Testing the general theory of relativity using gravitational wave propagation from dark standard sirens

Alternative theories of gravity predict modifications in the propagation of gravitational waves (GW) through space-time. One of the smoking-gun predictions of such theories is the change in the GW luminosity distance to GW sources as a function of redshift relative to the electromagnetic (EM) luminosity distance expected from EM probes. We propose a multi-messenger test of the theory of general relativity from the propagation of gravitational waves by combining EM and GW observations to resolve these issues from GW sources without EM counterparts (which are also referred to as dark standard sirens). By using the relation between the geometric distances accessible from baryon acoustic oscillation measurements, and luminosity distance measurements from the GW sources, we can measure any deviation from the general theory of relativity via the GW sources of unknown redshift that will be detectable by networks of GW detectors such as LIGO, Virgo, and KAGRA. Using this technique, the fiducial value of the frictional term can be measured to a precision $\Xi_0=0.98^{+0.04}_{-0.23}$ after marginalizing over redshift dependence, cosmological parameters, and GW bias parameters with $\sim 3500$ dark standard sirens of masses $30\,\rm M_\odot$ each distributed up to redshift $z=0.5$. For a fixed redshift dependence, a value of $\Xi_0=0.99^{+0.02}_{-0.02}$ can be measured with a similar number of dark sirens. Application of our methodology to the far more numerous dark standard sirens detectable with next generation GW detectors, such as LISA, Einstein Telescope and Cosmic Explorer, will allow achievement of higher accuracy than possible from use of bright standard sirens.