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Approximating neutron-star radii using gravitational-wave only measurements with symbolic regression

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A single symbolic expression approximates the neutron-star radius from only mass and tidal deformability, recovering radii to within a few hundred meters across a broad range of equations of state.

desk verdict A useful, honest symbolic-regression fit for R(M, Lambda) with real soft spots around non-uniqueness; deserves refereeing with softened claims and a baseline comparison. read the letter →

arxiv 2504.19962 v3 pith:SBL2B2UN submitted 2025-04-28 gr-qc astro-ph.HE

Micha{l} Bejger This is my paper · ORCID
classification gr-qcastro-ph.HE
keywords neutronstarsgravitationalwavestidaldeformabilityLovenumbersymbolicregressionequationofstateradiusinferenceGW170817
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that a single compact formula, discovered by symbolic regression, can recover a neutron star's radius from only two gravitational-wave observables: the component mass $M$ and tidal deformability $\Lambda$. If true, gravitational-wave detectors could estimate radii directly, without requiring a simultaneous electromagnetic measurement or a full equation-of-state reconstruction. The formula approximates the tidal Love number $k_2$ as a function of $M$ and $\Lambda$, then inverts the standard $k_2$–$\Lambda$–$R$ relation to obtain $R$. The author reports average errors of a few hundred meters on validation polytropes and on several realistic non-polytropic equations of state, and applying the formula to the GW170817 event yields radii consistent with previously reported inferences.

What carries the argument

The load-bearing object is the dimensionless tidal Love number $k_2$, defined through $k_2=\frac{3}{2}\Lambda\left(\frac{GM}{Rc^2}\right)^5$, which connects the measured deformability $\Lambda$ to the radius $R$. The paper uses a symbolic-regression search to find a simple closed-form surrogate $\tilde{k}_2(M,\Lambda)$ — Eq. (3) — by fitting stellar-structure data. Substituting $\tilde{k}_2$ into the inverted Love-number relation gives $\tilde{R}(M,\Lambda)$, turning radius estimation into the evaluation of one explicit algebraic expression and bypassing any need to assume a specific equation of state or solve the stellar-structure equations anew.

What would settle it

Evaluate Eqs. (2)–(3) on stellar-structure sequences for equations of state with strong first-order phase transitions over the mass range $1$–$2\,M_\odot$; if any sequence yields $|\Delta R|>0.5$ km at a mass where $\Lambda$ is well constrained, the claimed few-hundred-meter accuracy fails for that class of equations of state. An alternative is to find two realistic equations of state whose $(M,\Lambda)$ curves cross at a point where their radii differ by more than the reported error.

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Extended reading notes

Core claim

The paper's central claim is that, to good approximation, the radius $R$ of a non-rotating neutron star is a function of only $(M,\Lambda)$, the two quantities directly measured during a binary inspiral. Concretely, it proposes $\tilde{k}_2(M,\Lambda)=0.0351(\log_{10}\Lambda)^2\left(\frac{M}{M+2.05}-0.109\right)+0.0137$ for the tidal Love number, with $M$ in solar masses, and then $\tilde{R}(M,\Lambda)=\left(\frac{3\Lambda}{2\tilde{k}_2}\right)^{1/5}\frac{GM}{c^2}$. The relation is trained on solutions to the relativistic stellar-structure equations for piecewise polytropic equations of state and validated on both unseen polytropes and realistic dense-matter theories, with typical absolute radius differences of a few hundred meters. Applied to posterior samples from GW170817, it produces radii that fall inside the 50% and 90% credible regions of the published comparison distributions.

Load-bearing premise

The result relies on the assumption that one EOS-independent, single-valued function $k_2(M,\Lambda)$ exists, so that every $(M,\Lambda)$ pair corresponds to a unique radius; the paper's own test with intersecting sequences shows that two different equations of state can share the same mass and deformability while having different radii, giving the formula an intrinsic accuracy floor.

Editorial extensions

If this is right

  • Gravitational-wave-only radius estimates become a one-line evaluation, so individual binary-neutron-star events can yield radius measurements without costly equation-of-state sampling or structure integrations.
  • For the mass range roughly $1$–$2.5\,M_\odot$, the reported few-hundred-meter accuracy is competitive with typical equation-of-state-based inference, making the formula useful as a fast cross-check or as a summary statistic.
  • The expression extrapolates to non-polytropic equations of state and to sequences partially outside the training region, indicating that coverage of the physical equation-of-state space, rather than the specific training family, is the main driver of accuracy.
  • Applied to GW170817, the formula reproduces the published radius posterior distributions within the reported credible regions, so the relation can serve as a straightforward consistency test for future detections.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Augmenting the pointwise formula with a local slope, such as $d\Lambda/dM$, would likely resolve the intersection degeneracy the paper identifies, though it would give up the pure point-estimate character.
  • Because the fitted relation is cheap and explicit, it could seed hierarchical Bayesian analyses as an informed prior or act as a fast likelihood surrogate in parameter-estimation pipelines.
  • Retraining the same symbolic-regression pipeline on data that include hybrid stars or strong first-order phase transitions could extend the formula to the regimes where the current version shows its largest errors near the maximum mass.
  • A direct test against an independent electromagnetic radius measurement from X-ray pulse timing would be a natural next step; the paper does not include such a comparison.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper uses symbolic regression (PySR) on Tolman-Oppenheimer-Volkoff (TOV) solutions for piecewise-polytropic equations of state to derive a closed-form approximation for the tidal Love number k2 as a function of gravitational-wave measurable quantities M and log10 Lambda, Eq. (3). Inserting this into Eq. (2) yields an approximate neutron-star radius Rtilde(M, Lambda). The approximation is validated on a separate polytropic dataset, tested on sequences from six realistic non-polytropic EOSs, and applied to the GW170817 posterior samples, with reported average radius differences of a few hundred meters and maximum absolute differences up to 452.6 m. The paper claims the relation is EOS-independent and useful for multi-messenger radius inference.

