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Abstract

Realising the no-boundary proposal of Hartle and Hawking as a consistent
gravitational path integral has been a long-standing puzzle. In particular, it
was demonstrated by Feldbrugge et al. that the sum over all universes starting
from zero size results in an unstable saddle point geometry. Here we show that
in the context of gravity with a positive cosmological constant, path integrals
with a specific family of Robin boundary conditions overcome this problem.
These path integrals are manifestly convergent and are approximated by stable
Hartle-Hawking saddle point geometries. The price to pay is that the off-shell
geometries do not start at zero size. The Robin boundary conditions may be
interpreted as an initial state with Euclidean momentum, with the quantum
uncertainty shared between initial size and momentum.