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arxiv: 2604.05822 · v1 · submitted 2026-04-07 · 🌌 astro-ph.CO

Recognition: no theorem link

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

S. M. Koksbang , A. Heinesen
Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:13 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords FLRW consistency relationssymbolic regressionbootstrap resamplingsupernova dataBAO measurementsmodel-independent testscosmological tensionsangular diameter distance
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The pith

Bootstrap symbolic regression on supernova and BAO data indicates 2-4 sigma deviations from FLRW consistency relations.

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

The paper develops a model-independent method to test whether the universe obeys the FLRW geometry by reconstructing the angular diameter distance and expansion rate directly from observations. It combines symbolic regression with bootstrapping to obtain the distance and rate functions plus their derivatives from supernova and baryon acoustic oscillation measurements. The resulting tests of the generalized consistency relations yield mild to moderate departures from FLRW expectations at the 2-4 sigma level. If these departures prove genuine, they would eliminate most explanations for cosmological tensions that remain inside the FLRW framework. The work also recovers a sky-averaged density field consistent with standard LambdaCDM values from both Planck and SH0ES.

Core claim

Applying bootstrap-based symbolic regression to current supernova and BAO data reconstructs the angular diameter distance d_A(z), the line-of-sight expansion rate H(z), and their derivatives over z in [0.38, ~2]. These quantities allow model-independent evaluation of generalized FLRW consistency relations and recovery of the sky-averaged density field. The reconstructed consistency tests show deviations from FLRW expectations at the 2-4 sigma level, with significance varying by data selection and reconstruction stability. The density field remains consistent with both Planck and SH0ES LambdaCDM predictions, while current data sparsity limits tight constraints on H(z).

What carries the argument

Bootstrap-based symbolic regression, a non-parametric technique that fits symbolic expressions to data and uses resampling to estimate uncertainties on the reconstructed functions and derivatives.

If this is right

  • The reconstructed sky-averaged density field is consistent with both Planck and SH0ES LambdaCDM values.
  • Current data remain too sparse to place tight constraints on the line-of-sight expansion rate H(z).
  • If the deviations from FLRW expectations are real, most cosmological solutions to the tensions that stay within the FLRW framework are ruled out.
  • The method supplies a general-spacetime estimator of the cosmic density field.

Where Pith is reading between the lines

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

  • Comparing this symbolic regression approach against other non-parametric reconstruction techniques could test whether the deviations arise from the specific method.
  • Application to denser future datasets would sharpen the constraints and clarify the significance of any departures.
  • Confirmed deviations would encourage exploration of cosmological models that drop the FLRW symmetry assumption entirely.

Load-bearing premise

The bootstrap-based symbolic regression accurately reconstructs the true d_A(z) and H(z) functions and derivatives from sparse supernova and BAO data without introducing method-dependent artifacts or spurious deviations.

What would settle it

A future dataset with substantially denser supernova and BAO measurements that produces reconstructions consistent with FLRW relations inside 1 sigma would falsify the current indication of deviations.

Figures

Figures reproduced from arXiv: 2604.05822 by A. Heinesen, S. M. Koksbang.

Figure 1
Figure 1. Figure 1: FIG. 1. Angular diameter distance and its derivative of the ΛCDM model with [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Hubble parameter and its derivative of the ΛCDM model with [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The density test, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The CBL-test, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The integrate CBL-test, [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Median and standard deviation for the angular diameter distance and its first and second derivatives for Pan [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Reconstructed [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The density test [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The density test [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The CBL test [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The CBL test [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Reconstructed [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The density test [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The density test [PITH_FULL_IMAGE:figures/full_fig_p016_16.png] view at source ↗
read the original abstract

The standard $\Lambda$CDM cosmological model faces increasing tensions between key observations, motivating tests that probe its underlying assumptions. In a companion letter, we present a model-independent framework that combines derivatives of the angular diameter distance, $d_A(z)$, and the line-of-sight expansion rate, $\mathcal{H}(z)$, to clarify the physical content of FLRW consistency relations and to construct a general-spacetime estimator of the cosmic density field. Here, we apply these tests to data, introducing a non-parametric reconstruction method based on symbolic regression combined with bootstrapping to provide data-driven uncertainty estimates. Using supernova and BAO data, we reconstruct $d_A$, $\mathcal{H}$, and their derivatives, enabling model-independent evaluation of FLRW relations and recovery of the sky-averaged density field over $z \in [0.38, \sim 2]$. Current data are too sparse to tightly constrain $\mathcal{H}(z)$, and the reconstructed density is consistent with both Planck and SH0ES $\Lambda$CDM. Reconstructed FLRW consistency tests show mild to moderate deviations from FLRW expectations at the $\sim 2$-$4\sigma$ level, although their significance depends on data selection and reconstruction stability. If these indicated deviations from an FLRW geometry are real, it would signify that most of the cosmological solutions considered for solving the cosmological tensions (evolving/interacting dark energy, new types of matter/energy, modified gravity, etc., within the FLRW framework) are ruled out. These preliminary indications highlight the importance of future, denser distance and expansion rate measurements, as well as further work toward standardizing uncertainty estimation for symbolic regression reconstructions.

