Physical Foundations of Cosmology

5.5 Preheating and reheating 251

violation is largest fork=0. In this case the parametersg ̃,andmdrop from
(5.83) and the amplitudeχκ= 0 changes only by a numerical, parameter-independent
factor as a result of passing through the nonadiabatic region at|τ|<1. Because
the particle densitynis proportional to|χ|^2 , its growth from one oscillation to the
next can be written as
(
nj+^1
nj

)

k= 0

=exp( 2 πμk= 0 ), (5.84)

where the instability parameterμk= 0 does not depend ong ̃, andm. For modes with
k=0, the parameterμk= 0 is a function ofκ=k/k∗. In this case the adiabaticity
is not violated as strongly as for thek=0 mode, and henceμk= 0 is smaller than
μk= 0. To calculate the instability parameters we have to determine the change of
the amplitudeχin passing from theτ<−1 region to theτ>1 region. This can
be done using two independent WKB solutions of (5.83) in the asymptotic regions
|τ|1:

χ±=

1

(

κ^2 +τ^2

) 1 / 4 exp

(

±i

∫ √

κ^2 +τ^2 dτ

)

|τ|−

(^12) ± (^12) iκ 2
exp

(

±

iτ^2

2

)

. (5.85)

After passing through the nonadiabatic region the modeA+χ+becomes a mixture
of the modesχ+andχ−, that is,

A+χ+→B+χ++C+χ−, (5.86)

whereA+,B+andC+are the complex constant coefficients. Similarly, for the
modeA−χ−,wehave

A−χ−→B−χ−+C−χ+. (5.87)

Drawing an analogy with the scattering problem for the inverse parabolic potential,
we note that the mixture arises due to an overbarrier reflection of the wave. The
reflection is most efficient for the waves withk=0 which “touch” the top of the
barrier.
The quasi-classical solution is valid in the complex plane for|τ|1. Traversing
the appropriate contourτ=|τ|eiφin the complex plane fromτ−1toτ1,
we infer from (5.85), (5.86) and (5.87) that

B±=∓ie−

π 2 κ 2

A±. (5.88)

The coefficientsC±are not determined in this method. To find them we use the
Wronskian

W≡χχ ̇ ∗−χχ ̇∗, (5.89)