very specific predictions, which may allow us to test basic principles of the string inflation scenario and to study observational consequences of the decay of the metastable string theory vacua. The general requirement to this class of model is that $1-\Omega_0$ must be extremely small, which means that inflation after the tunneling must be very long.

The models to be studied in this paper will be quite generic, but we will keep in mind some of the expected features of tunneling in the landscape and the last stage of the slow-roll inflation. In particular, a typical vacuum energy in the landscape can approach Planck density or density close to the scale of the grand unification, even though of course there are many vacua with much smaller energy density. Meanwhile, in many models of slow-roll inflation based on the Kachru-Kallosh-Linde-Trivedi scenario of stabilization of stringy vacua [@Kachru:2003aw], the maximal Hubble constant at the last stage of the slow-roll inflation cannot be greater than the gravitino mass [@Kallosh:2004yh]. In such models, the typical vacuum energy of the decaying metastable vacuum in the open universe scenario is many orders of magnitude greater than the energy density during the last stage of the slow-roll 