Spherical perturbations of hairy black holes in designer gravity theories

TL;DR

This paper investigates the scalar quasi-normal mode spectrum of hairy Anti-de-Sitter black holes in higher-dimensional gravity theories, linking black hole stability to the dual field theory's effective potential and boundary conditions.

Contribution

It establishes a connection between the quasi-normal frequencies of hairy black holes and the effective potential of the dual field theory in certain models, especially at large condensates.

Findings

Least damped quasi-normal frequency correlates with the curvature of the effective potential.

Effective potential at large condensates can be identified with boundary deformation and conformal coupling.

Predictions of black hole damping rates match the effective potential analysis.

Abstract

We study the spectrum of scalar $l=0$ quasi-normal frequencies of Anti-de-Sitter hairy black holes in four and five dimensional designer gravity theories of the Einstein-scalar type, arising as consistent truncations of $ \mathcal{N} = 8$ gauged supergravity. In the dual field theory, such hairy black holes represent thermal states in which the operator corresponding to the bulk scalar field is condensed, due to the multi-trace deformation associated with non-standard boundary conditions. We show that, in a particular class of models, the effective potential describing the vacua of the deformed dual theory can be identified, at large values of the condensate, with the deformation plus the conformal coupling of the condensate to the curvature of the boundary geometry. In this limit, we show that the least damped quasi-normal frequency of the corresponding hairy black holes can be accurately predicted by the curvature of the effective potential describing the field theory at finite entropy.