A Bayesian Inference of Hybrid Stars with Large Quark Cores

This study employs Bayesian inference with hadronic and quark matter models to demonstrate that while the presence of large quark cores in 1.4 solar mass neutron stars depends on the specific quark model used, both frameworks predict quark matter in 2 solar mass stars, with the mass-radius curve slope serving as a key indicator of non-nucleonic matter.

The Gist

The Big Picture: The Ultimate Cosmic Pressure Cooker

Imagine a neutron star. It's a dead star that has collapsed under its own weight, packing the mass of our entire Sun into a city-sized sphere (about 12–14 kilometers wide). It is the densest thing in the universe.

Inside these stars, the pressure is so intense that it breaks the rules of normal physics. On Earth, matter is made of atoms, which are made of protons and neutrons. But inside a neutron star, the pressure is like a giant hydraulic press crushing a soda can until it's flat. The question scientists have been asking for decades is: What happens when you crush protons and neutrons hard enough?

Do they stay as squished balls? Or do they melt down into their fundamental ingredients: quarks?

This paper is a detective story. The authors are trying to figure out if neutron stars have a "core" made of pure quark soup (a Hybrid Star) or if they are just made of squished neutrons all the way through.

The Tools: Two Different Recipes

To solve this mystery, the scientists used a computer simulation method called Bayesian Inference. Think of this as a super-smart chef trying to find the perfect recipe.

1. The Chef's Constraints: The chef knows the final dish must taste a certain way (it must match real-world data from telescopes and gravitational wave detectors).
2. The Ingredients: The chef has two different "cookbooks" (theories) for how quarks behave:
   • Recipe A (NJL Model): This is like a strict recipe. It says quarks only start melting out of neutrons when the pressure is extremely high (about twice the density of the star's surface).
   • Recipe B (MFTQCD Model): This is a more flexible recipe. It suggests quarks might start melting out much earlier, even when the pressure is just slightly higher than the star's surface.

The scientists ran thousands of simulations, mixing these recipes with real data to see which "dish" (model) actually matches the universe we observe.

The Detective Work: Checking the Clues

The team didn't just guess; they checked their recipes against four major clues:

1. The "NICER" X-ray Photos: NASA's NICER telescope has taken pictures of neutron stars, measuring their size and weight. It's like weighing a watermelon and measuring its circumference to guess if it's full of water or sand.
2. The "GW170817" Crash: We once heard two neutron stars collide (a gravitational wave event). The way they smashed together tells us how "squishy" they are. If they were too stiff, they would bounce differently than if they were soft.
3. The "Two-Solar-Mass" Rule: We know some neutron stars are incredibly heavy (twice the mass of our Sun). Any recipe that can't support that much weight without collapsing into a black hole is thrown out.
4. The "Speed Limit" (Causality): Nothing can travel faster than light. The math must respect this rule.

The Results: What Did They Find?

After running their simulations, the scientists found some fascinating differences between the two recipes:

1. The "Early Melter" vs. The "Late Melter"

• The MFTQCD Model (The Early Melter): This model suggests that even in a "normal" sized neutron star (1.4 times the mass of the Sun), the core is already a soup of quarks. It's like a chocolate bar that starts melting the moment you hold it in your hand.
• The NJL Model (The Late Melter): This model says the core stays solid (made of neutrons) until the star gets very heavy (over 1.7 times the Sun's mass). The quark soup only appears in the heaviest stars.

2. The "Stiffness" of the Star

The paper found that to make a star heavy enough to be 2 solar masses, the "quark soup" inside needs to be stiff.

• Analogy: Imagine a mattress. If it's too soft, you sink right through. If it's very stiff (like a firm memory foam), it can support a heavy person.
• The scientists found that the quark matter acts like a super-firm mattress. This stiffness allows the star to be massive and have a larger radius (size) than expected. This might explain why some heavy stars look bigger than lighter ones.

3. The "Slope" Clue

The paper discovered a cool trick: If you look at a graph of Mass vs. Radius, the slope (the angle of the line) tells you what's inside.

• If the line goes down (heavier stars are smaller), it's likely just normal matter.
• If the line goes up (heavier stars are actually bigger), it's a strong hint that exotic quark matter is inside, acting like that stiff mattress.

