  Gravitational wave (GW) birefringence is a remarkable phenomenon that can be
used to test the parity violation in gravity. By coupling the fuzzy dark matter
(FDM) scalar to the gravitational Chern-Simons term, we explore the GW
birefringence effects in the FDM background. In particular, in light of the
highly oscillating granular FDM structure at the galactic scale, we are led to
investigating the GW propagation in the Chern-Simons gravity over the general
nontrivial scalar profile, which is a natural extension of previous studies on
the homogeneous and isotropic configurations. As a result, it is found that GWs
of both circularly polarized modes propagate in the straight line with the
speed of light, and does not show any velocity birefringence. However, when
considering the imaginary part of the dispersion relation, GWs exhibit the
amplitude birefringence in which one circular polarization is enhanced while
the other suppressed. Due to its local nature, the FDM-induced amplitude
birefringence only depends on the GW frequency without any reliance on the GW
event distance. More importantly, the birefringence factor shows a periodic
time variation with the period reflecting the FDM scalar mass, which is the
smoking gun for testing this new birefringence mechanism. Finally, we also
study the extra-galactic FDM contribution to the GW birefringence, which is
shown to be suppressed by the cosmological DM density and thus subdominant
compared with the galactic counterpart.
