Can primordial parity violation explain the observed cosmic birefringence?

Recently, the cross-correlation between $E$- and $B$-mode polarization of cosmic microwave background, which is well explained by birefringence with rotation angle $\ensuremath{\beta}\ensuremath{\approx}0.3\text{ }\text{ }\mathrm{deg}$, has been found in background data. We carefully investigate possibility explaining observed $EB$ correlation primordial chiral gravitational waves, can be generated parity-violating theories Universe. that wave scenario does not work due to overproduction $BB$ autocorrelation, far exceeds one SPTPol POLARBEAR.