Criterion for bubble formation in a de Sitter universe

The influence of the shape of the scalar field potential on the outcome of vacuum decay in a de Sitter universe is studied. The sufficient condition for vacuum decay via bubble formation, described by the Coleman-de Luccia instanton, is examined and a necessary condition is found. Both conditions require the curvature of the potential to be greater than 4 X (Hubble constant)(2), but while the sufficient condition states that this inequality is to be valid at the top of the barrier, the necessary condition requires that it holds at least somewhere within the barrier. The conditions leave a "gray zone" in the parameter space which, however, seems to be forbidden for a quartic potential.