Open inflation in the landscape

The open inflation scenario is attracting a renewed interest in the context of the string landscape. Since there are a large number of metastable de Sitter vacua in the string landscape, tunneling transitions to lower metastable vacua through the bubble nucleation occur quite naturally, which leads to a natural realization of open inflation. Although the deviation of Omega(0) from unity is small by the observational bound, we argue that the effect of this small deviation on the large-angle CMB anisotropies can be significant for tensor-type perturbation in the open inflation scenario. We consider the situation in which there is a large hierarchy between the energy scale of the quantum tunneling and that of the slow-roll inflation in the nucleated bubble. If the potential just after tunneling is steep enough, a rapid-roll phase appears before the slow-roll inflation. In this case the power spectrum is basically determined by the Hubble rate during the slow-roll inflation. On the other hand, if such a rapid-roll phase is absent, the power spectrum keeps the memory of the high energy density there in the large angular components. Furthermore, the amplitude of large angular components can be enhanced due to the effects of the wall fluctuation mode if the bubble wall tension is small. Therefore, although even the dominant quadrupole component is suppressed by the factor (1 - Omega(0))(2), one can construct some models in which the deviation of Omega(0) from unity is large enough to produce measurable effects. We also consider a more general class of models, where the false vacuum decay may occur due to Hawking-Moss tunneling, as well as the models involving more than one scalar field. We discuss scalar perturbations in these models and point out that a large set of such models is already ruled out by observational data, unless there was a very long stage of slow-roll inflation after the tunneling. These results show that observational data allow us to test various assumptions concerning the structure of the string theory potentials and the duration of the last stage of inflation.