Physical Foundations of Cosmology

5.5 Preheating and reheating 249

than the rate of their escape, thennk<1 and we can use the elementary theory
of reheating. The other important effect is the decrease of the amplitude (t)due
to both the expansion of the universe and particle production. Because the width
of the resonance layer is proportional to , it becomes more and more narrow.
As a result the particles can escape from this layer more easily and they do not
stimulate the subsequent production of particles. The rescattering of theχparticles
also suppresses the resonance efficiency by removing particles from the resonance
layer. Another effect is the change of the effective inflaton mass due to the newly
producedχparticles; this shifts the center of the resonance layer from its original
location.
To conclude, narrow parametric resonance is very sensitive to the interplay of
different complicating factors. It can be fully investigated only using numerical
methods. From our analytical consideration we can only say that the inflaton field
probably decays not as “slowly” as in the elementary theory, but not as “fast” as in
the case of pure narrow parametric resonance.

5.5.3 Broad resonance

So far we have considered only the case of a small coupling constant. Quantum
corrections to the Lagrangian are not very crucial ifg<mandg ̃<(m/)^1 /^2. They
can therefore be ignored when we consider inflaton decay in the strong coupling
regime:m>g>m^2 /for the three-leg interaction and (m/)^1 /^2 >g ̃>m/for
the quartic interaction (5.73). In this case the condition for narrow resonance is not
fulfilled and we cannot use the methods above. Perturbative methods fail because
the higher-order diagrams, built from the elementary diagrams, give comparable
contributions. Particle production can be treated only as a collective effect in which
many inflaton particles participate simultaneously. We have to apply the methods
of quantum field theory in an external classical background−as in Problem 5.11.
Let us consider quartic interaction (5.73). First, we neglect the expansion of the
universe. Forg ̃ mthe mode equation (see (5.74)):

χ ̈k+ω^2 (t)χk= 0 , (5.76)

where

ω(t)≡

(

k^2 +g ̃^22 cos^2 mt

) 1 / 2

, (5.77)

describes a broad parametric resonance. If the frequencyω(t)is a slowly vary-
ing function of time or, more precisely,|ω ̇|ω^2 , (5.76) can be solved in the