Impact of dark boson mediated feeble interaction between dark matter and hadronic matter on $f$-mode oscillation of neutron stars

We studied the possible presence of dark matter (DM) in neutron stars (NSs) and the structural properties of the DM admixed NSs (DMANSs) in one of our recent works \cite{Guha:2024pnn}. The feeble interaction between the fermionic DM ($\chi$) with the hadronic matter is introduced through a dark scalar ($\phi$) and a dark vector ($\xi$) boson as mediators. The allowed range of the mass of the fermionic DM ($m_{\chi}$), for a particular range of DM Fermi momentum ($k_F^{\chi}$), was obtained in the same work \cite{Guha:2024pnn} with respect to the various astrophysical constraints on the structural properties of compact stars viz. the mass, radius and tidal deformability. The present work is dedicated to the calculation and study of non-radial oscillation of the DMANSs using Cowling approximation. We particularly investigate the effect of presence of DM on the fundamental ($f$) mode oscillation frequencies of the DMANSs utilizing the previously obtained range of $m_{\chi}$ for four different hadronic models. In this work we thoroughly investigate how the individual and combined effects of $m_{\chi}$ and $k_F^{\chi}$ affect the $f$-mode oscillation frequency. Within the framework of our DMANS models, for a particular value of $k_F^{\chi}$, the range of $f_{max}^{DMANS}$ with respect to the allowed range of $m_{\chi}$, is also obtained in the present work for four different hadronic models. Since in the present era, the 1.4 and 2.01 $M_{\odot}$ NSs are of special interest after the detection of GW170817 and PSR J0740+6620, we particularly investigate, for the four hadronic models, the range of $f_{1.4}^{DMANS}$ and $f_{2.01}^{DMANS}$ with respect to the acceptable range of $m_{\chi}$ corresponding to the range of $k_F^{\chi}$.