the analytic continuation of the bounce solution. It is convenience to use the coordinate $\eta$ defined by ${{\rm d}}t =a(\eta ){{\rm d}}\eta$. The coordinates in the Lorentzian regime are given by (see e.g. [@Yamamoto:1996qq; @Sasaki:1994yt]) $$\begin{aligned}
&&\eta_{\rm E}=\eta_{\rm C}
=-\eta_{\rm R}-\frac{\pi}{2}i
=\eta_{\rm L}+\frac{\pi}{2}i
\,,
\label{eq:eta relation}
\\
&&\chi_{\rm E}=-i\chi_{\rm C}+\frac{\pi}{2}
=-i\chi_{\rm R}
=-i\chi_{\rm L}\,,\\
&&a_{\rm E}=a_{\rm C}=ia_{\rm R}=ia_{\rm L}\,.\end{aligned}$$ The Penrose diagram for this open Friedmann-Robertson-Walker universe is presented in Fig. \[fig:open\_conformal\_chart\].

![The Penrose diagram for the bubble nucleating universe when the inside of the bubble terminates in a de Sitter vacuum.[]{data-label="fig:open_conformal_chart"}](open_conformal_chart.eps){width="80mm"}

Typically, CDL instantons exist only if $V''>\kappa V$ [@Jensen:1983ac]. This condition, in combination with the requirement of the subsequent long stage of the slow-roll inflation, requires a very special choice of potentials in the single-field models [@Linde:1998iw; @Linde:1999wv]. Alternatively, one may consider multifield models [@Linde:1995xm; @Sasaki:1996qn; @Green:1996xe; @GarciaBellido:1997hy; @GarciaBellido:1997uh]. In this paper, we will concentrate on the single-field models and assume that the CDL instanton exists. Meanwhile, inflation typically requires $V''\ll \kappa V$. It is difficult to make these two conditions compatible. Therefore one may expect that the slow-roll inflation does not start immediately after the CDL tunneling, and there must be some intermediate stage of rapid rolling 