The effect of pasta phases on the quark-hadron phase transition is investigated for a set of relativistic mean-field equations of state for both hadron and quark matter. The results of the full numerical solution with pasta phases are compared with those of an interpolating construction used in previous works, for which we demonstrate an adequate description of the numerical results. A one-to-one mapping of the free parameter of the construction to the physical surface tension of the quark-hadron interface is obtained for which a fit formula is given. For each pair of quark and hadron matter models the critical value of the surface tension is determined, above which the phase transition becomes close to the Maxwell construction. This result agrees well with earlier theoretical estimates. The study is extended to neutron star matter in beta equilibrium with electrons and muons and is applied to investigate the effect of pasta phases on the structure of hybrid compact stars and the robustness of a possible third family solution.

We investigate the possibility that GW170817 was not the merger of two conventional neutron stars (NS), but involved at least one if not two hybrid stars with a quark matter core that might even belong to a third family of compact stars. To this end, we develop a Bayesian analysis method for selecting the most probable equation of state (EoS) under a set of constraints from compact star physics, which now also include the tidal deformability from GW170817 and the first result for the mass and radius determination for PSR J0030+0451 by the NICER Collaboration. We apply this method for the first time to a two-parameter family of hybrid EoS based on the DD2 model with nucleonic excluded volume for hadronic matter and the color superconducting generalized nlNJL model for quark matter. The model has a variable onset density for deconfinement and can mimic the effects of pasta phases with the possibility of producing a third family of hybrid stars in the mass-radius diagram. The main findings of this study are that: (1) the presence of multiple configurations for a given mass (twins or even triples) corresponds to a set of disconnected lines in the ${\mathsf{\Lambda }}_1$ – ${\mathsf{\Lambda }}_2$ diagram of tidal deformabilities for binary mergers, so that merger events from the same mass range may result in a probability landscape with different peak positions; (2) the Bayesian analysis with the above observational constraints favors an early onset of the deconfinement transition, at masses of $M_{\mathrm{onset}}\le 0.8M_{\odot }$ with an M–R relationship that in the range of observed neutron star masses is almost indistinguishable from that of a soft hadronic Akmal, Pandharipande, and Ravenhall (APR) EoS; (3) a few, yet fictitious measurements of the NICER experiment two times more accurate than the present value and a different mass and radius that would change the posterior likelihood so that hybrid EoS with a phase transition onset in the range $M_{\mathrm{onset}}$ = 1.1–1.6 $M_{\odot }$ would be favored.

We introduce a family of equations of state (EoS) for hybrid neutron star (NS) matter that is obtained by a two-zone parabolic interpolation between a soft hadronic EoS at low densities and a set of stiff quark matter EoS at high densities within a finite region of chemical potentials $$\mu _H< \mu < \mu _Q$$μ H < μ < μ Q . Fixing the hadronic EoS as the APR one and choosing the color-superconducting, nonlocal NJL model with two free parameters for the quark phase, we perform Bayesian analyses with this two-parameter family of hybrid EoS. Using three different sets of observational constraints that include the mass of PSR J0740+6620, the tidal deformability for GW170817, and the mass-radius relation for PSR J0030+0451 from NICER as obligatory (set 1), while set 2 uses the possible upper limit on the maximum mass from GW170817 as an additional constraint and set 3 instead of the possibility that the lighter object in the asymmetric binary merger GW190814 is a neutron star. We confirm that in any case, the quark matter phase has to be color superconducting with the dimensionless diquark coupling approximately fulfilling the Fierz relation $$\eta _D=0.75$$η D = 0.75 and the most probable solutions exhibiting a proportionality between $$\eta _D$$η D and $$\eta _V$$η V , the coupling of the repulsive vector interaction that is required for a sufficiently large maximum mass. We used the Bayesian analysis to investigate with the method of fictitious measurements the consequences of anticipating different radii for the massive $$2~M_\odot $$2 M ⊙ PSR J0740+6220 for the most likely equation of state. With the actual outcome of the NICER radius measurement on PSR J0740+6220 we could conclude that for the most likely hybrid star EoS would not support a maximum mass as large as $$2.5~M_\odot $$2.5 M ⊙ so that the event GW190814 was a binary black hole merger.

We investigate the robustness of third family solutions for hybrid compact stars with a quark matter core that correspond to the occurrence of high-mass twin stars against a softening of the phase transition by means of a construction that mimics the effects of pasta structures in the mixed phase. We consider a class of hybrid equations of state that exploits a relativistic mean-field model for the hadronic as well as for the quark matter phase. We present parametrizations that correspond to branches of high-mass twin star pairs with maximum masses between $2.05M_{\odot }$ and $1.48M_{\odot }$ having radius differences between 3.2 and 1.5 km, respectively. When compared to a Maxwell construction with a fixed value of critical pressure $P_c$, the effect of the mixed phase construction consists in the occurrence of a region of pressures around $P_c$ belonging to the coexistence of hadronic and quark matter phases between the onset pressure at $P_H$ and the end of the transition at $P_Q$. The maximum broadening which would still allow mass-twin compact stars is found to be ${(P_Q-P_H)}_{\max }\approx P_c$ for all parametrizations within the present class of models. At least the heavier of the neutron stars of the binary merger GW170817 could have been a member of the third family of hybrid stars. We present the example of another class of hybrid star equations of state for which the appearance of the third family branch is not as robust against mixed phase effects as that of the present work.

First-order phase transitions, such as the liquid-gas transition, proceed via formation of structures, such as bubbles and droplets. In strongly interacting compact star matter, at the crust-core transition but also the hadron-quark transition in the core, these structures form different shapes dubbed “pasta phases”. We describe two methods to obtain one-parameter families of hybrid equations of state (EoS) substituting the Maxwell construction that mimic the thermodynamic behaviour of pasta phase in between a low-density hadron and a high-density quark matter phase without explicitly computing geometrical structures. Both methods reproduce the Maxwell construction as a limiting case. The first method replaces the behaviour of pressure against chemical potential in a finite region around the critical pressure of the Maxwell construction by a polynomial interpolation. The second method uses extrapolations of the hadronic and quark matter EoS beyond the Maxwell point to define a mixing of both with weight functions bounded by finite limits around the Maxwell point. We apply both methods to the case of a hybrid EoS with a strong first order transition that entails the formation of a third family of compact stars and the corresponding mass twin phenomenon. For both models, we investigate the robustness of this phenomenon against variation of the single parameter: the pressure increment at the critical chemical potential that quantifies the deviation from the Maxwell construction. We also show sets of results for compact star observables other than mass and radius, namely the moment of inertia and the baryon mass.