Revisiting the Hubble constant, sound horizon and cosmography from late-time Universe observations

Tonghua Liu; Xiaolei Li; Zhiwei Yang

REVIEW 3 major objections 5 minor 1 cited by

Revisiting the Hubble constant, sound horizon and cosmography from late-time Universe observations

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper claims that model-independent late-universe data—DESI BAO, time-delay lenses, and Pantheon SNe—yield H0 ≈ 73 km/s/Mpc and rd ≈ 137–138 Mpc, favoring the SH0ES value over Planck flat-ΛCDM.

desk verdict A competent incremental combination of DESI BAO, H0LiCOW lensing, and Pantheon gives H0~73 and rd~138, but the 'model-independent' label is oversold given the silent flatness assumption and untested high-z expansion. read the letter →

arxiv 2505.01661 v1 pith:Y4KTREA3 submitted 2025-05-03 astro-ph.CO

Zhiwei Yang , Tonghua Liu , Xiaolei Li This is my paper · ORCID

classification astro-ph.CO MSC 85A4083F05 PACS 98.80.-k98.80.Es

keywords Hubbleconstantsoundhorizoncosmographybaryonacousticoscillationstime-delaylensingtypeIasupernovaetensionPadéapproximation

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 tries to measure the Hubble constant H0 and the sound horizon rd without assuming a cosmological model like ΛCDM. It combines DESI baryon acoustic oscillation data with six time-delay lensing distances and four deflector angular diameter distances from H0LiCOW, plus the Pantheon supernovae sample. Two flexible distance-redshift approximations are used: a fourth-order Taylor series and a Padé (2,1) rational function. Both give H0 ≈ 72.9–73.1 km/s/Mpc and rd ≈ 137–138 Mpc. Those values support the higher H0 measured by the SH0ES distance ladder and a smaller sound horizon than Planck's flat-ΛCDM prediction, so the Hubble tension persists even without assuming a cosmological model.

What carries the argument

The central machinery is a model-independent cosmographic parameterization of the distance-redshift relation. The Taylor series expands H(z) to fourth order in terms of the present-day deceleration q0, jerk j0, and snap s0, and integrates to give the luminosity distance DL(z); the Padé (2,1) approximation instead represents DL(z) as a rational function with the same cosmographic coefficients, chosen because Padé has a larger convergence radius at high redshift. These relations generate the angular diameter distance DA = DL/(1+z)^2, the transverse comoving distance DM = DL/(1+z), the Hubble distance DH = c/H(z), and the dilation scale DV = [z $DM^{2}$ DH]^{1/3}, so the DESI BAO measurements of DV/rd and DM/rd, DH/rd can be combined with the lensing distances that fix the absolute scale, with rd left free.

What would settle it

Redo the joint fit with spatial curvature Ω_k included as a free parameter and with a higher-order approximation such as Padé (3,2) or a fifth-order Taylor series; if the inferred H0 shifts by more than about 2 km/s/Mpc or rd by more than about 4 Mpc relative to the quoted errors, the reported values are artifacts of the truncation and the fixed flatness assumption.

Watch

Extended reading notes

Core claim

The authors claim that combining DESI-BAO with the six time-delay distances and four deflector angular diameter distances from H0LiCOW lenses together with the Pantheon SNe Ia gives H0 = 72.9 ± 1.8 km/s/Mpc for the Taylor series and H0 = 73.1 ± 1.8 km/s/Mpc for the Padé polynomials, with rd = 138.2 Mpc and 137.0 Mpc, respectively. These results are consistent with the SH0ES value of 73.04 ± 1.04 km/s/Mpc and in tension with Planck's 67.4 ± 0.5 km/s/Mpc under flat ΛCDM. The key point is that the absolute distances from strong lensing anchor the relative BAO distances, while supernovae track the expansion history and mainly sharpen the cosmographic parameters.

Load-bearing premise

The whole calibration rests on the assumption that the truncated fourth-order Taylor expansion (or the fixed Padé (2,1) form) is an accurate description of the distance-redshift relation across the full data range, including redshifts above z ≈ 2, and that spatial curvature can be held fixed rather than fitted.

Editorial extensions

If this is right

If the result is right, the Hubble tension is not an artifact of assuming ΛCDM in the late-universe data: a model-independent late-time calibration still lands at about 73 km/s/Mpc.
The inferred rd ≈ 137–138 Mpc is smaller than the Planck-ΛCDM expectation, the direction needed to ease the tension if new pre-recombination physics shrinks the sound horizon.
The combination of absolute lensing distances with relative BAO distances acts as a model-independent ladder; future DESI data releases and larger lens samples should push the H0 uncertainty below 1 km/s/Mpc.
The close agreement between the Taylor and Padé results (≈ 0.2 km/s/Mpc) suggests the H0 value is robust to the choice of distance approximation, although the Taylor series is explicitly less reliable at high redshift.
Supernovae contribute mainly to constraining the cosmographic parameters q0, j0, and s0 rather than to H0 and rd, so further tightening of H0 and rd will come mostly from additional BAO and lensing data.

