Astrophysical Constraints on Hadron--Quark Crossover in Hybrid Neutron Stars

This paper investigates astrophysical constraints on hybrid neutron-star equations of state (EoS) constructed by combining an extended linear sigma model (eLSM) description of quark matter with several representative hadronic equations of state through a smooth hadron–quark crossover. Four different hadronic models are examined to explore the model dependence of the resulting hybrid EoSs.

The quark phase is described by the eLSM with constituent quarks, incorporating chiral symmetry and its spontaneous breaking. The vector coupling strength gV is treated as a free parameter in the Bayesian analysis. The hadronic EoSs considered are SFHo (relativistic mean field), DD2 (relativistic mean field with density-dependent couplings), DNS(CMF) (chiral mean field with hyperons), and GMSR(BSK16) (nuclear meta-model with CLDM crust).

The hadron–quark crossover is constructed by interpolating the energy density with a fifth-order polynomial over a density interval defined by the central transition density ρ̄ and half-width Γ. The crossover parameters, along with gV, define the model space explored.

After imposing stability, causality, perturbative QCD matching, tidal deformability (Λ̃ 2.04 M⊙ or MTOV > 2.27 M⊙), only the eLSM+GMSR(BSK16) construction remains fully consistent with current astrophysical observations. The DD2 and DNS(CMF) models fail to simultaneously satisfy the maximum-mass and tidal deformability constraints, while SFHo shows tension with NICER radius measurements.

The surviving parameterizations are further constrained using Bayesian inference based on NICER mass–radius measurements of five pulsars (PSR J0030+0451, PSR J0740+6620, PSR J0614−3329, PSR J0437−4715, and PSR J1231−1411, using the updated posterior from Qi et al. [40]) and the GW170817 gravitational-wave event.

The posterior distributions favor a broad hadron–quark crossover occurring at relatively high baryon densities. The most probable parameterization is gV = 5.0, Γ = 3.6 ρ0, and ρ̄ = 5.8 ρ0, which is robust to the choice of maximum-mass constraint. This parameterization passes through the 68% credible regions of all NICER mass–radius constraints.

The formation of extended pure quark cores is strongly disfavored. While pQCD consistency alone allows configurations with extended pure quark cores, imposing the maximum-mass constraint MTOV > 2.04 M⊙ eliminates all finite-size pure quark cores from the remaining viable parameterizations. The tidal deformability constraint alone has little impact on this conclusion. Instead, the current observations favor neutron stars containing an extended mixed hadron–quark transition region. For the highest-posterior parameterization, the maximum-mass configuration has MTOV = 2.349 M⊙ and RTOV = 10.826 km, with no pure quark core (MQ = 0, RQ = 0), but a hadron–quark crossover region is present for stellar configurations with Mtr = 2.142 M⊙ and Rtr = 9.82 km.

The paper also investigates the implications of the inferred maximum-mass distribution for the secondary object of GW190814. The posterior distribution of MTOV, obtained without imposing a lower bound on MTOV to avoid circular reasoning, is largely determined by NICER measurements and favors maximum masses consistent with the heaviest observed pulsars. However, only a very small fraction of the posterior probability extends to maximum masses comparable to that required for a stable neutron star interpretation of the GW190814 secondary. The paper concludes that, within the present model and adopted observational constraints, its interpretation as a neutron star is unlikely.

Improvements to AI systems:

1. Bayesian-inference-driven equation-of-state selection: Implement a hierarchical Bayesian framework that automatically updates posterior probabilities for hybrid neutron-star EoS parameters (e.g., vector coupling g V, crossover width, transition density) using multi-messenger astrophysical data (NICER mass–radius, GW170817 tidal deformability). The improved AI can self-consistently reject model families (e.g., DD2, DNS(CMF)) that fail joint constraints, and identify the most probable parameterization without manual tuning.

2. Causal and stability-aware interpolation in physical models: Train a neural network or Gaussian process to generate smooth hadron–quark crossover EoSs using fifth-order polynomial interpolation in energy density, with built-in enforcement of causality (c s at most c), thermodynamic stability, and perturbative QCD matching at high densities. The AI can then rapidly explore the full parameter space and flag unphysical regions, reducing computational cost by orders of magnitude compared to brute-force grid searches.

3. Multi-constraint consistency checking with uncertainty propagation: Develop an AI system that ingests raw observational posteriors (e.g., from NICER and LIGO/Virgo) and automatically propagates systematic uncertainties (e.g., radius measurement biases, tidal deformability limits) into EoS predictions. This system can quantify the probability that a given EoS parameterization satisfies all current constraints simultaneously, and output credible intervals for stellar properties (mass, radius, core composition) with rigorous error bars.

4. Composition-aware stellar structure prediction: Build a deep learning surrogate model that, given an EoS, directly predicts the mass–radius relation, tidal deformability, and internal composition (e.g., presence and size of pure quark cores vs. mixed hadron–quark regions) for neutron stars. The improved AI can then invert this mapping to infer EoS parameters from observed pulsar data, and explicitly test hypotheses like extended pure quark cores are disfavored by computing posterior probabilities for core sizes.

5. Astrophysical-object classification under model uncertainty: Implement a probabilistic classifier that, using the posterior distribution of M TOV from the Bayesian analysis, estimates the likelihood that a compact object (e.g., GW190814's secondary) is a neutron star versus a black hole. The AI can incorporate the full EoS model uncertainty and observational priors, outputting a probability (e.g., <5% for neutron-star interpretation) rather than a binary decision, and can be retrained as new constraints emerge.

6. Automated model comparison and falsification: Create an AI system that performs Bayesian model selection across different hadronic EoSs (SFHo, DD2, DNS(CMF), GMSR(BSK16)) combined with quark models, automatically computing Bayes factors and evidence ratios. This system can rank models by their predictive power against astrophysical data, identify which components (e.g., hyperons, density-dependent couplings) are most incompatible, and suggest which physical ingredients to modify for future model improvements.

7. Real-time data assimilation for neutron-star physics: Develop an AI pipeline that continuously updates EoS posteriors as new observational data (e.g., from future NICER-like missions or gravitational-wave events) become available. Using sequential Monte Carlo or variational inference, the system can incrementally refine predictions for maximum mass, radius, and internal composition, and issue alerts when new data significantly shift the preferred parameter region (e.g., toward higher or narrower crossover).