Eikonal quasinormal modes of black holes beyond general relativity. II. Generalized scalar-tensor perturbations

Black hole spectroscopy, i.e., the identification of quasinormal mode frequencies via gravitational wave observations, is a powerful technique for testing the general relativistic nature of black holes. In theories of gravity beyond general relativity, perturbed black holes are typically described by a set of coupled wave equations for the tensorial field and the extra scalar/vector degrees of freedom, thus leading to a theory-specific quasinormal mode spectrum. In this paper we use the eikonal/geometric optics approximation to obtain analytic formulas for the frequency and damping rate of the fundamental quasinormal mode of a generalized, theory-agnostic system of equations describing coupled scalar-tensor perturbations of spherically symmetric black holes. Representing an extension of our recent work, the present model includes a massive scalar field, couplings through the field derivatives and first-order frame-dragging rotational corrections. Moving away from spherical symmetry, we consider the simple model of the scalar wave equation in a general stationary-axisymmetric spacetime and use the eikonal approximation to compute the quasinormal modes associated with equatorial and nonequatorial photon rings.