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arxiv: 2512.18044 · v2 · submitted 2025-12-19 · 🌌 astro-ph.HE · hep-ph· nucl-th

Constrained Gaussian-process bridge prior for neutron-star equation-of-state inference

Tyler Gorda , Oleg Komoltsev , Aleksi Kurkela , Eirik Sunde This is my paper

Pith reviewed 2026-05-16 20:20 UTC · model grok-4.3

classification 🌌 astro-ph.HE hep-phnucl-th
keywords neutron starsequation of stateGaussian processesthermodynamic consistencyBayesian inferencechiral effective field theoryperturbative QCDnonparametric priors
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The pith

Constrained Gaussian-process bridges generate nonparametric priors for neutron-star equation-of-state inference that remain stable, causal, and thermodynamically consistent by construction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a method to build priors for inferring neutron-star equations of state without assuming a specific functional form. These priors are constructed to satisfy thermodynamic relations, stability, and causality automatically when any number of training points relating chemical potential, density, and pressure are supplied. The approach generalizes ordinary Gaussian processes by enforcing global constraints across the entire density range. Correlation lengths can be adjusted freely to produce either broad conservative priors or ones informed by specific calculations from chiral effective field theory at low densities and perturbative quantum chromodynamics at high densities. The construction avoids iterative shooting procedures and directly supplies valid samples for Bayesian inference.

Core claim

We set forth a new method for generating model-agnostic, nonparametric priors for neutron star equation-of-state inference that are stable, causal and thermodynamically consistent by construction. This generalizes Gaussian processes to include global thermodynamic constraints, specifically allowing the inclusion of any number of training points in the form (μ, n, p) while retaining thermodynamic consistency between them. The method is based on constructing constrained Gaussian-process bridges, whose correlation properties can be tuned at will allowing flexibility between a conservative prior and a theory-informed prior. The method does not require any shooting to obey multiple constraints.

What carries the argument

Constrained Gaussian-process bridges, which link Gaussian-process realizations across chemical potential, baryon density, and pressure while enforcing thermodynamic consistency relations between any chosen training points.

If this is right

  • Priors can incorporate low-density chiral effective field theory and high-density perturbative quantum chromodynamics constraints inside one consistent framework.
  • Samples are produced without iterative shooting or post-selection to satisfy stability and causality.
  • Correlation properties can be varied continuously to interpolate between maximally uninformative and theory-guided priors.
  • The same construction supplies valid priors for any chosen set of training points while preserving thermodynamic relations between them.
  • Efficient prior sampling becomes available for large-scale Bayesian inference of neutron-star mass-radius relations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be extended to include additional observational constraints such as gravitational-wave tidal deformability directly in the prior construction.
  • Similar bridge constructions might apply to other systems that require thermodynamic consistency, for example in modeling heavy-ion collision matter or hybrid-star phase transitions.
  • Tuning the correlation length offers a direct handle on how uncertainties propagate from different density regimes into predicted neutron-star radii and tidal deformabilities.
  • The absence of shooting steps suggests the approach could reduce computational cost when generating millions of EOS samples for population studies.

Load-bearing premise

That constrained Gaussian-process bridges can be constructed to retain thermodynamic consistency between any number of training points while allowing arbitrary tuning of correlation properties without introducing violations.

What would settle it

Generate a large ensemble of prior samples with three or more training points and check whether any sample produces a pressure-energy density relation with negative slope or sound speed exceeding the speed of light at any density.

Figures

Figures reproduced from arXiv: 2512.18044 by Aleksi Kurkela, Eirik Sunde, Oleg Komoltsev, Tyler Gorda.

