Gravitational-wave versus X-ray tests of strong-field gravity

Electromagnetic observations of the radiation emitted by an accretion disk around a black hole, as well as gravitational wave observations of coalescing binaries, can be used to probe strong-field gravity. We here compare the constraints that these two types of observations can impose on theory-agnostic, parametric deviations from the Schwarzschild metric. On the gravitational wave side, we begin by computing the leading-order deviation to the Hamiltonian of a binary system in a quasi-circular orbit within the post-Newtonian approximation, given a parametric deformation of the Schwarzschild metrics of each binary component. We then compute the leading-order deviation to the gravitational waves emitted by such a binary in the frequency domain, assuming purely Einsteinian radiation-reaction. We compare this model to the LIGO-Virgo collaboration gravitational wave detections and place constraints on the metric deformation parameters, concluding with an estimate of the constraining power of aLIGO at design sensitivity. On the electromagnetic side, we first simulate observations with current and future X-ray instruments of an X-ray binary with a parametrically-deformed Schwarzschild black hole, and we then estimate constraints on the deformation parameters using these observations.We find that current gravitational wave observations have already placed constraints on the metric deformation parameters than are slightly more stringent than what can be achieved with current X-ray instruments. Moreover, future gravitational wave observations with aLIGO at design sensitivity by 2026 will be even more stringent, becoming stronger than constraints achievable with future ATHENA X-ray observations before it flies in 2034.