Reading between the lines

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

The paper never reports a fitted value for the curvature parameter Ω_k even though it appears explicitly in the Taylor luminosity distance; allowing Ω_k to float is a natural test that could shift the reported H0 and rd.
Given the known convergence problems of the Taylor series above z ≈ 1, the Padé (2,1) result is arguably the more reliable of the two, and a higher-order Padé check would show whether the central values are truly stable.
A confirmation of rd ≈ 137 Mpc by future DESI data would strengthen the motivation for early-universe models that reduce the sound horizon, tying this measurement directly to the wider Hubble-tension discussion.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Quick take: this is a competent, incremental late-time combination. The new piece is just the dataset combination—DESI DR1 BAO, six H0LiCOW time-delay distances, four deflector angular diameter distances, and Pantheon SNe—fitted with fourth-order Taylor and Padé (2,1) cosmographies. That exact combination hasn't been published, and the resulting H0 around 73 and rd around 137–138 Mpc are consistent with SH0ES and prior late-time analyses. The lensing KDE treatment of correlated posteriors is a nice touch, and the DIC comparison is honest about the fact that neither expansion wins consistently.

But the paper overstates model-independence. Two concrete issues. First, Omega_k appears in Eqs. (3)–(4) but is never fitted or reported; the analysis silently assumes flatness. That is exactly the kind of model dependence the title claims to avoid. Second, the Taylor series is a fourth-order expansion around z=0, and the data extend to z~4.2 through DESI Lyman-alpha BAO. The authors themselves note Taylor convergence problems at z>1 and recommend Padé, yet they still present the Taylor value as a headline. Agreement between Taylor and Padé does not bound shared truncation bias, since the Padé (2,1) coefficients come from the same low-z Taylor expansion. No convergence test, higher-order comparison, or residual check is shown. If the high-z distance ratios are mispredicted, the lensing anchor propagates that into H0 and rd, and the reported error bars likely understate this systematic.

Minor point: the Pantheon likelihood in Eq. (19) appears to use only diagonal errors; the full Pantheon covariance includes systematics. If they used the full covariance, the text should say so.

So the paper is fine as a dataset re-analysis, and the numbers are plausible, but I would not call the constraints model-independent. The authors are straight about the Taylor issue and the literature coverage is adequate. It deserves a serious referee, not a desk reject. The referee should ask for a treatment of Omega_k, a truncation-error estimate for the high-z BAO, and a clarification of the SNe covariance. Those fixes would make this a useful incremental contribution.

Referee Report

3 major / 5 minor

Summary. This paper combines DESI BAO measurements, six time-delay distances and four deflector angular-diameter distances from the H0LiCOW sample, and the Pantheon SNe Ia sample, using a fourth-order Taylor expansion and a Padé (2,1) approximant for the distance-redshift relation. The authors report H0 = 72.9 ± 1.8 km/s/Mpc and rd = 138.2 ± 3.9/3.3 Mpc for the Taylor cosmography, and H0 = 73.1 ± 1.8 km/s/Mpc and rd = 137.0 ± 3.7/3.2 Mpc for the Padé cosmography, concluding that these late-time values favor a larger H0 and a smaller rd than Planck flat-ΛCDM while agreeing with SH0ES.

Significance. If the result holds, the paper provides a useful late-time, cosmographic calibration of H0 and rd that is independent of early-universe physics and of a specific dark-energy model. The analysis is a direct likelihood fit to public data, uses a standard MCMC approach, and its full-data results are qualitatively consistent with earlier work by Wojtak & Agnello (2019) and with the H0LiCOW analyses. The main value is in showing that the DESI BAO + lensing + SNe combination can anchor an absolute distance scale without a CMB prior. However, the strength of the conclusion depends on the fidelity of the two low-order distance approximants over the full redshift range, and on the treatment of curvature, neither of which is adequately tested in the manuscript.

major comments (3)

