Constraining the time evolution of the propagation speed of gravitational waves with multimessenger astronomy

Several modified gravity theories predict a possible time variation of the propagation speed of gravitational waves (GW) which could be tested with multimessenger astronomy. For this purpose we derive the relation between the redshift dependence of the propagation speed of GWs and the time delay between the detection of GWs and electromagnetic waves (EMWs) emitted by the same source. For theories with Einstein frame minimal matter-gravity coupling (EMC) the propagation speed of GWs can be jointly constrained by the time delay between GWs and EWs and the GW-EMW luminosity distance ratio, allowing to derive a consistency relation between these two observables. A violation of this consistency relation could support a non-minimal coupling of matter to the Einstein frame metric, as in Chameleon models, allowing to test such theories with multimessenger astronomy.