332 Inflation II: origin of the primordial inhomogeneities
Problem 8.4Show that the scalar field perturbations in the synchronous gauge are
given by
δφs=δφ−φ ̇ 0∫
dt (8.38)(compare with (7.30)). Substituting (8.34) into (8.38), verify thatδφs=C 1 φ ̇ 0 ,
whereC 1 is the integration constant in (8.38) corresponding to the purely coordinate
mode. It is easy to understand why this mode is fictitious by simply considering the
homogeneous fieldφ 0 (η)and making the coordinate transformation (7.27) which
preserves the synchronous gauge. As we will see in Problem 8.8, the long-wave-
length physical perturbations of the scalar fields are suppressed in the synchronous
gauge by a factor(kη)^2.
Problem 8.5It is clear that large-scale metric fluctuations after inflation can depend
only on the few parameters characterizing them during the inflationary stage. Give
arguments for why the most natural candidates for these parameters areδφ, ̇φ 0 and
H. Out of these parameters, build the reasonable dimensionless combination which
could describe metric fluctuations after inflation. Substituting forδφthe amplitude
of the quantum fluctuations, compare the estimate obtained with (8.35). Which
questions remain open in this dimensional-based approach?
Problem 8.6Consider two slow-roll fieldsφ 1 andφ 2 with potentialV(φ 1 ,φ 2 )=
V(φ 1 )+V(φ 2 )and verify that the nondecaying mode of the long-wavelength per-
turbations is given by
=A
H ̇
H^2
+B
1
H
V 1 V ̇ 2 −V ̇ 1 V 2
V 1 +V 2
. (8.39)
The first term on the right hand side here is similar to (8.34) and it can be interpreted
as the adiabatic mode. The second term describes the entropic contribution which is
present when we have more than one field. When two or more scalar fields play an
important role during inflation we can get a variety of different spectra and inflation,
to a large extent, loses its predictive power. Therefore, we will not consider this case
any further. (HintIntroduce the new variablesy 1 ≡δφ 1 /V 1 ,φandy 2 ≡δφ 2 /V 2 ,φ.)
8.2.3 Why do we need inflation?
It is natural to ask whether quantum metric fluctuations can be substantially am-
plified in an expanding universe without an inflationary stage. Let us explain why
this is impossible.
Quantum metric fluctuations are large only near the Planckian scale. For ex-
ample, in Minkowski space the typical amplitude of vacuum metric fluctuations