Lorentzian Robin Universe of Gauss-Bonnet Gravity

TL;DR

This paper explores the quantum gravitational path integral in Gauss-Bonnet gravity with Robin boundary conditions, revealing stable universe configurations and connecting Euclidean and Lorentzian signatures through exact and saddle point analyses.

Contribution

It provides the first exact computation of the transition amplitude in Gauss-Bonnet gravity with Robin boundary conditions and analyzes the dominant boundary configurations using Picard-Lefschetz methods.

Findings

Exact transition amplitude computed in mini-superspace

Dominant configurations correspond to Hartle-Hawking no-boundary proposal

Picard-Lefschetz analysis clarifies Euclidean-Lorentzian crossover

Abstract

The gravitational path-integral of Gauss-Bonnet gravity is investigated and the transition from one spacelike boundary configuration to another is analyzed. Of particular interest is the case of Neumann and Robin boundary conditions which is known to lead to a stable Universe in Einstein-Hilbert gravity in four spacetime dimensions. After setting up the variational problem and computing the necessary boundary terms, the transition amplitude is computed \emph{exactly} in the mini-superspace approximation. The $\hbar\to0$ limit brings out the dominant pieces in the path-integral which is traced to an initial configuration corresponding to Hartle-Hawking no-boundary Universe. A deeper study involving Picard-Lefschetz methods not only allow us to find the integration contour along which the path integral becomes convergent but also aids in understanding the crossover from Euclidean to Lorentzian signature. Saddle analysis further highlights the boundary configurations giving dominant contribution to the path-integral which is seen to be those corresponding to Hartle-Hawking no-boundary proposal and agrees with the exact computation. To ensure completeness, a comparison with the results from Wheeler-DeWitt equation is done.