Significance. If the claimed accuracy is robust, the expression would be a compact and computationally inexpensive tool for extracting radius information from gravitational-wave-only measurements, complementing existing universal relations. The paper has clear strengths: the training and validation sets are separate, the code and data are openly available, the method is reproducible, and the application to GW170817 public posteriors is a useful demonstration. The author also explicitly acknowledges a key limitation in Sec. IV.B, namely that point values of M and Lambda may not uniquely determine the radius. However, the central claim of an EOS-independent pointwise map is stronger than the evidence because the non-uniqueness of the (M, Lambda) to R relation is not quantified. The reported error metrics are per-sequence averages rather than the conditional spread of R at fixed (M, Lambda), so the worst-case behavior at degeneracies remains uncharacterized.

major comments (2)
  1. [§IV.B, Fig. 4 and §IV.A, Table I] The intersecting-sequence test (test1/test2) demonstrates that identical or nearly identical (M, Lambda) pairs can arise from EOSs with different radii, and the text acknowledges that point values of M and Lambda may not be sufficient to recover R precisely. This non-uniqueness is load-bearing for the paper's central claim that Eqs. (2)-(3) provide an EOS-independent pointwise radius estimate. The error metrics reported in Table I and Fig. 2 (mean Delta R, mean |Delta R|, and maximum |Delta R| along each sequence) quantify per-sequence accuracy but do not measure the conditional distribution of R at fixed (M, Lambda). Please quantify this conditional spread over the full validation and realistic-EOS datasets (for example, the width of R in bins of (M, Lambda), or the error specifically at the crossing region), or explicitly qualify the claim as holding only away from such degeneracies.
  2. [§IV.A, Table I] The headline 'few hundred meters' accuracy is supported by the averaged quantities, but Table I shows |Delta R|max = 452.6 m for BM165 at M = 2.03 M_sun and a mean Delta R of +299.8 m for that entire sequence. Because radii near the maximum mass are less relevant for current gravitational-wave detections, reporting the error metrics separately for the astrophysically relevant mass window (e.g., 1.0-1.6 M_sun) would make the practical claim more precise and would clarify whether the larger discrepancies are confined to the high-mass tail of each sequence.
minor comments (4)
  1. [§IV.B, text after Fig. 4] The statement that Rtilde is proportional to (Lambda / log10^2(Lambda))^(1/5) and has a singularity at Lambda = 1 is misleading when applied to the full expression Eq. (3), because the +0.0137 constant term keeps k2 finite at log10 Lambda = 0. Please rephrase the sentence to refer explicitly to the leading term of the fitted expression rather than to Eqs. (2) and (3) as written.
  2. [§IV.A, Fig. 2 caption] The inset histogram of the mass at which |Delta R| is maximal is described in the text as belonging to the validation dataset, but the caption does not state this; please add that information for clarity.
  3. [§III, dataset description] The reason for the mass cuts M > 0.5 M_sun and Mmax > 1.9 M_sun in the training and validation data is not explained, while Sec. II notes that the approximation is valid down to M approximately 0.25 M_sun; please clarify the relationship between these choices.
  4. [Appendix A, Table A caption] The caption defines Loss but only implicitly describes Score; please state explicitly that a higher Score corresponds to a better balance of accuracy versus complexity, consistent with the 'best' expression being the chosen one.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. 2 is an exact algebraic inversion of the defining relation, and Eq. 3 is an openly fitted empirical relation validated on independent EOSs.

