A neutron star is usually treated as a single rotating fluid, with one moment of inertia and one angular velocity. If the star contains a second independently conserved fluid — superfluid neutrons, or dark matter — the rotation becomes richer. Two fluids sharing the same spacetime don't necessarily spin at the same rate, and the coupling between them comes only through gravity.
Kumar and Sotani (arXiv:2603.12613) develop the relativistic theory of slowly rotating two-fluid neutron stars and find that the system has two intrinsic rotational eigenmodes, each with its own moment of inertia. Even when the two fluids are co-rotating (no relative motion), the coupled system has two distinct inertial responses to perturbation. The eigenmodes arise from the gravitational frame-dragging coupling between the fluids — each fluid drags spacetime, and the other fluid responds to the dragged spacetime.
The application to dark-matter-admixed neutron stars is where the result becomes diagnostic. Neutron stars obey “universal relations” — empirical correlations between moment of inertia, tidal deformability, and quadrupole moment that are approximately independent of the equation of state. The key finding: whether these universal relations survive the addition of dark matter depends on dark-sector microphysics (the dark matter equation of state and interaction properties), not on the mere presence of a second component.
This means universal relations can test dark matter models. If future measurements of neutron star properties show deviations from the standard universal relations, the pattern of deviations encodes information about dark matter interactions. If the relations hold, it constrains how dark matter can couple to spacetime in the stellar interior.