Gravitational Faraday-Cartan effect beyond gravitomagnetism due to dark matter intrinsic spin

We show that the spin of dark matter induces a gravitational analog of the electromagnetic Faraday effect, where the polarization of gravitational waves undergoes a rotation as they propagate through a dark matter halo with a non-vanishing axial hypermomentum. An expression for the gravitational rotation angle is provided, which is analogous to the Faraday rotation in optics, and evaluate its significance in astrophysical settings. Although the effect is expected to be small under current observational constraints, we discuss its potential importance in the early universe. Importantly, this effect is distinct from the known gravitational Faraday rotation in gravitomagnetism, where the geometry of general relativity is split into a background and a low-frequency gravitomagnetic perturbation. In that framework, the polarization of an electromagnetic wave (or a high-frequency GW perturbation) rotates relative to the background geometry. In contrast, this gravitational Faraday-Cartan effect arises from a non-vanishing dark matter axial hypermomentum that breaks the parallel transport of GW polarization, without invoking any gravitomagnetic approximation. Notably, it only rotates gravitational wave polarization without affecting the electromagnetic wave one.