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 $F\equiv\Big(\frac{5}{256}\frac{1}{\tau_{obs}}\Big)^{3/8}\left(G_NM_c\right)^{-5/8}$ with $\tau_{obs}$ and $M_c$ being, respectively, the time to coalescence in the observer's frame and the chirp mass of the binary system.