be positive. One also obtains the Hubble parameter from the Eq.(\[SRdotH\]) $$H(t) \simeq H_{i} - \frac{M^2}{18 \left(2\alpha+3\beta+2\beta \ln x \right)} \left( t - t_{i} \right) \, . \label{Hit}$$ From this, one can estimate the expansion of the scale factor of the Universe during this region $$a(t_{\text{end}}) \simeq a_{i} \exp \left[ \frac{9 \left( 2\alpha + 3\beta + 2\beta \ln x_{i} \right) H_{i}^2}{M^2} \right] \label{SRa} \, ,$$ where we use the fact that $\ln x$ is almost constant in this region. Thus, one can constrain the $M^2$ to satisfy the e-foling number for the given values of $\alpha$ and $\beta$. We already do this numerically and do not need to probe any detail in this section. Second, we consider the oscillation period. During the slow-roll period, $\Bigl|\frac{1}{2} \frac{\dot{H}^2}{H} \Bigr| \ll \Bigl|3 \dot{H} H \Bigr|$ is satisfied. However, the magnitudes of these two terms become comparable as $H$ decreases and becomes small. And oscillatory phase is followed with $\Bigl|3 \dot{H} H \Bigr| \simeq 0$. $$\ddot{H} - \left( \frac{\alpha+\beta+\beta \ln x}{2\alpha+3\beta+2\beta \ln x} \right) \frac{\dot{H}^2}{H} + \frac{M^2}{6 \left( 2\alpha+3\beta+2\beta \ln x \right)} H + \frac{2\beta}{2\alpha+3\beta+2\beta \ln x } H^3 =  0 \, . \label{dotR3osc}$$ In order to obtain the oscillation period, $H^3$-term should be 