We study the mesonic nonlinear (NL) interaction equation of state (EOS) employing the relativistic mean-field model and investigate the effect of the $\sigma$-cut potential (NL-$\sigma$ cut) and dark matter (NL DM) on the nonradial and radial oscillation modes of neutron stars. For the NL-$\sigma$ cut, we include the $\sigma$-cut potential $U_{\mathrm{cut}}(\sigma )$ to study its effect. For dark matter, we use the neutron decay anomaly model. For each model, we investigate two extreme EOSs, stiff and soft, that cover the entire allowed parameter range from the given model, consistent with the current astrophysical constraints. The EOS and the stellar properties, such as mass and radius, are calculated, and the effect of the $\sigma$ cut and DM is discussed. Both nonradial and radial oscillation modes are computed in the general relativistic framework. We study the nonradial $f$ and $p_1$ mode frequency, damping time, and some quasiuniversal relations connecting the frequencies of the $f$ mode to the average density and compactness. The analysis shows that the $f$ and $p_1$ mode frequencies at both $1.4M_{\odot }$ and the maximum mass configuration are higher in the NL-DM model compared to the NL and NL-$\sigma$ models. The consistent alignment between our prior parametrizations and current calculations strongly supports the existence of quasiuniversal relations that hold true irrespective of the particular matter components involved. For the radial oscillations, we compute 10 lowest-order modes ($f$, $p$) and study the radial perturbations as well as the large frequency separation with the NL-$\sigma$ cut and NL-DM EOS, showing that the microphysics involved in the NS EOS is imprinted on the frequency separation between different nodes.

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• Dark matter
• Nuclear astrophysics
• Nuclear matter in neutron stars