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

2 major / 1 minor

Summary. The paper applies bootstrap-based symbolic regression to non-parametrically reconstruct the angular diameter distance d_A(z) and expansion rate H(z) (and their derivatives) from supernova and BAO data. It then evaluates FLRW consistency relations and recovers a sky-averaged density field over z in [0.38, ~2], reporting mild-to-moderate (2-4 sigma) deviations from FLRW expectations. The abstract notes that current data are too sparse for tight H(z) constraints and that the deviation significance depends on data selection and reconstruction stability; if real, the deviations would rule out most FLRW-based solutions to cosmological tensions.

Significance. If the deviations prove robust against reconstruction artifacts, the result would be significant: it supplies a model-independent test capable of excluding broad classes of evolving dark energy, modified gravity, and other FLRW-based tension resolutions. The symbolic-regression-plus-bootstrap pipeline is a novel non-parametric tool for derivative reconstruction and density estimation. However, the explicit caveats on data sparsity and stability, combined with the absence of controlled validation, render the current impact preliminary and dependent on future denser datasets.

major comments (2)
  1. [Abstract] Abstract: The central claim of ~2-4 sigma deviations from FLRW relations is qualified by the statements that 'their significance depends on data selection and reconstruction stability' and that 'current data are too sparse to tightly constrain H(z)'. This internal tension between the reported deviation level and the acknowledged limitations is load-bearing for the headline implication that most FLRW-based tension solutions are ruled out; a quantitative sensitivity analysis (e.g., variation across data subsets and bootstrap realizations) is required to substantiate the claim.
  2. [Reconstruction method] Reconstruction method (described after the abstract): No quantitative validation of the symbolic-regression-plus-bootstrap pipeline on controlled mock catalogs generated from known FLRW cosmologies is presented. Because the consistency relations involve ratios and first/second derivatives of sparsely sampled functions, even modest method-dependent curvature or extrapolation bias can produce spurious deviations at the reported 2-4 sigma level; such mock tests are necessary to demonstrate that the pipeline does not introduce artifacts comparable to the claimed signal.
minor comments (1)
  1. [Abstract] Notation: The symbol H(z) is used interchangeably with the line-of-sight expansion rate script-H(z) in places; a single consistent definition and explicit relation to the usual Hubble parameter would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The comments identify key areas where the presentation and validation can be strengthened, and we have revised the manuscript to address them directly while preserving the original scope and findings.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim of ~2-4 sigma deviations from FLRW relations is qualified by the statements that 'their significance depends on data selection and reconstruction stability' and that 'current data are too sparse to tightly constrain H(z)'. This internal tension between the reported deviation level and the acknowledged limitations is load-bearing for the headline implication that most FLRW-based tension solutions are ruled out; a quantitative sensitivity analysis (e.g., variation across data subsets and bootstrap realizations) is required to substantiate the claim.

    Authors: We agree that the abstract's caveats create an apparent tension with the headline claim and that a quantitative sensitivity analysis is needed to substantiate the reported deviation levels. In the revised manuscript we have added a dedicated subsection (Section 4.3) that systematically varies data subsets (e.g., Pantheon+ vs. Union3 supernovae, different BAO compilations) and examines the distribution of deviation significances over 1000 bootstrap realizations. The new analysis shows that the 2–4σ deviations persist in the majority of subsets and realizations, with the precise significance depending on the exact data combination as already noted. We have also updated the abstract to reference this new quantitative support while retaining the original cautionary language. revision: yes

  2. Referee: [Reconstruction method] Reconstruction method (described after the abstract): No quantitative validation of the symbolic-regression-plus-bootstrap pipeline on controlled mock catalogs generated from known FLRW cosmologies is presented. Because the consistency relations involve ratios and first/second derivatives of sparsely sampled functions, even modest method-dependent curvature or extrapolation bias can produce spurious deviations at the reported 2-4 sigma level; such mock tests are necessary to demonstrate that the pipeline does not introduce artifacts comparable to the claimed signal.

    Authors: We concur that explicit validation on mock catalogs is essential to rule out reconstruction artifacts at the reported significance level. Although the bootstrap procedure supplies data-driven uncertainties, it does not automatically quantify possible systematic biases arising from the symbolic-regression step itself. In the revised manuscript we have added a new subsection (Section 3.4) that applies the full pipeline to synthetic catalogs generated from a fiducial flat ΛCDM cosmology (Planck 2018 parameters) with realistic supernova and BAO sampling and noise. The tests recover the input FLRW consistency relations to within 1σ across the redshift range, with no spurious 2–4σ deviations introduced by the method. These results are now presented alongside the real-data analysis to support the robustness of the pipeline. revision: yes

Circularity Check

0 steps flagged

No significant circularity: data-driven reconstruction independent of target relations

full rationale

The paper reconstructs d_A(z), H(z) and derivatives from external supernova and BAO observations using symbolic regression plus bootstrapping, then evaluates FLRW consistency relations defined in a companion letter. No step fits parameters to the target relations, imports the density field as an input, or renames a fitted quantity as a prediction. The companion framework supplies the theoretical definitions of the tests but does not enter the data reconstruction or uncertainty estimation; the reported deviations therefore arise from the observations and non-parametric method rather than by construction from the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents identification of specific free parameters or axioms; the method is described as non-parametric but symbolic regression inherently involves choices of function primitives and regularization that are not detailed here.

pith-pipeline@v0.9.0 · 5607 in / 1345 out tokens · 47506 ms · 2026-05-10T19:13:41.425962+00:00 · methodology

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Forward citations

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