328 Inflation II: origin of the primordial inhomogeneities
V∼L^3. Assuming that the field is nearly homogeneous within this volume, we
can write its action (see (8.13)) in the form
S^12∫(
X ̇^2 +···
)
dt,whereX≡δφLL^3 /^2 and the dot denotes the derivative with respect to the physical
timet. It is clear thatXplays the role of the canonical quantization variable and
the corresponding conjugated momentum isP=X ̇∼X/L; in the latter estimate
we have assumed that the mass of the field is negligible and hence the field propa-
gates with the speed of light. The variablesXandPsatisfy the uncertainty relation
XP∼1(=1) and it follows that the minimal amplitude of the quantum fluc-
tuations isXm∼
√
LorδφL∼L−^1. Thus the amplitude of the minimal fluctuations
of a massless scalar field is inversely proportional to the physical scale. Taking into
account thatδφL∼|δφk|k^3 /^2 , wherek∼a/Lis thecomovingwavenumber, we
obtain
|δφk|∼
k−^1 /^2
a. (8.24)
Comparing this result with (8.23), we infer that|Ck|∼k−^1 /^2. The evolution of the
mode according to (8.23) preserves the vacuum spectrum.
The result obtained is not surprising and has a simple physical interpretation. On
scales smaller than the curvature scale one can always use the local inertial frame
in which spacetime can be well approximated by the Minkowski metric. Therefore
the short-wavelength fluctuations “think” they are in Minkowski space and the vac-
uum is preserved. Above, we have simply described this vacuum in an expanding
coordinate system, where the perturbations with given comoving wavenumbers are
continuously being stretched by the expansion. As a result, for a given physical
scale, they are replaced by perturbations which were initially on sub-Planckian
scales. This does not mean, however, that for a consistent treatment of quantum
perturbations we need nonperturbative quantum gravity. Given aphysical scale,
which is larger than the Planckian length, vacuum fluctuations with amplitudes
given above will always be present, irrespective of whether they are formally de-
scribed as “being stretched from sub-Planckian scales” in the expanding coordinate
system or as always existing at the given scale in the nonexpanding local inertial
frame.
We noted in Chapter 5 that inflation “washes out” all pre-existing classical in-
homogeneities by stretching them to very large scales. It is sometimes said that
inflation removes the “classical hairs.” However, it cannot remove quantum fluctu-
ations (“quantum hairs”). In place of the stretched quantum fluctuations, new ones
“are generated” via the Heisenberg uncertainty relation. For a given comoving