relativistic species. Reheating is terminated when the rate of expansion of the Universe becomes smaller than the total decay rate of the inflaton field into new fields. Even though this perturbative reheating process is not the full story, it is good enough to estimate the reheating epoch. Thus, we show the oscillatory epoch and reheating period based on the given model. The reheating process and reheating temperature based on the Starobinsky model have been investigated in the literature [@09091737; @12041472; @12124466; @160604346]. We probe the reheating process for the $R^2 \ln R$-correction model and investigate any difference compared to those of the Starobinsky model.

Oscillation
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At the end of the inflation, the scalar fields reach to the minimum of the potential and start to oscillate around it. In order to describe this period properly, one needs to describe this epoch semi-analytically. The field equation of the general $f(R)$-gravity theories is obtained from the action (\[Sfx\]) $$f'(R) \tensor{R}{_\mu_\nu} - \frac{1}{2} f(R) \tensor{g}{_\mu_\nu} + \left[ \tensor{g}{_\mu_\nu} - \nabla_{\mu} \nabla_{\nu} \right] f'(R) = \frac{1}{\Mp^2} \tensor{T}{_\mu_\nu} \, , \label{fEq}$$ where prime means the derivatives with respect to $R$. For the homogeneous and isotropic background, one can obtain two field equations from the above 