field, see e.g. [@Linde:1995xm; @Yamamoto:1996qq; @Sasaki:1996qn; @Green:1996xe; @GarciaBellido:1997te; @Garriga:1997ht; @GarciaBellido:1998wd; @GarciaBellido:1997uh; @GarciaBellido:1997hy]. However, synchronization of the tunneling in these models is only approximate. Therefore the geometry of the universe inside the bubble is not exactly described by the metric of an open homogeneous universe. That is why these models were called “quasiopen” [@Linde:1995xm].

In the simplest models of this type, the universe inside the bubble has an islandlike structure, since a sufficiently large number of e folds is possible only around the maximum of density formed by the statistical fluctuation. If we consider the simplest possibility that there is no direct interaction between the tunneling field and the inflaton field that drives inflation inside the bubble, these models are troubled with large supercurvature perturbations, especially when the energy scales before and after tunneling are significantly different. The angular power spectrum for the supercurvature perturbations is estimated as [@Sasaki:1996qn; @Yamamoto:1996qq; @GarciaBellido:1998wd] (see also [@Lyth:1995cw]) $$\begin{aligned}
\frac{\ell (\ell +1)}{2\pi}C_{\ell}^{(\Lambda )}
\approx\frac{\kappa}{\epsilon_{\rm inf}}
\left(\frac{H_{\rm L}}{2\pi}\right)^{2}
\left( 1-\Omega_{0}\right)^{\ell}
\,,\end{aligned}$$ where $\kappa =1/M_{\rm pl}^2=8\pi G$, $H_L$ is the Hubble rate on the false vacuum side, and $\epsilon_{\rm inf}$ denotes the slow-roll parameter during the standard slow-roll inflation. (In the following, we often use the Planck 