QUASIOPEN INFLATION

We show that a large class of two-field models of single-bubble open i nflation does not lead to infinite open universes, as was previously t hought, but to an ensemble of very large bur finite inflating ''island s.'' The reason is that the quantum tunneling responsible for the nucl eation of the bubble does not occur simultaneously along both field di rections and equal-time hypersurfaces in the open universe are not syn chronized with equal-density or fixed-field hypersurfaces. The most pr obable tunneling trajectory corresponds to a zero value of the inflato n field; large values, necessary for the second period of inflation in side the bubble, only arise as localized fluctuations. The interior of each nucleated bubble will contain an infinite number of such inflati ng regions of comoving size of order gamma(-1), where gamma is the sup ercurvature eigenvalue, which depends on the parameters of the model. Each one of these islands will be a quasi-open universe. Since the vol ume of the hyperboloid is infinite, inflating islands with all possibl e values of the held at their center will be realized inside of a sing le bubble. We may happen to live in one of those patches of comoving s ize d less than or similar to gamma(-1). where the universe appears to be open. In particular, we consider the ''supernatural'' model propos ed by Linde and Mezhlumian. There, an approximate U(1) symmetry is bro ken by a tunneling field in a first order phase transition, and slow-r oll inflation inside the nucleated bubble is driven by the pseudo Gold stone field. We find that the excitations of the pseudo Goldstone fiel d produced by the nucleation and subsequent expansion of the bubble pl ace severe constraints on this model. We also discuss the coupled and uncoupled two-field models.