Bayesian Constraints on the Neutron Star Equation of State with a Smooth Hadron-Quark Crossover

TL;DR

This study develops a Bayesian framework that unifies hadronic matter, a smooth hadron–quark crossover, and quark matter to constrain the neutron-star EOS from current data. The HM and QM sectors are connected by an infinitely differentiable crossover, with a trace-anomaly parameterization for QM and priors informed by nuclear theory, while observational likelihoods enforce causality, stability, a $M_{\rm TOV} \ge 1.97\,M_\odot$ limit, and NICER/GW170817 radii constraints. The results show strong constraints on low-to-intermediate density behavior, notably the symmetry-energy slope $L$ and curvature $K_{\rm sym}$, but only weak constraints on high-density and QM parameters; the crossover is favored at $\overline{\varepsilon} \approx (4$–$6)\varepsilon_0$ with $\Gamma \approx (0.5$–$1)\varepsilon_0$, while canonical radii sit around $R_{1.4} \approx 11.5$–$13.0$ km and $M_{\rm TOV} \approx 2.2\,M_\odot$. These findings imply that to probe quark matter, next-generation radius measurements with uncertainties near $\sim 0.2$ km or other inner-core observables are essential.

Abstract

We perform a Bayesian inference of the dense-matter equation of state (EOS) within a unified framework that incorporates hadronic matter, quark matter, and a smooth hadron--quark crossover. The EOS is constrained using physical consistency filters, gravitational-wave data from GW170817, NICER mass--radius measurements, and hypothetical future high-precision radius data. We find that current observations strongly constrain the density dependence of the nuclear symmetry energy, particularly its slope and curvature, while the highest-density hadronic parameters and quark-matter parameters remain only weakly constrained. The posterior distributions favor a crossover centered at an energy density $\overline{\varepsilon}\sim(4$--$6)\varepsilon_0$ with a width $Γ\sim(0.5$--$1.0)\varepsilon_0$, where $\varepsilon_0$ is the energy density of symmetric nuclear matter at saturation. The most probable radius of a canonical neutron star is $R_{1.4}\simeq 11.5-13.0$ km, and the maximum mass is $M_{\rm TOV}\simeq 2.2\,M_\odot$. Overall, present data primarily probe the low-to-intermediate density EOS and provide limited direct sensitivity to quark matter and genuinely high-density physics, highlighting the need for next-generation precision radius measurements.

Figures (8)

• Figure 1: PDFs of the HM parameters.
• Figure 2: PDFs of the QM parameters.
• Figure 3: Posterior PDFs of the hadron-quark crossover parameters.
• Figure 4: Posterior probability distributions for NS observables $R_{1.4}$ (upper), $R_{2.0}$ (middle), and $M_{\rm TOV}$ (bottom).
• Figure 5: The speed of sound squared and trace anomaly profiles with respect to energy density for accepted EOS using the most recent NICER data. The pQCD conformal limits are indicated by the dashed lines, while the GR limit for the trace anomaly is shown with the dotted line. Every bin was divided by the total number of accepted EOS. Note the logarithmic color scale.