Black hole perturbations in modified gravity

TL;DR

This paper investigates linear perturbations of nonrotating black holes within DHOST theories, analyzing asymptotic behaviors, computing quasi-normal modes, and identifying potential pathologies in specific black hole solutions.

Contribution

It introduces a systematic approach to study perturbations directly from the first-order system in DHOST theories, including both axial and polar modes, and applies it to specific black hole solutions.

Findings

Asymptotic behaviors of perturbations are characterized.

Quasi-normal modes are numerically computed for non-stealth solutions.

Pathologies are identified in the studied black hole solutions.

Abstract

We study the linear perturbations about nonrotating black holes in the context of degenerate higher-order scalar-tensor (DHOST) theories, using a systematic approach that extracts the asymptotic behaviour of perturbations (at spatial infinity and near the horizon) directly from the first-order radial differential system governing these perturbations. For axial (odd-parity) modes, this provides an alternative to the traditional approach based on a second-order Schr\"odinger-like equation with an effective potential, which we also discuss for completeness. For polar (even-parity) modes, which contain an additional degree of freedom in DHOST theories, and are thus more complex, we use a direct treatment of the four-dimensional first-order differential system (without resorting to a second order reformulation). We illustrate our study with two specific types of black hole solutions: 'stealth' Schwarzschild black holes, with a non trivial scalar hair, as well as a class of non-stealth black holes whose metric is distinct from Schwarzschild. The knowledge of the asymptotic behaviours of the perturbations enables us to compute numerically quasi-normal modes, as we show explicitly for the non-stealth solutions. Finally, the asymptotic form of the modes also signals some pathologies in the stealth and non-stealth solutions considered here.