  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.
