On the stability of scale-invariant black holes

TL;DR

This paper analyzes the stability of scale-invariant black hole solutions in quadratic gravity coupled to a scalar field, demonstrating classical and semi-classical stability properties and decay dynamics between solutions.

Contribution

It provides a comprehensive stability analysis of Schwarzschild-de Sitter solutions in a scale-invariant gravity model using perturbation theory and Euclidean path integrals.

Findings

One solution is classically stable, the other unstable.

The unstable solution is metastable and decays into the stable one.

Semi-classical analysis favors the stable fixed point.

Abstract

Quadratic scale-invariant gravity non minimally coupled to a scalar field provides a competitive model for inflation, characterized by the transition from an unstable to a stable fixed point, both characterized by constant scalar field configurations. We provide a complementary analysis of the same model in the static, spherically symmetric setting, obtaining two Schwarzschild-de Sitter solutions, which corresponds to the two fixed points existing in the cosmological scenario. The stability of such solutions is thoroughly investigated from two different perspectives. First, we study the system at the classical level by the analysis of linear perturbations. In particular, we provide both analytical and numerical results for the late-time behavior of the perturbations, proving the stable and unstable character of the two solutions. Then we perform a semi-classical, non-linear analysis based on the Euclidean path integral formulation. By studying the difference between the Euclidean on-shell actions evaluated on both solutions, we prove that the unstable one has a meta-stable character and is spontaneously decaying into the stable fixed point which is always favoured.