Composition g-modes and f-modes of neutron stars with gravitationally coupled dark matter: degeneracy with the symmetry energy

Composition gravity (g) modes have been proposed as a probe of dark matter in neutron stars, on the basis of one-fluid models in which the dark matter is in chemical equilibrium with the neutrons. We computed in full general relativity the g - and f-modes of stars in which fermionic dark matter couples to the nucleons only through gravity, and inferred the dark mass fraction from NICER and GW170817 data with a prior that comprises 63 nucleonic equations of state (EOSs) from Skyrme and relativistic mean-field functionals, among them the models of Dutra et al. and of the compilation of Sun, Bhattiprolu and Lattimer. The data do not prefer dark matter and bound its mass fraction at 0.12 - 0.26 (95%) for particle masses of 0.5 -- 2 GeV. Coupled only through gravity, the dark matter reaches the buoyancy that restores a g-mode only through the gravitational field: a dark core holding 10% of the mass raises the fundamental g-mode by 10-15% at fixed EOS, several times less than in the one-fluid picture. The larger effect is indirect: a star with a dark core needs a stiffer nucleonic EOS, and since the g-mode is set by the density dependence of the symmetry energy, its frequency reflects the EOS that the star requires rather than the dark matter it contains; at fixed radius and symmetry energy a core holding 10% of the mass is equivalent to a change of the slope L of 9-12 MeV. The f-mode, which follows the mean density, behaves differently: a dark core breaks the relation between the f-mode and the tidal deformability by about one percent, far outside the posterior spread of nucleonic stars.