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arxiv: 2512.06454 · v2 · submitted 2025-12-06 · 🌌 astro-ph.CO

Testing the Distance Duality Relation with Cosmological Observations at high Redshift using Artificial Neural Network

Yukang Xie , Yang Liu , Puxun Wu , Xiangyun Fu , Nan Liang This is my paper

Pith reviewed 2026-05-17 00:38 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords distance duality relationtype Ia supernovaegamma-ray burstsbaryon acoustic oscillationsstrong gravitational lensingartificial neural networkhigh redshiftcosmology
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The pith

High-redshift observations from supernovae, gamma-ray bursts, BAO and lensing are consistent with the standard distance duality relation within about 2 sigma.

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

The paper conducts a model-independent test of the cosmic distance duality relation, which predicts that luminosity distance and angular diameter distance are related by a simple factor involving redshift under metric gravity and photon conservation. It combines Pantheon+ type Ia supernovae, Fermi gamma-ray bursts, DESI baryon acoustic oscillation measurements, and galaxy-scale strong lensing systems reaching redshifts up to about 8. An artificial neural network reconstructs the distances without assuming a specific cosmological model. The analysis concludes that the standard form of the relation holds within roughly 2 sigma confidence level across these probes.

Core claim

The standard DDR is consistent with cosmological observations at high redshift within the ∼2σ confidence level.

What carries the argument

Artificial neural network reconstruction that converts observed distance indicators into model-independent luminosity and angular diameter distances for direct comparison against the duality prediction.

Load-bearing premise

The artificial neural network reconstruction of distances introduces no systematic bias that would artificially pull the test toward consistency with the standard duality relation.

What would settle it

A new reconstruction method or larger high-redshift sample that yields a deviation from the standard duality relation larger than 3 sigma would falsify the reported consistency.

Figures

Figures reproduced from arXiv: 2512.06454 by Nan Liang, Puxun Wu, Xiangyun Fu, Yang Liu, Yukang Xie.

Figure 1
Figure 1. Figure 1: ANN reconstructions: left panel for m(z) from Pantheon+ at 0.01 < z ≤ 1.4; middle and right panels for GRB distance modulus µ(z) from Fermi FULL and GOLD samples at z > 1.4, respectively. Since the χ 2 loss depends on the full Pantheon+ covariance matrix, which has been veri￾fied to be positive definite and remains unchanged throughout training, we employ full-batch gradient descent; therefore, for both Pa… view at source ↗
Figure 2
Figure 2. Figure 2: Constraints on the cosmological parameters [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The 1D marginalized posterior distributions of the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

The cosmic Distance Duality Relation (DDR) is a fundamental prediction of metric gravity under photon number conservation. In this work, we perform a model-independent test of the DDR using Pantheon+ type Ia supernovae (SN Ia), \emph{Fermi} gamma-ray bursts (GRBs) with the FULL and GOLD samples, the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) baryon acoustic oscillation (BAO) measurements, and the galaxy-scale strong gravitational lensing (SGL) system samples at high redshift $0.01 < z \lesssim 8$ using an artificial neural network (ANN) approach. Our results show that the standard DDR is consistent with cosmological observations at high redshift within the $\sim 2 \sigma$ confidence level.

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 / 2 minor

Summary. The paper performs a model-independent test of the cosmic Distance Duality Relation (DDR) at high redshifts (0.01 < z ≲ 8) by using an artificial neural network to reconstruct luminosity distances from Pantheon+ SN Ia and Fermi GRB (FULL and GOLD) samples and angular-diameter distances from DESI DR2 BAO and galaxy-scale strong gravitational lensing (SGL) systems. The central result is that the standard DDR (η(z) = 1) remains consistent with these combined observations within approximately 2σ.

Significance. If the ANN reconstructions prove unbiased, the work provides a valuable high-redshift extension of DDR tests that combines SN Ia, GRBs, BAO, and SGL in a non-parametric framework. This could tighten constraints on photon-number conservation in metric theories. The use of GRB and SGL data at z ≳ 2 is a clear strength, though the overall impact hinges on demonstrating that the network does not introduce systematic smoothing that artificially favors consistency.

