in the region C to obtain $\rho_{+}^p$ and the evolution equation  in the region R to obtain ${\mathbb H}_p$ and $\eta_{\rm T}^p$.

Our main focus in this paper is on the latter process. Investigating the various possibilities for the tunneling potential is beyond the scope of this paper. We simply use the known results under thin-wall approximation [@Sasaki:1997ex]. For a thin-wall bubble, the effective potential $U_{\rm T}$ is characterized by two constant parameters, $\eta_{\rm W}$ and $\Delta s$: $$\begin{aligned}
U_{\rm T}(\eta_{\rm C})
\approx \Delta s\delta (\eta_{\rm C}-\eta_{\rm W})\,,
\label{eq:thin-wall potential}\end{aligned}$$ It can be shown that $\eta_{\rm W}$ and $\Delta s$ are expressed in terms of the vacuum energies and the surface tension of the wall, $S_1=\int{{\rm d}}t_{\rm C}\dot{\phi}^2$, as $$\begin{aligned}
&&\Delta s=\frac{2}{\sqrt{(\alpha +1)^2+4\beta}}
\,,\label{eq:Delta s eq}\\
&&\eta_{\rm W}=\frac{1}{2}\ln
\left(\frac{[\sqrt{(\alpha +1)^2+4\beta}+\alpha +1]^{2}}
{4\beta}\right)
\,,\label{eq:eta_W eq}\end{aligned}$$ with $$\begin{aligned}
\alpha =\frac{4(H_{\rm L}^2-H_*^2)}{\kappa^{2} S_1^2}
\,,\  
\beta =\left(\frac{2H_*}{\kappa S_1}\right)^{2}\,,
\label{eq:alpha beta}\end{aligned}$$ where $H_{\rm L}=\sqrt{\kappa V(\phi_{\rm L})/3}$ and $H_*=\sqrt{\kappa V(\phi_{*})/3}$.

In terms of $\eta_{\rm W}$ and $\Delta s$, we can solve the equation for the mode function to obtain the reflection coefficient $\rho_{+}$: $$\begin{aligned}
\rho_{+}^p=\frac{-ie^{2ip\eta_{\rm W}}\Delta s}{2p+i\Delta s}
\,.
\label{eq:reflection}\end{aligned}$$ From Eq. , the transmission probability from the false vacuum is given by $\bigl|\sigma_{+}^p\bigl|^{2}=1/[1+(\Delta s/2p)^2]$. From 