Is the coexistence of strange quark stars and hadronic stars favored by astrophysical data? A Bayesian analysis

Abstract: Hadronic stars and strange quark stars could coexist within the so-called two-families scenario. In this respect, hadronic matter and strange quark matter correspond to two distinct equilibrium phases described by two different equations of state. We perform here the first detailed Bayesian analysis that makes use of astrophysical and laboratory data in order to constrain the equations of state adopted within the two-families scenario for hadronic and strange quark matter. In particular, in hadronic matter we consider the possible formation of hyperons and delta resonances (beside nucleons) within a class of non linear relativistic mean field models and in quark matter we consider the possible formation of a color-superconducting phase within a bag-like model. Results of the analysis indicate that while at the moment both the one-family and the two-families scenarios are compatible with the data, by comparing the Bayes factors of both models, the two-families scenario is favored with respect to the one-family scenario. Specifically, the two-families framework naturally relieves the tension between the intermediate-density softness of the equation of state required by small-radius objects and the high-density stiffness needed to support massive pulsars. Ultimately, future detections of even more massive compact objects, very compact ordinary-mass objects, or precise measurements of two distinct masses with the same radius, will provide strong indications in favor of the two-families scenario.

• The paper demonstrates that Bayesian analysis yields a decisive log Bayes factor (~12) favoring a two-families scenario over a single hadronic model.
• It employs a comprehensive model using NICER, GW, and heavy-ion collision data to parameterize both hadronic and quark matter equations of state.
• The study implies that the coexistence of compact star families naturally reconciles low-mass small-radius objects with the support for high-mass pulsars.

Bayesian Inference of Compact Star Structure: Testing the Two-Families Scenario

Introduction

The equation of state (EOS) for strongly interacting matter at supranuclear densities remains an unsettled question in nuclear physics and astrophysics. Compact objects such as neutron stars (NSs) offer unique empirical access to this regime. With inputs from gravitational wave (GW) events (notably GW170817), precision radio and X-ray mass–radius (M–R) measurements (e.g., via NICER), and mass determinations for high-mass pulsars (e.g., PSR J0952-0607), it has become feasible to statistically constrain the EOS utilizing Bayesian inference. However, the microphysical nature of the composition of these stars—whether they are purely hadronic, hybrid, or contain absolutely stable strange quark matter (SQM)—remains contested.

This work implements a comprehensive Bayesian analysis to directly compare the one-family (1F) scenario (purely hadronic stars, possibly with hyperons and delta resonances) and the two-families (2F) scenario, where hadronic stars (HSs) and self-bound strange quark stars (QSs) coexist as disconnected sequences, drawing on a broad set of astrophysical and laboratory data including multimessenger constraints and heavy-ion collision (HIC) experiments. The analysis quantifies the statistical preference for each scenario and elucidates the macroscopic and microphysical implications for compact star structure.

Theoretical Model Framework

The 1F scenario assumes all observed compact stars are HSs, modeled using a non-linear relativistic mean-field (RMF) Lagrangian calibrated by nuclear empirical parameters (nuclear saturation density, symmetry energy, effective Dirac mass, etc.), and incorporating both hyperons and delta isobars with parameterized meson couplings. The inclusion of hyperons and delta resonances is critical, as these softening agents are expected at higher densities and substantially impact both the maximum mass and radius predictions.

The 2F scenario considers, in addition to the above hadronic branch (which is only metastable in this picture), a second branch of absolutely stable self-bound QSs composed of color-flavor-locked (CFL) SQM. The quark-matter EOS is given by a three-flavor extended bag model with a leading-order perturbative QCD correction ($a_4$) and a variable superconducting pairing gap ($\Delta_\mathrm{CFL}$), implemented while enforcing the Bodmer-Witten hypothesis (SQM is more bound than nuclear matter at zero pressure).

Parameter estimation employs up to 14 EOS parameters (hadronic and quark branches together in 2F), with physically motivated broad uniform priors, and hyperon/delta sector informed by both empirical optical potential constraints and SU(6) symmetry.

Bayesian Methodology

The analysis leverages a fully parameterized likelihood incorporating the following data:

• Pulsar M–R posteriors from NICER (PSR J0030+0451, PSR J0740+6620, PSR J0614-3329), the HESS J1731-347 compact central object, and high-precision masses (e.g., PSR J0952-0607).
• Constraints from GW170817 in the form of a four-dimensional joint posterior in component masses and tidal deformabilities.
• Laboratory constraints: low-density chiral EFT bands for pure neutron matter (PNM) and symmetric nuclear matter (SNM), plus high-density HIC flow measurements.

Marginalization over branch population weights, normalization of the mass prior, and mixture modeling for unidentified family assignment are performed rigorously. In the 2F scenario, the family label is a latent discrete variable, handled via summation and marginalization in the likelihood. Bayes factors (model evidences) are computed using dynamic nested sampling and allow for robust model selection that penalizes increased parameter volume.