[Section 2.1, Eqs. (3)-(4), Table 1] The curvature parameter Ω_k appears explicitly in the distance expansion at third and fourth order in z, yet it is absent from the list of fitted parameters and is never reported in Table 1. Setting Ω_k = 0 is an unstated flatness assumption, and it conflicts with the paper's claim of model independence. Because the DESI BAO data include high-redshift Lyman-alpha bins, a nonzero Ω_k would change the model predictions for D_M/rd and D_H/rd and would propagate into the inferred H0 and rd. Please either fit Ω_k with a prior, or explicitly state the flat assumption and demonstrate insensitivity by repeating the analysis for fixed Ω_k values such as ±0.01 or by adding a geometric curvature probe.
[Section 2.2, Eqs. (2), (3), (7), Table 1] The central calibration uses the fourth-order Taylor expansion and the Padé (2,1) approximant out to the highest DESI BAO redshifts, including Lyman-alpha bins with effective redshifts above 2, while Section 2.2 itself concedes that the Taylor series has convergence problems for z > 1. No truncation-error estimate, residual plot, or higher-order comparison is provided. The agreement between the Taylor and Padé results at the ~0.2 km/s/Mpc level is not a sufficient robustness check, because both approximants are constructed from the same low-order Taylor coefficients and can share a common high-z bias. Please add quantitative convergence tests, for example a Padé (3,2) or (2,2) fit, or a repeat of the analysis with the BAO data truncated at z < 1.5, together with residuals as a function of redshift.
[Section 3.3, Eq. (19)] The Pantheon χ² in Eq. (19) sums only the diagonal variance σ²_μ,SN. The Pantheon release includes a full covariance matrix with systematic contributions, and standard analyses use the inverse covariance matrix. Restricting to the diagonal underestimates the SNe uncertainties and can bias the inferred q0, j0, and s0, which enter the distance model and therefore the calibration of H0 and rd. Please use the full Pantheon covariance matrix, or clearly justify the diagonal-only approximation and quantify its effect on the reported parameters.

minor comments (5)

[Section 2.2 and Section 5] The redshift coverage is described inconsistently: Section 3.1 states DESI BAO spans 0.1 < z < 4.2, while Section 2.2 and the conclusion refer to data extending to z ≈ 2.3. Please state the effective redshifts of the highest DESI bins (e.g., the Lyman-alpha effective redshift) consistently throughout.
[Section 4, Eq. (21)] The DIC definition contains an algebraic typo: DIC = D(θbar) + 2p_D = Dbar + p_D, not D(θbar) + p_D. Additionally, 'deending' should read 'depending'.
[Section 4 and Table 1] The paper reports the Taylor-series value as a headline result in the abstract and conclusion even though it recommends using the Padé approximation and discarding the Taylor expansion in future analyses; please state explicitly which of the two results is considered fiducial.
[References] Several references are duplicated in the bibliography, including Jee et al. 2019, Liao et al. 2020, Poulin et al. 2019, and Wojtak & Agnello 2019; these should be merged.
[Throughout] There are minor typographical errors, including 'Supernavae' in the Section 3.3 title, 'Talor' after Eq. (8), and 'constraining chances' in Section 4; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: H0 and rd are free parameters fitted to independent external data, with cosmographic forms transparently adopted from external literature.

full rationale

The paper is a direct likelihood analysis rather than a derivation: H0, rd, q0, j0 and s0 are free parameters with flat priors, fitted to DESI BAO, H0LiCOW time-delay distances, deflector angular-diameter distances, and Pantheon supernovae via Eqs. (13), (15), (16) and (19). The Taylor series (Eqs. 2-4) and the Padé (2,1) form (Eq. 7) are explicit standard parameterizations (Visser 2004; Gruber & Luongo 2014; Capozziello et al. 2020) whose coefficients are the fitted parameters; nothing in these equations algebraically forces H0 ≈ 73 km/s/Mpc or rd ≈ 137-138 Mpc. The lensing distances supply the absolute scale, BAO supply r_d-scaled distances, and supernovae constrain the expansion shape, which is the stated data-combination strategy rather than a circular reuse of the target values. The self-citations (Liu et al. 2022a,b; Liu et al. 2024) are context references for cosmographic methods and Hubble-tension discussion; they are not used to set priors, fix parameters, or justify a uniqueness claim. The acknowledged Taylor/Padé convergence concern at z > 1 (Sec. 2.2) and the unfitted curvature parameter Ω_k are potential systematic-error or model-risk issues, not circularity: the headline quantities are not inputs by construction, and the final values are compared against, not derived from, SH0ES and Planck.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The paper's central values depend on standard cosmological assumptions plus several unstated or weakly justified choices. The free parameters H0, rd, q0, j0, s0, and M_B are all fitted to the data. The key axioms are FLRW geometry, the validity of truncated Taylor and Pade approximations to z = 4.2, the treatment of Omega_k, and the reliability of the H0LiCOW distance posteriors. No new particles or forces are introduced. The most fragile entry is the treatment of Omega_k, which enters the distance formulas but is absent from the parameter table.

free parameters (6)

H0 = 72.9 km/s/Mpc (Taylor), 73.1 km/s/Mpc (Pade)
It is the primary fitted output of the combined dataset.
rd = 138.2 Mpc (Taylor), 137.0 Mpc (Pade)
It is the sound horizon fitted simultaneously with H0 and the cosmographic parameters.
q0 = -0.460 +/- 0.013 (Taylor full data)
It is the deceleration parameter in the cosmographic expansion.
j0 = 0.656 +/- 0.065 (Taylor full data)
It is the jerk parameter in the cosmographic expansion.
s0 = -0.437 +/- 0.050 (Taylor full data)
It is the snap parameter in the Taylor expansion; it is not constrained in the Pade (2,1) model.
M_B = unknown (marginalized)
It is the absolute magnitude of SNe Ia, marginalized over in the likelihood.

