Physical Foundations of Cosmology

250 Inflation I: homogeneous limit

quasiclassical (WKB) approximation:

χk∝

1

√

ω

exp

(

±i

∫

ωdt

)

. (5.78)

In this case the number of particles,nχ∼εχ/ω, is an adiabatic invariant and is
conserved. For most of the time the condition|ω ̇|ω^2 is indeed fulfilled. However,
every time the oscillating inflaton vanishes attj=m−^1 (j+ 1 / 2 )π, the effective
mass of theχfield, proportional to|cos(mt)|, vanishes. It is shortly before and after
tjthat the adiabatic condition is strongly violated:

|ω ̇|

ω^2

=

mg ̃^22 |cos(mt)sin(mt)|

(

k^2 +g ̃^22 cos^2 (mt)

) 3 / 2 ≥^1. (5.79)

Considering a small time intervaltm−^1 in the vicinity oftj, we can rewrite
this condition as
t/t∗
(
k^2 t∗^2 +(t/t∗)^2

) 3 / 2 ≥^1 , (5.80)

where

t∗(g ̃ m)−^1 /^2 =

1

m

(g ̃/m)−^1 /^2. (5.81)

It follows that the adiabatic condition is broken only within short time intervals
t∼t∗neartjand only for modes with

k<k∗t∗−^1 m(g ̃/m)^1 /^2. (5.82)

Therefore, we expect thatχparticles with the corresponding momenta are created
only during these time intervals. It is worth noting that the momentum of the
created particle can be larger than the inflaton mass by the ratio(g ̃/m)^1 /^2 >1; the
χparticles are produced as a result of a collective process involving many inflaton
particles. This is the reason why we cannot describe the broad resonance regime
using the usual methods of perturbation theory.
To calculate the number of particles produced in a single inflaton oscillation we
consider a short time interval in the vicinity oftjand approximate the cosine in
(5.76) by a linear function. Equation (5.76) then takes the form

d^2 χκ

dτ^2

+

(

κ^2 +τ^2

)

χκ= 0 , (5.83)

where the dimensionless wavenumberκ≡k/k∗and timeτ≡

(

t−tj

)

/t∗have
been introduced. In terms of the new variables the adiabaticity condition is broken
at|τ|<1 and only forκ<1. It is remarkable that the coupling constantg ̃, the
mass and the amplitude of the inflaton enter explicitly only inκ^2. The adiabaticity