We combine the equation of state of dense matter up to twice nuclear saturation density $n_{\mathrm{sat}}$ obtained using chiral effective field theory $\left(\chi \mathrm{EFT}\right)$ and recent observations of neutron stars to gain insights about the high-density matter encountered in their cores. A key element in our study is the recent Bayesian analysis of correlated EFT truncation errors based on order-by-order calculations up to next-to-next-to-next-to-leading order in the $\chi \mathrm{EFT}$ expansion. We refine the bounds on the maximum mass imposed by causality at high densities and provide stringent limits on the maximum and minimum radii of $\sim 1.4{\mathrm{M}}_{\odot }$ and $\sim 2.0{\mathrm{M}}_{\odot }$ stars. Including $\chi \mathrm{EFT}$ predictions from $n_{\mathrm{sat}}$ to $2n_{\mathrm{sat}}$ reduces the permitted ranges of the radius of a $1.4{\mathrm{M}}_{\odot }$ star, $R_{1.4}$, by $\sim 3.5\text{km}$. If observations indicate $R_{1.4}<11.2\text{km}$, then our study implies that either the squared speed of sound $c_s^2>1/2$ for densities above $2n_{\mathrm{sat}}$ or that $\chi \mathrm{EFT}$ breaks down below $2n_{\mathrm{sat}}$. We also comment on the nature of the secondary compact object in GW190814 with mass $\simeq 2.6{\mathrm{M}}_{\odot }$ and discuss the implications of massive neutron stars $>2.1{\mathrm{M}}_{\odot }\left(2.6{\mathrm{M}}_{\odot }\right)$ in future radio and gravitational-wave searches. Some form of strongly interacting matter with $c_s^2>0.35\left(0.55\right)$ must be realized in the cores of such massive neutron stars. In the absence of phase transitions below $2n_{\mathrm{sat}}$, the small tidal deformability inferred from GW170817 lends support for the relatively small pressure predicted by $\chi \mathrm{EFT}$ for the baryon density $n_{\mathrm{B}}$ in the range $1\text{–}2n_{\mathrm{sat}}$. Together they imply that the rapid stiffening required to support a high maximum mass should occur only when $n_{\mathrm{B}}\gtrsim 1.5\text{–}1.8n_{\mathrm{sat}}$.

Quark matter may appear due to a hadronic-quark transition in the core of a hybrid star. Quarkyonic matter is an approach in which both quarks and nucleons appear as quasiparticles in a crossover transition, and provides an explicit realization of early ideas concerning quark matter (e.g., the MIT bag model). This description has recently been employed by McLerran and Reddy to model chargeless (pure neutron) matter with an approach that has the virtue that the speed of sound rises quickly at a neutron-quark transition so as to satisfy observational constraints on the neutron star maximum mass ($\gtrsim 2M_{\odot }$) and the radius of a $1.4M_{\odot }$ star ($R_{1.4}\lesssim 13.5\mathrm{km}$). Traditional models involving first-order transitions result in softer pressure-energy density relations that have difficulty satisfying these constraints except with very narrow choices of parameters. We propose a variation of quarkyonic matter involving protons and leptons whose energy can be explicitly minimized to achieve both chemical and beta equilibrium, which cannot be done in the chargeless formulation. Quarkyonic stellar models are able to satisfy observed mass and radius constraints with a wide range of model parameters, avoiding the obligatory fine-tuning of conventional hybrid star models, including requiring the transition density to be very close to the nuclear saturation density. Our formulation fits experimental and theoretical properties of the nuclear symmetry energy and pure neutron matter, and contains as few as three free parameters. This makes it an ideal tool for the study of high-density matter that is an efficient alternative to piecewise polytrope or spectral decomposition methods.