The Conclusion: A Cosmic Compromise

The paper concludes that Hybrid Stars are very likely real.

• For heavy stars (2 solar masses): Both recipes agree that they almost certainly have a giant core of quark matter.
• For lighter stars (1.4 solar masses): It depends on which "recipe" is right. If the MFTQCD model is correct, even these smaller stars have quark cores. If the NJL model is correct, they are still just squished neutrons.

The Takeaway:
The universe is playing a game of cosmic Jenga. The scientists are trying to figure out how many blocks (neutrons) can be stacked before the tower melts into a different substance (quarks). Their work suggests that the "melting" happens, and it changes the shape and size of the star in ways we can now detect.

In short: Neutron stars are likely not just giant atomic nuclei; they are cosmic laboratories where matter transforms into a strange, exotic fluid, and we are finally learning how to read the menu.

Technical Summary

1. Problem Statement

The internal composition of neutron stars (NSs), particularly the existence of a phase transition from hadronic matter to deconfined quark matter in their cores, remains a fundamental open question in nuclear astrophysics. While Quantum Chromodynamics (QCD) predicts such a transition at high densities, the nature of this transition (crossover vs. first-order) and the resulting Equation of State (EOS) are uncertain due to the limitations of terrestrial experiments and lattice QCD calculations at finite baryon density.

The specific challenges addressed in this work include:

• Determining if hybrid EOSs can satisfy current multi-messenger observational constraints (NICER X-ray data, GW170817 gravitational waves) while containing large quark cores.
• Understanding how different quark matter models (NJL vs. MFTQCD) affect the onset density of deconfinement and the resulting stellar properties (mass, radius, tidal deformability).
• Identifying observable signatures (e.g., mass-radius slope, compactness) that distinguish purely hadronic stars from hybrid stars.

2. Methodology

The authors employ a Bayesian inference framework to sample the parameter space of hybrid EOSs, combining theoretical models with observational and theoretical constraints.

Theoretical Models

• Hadronic Phase: Described using the Relativistic Mean Field (RMF) model with non-linear terms. The Lagrangian includes interactions mediated by $\sigma$, $\omega$, and $\rho$ mesons. Parameters ($g_\sigma, g_\omega, g_\rho$, and non-linear couplings) are sampled via Bayesian analysis.
• Quark Phase: Two distinct models are utilized to describe the quark core:
  1. Nambu–Jona-Lasinio (NJL) Model: A three-flavor effective model incorporating chiral symmetry breaking, dynamical quark masses, and vector interactions (including 8-quark vertices) to stiffen the EOS.
  2. Mean Field Theory of QCD (MFTQCD): A model derived by decomposing gluon fields into soft and hard components. It behaves similarly to a vectorial MIT bag model but allows for a phase transition at lower densities.
• Phase Transition: The Maxwell construction is used to connect the hadronic and quark phases, assuming a sharp interface with equal pressures but a discontinuity in energy density.

Bayesian Setup

• Priors: Uniform distributions are defined for model parameters (coupling constants, bag constants, etc.). Five distinct datasets (EOS sets) are generated:
  1. RMF: Purely hadronic.
  2. NJL: Hybrid with NJL quark phase.
  3. NJL-GW: Hybrid with NJL, constrained by GW170817 tidal deformability.
  4. r-NJL: Hybrid with NJL, but with hadronic priors restricted to match the RMF set.
  5. MFTQCD: Hybrid with MFTQCD quark phase.
• Constraints (Likelihoods):
  • Nuclear Matter Properties (NMP): Saturation density, binding energy, incompressibility ($K_0$), and symmetry energy ($J_{sym}$).
  • Pure Neutron Matter (PNM): Constraints from Chiral Effective Field Theory ($\chi$EFT).
  • Observational Data: NICER mass-radius measurements for PSR J0030+0451, J0740+6620, and J0437+4715; HESS J1731-347 (low-mass compact object).
  • Gravitational Waves: Tidal deformability constraints from GW170817 (applied to specific sets).
  • Theoretical Limits: Causality ($c_s < c$) and perturbative QCD (pQCD) constraints at high densities ($7\rho_0$).
• Sampler: The PyMultiNest nested sampling algorithm (via BILBY) is used to explore the posterior distributions.

