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.
