Rotations of the polarization of a gravitational wave propagating in Universe

In this paper, we study the polarization of a gravitational wave (GW) emitted by an astrophysical source at a cosmic distance propagating through the Friedmann-Lema^itre-Robertson-Walk universe. By considering the null geodesic deviations, we first provide a definition of the polarization of the GW in terms of the Weyl scalars with respect to a parallelly-transported frame along the null geodesics, and then show explicitly that, due to different effects of the expansion of the universe on the two polarization modes, the so-called + and $times$ modes, the polarization angle of the GW changes generically, when it is propagating through the curved background. By direct computations of the polarization angle, we show that different epochs, radiation-, matter- and $Lambda$-dominated, have different effects on the polarization. In particular, for a GW emitted by a binary system, we find explicitly the relation between the change of the polarization angle $|Delta varphi|$ and the redshift $z_s$ of the source in different epochs. In the $Lambda$CDM model, we find that the order of $|Delta varphi|{eta_0 F}$ is typically $O(10^{-3})$ to $O(10^3)$, depending on the values of $z_s$, where $eta_0$ is the (comoving) time of the current universe, and $FequivBig(frac{5}{256}frac{1}{tau_{obs}}Big)^{3/8}left(G_NM_cright)^{-5/8}$ with $tau_{obs}$ and $M_c$ being, respectively, the time to coalescence in the observers frame and the chirp mass of the binary system.