“…Current and future GW observations can constrain these gravitational theories with high accuracy [43][44][45][46][47][48][49][50][51][52]. Some kinds of modified theories, including parity-violating effects, have been studied in recent years [53][54][55][56][57][58][59][60][61][62][63][64][65][66], like Chern-Simons (CS) gravity [26,67] (consisting of dynamical and non-dynamical cases), ghost-free parity-violating theory [68][69][70][71][72], Nieh-Yan modified teleparallel gravity [73,74], parity-violating symmetric teleparallel gravities [75] and the general spatial covariant gravities [31,32,60,61,63,[76][77][78]]. As a typical example, CS theory introduces a coupling term between the pseudo-scalar and Pontryagin density in the Einstein-Hilbert action, which causes the nonconservation of the CS topological current.…”