248 Inflation I: homogeneous limit
nχ∝|χk|^2. Show that in the center of the first instability band,
nχ∝exp
(
4 πg
m^2
N
)
, (5.72)
whereNis the number of oscillations. Compare this result with (5.70) and explain
why they are different by a numerical factor in the exponent. Thus, Bose conden-
sation can be interpreted as a narrow parametric resonance in the first instability
band, and vice versa. Give a physical interpretation of the higher-order resonance
bands in terms of particle production.
Using the results of this problem we can reduce the investigation of the inflaton
decay due to the coupling
Lint=−^12 g ̃^2 φ^2 χ^2 , (5.73)
to the case studied above. In fact, the equation for a massless scalar fieldχ, coupled
to the inflatonφ= cosmt, takes the form
χ ̈k+
(
k^2 +g ̃^22 cos^2 mt
)
χk= 0 , (5.74)
which coincides with (5.71) form^2 χ= 2 g after the substitutionsg ̃^22 → 4 g
andm→m/2. Thus, the two problems are mathematically equivalent. Using this
observation and making the corresponding replacements in (5.72), we immediately
find that
nχ∝exp
(
πg ̃^22
4 m^2
N
)
. (5.75)
The condition for narrow resonance isg ̃ mand the width of the first resonance
band can be estimated from (5.65) ask∼m(g ̃^22 /m^2 ).
In summary, we have shown that even for a small coupling constant the elemen-
tary theory of reheating must be modified to take into account the Bose condensation
effect, and that this can lead to an exponential increase of the reheating efficiency.
Problem 5.12Taking a few concrete values forgandm, compare the results of
the elementary theory with those obtained for narrow parametric resonance.
So far we have neglected the expansion of the universe, the back-reaction of
the produced particles and their rescatterings. All these effects work to suppress
the efficiency of the narrow parametric resonance. The expansion shifts the mo-
menta of the previously created particles and takes them out of the resonance layer
(Figure 5.6(a)). Thus, the occupation numbers relevant for Bose condensation are
actually smaller than what one would expect according to the naive estimate (5.66).
If the rate of supply of newly created particles in the resonance layer is smaller