We present a systematic study of $f$-mode oscillations in neutron stars containing hyperons, extending recent results obtained within the Cowling approximation to linearized general relativity. Employing a relativistic mean-field model, we find that the Cowling approximation can overestimate the quadrupolar $f$-mode frequency of neutron stars by up to 30% compared to the frequency obtained in the linearized general relativistic formalism. Imposing current astrophysical constraints, we derive updated empirical relations for gravitational wave asteroseismology. The frequency and damping time of quadrupole $f$-mode oscillations of hyperonic stars are found to be in the range of 1.47–2.45 kHz and 0.13–0.51 s, respectively. Our correlation studies demonstrate that among the various parameters of the nucleonic and hyperonic sectors of the model, the nucleon effective mass shows the strongest correlation with mode characteristics and neutron star observables. Estimates for the detectability of $f$ modes in a transient burst of gravitational waves from isolated hyperonic stars is also provided.

We study the principal core $g$-mode oscillation in hybrid stars containing quark matter and find that they have an unusually large frequency range ($\approx 200–600\mathrm{Hz}$) compared to ordinary neutron stars or self-bound quark stars of the same mass. Theoretical arguments and numerical calculations that trace this effect to the difference in the behavior of the equilibrium and adiabatic sound speeds in the mixed phase of quarks and nucleons are provided. We propose that the sensitivity of core $g$-mode oscillations to non-nucleonic matter in neutron stars could be due to the presence of a mixed quark-nucleon phase. Based on our analysis, we conclude that for binary mergers where one or both components may be a hybrid star, the fraction of tidal energy pumped into resonant $g$-modes in hybrid stars can exceed that of a normal neutron star by a factor of 2 to 3, although resonance occurs during the last stages of inspiral. A self-bound star, on the other hand, has a much weaker tidal overlap with the $g$-mode. The cumulative tidal phase error in hybrid stars, $\mathrm{\Delta }\varphi \cong 0.5\mathrm{rad}$, is comparable to that from tides in ordinary neutron stars, presenting a challenge in distinguishing between the two cases. However, should the principal $g$-mode be excited to sufficient amplitude for detection in a postmerger remnant with quark matter in its interior, its frequency would be a possible indication for the existence of non-nucleonic matter in neutron stars.

Oscillation modes of isolated compact stars can, in principle, be a fingerprint of the equation of state (EoS) of dense matter. We study the non-radial high-frequency $l=2$ spheroidal modes of neutron stars and strange quark stars, adopting a two-component model (core and crust) for these two types of stars. Using perturbed fluid equations in the relativistic Cowling approximation, we explore the effect of a strangelet or hadronic crust on the oscillation modes of strange stars. The results differ from the case of neutron stars with a crust. In comparison to fluid-only configurations, we find that a solid crust on top of a neutron star increases the $p$-mode frequency slightly with little effect on the $f$-mode frequency, whereas for strange stars, a strangelet crust on top of a quark core significantly increases the $f$-mode frequency with little effect on the $p$-mode frequency.

We perform the first study of the principal core $g$-mode oscillation in hybrid stars containing quark matter, utilizing a crossover model for the hadron-to-quark transition inspired by lattice QCD. The ensuing results are compared with our recent findings of $g$-mode frequencies in hybrid stars with a first-order phase transition using Gibbs constructions. We find that models using Gibbs construction yield $g$-mode amplitudes and the associated gravitational energy radiated that dominate over those of the chosen crossover model owing to the distinct behaviors of the equilibrium and adiabatic sound speeds in the various models. Based on our results, we conclude that were $g$ modes to be detected in upgraded LIGO and Virgo detectors it would indicate a first-order phase transition akin to a Gibbs construction.

Quasinormal oscillation modes of neutron stars provide a means to probe their interior composition using gravitational wave astronomy. We compute the frequencies and damping times of composition-dependent core $g$-modes of neutron stars containing quark matter employing linearized perturbative equations of general relativity. We find that ignoring background metric perturbations due to the oscillating fluid, as in the Cowling approximation, underestimates the $g$-mode frequency by up to 10% for higher mass stars, depending on the parameters of the nuclear equation of state and how the mixed phase is constructed. The $g$-mode frequencies are well described by a linear scaling with the central lepton (or combined lepton and quark) fraction for nucleonic (hybrid) stars. Our findings suggest that neutron stars with and without quarks are manifestly different with regards to their quasinormal $g$-mode spectrum, and may thus be distinguished from one another in future observations of gravitational waves from merging neutron stars.

Compact stars containing quark matter may masquerade as neutron stars in the range of measured mass and radius, making it difficult to draw firm conclusions on the phase of matter inside the star. The sensitivity of core g-mode oscillations to the presence of a mixed phase may alleviate this difficulty. In hybrid stars that admit quark matter in a mixed phase, the g-mode frequency rises sharply due to a marked decrease in the equilibrium sound speed. Resonant excitation of g-modes can leave an imprint in the waveform of coalescing binary compact stars. We present analytic and numeric results to assess the sensitivity displayed by g-mode oscillations to quark matter in a homogeneous or mixed phase and also compute relevant damping times in quark matter due to viscosity.