The recent direct observation of gravitational wave event GW170817 and its GRB 170817 A signal has opened up a new window to studying neutron stars and heralds a new era of astronomy referred to as the multimessenger astronomy. Both gravitational and electromagnetic waves from a single astrophysical source have been detected for the first time. This combined detection offers an unprecedented opportunity to place constraints on the neutron star matter equation of state (EoS). The existence of a possible hadron–quark phase transition in the central regions of neutron stars is associated with the appearance of g modes, which are extremely important as they could signal the presence of a pure quark matter core in the centers of neutron stars. Observations of g modes with frequencies between 1 and 1.5 kHz could be interpreted as evidence of a sharp hadron–quark phase transition in the cores of neutron stars. In this article, we shall review the description of the dense matter composing neutron stars, the determination of the EoS of such matter, and the constraints imposed by astrophysical observations of these fascinating compact objects.

We investigate the hadron-quark phase transition in cold neutron stars in light of (i) the observed limits on the maximum-mass of heavy pulsars, (ii) constraints on the tidal properties inferred from the gravitational waves emitted in binary neutron-star mergers, and (iii) mass and radius constraints derived from the observation of hot spots on neutron star observed with the Neutron Star Interior Composition Explorer instrument. Special attention is directed to the possible presence of $\mathrm{\Delta }(1232)$ baryons in neutron star matter. Our results indicate that this particle could make up a large fraction of the baryons in neutron stars and thus have a significant effect on the properties of such objects, particularly on their radii. This is partially caused by the low density appearance of $\mathrm{\Delta }\mathrm{s}$ for a wide range of theoretically defensible sets of meson-hyperon, SU(3) ESC08 model, and meson-$\mathrm{\Delta }$ coupling constants. The transition of hadronic matter to quark matter, treated in the $2\mathrm{SC}+\mathrm{s}$ condensation phase, is found to occur only in neutron stars very close to the mass peak. Nevertheless, quark matter may still constitute an appreciable fraction of the stars’ total matter if the phase transition is treated as Maxwell-like (sharp), in which case the neutron stars located beyond the gravitational mass peak would remain stable against gravitational collapse. In this case, the instability against gravitational collapse is shifted to a new (terminal) mass different from the maximum-mass of the stellar sequence, giving rise to stable compact objects with the same gravitational masses as those of the neutron stars on the traditional branch, but whose radii are smaller by up to 1 km. All models for the equation of state of our study fall comfortably within the bound established very recently by Annala et al. [ Nat. Phys., 10.1038/s41567-020-0914-9 (2020)].

In the first part of this paper, we use a nonlocal extension of the three-flavor Polyakov-Nambu-Jona-Lasinio model, which takes into account flavor-mixing, momentum dependent quark masses, and vector interactions among quarks, to investigate the possible existence of a spinodal region (determined by the vanishing of the speed of sound) in the QCD phase diagram and determine the temperature and chemical potential of the critical end point. In the second part of the paper, we investigate the quark-hadron composition of baryonic matter at zero as well as nonzero temperature. This is of great topical interest for the analysis and interpretation of neutron star merger events such as GW170817. With this in mind, we determine the composition of proto-neutron star matter for entropies and lepton fractions that are typical of such matter. These compositions are used to delineate the evolution of proto-neutron stars to neutron stars in the baryon-mass versus gravitational-mass diagram. The hot stellar models turn out to contain significant fractions of hyperons and $\mathrm{\Delta }$ isobars but no deconfined quarks. The latter are found to exist only in cold neutron stars.

We study local and non-local Polyakov Nambu-Jona-Lasinio models and analyze their respective phase transition diagram. We construct hybrid stars using the zero temperature limit of the local and non-local versions of Nambu-Jona-Lasinio model for quark matter and the modern GM1(L) parametrization of the non-linear relativistic mean field model for hadronic matter. We compare our models with data from PSR J1614-2230 and PSR J0343+0432 and also from GW170817 and its electromagnetic counterpart GRB170817A and AT2017gfo. We study observational signatures of the appearance of a mixed phase as a result of modeling a phase transition that mimics the Gibbs formalism and compare the results with the sharp first-order phase transition obtained using the Maxwell construction. We also study in detail the g-mode associated with discontinuities in the equation of state, and calculate non-radial oscillation modes using relativistic Cowling approximation.

In the first part of this paper, we investigate the possible existence of a structured hadron-quark mixed phase in the cores of neutron stars. This phase, referred to as the hadron-quark pasta phase, consists of spherical blob, rod, and slab rare phase geometries. Particular emphasis is given to modeling the size of this phase in rotating neutron stars. We use the relativistic mean-field theory to model hadronic matter and the non-local three-flavor Nambu–Jona-Lasinio model to describe quark matter. Based on these models, the hadron-quark pasta phase exists only in very massive neutron stars, whose rotational frequencies are less than around 300 Hz. All other stars are not dense enough to trigger quark deconfinement in their cores. Part two of the paper deals with the quark-hadron composition of hot (proto) neutron star matter. To this end we use a local three-flavor Polyakov–Nambu–Jona-Lasinio model which includes the ’t Hooft (quark flavor mixing) term. It is found that this term leads to non-negligible changes in the particle composition of (proto) neutron stars made of hadron-quark matter.