Search for the birefringence of gravitational waves with the third observing run of Advanced LIGO-Virgo

Gravitational waves would attain birefringence during their propagation from distant sources to the Earth, when the CPT symmetry is broken. If it was sizeable enough, such birefringence could be measured by the Advanced LIGO, Virgo and KAGRA detector network. In this work, we place constraints on the birefringence of gravitational waves with the third observing run of this network, i.e. two catalogues GWTC-2 and GWTC-3. For the dispersion relation $\omega^{2}=k^{2}\pm2\zeta k^{3}$, our analysis shows the up-to-date strictest limit on the CPT-violating parameter, i.e. \textcolor{black}{{$\zeta=4.07^{+5.91}_{-5.79}\times10^{-17}\mathrm{m}$}}, at $68\%$ confidence level. This limit is stricter by $\sim$5 times than the existing one {($\sim 2\times10^{-16}$m)} and stands for the first $\sim$10GeV-scale test of the CPT symmetry in gravitational waves. The results of Bayes factor strongly disfavor the birefringence scenario of gravitational waves.