Physical Foundations of Cosmology

5.5 Preheating and reheating 247

The occupation numbersnkexceed unity, and hence the Bose condensation effect
is essential only if

nχ>

g

π^2

nφ. (5.67)

Taking into account that at the end of inflation ∼1, we infer that the occupation
numbers begin to exceed unity as soon as the inflaton converts a fractiongof its
energy toχparticles. The derivation above is valid only forg m^2 /8. Therefore,
ifm∼ 10 −^6 , then at most a fractiong∼m^2 ∼ 10 −^12 of the inflaton energy can be
transferred toχparticles in the regime wherenk<1. Thus, the elementary theory
of reheating, which is applicable fornk1, fails almost immediately after the
beginning of reheating. Given the result in (5.66), the effective decay rate (5.63)
becomes

eff

g^2

8 πm

(

1 +

2 π^2

g

nχ

nφ

)

, (5.68)

where we have used (5.58) forχ. Substituting this expression into the second
equation in (5.61), we obtain

1

a^3

d

(

a^3 nχ

)

dN

=

g^2

2 m^2

(

1 +

2 π^2

g

nχ

nφ

)

nφ, (5.69)

whereN≡mt/ 2 πis the number of inflaton oscillations. Let us neglect for a
moment the expansion of the universe and disregard the decrease of the inflation
amplitude due to particle production. In this case =const and fornk1 (5.69)
can be easily integrated. The result is

nχ∝exp

(

π^2 g

m^2

N

)

∝exp( 2 πμN), (5.70)

whereμ≡πg/

(

2 m^2

)

is the parameter of instability.

Problem 5.11Derive the following equation for the Fourier modes of the fieldχ
in Minkowski space:

χ ̈k+

(

k^2 +m^2 χ+ 2 g cosmt

)

χk= 0. (5.71)

Reduce it to the well known Mathieu equation and, assuming thatm^2 m^2 χ≥
2 |g |, investigate the narrow parametric resonance. Determine the instability bands
and the corresponding instability parameters. Compare the width of the first insta-
bility band with (5.65). Where is this band located? The minimal value of the initial
amplitude ofχkis due to vacuum fluctuations. The increase ofχkwith time can be
interpreted as the production ofχparticles by the external classical fieldφ, with