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REVIEW 3 major objections 4 minor 1 cited by

The Hubble-Lema\^{i}tre constant and sound horizon from low-redshift probes

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Leaving the sound horizon free, a lens-calibrated low-redshift ladder returns H0 = 72 ± 7 and r_s = 137 ± 4.5 Mpc, below the CMB value.

desk verdict Useful, honest inverse-distance-ladder analysis, but the advertised sound-horizon tension is largely inherited from the H0LiCOW H0 prior rather than independently established by the low-redshift data. read the letter →

arxiv 1908.02401 v1 pith:6QIOE2HY submitted 2019-08-06 astro-ph.CO

Rados{l}aw Wojtak , Adriano Agnello This is my paper · ORCID
classification astro-ph.CO
keywords Hubbleconstanttensionsoundhorizoninversedistanceladdertime-delaylensesbaryonacousticoscillationsTypeIasupernovaecosmography
topics Dark Energy
open problems Dark Energy
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

Measuring the expansion rate H0 from low-redshift data usually requires assuming the sound horizon scale r_s from the cosmic microwave background. This paper removes that assumption. Using angular-diameter distances to three gravitationally lensed quasars to set the absolute scale, supernova and BAO data to trace relative distances, and a fourth-order cosmographic fit independent of any cosmological model, it obtains H0 = 72 ± 7 km/s/Mpc and H0 r_s = 9895 ± 161 km/s. Combined with the lens-based H0 measurement, this gives r_s = 137.0 ± 4.5 Mpc, systematically lower than the CMB-inferred value of about 147 Mpc. If correct, the early-versus-late universe tension is not an artifact of fixing r_s from the CMB; it is a genuine discrepancy in the cosmological model or in the distance calibration.

What carries the argument

The machinery is the inverse distance ladder built on a fourth-order polynomial expansion of the expansion history H(z)/H0 in redshift, with free coefficients for deceleration, jerk, snap, and curvature, together with the distance duality relations D_lum = (1+z)^2 D_ang = (1+z) D_M. Supernova and BAO relative distances set the shape of this expansion, while three lens angular-diameter distances fix the absolute scale; because r_s is left as a free parameter, the BAO data constrain the product H0 r_s rather than H0 alone. The lens distances carry about 80 percent of the error budget, so this absolute calibration is the load-bearing part of the argument.

What would settle it

Obtain spatially-resolved stellar kinematics for the three lens galaxies to determine their velocity-anisotropy profiles without the assumed Osipkov-Merritt form, then recompute the three angular-diameter distances; if the corrected distances move H0 r_s to a CMB-compatible value around 147 Mpc, the claimed low r_s dissolves.

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Extended reading notes

Core claim

The paper shows that when r_s is left free and the distance ladder is calibrated by three time-delay lens angular-diameter distances, the low-redshift data independently prefer H0 = (72 ± 7) km/s/Mpc and H0 r_s = (9895 ± 161) km/s. Combining the latter with the lens-based H0 = 72.$5^{{+2.1}}$_{-2.3} km/s/Mpc yields r_s = (137.0 ± 4.5) Mpc, roughly 10 Mpc below the CMB-inferred value of about 147 Mpc. Because the analysis uses a cosmographic expansion and distance duality rather than a specific dark-energy model or a Cepheid calibration, the authors conclude that the H0/r_s tension reflects either new physics beyond standard cosmology or systematic errors in the low-redshift distance calibration, not the choice of r_s prior.

Load-bearing premise

The three lens angular-diameter distances are unbiased absolute calibrators, which requires the assumed stellar-orbit anisotropy in each lens to be correct and lensing mass to equal dynamical mass; these distances dominate the error budget.

Editorial extensions

If this is right

  • The early-to-late universe tension persists even when r_s is not pinned to the CMB, so it cannot be dismissed as a prior artifact.
  • Percent-level low-redshift H0 measurements, for example from future time-delay lens samples of ten to forty systems, would discriminate between r_s around 137 Mpc and around 147 Mpc.
  • The analysis is independent of the assumed dark-energy model because the fourth-order cosmographic expansion and distance duality are the only geometry assumptions.
  • Removing supernovae with z < 0.1 changes the results negligibly, so low-redshift supernova systematics and local peculiar velocities are not driving the outcome.
  • The low-redshift 6dFGS BAO distance is inconsistent with other distance scalings unless H0 is around 60 km/s/Mpc, which may explain the lower H0 seen in earlier inverse-ladder analyses.

