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Late-time modifications to cosmic expansion can raise H0 by at most ~2–4 percent, far short of the ~9 percent needed to resolve the Hubble tension.

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2026-08-01 08:01 UTC pith:X6NQWDKD

load-bearing objection A careful, honest paper that turns the late-time no-go into a shape wall; the low-redshift wall is solid, the headline high-redshift bound leans on a model-guided tail. the 2 major comments →

arxiv 2607.21244 v1 pith:X6NQWDKD submitted 2026-07-23 astro-ph.CO gr-qchep-ph

Hubble tension: the shape wall

Miguel A. Sabogal , Luis A. Escamilla , Davide Pedrotti , Edvig Selimaj , Sunny Vagnozzi This is my paper
classification astro-ph.CO gr-qchep-ph
keywords Hubble tensionlate-time cosmologydimensionless expansion historydistance integralbaryon acoustic oscillationstype Ia supernovaeGaussian process reconstructionshape of expansion
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that late-time solutions to the Hubble tension are blocked not only by a normalization wall but by a 'shape wall': the requirement that any increase in H0 be matched by an equal fractional increase in the dimensionless distance integral I = ∫ dz/E(z). Reconstructing the dimensionless expansion history E(z) — the expansion rate relative to today's value — nonparametrically from supernova and BAO data, treating the BAO scale as an uncalibrated nuisance, the authors find that the directly probed part of the integral allows well under a percent increase in H0, and the total, including a model-guided estimate of the unprobed high-redshift tail, allows only ~2 percent (most conservatively) to ~3.7 percent. Since fully resolving the tension needs a ~9 percent shift, a purely late-time fix is ruled out, and even hybrid early-plus-late proposals are capped by this wall. The result matters because it converts vague skepticism about late-time solutions into a quantitative, shape-based ceiling.

Core claim

The central discovery is a simple ratio identity: with the CMB acoustic angular scale and the sound horizon fixed, a fractional increase in H0 equals the fractional increase in the dimensionless distance integral I ≡ ∫_0^{z★} dz/E(z). The paper splits I into a low-redshift part I<, directly constrained by data, and a high-redshift part I> that is not, and shows that Gaussian Process reconstructions of E(z) — built only from relative distances, with the BAO normalization marginalized — keep δI_</I_< below about 1.5 percent at 95% confidence for the combination of supernova and BAO data. Accounting for I> with smooth-model estimates yields δH0/H0 ≲ 2.07% (conservative) or ≲ 3.67% (maximally pe

What carries the argument

The dimensionless distance integral I = ∫ dz/E(z), whose perturbation at fixed acoustic scale and sound horizon obeys δH0/H0 ≃ δI/I (Eq. 8). The argument is carried by splitting I into a directly probed part I< (z < z_max ≈ 2.3) and an unprobed tail I>, weighting them by f< ≈ 0.42 and f> ≈ 0.58, and reconstructing I< nonparametrically from the derivative of the dimensionless comoving distance, 1/E(z), using Gaussian Process regression with a squared-exponential kernel and the BAO amplitude treated as a marginalized nuisance. This isolates pure shape information, so the bound is independent of any absolute calibration of distances.

Load-bearing premise

The ceiling on H0 depends on estimating the part of the distance integral above redshift ~2.3 — where we have no direct data — using smooth ΛCDM and wCDM models; if the real expansion history at those redshifts deviates more than those models allow, the ~2 percent bound no longer holds.

What would settle it

Measure the distance-redshift relation across z ≈ 2.3–3 (for example with high-redshift BAO, CMB lensing, or angular-diameter distances) at a few percent precision; if the implied δI_>/I_> exceeds ~5% at 95% confidence while still fitting low-redshift data, the shape wall is breached and a late-time resolution becomes viable again.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A purely late-time modification of the expansion history cannot resolve the Hubble tension: the allowed increase in H0 is roughly 2–4 percent, not the ~9 percent needed.
  • Even models that combine early-time new physics (which lowers the sound horizon) with late-time modifications are capped: the late-time piece cannot contribute more than the shape wall allows.
  • The wall cannot be evaded by a redshift-independent recalibration of BAO or supernova distances, because the analysis marginalizes over the absolute normalization.
  • Remaining loopholes narrow to new physics at redshifts z ≪ 0.01, redshift-dependent violations of the distance-duality relation, or sharp expansion features too rapid for the reconstruction to resolve.
  • The constraint is stable across three independent supernova samples, with the tightest limits coming from data combinations that include BAO.

