Propagation effects of Lorentz violation in gravitational waves

We investigate the propagation of gravitational waves in the presence of Lorentz- and diffeomorphism-violating operators within the linearized gravitational sector of the Standard Model Extension. Focusing on isotropic contributions, we analyze the combined effects of the nondispersive CPT-even dimension-four coefficient $\mathring{k}^{(4)}_{(I)}$ and the CPT-odd dimension-five coefficient $\mathring{k}^{(5)}_{(V)}$ on tensorial gravitational radiation. The modified dispersion relation induces both a rescaling of the propagation speed and helicity-dependent dispersive corrections, leading to birefringence and polarization mixing without introducing additional propagating degrees of freedom. We derive the retarded Green function associated with the modified wave operator and obtain explicit expressions for the gravitational waveform generated by matter sources. As a concrete application, we examine a binary black hole system and show how Lorentz violation alters the observed strain through shifted retarded times, amplitude rescaling, and higher derivative corrections to the quadrupole formula. The CPT-odd term produces characteristic attenuation and distortions in the waveform, which can be interpreted as energy exchange between the gravitational wave and the Lorentz-violating background rather than a violation of energy conservation. Using published LIGO-Virgo-KAGRA propagation tests and polarization consistency arguments, we translate current observational constraints into bounds on $\mathring{k}^{(4)}_{(I)}$ and $\mathring{k}^{(5)}_{(V)}$.