Gravitational wave propagation beyond General Relativity: geometric optic expansion and lens-induced dispersion

Abstract: The nature of gravity can be tested by how gravitational waves (GWs) are emitted, detected, and propagate through the universe. Propagation tests are powerful, as small deviations compound over cosmological distances. However, GW propagation tests of theories beyond Einstein's general relativity (GR) are limited by the high degree of symmetry of the average cosmological spacetime. Deviations from homogeneity, i.e. gravitational lenses, allow for new interactions, e.g., between standard GW polarization and new scalar or vector fields, with different spin. Therefore, GW lensing beyond GR offers novel tests of cosmological gravity. Here we present the theory of GW propagation beyond GR in the short-wave expansion, including corrections to the leading-order amplitude and phase for the first time. As an example, we compute the dispersive (frequency-dependent) corrections to all metric and scalar field perturbations in Brans-Dicke, the simplest modified theory exhibiting GW dispersion. GW lensing effects are too small to observe in Brans-Dicke theories compatible with solar system and binary pulsar limits. Nevertheless, our formalism opens the possibility of novel tests of gravity, including dark-energy theories and screening mechanisms.