assumptions (4)

domain assumption FLRW metric and homogeneity and isotropy underlie the cosmographic parameterization.
The paper's distance formulas in Section 2 assume a FLRW universe, which is standard in cosmology but still a physical assumption.
ad hoc to paper The fourth-order Taylor expansion and Pade (2,1) approximant are faithful over 0.1 < z < 4.2.
The expansion order is chosen following prior cosmography papers, not derived from the data, and the paper itself warns about Taylor convergence at high z in Section 2.2.
ad hoc to paper Curvature Omega_k is either fixed or irrelevant; it is not marginalized.
Omega_k appears in Eqs. 3-4 but is absent from Table 1 and the text, so this is an unstated assumption that affects the model-independence claim.
domain assumption The H0LiCOW distance posteriors and DESI BAO covariance are used as published without additional systematic adjustment.
The analysis adopts the public lensing posterior chains and the DESI covariance; if those contain cosmology-dependent or incomplete error modeling, the final errors shift.

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Pith. "Pith review of Revisiting the Hubble constant, sound horizon and cosmography from late-time Universe observations." pith.science (2026). https://pith.science/paper/Y4KTREA3

@misc{pith2026250501661,
  author       = {Pith},
  title        = {Pith review of: Revisiting the Hubble constant, sound horizon and cosmography from late-time Universe observations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y4KTREA3}},
  note         = {Machine review of arXiv:2505.01661}
}

abstract

The Hubble tension has become one of the central problems in cosmology. In this work, we determine the Hubble constant $H_0$ and sound horizon $r_d$ by using the combination of Baryon Acoustic Oscillations (BAOs) from DESI surveys, time-delay lensed quasars from H0LiCOW collaborations and the Pantheon supernovae observations. We consider two cosmological approaches, i.e., Taylor series and Pad\'{e} polynomials, to avoid cosmological dependence. The reason for using this combination of data is that the absolute distance provided by strong gravitational lensing helps anchor the relative distance of BAO, and supernovae provide a robust history of universe evolution. {{Combining the 6 time-delay distance (6$D_{\Delta t}$) plus 4 angular diameter distance to the deflector (4$D_d$) measurements of time-delay lensed quasars,}} the BAO and the type Ia of supernovae (SNe Ia) datasets, we obtain a model-independent result of $r_d = 138.2_{-3.9}^{+3.3}$ Mpc and $H_0 = 72.9^{+1.8}_{-1.8}$ ${\mathrm{~km~s^{-1}~Mpc^{-1}}}$ for the Taylor series cosmography and $r_d = 137.0_{-3.7}^{+3.2}$ Mpc and $H_0 = 73.1_{-1.7}^{+1.8}$ ${\mathrm{~km~s^{-1}~Mpc^{-1}}}$ for the Pad\'{e} polynomials cosmography. The determination of $r_d$ and $H_0$ prefers larger $H_0$ and smaller $r_d$ than Planck data under the assumption of flat-$\Lambda$CDM model. However, the values of $H_0$ are consistent with the $H_0$ determination from SH0ES collaboration.

Figures

Figures reproduced from arXiv: 2505.01661 by the authors.

**Figure 1.** The constraints on 𝐻0, 𝑟𝑑 and the Taylor expansion cosmographic parameters using the combined DESI-BAO, Time-Delay Lensing and SN Ia. Contours represent the 68.3% and 95.5% confidence intervals. We show the results from different data combinations in different colors. Now let us remark on the sound horizon 𝑟𝑑 measurements from the different data combinations. The final result obtained combining all data gives 𝑟𝑑 = … view at source ↗

**Figure 2.** Same as [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** A comparison of Hubble constant values from different data combinations. using Padé approximation and discarding Taylor expansion in future analyses. 5. CONCLUSION In this work, we determine the Hubble constant 𝐻0, sound horizon 𝑟𝑑 and other cosmological parameter such as deceleration parameter from observations including recent DESIBAO, Time-Delay lensing and SN Ia without assuming particular cosmogical model. T… view at source ↗

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

Late-time cosmological constraints on three holographic dark energy models with DESI DR2 BAO and Type Ia supernovae
physics.gen-ph 2026-07 accept novelty 4.0 of 10

DESI DR2 and late-time data constrain HDE, ADE and RDE, yielding H0≈67–68 km/s/Mpc, c≈1, n≈2.8, γ≈0.54, with none resolving the Hubble tension or decisively beating ΛCDM.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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