Finite size effects in a neutron star merger are manifested, at leading order, through the tidal deformabilities of the stars. If strong first-order phase transitions do not exist within neutron stars, both neutron stars are described by the same equation of state, and their tidal deformabilities are highly correlated through their masses even if the equation of state is unknown. If, however, a strong phase transition exists between the central densities of the two stars, so that the more massive star has a phase transition and the least massive star does not, this correlation will be weakened. In all cases, a minimum deformability for each neutron star mass is imposed by causality, and a less conservative limit is imposed by the unitary gas constraint, both of which we compute. In order to make the best use of gravitational wave data from mergers, it is important to include the correlations relating the deformabilities and the masses as well as lower limits to the deformabilities as a function of mass. Focusing on the case without strong phase transitions, and for mergers where the chirp mass $\mathcal{M}\le 1.4M_{\odot }$, which is the case for all observed double neutron star systems where a total mass has been accurately measured, we show that the ratio of the dimensionless tidal deformabilities satisfy ${\mathrm{\Lambda }}_1/{\mathrm{\Lambda }}_2\sim q^6$, where $q=M_2/M_1$ is the binary mass ratio; $\mathrm{\Lambda }$ and $M$ are the dimensionless deformability and mass of each star, respectively. Moreover, they are bounded by $q^{n_{-}}\ge {\mathrm{\Lambda }}_1/{\mathrm{\Lambda }}_2\ge q^{n_{0+}+q{n}_{1+}}$, where $n_{-}<n_{0+}+q{n}_{1+}$; the parameters depend only on $\mathcal{M}$, which is accurately determined from the gravitational-wave signal. We also provide analytic expressions for the wider bounds that exist in the case of a strong phase transition. We argue that bounded ranges for ${\mathrm{\Lambda }}_1/{\mathrm{\Lambda }}_2$, tuned to $\mathcal{M}$, together with lower bounds to $\mathrm{\Lambda }(M)$, will be more useful in gravitational waveform modeling than other suggested approaches.

Among the various oscillation modes of neutron stars, $f$- and $g$- modes are the most likely to be ultimately observed in binary neutron star mergers due to their relatively large coupling and shared frequencies with tidal excitations. The $f$-mode frequency and damping time are known to correlate in normal neutron stars with their compactness, and previous fits to hadronic stars are extended and shown to be valid for an extremely broad sampling of equations of state using a piecewise polytropic parametrization scheme for hadrons and a constant sound-speed parametrization for quark matter. Separate fits applicable to quark (self-bound) stars are improved. Much more significant correlations exist with tidal deformability, and therefore with moment of inertia and quadrupole moment. It is conclusively demonstrated that these correlations are the same for all types of stars, whether hadronic, hybrid, or pure quark, and its accuracy is quantified. A novel 1-node branch of the $f$-mode that occurs in low-mass hybrid stars in a narrow mass range just beyond the critical mass necessary for a phase transition to appear is identified. This 1-node branch shows the largest, but still small, deviations from the universal correlation for any configuration. It is characterized by a nonmonotonic relation between neutron star mass and $f$-mode frequency, in contrast to the behavior otherwise observed in normal, quark and hybrid stars. The $g$-mode only exists in matter with a nonbarotropic equation of state involving temperature, chemical potential or composition (such as being out of beta equilibrium), or a phase transition in barotropic matter. The $g$-mode therefore could serve as a probe for studying phase transitions in hybrid stars. In contrast with the $f$-mode, $g$-mode frequencies do not correlate well with tidal deformability, but depend strongly on properties of the transition (the density and the magnitude of the discontinuity) at the transition. Imposing causality and maximum mass constraints, a fit involving neutron star and phase transition properties is found and the $g$-mode frequency is determined to have an upper bound of about 1.25 kHz. However, if the sound speed $c_s$ in the inner core at densities above the phase transition density is restricted to $c_s^2\le 1/3$, $g$-mode frequencies can only reach about 0.8 kHz, which are significantly lower than $f$-mode frequencies (1.3–2.8 kHz). $g$-mode gravitational wave damping times are found to be extremely long, $>{10}^4\mathrm{s}$ (${10}^2\mathrm{s}$) in the inner core with $c_s^2\le 1/3$ (1), in comparison with $f$-mode damping times (0.1–1 s).

We investigate the impact of the neutron-star matter equation of state on the $f$- and $p_1$-mode oscillations of neutron stars obtained within the Cowling approximation and linearized general relativity. The $f$- and $p_1$-mode oscillation frequencies, and their damping times are calculated using representative sets of Skyrme Hartree-Fock and relativistic mean-field models, all of which reproduce nuclear systematics and support $2M_{\odot }$ neutron stars. Our study shows strong correlations between the frequencies of $f$- and $p_1$-modes and their damping times with the pressure of $\beta$-equilibrated matter at densities equal to or slightly higher than the nuclear saturation density ${\rho }_0$. Such correlations are found to be almost independent of the composition of the stars. The frequency of the $p_1$-mode of $1.4M_{\odot }$ star is strongly correlated with the slope of the symmetry energy $L_0$ and $\beta$-equilibrated pressure at density ${\rho }_0$. Compared to GR calculations, the error in the Cowling approximation for the $f$-mode is about 30% for neutron stars of low mass, whereas it decreases with increasing mass. The accuracy of the $p_1$-mode is better than 15% for neutron stars of maximum mass, and improves for lower masses and higher number of radial nodes.

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.