Strong Claims and EOS Constraints

The key numerical result is a decisive preference for the 2F scenario over purely hadronic models when the full dataset is employed, with log Bayes factors $\ln B_\text{2F,1F} \sim 12$ (see Table III in the original manuscript)—even accounting for the larger prior volume of 2F. This preference holds robustly across variations in hyperon sector priors, quark model parameter ranges, and plausible alternate interpretations of GW170817 family assignments.

Notably, the 2F scenario resolves the tension between the need for an intermediate-density soft EOS, as implied by small-radius objects (e.g., PSR J0614-3329 and HESS J1731-347) and the requirement for high-density stiffness to support $\gtrsim2\,M_{\odot}$ pulsars (e.g., PSR J0740+6620, PSR J0952-0607). In the 1F scenario, this tension is only reconciled with fine-tuned parameterizations yielding an early onset of delta resonances (at $n_B \simeq 1.8\,n_\mathrm{sat}$), which soften the EOS at intermediate densities but necessitate subsequent extreme stiffening at larger densities; such behavior is imprinted as a quasi-inflection in the corresponding M-R curves. However, this single-sequence hadronic solution is increasingly incompatible as more precise or extreme constraints are added (see cumulative Bayes factor increases with added data).

By contrast, the 2F scenario allows for a compact hadronic branch (with lower maximum mass), and a QS branch that naturally accommodates the observed high-mass objects due to the intrinsic stiffness of color-superconducting SQM at high densities. Marginalized posteriors indicate the hadronic branch has $M_\text{max,HS} \sim 1.54\,M_{\odot}$, $R_{1.4,HS} \sim 11.1$ km, and $\Lambda_{1.4,HS} \sim 186$; the QS branch supports up to $M_\text{max,QS} \sim 2.33\,M_{\odot}$ and matches the observed radii and deformabilities of heavy pulsars. The separation is also evident in the composition: the hadronic branch's threshold for hyperon appearance is delayed due to the early onset of deltas, and only the QS branch is compatible with particularly compact, low-mass objects such as HESS J1731-347.

Implications and Theoretical Significance

The outcomes substantiate several important theoretical and phenomenological points:

• EOS flexibility: The two-families (2F) paradigm provides a natural, less fine-tuned means to simultaneously account for the existence of very massive pulsars and radius/tidal deformability constraints indicating a soft EOS at lower/intermediate densities, as highlighted by the superior Bayesian evidence.
• Resolution of the hyperon puzzle: The meta-stability of the hadronic branch in 2F enables the appearance of hyperons and deltas without violating the observed maximum mass constraint, thus providing a compelling solution to the hyperon puzzle without recourse to unrealistically strong repulsive interactions.
• Population-level bimodality: While not modeled explicitly, the emergent disconnected mass ranges for HSs and QSs suggest that observed mass bimodality in recycled pulsars could be at least partially explained by two coexisting families, a hypothesis testable by future high-precision M–R catalogs.
• Observational discriminants: The Bayesian evidence for 2F is strengthened systematically by the addition of small-radius/low-mass objects and HIC constraints. Future decisive support for 2F would arise from observations of compact objects with extremely small radii ($<11$ km at canonical masses), ever more massive ($\Delta_\mathrm{CFL}$0) stars, or the direct observation of two stars with identical radii but widely different masses.

Limitations and Future Directions

Current results are subject to the assumptions of the adopted EOS parameterization, particularly the fixed form of the hadronic RMF and the bag-model representation of CFL quark matter. The analysis does not include potential hybrid stars (HSs with deconfined quark cores via first-order transitions) as a competing third hypothesis; such an analysis is a logical extension for future work, especially as Bayesian hybrid-star studies have found similar patterns of tension and relief.

Additionally, improved multimessenger data—e.g., more precise GW event constraints, further NICER and X-ray burst sources, as well as enhanced low- and high-density laboratory and theoretical constraints—will be critical in further disfavouring or supporting the existence of multiple compact star families.

Conclusion

The first detailed Bayesian model selection analysis incorporating the two-families scenario shows that existing astrophysical and laboratory data statistically favour the coexistence of hadronic stars and self-bound strange quark stars over a single hadronic sequence, as the tension between the central compact object radii, observed tidal deformabilities, and high-mass pulsar support is naturally relieved in the 2F framework. The imprints on the EOS (early delta resonance softening, smaller radii at intermediate masses, disconnected maximum mass scales) and the clear separation in population properties are robust features. Future work incorporating hybrid stars, improved EOS flexibility, and new observations will further clarify the phase structure of dense QCD matter realized in nature.

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Reference: "Is the coexistence of strange quark stars and hadronic stars favored by astrophysical data? A Bayesian analysis" (2606.21435)