Let us estimate the evolution of the Hubble parameter and the scale factor during reheating in more detail. If we neglect the r.h.s. of Eq. (3.7), we get the solution . Setting to derive the solution of Eq. (3.7), we obtain 
In order to study particle production during reheating, we consider a scalar field with mass . We also introduce a nonminimal coupling between the field and the Ricci scalar . Then the action is given by for the detail of quantum field theory in curved spacetime. Then each Fourier mode obeys the following equation of motion [565, 606, 426]. We then study the case in which the nonminimal coupling is larger than the order of 1. In this case the non-adiabatic particle production preheating [584, 353, 538, 354] can occur via parametric resonance.
In this case there is no explicit coupling among the fields and . Hence the particles are produced only gravitationally. In fact, Eq. (3.38) reduces to: . The presence of the time-dependent term leads to the creation of the particle . We can write the solution of Eq. (3.39) iteratively, as 
After the universe enters the radiation-dominated epoch, the term becomes small so that the flat-space solution is recovered. The choice of decomposition of into and is not unique. In curved spacetime it is possible to choose another decomposition in term of new ladder operators and , which can be written in terms of and , such as . Provided that , even though , we have . Hence the vacuum in one basis is not the vacuum in the new basis, and according to the new basis, the particles are created. The Bogoliubov coefficient describing the particle production is
During the oscillating phase of the Ricci scalar the time-dependence of is given by , where and ( is a constant). When we evaluate the term in Eq. (3.42), the time-dependence of can be neglected. Differentiating Eq. (3.42) in terms of and taking the limit , it follows that
In the regime the evolution of the scale factor is given by , and hence it was found that the transition from the oscillating phase to the radiation-dominated epoch occurs slower compared to the estimation given above. Since the epoch of the transient matter-dominated era is about one order of magnitude longer than the analytic estimation , we take the value to estimate the reheating temperature . Since the particle energy density is converted to the radiation energy density , the reheating temperature can be estimated as4
If is larger than the order of unity, one can expect the explosive particle production called preheating prior to the perturbative regime discussed above. Originally the dynamics of such gravitational preheating was studied in [70, 592] for a massive chaotic inflation model in Einstein gravity. Later this was extended to the f (R) model (3.6) .
Introducing a new field , Eq. (3.37) reads
The oscillating term gives rise to parametric amplification of the particle . In order to see this we introduce the variable defined by , where the plus and minus signs correspond to the cases and respectively. Then Eq. (3.52) reduces to the Mathieu equation[419, 353, 591]. In the Minkowski spacetime the parameters and are constant. If and are in an instability band, then the perturbation grows exponentially with a growth index , i.e., . In the regime the resonance occurs only in narrow bands around , where , with the maximum growth index . Meanwhile, for large , a broad resonance can occur for a wide range of parameter space and momentum modes .
In the expanding cosmological background both and vary in time. Initially the field is in the broad resonance regime () for , but it gradually enters the narrow resonance regime (). Since the field passes many instability and stability bands, the growth index stochastically changes with the cosmic expansion. The non-adiabaticity of the change of the frequency can be estimated by the quantity
For the model (3.6) it was shown in  that massless particles are resonantly amplified for . Massive particles with of the order of can be created for . Note that in the preheating scenario based on the model the parameter decreases more rapidly () than that in the model (3.6) . Hence, in our geometric preheating scenario, we do not require very large initial values of [such as ] to lead to the efficient parametric resonance.
While the above discussion is based on the linear analysis, non-linear effects (such as the mode-mode coupling of perturbations) can be important at the late stage of preheating (see, e.g., [354, 342]). Also the energy density of created particles affects the background cosmological dynamics, which works as a backreaction to the Ricci scalar. The process of the subsequent perturbative reheating stage can be affected by the explosive particle production during preheating. It will be of interest to take into account all these effects and study how the thermalization is reached at the end of reheating. This certainly requires the detailed numerical investigation of lattice simulations, as developed in [255, 254].
At the end of this section we should mention a number of interesting works about gravitational baryogenesis based on the interaction between the baryon number current and the Ricci scalar ( is the cut-off scale characterizing the effective theory) [179, 376, 514]. This interaction can give rise to an equilibrium baryon asymmetry which is observationally acceptable, even for the gravitational Lagrangian with close to 1. It will be of interest to extend the analysis to more general f (R) gravity models.
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