Stability of saddles and choices of contour in the Euclidean path integral for linearized gravity: Dependence on the DeWitt Parameter

Due to the conformal factor problem, the definition of the Euclidean gravitational path integral requires a non-trivial choice of contour. The present work examines a generalization of a recently proposed rule-of-thumb \cite{Marolf:2022ntb} for selecting this contour at quadratic order about a saddle. The original proposal depended on the choice of an indefinite-signature metric on the space of perturbations, which was taken to be a DeWitt metric with parameter $\alpha =-1$. This choice was made to match previous results, but was otherwise admittedly {\it ad hoc}. To begin to investigate the physics associated with the choice of such a metric, we now explore contours defined using analogous prescriptions for $\alpha \neq -1$. We study such contours for Euclidean gravity linearized about AdS-Schwarzschild black holes in reflecting cavities with thermal (canonical ensemble) boundary conditions, and we compare path-integral stability of the associated saddles with thermodynamic stability of the classical spacetimes. While the contour generally depends on the choice of DeWitt parameter $\alpha$, the precise agreement between these two notions of stability found at $\alpha =-1$ continues to hold over the finite interval $(-2,-2/d)$, where $d$ is the dimension of the bulk spacetime. This agreement manifestly fails for $\alpha > -2/d$ when the DeWitt metric becomes positive definite. However, we also find dramatic failures for $\alpha< -2$ that correlate with breakdowns of the de Donder-like gauge condition defined by $\alpha$, and at which the relevant fluctuation operator fails to be diagonalizable. This provides criteria that may be useful in predicting metrics on the space of perturbations that give physically-useful contours in more general settings. Along the way, we also identify an interesting error in \cite{Marolf:2022ntb}, though we show this error to be harmless.