Where Pith is reading between the lines

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

  • A sharper version of the wall could be derived from future data that extends the directly probed redshift range: adding a single percent-level distance measurement at z ≳ 2.5 would turn the model-guided tail estimate into a nonparametric one.
  • If the shape wall survives, the Hubble tension becomes essentially an early-universe problem; late-time physics, at most, can shave off a fraction of the discrepancy rather than close it.
  • The same δH0/H0 ≃ δI/I machinery could be turned around as a consistency test: calibrate H0 from the distance shape alone and compare with direct local measurements, yielding a model-agnostic cross-check of the tension's magnitude.
  • The treatment of I> is the main frontier: CMB geometric information at z★ could be injected as a boundary anchor, something the paper notes but leaves to follow-up work.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper derives a clean geometric identity: at fixed acoustic scale θ_s and fixed sound horizon r_s^*, a fractional increase in H_0 must be matched by the same fractional increase in the dimensionless distance integral I = ∫ dz/E(z) [Eq. (8)]. Splitting I into a directly probed low-redshift part I_< (z<z_max≈2.3) and an unprobed high-redshift tail I_>, the authors use Gaussian Process reconstructions of E(z) from unanchored PantheonPlus/DESY5Dvk/Union3 SNeIa and DESI DR2 BAO data, with the BAO normalization marginalized over, to obtain tight nonparametric upper limits on δI_</I_< (95% upper ≈1.5% for PantheonPlus+DESI). They then estimate the allowed variation of I_> by fitting ΛCDM and wCDM to the same late-time data, leading to total bounds δH_0/H_0 ≲2.07% (95%, conservative) and ≲3.67% (95%, maximally permissive), far below the ≈9% shift needed to fully resolve the Hubble tension. The paper concludes that a percent-level 'shape wall' prevents fully late-time solutions to the tension.

Significance. The central relation δH_0/H_0 ≃ δI/I is a useful and elegant reformulation of the constraints imposed by fixed θ_s and r_s^*. The nonparametric GP reconstruction is careful and well cross-checked: the authors verify kernel choice, propagate hyperparameter uncertainties, preserve correlations across redshifts, and confirm the low-z results with an independent node-based method (Appendix B). The directly constrained low-redshift wall, (δH_0/H_0)_< ≲0.8% at 95% for the baseline PantheonPlus+DESI combination, is a robust and important result that strengthens the case against late-time resolutions. However, the headline full-integral bounds depend on model-guided estimates of the high-redshift contribution I_>, and the paper's own Sec. V acknowledges that these may be evaded. The paper is transparent about this limitation, but the abstract and conclusions present the percent-level wall for the full post-recombination epoch as a stronger statement than the nonparametric analysis alone supports.

