Generative artificial intelligence for reconstructing neutron-star matter

Abstract: Neutron-star cores hold the only known matter in the universe that is simultaneously cold and strongly interacting, compressed beyond nuclear density into a state of unknown composition. The equation of state links stellar masses, radii and tidal deformabilities to this regime, but recovering this key quantity from sparse observations is an ill-posed inverse problem. Existing analyses bury a prior in a fixed functional form, unevenly weighting admissible solutions and biasing the result. We reconstruct the equation of state with a denoising diffusion model that keeps prior, physics and data separate: it learns an inspectable, physically motivated prior anchored to first-principles nuclear theory, while perturbative-QCD and astrophysical constraints are imposed exactly. Future measurements therefore will update the posterior by reweighting alone, without retraining or resampling. The inferred radius of 12.6 km and tidal deformability of 469 at 1.4 solar masses reproduce Gaussian-process and heavy-ion-informed inferences despite a far broader prior. We find near-conformal but still stiff matter in the heaviest stars, consistent with a gradual hadron-quark crossover and disfavouring a strong first-order phase transition. More broadly, coupling a learned prior to exactly enforced physics establishes a template for ill-posed inverse problems where theory and data constrain different regions.

• The paper combines a denoising diffusion prior, exact chiral effective field theory inpainting, and likelihood reweighting of pQCD and multimessenger data to reconstruct the neutron-star equation of state without retraining.
• The reconstruction predicts R(1.4 M☉) = 12.56^{+0.36}_{-0.45} km, Λ(1.4 M☉) = 469^{+97}_{-100}, and near-conformal but still stiff matter in maximum-mass stars, with c_s² ≈ 0.47 at the centre.
• The posterior reduces the probability of a strong first-order-transition signature from 8.8% to 1.4%—about 7:1 odds against it—while emphasizing that conclusions remain dependent on the learned prior and current observational coverage.

The paper presents a reconstruction of the cold, $\beta$-equilibrated neutron-star-matter equation of state (EOS) using a denoising diffusion model as an explicitly learned, physically motivated prior, combined with exact enforcement of theory and observational constraints (2608.17457). The central methodological claim is that the ill-posed inverse problem of recovering $P(\varepsilon)$ from sparse multimessenger data can be structured so that prior, physics, and data remain separable: the diffusion network supplies an inspectable prior over speed-of-sound profiles, chiral effective field theory ($\chi$EFT) is imposed during generation by inpainting, and perturbative-QCD (pQCD) and astrophysical likelihoods are applied afterwards by importance reweighting. Because reweighting acts on raw prior samples, future measurements update the posterior without retraining or resampling.

Inference framework

The target quantity is the squared speed of sound $c_s^2(n)$ on a 200-point grid spanning $n/n_0 \in [0.5, 8]$, with $n_0 \approx 0.16~\mathrm{fm}^{-3}$. A DDPM with $v$-prediction, a cosine variance schedule, and a ResNet–attention denoiser ($1.94 \times 10^7$ parameters) is trained on $10^6$ synthetic profiles drawn from ten functional classes: piecewise polytropes, spectral representations, logistic-plus-Gaussian sound-speed parametrizations, relativistic mean-field (RMF) models, constant-sound-speed (CSS) first-order-transition constructions, quarkyonic matter, NJL-inspired crossovers, and pQCD-motivated asymptotic forms, plus two convex-combination families. This training set deliberately contains sound-speed peaks, first-order-transition plateaus, and near-conformal tails without committing to any of them.

Conditioning proceeds in two stages. At generation, five low-density $\chi$EFT anchors—Monte-Carlo moments of $P(\varepsilon)$0 charge-neutral $P(\varepsilon)$1 realizations built from the BUQEYE N$P(\varepsilon)$2LO pure-neutron- and symmetric-matter ensembles with full correlated covariance—are imposed by the RePaint inpainting rule at every reverse step, with truncation uncertainty propagated by per-sample anchor jitter. After generation, each surviving profile is converted to a full EOS, integrated through the Tolman–Oppenheimer–Volkoff and tidal equations, and reweighted by the marginalized pQCD likelihood at the termination point, three NICER mass–radius posteriors (PSR J0030+0451, J0740+6620, J0437$P(\varepsilon)$34715), the GW170817 tidal deformability, and the PSR J1614$P(\varepsilon)$42230 maximum-mass bound. Of $P(\varepsilon)$5 drawn profiles, 89,513 satisfy causality and thermodynamic stability; the full posterior retains an effective sample size of $P(\varepsilon)$6.

A data ablation isolates each constraint's action. The pQCD likelihood operates almost exclusively above $P(\varepsilon)$7, lifting the high-density band toward $P(\varepsilon)$8 at modest statistical cost ($P(\varepsilon)$9 alone); the astrophysical likelihoods constrain the density range probed by observed stars and reweight far more strongly ($\chi$0 alone). The two are complementary: astrophysical data fix the canonical-mass region while pQCD trims the stiff tail back toward the conformal value.

