Physical Foundations of Cosmology

252 Inflation I: homogeneous limit

whereχis an arbitrary complex solution of (5.83). Taking the derivative ofWand

using (5.83) to express ̈χin terms ofχ,wefind

W ̇ = 0 (5.90)

and henceW=const. From this we infer that the coefficientsA,BandCin (5.86)

and (5.87) satisfy the “probability conservation” condition

|C±|^2 −|B±|^2 =|A±|^2. (5.91)

SubstitutingBfrom (5.88), we obtain

C±=

√

1 +e−πκ^2 |A±|eiα±, (5.92)

where the phasesα±remain undetermined.

At|τ|1 the modes of fieldχsatisfy the harmonic oscillator equation with a

slowly changing frequencyω∝|τ|. In quantum field theory the occupation number

nkin the expression for the energy of the harmonic oscillator,

εk=ω(nk+ 1 / 2 ), (5.93)

is interpreted as the number of particles in the corresponding modek. In the adiabatic

regime (|τ|1) this number is conserved and it changes only when the adiabatic

condition is violated. Let us consider an arbitrary initial mixture of the modesχ+

andχ−. After passing through the nonadiabatic region att∼tj, it changes as

χj=A+χ++A−χ−→χj+^1 =(B++C−)χ++(B−+C+)χ−. (5.94)

Taking into account that

n+

1

2

=

ε

ω

ω|χ|^2 , (5.95)

we see that as a result of this passage the number of particles in the modekincreases

(
nj+^1 + 1 / 2
nj+ 1 / 2

)

k



ω

∣∣

χj+^1

∣∣ 2

ω

∣∣

χj

∣∣ 2 

|B++C−|^2 +|B−+C+|^2

|A+|^2 +|A−|^2

(5.96)

times, where we have averaged|χ|^2 over the time intervalm−^1 >t>ω−^1. With

BandCfrom (5.88) and (5.92), this expression becomes

(

nj+^1 + 1 / 2

nj+ 1 / 2

)

k



(

1 + 2 e−πκ

2 )

+

4 |A−||A+|

|A+|^2 +|A−|^2

cosθe−

π 2 κ 2 √

1 +e−πκ^2.

(5.97)