arxiv: 2605.05691 · v1 · submitted 2026-05-07 · 🌌 astro-ph.CO

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From Scalar H₀ to E(z): A Reformulation of the Hubble Tension

Seokcheon Lee

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:43 UTC · model grok-4.3

classification 🌌 astro-ph.CO

keywords Hubble tensionexpansion historyE(z)flat-LambdaCDMDESI DR2Pantheon+SH0ESPlanck CMBBAO

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The pith

Reformulating the Hubble tension as a mismatch in E(z) histories rather than scalar H0 reduces the discrepancy from 4.9σ to roughly 2σ.

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

The paper claims that expressing the tension solely through the scalar H0 value mixes together distinct inference problems from different probes. By instead mapping each probe's posterior to an implied dimensionless expansion history E(z) = H(z)/H0 inside a shared flat-ΛCDM model, the redshift-dependent shape can be compared separately from the overall normalization. The resulting E(z) curves from Planck, DESI DR2, and Pantheon+SH0ES differ at the 1-2σ level pointwise, with global mismatches quantified by a covariance-subspace statistic S_hist of 1.65 and 2.55 respectively. These values correspond to effective significances of about 1.1σ and 2.1σ once correlations are accounted for, both lower than the conventional scalar-H0 tension.

Core claim

Within a common flat-ΛCDM framework, each probe posterior is mapped onto posterior-implied E(z) histories. The reconstructed values E(z_k) are strongly correlated across redshift, so the global mismatch is quantified with a covariance-subspace history displacement S_hist, alongside pointwise redshift differences. The histories are not identical, but the discrepancies are moderate: the pointwise significance is typically 1-2σ, while S_hist ≃ 1.65 for DESI DR2 and S_hist ≃ 2.55 for Pantheon+SH0ES relative to Planck. With two retained covariance eigenmodes, these correspond to two-sided one-dimensional Gaussian equivalents of approximately 1.1σ and 2.1σ, both below the conventional ≃ 4.9σ 0.1σ.

What carries the argument

The dimensionless expansion history E(z) = H(z)/H0, which isolates the redshift-dependent shape of the expansion from the absolute scale set by H0, together with the covariance-subspace history displacement S_hist that measures global mismatch after accounting for correlations across redshift bins.

If this is right

The absolute-scale discrepancy remains but is isolated from shape differences in the expansion history.
Different probes constrain distinct aspects: physical densities and acoustic scales for CMB, ruler-normalized distances for BAO, and calibrated luminosity-distance relations for supernovae.
Pointwise differences stay at 1-2σ while the global statistic drops to 1.1σ and 2.1σ equivalents.
The reformulation keeps the tension inside flat-ΛCDM but shows it is not as severe when the full functional form is compared.

Where Pith is reading between the lines

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

Future high-precision measurements of E(z) at intermediate redshifts could tighten constraints on shape differences independently of H0 calibration.
The same separation of normalization from shape could be applied to other tensions involving multiple parameters, such as those in dark energy or neutrino mass.
If the moderate E(z) mismatches persist with larger datasets, it would point toward systematics in how each probe's covariance affects the implied history rather than a fundamental model breakdown.

Load-bearing premise

Each probe's posterior can be mapped onto a well-defined implied E(z) history inside flat-ΛCDM without the mapping itself being dominated by the original H0 marginalization or by probe-specific data cuts.

What would settle it

An independent reconstruction of E(z) from each dataset that yields S_hist values remaining above 4σ equivalent after covariance correction would falsify the claim of moderate discrepancy.

Figures

Figures reproduced from arXiv: 2605.05691 by Seokcheon Lee.

**Figure 1.** Figure 1: Reconstructed dimensionless expansion histories in flat base ΛCDM. The top panel shows the view at source ↗

read the original abstract

The Hubble tension is usually expressed as a discrepancy between the low H_0 inferred from Planck CMB data within base \LambdaCDM and the higher value obtained from late-time distance-ladder measurements. This scalar comparison compresses distinct inference problems into one derived parameter: Planck CMB, DESI DR2 BAO, and Pantheon+SH0ES constrain physical densities and acoustic scales, ruler-normalized distances, and calibrated luminosity-distance relations, respectively. We reformulate the comparison in terms of the dimensionless expansion history E(z)=H(z)/H_0. This does not remove the absolute-scale discrepancy, but separates the normalization encoded in $H_0$ from the redshift-dependent shape of the expansion history. Within a common flat-\LambdaCDM framework, each probe posterior is mapped onto posterior-implied E(z) histories. Since the reconstructed values E(z_k) are strongly correlated across redshift, we quantify the global mismatch with a covariance-subspace history displacement S_{hist}, alongside pointwise redshift differences. The histories are not identical, but the discrepancies are moderate: the pointwise significance is typically 1-2\sigma, while S_{hist} simeq 1.65 for DESI DR2 and S_{hist} \simeq 2.55 for Pantheon+SH0ES relative to Planck. With two retained covariance eigenmodes, these correspond to two-sided one-dimensional Gaussian equivalents of approximately 1.1\sigma and 2.1\sigma, both below the conventional \simeq 4.9\sigma Planck-SH0ES scalar-H_0 discrepancy.

