Physical Foundations of Cosmology

5.5 Preheating and reheating 245

coupling is not so large, the decay can still be very efficient. The reason is that the
effective decay rate into bosons,eff, is equal toχ, given in (5.58), only if the
phase space ofχparticles is not densely populated by previously createdχparti-
cles. Otherwiseeffcan be made much larger by the effect of Bose condensation.
This amplification of the inflaton decay is discussed in the next section.
Taking into account the expansion of the universe, the equations for the number
densities of theφandχparticles can be written as

1

a^3

d

(

a^3 nφ

)

dt

=−effnφ;

1

a^3

d

(

a^3 nχ

)

dt

= 2 effnφ, (5.61)

where the coefficient 2 in the second equation arises becauseoneφparticle decays
intotwoχparticles.

Problem 5.10Substituting (5.60) into the first equation in (5.61), derive theap-
proximateequation

φ ̈+( 3 H+eff)φ ̇+m^2 φ 0 , (5.62)

which shows that the decay of the inflaton amplitude due to particle production
may be roughly taken into account by introducing an extra friction termeffφ ̇.Why
is this equation applicable only during the oscillatory phase?

5.5.2 Narrow resonance

The domain of applicability of elementary reheating theory is limited. Bose con-
densation effects become important very soon after the beginning of the inflaton
decay. Because the inflaton particle is “at rest,” the momenta of the two producedχ
particles have the same magnitudekbut opposite directions. If the corresponding
states in the phase space ofχparticles are already occupied, then the inflaton decay
rate is enhanced by a Bose factor. The inverse decay processχχ→φcan also take
place. The rates of these processes are proportional to
∣
∣〈nφ− 1 ,nk+ 1 ,n−k+ 1

∣

∣aˆk+aˆ+−kaˆ−φ

∣

∣nφ,nk,n−k〉

∣

∣^2 =(nk+ 1 )(n−k+ 1 )nφ

and
∣
∣〈nφ+ 1 ,nk− 1 ,n−k− 1

∣

∣aˆ+φaˆk−aˆ−−k

∣

∣nφ,nk,n−k〉

∣

∣^2 =nkn−k(nφ+ 1 )

respectively, whereaˆk±are the creation and annihilation operators forχparticles
andn±kare their occupation numbers. To avoid confusion the reader must always
distinguish the occupation numbers from the number densities keeping in mind
that the occupation number refers to a density per cell of volume( 2 π)^3 (in the
Planckian units) in the phase space, while the number density is the number of