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 [426]

where is the time at the onset of reheating. The constant is determined by matching Eq. (3.30) with the slow-roll inflationary solution at . Then we get and Taking the time average of oscillations in the regime , it follows that . This corresponds to the cosmic evolution during the matter-dominated epoch, i.e., . The gravitational effect of coherent oscillations of scalarons with mass is similar to that of a pressureless perfect fluid. During reheating the Ricci scalar is approximately given by , i.e. In the regime this behaves asIn 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 [88]. Then the action is given by

where . Taking the variation of this action with respect to gives We decompose the quantum field in terms of the Heisenberg representation: where and are annihilation and creation operators, respectively. The field can be quantized in curved spacetime by generalizing the basic formalism of quantum field theory in the flat spacetime. See the book [88] for the detail of quantum field theory in curved spacetime. Then each Fourier mode obeys the following equation of motion where is a comoving wavenumber. Introducing a new field and conformal time , we obtain where the conformal coupling correspond to . This result states that, even though (that is, the field is minimally coupled to gravity), still gives a contribution to the effective mass of . In the following we first review the reheating scenario based on a minimally coupled massless field ( and ). This corresponds to the gravitational particle production in the perturbative regime [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

where . Since is of the order of , one has for the mode deep inside the Hubble radius. Initially we choose the field in the vacuum state with the positive-frequency solution [88]: . 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 [626]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

The typical wavenumber in the -coordinate is given by , whereas in the -coordinate it is . Then the energy density per unit comoving volume in the -coordinate is [426] where in the last equality we have used the fact that the term approaches 0 in the early and late times.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

where we used the relation . Shifting the phase of the oscillating factor by , we obtain The proper energy density of the field is given by . Taking into account relativistic degrees of freedom, the total radiation density is which obeys the following equation Comparing this with the continuity equation (2.17) we obtain the pressure of the created particles, as Now the dynamical equations are given by Eqs. (2.15) and (2.16) with the energy density (3.45) and the pressure (3.47).In the regime the evolution of the scale factor is given by , and hence

where we have neglected the backreaction of created particles. Meanwhile the integration of Eq. (3.45) gives where we have used the averaged relation [which comes from Eq. (3.33)]. The energy density evolves slowly compared to and finally it becomes a dominant contribution to the total energy density () at the time . In [426] 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 [426], 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 as

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) [591].

Introducing a new field , Eq. (3.37) reads

As long as is larger than the order of unity, the last two terms in the bracket of Eq. (3.51) can be neglected relative to . Since the Ricci scalar is given by Eq. (3.33) in the regime , it follows thatThe 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

where The strength of parametric resonance depends on the parameters and . This can be described by a stability-instability chart of 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 [353]. Meanwhile, for large , a broad resonance can occur for a wide range of parameter space and momentum modes [354].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

where the non-adiabatic regime corresponds to . For small and we have around , where are positive integers. This corresponds to the time at which the Ricci scalar vanishes. Hence, each time crosses 0 during its oscillation, the non-adiabatic particle production occurs most efficiently. The presence of the mass term tends to suppress the non-adiabaticity parameter , but still it is possible to satisfy the condition around .For the model (3.6) it was shown in [591] 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) [354]. 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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