(P)reheating after inflation 177Figure 4.4.Early stages of parametric resonance in the theory^12 m^2 φ^2 2 in an expanding
universe with scale factora ∼t^2 /^3 forg= 5 × 10 −^4 ,m= 10 −^6 MP. Note that the
number of particlesnkin this process typically increases, but it may occasionally decrease
as well. This is a distinctive feature of stochastic resonance in an expanding universe. A
decrease in the number of particles is a purely quantum mechanical effect which would be
impossible if these particles were in a state of thermal equilibrium.
shows that it increases three times more often than it decreases, so the total
number of produced particles grows exponentially, though in a rather specific
way, see figure 4.4. We called this regimestochastic resonance.
In the course of time the amplitude of the oscillations of the fieldφdecreases,
and whengφbecomes smaller thanm, particle production becomes inefficient
and their number stops growing.
In reality the situation is even more complicated. First of all, created particles
change the frequency of oscillations of the fieldφbecause they give a contribution
∼g^2 〈χ^2 〉to the effective mass squared of the inflaton field [16]. Also, these
particles scatter on each other and on the oscillating scalar fieldφ, which leads
to additional particle production. As a result, it becomes extremely difficult to
describe analytically the last stages of the process of the parametric resonance,