In this work, we investigate the gravitational wave emission from the periodic orbital motion of a test particle around two specific types of black holes in Einstein-\AE{}ther theory, a modified gravity that locally breaks Lorentz symmetry while remaining consistent with theoretical and observational constraints through a careful selection of its four coupling constants ci. Focusing on the impact of the \ae{}ther field, we examine the properties of periodic orbits, which are characterized by a set of three topological integers (z,w,v) that uniquely classify their trajectories. We then calculate the gravitational waveforms generated by these periodic orbits, identifying potential observational signatures. Our analysis reveals a direct connection between the zoom-whirl orbital behavior of the small compact object and the gravitational waveforms it emits: higher zoom numbers lead to increasingly intricate waveform substructures. Moreover, the presence of the \ae{}ther field introduces significant modifications to these waveforms, imprinting measurable deviations that could be potentially tested or constrained by future space-based gravitational wave detectors.
@article{arxiv.2505.00294,
title = {Gravitational radiations from periodic orbits around Einstein-\AE{}ther black holes},
author = {Shuo Lu and Tao Zhu},
journal= {arXiv preprint arXiv:2505.00294},
year = {2025}
}