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abstract: 'We study the stability under linear odd-parity perturbations of static spherically symmetric black holes in Horndeski gravity. We derive the master equation for these perturbations and obtain the conditions of no-ghost and Laplacian instability. In order for the black hole solutions to be stable, we study their generalized “Regge-Wheeler potential”. It turns out that the problem is reduced to an algebraic problem where three functions characterizing the black hole should be positive outside the horizon to prove the stability. We found that these conditions are similar to the no-ghost and Laplacian instability conditions. We apply our results to various known solutions.'
author:
- Apratim Ganguly
- Radouane Gannouji
- 'Manuel Gonzalez-Espinoza'
- 'Carlos Pizarro-Moya'
title: 'Black hole stability under odd-parity perturbations in Horndeski gravity'
---

Introduction
============

Extended models of gravity became very popular in cosmology and black hole physics. In cosmology, these models are used to explain inflation, dark energy or dark matter, while hairy black hole solutions are studied in the case of compact objects. They can have a very rich phenomenology (see [@Berti:2015itd] for a recent review). In this paper, we focus on static black hole solutions and their stability.

In fact, since the seminal 