3. Key Contributions

• Comparative Model Analysis: A rigorous comparison of two quark models (NJL and MFTQCD) within the same Bayesian framework, revealing how the choice of quark model dictates the onset density of quark matter.
• Large Quark Cores: Demonstration that large quark cores are possible in NSs, but their existence depends heavily on the stiffness of the hadronic phase and the specific quark model used.
• Mass-Radius Slope as a Diagnostic: Identification of the slope of the mass-radius curve ($dM/dR$) as a potential indicator of exotic matter. The authors show that a positive slope at high masses ($\sim 1.8 M_\odot$) may signal the presence of quark matter.
• Impact of GW170817: Quantification of how gravitational wave constraints restrict the parameter space, specifically reducing the maximum radius of $1.4 M_\odot$ stars in hybrid models.

4. Key Results

Phase Transition Density

• MFTQCD: Predicts a phase transition at low densities ($\rho_{trans} \approx 0.17 - 0.30 \, \text{fm}^{-3}$), just above nuclear saturation density. This allows quark matter to exist even in $1.4 M_\odot$ neutron stars.
• NJL: Predicts a phase transition at higher densities ($\rho_{trans} \approx 0.30 - 0.40 \, \text{fm}^{-3}$, roughly $>2\rho_0$). Consequently, quark matter is unlikely to appear in $1.4 M_\odot$ stars but is present in stars $>1.7 M_\odot$.

Mass and Radius

• Maximum Mass: All hybrid models support stars up to $\sim 2.1 - 2.4 M_\odot$. The MFTQCD model yields the highest maximum masses ($\sim 2.44 M_\odot$).
• Radii:
  • NJL sets: Predict larger radii for $1.4 M_\odot$ stars ($R_{1.4} \approx 13.4 - 13.7$ km) due to a stiffer hadronic phase required to support the transition.
  • MFTQCD set: Predicts smaller radii ($R_{1.4} \approx 11.6 - 12.6$ km) and is the only set compatible with the HESS J1731-347 data (a low-mass compact object), suggesting a low-density transition.
• GW170817 Constraint: The standard NJL set (without GW constraints) predicts radii too large to satisfy GW170817 tidal deformability limits. The NJL-GW and r-NJL sets successfully satisfy these constraints by softening the hadronic phase, though this reduces the maximum radius of $1.4 M_\odot$ stars.

Mass-Radius Slope

• Purely hadronic (RMF) models predominantly show a negative slope ($dM/dR < 0$) across all masses.
• Hybrid models (NJL and MFTQCD) frequently exhibit a positive slope ($dM/dR > 0$) at lower masses ($1.2 - 1.6 M_\odot$).
• Crucially, a significant fraction of hybrid EOSs maintain a positive slope even at $1.8 M_\odot$, which the authors propose as a signature of non-nucleonic (quark) matter.

Compactness and Trace Anomaly

• All models satisfy the maximum compactness limit $C = M/R \leq 1/3$.
• The study challenges previous assumptions that a polytropic index $\gamma < 1.75$ or trace anomaly measure $d_c < 0.2$ are definitive signatures of quark matter, as some hadronic EOSs also satisfy these conditions at high densities.

5. Significance

This work provides a comprehensive statistical assessment of hybrid star scenarios, moving beyond single-EOS comparisons to a probabilistic landscape.

• Model Discrimination: It highlights that the MFTQCD model is uniquely capable of explaining low-mass compact objects (like HESS J1731-347) as hybrid stars, whereas NJL models generally require higher masses for quark cores.
• Observational Guidance: The finding that a positive slope in the mass-radius curve at $1.8 M_\odot$ could indicate a quark core offers a new, testable prediction for future NICER observations and gravitational wave detections.
• Future Prospects: The authors emphasize that while current data allows for multiple scenarios, next-generation telescopes (measuring radii with $<100$ m precision) will be essential to distinguish between these models and confirm the existence of large quark cores.

In conclusion, the paper demonstrates that hybrid stars with large quark cores are a viable solution consistent with current data, but the specific properties of the core (size, onset density) are highly sensitive to the underlying quark matter model and the stiffness of the hadronic phase.