\omega \Bigr) \cos \left[\left(t-t_{\text{osc}}\right) \omega \right] & \\ 
                                              + 3 \cos \left[3 \left(t-t_{\text{osc}}\right) \omega \right]+8 \omega  \Bigl( 2+\left(t -t_{\text{osc}}\right) \omega \Bigr) \sin [(t-\tosc) \omega ] \Biggr) &, \,\, \tosc \leq t  \, . \end{dcases}$$

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We show the evolution of $H$ and $R$ in Fig.\[fig-2\]. One should note that the shape and size of the oscillatory phase depend only on the $M$ and the $\alpha$ because one should use $\alpha \gg \beta$ in order to obtain the oscillatory behavior on the Hubble parameter as shown in the above.

Reheating
---------

The scalaron reaches to its minimum potential at $\phi = 0$ ([*i.e.*]{} $R=0$) and starts to oscillate. These oscillations excite the fields and reheat the Universe. To estimate the reheating, one can add the simple case of a scalar field $\chi$ to the original action in Eq.(\[Sfx\]) $$\begin{aligned}
 S &\equiv \int d^{4} x \sqrt{-g} \left( \frac{\bMp^2}{2} f \left(R \right) + {\cal L}_{\chi} + {\cal L}_{r} \right) \label{SfRchi}  \\
&= \frac{1}{2} \int d^{4} x \sqrt{-g} \left( \bMp^2 f \left(R \right) -  \tensor{g}{^\mu^\nu} \partial_{\mu} \chi \partial_{\nu} \chi - m_{\chi}^2 \chi^2 \right) + S_{r} \, , \nonumber \end{aligned}$$ where $S_{r}$ is the action of the radiation which is produced by 