Birefringence tests of gravity with multi-messenger binaries

Extensions to General Relativity (GR) allow the polarization of gravitational waves (GW) from astrophysical sources to suffer from amplitude and velocity birefringence, which respectively induce changes in the ellipticity and orientation of the polarization tensor. We introduce a multi-messenger approach to test this polarization behavior of GWs during their cosmological propagation using binary sources, for which the initial polarization is determined by the inclination and orientation angles of the orbital angular momentum vector with respect to the line of sight. In particular, we use spatially-resolved radio imaging of the jet from a binary neutron star (BNS) merger to constrain the orientation angle and hence the emitted polarization orientation of the GW signal at the site of the merger, and compare to that observed on Earth by GW detectors. For GW170817 we constrain the deviation from GR due to amplitude birefringence to $\kappa_A = -0.12^{+0.60}_{-0.61}$, while the velocity birefringence parameter $\kappa_V$ remains unconstrained. The inability to constrain $\kappa_V$ is due to the fact that Virgo did not detect GW170817, and measurements of the polarization orientation require information from a combination of multiple detectors with different alignments. For this reason, we also mock future BNS mergers with resolved afterglow proper motion and project that $\kappa_V$ could be constrained to a precision of $5\,$rad (corresponding to an angular shift of the GW polarization of $\delta\phi_V\approx 0.2\,$rad for a BNS at $100\,$Mpc) by a future network of third-generation ground-based GW detectors such as Cosmic Explorer and the radio High Sensitivity Array. Crucially, this velocity birefringence effect cannot be constrained with dark binary mergers as it requires polarization information at the emission time, which can be provided only by electromagnetic emission.