Dark matter in neutron stars: two-fluid f-mode oscillations in full general relativity

A dark-matter core and a soft nuclear equation of state can produce similar mass--radius signatures, motivating a complementary probe through the f-mode spectrum. We compute quadrupolar f-mode frequencies and gravitational-wave damping times in full general relativity for cold, nonrotating neutron stars admixed with self-interacting bosonic dark matter (DM). Our two-fluid survey spans 21 nuclear equations of state covering nucleonic, hyperonic, Δ-admixed, and combined hyperon-- Δ compositions at F DM=1 -- 20%. The main sequences use λ=π and boson masses of 500 and 100, MeV for the core and halo studies, respectively. Independent normal-matter (NM) and DM displacements couple through the metric, yielding NM-led and DM-led f-mode branches whose radiative damping is determined by outgoing-wave boundary conditions. At 1.4,M, the core DM-led damping time decreases from order 10 cubed, s at F DM=1% to order 1, s at 20%, approximately following an empirical F DM-2 dependence, while NM-led damping times remain within 0.12 -- 0.33, s. Halo DM-led damping instead increases with dark fraction: at the same stellar mass and F DM=20%, dark radii approach 88, km, frequencies fall to 0.14, kHz, and damping times reach approximately 450, s. Selected high-mass core sequences exhibit strong DM-led damping-time upturns not reproduced by the leading mass-quadrupole diagnostic. Representative core calculations show greater DM-led sensitivity to the bosonic self-coupling and retain both branches at =3. Under the adopted nonrotating, circular-binary prescriptions, no evaluated configuration satisfies all geometric and perturbative conditions for a controlled linear tidal resonance.