Hyperonic equation of state for neutron stars: A systematic Bayesian comparison of density-dependent and non-linear relativistic mean-field models

Abstract: A systematic Bayesian inference study of the equation of state (EOS) of dense matter with strangeness is presented, extending five relativistic mean-field (RMF) models with both constant and density-dependent couplings to include the full baryon octet. The hyperon-nucleon couplings in the scalar channel are varied within ranges informed by hypernuclear data, while vector isoscalar couplings are fixed by the SU(6) symmetry quark model. Observational constraints from NICER (PSR J0030, J0437, J0740) and GW170817, theoretical constraints from chiral effective field theory ($χ$EFT) and perturbative QCD (pQCD), and experimental constraints from nuclear saturation properties are imposed simultaneously. We find that the inclusion of hyperons systematically reduces the maximum neutron star mass by $0.05$-$0.10,M_\odot$ across all models while increasing the radius at $1.4,M_\odot$ by $0.5$-$0.8$ km. The speed of sound exhibits a characteristic softening at densities $2$-$3,ρ<em>{\rm sat}$ coinciding with hyperon onset. All hyperonic models remain consistent with the $2,M</em>\odot$ constraint. Models with a more flexible isovector channel span a larger proton fraction when only nucleons are included. However, the extra flexibility is visibly suppressed by hyperons, meaning that the average proton distribution is independent of model flexibility when hyperons are included. Less flexible models show comparable or slightly increased proton fractions due to EOS stiffening when hyperons are included. Only a residual number of hyperonic equations of state give rise to a mass-radius curve with a negative slope at low masses. A $1.8,M_\odot$ neutron star with a radius larger than or similar to the radius of a $1.2,M_\odot$ star would provide strong evidence that the star contains baryonic degrees of freedom beyond nucleons.

• The paper demonstrates that hyperon inclusion systematically softens the EOS, reducing neutron star maximum mass by 0.05–0.10 M☉.
• It employs Bayesian inference integrating multi-messenger astrophysical and laboratory data to calibrate density-dependent and non-linear RMF models.
• The study finds that repulsive hyperon-hyperon φ-meson interactions enable compliance with 2 M☉ constraints while increasing the 1.4 M☉ radius by 0.5–0.8 km.

Systematic Bayesian Analysis of Hyperonic Equations of State in Neutron Stars

Introduction

The study systematically investigates the impact of hyperonic degrees of freedom on the equation of state (EOS) of neutron star (NS) matter using five relativistic mean-field (RMF) models, spanning both density-dependent (DDH) and non-linear (NL) formulations. A core aim is to resolve how the inclusion of the full baryon octet—especially hyperons—affects neutron star maximum masses, radii, and underlying nuclear matter properties, under rigorous constraints from recent multi-messenger observations, theoretical low- and high-density nuclear physics, and hypernuclear data. The analysis applies a robust Bayesian inference framework, allowing both astrophysical and laboratory constraints to transparently shape model posteriors.

Relativistic Mean-Field Models and Hyperonic Couplings

Five RMF model families are explored: DDB, DDBy, DDME, DDMEy (all DDH), and RMFNL (NL). DDH models utilize meson-nucleon couplings that scale with density, while NL models implement constant meson couplings supplemented with non-linear self- and mixed-interaction terms. In each model, the nucleonic sector is extended to include the entire baryon octet (Λ, Σ, Ξ), with hyperon-meson couplings constrained by SU(6) quark model relations for vector couplings and empirical hypernuclear potentials for scalar couplings.

The Bayesian treatment samples over scalar coupling ratios ($x_s$, $x_{s\Sigma}$, $x_{s\Xi}$) with broad, physically motivated priors. The vector isoscalar couplings ($x_{\omega i}, x_{\phi i}$) are fixed by SU(6), and the isovector dependence is modulated in some models by the $y$ parameter, which controls density dependence in the $\varrho$-meson channel. Hyperon-hyperon repulsion, mediated by the $\phi$ meson, is crucial for stiffening the EOS at high density and enabling massive NSs even when hyperonic softening is present.

Bayesian Inference and Multi-Observable Constraints

The inference workflow integrates:

• Laboratory nuclear matter properties at saturation (binding energy, incompressibility $K_0$, symmetry energy $J_{\rm sym}$),
• Sub-saturation energy density predictions from chiral EFT (for neutron matter),
• NS mass-radius data from NICER (J0030+0451, J0437–4715, J0740+6620),
• Tidal deformability measurements from GW170817,
• High-density pQCD constraints,
• Hypernuclear data for scalar optical potentials.

This multimodal likelihood ensures consistency of the EOS parameters with both terrestrial experiments and neutron-star observations.

Equation of State Softening and Macroscopic Stellar Properties

The introduction of hyperons leads to a systematic softening of the EOS at super-saturation densities, visible in a sharp reduction of sound speed $c_s^2$ at $x_{s\Sigma}$0–$x_{s\Sigma}$1 Figure 1.

Figure 1: Posterior domains for EOS, mass-radius relation, and speed of sound for all model families, comparing nucleonic (gray) and hyperonic (colored) versions.

