# The No-Boundary Proposal as a Path Integral with Robin Boundary Conditions

## TL;DR

This paper demonstrates that using Robin boundary conditions in gravitational path integrals with a positive cosmological constant yields stable, convergent solutions that align with the Hartle-Hawking proposal, resolving previous instability issues.

## Contribution

It introduces a novel approach with Robin boundary conditions to stabilize the gravitational path integral in the no-boundary proposal, avoiding zero-size initial geometries.

## Key findings

- Path integrals with Robin boundary conditions are convergent.
- Stable Hartle-Hawking saddle points are obtained.
- Initial geometries do not start at zero size.

## 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.