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driven by the remnant oscillations of the inflaton conden- sate. The resulting distribution of fluctuations is charac- terized by the spectrum 

which is non-thermal (for a thermal distribution we would have n ( k ) ∼ k − $^{1}$). The evolution during the phase of turbulence has been 

shown [11] to be self-similar in the sense that as a function of time the spectrum scales as 

where τ is a rescaled time ( τ = t/t$_{0}$ , where t$_{0}$ is the time when the turbulent scaling regime begins), n$_{0}$ gives the initial distribution of the particles, and p and q are positive rational numbers whose values are determined in numerical simulations. This equation (53) describes the overall growth in the number of fluctuation quanta and at the same time gives the increase of the UV cutoff frequency as a function of time. 

The phase of turbulence ends once most of the energy has been drained from the inflaton field. At this time quantum processes take over and lead to the thermaliza- tion of the spectrum. In the case of an O ( N ) scalar field model, the preheating and thermalization was considered analytically in the large N approximation [56], and ap- plied to thermalization of fermions and gauge fields in hybrid inflation in [57] (see also [58] for a more general study of thermalization of quantum fields in an expand- ing universe). 

Note that the time interval of the resonant preheat- ing phase is given by the inflaton mass m and hence is very short on the Hubble time scale. The period when the intial re-scatterings take place and which ends when the turbulent scaling distribution becomes established is longer than the period of the intial resonance, but not by a large factor [11]. Thus, the time t$_{0}$ is of the same order of magnitude as the time when inflation ends. However, the period of driven turbulence is very long, in particular if all of the coupling constants in the field theory model are small. If c$_{r}$ is the fraction of the energy density ρ$_{I}$ at the end of inflation which remains in the inflaton at the beginning of the phase of turbulence, then a rough estimate of the time τ$_{th}$ of thermalization is [11] 

A typical value of p is p = 1 / 7. Given the normalization of m from the observed magnitude of the cosmological fluctuations we find τ$_{th}$ ∼ c 7 / 4 r 10 $^{21}$. Hence, we see that the temperature at which thermal equilibrium finally be- comes established is low. The resulting value of the re- heating temperature in fact agrees roughly with what is obtained from the perturbative arguments given in the previous subsection. 

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As an important application of the theory of reheating described in the previous sections we consider reheat- ing in supersymmetric models. Supersymmetry (SUSY) introduces new degrees of freedom and new parameters, and a large number of scalar fields that may acquire large VEVs during inflation. These elements can affect various aspects of reheating that we discussed in Sections III and IV. Here we demonstrate some of the possible effects in the context of a well-motivated SUSY model. 

The minimal supersymmetric SM (MSSM) is a well- motivated extension of the SM (for reviews see e.g. [59, 60]). The new fields in the MSSM are scalar partners of leptons and quarks, called sleptons and squarks respec- tively, and fermionic partners of gauge and Higgs fields called gauginos and Higgsinos respectively. The MSSM superpotential is given by 

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where H$_{u}$, H$_{d}$, Q, L, u, d, e in Eq. (55) are chiral super- fields representing the two Higgs fields (and their Hig- gsino partners), left-handed (LH) (s)quark doublets, right-handed (RH) up- and down-type (s)quarks, LH (s)lepton doublets and RH (s)leptons respectively. The dimensionless Yukawa couplings h$_{u}$, h$_{d}$, h$_{e}$ are 3 × 3 ma- trices in the flavor space, and we have omitted the gauge and flavor indices. The last term is the µ term, which is a SUSY version of the SM Higgs boson mass. 

Now we consider inflaton couplings to the MSSM fields. For a gauge singlet inflaton, the only renormalizable cou- pling occurs through the superpotential term 2 g Φ H$_{u}$H$_{d}$ , with Φ being the inflaton superfield (for details, see [61]). Taking into account the inflaton superpotential mass term ( m/ 2) ΦΦ, the renormalizable part of the scalar po- tential that is relevant for the inflaton decay into MSSM scalars is given by: 

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where χ$_{1}$$_{,}$$_{2}$ denotes the scalar component of ( H$_{u}$ ± H$_{d}$ ) $_{/}$√$_{2 superfields, and we have only considered the real}$ parts of the inflaton, φ , and χ$_{1}$$_{,}$$_{2}$ fields. 

We see that even in the simplest SUSY set up, the scalar potential is more involved than the non-SUSY case given in Eq. (13), which in turn can alter the picture of preheating presented in Section III (see the detailed discussion in Refs. [48, 61]). An interesting feature of Eq. (56) is that both the cubic φχ 2 and quartic φ $^{2}$χ2 interactions appear and SUSY naturally relates their 

