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.

The central compact object within HESS J1731-347 possesses unique mass and radius properties that renders it a compelling candidate for a self-bound star. In this research, we examine the capability of quark stars composed of color superconducting quark matter to explain the latter object by using its marginalized posterior distribution and imposing it as a constraint on the relevant parameter space. Namely, we investigate quark matter for $N_f=2,3$ in the color superconducting phase, incorporating perturbative QCD corrections, and we derive their properties accordingly. The utilized thermodynamic potential of this work possesses an MIT bag model formalism with the parameters being established as flavor-independent. In this instance, we conclude the favor of the 3-flavor over the 2-flavor color superconducting quark matter and we isolate our interest on the former, without neglecting the possible favor of the latter in a different framework. The parameter space is further confined due to the additional requirement for a high maximum mass ($M_{\mathrm{TOV}}\ge 2.6M_{\odot }$), accounting for GW190814’s secondary companion. Our study places significant emphasis on the speed of sound and the trace anomaly which was proposed as a measure of conformality [Y. Fujimoto et al., Phys. Rev. Lett. 129, 252702 (2022).]. We conclude that it is possible for color-flavor locked quark stars to reach high masses without violating the conformal bound or the $⟨\mathrm{\Theta }{⟩}_{{\mu }_B}\ge 0$, provided that the quartic coefficient ${\alpha }_4$ does not exceed an upper limit which depends on the established $M_{\mathrm{TOV}}$. For $M_{\mathrm{TOV}}=2.6M_{\odot }$, we find that the limit reads ${\alpha }_4\le 0.594$. Lastly, a further study takes place on the agreement of color-flavor locked quark stars with additional astrophysical objects including the GW170817 and GW190425 events, followed by a relevant discussion.

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.