\left(1 - e^{-\sqrt{\frac{2}{3}} \bphi} \right)^{2} \frac{1+2W_{k}(X)}{16 \beta W_{k}(X)^{2}} = \left( \frac{1 - \bff_{,y}}{\bff_{,y}} \right)^2  \frac{1+2W_{k}(X)}{16 \beta W_{k}(X)^{2}}  \label{VforR2} \, , \end{aligned}$$ where $W_{k}$ is the ProductLog (Lambert function) of branch $k=0$ for the positive value of $\beta$ and $k=-1$ for the negative one. Even though there exist these exact analytic solutions, the numerical results for the dynamics of the system are more clear than the analytic ones due to the complexity of the Lambert function. Thus, we numerically solve the evolutions of the inflaton field. The behaviors of the inflaton potential for the different values of $\beta$ are shown in the Fig.\[fig-1\]. $M = 6 \times 10^{12}$ GeV is used in this figure. The solid, dotted, and dashed lines depict the evolution of the inflaton potential when $\beta = 0, -0.02$, and 0.02, respectively. The requirement for the attractive gravity and ghost free graviton is given by $\partial f/ \partial R \, (\partial \bff/ \partial y) > 0$. In order to avoid a curvature singularity and Dolgov-Kaasaki instability, one also needs the condition $\partial^2 f/ \partial R^2 \, (\partial^2 \bff/ \partial y^2) > 0$ [@09091737]. These conditions provide the upper bound on the values of $\beta$ for the given value 