Gravitational-wave luminosity distance in modified gravity theories

In modified gravity the propagation of gravitational waves (GWs) is in general different from that in general relativity. As a result, the luminosity distance for GWs can differ from that for electromagnetic signals, and is affected both by the dark energy equation of state w(DE)(z) and by a function delta(z) describing modified propagation. We show that the effect of modified propagation in general dominates over the effect of the dark energy equation of state, making it easier to distinguish a modified gravity model from.CDM. We illustrate this using a nonlocal modification of gravity that has been shown to fit remarkably well cosmic microwave background, supernovae, baryon acoustic oscillation, and structure formation data, and we discuss the prospects for distinguishing nonlocal gravity from Lambda CDM with the Einstein Telescope. We find that, depending on the exact sensitivity, a few tens of standard sirens with measured redshift at z similar to 0.4, or a few hundreds at 1 <= z <= 2, could suffice.