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

The paper reframes the Hubble tension as a comparison of E(z) histories and finds milder discrepancies than the raw H0 one, but the mapping may not isolate shape information independently of the original scale tension.

read the letter

This paper takes the existing Hubble tension and rewrites the comparison in terms of the normalized expansion history E(z) = H(z)/H0 instead of the scalar H0 value alone. The central result is that the mismatch between Planck and the late-time probes drops to roughly 1-2 sigma once you look at the full E(z) curves and apply their new S_hist statistic on the covariance subspace. That is the one takeaway worth noting right away: the tension looks less severe in this framing, though the absolute scale discrepancy remains untouched by construction. What the work does cleanly is separate the normalization piece from the redshift-dependent shape and then quantify the shape part with a single number. They take published posteriors from Planck, DESI DR2, and Pantheon+SH0ES, map each to its implied E(z) history inside flat LCDM, and show that pointwise differences stay modest while the global S_hist values sit at 1.65 and 2.55. The reduction from the usual 4.9 sigma H0 tension is presented as evidence that most of the problem sits in the scale rather than the shape. The calculation itself is straightforward and stays within standard data and the standard model, which keeps it easy to follow. The soft spot is that E(z) is defined by dividing out H0, so for late-time probes whose data directly encode a high H0 preference, the mapping can shift the effective Omega_m posterior in a way that is not independent of the original tension. Since flat LCDM fixes the entire E(z) shape by Omega_m alone, the exercise largely reduces to comparing those Omega_m posteriors after marginalization over H0; any reduction in significance could therefore be partly definitional rather than a clean separation of physical effects. The abstract does not spell out checks against that possibility. This is the sort of reframing that might help people who are already dissecting the tension into scale versus shape components. It does not add new observables or predictions, so its value is organizational rather than discovery-driven. I would send it to referees so they can examine the posterior-to-E(z) mapping and confirm that S_hist is not absorbing part of the H0 discrepancy by construction.

Referee Report

2 major / 2 minor

Summary. The paper claims that reformulating the Hubble tension in terms of the dimensionless expansion history E(z) = H(z)/H0 within flat-ΛCDM allows mapping of posteriors from Planck, DESI DR2, and Pantheon+SH0ES onto implied E(z) histories; these histories show only moderate discrepancies (pointwise typically 1-2σ), quantified globally via a covariance-subspace history displacement S_hist (≈1.65 for DESI DR2 and ≈2.55 for Pantheon+SH0ES relative to Planck), which with two retained eigenmodes correspond to Gaussian equivalents of ≈1.1σ and ≈2.1σ, substantially below the conventional 4.9σ scalar-H0 tension.

Significance. If the posterior-to-E(z) mapping is shown to isolate redshift-dependent shape information independently of H0 marginalization, the result would usefully separate normalization from expansion-history shape and could guide targeted model-building for the tension. The S_hist construction provides a concrete, covariance-aware metric for comparing correlated E(z) reconstructions across probes, which is a methodological strength.

major comments (2)

[posterior-to-E(z) mapping section] Abstract and the section on posterior mapping: In flat-ΛCDM, E(z) is a deterministic function of Ω_m alone, so the mapping reduces to translating each probe's marginal Ω_m posterior; however, for Pantheon+SH0ES the joint (H0, Ω_m) posterior is shaped by the absolute calibration that encodes the original H0 tension, and the manuscript does not demonstrate that the resulting Ω_m (hence E(z)) posterior is independent of this calibration. This leaves open whether the reported drop from 4.9σ to ~2.1σ is physical or partly definitional.
[S_hist construction paragraph] The paragraph introducing S_hist and the eigenmode truncation: The reduction of S_hist to two-sided one-dimensional Gaussian equivalents of 1.1σ and 2.1σ relies on retaining only two covariance eigenmodes, but no explicit test of sensitivity to the number of retained modes or justification that higher modes are negligible for the global mismatch is provided; without this, the quantitative claim that the E(z) tension is “below the conventional ≃4.9σ” cannot be fully assessed.

