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Abstract

We elaborate on the correspondence between the canonical partition function
in asymptotically AdS universes and the no-boundary proposal for positive
vacuum energy. For the case of a pure cosmological constant, the analytic
continuation of the AdS partition function is seen to define the no-boundary
wave function (in dS) uniquely in the simplest minisuperspace model. A
consideration of the AdS gravitational path integral implies that on the dS
side, saddle points with Hawking-Moss/Coleman-De Luccia-type tunnelling
geometries are irrelevant. This implies that simple topology changing
geometries do not contribute to the nucleation of the universe. The analytic
AdS/dS equivalence holds up once tensor fluctuations are added. It also works,
at the level of the saddle point approximation, when a scalar field with a mass
term is included, though in the latter case, it is the mass that must be
analytically continued. Our results illustrate the emergence of time from space
by means of a Stokes phenomenon, in the case of positive vacuum energy.
Furthermore, we arrive at a new characterisation of the no-boundary condition,
namely that there should be no momentum flux at the nucleation of the universe.