Gravitational Wave Signatures of Periodic Motion near Higher-Derivative Einstein-\AE ther Black Holes

Higher-derivative modifications of general relativity are generically expected from effective field theory approaches to quantum gravity, and they arise naturally in Lorentz-violating theories such as Einstein-Ether gravity. In this work, we investigate black hole spacetimes within Einstein-Ether theory supplemented by quadratic curvature corrections, including terms proportional to $R^2$, $R_{\mu\nu} R^{\mu\nu}$, and $R_{\mu\nu\lambda\rho} R^{\mu\nu\lambda\rho}$. We derive the corrected static, spherically symmetric metric perturbatively and examine its effects on the geodesic structure and gravitational wave emission. In particular, we analyze periodic timelike orbits in this background and compute the associated tensor-mode gravitational waveforms using the quadrupole approximation. Our results demonstrate that even small higher-derivative corrections can induce distinguishable shifts in the orbital dynamics and imprint characteristic phase modulations and harmonic deformations in the gravitational wave signal. These effects modify the frequency spectrum and amplitude envelope of $h_{+}$ and $h_{\times}$ in a manner sensitive to the coupling constants $\alpha$, $\beta$, and $\gamma$, and the Ether parameter $c_{13}$. The resulting signatures provide a potential observational window into ultraviolet deviations from general relativity and Lorentz symmetry in the strong-field regime.