Eikonal gravitational-wave lensing in Einstein-aether theory

Einstein-aether theory provides a model to test the validity of local Lorentz invariance in gravitational interactions. The speed of gravitational waves as measured from the binary neutron star event GW170817 sets stringent limits on Einstein-aether theory, but only on a combination of the theory's free parameters. For this reason, a significant part of the theory's parameter space remains unconstrained by observations. Motivated by this, we explore the propagation of gravitational waves in Einstein-aether theory over an inhomogeneous background (i.e., gravitational wave lensing) as a potential mechanism to break the degeneracies between the theory's free parameters, and hence enable new constraints on the theory to be obtained. We reduce our analysis to gravitational waves that pass far from the lens' center and short wavelength signals, both compared to the lens' gravitational radius (eikonal limit). By applying these approximations and bringing the field equations into the form of the so-called kinetic matrix and applying a formalism known as the propagation eigenstate framework, we find that the speed of gravitational waves is modified by inhomogeneities in the aether field. However, the modification is common to both gravitational polarizations and vanishes in the limit in which gravitational waves propagate with luminal speed. This lens-dependent gravitational wave speed contrasts with the lens-induced birefringence observed in other theories beyond general relativity, like Horndeski's theory. While the potential to improve tests based on gravitational-wave speed is limited, our formalism sets the basis to fully describe signal propagation over inhomogeneous spacetimes in Einstein-aether theory and other extensions of general relativity.