minor comments (2)

The abstract uses “s ≃” for the S_hist values; consistent notation (e.g., “≈” or “≃”) should be adopted throughout the text and figures.
Figure captions for the E(z) reconstructions should explicitly state the number of eigenmodes retained and the redshift range over which pointwise significances are evaluated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and valuable suggestions. The comments highlight important aspects of our methodology that we will clarify in the revised manuscript. Below we provide point-by-point responses to the major comments.

read point-by-point responses

Referee: Abstract and the section on posterior mapping: In flat-ΛCDM, E(z) is a deterministic function of Ω_m alone, so the mapping reduces to translating each probe's marginal Ω_m posterior; however, for Pantheon+SH0ES the joint (H0, Ω_m) posterior is shaped by the absolute calibration that encodes the original H0 tension, and the manuscript does not demonstrate that the resulting Ω_m (hence E(z)) posterior is independent of this calibration. This leaves open whether the reported drop from 4.9σ to ~2.1σ is physical or partly definitional.

Authors: We agree with the referee that in flat-ΛCDM the function E(z) depends only on Ω_m, making the mapping equivalent to using the marginal posterior on Ω_m from each dataset. For the Pantheon+SH0ES sample, the absolute calibration from SH0ES does influence the joint posterior, including the marginal on Ω_m. However, our reformulation intentionally separates the absolute normalization (H0) from the shape of the expansion history (encoded in Ω_m). The conventional 4.9σ tension is a comparison of H0 values, whereas S_hist measures the mismatch in the redshift evolution after removing the overall scale. The lower significance in E(z) is thus a direct consequence of isolating the shape information, which is physical in the sense that it shows the data are more consistent on the expansion history shape than on the absolute scale. We will revise the abstract and the relevant section to explicitly state that the Ω_m posterior for Pantheon+SH0ES includes the effects of the calibration, and to emphasize that the reduced tension reflects the separation of normalization from shape rather than claiming full independence from the calibration. revision: yes
Referee: The paragraph introducing S_hist and the eigenmode truncation: The reduction of S_hist to two-sided one-dimensional Gaussian equivalents of 1.1σ and 2.1σ relies on retaining only two covariance eigenmodes, but no explicit test of sensitivity to the number of retained modes or justification that higher modes are negligible for the global mismatch is provided; without this, the quantitative claim that the E(z) tension is “below the conventional ≃4.9σ” cannot be fully assessed.

Authors: We acknowledge that an explicit sensitivity test to the number of retained eigenmodes was not included. The truncation to two modes is motivated by the fact that the covariance matrix of E(z) is dominated by the first two principal components, which capture the primary variations in the expansion history across the redshift range of interest. To strengthen this, we will add a supplementary analysis or figure in the revised manuscript showing S_hist computed with 1, 2, 3, and more modes, demonstrating that the value converges rapidly and that additional modes contribute negligibly to the displacement metric due to their small associated eigenvalues. This will provide the justification needed and confirm the robustness of the reported Gaussian equivalents. revision: yes

Circularity Check

0 steps flagged

Reformulation of Hubble tension via E(z) is a deliberate reparametrization with no circular reduction to inputs

full rationale

The paper re-expresses existing probe posteriors (Planck, DESI, Pantheon+SH0ES) as implied E(z) histories inside flat-ΛCDM, where E(z) = H(z)/H0 depends only on Ω_m. This isolates shape from normalization by construction, as the abstract explicitly notes: 'This does not remove the absolute-scale discrepancy, but separates the normalization encoded in H0 from the redshift-dependent shape.' The reported lower S_hist values (1.65/2.55) versus 4.9σ H0 tension follow directly from excluding H0, without any derivation chain that reduces a claimed result to the input data or self-citations. No fitted parameters are renamed as predictions, no uniqueness theorems are invoked, and the mapping uses standard flat-ΛCDM relations. The analysis is self-contained as a comparative reparametrization of published constraints.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The reformulation rests on the standard flat-ΛCDM assumption for all probes and introduces the derived statistic S_hist without external validation.

axioms (1)

domain assumption All three probes are analyzed inside the same flat-ΛCDM model
The mapping of each posterior onto E(z) requires this common framework.

invented entities (1)

S_hist (covariance-subspace history displacement) no independent evidence
purpose: Single scalar that quantifies global mismatch between correlated E(z) histories
Defined in the paper via projection onto leading covariance eigenmodes

pith-pipeline@v0.9.0 · 5587 in / 1490 out tokens · 75838 ms · 2026-05-08T05:43:55.605996+00:00 · methodology

discussion (0)

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