the decay of $\chi$ field after the inflation. In this scenario, we do not consider the coupling between $R$ and $\chi$ ([*i.e.*]{} no decay of the scalaron). One can use the same field equation given in Eq.(\[fEq\]) by specifying the energy momentum tensor as $$\tensor{T}{^\mu^\nu} \equiv {T_{\chi}}\tensor{\vphantom{T}}{^\mu^\nu} + {T_{r}}\tensor{\vphantom{T}}{^\mu^\nu}  = \partial^{\mu} \chi \partial^{\nu} \chi - \tensor{g}{^\mu^\nu} {\cal L}_{\chi} + \left( \rho_{r} + P_{r} \right) U^{\mu} U^{\nu} + P_{r} \tensor{g}{^\mu^\nu} \, , \label{Tmunu}$$ where $U^{\mu}$ is the four-velocity and $\rho_{r}$ and $P_{r}$ are the energy density and pressure of the radiation, respectively. In the FRW metric, one obtains the field equations for the $\chi$ and radiation, $$\begin{aligned}
 \ddot{\chi} + 3H \dot{\chi} + m_{\chi}^2 \chi &= -\Gamma_{\chi} \dot{\chi} \,\, , \label{chifeq} \\
\dot{\rho}_{r} + 4 H \rho_{r} &= \Gamma_{\chi} \dot{\chi}^{2} \,\, . \label{dotrho} \end{aligned}$$ Because one is interested in the oscillatory region in the background produced by the scalaron, the background evolutions are given by Eqs.(\[Htiosc\])-(\[Rtiosc\]). It means that one can ignore any backreaction on the background evolutions from $\chi$ and $\rho_{r}$ during this period. Thus, one can solve the field equations (\[chifeq\]) and (\[dotrho\]) both analytically and numerically. For this purpose, one can rewrite these equations as a function of e-folding 