[**Seokcheon Lee$^{\,\dagger}$**]{}

$^\dagger$[*Research institute of natural science, Gyeongsang national university, 501 Jinju Daero, Jinju city, 52828, Korea*]{}

0.6in

**Abstract**

Starobinsky has suggested an inflation model which is obtained from the vacuum Einstein’s equations modified by the one-loop corrections due to quantized matter fields. Although the one-loop gravitational action is not known for a general FRW background, it can be obtained in a de Sitter space to give $\Mp^2 R + \alpha R^2 + \beta R^2 \ln (R/M^2)$. Thus, one needs to investigate the inflationary behavior of this model compared to the Starobinsky model ([*i.e.*]{} $\beta = 0$). The coefficient $\alpha$ can be changed by varying the renormalization scale $M^2$ and $\beta$ is obtained from the quantum anomaly which is related to the numbers of quantum fields. It has been assumed that $\alpha \gg \beta$. We investigate the viable values of $\alpha$ and $\beta$ based on the CMB observation. We also scrutinize the reheating process in this model.

Introduction
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In semi-classical gravity, one treats the background as classical, but with taking the back-reaction of the matter into account. In a curved space-time, even in the absence of classical matter or radiation, quantum fluctuations of matter fields give nontrivial contribution to 