Effects of Dark Matter on $f$-mode oscillations of Neutron Stars

The effect of dark matter (DM) on $f$-mode oscillations in DM admixed neutron stars (NSs) is investigated in a comprehensive analysis with particular attention to the role of the nuclear equation of state. Hadronic matter is modeled by the relativistic mean field model and the DM model is based on the neutron decay anomaly. The non-radial $f$-mode oscillations for such DM admixed NS are studied in a full general relativistic framework. We investigate the impact of DM, DM self-interaction, and DM fraction on the $f$-mode characteristics. We derive relations encoding the effect of DM on $f$-mode parameters. We then perform a systematic study by varying all the model parameters within their known uncertainty range and obtain a universal relation for the DM fraction based on the total mass of the star and DM self-interaction strength. We also perform a correlation study among model parameters, NS observables, in particular, $f$-mode parameters. Finally, we check the $f$-mode universal relations (URs) for the case of DM admixed NSs and demonstrate the existence of a degeneracy between purely hadronic NSs and DM admixed NSs.