Reading between the lines

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

  • If future spatially-resolved lens kinematics remove the mass-anisotropy degeneracy and shift the lens distances upward, as the two-parameter anisotropy reassessment suggests, the inferred H0 would rise and the tension with the CMB would strengthen rather than dissolve.
  • The same distance-ratio test shown in the paper could be applied to gravitational-wave standard sirens, turning catalog-level data into a redshift-resolved check for distance-dependent systematics.
  • A robust low r_s from this ladder would favor early-universe solutions such as extra relativistic species or a smaller recombination scale, while a systematics explanation would point to anisotropic stellar distributions in the lens galaxies.
  • The product H0 r_s ≈ 9895 km/s, being independent of absolute calibration, is the sharpest number in the paper; improving BAO shape measurements could pin it down even before lens distances improve.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper revisits the H0 and sound-horizon tension using an inverse distance ladder calibrated with angular-diameter distances to three H0LiCOW lenses, combined with JLA Type Ia supernovae and BAO measurements from BOSS, 6dFGS+SDSS-MGS, and WiggleZ, while leaving rs free. The expansion history is parameterized with a fourth-order cosmographic series, making the analysis independent of the Einstein field equations. The authors find H0rs = (9895 ± 161) km/s and H0 = (72 ± 7) km/s/Mpc from the lens-calibrated ladder, and, after combining H0rs with the H0LiCOW time-delay H0 = (72.5 ± 2.3) km/s/Mpc, they obtain rs = (137.0 ± 4.5) Mpc, which they interpret as systematically lower than the CMB-inferred value of about 147 Mpc. The paper also examines robustness to removing low-redshift supernovae and to BAO reconstruction choices.

Significance. If the claimed result were fully supported, it would be important because it would locate the late-time/early-time discrepancy in the absolute distance scale and the sound horizon rather than in the CMB-fixed value of rs, with implications for new physics or systematics in distance calibrations. The paper is transparent about the lens-anisotropy assumption and provides a useful comparison of results from different BAO data combinations. The likelihood treatment is standard, the data choices are clearly described, and the paper explicitly separates the model-independent combination H0rs from the absolute calibration step. However, as discussed in the major comments, the central claim of a tension in rs is not established by the inverse distance ladder alone: the paper's own H0 uncertainty of ±7 km/s/Mpc propagates to a ±14 Mpc uncertainty on rs, which is consistent with the CMB value, and the headline 4.5 Mpc error is achieved only by importing the H0LiCOW H0, whose lens-model systematics are not included in the error budget.

major comments (3)
  1. [§3, Table 2] The central claim that low-redshift data yield a sound horizon systematically lower than the CMB value is not supported by the inverse distance ladder alone. Table 2 gives H0rs = (9895 ± 161) km/s and H0 = (72 ± 7) km/s/Mpc from the lens-calibrated ladder; dividing the former by the latter gives rs ≈ 137 ± 14 Mpc, which is within about 0.7σ of the Planck value rs = (147.05 ± 0.30) Mpc. The quoted rs = (137.0 ± 4.5) Mpc is obtained by instead dividing by the external H0LiCOW H0 = (72.5 ± 2.3) km/s/Mpc, whose lens-model assumptions are the same as those entering the three calibrating angular-diameter distances. The manuscript should state this dependence explicitly and present the propagated rs from Table 2 alongside the H0LiCOW-combined value.
  2. [§2.3, Table 1] The three angular-diameter distances are the sole absolute calibrators and dominate the error budget, but the quoted uncertainties do not include the model-form systematic of the assumed Osipkov-Merritt stellar-anisotropy profile or the assumed equality of lensing and dynamical mass. The paper itself notes that Jee et al. (submitted) obtain slightly smaller D_ang with more general anisotropy families, which would increase H0 and decrease rs. Because the external H0LiCOW H0 used for the headline rs shares these same assumptions, a bias of only 2–3 km/s/Mpc in H0 (corresponding to Δrs ≈ 4–5 Mpc) would erase the claimed tension. A quantitative estimate of this systematic needs to be included before the 4.5 Mpc error can be treated as the dominant uncertainty.
  3. [§2, Eqs. (1)–(4)] The fourth-order cosmographic expansion is fitted to BOSS BAO at z = 0.61 and to a lens at z = 0.745 without any estimate of truncation error. At these redshifts the expansion converges slowly, and the claim that the results are independent of the adopted expansion history requires a demonstration that higher-order terms (or a different model of H(z)) do not shift H0rs and H0 by more than the quoted errors. Please add a quantitative truncation test, for example by comparing fits with fifth- and sixth-order terms or by examining residuals.
minor comments (4)
  1. [Introduction and Discussion] The reference to 'Barnal et al. (2016)' should be 'Bernal et al. (2016)' in both occurrences.
  2. [§2.4] The prior ranges on the model parameters θ = (H0, rs, Ωk, q0, j0, s0, M) are not stated; please specify them, since the text reports uniform priors on rs but not the bounds on the other parameters.
  3. [Figure 1] The y-axis label 'cln(1+z)/DM' should be typeset as 'c ln(1+z)/D_M' for clarity.
  4. [§4.2] The statement that consistency of the 6dFGS DV point with other distance scalings would require H0 = (60 ± 6) km/s/Mpc is presented without derivation; given that this point is included in the fiducial fit, a short explanation of how this number was obtained would help the reader assess its impact.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the analysis is a direct fit to published distance catalogs, and the sound-horizon combination is arithmetic rather than a self-imported result.