The hyperonic softening reduces the maximum NS mass by $x_{s\Sigma}$2–$x_{s\Sigma}$3 across all models (cf. Table 1 in the paper), but with the inclusion of repulsive hyperon-hyperon $x_{s\Sigma}$4-meson interactions, all parameterizations remain compatible with the $x_{s\Sigma}$5 observational constraint. A robust feature is an increase of the radius at $x_{s\Sigma}$6 by $x_{s\Sigma}$7–$x_{s\Sigma}$8 km in hyperonic stars.

Particle Composition and Hyperonization

Hyperonization initiates at $x_{s\Sigma}$9–$x_{s\Xi}$0, with the $x_{s\Xi}$1 hyperon always appearing first, followed closely by $x_{s\Xi}$2 and $x_{s\Xi}$3 (Figure 2, Figure 3).

Figure 2: Particle fractions as a function of baryon density; the onset and growth of $x_{s\Xi}$4, $x_{s\Xi}$5, and $x_{s\Xi}$6 are evident.

Hyperon onset densities are tightly clustered between $x_{s\Xi}$7–$x_{s\Xi}$8 fm$x_{s\Xi}$9 for $x_{\omega i}, x_{\phi i}$0, $x_{\omega i}, x_{\phi i}$1–$x_{\omega i}, x_{\phi i}$2 fm$x_{\omega i}, x_{\phi i}$3 for $x_{\omega i}, x_{\phi i}$4, and $x_{\omega i}, x_{\phi i}$5–$x_{\omega i}, x_{\phi i}$6 fm$x_{\omega i}, x_{\phi i}$7 for $x_{\omega i}, x_{\phi i}$8, with some model dependence Figure 3. Notably, in all hyperonic EOSs, the $x_{\omega i}, x_{\phi i}$9 fraction exceeds that of protons at $y$0 fm$y$1.

Figure 3: Posterior distributions of onset densities for $y$2, $y$3, and $y$4 hyperons in all models.

Impact on Proton Fraction and Isovector Physics

A central result is the sharp reduction and narrowing of the proton-fraction posterior in models with a flexible isovector channel (parameter $y$5) upon the inclusion of hyperons Figure 4.

Figure 4: The proton fraction as a function of baryon density; hyperonic posteriors (colored) are substantially constrained compared to nucleonic cases (gray), especially in $y$6-models.

For models without $y$7, responding to the $y$8 constraint requires an EOS stiffening that increases high-density symmetry energy, and thus the proton fraction, offsetting (in part) the charge-compensation effects from hyperons.

A key claim is that once hyperons are present, the average proton fraction becomes essentially independent of the isovector-channel flexibility, being determined dominantly by the hyperon content.

Model Calibration with Hypernuclear Optical Potentials

The posterior optical potentials for hyperons in symmetric nuclear matter at saturation are consistent with empirical values: $y$9 to $\varrho$0 MeV, $\varrho$1 to $\varrho$2 MeV, $\varrho$3 to $\varrho$4 MeV Figure 5. This alignment validates both the selection of coupling priors and the ability of the RMF models to interpolate faithfully between laboratory and astrophysical environments.

Figure 5: Posterior distributions for $\varrho$5, $\varrho$6, $\varrho$7 for all hyperonic RMF models.

Mass-Radius Topology and the Slope Diagnostic

The slope of the mass-radius curve ($\varrho$8) offers a diagnostic for the presence of non-nucleonic degrees of freedom Figure 6. In nucleonic EOSs, the radius typically decreases with increasing mass for $\varrho$9 (i.e., $\phi$0), whereas in hyperonic models, $\phi$1 even for low masses due to the requirement for a stiffer low-density EOS to counteract high-density softening.

Figure 6: Posterior distributions of $\phi$2 at fixed NS masses; the sign change to $\phi$3 at low mass can distinguish hyperonic from nucleonic EOS.

Observation of a massive NS with a radius equal to or larger than a lighter star would indicate significant non-nucleonic content, serving as a discriminant for hyperonic or hybrid interiors.

Trends in Nuclear Matter Parameters

The nuclear incompressibility $\phi$4 and symmetry energy curvature $\phi$5 in hyperonic EOS samples are systematically larger and less negative than in nucleonic cases, as a direct consequence of the need to counteract the EOS softening from hyperonization (Table in the paper). Most other saturation parameters are insensitive to the inclusion of hyperons.

Theoretical and Practical Implications

The findings demonstrate that, with appropriately calibrated RMF interactions—specifically, models incorporating repulsive hyperon-hyperon dynamics—all theoretical and observational constraints on NS structure can be satisfied in the presence of hyperons. The results emphasize that hyperonization is likely encoded in subtle structural features (mass-radius topology, proton fraction evolution, onset densities) and cannot be excluded by $\phi$6 or $\phi$7 alone.

These trends suggest that statistical inference using future, more precise NS radii and tidal deformability (especially for high-mass stars), possibly augmented by direct determination of the mass-radius slope for multiple NS masses, could enable decisive evidence for or against the presence of hyperons in the core.

Conclusion

This study provides a comprehensive, constraint-driven assessment of the role of hyperons in neutron-star interiors using RMF density functionals. The inclusion of hyperon degrees of freedom, with empirically calibrated couplings and repulsive $\phi$8-mediated interactions, naturally resolves the hyperon puzzle within current astrophysical constraints, while generating distinctive theoretical signatures—most notably in the behavior of the mass-radius relation and particle fractions. Future developments in observational accuracy and inference techniques are likely to transform the theoretical diagnosis of multi-baryonic composition in neutron stars.