Propagation and polarization of gravitational waves on curved spacetime backgrounds in Einstein-\AE ther theory

Abstract

We analyze the propagation and polarization properties of high-frequency gravitational waves in Einstein-\AE ther theory on vorticity-free and slowly-varying backgrounds at both leading and next-to-leading orders within the geometric optics approximation. The linear perturbation analysis is performed in the background \AE ther-orthogonal frame, in which the axes of the gravitational wave sound cones remain perpendicular to these hypersurfaces, thereby simplifying the analysis. The leading-order results show that Einstein-\AE ther theory admits two tensor modes, two vector modes, and one scalar mode, consistent with the findings in the flat spacetime background. We further derive the dispersion relations and linear stability conditions for these modes in curved backgrounds. At next-to-leading order, we obtain the amplitude evolution equations, finding that the graviton number is conserved for the tensor modes but not for the vector and scalar modes. Next-to-leading-order effects also induce mixing among polarization modes. Our study demonstrates that, after imposing the GW170817 constraint on the propagation speed of gravitational waves, the vector modes mixed with the leading-order tensor modes cannot be used to distinguish between general relativity and Einstein-\AE ther theory. On the other hand, the mixing between scalar modes and the leading-order tensor modes leads to distinct predictions in the two theories, providing a promising avenue to test Einstein-\AE ther gravity through the detection of polarization mixing in gravitational waves.