Nonradial oscillation modes in hybrid stars with hyperons and delta baryons: Full general relativity formalism vs. the Cowling approximation

We study the effects of hyperons, delta baryons, and quark matter phase transitions on $f$-mode oscillations in neutron stars. Using the density-dependent relativistic mean-field model (DDME2) for the hadronic phase and the density-dependent quark mass (DDQM) model for the quark phase, we construct hadronic and hybrid equations of state (EoSs) consistent with astrophysical constraints. Including hyperons and delta baryons soften the EoS, reducing maximum mass, while phase transition to the quark matter further softens the EoS, decreasing the speed of sound and hence the maximum mass. We confirm the well-known overestimation of $f$-mode frequencies by the Cowling approximation (by about 10-30\%) compared to full General Relativity calculation, and show that this discrepancy persists across models including hyperons, $\Delta$ baryons, and a phase transition to quark matter. While the discrepancy generally decreases with stellar mass, it increases near the maximum mass in the presence of a phase transition compared to EoSs without this phenomenology. We derive universal relations connecting the frequencies of the $f$-mode to the average density, compactness, and tidal deformability, finding significant deviations due to hyperons and delta baryons. These deviations could provide distinct observational signatures in gravitational wave data, offering new insights into dense matter physics and advancing gravitational wave asteroseismology of neutron star interiors. Empirical relations for mass-scaled and radius-scaled frequencies are also provided, highlighting the importance of GR calculations for accurate modeling.