full rationale

The derivation chain is a standard inverse-distance-ladder fit. Angular-diameter distances to three lenses (Table 1, from Jee et al. 2015 and Birrer et al. 2019) set the absolute scale; BAO and JLA SNe constrain relative distance scaling; and the polynomial H(z) parameterization (Eqs. 1-4, following Visser and Macaulay et al.) is an independent cosmographic ansatz, not one that encodes H0 or rs. The paper openly fits H0 and H0rs as free parameters (Eq. 6), so H0 = (72 +/- 7) and H0rs = (9895 +/- 161) are outputs, not inputs renamed as predictions. The subsequent rs = (137.0 +/- 4.5) Mpc is obtained by dividing the fitted H0rs by the external H0LiCOW H0 value; this is arithmetic, not an equation-level identity, and the paper does not dress it up as a first-principles prediction. The main statistical caveat is that the three D_ang calibrators and the H0LiCOW H0 are not fully independent, since both come from the same lens systems and share lensing/dynamical modeling assumptions; the paper itself flags the mass-anisotropy sensitivity in Section 2.3 and notes that updated distances from Jee et al. (subm.) would shift H0. That caveat concerns systematic independence, not definitional circularity. Self-citations (e.g., Shajib et al. 2018, Wojtak et al. 2014) appear only as auxiliary forecasts or robustness checks, not as load-bearing inputs. No fitted parameter is renamed as a prediction, no prior work by the authors is invoked to forbid alternatives, and no calibration parameter is defined in terms of the target quantity. Therefore the circularity score is 0.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The analysis rests on a modest number of standard cosmological assumptions (distance duality, lens-dynamical mass equality, anisotropy profile) and fits seven free parameters to public data. No new physical entities are introduced. The main burdens are the cosmographic truncation and the lens calibration assumptions, which dominate the final uncertainty.

