A Bayesian Inference of Hybrid Stars with Large Quark Cores

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 and one set with only the hadron phase. We impose constraints regarding the properties of nuclear matter, X-ray observational data from NICER, gravitational wave data from the binary neutron star merger GW170817, perturbative QCD (pQCD) calculations, and causality. The MFTQCD allows for a phase transition to quark matter at low densities, just above saturation density, while for the NJL sets, the phase transition occurs above twice the saturation density. As a result, the MFTQCD model predicts the presence of quark matter in the inner core of 1.4 M$\odot$ NSs, while NJL models suggest a low probability of quark matter in the interior of a 1.4 M$\odot$ NS. Both models predict the existence of quark matter in 2 M$\odot$ NSs. The slope of the mass-radius curve has been shown to carry information about the presence of quark matter. In particular, a positive slope at 1.8 M$\odot$ indicates the presence of non-nucleonic matter. A hybrid star with a stiff quark equation of state could explain a larger radius in more massive stars, such as two solar mass stars, compared to canonical NSs.

💡 Research Summary

This paper presents a comprehensive Bayesian analysis of the equation of state (EOS) of neutron stars (NSs) that may contain sizable quark cores, i.e., hybrid stars. The authors model the hadronic phase with a relativistic mean‑field (RMF) theory that includes nonlinear σ, ω, and ρ meson self‑interactions. The RMF coupling constants (gσ, gω, gρ) and nonlinear coefficients (b, c, ξ, Λω) are treated as free parameters with broad uniform priors, constrained later by nuclear saturation properties (energy per nucleon, symmetry energy, incompressibility) and causality.

For the quark phase two distinct effective QCD models are employed. The first is the three‑flavor Nambu–Jona‑Lasinio (NJL) model, which respects global QCD symmetries, implements dynamical chiral symmetry breaking, and incorporates four‑quark and eight‑quark interaction channels (vector, axial‑vector, scalar‑vector mixing). Fixed parameters (current quark masses, cutoff Λ, coupling G, ’t Hooft term κ) are calibrated to meson phenomenology, while the additional vector‑type couplings (Gω, Gρ, Gωω, Gσω, Gρω) are sampled in the Bayesian inference. The second quark model is the Mean‑Field Theory of QCD (MFTQCD), derived by separating soft and hard gluon fields, treating the soft component as a constant condensate and the hard component as a classical background. This yields an EOS of the form
P = (27/2) ξ² ρ² − B + PF,
ε = (27/2) ξ² ρ² + B + εF,
where ξ = g/mG controls a repulsive vector term and B plays the role of a bag constant. Both ξ and B are assigned uniform priors.

The phase transition between hadronic and quark matter is modeled with a Maxwell construction, i.e., a sharp interface at equal pressure but discontinuous energy density. The authors justify this choice by (i) assuming a sufficiently large surface tension that suppresses mixed‑phase “pasta” structures, (ii) the stronger observational signatures a sharp transition imprints on tidal deformabilities, and (iii) computational efficiency for the nested‑sampling algorithm.

Bayesian inference is performed using PyMultiNest (nested sampling) within the Bilby framework. The likelihood combines five independent constraints: (1) nuclear matter properties at saturation and pure neutron matter from chiral effective field theory; (2) NICER mass‑radius measurements of PSR J0030+0451 and PSR J0740+6620; (3) tidal deformability limits Λ1.4 and Λ1.6 from GW170817; (4) perturbative QCD (pQCD) pressure bounds at densities ≳5 ρ0; and (5) the causality condition (sound speed ≤ c).

The sampling yields five EOS families: four hybrid sets (three NJL‑based, one MFTQCD‑based) and one purely hadronic RMF set. The NJL families (labeled NJL‑A, B, C) consistently place the deconfinement transition at 2–3 ρ0. Consequently, a 1.4 M⊙ star remains essentially nucleonic; only for masses ≳2 M⊙ does a quark core of ≈0.5–1 km radius appear. The posterior distributions for the NJL vector couplings are modest, leading to a relatively soft quark EOS.

In contrast, the MFTQCD family (MFTQCD‑D) prefers larger ξ (≈0.5) and smaller bag constants (≈30 MeV fm⁻³). This produces a transition just above saturation density (≈1.1 ρ0). The result is a high probability (≈70 %) that even a canonical 1.4 M⊙ star hosts a quark core occupying ~30 % of its radius. Moreover, the stiff vector term keeps the pressure high at large densities, yielding radii ≳12.5 km for 2 M⊙ stars—larger than typical purely nucleonic predictions.

A novel diagnostic introduced is the slope of the mass‑radius curve, dR/dM, evaluated at 1.8 M⊙. The authors find that a positive slope is a robust indicator of non‑nucleonic matter (quark or hyperonic) in the core. This provides a potentially observable signature: future high‑precision NICER or next‑generation X‑ray missions could infer the presence of a phase transition simply from the curvature of the M‑R relation, without requiring direct tidal‑deformability measurements.

The study concludes that Bayesian model selection, when fed with current multi‑messenger data, does not rule out hybrid stars with large quark cores. In particular, the MFTQCD framework, which allows an early transition, yields EOSs fully compatible with NICER radii, GW170817 tidal constraints, pQCD high‑density limits, and causality. The NJL models, while still viable, predict quark matter only in the most massive stars. The work highlights the importance of combining nuclear theory, astrophysical observations, and statistical inference to narrow down the dense‑matter EOS and to identify observable fingerprints of deconfinement in neutron stars.