negligible ([*i.e.*]{} $\beta \simeq 0$). With this condition, the solution for Eq.(\[dotR3osc\]) is given by $$\begin{aligned}
 & H(t) = c_{1} \times \cos^{\frac{M^2}{6 \omega^2}} \left[ \omega (t - t_{\text{osc}}) \right] \, , \nonumber \\ 
 & \text{where} \,\, \omega = \frac{M \sqrt{\alpha+2\beta+\beta \ln x}}{\sqrt{6} (2\alpha+3\beta+2\beta \ln x)} \label{Hosc} \,\,  \text{and} \,\, t_{\text{osc}} =  6 c_{2} \left( 2\alpha+3\beta+2\beta \ln x \right) \, . \end{aligned}$$ $c_1$ and $c_2$ are integral constants. $t_{\text{osc}}$ is the time when the oscillation period begins with $\Bigl|\frac{1}{2} \frac{\dot{H}^2}{H} \Bigr| = \Bigl|3 \dot{H} H \Bigr|$. One can estimate the order of the frequency, $\omega$ at this epoch. ${\cal O} (\omega) =  {\cal O} (M/\hbar) \sim 10^{37}$ Hz. We point out the oscillatory behavior before derive the approximate solution. The oscillation is powered by $\frac{M^2}{6\omega^2}$ and thus it should be an integer. This means $\beta \simeq 0$. Also, the oscillation should be damped and thus, one needs to replace $c_{1}$ with some function $g(t)$ in Eq.(\[Hosc\]). Now, one needs to specify the $g(t)$ in the $\alpha \gg \beta$ limit. For this we compare the $H(t)$s in both before and during the oscillation region. $$\label{Htiosc}
        H(t) = \begin{dcases} H_{i} - \dfrac{2\omega^2}{3} \left(t-t_{i} \right) = \dfrac{\omega}{3} \left( 1 - 2 \omega \left( t 