predicted from the models for $N_{\text{e-foldings}} = 50$.[]{data-label="tab-2"}

Thus, even though the logarithmic corrections to the Starobinsky model is natural if one accepts the fact that $f(R)$-gravity is originated from the quantum corrections of the matter in the curved space-time, the correction is strongly constrained in order to satisfy the observation. In semi-classical theory, one can obtain the $R^2$-term from the local action and $R^2 \ln R$-term from the non-local action. The coupling constant $\alpha$ can be changed by varying the renormalization scale $M$, while the coefficient $\beta$ is related to the trace anomaly. $\beta$-value can be determined by the numbers of quantum fields for the different spins. And usually $\alpha \gg \beta$ is satisfied [@Vilenkin85]. Thus, if one can obtain the constrain of $\beta$ accurately, then one might be able to obtain the number of fields during the early universe. Even though,$\beta$ should be less than 1% compared to $\alpha$, its effect on the cosmological parameters can be distinguishable if the accuracies of observations reach to percentage level.

Reheating
=========

The basic reheating process of the Universe at the end of the inflation is that the oscillating inflaton fields produce radiations via the tree-level decay of inflaton particles into 