Black hole perturbations in modified gravity theories

The recent first detection of gravitational waves (GWs) from binary black hole mergers has spurred a renewed interest in possible deviations from General Relativity (GR), since they could be detected in the GWs emitted by such systems. Of particular interest is the ringdown phase of a binary black hole merger, which can be described by linear perturbations about a background stationary black hole solution. These perturbations mainly correspond to a superposition of 'quasi-normal modes' (QNMs), whose frequencies form a discrete set. One expects that modified gravity models could predict QNMs that differ from their GR counterpart: the detailed analysis of the GW signal represents an invaluable window to test GR and to look for specific signatures of modified gravity. The work done in this thesis takes place in the context of scalar-tensor theories of gravity, and more particularly the Degenerate Higher-Order Scalar-Tensor theories. We start by a review of these theories and their properties, and describe a way to reformulate them in a framework with a clear geometrical interpretation. We then study linear perturbations about several existing nonrotating black hole solutions of such theories, and show why the perturbation equations obtained are very hard to decouple in general. When it is possible, in the case of odd parity perturbations, we describe the propagation of waves and relate it to the stability of the underlying spacetime. When it is not, we circumvent the difficulty by making use of an algorithm proposed recently in the mathematical literature that allows us to decouple the equations both at the black hole horizon and at infinity. This allows us to get the asymptotic behaviour of waves on such spacetimes, yielding valuable information that can allow us to rule some of them out. Finally, we use the asymptotic behaviours obtained to compute QNMs numerically.