Neutron stars are among the densest known objects in the universe and an ideal laboratory for the strange physics of supercondensed matter. While the simultaneous measurements of mass and radius of nonrotating neutron stars may impose constraints on the properties of the dense nuclear matter, the observation and study of maximally rotating ones, close to the mass-shedding limit, may lead to significantly further constraints. Theoretical predictions allow neutron stars to rotate extremely fast (even more than $2000\mathrm{Hz}$). However, until this moment, the fastest observed rotating pulsar has a frequency of $716\mathrm{Hz}$, much lower compared to the theoretical predictions. There are many suggestions for the mechanism which lead to this situation. In any case, the theoretical study of uniformly rotating neutron stars, along with accurate measurements, may offer rich information concerning the high-density part of the equation of state. In addition, neutron stars through their evolution may provide us with a criterion to determine the final fate of a rotating compact star. Sensitivity of bulk neutron stars properties on the equation of state at the mass-shedding limit are the main subject of the present study.

The upper bound of the speed of sound in dense nuclear matter is one of the most interesting but still unsolved problems in nuclear physics. Theoretical studies in connection with recent observational data of isolated neutron stars as well as binary neutron stars systems offer an excellent opportunity to shed light on this problem. In the present work, we suggest a method to directly relate the measured tidal deformability (polarizability) of binary neutron stars system (before merger) to the maximum neutron star mass scenario and possible upper bound on the speed of sound. This method is based on the simple but efficient idea that while the upper limit of the effective tidal deformability favors soft equations of state, the recent high measured values of neutron star mass favor stiff ones. In the present work, first, using a simple well established model we parametrize the stiffness of the equation of state with the help of the speed of sound. Second, in comparison with the recent observations by LIGO/VIRGO collaboration of two events, GW170817 and GW190425, we suggest possible robust constraints. Moreover, we evaluate and postulate, in the framework of the present method, what kind of future measurements could help us to improve the stringent of the constraints on the neutron star equation of state.

On 14 August 2019, the LIGO/Virgo collaboration observed a compact object with mass $\sim {2.59}_{-0.09}^{+0.08}{M}_{\odot }$, as a component of a system where the main companion was a black hole with mass $\sim 23M_{\odot }$. A scientific debate initiated concerning the identification of the low mass component, as it falls into the neutron star–black hole mass gap. The understanding of the nature of GW190814 event will offer rich information concerning open issues, the speed of sound and the possible phase transition into other degrees of freedom. In the present work, we made an effort to probe the nuclear equation of state along with the GW190814 event. Firstly, we examine possible constraints on the nuclear equation of state inferred from the consideration that the low mass companion is a slow or rapidly rotating neutron star. In this case, the role of the upper bounds on the speed of sound is revealed, in connection with the dense nuclear matter properties. Secondly, we systematically study the tidal deformability of a possible high mass candidate existing as an individual star or as a component one in a binary neutron star system. As the tidal deformability and radius are quantities very sensitive on the neutron star equation of state, they are excellent counters on dense matter properties. We conjecture that similar isolated neutron stars or systems may exist in the universe and their possible future observation will shed light on the maximum neutron star mass problem.

In agreement with the constantly increasing gravitational wave events, new aspects of the internal structure of compact stars can be considered. A scenario in which a first-order transition takes place inside these stars is of particular interest, as it can lead, under certain conditions, to a third gravitationally stable branch (besides white dwarfs and neutron stars), the twin stars. The new branch yields stars with the same mass as normal compact stars but quite different radii. We focus on hybrid stars undergoing a hadron-to-quark phase transition near their core and how this new stable configuration arises. Emphasis is given on the aspects of the phase transition and its parametrization in two different ways—namely, with the Maxwell and Gibbs constructions. We systematically study the gravitational mass, the radius, and the tidal deformability, and we compare them with the predictions of the recent observation by the LIGO/VIRGO Collaboration, the GW170817 event, and the mass and radius limits, suggesting possible robust constraints. Moreover, we extend the study to include rotation effects on the twin star configurations. The recent discovery of the fast rotating supermassive pulsar PSR J0952-0607 triggered the efforts to constrain the equation of state and, moreover, to examine possible predictions related to the phase transition in dense nuclear matter. We pay special attention to relate the PSR J0952-0607 pulsar properties to the twin star predictions, and mainly to explore the possibility that the existence of such a massive object would rule out the existence of twin stars. Finally, we discuss the constraints on the radius and mass of the recently observed compact object within the supernova remnant HESS J1731-347. The estimations imply that this object is either the lightest known neutron star or a star with a more exotic equation of state.

The prediction of the equation of state of hot, dense nuclear matter is one of the most complicated and interesting problems in nuclear astrophysics. At the same time, knowledge of it is the basic ingredient for some of the most interesting studies. In the present work, we concentrate our study on the construction of the equation of state of hot, dense nuclear matter, related mainly to the interior of the neutron star. We employ a theoretical nuclear model, which includes momentum-dependent interaction among the nucleons, along with state-of-the-art microscopic calculations. Thermal effects are introduced in a self-consistent way, and a set of isothermal and isentropic equations of state are predicted. The predicted equations of state are used in order to acquire and extend the knowledge of the thermal effect on both nonrotating and rapidly rotating with the Kepler frequency neutron stars. The simultaneous study of thermal and rotation effects provides useful information on some of the most important quantities, including the mass (gravitational and baryon) and radius, the Kepler frequency and Kerr parameter, the moment of inertia, etc. These quantities are directly related to studies of protoneutron stars and mainly the hot and rapidly rotating remnant of a binary neutron star merger. Data from the late observations of binary neutron star mergers and the present study may offer useful tools for investigation and help in providing possible constraints on the equation of state of nuclear matter.

The recent analysis on the central compact object in the HESS J1731-347 remnant suggests interestingly small values for its mass and radius. Such an observation favors soft nuclear models that may be challenged by the observation of massive compact stars. In contrast, the recent PREX-II experiment, concerning the neutron skin thickness of ${}^{208}\mathrm{Pb}$, points toward stiff equations of state that favor larger compact star radii. In the present study, we aim to explore the compatibility between stiff hadronic equations of state (favored by PREX-II) and the HESS J1731-347 remnant in the context of hybrid stars. For the construction of hybrid equations of state we use three widely employed Skyrme models combined with the well-known vector MIT bag model. Furthermore we consider two different scenarios concerning the energy density of the bag. In the first case, that of a constant bag parameter, we find that the resulting hybrid equations of state (which satisfy the HESS J1731-347 constraints) are strongly disfavored by the observation of $\sim 2M_{\odot }$ pulsars. However, the introduction of a Gaussian density dependence yields results that are compatible with the conservative $2M_{\odot }$ constraint. The utilization of recent data based on the observation of PSR $\mathrm{J}0030+0451$, PSR J0952-0607, and GW190814 allows for the imposition of additional constraints on the relevant parameters and the stiffness of the two phases. Interestingly, we find that the derived hybrid equations of state do not satisfy the PSR $\mathrm{J}0030+0451$ constraints in $1\sigma$ and only marginally agree with the $2\sigma$ estimations. In addition, we estimate that the observation of massive pulsars, like PSR J0952-0607, in combination with the existence of HESS J1731-347, may require a strong phase transition below $\sim 1.7n_0$. Finally, we show that the supermassive compact object ($2.5-2.67M_{\odot }$) involved in GW190814 could potentially be explained as a rapidly rotating hybrid star.