Constraining the effective field theory of dark energy with multimessenger astronomy

The effective field theory of dark energy predicts 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. According to the EFT the friction term of the GW propagation equation depends on the effective Planck mass and GW speed time variation, affecting the GW-EMW luminosity distance ratio. We compute the general form of the GW-EMW luminosity distance ratio in terms of the effective GW speed and effective Planck mass, and then focus on theories with constant Planck mass (CPM) and time varying GW speed. For CPM theories the GW speed can be jointly constrained by the GW-EMW detection time delay and luminosity distance ratio, allowing to derive a consistency relation between these two observables. The event GW170817 and its EM counterpart satisfy the CPM consistency condition, and allows to set constraints on the time variation of the GWs speed, and consequently on the coefficients of the effective theory.