major comments (2)
  1. [§V, Tab. II, Eq. (10)] The headline bounds δH_0/H_0 ≲2.07% (conservative) and ≲3.67% (maximally permissive) are not nonparametric: they rely on ΛCDM and wCDM fits to constrain δI_>/I_> for the z>2.33 tail. Because f_>≈0.58, a smooth modification of E(z) that lowers it only for z>2.33 could increase I_> by ≈15% and thereby produce δH_0/H_0≈9% while leaving the z<2.33 data essentially unchanged; CMB θ_s fixes only the combination r_sH_0/I, not I_> independently. The abstract's claim that the shape wall 'prevents a fully late-time solution' is therefore too strong unless explicitly conditioned on the assumed smooth two-parameter families. I recommend either softening the abstract/conclusions to state the model-dependence, or adding an analysis using a broader class of smooth high-z extensions (or CMB geometrical information) to bound I_> in a less model-dependent way.
  2. [§V, Tab. II] The 'maximally permissive' benchmark still assumes wCDM with constant w. This two-parameter family does not span the space of smooth high-z expansion histories; a smooth transition in w(z), a mild curvature term, or a redshift-dependent modification to the radiation/matter content above z_max could produce larger δI_> than allowed by these fits. The text labels the I_> estimates as model-guided, but the Abstract and Conclusions do not carry this caveat. The distinction between a direct low-z wall (robust) and a conditional full-post-recombination wall (model-dependent) should be made prominent in the summary of results.
minor comments (5)
  1. [§V (text around Tab. II)] Typo: in the paragraph introducing the conservative benchmark, the second inequality reads 'δI_</I_< ≲1.22% (2.48%)' but should refer to δI_>/I_>.
  2. [Table II caption] Typo: 'with the with the wCDM-based model-guided limits' should read 'with the wCDM-based model-guided limits'.
  3. [Abstract] The abstract uses '≲2%' while the body quotes 2.07% at 95% and 3.67% in the maximally permissive case; the abstract should either quote the range or explicitly state which benchmark is being cited.
  4. [Appendix A, Eq. (A6)] The redshift uncertainty propagation for SNeIa distances neglects the derivative of D_L with respect to z; for consistency with the stated first-order propagation, the expression should include the full Jacobian or state the approximation explicitly.
  5. [§III.B / Appendix A] The use of a zero-mean GP prior for the dimensionless distance D_M(z) is unusual because D_M(z) is positive and increasing; the authors should justify more explicitly why this choice does not bias the reconstruction, particularly at high z where data are sparse, given that the derived E(z) is obtained by differentiating the GP.

Circularity Check

0 steps flagged

No significant circularity: Eq. (8) is an exact identity and the low-redshift wall is nonparametric; the high-redshift I> estimate is an explicitly acknowledged model-guided extrapolation, not a circular reduction.

full rationale

The central relation δH0/H0 ≃ δI/I (Eq. 8) is derived from the exact identity θs = rsH0/I (Eq. 5) by setting δθs = δrs = 0 (Eq. 7). This is algebraic, not a fit, and does not define H0 in terms of I by construction beyond the physical constraint that a fixed sound horizon and fixed acoustic scale force the relative changes to match. The low-redshift constraint δI</I< is obtained nonparametrically: PantheonPlus/DESY5Dvk/Union3 SNeIa are used as relative distances, the BAO normalization rdh is explicitly marginalized over (Sec. III and App. A), and the GP reconstruction is conditioned only on z<zmax data. Thus the sub-percent low-z wall (Tab. I) is self-contained and not fitted to the target quantity. The only potentially load-bearing step is the high-redshift contribution δI>/I>, which is estimated by fitting ΛCDM/wCDM to the same PantheonPlus+DESI data (Sec. V) and then combined through the identity Eq. (10) to produce the headline bounds 2.07% (conservative) and 3.67% (maximally permissive). The paper explicitly flags this: 'our limits on I> are model-guided, and may in principle be evaded' (Sec. V). This is a model-dependence/extrapolation limitation, not a claim that a fitted parameter predicts itself: the I> posterior is a derived quantity from an assumed smooth parametric family, and the paper does not present it as a nonparametric first-principles bound. No self-citation uniqueness theorem is invoked, no ansatz is smuggled via a citation, and no known result is merely renamed. The self-citations (e.g. Refs. [198, 200, 295]) are contextual and do not carry the derivation. Therefore there is no circular step; the honest finding is that the low-z wall is robust, while the full-integral wall inherits the acknowledged model-dependence of the I> estimate.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The paper introduces no new physical entities. Its free parameters are the GP hyperparameters, the BAO normalization nuisance parameter, and the parameters of the ΛCDM/wCDM fits used for the I_> estimate. The main result depends on standard cosmological assumptions (flatness, distance duality, smoothness) plus the model-guided treatment of the unobserved high-redshift part of the integral.

