Maximal mass of neutron stars constrained by neutron star observations

By employing a Bayesian weighting framework that integrates multimessenger observations (including GW170817, NICER, and candidate compact objects) with hybrid equations of state, this study determines that the maximum neutron star mass is robustly constrained to approximately 2.2–2.3 solar masses while the corresponding radius depends more strongly on the underlying hadronic model, typically falling near 12 km.

The Gist

Imagine the universe is filled with a mysterious, super-dense material called "neutron star matter." It's so heavy that a single teaspoon would weigh as much as a mountain. For a long time, physicists have been trying to figure out the "rules of the game" for this material—specifically, how stiff or squishy it is. This set of rules is called the Equation of State (EOS).

The big question this paper tries to answer is: What is the absolute heaviest a neutron star can get before it collapses into a black hole?

Here is the story of how the authors solved this puzzle, explained in simple terms:

1. The Two Starting Points (The Recipes)

To figure out the rules, the scientists started with two different "recipes" for how this dense matter behaves at lower densities. Think of these as two different theories about how the ingredients mix:

• Recipe A (SFHo): A "softer" recipe, meaning the matter is a bit easier to squeeze.
• Recipe B (DD2): A "stiffer" recipe, meaning the matter resists being squeezed more.

They knew these recipes worked well at the "beginning" of the density scale, but they didn't know what happened at the extreme, super-high densities found in the center of a neutron star. To fill in the gap, they used a mathematical "bridge" to connect their recipes to what we know about particle physics at the highest possible energies.

2. The Detective Work (Using Real Clues)

Instead of just guessing, the authors acted like detectives. They took their two recipes and tested them against real-world clues gathered by telescopes and gravitational wave detectors. They used a special statistical method (called Bayesian weighting) to see which versions of their recipes survived the test.

Here are the clues they used:

• The "Big Crash" (GW170817): When two neutron stars crashed into each other, they sent ripples through space. The way these ripples behaved told the scientists how "squishy" the stars were.
• The "Flashlight" (NICER): A space telescope took pictures of hot spots on spinning neutron stars. By measuring how big the stars looked and how heavy they were, they got a direct size-to-weight ratio.
• The "Lightweight" Candidate (HESS J1731–347): A very small, light object that might be a neutron star.
• The "Heavyweight" Candidate (GW190814): A mysterious object that is heavier than most neutron stars but lighter than most black holes. The scientists asked: Could this actually be a super-heavy neutron star?

3. The Results: What the Clues Told Them

The scientists ran their two recipes through these clues and looked at the results.

The Weight Limit (Maximum Mass):

• The Surprise: It didn't matter much which starting recipe (Soft or Stiff) they used. The real-world clues were so strong that they forced both recipes to agree on the same answer.
• The Verdict: When they used the most reliable clues (the "Big Crash" and the "Flashlight"), the maximum weight a neutron star can hold is about 2.2 to 2.3 times the mass of our Sun.
• The "Heavyweight" Twist: If they assume that mysterious heavy object (GW190814) is a neutron star, the limit jumps up to about 2.6 to 2.7 times the Sun's mass. However, this creates a conflict with the "squishiness" clues from the Big Crash, making it a tricky situation.

The Size Limit (Radius):

• The Difference: Unlike the weight, the size of the star did depend on which starting recipe they used.
• The Verdict: The "Soft" recipe predicted a radius of about 11.8 km, while the "Stiff" recipe predicted about 12.4 km.
• The Sweet Spot: When all the best clues are combined, the most likely size for these stars is around 12 kilometers (give or take 1 km).

4. The Big Picture

The paper concludes that by looking at the "endpoints" (the heaviest and largest possible stars) and using a mix of real astronomical data, we can narrow down the rules of the universe's densest matter.

• The Weight: The universe seems to have a "speed limit" for how heavy a neutron star can be, sitting comfortably around 2.2 to 2.3 solar masses. This matches the heaviest neutron star we have actually seen so far.
• The Size: They are roughly the size of a small city, about 12 km across.
• The Takeaway: The real-world observations (the clues) are much more powerful than the theoretical starting guesses. No matter which theory you start with, the data from the stars themselves forces the answer to converge on the same numbers.

In short, the universe has given us a very clear answer: Neutron stars can get incredibly heavy, but there is a hard ceiling, and they are surprisingly small for how much they weigh.

Technical Summary

1. Problem Statement

The central challenge in nuclear and heavy-ion physics is determining the Equation of State (EOS) of strongly interacting matter at intermediate densities and chemical potentials. While low-density matter is constrained by nuclear experiments and lattice QCD, and high-density matter by perturbative QCD (pQCD), the region relevant to neutron star cores (several times nuclear saturation density, $\rho_0$) remains poorly understood.

The specific problem addressed is determining the maximum mass ($M_{TOV}$) and the corresponding radius ($R_{TOV}$) of neutron stars. These endpoints of the mass-radius ($M(R)$) sequences are critical for understanding the stiffness of the EOS at supranuclear densities. The authors aim to quantify how current multimessenger observations constrain these endpoints and to what extent the results depend on the choice of the underlying hadronic baseline model.

