Norm of the no-boundary state

Abstract

We consider Einstein gravity with positive cosmological constant coupled to matter in an asymptotically de Sitter universe with sphere boundary at timelike infinity. In this setting we show that, to one-loop order and at late time, the norm of the no-boundary state vanishes, going as $e^{S_0}\frac{Z{S}_0^{-d(d+1)/4}}{\text{vol}(SO(d,1))}$ with $S_0$ the tree-level entropy of the static patch, $d$ the spacetime dimension, and $Z$ non-negative. We show that the presence of an observer stabilizes the norm to a large, positive value.

38+10 pages, 6 figures; v2: results generalized, typos corrected, references added