free parameters (6)
  • GP kernel amplitude σ_f
    Fitted jointly with the GP reconstruction via MCMC with flat priors; posterior not reported in text. Controls the flexibility of the reconstructed expansion history.
  • GP correlation length ℓ
    Fitted jointly with the GP reconstruction via MCMC; controls the smoothness scale of E(z).
  • BAO normalization r_dh = marginalized
    Nuisance parameter for the BAO amplitude (α = H0 r_d / c), sampled and marginalized over to isolate shape information; no external prior used.
  • Ωm (ΛCDM/wCDM fits for I_>)
    Fit to PantheonPlus+DESI data to estimate the allowed variation of I_>; posterior width is used as an upper limit.
  • w (wCDM fit for I_>)
    Constant dark-energy equation of state fit to PantheonPlus+DESI data for the 'maximally permissive' estimate of δI_>/I_>.
  • Fiducial Ωm = 0.31 = 0.31 (from Planck ΛCDM)
    Assumed fiducial cosmology used to define δI_</I_< and the weights f_<, f_>; chosen by hand from Planck, not fitted in this analysis.
axioms (6)
  • domain assumption Spatially flat FLRW metric
    Assumed throughout (Sec. II). Non-zero spatial curvature could weaken the constraints, as noted in Sec. V.
  • domain assumption Distance-duality relation holds
    Assumed when converting SNeIa luminosity distances to comoving distances (Sec. III.B, App. A). Violations would alter the comparison of SNeIa and BAO data.
  • domain assumption The expansion history is sufficiently smooth
    GP reconstruction assumes smoothness; sharp features below the data redshift resolution could evade the shape wall (Sec. V).
  • domain assumption θ_s is fixed to extremely high precision
    Required for δθ_s/θ_s ≈ 0 in Eq. (7); motivated by CMB observations.
  • domain assumption r_s (and r_d) unchanged for pure late-time solutions
    Definition of a late-time solution; the analysis constrains only the late-time contribution in hybrid models.
  • ad hoc to paper ΛCDM and wCDM provide reasonable models for the high-redshift tail I_>
    Used to estimate δI_>/I_> since no direct data exist above z_max. The paper acknowledges this is model-guided, not nonparametric.

pith-pipeline@v1.3.0-alltime-deepseek · 40896 in / 12456 out tokens · 120979 ms · 2026-08-01T08:01:26.791281+00:00 · methodology

0 comments
read the original abstract

The standard "no-go theorem" against late-time solutions to the Hubble tension is essentially a normalization wall, since Baryon Acoustic Oscillation (BAO) measurements constrain the product $H_0r_d$, with $r_d$ the sound horizon at baryon drag. However, late-time solutions (which keep $r_d$ fixed) are tightly constrained not only by the BAO normalization $H_0r_d$, but also by the shape of the expansion history, i.e. the dimensionless expansion rate $E(z) \equiv H(z)/H_0$. We show that, if $r_d$ and the acoustic angular scale $\theta_s$ are fixed, an increase in $H_0$ needs to be matched by an equal fractional increase in the dimensionless distance integral $I \equiv \int dz/E(z)$: $\delta H_0/H_0 \simeq \delta I/I$. We use this to quantify the "shape wall" set by relative distance constraints on $E(z)$, which we reconstruct nonparametrically, using Gaussian Processes and the latest unanchored Type Ia Supernovae (SNeIa) and BAO data. In our most conservative analysis using PantheonPlus SNeIa and DESI DR2 BAO data, we find a maximum fractional increase in $H_0$ of $\lesssim 2\%$, falling well short of the $\gtrsim 8\%$ required to solve the tension. This shape wall holds even in the presence of early-time new physics, and limits the maximum increase in $H_0$ which can be contributed by late-time modifications to $E(z)$ at fixed $\theta_s$. Therefore, late-time modifications, whether invoked alone or alongside early-time new physics, face not only the well-known normalization wall, but also a stringent percent-level shape wall.

Figures

Figures reproduced from arXiv: 2607.21244 by Davide Pedrotti, Edvig Selimaj, Luis A. Escamilla, Miguel A. Sabogal, Sunny Vagnozzi.

Figure 1
Figure 1. Figure 1: FIG. 1. Posterior reconstructions of the dimensionless expansion rate relative to the fiducial ΛCDM model, [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. 1D marginalized posterior distributions for [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. As in the three upper sub-panels of Fig. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. As in the three upper sub-panels of Fig. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

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