2. Methodology

A. Construction of Hybrid EOSs

The authors construct families of causal hybrid EOSs by connecting low-density hadronic matter to high-density quark matter:

• Low-Density Regime: Two representative hadronic models are used as baselines:
  • SFHo: A relativistic mean-field model (softer EOS, incompressibility $K=245$ MeV).
  • DD2: A stiffer hadronic EOS.
• High-Density Regime: An extended Linear Sigma Model (eLSM) based on $U(3) \times U(3)$ chiral symmetry, incorporating constituent quarks and meson nonets. This model is constrained to match lattice QCD thermodynamics at low chemical potential and pQCD results at high chemical potential.
• Interpolation: To bridge the two regimes, a fifth-order polynomial interpolation is applied to the energy density as a function of baryon density. This ensures thermodynamic consistency and continuity of pressure and the speed of sound ($c_s$) across the transition region, avoiding large first-order phase transitions.
• Asymptotic Constraint: The quark-meson EOS is required to smoothly connect to pQCD results at asymptotically high densities ($\mu \approx 2.6$ GeV).

B. Bayesian Framework

The study employs a Bayesian weighting framework to analyze the probability distributions of the $M(R)$ sequence endpoints.

1. Prior: A broad ensemble of causal hybrid EOSs is generated spanning a range of stiffness at supranuclear densities.
2. Likelihood Weighting: The ensemble is weighted against observational data to generate posterior probability distributions for $M_{TOV}$ and $R_{TOV}$.

C. Observational Constraints

The following constraints are applied sequentially and in combination:

1. pQCD Constraint: Ensures the EOS approaches the N3LO pQCD reference point ($\mu=2.6$ GeV, $\rho=6.47$ fm$^{-3}$, $p=3823$ MeV/fm$^3$) while maintaining causality ($c_s < 1$).
2. GW170817: Tidal deformability constraint ($\tilde{\Lambda} < 720$) and a stricter combined constraint ("$\Lambda_{190}$": $70 < \Lambda_{1.4} < 580$ and $9.1 < R_{1.4} < 12.8$ km).
3. NICER: Mass-radius measurements for five pulsars (J0030+0451, J0740+6620, J0614–3329, J1231–1411, J0437–4715) with $\sim 10\%$ precision.
4. HESS J1731–347: A candidate for a very low-mass neutron star ($M \approx 0.77 M_\odot$), treated with caution due to interpretation uncertainties.
5. GW190814: The "mass-gap" object ($M \approx 2.59 M_\odot$). The authors test the hypothesis that this object is a neutron star by applying a Gaussian constraint centered at $2.59 M_\odot$.

3. Key Contributions

• Unified Diagnostic: Demonstrates that the probability distributions of the $M(R)$ sequence endpoints serve as a sensitive, complementary diagnostic for constraining high-density EOS behavior within a multimessenger framework.
• Baseline Independence vs. Dependence: Systematically disentangles the influence of observational data from the theoretical choice of the hadronic baseline (SFHo vs. DD2).
• Tension Identification: Identifies a specific tension between the stiff DD2 EOS, the "mass-gap" neutron star hypothesis, and the strict tidal deformability constraints ($\Lambda_{190}$).

4. Key Results

Maximum Mass ($M_{TOV}$) Distributions

• Dominance of Observations: The peak of the $M_{TOV}$ distribution is primarily driven by observational constraints rather than the choice of the baseline hadronic EOS (SFHo or DD2).
• Progression of Constraints:
  • pQCD only: Yields a peak slightly above $1.5 M_\odot$.
  • GW170817/HESS: No significant shift in the peak position.
  • NICER: Shifts the peak to 2.1–2.2 $M_\odot$ and sharpens the distribution.
  • Mass-Gap (GW190814): Shifts the peak significantly to 2.6–2.7 $M_\odot$.
• Robust Conclusion: When applying the most robust constraints (pQCD + GW170817 + NICER), both EOS models converge on a preferred maximum mass of 2.2–2.3 $M_\odot$. This is consistent with the observed mass of PSR J0952–0607 ($2.35 \pm 0.17 M_\odot$).

Radius ($R_{TOV}$) Distributions

• Baseline Dependence: Unlike mass, the radius distribution shows a strong dependence on the underlying hadronic EOS.
  • SFHo+eLSM: Peaks near 11.8 km (with robust constraints).
  • DD2+eLSM: Peaks near 12.4 km (with robust constraints).
• General Preference: Combined constraints favor a radius range of $12 \pm 1$ km.
• Tidal Deformability Impact: Applying the stricter $\Lambda_{190}$ constraint narrows the radius distributions and lowers the preferred values, as lower tidal deformability correlates with smaller radii.

The "Mass-Gap" Tension

• If the GW190814 object is interpreted as a neutron star, the DD2-based EOS fails to provide a consistent description when combined with the $\Lambda_{190}$ tidal constraint. The stiffness required to support a $2.6 M_\odot$ star in the DD2 model results in tidal deformabilities that are too large to satisfy the GW170817 bounds.
• The SFHo-based EOS (softer) handles this combination slightly better but still faces challenges.

5. Significance

This work provides a rigorous statistical assessment of the neutron star maximum mass, moving beyond single-point estimates to probability distributions.

1. Validation of Current Models: The results support the existence of neutron stars up to $\sim 2.2-2.3 M_\odot$, validating current nuclear physics models against astrophysical data.
2. Constraint on Phase Transitions: The smooth matching procedure and the resulting distributions suggest that large, first-order phase transitions are unlikely to occur in the density range probed by current observations, or at least are tightly constrained.
3. Future Directions: The study highlights that future precision in radius measurements (e.g., from NICER or future X-ray missions) and the definitive nature of the GW190814 object (neutron star vs. black hole) are critical for resolving the remaining uncertainties in the high-density EOS. The tension between the mass-gap hypothesis and tidal deformability constraints suggests that if GW190814 is indeed a neutron star, the EOS must be softer than the DD2 model predicts, or the tidal constraints require re-evaluation.