178 Inflationary cosmology and creation of matter in the universe
Figure 4.5. Development of the resonance in the theory^12 m^2 φ^2 +^14 λφ^4 +^12 g^2 φ^2 χ^2
forg^2 /λ=5200. The upper curve corresponds to the massless theory, the lower curve
describes stochastic resonance with a theory with a massmwhich is chosen to be much
smaller than
√
λφduring the whole period of calculations. Nevertheless, the presence of a
small mass term completely changes the development of the resonance.
even though in many cases it is possible to estimate the final results. In particular,
one can show that the whole process of parametric resonance typically takes only
few dozen of oscillations, and the final occupation numbers of particles grow up
tonk∼ 102 g−^2 [16]. But a detailed description of the last stages of pre-heating
requires lattice simulations, as proposed by Khlebnikov and Tkachev [18].
The theory of pre-heating is very sensitive to the choice of the model. For
example, in the theory^14 λφ^4 +^12 g^2 φ^2 χ^2 the resonance does not become stochastic
despite expansion of the universe. However, if one adds to this theory even a very
small termm^2 φ^2 , the resonance becomes stochastic [17].
This conclusion is illustrated by figure 4.5, where we show the development
of the resonance both for the massless theory withg^2 /λ∼5200, and for the
theory with a small massm. As we see, in the purely massless theory the
logarithm of the number densitynk for the leading growing mode increases
linearly in timex, whereas in the presence of a massm, which we took to be much
smaller than
√
λφduring the whole process, the resonance becomes stochastic.
In fact, the development of the resonance is rather complicated even for
smallerg^2 /λ. The resonance for a massive field withm
√
λφin this case
is not stochastic, but it may consist of stages of regular resonance separated by
the stages without any resonance, see figure 4.6.
Thus we see that the presence of the mass term^12 m^2 φ^2 can modify the
nature of the resonance even if this term is much smaller than^14 λφ^4 .Thisisa
rather unexpected conclusion, which is an additional manifestation of the non-
perturbative nature of pre-heating.