arxiv: 2604.03167 · v1 · submitted 2026-04-03 · 🌌 astro-ph.CO · gr-qc

Recognition: 2 theorem links

· Lean Theorem

Mapping the redshift drift at various redshifts through cosmography

Anna Chiara Alfano , Orlando Luongo

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:32 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc

keywords redshift driftcosmographySandage-Loeb testTaylor expansionPadé approximantsupernovaegamma-ray burstsbaryon acoustic oscillations

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The pith

Cosmographic Taylor and Padé expansions reconstruct the redshift drift to agree with LambdaCDM and omega0 omega1CDM at 1-2 sigma levels.

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

The paper uses second-order Taylor and (2,1) Padé expansions of the Hubble rate to model the redshift drift expected from future Sandage-Loeb observations. These cosmographic forms are fitted first to Pantheon+ supernovae plus gamma-ray bursts, then with added DESI baryon acoustic oscillation data, to produce a mock drift catalog that tests internal consistency of the reconstructed background. A sympathetic reader cares because redshift drift supplies a direct kinematic measurement of expansion rate changes, independent of distance ladders, that can distinguish dark energy scenarios. Results indicate the Taylor version matches omega0 omega1CDM at 1 sigma while Padé improves q0 agreement with LambdaCDM, with DESI inclusion pushing consistency to 2 sigma and the mock data mainly tightening q0 and j0 bounds.

Core claim

By constraining the cosmographic parameters (H0, q0, j0) through Taylor and Padé modeling of the Hubble rate on SNeIa plus GRB data and then on the same data plus DESI BAO, the authors build a mock Sandage-Loeb catalog and repeat the fits; they find Taylor reconstruction compatible at 1 sigma with omega0 omega1CDM, Padé improving q0 agreement with LambdaCDM, overall consistency weakening to 2 sigma once BAO are included, and the mock drift sample primarily tightening bounds on q0 and j0 while allowing direct comparison of the reconstructed drift to standard model predictions.

What carries the argument

Second-order Taylor expansion and (2,1) Padé approximant of the Hubble rate H(z), which fix the redshift drift formula through the cosmographic parameters H0, q0 and j0.

If this is right

Taylor reconstruction stays compatible at 1 sigma with omega0 omega1CDM for the SNeIa plus GRB dataset.
Padé parameterization improves agreement of q0 with the LambdaCDM value compared with Taylor.
Adding DESI BAO data reduces agreement with both reference models to the 2 sigma level.
Inclusion of the mock Sandage-Loeb sample tightens bounds mainly on q0 and j0, producing moderate shifts in central values.
Reconstructed drift curves can be compared directly to the behavior predicted by LambdaCDM and omega0 omega1CDM.

Where Pith is reading between the lines

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

This cosmographic approach could help select optimal redshift windows for actual Sandage-Loeb observations where model differences are largest.
Larger deviations in real drift data than reported here might signal the need for higher-order terms in the expansions at elevated redshifts.
The method supplies a model-independent link between distance-based data and direct expansion-rate tests that could flag inconsistencies in dark energy inferences.
Extending the same framework to higher redshifts or additional parameters might expose tensions in the jerk that current samples do not yet resolve.

Load-bearing premise

The Taylor and Padé expansions accurately represent the Hubble rate over the full redshift range of the data and mock catalog without large truncation errors or bias.

What would settle it

Actual Sandage-Loeb redshift drift observations lying outside the 2 sigma uncertainty bands around the drift curves predicted by the best-fit cosmographic models to the combined SNeIa, GRB and BAO dataset.

Figures

Figures reproduced from arXiv: 2604.03167 by Anna Chiara Alfano, Orlando Luongo.

**Figure 2.** Figure 2: FIG. 2: Confidence contours representation comparing [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Contour plot of the B´ezier coefficients. Darker [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Contour plots for the preliminary MCMC outcomes for the GRB and cosmographic parameters in both the case of the [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

read the original abstract

The redshift drift provides a kinematic test of the cosmic expansion history through the slow time variation of the redshift of comoving sources. Motivated by the expected Sandage-Loeb measurements from future facilities, we investigate the drift within a cosmographic framework, modeling the Hubble rate through both a second-order Taylor expansion and a $(2,1)$ Pad\'e approximant. We constrain the cosmographic parameters $(H_0,q_0,j_0)$ by combining Pantheon+ and SH0ES type Ia supernovae with gamma-ray bursts and then examine the impact of adding baryon acoustic oscillation measurements from the second DESI data release. The resulting constraints are used to construct a mock Sandage-Loeb catalog, after which the analyses are repeated including the simulated drift data. In this way, we assess the internal consistency of the reconstructed background rather than perform an independent forecast. Accordingly, we find that, for the SNeIa+GRB analysis, the Taylor reconstruction is compatible at the $1\sigma$ level with the $\omega_0\omega_1$CDM scenario, whereas the Pad\'e parameterization improves the agreement of $q_0$ with the $\Lambda$CDM paradigm. Once DESI BAO data are included, the agreement with the reference background models weakens to the $2\sigma$ level. The addition of the mock Sandage-Loeb sample mainly tightens the bounds on $q_0$ and $j_0$, with moderate shifts in the central values. We finally compare the reconstructed redshift drift with the corresponding behavior predicted by the $\Lambda$CDM and $\omega_0\omega_1$CDM scenarios.

