Lorentzian vacuum transitions: Open or closed universes?

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We consider the generalisation of quantum tunneling transitions in the WKB approximation to the time-independent functional Schr\"odinger and Wheeler-DeWitt equations. Following a Lorentzian approach, we compute the transition rates among different scalar field vacua and compare with those performed by Coleman and collaborators using the Euclidean approach. For gravity, we develop a general formalism for computing transition rates in Wheeler's superspace. This is then applied to computing decays in flat space and then to transitions in the presence of gravity. In the latter case we point out the complexities arising from having non-positive definite kinetic terms illustrating them in the simplified context of mini-superspace. This corresponds to a generalisation of the well-known `tunneling from nothing' scenarios. While we can obtain the leading term for the transitions obtained by Euclidean methods we also point out some differences and ambiguities. We show that there is no obstruction to keeping the spherically ($SO(4)$) symmetric closed slicing for the new vacuum after a de Sitter to de Sitter transition. We argue that this is the natural Lorentzian realisation of the Coleman-De Luccia instanton and that a closed universe is also obtained if the mini-superspace assumption is relaxed. This is contrary to the open universe predicted by Coleman-De Luccia which relies on an analytic continuation performed after bubble nucleation. Our findings may have important cosmological implications related to the origin of inflation and to the string landscape. In particular, they question the widespread belief that evidence for a closed universe would rule out the string landscape.