horizon crossing. The
natural question arises: whichδφplays the role of a canonical quantization variable?
We found in Problem 8.8 that one can get results differing by a numerical factor
depending on whether we relate the quantum perturbation toδφsorδφ. This is
not surprising because at the moment of horizon crossing the metric fluctuations
may become relevant and the Minkowski space approximation fails. To resolve the
gauge ambiguity and derive the exact numerical coefficients we need a rigorous
quantum theory.
8.3.3 Quantizing perturbations
ActionIn order to construct a canonical quantization variable and properly normal-
ize the amplitude of quantum fluctuations, we need the action for the cosmological
perturbations. To obtain it one expands the action for the gravitational and scalar
fields to second order in perturbations. After use of the constraints, the result is
reduced to an expression containing only the physical degrees of freedom. The
steps are very cumbersome but fortunately they can be avoided. This is because
the action for the perturbations can be unambiguously inferred directly from the
equations of motion (8.57) up to an overall time-independent factor. This factor can
then be fixed by calculating the action in some simple limiting case. The first order
action reproducing the equations of motion (8.57) is
S=∫ [(
v
z)′
Oˆ
(u
θ)
−
1
2
c^2 s(u)Ouˆ +1
2
c^2 svOˆv]
dηd^3 x, (8.77)