Quasinormal modes of black holes in Horndeski gravity

We study the perturbations to general relativistic black holes (i.e., those without scalar hair) in Horndeski scalar-tensor gravity. First, we derive the equations of odd and even parity perturbations of both the metric and scalar field in the case of a Schwarzschild black hole, and show that the gravitational waves emitted from such a system contain a mixture of quasinormal mode frequencies from the usual general relativistic spectrum and those from the new scalar field spectrum, with the new scalar spectrum characterized by just two free parameters. We then specialize to the subfamily of Horndeski theories in which gravitational waves propagate at the speed of light c on cosmological backgrounds; the scalar quasinormal mode spectrum of such theories is characterized by just a single parameter μ acting as an effective mass of the scalar field. Analytical expressions for the quasinormal mode frequencies of the scalar spectrum in this subfamily of theories are provided for both static and slowly rotating black holes. In both regimes comparisons to quasinormal modes calculated numerically show good agreement with those calculated analytically in this work.