full rationale

The derivation chain is: TOV solutions generate (M, R, k2, Lambda) tuples; symbolic regression fits k2(M, Lambda) as Eq. 3; Eq. 2 then solves Eq. 1 algebraically for R. Eq. 2 is an exact rearrangement of the definition k2 = (3Lambda/2)(GM/R c^2)^5, so the radius estimate is a transformed version of the fitted k2 relation, not an independent prediction built from the same target in disguised form. R is not a feature used to fit Eq. 3, the validation data are disjoint from the training data, and the realistic non-polytropic EOSs (WFF1, APR, SLy4, BSk21, BM165, NL3) lie partly outside the training family, providing an external test. The GW170817 application compares posterior samples propagated through Eq. 3 with previously published inferred radii, which is a comparison, not a circular re-use of fitted outputs. The admitted non-uniqueness of the pointwise (M, Lambda) to R map in Sec. IV.B is an honest physical limitation, and the author explicitly proposes adding dLambda/dM in future work; it does not make the fitted relation equivalent to its inputs by construction. The self-citation to [17] supplies the training dataset and EOS parametrization but is not used to justify the fitted formula itself, and the data-generation procedure is described in the paper, so no load-bearing step reduces to a self-citation or to a definition of the predicted quantity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a framework where TOV solutions with a parameterized piecewise-polytropic EOS family generate the training data, and where a single fitted expression generalizes across EOSs. No new physical entities are introduced. The main free parameters are the four constants in Eq. 3, fitted by symbolic regression; the PySR hyperparameters chosen via Optuna on the validation set are meta-parameters. The weakest assumption is that (M, Lambda) determines R uniquely, which the paper's own intersecting-sequence test contradicts.

free parameters (4)
  • Fitted constant 0.0351 = 0.0351
    Coefficient of (log10 Lambda)^2 in Eq. 3, fitted by symbolic regression to minimize training MSE.
  • Fitted constant 2.05 = 2.05
    Mass scale in the denominator M/(M+2.05) in Eq. 3, fitted to training data.
  • Fitted constant 0.109 = 0.109
    Offset in the mass term of Eq. 3, fitted to training data.
  • Fitted constant 0.0137 = 0.0137
    Additive constant in Eq. 3, fitted to training data.
assumptions (5)
  • standard math The TOV equations correctly describe the structure of non-rotating neutron stars.
    All training and validation data are generated by TOV solutions; used throughout Sec. II and III.
  • domain assumption The piecewise-polytropic EOS family from [17] adequately spans the astrophysically relevant region of the M-R plane.
    Training and validation data are drawn from this family; Sec. III and Appendix B.
  • domain assumption The low-density crust EOS is fixed to the SLy4 crust for all parametric EOSs.
    Sec. III states the crust EOS is represented by the SLy4 low-density part for all models.
  • standard math The static tidal deformability in the quadrupole approximation and the Love number relation in Eq. 1 are valid.
    Eq. 1 defines k2 in terms of Lambda, M, and R; this relation is the basis of the radius reconstruction in Eq. 2.
  • domain assumption The selected realistic EOSs (WFF1, APR, SLy4, BSk21, BM165, NL3) are representative enough to test extrapolation.
    These six EOSs are used in Sec. IV.A as post-training validation; their representativeness is assumed.

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Pith. "Pith review of Approximating neutron-star radii using gravitational-wave only measurements with symbolic regression." pith.science (2026). https://pith.science/paper/SBL2B2UN

@misc{pith2026250419962,
  author       = {Pith},
  title        = {Pith review of: Approximating neutron-star radii using gravitational-wave only measurements with symbolic regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SBL2B2UN}},
  note         = {Machine review of arXiv:2504.19962}
}
read the original abstract

Gravitational waves emitted by binary neutron-star inspirals carry information on components' masses and tidal deformabilities, but not directly radii, which are measured by electromagnetic observations of neutron stars. To improve the multi-messenger astronomy studies of neutron stars, an expression for neutron-star radii as a function of gravitational-wave only data would be advantageous, as it would allow to compare information from two different channels. In order to do so, a symbolic regression method, pySR, is trained on TOV solutions to piecewise polytropic EOS input to discover an approximate symbolic expression for the neutron-star radius as a function of gravitational-wave measurements only. The approximation is tested on piecewise polytropic EOS NS data, as well as on NS sequences based on selected realistic (non-polytropic) dense-matter theory EOSs, achieving consistent agreement between the ground truth values and the symbolic approximation for a broad range of NS parameters covering current astrophysical observations, with average radii differences of few hundred meters. Additionally, the approximation is applied to the GW170817 gravitational-wave mass and tidal deformability posteriors, and compared to reported inferred radius distributions.

Figures

Figures reproduced from arXiv: 2504.19962 by the authors.

Figure 1
Figure 1. FIG. 1. Training and validation datasets, and NS sequences [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. shows the distribution of ∆k2 = k2 − ˜k2 (top panel) for the training and validation datasets, and ∆R = R − R˜ (bottom panel) for the validation dataset only, i.e. 4-tuples of (M, R, k2,Λ), and resulting ˜k2(M,Λ) from Eq. 3 and R˜(M,Λ) from Eq. 2. Top panel presents distribution of ∆k2 for the two datasets in the full range of masses M ∈ (0.5 M⊙, Mmax) - the distributions are centred around zero difference, which is… view at source ↗
Figure 3
Figure 3. FIG. 3. Ground truth [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of the results from Fig. 3 of the LIGO-Virgo Collaboration [ [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Histograms of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

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Pith tools

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