Non-radial oscillations of hadronic neutron stars, quark stars, and hybrid stars : Calculation of $f$, $p$, and $g$ mode frequencies

The composition and equation of state (EoS) of dense matter relevant to compact stars are quite inconclusive. However, certain observational constraints on the structural properties of compact stars help us constrain the EoS to a fair extent. Moreover, gravitational asteroseismology gives a notion of the composition and EoS of compact stars. The next generation gravitational wave (GW) detectors are likely to detect several oscillation mode frequencies of the GWs. In this work we compute the fundamental ($f$) and the first pressure ($p_1$) mode frequencies ($f_f$ and $f_{p1}$, respectively) with different compositions viz., hadronic, quark, and hybrid star (HS) matter. For HSs, we also study the gravity ($g$) mode frequency ($f_g$). For each phase we also study the correlation between the oscillation frequencies of 1.4 $M_{\odot}$ and 2.01 $M_{\odot}$ compact stars with other different properties. We find that various possible composition of compact stars substantially affects the oscillation frequencies. However, the mass-scaled angular $f$ mode frequency ($\omega_f M$) varies universally with compactness ($C$) for all hadronic, quark and hybrid stars. The $f$ mode frequency ($f_{f_{1.4}}$) of the canonical 1.4 $M_{\odot}$ compact star, obtained with different composition, is quite correlated with the canonical radius ($R_{1.4}$) and tidal deformability ($\Lambda_{1.4}$) while $f_{p_{1.4}}$ is well correlated with slope parameter of the symmetry energy. We also show that $f_{g_{1.4}}$ of the HSs varies almost linearly with $\Lambda_{1.4}$. Should $g$ modes be detected, they could not only support the existence of HSs, but $f_g$ could be useful to understand the strength of quark repulsion in HSs.