  We initiate a general investigation into gravitational wave signatures of
modifications to scattering of gravitational radiation from black holes. Such
modifications may be present due to the quantum dynamics that makes black holes
consistent with quantum mechanics, or in other models for departures from
classical black hole behavior. We propose a parameterization of the corrections
to scattering as a physically meaningful, model-independent, and practical
bridge between theoretical and observational aspects of the problem; this
parameterization can incorporate different models in the literature. We then
describe how these corrections influence the gravitational wave signal, e.g. of
a body orbiting a much more massive black hole. In particular, they generically
change the rate of energy emission; this effect can be leveraged over many
orbits of inspiral to enhance the sensitivity to small corrections, as has been
noticed in simple models. We provide preliminary estimates of the sensitivity
of future gravitational wave observations to these corrections, and outline
further work to be done to connect both to a more fundamental theory of quantum
black holes, and to realistic observational situations.
