Revisiting the no-boundary proposal with a scalar field

TL;DR

This paper refines the no-boundary proposal by including scalar fields, showing how to eliminate unstable saddle points and exploring implications for different potential signs, with relevance to inflation and AdS spaces.

Contribution

It demonstrates a method to implement the no-boundary proposal with scalar fields, introducing regularity conditions that remove unstable saddle points and analyzing resulting geometries.

Findings

Unstable saddle points can be eliminated with regularity conditions.

Additional saddle points correspond to unclosed geometries.

Classical spacetime emerges only for inflationary potentials.

Abstract

Recent works have suggested that the no-boundary proposal should be defined as a sum over regular, not necessarily compact, metrics. We show that such a prescription can be implemented in the presence of a scalar field. For concreteness, we consider the model of Garay et al., in which the potential is a sum of exponentials, and which lends itself to an analytical treatment. Compared to the earlier implementation, we find that saddle points with unstable fluctuations can be eliminated by imposition of an appropriate regularity condition. This leads to the appearance of additional saddle points, corresponding to unclosed geometries. We argue that such saddles will occur generically, though we also find in our example that they are subdominant to the closed, Hartle-Hawking, saddle points. When the potential is positive, classical spacetime is only predicted for inflationary histories. When the potential is negative, we recover the AdS gravitational path integral, with a stable scalar field included. One puzzle that we find is that in general the path integral must be restricted to sum only over specific, discrete and late time dependent initial values of the scalar field. Only when the scalar is required to take real values is this puzzle eliminated, a situation that moreover leads to advantageous phenomenological characteristics.