Physical Foundations of Cosmology

8.2 Perturbations on inflation (slow-roll approximation) 327

λph∼a/k, grows. For the modes we will be interested in, the physical wavelength
starts smaller than the Hubble radius but eventually exceeds it.
Our strategy will be the following. We start with a short-wavelength perturbation
and fix its amplitude at the minimal possible level allowed by the uncertainty
principle (vacuum fluctuations). We then study how the perturbation evolves after
it crosses the Hubble radius.
We occasionally adopt the widespread custom of referring to the curvature
(Hubble) radius as the (event) horizon scale. To avoid confusion the reader should
be sure to distinguish the curvature radius from theparticlehorizon scale, which
grows exponentially during inflation. What is relevant for the dynamics of the pertur-
bations is the curvature scale, not the particle horizon size which has a kinematical
origin.

8.2.1 Inside the Hubble scale

The gravitational field is not crucial to the evolution of the short-wavelength pertur-
bations withλphH−^1 or, equivalently, withkHa∼|η|−^1. In fact, for very
largek|η|the spatial derivative term dominates in (8.17) and its solution behaves as
exp(±ikη)to leading order. The gravitational field also oscillates, so that ′∼k ,
and can be estimated from (8.20) as ∼k−^1 φ 0 ′δφ. Using this estimate and taking
into account that during inflationV,φφV∼H^2 , we find that only the first three
terms in (8.17) are relevant. Thus, for a plane wave perturbation with comoving
wavenumberk, this equation reduces to

δφ

′′

k+^2 Hδφ

′

k+k

(^2) δφ
k^0 , (8.21)
and with the substitutionδφk=uk/a, it becomes
u′′k+

(

k^2 −

a′′

a

)

uk= 0. (8.22)

Fork|η|1 the last term in (8.22) can be neglected and the resulting solution for
δφkis

δφk

Ck

a

exp(±ikη), (8.23)

whereCkis a constant of integration which has to be fixed by the initial conditions.
The physical ingredient is that the initial scalar field modes arise as vacuum quantum
fluctuations.

Quantum fluctuationsTo make a rough estimate for the typical amplitude of the
vacuum quantum fluctuationsδφLon physical scalesL, we consider a finite volume