However, on the basis of our present understanding of the situation, one should not be surprised to have a very long-lasting noninflationary stage after tunneling, which should last until the large energy density of the metastable vacuum state is reduced by many orders of magnitude and the slow-roll inflation begins. We will show that the behavior in the phase of the rapid rolling down significantly affects the CMB power spectrum. In the next subsection, we will investigate the field evolution inside the open universe in the region R in Fig. \[fig:open\_conformal\_chart\].

Inflation in the open universe {#sec:evol}
------------------------------

In order to study the field dynamics inside the light cone emanating from the center of the bubble (region R), it is useful to use the following identity: $$\begin{aligned}
\frac{{{\rm d}}\ln\rho_{\phi}}{{{\rm d}}\ln a_{\rm R}}
=-3\left( 1+w_{\phi}\right)
\,,\end{aligned}$$ where $\rho_{\phi}=\dot{\phi}^2/2+V$, $p_{\phi}=\dot{\phi}^2/2-V$ and $w_{\phi}\equiv p_{\phi}/\rho_{\phi}$. The asymptotic boundary conditions at the nucleation point are given by $$\begin{aligned}
&&a_{\rm R}(t)=t\,,\ \ \ 
\dot{\phi}(t)=-\frac{V'(\phi_{*})}{4}t\,.\end{aligned}$$ Thus, we have $$\begin{aligned}
1+w_{\phi}={\cal O}\biggl(\frac{\dot{\phi}^2}{V}\biggr)
={\cal O}\left(\epsilon_{*}H_*^2t^2\right)
\,.\label{eq:CD 1+w behavior}\end{aligned}$$ where we have introduced $$\epsilon\equiv {1\over 2\kappa}\left(\frac{V'}{V}\right)^{2}\,,
\label{slow-roll0}$$ and $\epsilon_{*}=\epsilon (\phi_{*})$. We have also introduced $H_*^2\equiv \kappa V(\phi_{*})/3$, which is not the expansion rate at $t=0$. This suppressed decreased rate of 