David Langlois, Karim Noui, Hugo Roussille

The traditional approach to perturbations of nonrotating black holes in General Relativity uses the reformulation of the equations of motion into a radial second-order Schr\"odinger-like equation, whose asymptotic solutions are elementary. Imposing specific boundary conditions at spatial infinity and near the horizon defines, in particular, the quasi-normal modes of black holes. For more complicated equations of motion, as encountered for instance in modified gravity models with different background solutions and/or additional degrees of freedom, such a convenient Schr\"odinger-like reformulation might be unavailable, even in a generalised matricial form. In order to tackle such cases, we present a new approach that analyses directly the first-order differential system in its original form and extracts the asymptotic behaviour of perturbations. As a pedagogical illustration, we apply this treatment to the perturbations of Schwarzschild black holes and then show that the standard quasi-normal modes can be obtained numerically by solving this first-order system with a spectral method. This new approach paves the way for a generic treatment of the asymptotic behaviour of black hole perturbations and the identification of quasi-normal modes in theories of modified gravity.

David Langlois, Karim Noui, Hugo Roussille

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. By contrast, for polar (even-parity) modes, which contain an additional degree of freedom in DHOST theories, a similar generalised second-order Schr\"odinger-like matricial system does not seem available in general, which leaves only the option of a direct treatment of the four-dimensional first-order differential system. 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 pertubations 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.