A Bayesian Inference of Hybrid Stars with Large Quark Cores

TL;DR

This study investigates whether neutron stars can host large quark cores by constructing hybrid equations of state (EOS) that couple a relativistic mean-field hadronic phase to either the Nambu–Jona-Lasinio (NJL) or mean-field theory of QCD (MFTQCD) quark phases, connected via Maxwell construction. A Bayesian framework, implemented with PyMultiNest, samples hadronic and quark-sector parameters under priors and a suite of observational and theoretical constraints from nuclear matter properties, NICER X-ray data, GW170817 tidal information, and perturbative QCD bounds. The analysis yields five EOS sets (four hybrids and one hadronic RMF) and reveals a clear model dependence: MFTQCD tends to induce phase transitions at lower densities and can produce quark cores in 1.4 solar-mass stars, while NJL-type hybrids push the transition to higher densities and radii. All sets can describe current NICER and pQCD constraints and accommodate 2-solar-mass neutron stars; however, GW170817 and radius measurements favor certain parameter regions, with the MFTQCD scenario remaining most compatible with a broader range of observations. The work highlights how the onset density and stiffness of the quark EOS influence NS observables such as mass, radius, and tidal deformability, and it emphasizes that upcoming high-precision radius measurements will be crucial to discerning quark matter in neutron-star interiors.

Abstract

Neutron stars (NSs) are interesting objects capable of reaching densities unattainable on Earth. The properties of matter under these conditions remain a mystery. Exotic matter, including quark matter, may be present in the NS core. In this work, we explore the possible compositions of NS cores, in particular, the possible existence of large quark cores. We use the Relativistic Mean Field (RMF) model with nonlinear terms for the hadron phase and the Nambu-Jona-Lasinio (NJL) model and Mean Field Theory of Quantum Chromodynamics (MFTQCD) for the quark phase. Through Bayesian inference, we obtain different sets of equations: four sets with hybrid equations (three using the NJL model and the other using the MFTQCD model), and one set with only the hadron phase. We impose constraints regarding the properties of nuclear matter, X-ray observational data from NICER, perturbative QCD (pQCD) calculations, and causality on all sets. One set of hybrid NJL equations of state was also constrained by adding the GW170817 detection. All sets can describe observational data and theoretical restrictions. The MFTQCD allows for a phase transition to quark matter at lower densities compared to the NJL models. The MFTQCD model indicates that NSs with 1.4 solar mass have quark matter in their inner core. However, NJL models suggest that it is more probable that 1.4 solar mass NSs do not contain quark matter. Both the MFTQCD and NJL models agree that there is quark matter in 2 solar mass NSs. It is discussed that hybrid stars with a stiff quark equation of state could explain a larger radius of more massive stars, such as two solar mass stars, with respect to the canonical NS.

Figures (23)

• Figure 1: Example of a physical (top) and non-physical (bottom) phase transition. Cyan solid, black dashed and black dotted lines represent the hybrid, quark and hadron equations.
• Figure 2: Example of a physical (top) and non-physical (bottom) phase transition. The red dot shows the maximum mass, and the green dot shows the phase transition.
• Figure 3: Pressure versus energy density of the 90% of CI. Sets NJL (hybrid), MFTQCD (hybrid) and RMF (hadron) are represented in orange, cyan and hatched bands, respectively. Vertical lines indicate the 90% CI maximum for $\epsilon$. Band in gray represents the full (solid) and 90% of CI (dashed) of model-independent results from Altiparmak:2022bke.
• Figure 4: Pressure (top) and speed of sound squared in units of $c^2$ (bottom) 90% CI distributions versus the baryonic density. Same color code from Fig. \ref{['fig:paper2_PxE']}. Vertical bands represent the central density at the maximum NS mass of the 90% CI.
• Figure 5: Mass-radius diagram of the 90% of CI. Same color code from Fig. \ref{['fig:paper2_PxE']}. Observational data are shown as PSR J0030+0451 (blue) Vinciguerra_2024Miller_2019, PSR J0740+6620 (orange) Salmi_2024Miller:2021qha, PSR J0437+4715 (green) Choudhury_2024 and HESS J1731-347 (purple) hess with $1 \sigma$, $2 \sigma$ and $3 \sigma$ represented by solid, dashed and dotted lines, respectively.