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arxiv: 2606.21435 · v1 · pith:6G57JVVYnew · submitted 2026-06-19 · ⚛️ nucl-th · astro-ph.HE

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

Luca Passarella , Mirco Guerrini , Giuseppe Pagliara , Andrea Lavagno , Alessandro Drago This is my paper

Pith reviewed 2026-06-26 12:48 UTC · model grok-4.3

classification ⚛️ nucl-th astro-ph.HE
keywords two-families scenariostrange quark starshadronic starsBayesian analysisequation of statehyperonsdelta resonancescolor superconductivity
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The pith

Bayesian analysis favors the two-families scenario of hadronic and strange quark stars over the one-family scenario.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper performs the first detailed Bayesian analysis that combines astrophysical and laboratory data to constrain the equations of state in the two-families scenario, where hadronic stars and strange quark stars represent two distinct equilibrium phases. It employs non-linear relativistic mean field models for hadronic matter that include hyperons and delta resonances, together with bag-like models for quark matter that allow color superconductivity. While both the one-family and two-families pictures remain compatible with existing data, comparison of Bayes factors shows preference for the two-families case. This preference arises because the two-families framework removes the conflict between an intermediate-density soft equation of state required by small-radius objects and a high-density stiff equation of state needed to support massive pulsars.

Core claim

Hadronic stars and strange quark stars could coexist within the so-called two-families scenario, with hadronic matter and strange quark matter as two distinct equilibrium phases described by two different equations of state. The Bayesian analysis finds that while both the one-family and two-families scenarios are compatible with the data, comparison of the Bayes factors favors the two-families scenario. 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. Future detections of even more massive compact objects, very compact

What carries the argument

Bayesian comparison of Bayes factors between one-family and two-families equations of state constructed from non-linear relativistic mean field hadronic models (with hyperons and deltas) and bag-like quark models (with possible color superconductivity).

If this is right

  • Both one-family and two-families scenarios remain compatible with current astrophysical and laboratory data.
  • Bayes factors indicate that the two-families scenario is favored over the one-family scenario.
  • The two-families picture removes the requirement that one equation of state be simultaneously soft at intermediate densities and stiff at high densities.
  • Detections of even more massive compact objects or very compact ordinary-mass objects would strengthen the case for two families.
  • Precise measurements showing two distinct masses at the same radius would provide strong support for the two-families scenario.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the two-families picture holds, radius measurements of individual objects could belong to one of two separate branches rather than tracing a single equation of state.
  • Population studies of compact objects might reveal a bimodal distribution in radii at fixed mass once selection effects are accounted for.
  • Gravitational-wave signals from mergers could show distinct tidal deformability patterns depending on whether the objects belong to the hadronic or quark family.

Load-bearing premise

The specific non-linear relativistic mean field models for hadronic matter and bag-like models for quark matter are adequate representations of the physics across the relevant density range.

What would settle it

A future observation of a single continuous mass-radius relation for all compact objects that simultaneously accommodates both small radii at ordinary masses and very high masses without requiring two separate branches would falsify the preference for the two-families scenario.

Figures

Figures reproduced from arXiv: 2606.21435 by Alessandro Drago, Andrea Lavagno, Giuseppe Pagliara, Luca Passarella, Mirco Guerrini.

Figure 1
Figure 1. Figure 1: FIG. 1: Corner plot showing the posterior probability distributions and correlations for the one-family scenario. The [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Pressure [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Probability density distributions of the central baryon density of the maximum TOV mass configuration [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Particle fractions [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Mass–radius relations for the 1F and 2F scenarios obtained from the TOV equations. Shaded envelopes and [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Mass-tidal deformability Λ relations for the 1F [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Macroscopic observable posteriors for the HSs [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Macroscopic observable posteriors for the HSs (panel a) and QSs (panel b) in the 2F scenario. [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Compactness-tidal deformability Λ relations [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript performs the first detailed Bayesian analysis constraining equations of state in the two-families scenario, employing nonlinear relativistic mean-field models (with hyperons and deltas) for hadronic matter and bag-like models (with color superconductivity) for strange quark matter. It concludes that both the one-family and two-families scenarios remain compatible with current astrophysical and laboratory data, but that Bayes factors favor the two-families scenario because it naturally relieves the tension between intermediate-density softness (required by small-radius objects) and high-density stiffness (required by massive pulsars).