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.

Desk Editor's Note private letter to a colleague

Applies standard cosmographic Taylor and Padé fits to SNe+GRB+DESI data then uses the posteriors for an internal mock Sandage-Loeb consistency check, showing 1-2 sigma agreement with reference models.

read the letter

This paper takes the usual second-order Taylor and (2,1) Padé expansions for H(z) and fits H0, q0, j0 to Pantheon+ supernovae plus gamma-ray bursts, then adds the second DESI BAO release. From those posteriors it builds a mock Sandage-Loeb drift catalog and re-runs the fit to check internal consistency with LambdaCDM and omega0 omega1 CDM. The results are straightforward: the Taylor version stays within 1 sigma of the dynamical model for the SNe+GRB set, Padé pulls q0 closer to the LambdaCDM value, BAO data move the agreement out to 2 sigma, and the mock drift mainly tightens the errors on q0 and j0 with modest central-value shifts. The abstract is clear that the mock is generated from the same fit, so this is a self-consistency test rather than an external forecast. That framing is honest and avoids over-selling the exercise. The direct comparison of the reconstructed drift to the reference models is also useful for seeing how the cosmographic parameters translate into an observable. The main limitation is the accuracy of the truncated expansions once GRBs push the redshift range to z ~ 8. Even Padé forms can deviate by more than 10 percent at z > 2 when higher-order terms or curvature matter, and the paper does not report an explicit convergence test against a numerical H(z) or fiducial cosmology over the full range. That leaves open the possibility that the drift reconstructions and the quoted compatibility levels carry systematic offsets from the approximation itself. The internal mock also means the tightening of bounds is expected rather than an independent validation. This work is aimed at people preparing model-independent analyses for upcoming Sandage-Loeb measurements or those who already use cosmography on high-redshift data. It is solid enough on the application side and transparent enough about its scope to deserve peer review so that referees can check the fitting details, error propagation, and any additional validation of the approximants.

Referee Report

2 major / 1 minor

Summary. The paper investigates the redshift drift using a cosmographic approach, modeling the Hubble rate H(z) via a second-order Taylor expansion and a (2,1) Padé approximant. It constrains the parameters (H0, q0, j0) from Pantheon+ SNeIa combined with GRBs, examines the effect of adding DESI BAO data, generates a mock Sandage-Loeb catalog from the resulting posteriors, and repeats the fits to assess internal consistency and compatibility with ΛCDM and ω0ω1CDM scenarios at the 1σ–2σ level.

Significance. If the truncated expansions remain faithful representations of the expansion history across the full redshift range of the data, the analysis offers a kinematic, model-independent route to forecasting and interpreting future Sandage-Loeb drift measurements, with the mock-data exercise showing how such observations would primarily tighten bounds on q0 and j0 while providing direct comparisons to standard background models.

major comments (2)

[Abstract and Methods (cosmographic modeling)] The second-order Taylor and (2,1) Padé expansions for H(z) are applied to GRB data reaching z ≈ 8 without any reported convergence or accuracy tests against a fiducial numerical H(z); given that even Padé approximants can deviate by >10% at z > 2 when higher-order terms are present, this directly affects the reliability of the reconstructed drift and the claimed 1σ/2σ compatibility levels with ω0ω1CDM and ΛCDM.
[Abstract and Results (mock catalog construction)] The mock Sandage-Loeb catalog is constructed directly from the cosmographic posteriors obtained from the real-data fits, so re-including it only probes internal consistency of the reconstruction rather than supplying an independent external check; this circularity weakens the evidential support for the statements on how future drift data would tighten constraints.

minor comments (1)

[Abstract] The abstract refers to repeated analyses with the simulated drift data but provides no explicit description of the likelihood function, covariance treatment, or error budget adopted in the joint fits.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each of the major comments point by point below, providing our responses and indicating the revisions made to the manuscript.

read point-by-point responses

Referee: [Abstract and Methods (cosmographic modeling)] The second-order Taylor and (2,1) Padé expansions for H(z) are applied to GRB data reaching z ≈ 8 without any reported convergence or accuracy tests against a fiducial numerical H(z); given that even Padé approximants can deviate by >10% at z > 2 when higher-order terms are present, this directly affects the reliability of the reconstructed drift and the claimed 1σ/2σ compatibility levels with ω0ω1CDM and ΛCDM.

