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Abstract: 

We derive an effective equation and action for the propagation of gravitational waves (GW), encoding the effects of interaction and self-interaction in a time, frequency and polarization dependent effective speed.This effective approach generalizes the effective theory of dark energy and allows to make model independent predictions of observable quantities. We discuss how the frequency and polarization dependence of the GW-EM distance ratio and time delay provide a new test of general relativity and its modifications, and more in general of the interaction of GWs with other fields. As an application, consistent with the effective theory of dark energy, we show that for a luminal modified gravity theory, the gravitational-wave propagation and luminosity distance are the same as in general relativity, but depending on the matter gravity coupling, the electromagnetic luminosity distance can be modified w.r.t general relativity.