Significance. If the Bayes-factor comparison holds under the stated modeling assumptions, the work supplies a quantitative argument that the two-families picture is preferred by existing data and offers a resolution to an apparent EOS tension. The explicit use of both laboratory and astrophysical constraints within a Bayesian framework is a methodological strength.

major comments (2)
  1. [Abstract] Abstract: the central claim that 'by comparing the Bayes factors of both models, the two-families scenario is favored' is load-bearing for the paper's conclusion, yet the abstract (and the provided text) supplies no numerical Bayes-factor values, no list of the data sets entering the likelihood, and no description of prior ranges or convergence diagnostics; without these elements the quantitative preference cannot be verified.
  2. [Model description] Model construction (hadronic and quark sectors): the Bayes-factor ratio is obtained exclusively inside the restricted class of nonlinear RMF parametrizations (hyperons + deltas) and bag-like quark models (with color superconductivity). The claim that the two-families framework 'naturally relieves the tension' is therefore conditional on these parametrizations being representative across the relevant density range; no robustness test against alternative EOS families is reported, which directly affects the evidential force of the model comparison.
minor comments (2)
  1. The notation used for the meson-nucleon coupling constants and the bag constant should be collected in a single table for clarity.
  2. Figure captions should explicitly state which data sets are shown in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's insightful comments on our manuscript. Below we address each major comment in detail, indicating where revisions will be made to strengthen the paper.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'by comparing the Bayes factors of both models, the two-families scenario is favored' is load-bearing for the paper's conclusion, yet the abstract (and the provided text) supplies no numerical Bayes-factor values, no list of the data sets entering the likelihood, and no description of prior ranges or convergence diagnostics; without these elements the quantitative preference cannot be verified.

    Authors: We agree with the referee that the abstract and main text should provide these quantitative details to support the claim. In the revised manuscript, we will include the specific Bayes factor value, list the data sets used in the likelihood, describe the prior ranges, and report on convergence diagnostics. revision: yes

  2. Referee: [Model description] Model construction (hadronic and quark sectors): the Bayes-factor ratio is obtained exclusively inside the restricted class of nonlinear RMF parametrizations (hyperons + deltas) and bag-like quark models (with color superconductivity). The claim that the two-families framework 'naturally relieves the tension' is therefore conditional on these parametrizations being representative across the relevant density range; no robustness test against alternative EOS families is reported, which directly affects the evidential force of the model comparison.

    Authors: Our analysis is conducted within this established class of models, which are standard for incorporating hyperons, deltas, and color superconductivity in the two-families context. We will add text to the manuscript explaining the rationale for these choices and acknowledging that the preference is within this model class. A full exploration of alternative EOS families is left for future work as it would constitute a separate extensive study. revision: partial

Circularity Check

0 steps flagged

No circularity: standard Bayesian model comparison with independent evidence ratio

full rationale

The paper conducts a Bayesian parameter estimation and model comparison between one-family and two-families EOS scenarios using astrophysical and laboratory data. The Bayes factor is computed from the marginal likelihoods under each model's prior volume on parameters (nonlinear RMF with hyperons/deltas for hadronic matter; bag model with color superconductivity for quark matter). This is a standard, non-circular procedure: parameters are constrained by data, then evidence ratios quantify relative support. The statement that two-families 'naturally relieves the tension' follows directly from the two-EOS structure allowing separate softness/stiffness regimes, but the quantitative preference is data-driven rather than tautological. No self-citation load-bearing steps, no fitted inputs renamed as predictions, and no ansatz or uniqueness theorems imported from prior author work appear in the derivation. The result remains conditional on the chosen model class (a correctness issue, not circularity).

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

Abstract-only review; the models contain multiple unspecified parameters that are constrained by the Bayesian procedure, and the two-families framework rests on the domain assumption that hadronic and strange quark matter constitute distinct equilibrium phases.

free parameters (2)
  • coupling constants and meson masses in non-linear RMF models
    These parameters are adjusted within the Bayesian analysis to reproduce nuclear and astrophysical data.
  • bag constant and superconducting gap parameters in the quark-matter model
    These parameters are likewise varied to fit the same data sets.
axioms (1)
  • domain assumption Hadronic matter and strange quark matter correspond to two distinct equilibrium phases described by different equations of state.
    This premise is required for the two-families scenario to be defined and is stated in the abstract.
invented entities (1)
  • strange quark stars as a distinct family coexisting with hadronic stars no independent evidence
    purpose: To provide an additional degree of freedom that relieves the tension between soft and stiff equation-of-state requirements.
    The entity is postulated within the two-families framework; no independent falsifiable signature outside the present analysis is supplied in the abstract.

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discussion (0)

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