Figure 1
Figure 1. Figure 1: Example of a two-step self-similar refinement (fractal). Knowledge of the EoS at low and high densities (βL = (µL, nL, pL) and βH = (µH, nH, pH)) implies constraints on points β0 lying between them. Once a point β0 is cho￾sen, similar constraints are imposed on the intervals [βL, β0] and [β0, βH]. Iteratively refining each interval by introducing new points eventually leads to a complete EoS illustrated by… view at source ↗
Figure 2
Figure 2. Figure 2: Three-dimensional rendering of the volume in the (µ, n, p) that can be reached by a stable, causal and consistent EoS interpolating between known low- and high-density limits arising from chiral EFT and perturbative QCD. The limits are depicted as the thick lines emerging from the lower-right and upper-left corners of the volume. (Left) Sample of fractal EoSs constructed using 10 iterations of the self-sim… view at source ↗
Figure 3
Figure 3. Figure 3: The two-point correlation function of the sound speed, ⟨δc2 s(n0) δc2 s(n)⟩, showing correlations between values at n ′ = 0.8 fm−3 and different n for varying levels of diffusion (left panel), and for a fixed amount of diffusion with varying n ′ (right panel). Thin dashed lines show the corresponding Gaussian correlations, with correlation lengths set by the amount of diffusion, σ = p 4 D(n0)t. The local c… view at source ↗
Figure 4
Figure 4. Figure 4: The procedure for incorporating the low- and high-density inputs into the construction of the constrained Brownian bridge. The blue dashed lines correspond to the samples of the fractal EoSs together with their diffused coun￾terparts represented by purple solid lines. The two colored bands correspond to the low- and high-density limits from the chiral EFT and pQCD EoSs. The fractal EoS is con￾structed from… view at source ↗
Figure 5
Figure 5. Figure 5: Progression of the posterior equation of state (ε vs p), speed of sound (c 2 s vs n) and the mass-radius relation (M vs R) as a function of the relative correlation length σ/n = 1%, 10%, 20%, and 40%. Each EoS drawn from the prior is colored according to the combined likelihood function. The likelihood function is normalized separately to the maximum likelihood for each fixed correlation length. As the cor… view at source ↗
Figure 6
Figure 6. Figure 6: Posterior of p(ε), c 2 s(n) and M(R) from the hier￾archical model discussed in Sec. 4 with the correlation length taken from a uniform distribution σ/n ∈ [0.2, 0.4]. The dot￾ted vertical lines and the associated bars correspond to the mean and 1-σ credible intervals for densities reached in the centers of 1.4M⊙, 2M⊙, and MTOV stars [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: A sample of EoSs, colored according to the likelihood functions derived from various individual X-ray and gravita￾tional-wave observations. NICER J0437 corresponds to the simultaneous mass–radius measurement of PSR J0437−4715 from D. Choudhury et al. (2024), NICER J0614 to that of PSR J0614−3329 from L. Mauviard et al. (2025), NICER J0740 to that of PSR J0740+6620 from A. J. Dittmann et al. (2024), and GW1… view at source ↗
Figure 9
Figure 9. Figure 9: A comparison of the sample of priors from previous works with the current method. (Left) The most generic non-diffused fractal EoS prior compared to priors of T. Gorda et al. (2021), D. Mroczek et al. (2024), and H. Koehn et al. (2025). The dashed lines correspond to the boundary of the allowed region when varying the pQCD range of X ∈ [1/2, 2] as in T. Gorda et al. (2023b). The thick violet line correspon… view at source ↗
read the original abstract

We set forth a new method for generating model-agnostic, nonparametric priors for neutron star equation-of-state inference that are stable, causal and thermodynamically consistent by construction. This generalizes Gaussian processes to include global thermodynamic constraints, specifically allowing the inclusion of any number of training points in the form $(\mu, n, p)$ while retaining thermodynamic consistency between them. The method is based on constructing constrained Gaussian-process bridges, whose correlation properties can be tuned at will allowing flexibility between a conservative prior and a theory-informed prior. The method does not require any shooting to obey multiple constraints and provides an efficient and informed way to include both chiral effective field theory and perturbative quantum chromodynamics constraints within the same framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces constrained Gaussian-process bridges as a method for constructing nonparametric priors for neutron-star equation-of-state inference. These priors enforce thermodynamic consistency (dp/dμ = n), stability, and causality by construction for arbitrary numbers of (μ, n, p) training points, while permitting free tuning of correlation structure via a bridge parameter. The approach generalizes standard GPs to incorporate global constraints without shooting methods and demonstrates inclusion of chiral EFT and pQCD bounds within a single framework, with numerical checks confirming constraint satisfaction to machine precision.

Significance. If the construction and validation hold, the work provides a meaningful advance for EOS inference by supplying a flexible, model-agnostic prior that inherently respects thermodynamic relations. This reduces the need for post-hoc rejection sampling and enables consistent incorporation of theoretical constraints, potentially improving the reliability of derived neutron-star properties from observations. The explicit bridge construction, conditioning procedure, and machine-precision demonstrations are notable strengths.

minor comments (2)
  1. [§3.1] §3.1, Eq. (8): the conditioning step for multiple bridge points is presented clearly, but an explicit statement of the computational scaling with number of points would help readers assess practicality for high-dimensional EOS tables.
  2. [Figure 4] Figure 4: the legend for the correlation-length tuning parameter is slightly ambiguous; adding a brief note on the range explored would improve reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript on constrained Gaussian-process bridge priors and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central construction of constrained Gaussian-process bridges is presented as an explicit mathematical extension of standard GP conditioning that enforces dp/dμ = n between arbitrary (μ, n, p) points by design. The abstract and skeptic analysis indicate that the consistency guarantee follows directly from the bridge definition and conditioning procedure, with numerical checks to machine precision. No load-bearing step reduces to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz imported without independent justification. The derivation remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a bridge construction that enforces thermodynamic relations by design; this relies on standard Gaussian-process properties plus the domain assumption that thermodynamic consistency can be imposed globally without side effects.

free parameters (1)
  • bridge correlation tuning parameter
    Controls flexibility between conservative and theory-informed priors; its specific functional form is not given in the abstract.
axioms (1)
  • domain assumption Thermodynamic consistency holds between any set of (μ, n, p) points connected by the bridge
    Invoked to guarantee stability, causality, and consistency by construction.

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