334 Inflation II: origin of the primordial inhomogeneities
In a universe which undergoes a stage of accelerated expansion, the Hubble
scaleH−^1 =a/a ̇grows more slowly than the scale factorabecause the rate of the
expansiona ̇increases. Hence a perturbation which was initially inside the horizon
soon leaves it (see Figure 8.3) and starts to “feel” the curvature effects which
preserve its amplitude from decay. The amplitude even grows slightly. We will see
later that this growth of the amplitude is a rather general property of inflationary
scenarios and it results in the deviation of the spectrum from a flat one. Thus, the
initial amplitude of the subcurvature perturbation decays only until the moment
of horizon crossing. After that it freezes out and the perturbation is stretched to
galactic scales withnearlyunchanged amplitude. Because the curvature scale does
not change significantly during inflation, the freeze-out amplitude is nearly the
same for different scales and this leads to a nearly flat spectrum for the produced
inhomogeneities.
The initial quantum fluctuations are Gaussian. Subsequent evolution influences
only their power spectrum and preserves the statistical properties of the fluctuations.
As a consequence, simple inflation predictsGaussian adiabaticperturbations.
8.3 Quantum cosmological perturbations
In this section we develop a consistent quantum theory of cosmological perturba-
tions. We consider a flat universe filled by a scalar field condensate described by
the action
S=∫
p(X,φ)√
−gd^4 x, (8.40)where
X≡^12 gαβφ,αφ,β. (8.41)The Lagrangianp(X,φ)plays the role of pressure. Indeed, by varying action (8.40)
with respect to the metric, we obtain the energy–momentum tensor in the form of
an ideal hydrodynamical fluid (see Problem 5.17):
Tβα=(ε+p)uαuβ−pδβα. (8.42)Hereuν≡φ,ν/
√
2 Xand the energy densityεis given by the expression
ε≡ 2 Xp,X−p, (8.43)wherep,X≡∂p/∂X. Thus, a scalar field can be used to describe potential flow of an
ideal fluid. Conversely, hydrodynamics provides a useful analogy for a scalar field
with an arbitrary Lagrangian. Action (8.40) is enough to describe all single-field
inflationary scenarios, includingkinflation. Ifpdepends only onX, thenε=ε(X),