free parameters (7)
  • H0 = 72.0 ± 6.7 km/s/Mpc (main free-r_s fit)
    Central parameter of the fit; absolute normalization inferred from lens angular-diameter distances extrapolated to z=0.
  • r_s = 138.6 ± 13.0 Mpc (free fit; also reported as H0 r_s = 9895 ± 161 km/s)
    Sound horizon left free; the product H0 r_s is effectively constrained by BAO and SNe, while r_s alone is weakly constrained without CMB input.
  • Omega_k = 0.26 ± 0.33
    Curvature term in the cosmographic distance expansion (Eq. 3).
  • q0 = -0.47 ± 0.15
    Deceleration parameter in the fourth-order H(z) expansion (Eq. 1).
  • j0 = 1.01 ± 1.41
    Jerk parameter in the expansion (Eq. 1).
  • s0 = 1.35 ± 3.3
    Snap parameter in the expansion (Eq. 1).
  • M (SN absolute magnitude normalization) = not quoted
    Free normalization of the SN Hubble diagram; absorbs the absolute magnitude and the log of H0.
assumptions (6)
  • domain assumption The fourth-order cosmographic expansion (Eqs. 1-4) accurately represents H(z) and distances over the redshift range used (z < ~0.8).
    The paper adopts the Macaulay et al. (2018) parameterization and does not quantify truncation error at higher order; convergence of the series at z=0.8 is assumed.
  • domain assumption Distance duality relation Dlum = (1+z) Dang = (1+z)^2 DM holds.
    Section 2: the authors invoke Etherington (1933) and state it holds for GR and a wide class of metric theories; if violations exist (e.g., modified gravity with photon non-conservation), the extrapolation changes.
  • domain assumption Equality of lensing mass and dynamical mass in the deflector galaxies.
    Section 4.3: the Dang distances from lensing plus kinematics rely on this; GR ensures it, but f(R) and other theories violate it. Current systematic uncertainty is about 20% per system.
  • domain assumption Osipkov-Merritt anisotropy profile for stellar kinematics in the lens galaxies.
    Section 2.3: the published lens distances assume this form; more general two-parameter anisotropy families change the inferred distances and hence H0 (Jee et al., subm.).
  • domain assumption Independence of BAO surveys (BOSS, 6dF+SDSS MGS, WiggleZ) and no correlation between their reported covariance matrices.
    Section 2.4: the likelihood sums the chi-square contributions assuming independent data sets; any overlooked systematics would alter the error budget.
  • domain assumption JLA SN light-curve fitting and systematic corrections from Betoule et al. (2014) are valid.
    The SN distance moduli are taken from JLA with their published covariance matrix; this is a standard input.

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Cite this review

Pith. "Pith review of The Hubble-Lema\^{i}tre constant and sound horizon from low-redshift probes." pith.science (2026). https://pith.science/paper/6QIOE2HY

@misc{pith2026190802401,
  author       = {Pith},
  title        = {Pith review of: The Hubble-Lema\^itre constant and sound horizon from low-redshift probes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6QIOE2HY}},
  note         = {Machine review of arXiv:1908.02401}
}
abstract

We revisit the claimed tension, or lack thereof, of measured values of the Hubble-Lema\^{i}tre parameter $H_0$ from Cosmic Microwave Background (CMB) data and low-redshift indicators. Baryon Acoustic Oscillations (BAO) rely on the scale of the sound horizon at recombination $r_s$ to convert angular measurements into angular-diameter distances, so fixing $r_s$ from CMB measurements already constrains $H_0.$ If departures from concordance cosmology are to be constrained, truly independent measurements of $H_0$ are needed. We use the angular-diameter distances to three time-delay lenses from the H0LiCOW collaboration to calibrate the distance ladder, combine them with relative distances from Supernovae Ia and BAO, leaving $r_s$ completely free, and provide the inferred coefficients ($q_{0},j_{0},s_{0}$) in the polynomial expansion of H(z). We obtain $H_{0}r_{s}=(9895\pm161)$km/s and $H_0=(72\pm7)$km/s/Mpc. Combined with $H_0$ from H0LiCOW, then $r_s=(137\pm4.5)$Mpc is consistent with previous work and systematically lower than the CMB-inferred value. Our results are independent of the adopted cosmology, and removing Supernovae with z<0.1 has a negligible effect.

Figures

Figures reproduced from arXiv: 1908.02401 by the authors.

Figure 1
Figure 1. Cosmographic inference from the inverse distance ladder, cali￾brated on three angular-diameter distances from lensing and extrapolated via BAO and SNe Ia. The top panel shows the combined inference on H0 and rs, without priors on rs from the CMB. Both the CMB (green, rs ≈ 147 Mpc) and low-redshift measurements H0LiCOW (grey bands) and SH0ES (orange lines) are compatible within the current uncertainty level. The bott… view at source ↗
Figure 2
Figure 2. Consistency between relative luminosity distances from Type-Ia SNe, angular diameter distances from lensing observations (left) and comoving distances from BAO observations. The dashed lines indicate equal values of the quantities on both axes. 4.2 Reliability of Low-Redshift Probes All of the above measurements rely on fundamental relations be￾tween cosmological distances. In particular, in utter generality Dlum = … view at source ↗

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Cited by 1 Pith paper

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

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