Testing Lorentz Invariance of Gravity in the Standard Model Extension with GWTC-3

Successful detection of gravitational waves has presented a new avenue to explore the nature of gravity. With the cumulative catalog of detected events, we can perform tests on General Relativity from various aspects with increasing precision. In this work, we focus on Lorentz symmetry during propagation of gravitational waves. Considering the dispersion relation in the gauge-invariant linearized gravity sector of the Standard-Model Extension, the anisotropy, birefringence, and dispersion effects will be induced during propagation of gravitational waves because of the Lorentz violating modification, and cause dephasings in waveform received by detectors. With the distorted waveform, we perform full Bayesian inference with confident events in the last gravitational wave catalog. We consider two cases associated with the lowest mass dimension $d=5,6$ which are supposed to have the most significant effects, and place the constraints on the expansion coefficients characterizing the Lorentz violating behavior which have 16 independent components for $d=5$ and 18 components for $d=6$. We do not find any evidence for Lorentz violation in the gravitational wave data, the constraints on the coefficients are on the order of $10^{-15}{\rm m}$ for $d=5$ and $10^{-10}{\rm m^2}$ for $d=6$ respectively.