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arxiv: 2511.23073 · v1 · submitted 2025-11-28 · 🌌 astro-ph.CO · astro-ph.GA· astro-ph.IM· cs.LG

Constraining dark matter halo profiles with symbolic regression

Alicia Mart\'in , Tariq Yasin , Deaglan J. Bartlett , Harry Desmond , Pedro G. Ferreira This is my paper

Pith reviewed 2026-05-17 04:26 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GAastro-ph.IMcs.LG
keywords dark matter halossymbolic regressionweak lensingNFW profilegalaxy clustershalo density profilesmodel selectionexcess surface density
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The pith

Exhaustive symbolic regression recovers the NFW profile from weak lensing data when fractional errors are around 5 percent.

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

This paper develops a method using exhaustive symbolic regression to determine the analytic form of dark matter halo density profiles straight from observational data instead of relying on forms suggested by simulations. The authors generate mock weak lensing excess surface density measurements assuming NFW halos and add constant fractional errors of varying sizes to see what functions the regression prefers. When the error is about 5 percent, the method reliably finds the NFW profile itself even using only 20 clusters. With the larger errors that match current surveys, it instead picks simpler expressions, mainly because the data are most accurate at large radii where many profiles look similar. The result is a data-driven, simulation-free way to identify which parts of the halo profile the observations actually constrain.

Core claim

Exhaustive symbolic regression applied to mock excess surface density data drawn from NFW halos selects the NFW function for fractional uncertainties of approximately 5 percent, even when the sample contains only 20 clusters. At the higher uncertainty levels that characterize present-day weak lensing surveys, simpler analytic functions are preferred although NFW remains competitive; this occurs because the measurement errors are smallest in the cluster outskirts, so the fit is driven by the outer density slope.

What carries the argument

Exhaustive Symbolic Regression (ESR), which searches over a space of possible analytic functions to identify the expression that fits the data while remaining as simple as possible.

If this is right

  • For fractional errors near 5 percent, the NFW profile is recovered from samples as small as 20 clusters.
  • At uncertainties typical of current surveys, simpler functions are selected over NFW though it stays competitive.
  • The selection of simpler functions is driven by weak lensing errors being smallest at large radii.
  • The approach supplies a simulation-independent framework for testing mass models and determining which profile features data actually constrain.

Where Pith is reading between the lines

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

  • Applying the same regression to actual observed cluster lensing data could show whether real measurements support NFW or require even simpler descriptions.
  • Comparing results across different cluster samples might reveal if the preferred profile depends on mass or redshift in ways not captured by fixed forms.
  • Using mock data generated from non-NFW profiles would test how well the method identifies the true underlying shape under realistic noise.
  • The technique could be extended to joint fits with other probes such as X-ray or Sunyaev-Zel'dovich observations to tighten constraints on the density profile.

Load-bearing premise

Constant fractional uncertainty assigned to each excess surface density point together with mock data generated exactly from NFW profiles is enough to represent the error properties and selection biases present in real weak lensing observations of galaxy clusters.

What would settle it

Running exhaustive symbolic regression on real weak lensing excess surface density measurements from a large sample of galaxy clusters and checking if the best-fit simple functions match the NFW form or deviate toward power laws or other simpler expressions.

Figures

Figures reproduced from arXiv: 2511.23073 by Alicia Mart\'in, Deaglan J. Bartlett, Harry Desmond, Pedro G. Ferreira, Tariq Yasin.

