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Modifications of General Relativity leave their imprint both on the cosmicexpansion history through a non-trivial dark energy equation of state, and onthe evolution of cosmological perturbations in the scalar and in the tensorsectors. In particular, the modification in the tensor sector gives rise to anotion of gravitational-wave (GW) luminosity distance, different from thestandard electromagnetic luminosity distance, that can be studied with standardsirens at GW detectors such as LISA or third-generation ground basedexperiments. We discuss the predictions for modified GW propagation from someof the best studied theories of modified gravity, such as Horndeski or the moregeneral degenerate higher order scalar-tensor (DHOST) theories, non-localinfrared modifications of gravity, bigravity theories and the correspondingphenomenon of GW oscillation, as well as theories with extra or varyingdimensions. We show that modified GW propagation is a completely genericphenomenon in modified gravity. We then use a simple parametrization of theeffect in terms of two parameters $(\Xi_0,n)$, that is shown to fit well theresults from a large class of models, to study the prospects of observingmodified GW propagation using supermassive black hole binaries as standardsirens with LISA. We construct mock source catalogs and perform detailed MarkovChain Monte Carlo studies of the likelihood obtained from LISA standard sirensalone, as well as by combining them with CMB, BAO and SNe data to reduce thedegeneracies between cosmological parameters. We find that the combination ofLISA with the other cosmological datasets allows one to measure the parameter$\Xi_0$ that characterizes modified GW propagation to the percent levelaccuracy, sufficient to test several modified gravity theories. [Abridged]

Testing modified gravity at cosmological distances with LISA standard sirens