⚛️ nuclear theory

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

This paper presents a Bayesian analysis using astrophysical and laboratory data which finds that the two-families scenario, proposing the coexistence of hadronic and strange quark stars, is favored over the standard one-family model because it naturally resolves the tension between the soft equation of state required for small-radius objects and the stiff equation of state needed to support massive pulsars.

Original authors: Passarella Luca, Guerrini Mirco, Pagliara Giuseppe, Lavagno Andrea, Drago Alessandro

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Passarella Luca, Guerrini Mirco, Pagliara Giuseppe, Lavagno Andrea, Drago Alessandro

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is filled with cosmic "weights" called neutron stars. These are the crushed remnants of dead stars, so dense that a teaspoon of their material would weigh a billion tons on Earth. For decades, physicists have been trying to figure out exactly what these stars are made of and how they hold together under such extreme pressure.

This paper is like a massive detective story where the authors use a sophisticated statistical tool (called Bayesian analysis) to solve a mystery: Are all these heavy stars made of the same stuff, or are there two completely different types of stars living in the same neighborhood?

Here is the breakdown of their investigation in simple terms:

The Two Suspects: One Family vs. Two Families

The researchers tested two main theories about the "recipe" for these stars:

The "One-Family" Theory (1F):
Imagine a single, giant family where every star is made of the same ingredients: protons and neutrons (nuclear matter), perhaps with some extra "spicy" particles called hyperons or delta resonances mixed in. In this scenario, there is only one type of star. If you squeeze this material too hard, it might turn into something else, but it's all part of one continuous line.
The "Two-Families" Theory (2F):
This theory suggests there are two distinct species of stars that don't mix.
- Family A (Hadronic Stars): These are the "normal" heavy stars made of protons and neutrons.
- Family B (Strange Quark Stars): These are made of "strange quark matter," a soup of fundamental particles that is so stable it could exist on its own.
- The Twist: In this scenario, the "normal" stars are actually unstable. If they get too heavy or dense, they might suddenly transform into the "strange" stars. It's like having two different species of animals living in the same forest, but one species is only stable if it stays small, while the other can grow huge.

The Puzzle: The "Goldilocks" Problem

The scientists had a problem. The data they collected from telescopes and gravitational wave detectors (like LIGO) sent them two conflicting messages:

Message 1: Some stars are very small and compact (like a heavy object squeezed into a tiny box). This suggests the material inside is "soft" and squishy.
Message 2: Other stars are extremely massive (twice the mass of our Sun). This suggests the material inside must be "stiff" and strong, like a steel beam, to hold up that much weight without collapsing.

The "One-Family" Theory struggled here. It was like trying to build a single bridge that is soft enough to bounce like a trampoline for small cars, but stiff enough to hold a tank. It's very hard to make one material do both things perfectly. The "One-Family" models had to twist and turn their physics to try to fit both the small, squishy stars and the massive, heavy ones into a single recipe.

The "Two-Families" Theory solved the puzzle naturally.

The small, squishy stars belong to the "normal" family. Their recipe is soft, which explains why they are small.
The massive, heavy stars belong to the "strange quark" family. Their recipe is incredibly stiff, allowing them to hold up huge weights.
Because they are two different families, they don't have to compromise. Each can be exactly what it needs to be.

The Investigation: How They Decided

The authors didn't just guess; they ran a massive computer simulation using real data from:

Telescopes (NICER): Measuring the size and weight of specific pulsars.
Gravitational Waves (GW170817): Listening to the "chirp" of two stars crashing together to see how squishy they were.
Particle Accelerators (Heavy Ion Collisions): Smashing atoms together on Earth to see how matter behaves under pressure.

They used a statistical method called Bayesian analysis. Think of this as a judge weighing the evidence. They asked: "If the universe works like the 'One-Family' theory, how likely is it that we would see this specific data? And if it works like the 'Two-Families' theory, how likely is it?"

The Verdict

The results were clear: The "Two-Families" theory is the winner.

The data fits the "Two-Families" scenario much better.
The "One-Family" theory was forced into a difficult compromise, trying to be both soft and stiff at the same time, which made it less likely to be true.
The "Two-Families" theory naturally explained why we see tiny, dense stars and giant, heavy stars without needing to stretch the rules of physics.

The Bottom Line

The paper concludes that the universe likely hosts two distinct types of dense stars coexisting side-by-side. One type is made of standard nuclear matter (which can be small and compact), and the other is made of strange quark matter (which can grow to be incredibly massive).

The authors note that while the current evidence strongly favors this "Two-Families" idea, they will need even more precise measurements in the future—like finding a star that is very heavy but has the exact same size as a light one—to make this conclusion absolutely undeniable. For now, however, the evidence points to a cosmic neighborhood with two very different kinds of residents.

Technical Summary: Coexistence of Strange Quark Stars and Hadronic Stars

Problem Statement
The microscopic composition of compact stars and the equation of state (EOS) of strongly interacting dense matter remain central open problems in nuclear physics and astrophysics. A specific tension exists between observational constraints requiring a "soft" EOS at intermediate densities (to explain small radii of ordinary-mass objects and tidal deformability from GW170817) and the need for a "stiff" EOS at high densities (to support pulsars with masses exceeding $2 M_\odot$ ). Furthermore, the inclusion of hyperons and $\Delta$ resonances in hadronic matter typically softens the EOS, leading to the "hyperon puzzle" where maximum masses fall below observed values.