Figure 1
Figure 1. Figure 1: An example result of the ESR fitting procedure for one mock galaxy cluster, generated with a fractional uncertainty of f = 0.05. Left panel: The mock weak lensing ESD data (black points) are compared to the best-fit NFW profile (red) and the next two best-fitting functions discovered by ESR, |θ0| |θ1| r /r (purple) and (|θ0|/r) |θ1|−r (blue). Right panel: The corresponding 3D density profiles (ρ) for the t… view at source ↗
Figure 2
Figure 2. Figure 2: Optimal values of the objectives at each complexity for a fractional uncertainty of f = 0.05. The red curve (left axis) shows the per-complexity minimum description length relative to the global best description-length (NFW in this case). The blue curve (right axis) shows the per-complexity best likelihood relative to the global best likelihood across all models. Stars mark NFW in both metrics. For both ax… view at source ↗
Figure 3
Figure 3. Figure 3: Breakdown of the total Description Length (L(D)) for NFW as a function of the fractional uncertainty (f) in the mock data. The Total L(D) (black) is the sum of the Residual component (− log Lˆ) (red), the Parameter length (blue) and the function complexity. The plot shows that the parameter cost (blue line) dominates the total cost at low noise and drops significantly as the data becomes less constraining.… view at source ↗
Figure 4
Figure 4. Figure 4: The performance of the NFW profile relative to the other ESR-discovered functions, shown as a function of the fractional uncertainty (f) in the mock data. Top panel: The difference in total Description Length, ∆L(D) = L(D)Alt − L(D)NFW, where L(D)Alt is the description length of the best function other than NFW (black). The equivalent difference for likelihood (red) and parameter length (blue) also shown. … view at source ↗
Figure 5
Figure 5. Figure 5: Change in description length (∆L(D)) between NFW and the best alternative dark matter halo model. The curves show results for 2 different levels of Gaussian noise in the ESD data: f ∈ {0.01, 0.05} as a function of the number of clusters. Any point higher than 0 shows a preference for NFW. Nevertheless, the fact that NFW remains a leading candidate (top 10) even at higher noise levels is encouraging. It ind… view at source ↗
read the original abstract

Dark matter haloes are typically characterised by radial density profiles with fixed forms motivated by simulations (e.g. NFW). However, simulation predictions depend on uncertain dark matter physics and baryonic modelling. Here, we present a method to constrain halo density profiles directly from observations using Exhaustive Symbolic Regression (ESR), a technique that searches the space of analytic expressions for the function that best balances accuracy and simplicity for a given dataset. We test the approach on mock weak lensing excess surface density (ESD) data of synthetic clusters with NFW profiles. Motivated by real data, we assign each ESD data point a constant fractional uncertainty and vary this uncertainty and the number of clusters to probe how data precision and sample size affect model selection. For fractional errors around 5%, ESR recovers the NFW profile even from samples as small as 20 clusters. At higher uncertainties representative of current surveys, simpler functions are favoured over NFW, though it remains competitive. This preference arises because weak lensing errors are smallest in the outskirts, causing the fits to be dominated by the outer profile. ESR therefore provides a robust, simulation-independent framework both for testing mass models and determining which features of a halo's density profile are genuinely constrained by the data.

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

1 major / 2 minor

Summary. The paper introduces Exhaustive Symbolic Regression (ESR) as a method to derive analytic expressions for dark matter halo density profiles directly from weak-lensing excess surface density (ESD) data, without assuming simulation-motivated forms such as NFW. On mock ESD datasets generated from synthetic NFW clusters, with each point assigned a constant fractional uncertainty, the approach recovers the input NFW profile for fractional errors around 5% even with samples as small as 20 clusters. At higher uncertainties representative of current surveys, simpler functional forms are selected over NFW, which the authors attribute to the radial dependence of weak-lensing errors.