major comments (2)
  1. [§3] §3 (ANN methodology and training): The manuscript does not report hold-out validation or recovery tests on mock catalogs that inject controlled DDR violations (e.g., η(z) = 1 + εz with ε chosen to produce 3–5σ deviations). In the sparse high-z regime (z ≳ 2) where GRB and SGL points are few, standard ANN architectures with typical regularization can suppress localized deviations; without such tests the ~2σ consistency result cannot be shown to be free of reconstruction bias.
  2. [§4.2] §4.2 (uncertainty propagation and error budget): The final η(z) error bars combine ANN epistemic/aleatoric uncertainties with observational errors, yet the text does not specify how these are added in quadrature or via Monte Carlo sampling, nor whether the network loss function penalizes large residuals. This directly affects whether the reported 2σ threshold is robust or underestimated.
minor comments (2)
  1. [Figure 3] Figure 3: The legend and caption should explicitly state the redshift bins used for the binned η(z) points and whether the shaded band includes only statistical or total (stat+sys) uncertainty.
  2. [Table 1] Table 1: The GRB GOLD sample selection criteria (e.g., fluence or duration cuts) are referenced but not tabulated; adding a short column or footnote would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which have helped us improve the presentation and robustness of the analysis. We address each major comment below and have revised the manuscript to incorporate additional validation and clarification.

read point-by-point responses

  1. Referee: [§3] §3 (ANN methodology and training): The manuscript does not report hold-out validation or recovery tests on mock catalogs that inject controlled DDR violations (e.g., η(z) = 1 + εz with ε chosen to produce 3–5σ deviations). In the sparse high-z regime (z ≳ 2) where GRB and SGL points are few, standard ANN architectures with typical regularization can suppress localized deviations; without such tests the ~2σ consistency result cannot be shown to be free of reconstruction bias.

    Authors: We agree that explicit recovery tests on mock catalogs with injected DDR violations are important to demonstrate that the ANN does not suppress deviations through regularization, especially given the sparse sampling at z ≳ 2. The original manuscript described the network architecture, training, and application to real data but did not include such controlled recovery tests. We have now generated mock catalogs based on a fiducial cosmology, injected controlled violations of the form η(z) = 1 + εz at levels producing 3–5σ deviations, and performed recovery tests. These tests show that the ANN recovers the injected signals without significant bias or excessive smoothing. We have added a new subsection in §3 detailing the mock generation procedure, the recovery results, and a new figure illustrating the recovered η(z) curves. This directly addresses the concern about potential reconstruction bias. revision: yes

  2. Referee: [§4.2] §4.2 (uncertainty propagation and error budget): The final η(z) error bars combine ANN epistemic/aleatoric uncertainties with observational errors, yet the text does not specify how these are added in quadrature or via Monte Carlo sampling, nor whether the network loss function penalizes large residuals. This directly affects whether the reported 2σ threshold is robust or underestimated.

    Authors: We thank the referee for highlighting the need for greater clarity on uncertainty propagation. The total uncertainty on η(z) was computed by first obtaining the ANN epistemic uncertainty from an ensemble of independently trained networks and the aleatoric component via Monte Carlo dropout, then combining these with the observational errors through Monte Carlo sampling of the full error distributions rather than simple quadrature. The loss function employed during training includes both mean-squared error and L2 regularization terms to penalize large residuals. However, these steps were described only briefly in the original text. We have revised §4.2 to provide a detailed, step-by-step description of the Monte Carlo propagation procedure, the combination method, and the explicit form of the loss function. This revision ensures the robustness of the reported ~2σ consistency is more transparently justified. revision: yes

Circularity Check

0 steps flagged

ANN reconstructions from external catalogs enable independent DDR test with no definitional reduction

full rationale

The paper trains an ANN on Pantheon+ SN Ia, Fermi GRBs, DESI DR2 BAO, and high-z SGL samples to reconstruct luminosity and angular-diameter distances in a model-independent manner, then forms the duality parameter η(z) as their ratio to test consistency with the standard DDR. No load-bearing step equates the reported ~2σ consistency to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same authors. The test remains falsifiable against the input catalogs; external benchmarks (observed distances) are not redefined by the output.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on the standard assumption that metric gravity plus photon conservation implies the distance duality relation; no new free parameters, axioms, or invented entities are introduced beyond those already present in the input catalogs.

axioms (1)
  • domain assumption Metric gravity together with photon-number conservation implies the distance duality relation eta(z) = 1.
    Explicitly stated in the abstract as the fundamental prediction being tested.

pith-pipeline@v0.9.0 · 5443 in / 1264 out tokens · 35193 ms · 2026-05-17T00:38:11.304261+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

  1. Model-independent test of the cosmic distance duality relation with recent observational data

    astro-ph.CO 2026-03 conditional novelty 6.0

    Two model-independent methods applied to latest SN and BAO data find the cosmic distance duality relation consistent with observations within 1 sigma and no evidence of violation.

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