Sound speed and conformality

Compared with the Gaussian-process (GP) and four-segment interpolation (C4) posteriors of Annala et al., the diffusion reconstruction develops no sharp, localized sound-speed peak: where individual samples peak, they peak lower and at higher density, and the median rises smoothly to a plateau. Since all three analyses use comparable constraints, this difference reflects the prior breadth. The derived diagnostics approach conformality—the polytropic index falls through the quark-matter criterion $\chi$1 near $\chi$2, the trace anomaly $\chi$3 reaches a median consistent with zero at the TOV point, and the conformal distance $\chi$4 drops below 0.2 above $\chi$5—yet the speed of sound remains well above its conformal value, $\chi$6 at the centre of the maximum-mass star. Dense matter is thus stiff while its bulk thermodynamics conformalize; the two need not conformalize together. Notably, for 45% of the posterior weight $\chi$7 has not attained its maximum by the centre of the maximum-mass star, so current observables cannot locate the peak; a fixed functional form would place it inside a star by construction.

Neutron-star structure

The reconstruction yields $\chi$8 km and $\chi$9, consistent with GP, C4, and several literature inferences at the canonical mass. At high mass the results depart from prior work: $c_s^2(n)$0 km and $c_s^2(n)$1 fall below all benchmark ensembles, so the mass–radius relation is tilted rather than uniformly shifted. At the TOV central density, $c_s^2(n)$2, three independent diagnostics agree on near-conformality: $c_s^2(n)$3, $c_s^2(n)$4, and $c_s^2(n)$5. Because each indicator weights $c_s^2(n)$6, $c_s^2(n)$7, and $c_s^2(n)$8 differently, their joint agreement is stronger than any single criterion. The pressure at the TOV chemical potential $c_s^2(n)$9 GeV is $n/n_0 \in [0.5, 8]$0, consistent with Annala et al.'s $n/n_0 \in [0.5, 8]$1.

Phase transition and symmetry energy

Testing the necessary (not sufficient) signature of a strong first-order transition—$n/n_0 \in [0.5, 8]$2 dropping below 0.1 above its in-window peak—the probability falls from a prior value of 0.088 to a posterior value of 0.014, giving a Bayes factor of 0.14, odds of about 7:1 against the signature. The paper is careful to note this is substantial but not decisive, and that the ratio is not an artefact of prior frequencies: ensembles with nearly identical prior frequencies yield Bayes factors differing by a factor of two. An independent heavy-ion-informed analysis finds a value below unity on a different prior and criterion. The overall reading is a gradual hadron–quark crossover rather than a sharp transition, though the authors concede this cannot be established by thermodynamic diagnostics alone.

The same reconstruction yields a symmetry-energy slope of $n/n_0 \in [0.5, 8]$3 MeV, sitting between the Koehn et al. inference ($n/n_0 \in [0.5, 8]$4 MeV) and the experiment-driven Tsang et al. value ($n/n_0 \in [0.5, 8]$5 MeV), and consistent with the $n/n_0 \in [0.5, 8]$6EFT anchoring band.

Limitations and open questions

The paper states plainly that the reconstruction remains prior-dependent: the effective sample size of 4.2% indicates the likelihood does not overwhelm the prior, and the predictive median $n/n_0 \in [0.5, 8]$7 rises only from 12.0 to 12.6 km upon conditioning, so the radius scale is inherited substantially from the $n/n_0 \in [0.5, 8]$8EFT-anchored prior. Robustness checks—alternative regulator cutoff ($n/n_0 \in [0.5, 8]$9 MeV), alternative J0437 analyses, an additional maximum-mass threshold from PSR J0952$n_0 \approx 0.16~\mathrm{fm}^{-3}$00607, and augmented training ensembles including free-form Gaussian-process and piecewise-linear families—leave all fiducial observables within their 68% credible intervals and the Bayes factor between 0.04 and 0.24, never approaching unity. The most cutoff-sensitive quantity is $n_0 \approx 0.16~\mathrm{fm}^{-3}$1, shifting from 60.5 to 66.8 MeV, exceeding the quoted truncation systematic and reported separately. The EOS is a one-dimensional cold, charge-neutral slice; extension to $n_0 \approx 0.16~\mathrm{fm}^{-3}$2 for mergers, proto-neutron stars, and supernovae is identified as the substantial open extension, and locating the reconstruction relative to the deconfined regime requires composition-resolved input. Whether the sound-speed peak lies inside a star is left to post-merger gravitational-wave signals probing transiently higher densities. The paper also explains why per-class Bayes factors are not constructible within the joint-training framework, and why distributional-similarity rankings would be structurally misleading.

Conclusion

This work establishes a template for ill-posed inverse problems in which theory and data constrain different regions: a learned, theory-anchored diffusion prior, exact inpainting of controlled low-density input, and likelihood reweighting that preserves multimodality and permits shape-level tests such as the phase-transition Bayes factor. The resulting picture—an EOS stiffer at high density and softer at intermediate density than previous reconstructions, with near-conformal but stiff matter in the heaviest stars and odds of roughly 7:1 against a strong first-order transition—is robust to the composition of the prior and to bounded changes in the data, while remaining explicitly conditional on the prior's breadth.