The "two-families" (2F) scenario proposes a resolution: the coexistence of two disconnected branches of compact stars. In this framework, hadronic stars (HSs) are metastable and populate a branch with smaller radii, while absolutely stable strange quark matter (SQM) forms a self-bound branch of strange quark stars (QSs) capable of supporting higher masses. While this scenario has been explored theoretically, it had not previously been subjected to a rigorous Bayesian analysis comparing it directly against the standard "one-family" (1F) hypothesis (where all stars are hadronic) using a unified set of astrophysical and laboratory data.

Methodology
The authors perform the first detailed Bayesian analysis to constrain the EOS parameters for both the 1F and 2F scenarios simultaneously.

Models:
- Hadronic Sector: A non-linear relativistic mean-field (RMF) model including nucleons, hyperons ( $\Lambda, \Sigma, \Xi$ ), and $\Delta$ resonances. The model utilizes nuclear empirical parameters at saturation and density-dependent couplings for the $\rho$ meson to control the symmetry energy slope ( $L_{sym}$ ).
- Quark Sector (2F only): A color-flavor-locked (CFL) bag-model EOS describing absolutely stable SQM. This sector includes parameters for the bag constant ( $B$ ), the CFL pairing gap ( $\Delta_{CFL}$ ), and perturbative QCD corrections ( $a_4$ ).
Data and Likelihoods: The analysis incorporates:
- Mass-Radius (M-R) Constraints: Joint posteriors from NICER (PSR J0030+0451, PSR J0740+6620, PSR J0614–3329) and X-ray observations of HESS J1731–347.
- Mass-Only Constraints: PSR J0952–0607 and the secondary component of GW190814 (treated as a potential compact star).
- Gravitational Waves: The full joint posterior for GW170817 (masses and tidal deformabilities), treated with marginalization over family labels (HS-HS, HS-QS, QS-HS, QS-QS) in the 2F scenario.
- Laboratory/Theoretical Constraints: Chiral effective field theory ( $\chi$ EFT) bands for pure neutron matter and heavy-ion collision (HIC) flow data for symmetric nuclear matter.
Statistical Framework:
- The study uses dynamic nested sampling (via the dynesty package) to compute posterior distributions and Bayesian evidences ( $Z$ ).
- The Bayes factor ( $B_{2F,1F} = Z_{2F}/Z_{1F}$ ) is used to compare the models, accounting for the different parameter space dimensionalities (11 parameters for 1F vs. 14 for 2F).
- In the 2F scenario, the family assignment of each observed source is treated as a latent variable and marginalized over, with priors defined based on structural stability and astrophysical formation channels.

Key Contributions

First Bayesian Comparison: This work provides the first rigorous Bayesian comparison between the 1F and 2F scenarios using a comprehensive dataset that includes both low-density nuclear constraints and high-mass astrophysical observations.
Unified Framework: It constrains the metastable hadronic EOS and the absolutely stable SQM EOS simultaneously within the 2F hypothesis, rather than assuming the nature of the observed sources a priori.
Resolution of Tension: The analysis demonstrates how the 2F scenario naturally resolves the tension between intermediate-density softness and high-density stiffness by separating these requirements into two distinct stellar branches.

Results

Model Preference: The 2F scenario is favored over the 1F scenario across all dataset combinations. For the full dataset (including HIC and all astrophysical constraints), the log-Bayes factor is $\ln B_{2F,1F} = 11.9$ , indicating "decisive" support for the two-families scenario.
EOS Characteristics:
- 1F Scenario: To satisfy both small-radius constraints and the $2 M_\odot$ limit, the 1F EOS requires a finely balanced compromise. This is achieved through an early onset of $\Delta$ resonances ( $\sim 1.8 n_{sat}$ ) which softens the EOS at intermediate densities, followed by strong stiffening at high densities. The maximum mass is constrained to $M_{max} \approx 2.05 M_\odot$ .
- 2F Scenario: The tension is relieved. The hadronic branch remains soft (supported by $\Delta$ resonances and hyperons) with a maximum mass of $M_{HS,max} \approx 1.54 M_\odot$ and a radius of $R_{1.4} \approx 11.1$ km. The heavy pulsars are supported by the stiffer QS branch, reaching $M_{QS,max} \approx 2.33 M_\odot$ .
Parameter Constraints: The 2F posterior prefers a larger Dirac effective mass ( $m^*/m$ ) and less extreme tuning of $\Delta$ couplings compared to the 1F case. The quark sector parameters ( $B, \Delta_{CFL}, a_4$ ) are moderately constrained, with a positive correlation between the bag constant and the pairing gap.
GW170817 Interpretation: In the 2F scenario, the GW170817 event is most likely a HS-HS merger, but mixed channels (HS-QS) contribute significantly (~35% of the likelihood), whereas the QS-QS channel is disfavored for the best-fit EOS.

Significance and Claims
The paper claims that current astrophysical and laboratory data are more naturally organized by two disconnected compact-star families than by a single purely hadronic sequence. The 2F scenario offers a natural explanation for the coexistence of very compact ordinary-mass objects (like HESS J1731–347) and very massive pulsars, while effectively removing the hyperon puzzle.

The authors emphasize that their conclusion is robust against variations in priors (hyperon potentials, SQM parameter ranges) and source-mass assumptions, provided the full dataset is used. However, they note that the statistical preference is less definitive if restricted datasets (excluding HESS J1731–347, PSR J0614–3329, and HIC data) are considered.

The study explicitly excludes hybrid stars (stars with quark cores) from the current comparison, noting that a complete model selection should include them as a third hypothesis in future work. The authors conclude that future detections of more massive compact objects, very compact ordinary-mass objects, or precise measurements of two distinct masses with the same radius would provide even stronger indications in favor of the two-families scenario.

The Two Suspects: One Family vs. Two Families

The Puzzle: The "Goldilocks" Problem

The Investigation: How They Decided

The Verdict

The Bottom Line

More like this