inflationary models [@Linde:2005ht]. However, in the context of string landscape scenario [@Kachru:2003aw; @Susskind:2003kw; @Freivogel:2004rd] it is quite possible that our part of the universe appeared as a result of quantum tunneling, after being trapped in one of the many metastable vacua of string theory. In many cases, tunneling can be described by a Euclidean $O(4)$-symmetric bounce solution called Coleman-De Luccia (CDL) instanton [@Coleman:1977py; @Coleman:1980aw]. Because of the symmetry of the bounce solution, the expanding bubble has $O(3,1)$ symmetry. The bubble formed by the CDL instanton looks from the inside like an infinite open universe [@Coleman:1977py; @Coleman:1980aw]. If there is no inflation after tunneling, the interior of the bubble becomes an almost empty curvature dominated universe with $\Omega_{0} \ll 1$, which is ruled out by observations. However, if the universe experienced a sufficiently long stage of slow-roll inflation inside the light cone emanating from the center of the bubble, the universe almost exactly flat, with $\Omega_{0} \approx
1$ [@Gott:1982zf; @Gott:1984ps; @Sasaki:1993ha; @Ratra:1994vw; @Bucher:1994gb; @Sasaki:1994yt; @Lyth:1995cw; @Yamamoto:1995sw; @Bucher:1995ga].

If inflation inside the bubble is too long, the universe will be absolutely flat, and hence we would be unable to tell whether our part of the universe was produced by this mechanism. Therefore 