The Lorentzian Geometry of Tunneling in Global de Sitter at Late Time

Abstract

It is widely believed that Coleman--De Luccia (CDL) instantons characterize tunneling transitions in a de Sitter multiverse. Their most naive interpretation uses analytic continuation to Lorentzian de Sitter at the minimal size of the spatial three-sphere. However, what one really wants is a geometry where a small bubble of new vacuum forms within the huge spatial sphere of an old parent de Sitter. Even by applying de Sitter isometries to the original CDL solutions, this cannot in general be achieved. In particular, it fails in the gravity-dominated regime, i.e. for up-tunneling and for transitions with heavy domain walls. These cases remain pathological in that the whole multiverse is in the causal future of every single up-tunneling event. To solve this problem, we develop a Hamiltonian description of how a small off-shell bubble grows and eventually goes on shell within the large spatial sphere of late-time de Sitter. We provide the corresponding WKB analysis, recovering the CDL rate. We explain that our tunneling process and that of CDL are described by two different analytic continuations of a unique on-shell trajectory in the Hamiltonian treatment. Our analysis has crucial implications for the Larfors--Johnson problem, which questions the standard mechanism for populating the string-theoretic flux landscape on the basis of an instability of the relevant domain walls.