Gravitational wave propagation in Ho\v{r}ava-Lifshitz gravity

We investigate the generation and propagation of gravitational waves in the leading parity-even infrared truncation of Ho\v{r}ava-Lifshitz gravity, characterized by the modified tensor dispersion relation $\omega^{2}=k^{2}+\alpha k^{4}$. Working in the transverse-traceless sector, we show that the higher-spatial-derivative correction preserves the conventional plus and cross polarizations and introduces neither polarization mixing, helicity splitting, nor gravitational birefringence. We construct the retarded Green function of the modified wave operator and derive the radiation-zone waveform to first order in $\alpha$. The resulting signal exhibits a frequency-dependent amplitude renormalization together with a dispersive propagation phase that accumulates over the source-observer distance. We apply the formalism to a binary black hole system in a quasi-circular orbit and obtain the polarization waveforms for an arbitrary observation direction. We further derive the corresponding energy flux, total luminosity, and adiabatic chirp evolution. In terms of the observed gravitational wave frequency $f$, the leading corrections satisfy $\Delta h_{A}/h_{A}^{\mathrm{GR}}=-8\pi^{2}\alpha f^{2}$ and $\Delta P/P_{\mathrm{GR}} =\Delta\dot{f}/\dot{f}_{\mathrm{GR}} =-16\pi^{2}\alpha f^{2}$, while the accumulated generation phase has the frequency dependence of a relative third post-Newtonian contribution. By mapping the Ho\v{r}ava-Lifshitz coefficient to the LIGO-Virgo-KAGRA modified-dispersion parametrization, we obtain $-6.2\times10^{2}\,\mathrm{eV}^{-2} <\alpha< 1.9\times10^{2}\,\mathrm{eV}^{-2}$ at $90\%$ credibility from the GWTC-4.0 posterior.