Significance. If the central results hold, the work offers a simulation-independent framework for testing halo-profile assumptions and identifying which radial features are actually constrained by data. The mock tests provide a clear demonstration of recovery behavior under controlled error levels and sample sizes, which strengthens the methodological contribution. The approach could be valuable for future surveys if the model-selection outcomes can be shown to reflect genuine observational constraints rather than specifics of the mock error model.

major comments (1)
  1. [Abstract] Abstract: the statement that 'simpler functions are favoured over NFW... because weak lensing errors are smallest in the outskirts, causing the fits to be dominated by the outer profile' is not supported by the described mock-data procedure, which assigns a constant fractional uncertainty to every ESD data point. Constant fractional errors lack the radial variation (smaller uncertainties at large radii) invoked in the explanation; therefore the observed preference for simpler expressions may be an artifact of the uniform-error assumption rather than a reflection of physical weak-lensing error distributions. This directly affects the interpretation of the higher-uncertainty results and the claim that the method reveals genuine data constraints on halo-profile features.
minor comments (2)
  1. [Abstract / Methods] The abstract and methods section would benefit from explicit statements of the ESR search space, complexity penalty functional form, and quantitative model-comparison statistics (e.g., exact AIC or BIC thresholds) used to declare one expression 'favoured'.
  2. [Figures / Results] Figure captions and text should clarify whether the reported 'recovery' is based on exact functional match, parameter recovery within uncertainties, or a statistical model-selection criterion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the work and for the constructive comment on the abstract. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that 'simpler functions are favoured over NFW... because weak lensing errors are smallest in the outskirts, causing the fits to be dominated by the outer profile' is not supported by the described mock-data procedure, which assigns a constant fractional uncertainty to every ESD data point. Constant fractional errors lack the radial variation (smaller uncertainties at large radii) invoked in the explanation; therefore the observed preference for simpler expressions may be an artifact of the uniform-error assumption rather than a reflection of physical weak-lensing error distributions. This directly affects the interpretation of the higher-uncertainty results and the claim that the method reveals genuine data constraints on halo-profile features.

    Authors: We appreciate the referee drawing attention to this detail. Assigning a constant fractional uncertainty to each ESD point does produce radially varying absolute uncertainties, since the ESD signal itself declines with radius. The absolute errors (fractional uncertainty multiplied by the local ESD value) are therefore smallest in the outskirts. When performing the fits, these outer points carry greater weight in the likelihood, causing the model selection to be dominated by the outer profile. This directly supports the explanation given in the abstract. To prevent any ambiguity, we will revise the abstract and the relevant methods paragraph to state explicitly that the constant fractional error model implies smaller absolute uncertainties at large radii. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results are obtained by generating mock ESD data from known NFW profiles, assigning constant fractional uncertainties to each point, and then running ESR to identify the best-fitting analytic expressions as a function of error level and sample size. These outcomes follow directly from the simulation protocol and the ESR search procedure without any step reducing by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation. The abstract's explanatory remark about radial error variation is an interpretive comment on real data rather than a premise that the mock results depend upon; the reported recovery rates and model preferences are independent of that remark. No equations or derivation chain in the described workflow collapses to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions about weak lensing data generation and the behavior of symbolic regression search; no free parameters are fitted to produce the reported recovery or selection results, and no new entities are postulated.

axioms (1)
  • domain assumption Mock ESD data generated from NFW profiles with constant fractional uncertainty per point is representative for testing profile recovery and model selection.
    This setup is used to generate the synthetic clusters and to interpret why simpler functions are selected at higher uncertainties.

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Exhaustive Symbolic Integration: Integration by Differentiation and the Landscape of Symbolic Integrability

    cs.SC 2026-05 unverdicted novelty 8.0

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  2. The functional form of galaxy and halo luminosity and mass functions

    astro-ph.GA 2026-04 unverdicted novelty 7.0

    Exhaustive symbolic regression identifies low-complexity functional forms for luminosity and mass functions that outperform Schechter and Press-Schechter parametrizations while satisfying physical extrapolation and in...

  3. Model-independent constraints on generalized FLRW consistency relations with bootstrap-based symbolic regression

    astro-ph.CO 2026-04 unverdicted novelty 6.0

    Bootstrap-based symbolic regression on supernova and BAO data finds mild 2-4 sigma deviations from FLRW consistency relations, which if real would rule out most FLRW-based solutions to cosmological tensions.

Reference graph

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