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Negative modes of Coleman-De Luccia bounces

We investigate the negative modes about Coleman-De Luccia bounces governing vacuum transitions in de Sitter space, with the goal of gaining physical insight into the various anomalous results associated with these that have been reported in the literature. For the case of bounces with radii much less that the horizon distance $H^{-1}$ we find two distinct regimes, distinguished by the magnitude of the bubble nucleation rate $Γ$. If $Γ/H^4 \gg 1$, then the behavior of the modes contributing to the determinant factors in $Γ$ is much as in flat spacetime, and the calculation of $Γ$ goes over smoothly to the flat spacetime calculation as the gravitational coupling is taken to zero. This is not the case if $Γ/H^4 \ll 1$. These two regimes correspond to the two possible outcomes of de Sitter vacuum decay --- either a rapidly completed transition or non-percolation and eternal inflation. For bounces with radii comparable to the horizon length, we confirm previous results concerning anomalous negative modes with support on the bounce wall. We also find further evidence supporting previous claims, based on thin-wall arguments, of the absence of expected negative modes for a class of bounces that arises when the initial and final vacua are nearly degenerate.