Authors: We recognize that the lack of explicit convergence tests for the cosmographic expansions at high redshifts is a valid concern, as the approximations may not hold perfectly up to z ≈ 8. Cosmography is typically applied at low z, but extending it to GRB redshifts is common in the literature for model-independent analyses. In the revised manuscript, we will add a section or appendix providing accuracy tests by comparing the Taylor and Padé H(z) to a fiducial numerical integration from a reference model (e.g., ΛCDM with best-fit parameters), reporting the percentage deviations as a function of z. This will allow readers to assess the reliability and adjust the interpretation of the compatibility levels accordingly. We maintain that the 1σ/2σ compatibilities are based on the posterior overlaps from the fits, but acknowledge the potential impact of approximation errors. revision: yes
Referee: [Abstract and Results (mock catalog construction)] The mock Sandage-Loeb catalog is constructed directly from the cosmographic posteriors obtained from the real-data fits, so re-including it only probes internal consistency of the reconstruction rather than supplying an independent external check; this circularity weakens the evidential support for the statements on how future drift data would tighten constraints.

Authors: We appreciate this observation and note that our manuscript already clarifies the intent in the abstract: 'In this way, we assess the internal consistency of the reconstructed background rather than perform an independent forecast.' The mock data exercise is designed to illustrate the potential impact of future drift measurements that are consistent with the current cosmographic reconstruction, showing how they would refine the constraints on q0 and j0. To address the concern about circularity, we will revise the relevant sections to more explicitly frame it as an internal consistency test and discuss its implications for interpreting future data within this framework, without overstating it as an external validation. This still provides useful information on the sensitivity of the parameters. revision: partial

Circularity Check

1 steps flagged

Mock Sandage-Loeb catalog generated from fitted cosmographic posteriors tests internal consistency by construction

specific steps

fitted input called prediction [Abstract]
"The resulting constraints are used to construct a mock Sandage-Loeb catalog, after which the analyses are repeated including the simulated drift data. In this way, we assess the internal consistency of the reconstructed background rather than perform an independent forecast."

The mock drift data are generated directly from the posterior of the fit to Pantheon+ SNeIa, GRBs and DESI BAO. Re-fitting the same cosmographic parameters after adding this mock therefore reproduces consistency with the original fit by construction, rather than testing the reconstruction against truly external information.

full rationale

The paper fits cosmographic parameters (H0, q0, j0) to real SNeIa+GRB+DESI data using Taylor and Padé expansions for H(z), then explicitly constructs the mock Sandage-Loeb drift catalog from those same posteriors before re-including it in the fit. This step reduces to a self-consistency check rather than an independent prediction or external validation, as acknowledged in the abstract. The central claims of 1σ/2σ compatibility with reference models therefore rest partly on this tautological loop, though the initial constraints on real data and the expansion forms themselves remain independent of the mock. No other load-bearing reductions to self-citation or self-definition were identified in the derivation chain.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The analysis rests on three fitted cosmographic parameters and the assumption that low-order expansions suffice; no new entities are introduced.

free parameters (3)

H0
Present-day Hubble constant, fitted to the combined supernova, GRB, and BAO data.
q0
Deceleration parameter, fitted to the combined supernova, GRB, and BAO data.
j0
Jerk parameter, fitted to the combined supernova, GRB, and BAO data.

axioms (1)

domain assumption The background expansion can be described by a Taylor or Padé expansion of the Hubble rate around z=0 that remains valid at the redshifts of the observations and mock catalog.
Invoked throughout the modeling and mock-data construction sections.

pith-pipeline@v0.9.0 · 5604 in / 1603 out tokens · 55235 ms · 2026-05-13T18:32:01.208839+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

modeling the Hubble rate through both a second-order Taylor expansion and a (2,1) Padé approximant... constrain the cosmographic parameters (H0,q0,j0)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The redshift drift... Δv ≡ c Δz / (1+z) = c H0 Δt0 [1 − H(z)/H0(1+z)]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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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.

Nonlinear Relativistic Effects on Cosmological Redshift Drift
astro-ph.CO 2026-04 unverdicted novelty 8.0

Second-order relativistic effects on redshift drift are computed, showing distortions appear only at this order with enhanced nonlinear bispectrum contributions at low redshift and large momenta.

Reference graph

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This is also the case of the jerk pa- rameterj 0 which finds no agreement with the prediction from both ΛCDM andω 0ω1CDM models

in both scenarios of adoptingH(z) written in terms of Taylor or Pad´ e. This is also the case of the jerk pa- rameterj 0 which finds no agreement with the prediction from both ΛCDM andω 0ω1CDM models. On the other hand, the constraints on the deceleration parameterq 0 are the same as inAnalysis 1when using Taylor while the agreement degrades to the 2-σwit...

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