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A head-to-head comparison of 14 proposed Hubble-tension solutions finds that only transient early-universe energy injections both shift the inferred Hubble constant to about 70 km/s/Mpc and cut the residual discrepancy with local measuremen

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-02 05:37 UTC pith:R7DHGOWV

load-bearing objection Solid, well-executed H0 Olympics update that deserves refereeing; the Cold NEDE Bayesian qualification is knife-edge and should be checked before the 'all four' claim is trusted. the 3 major comments →

arxiv 2607.13282 v1 pith:R7DHGOWV submitted 2026-07-14 astro-ph.CO hep-ph

The H₀ World Cup. I. Summary of the baseline group stage results

Nils Sch\"oneberg , Vivian Poulin , Angelo G. Ferrari , Fabio Finelli , Julien Lesgourgues , Luca Morelli , Markus R. Mosbech , Ravi Kumar Sharma
show 1 more author
Th\'eo Simon
This is my paper
classification astro-ph.CO hep-ph PACS 98.80.-k
keywords Hubble tensionearly dark energyH0cosmological model comparisonBayesian evidenceAkaike information criterionsound horizonrecombination
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 asks whether any of the fourteen most studied fixes for the Hubble tension genuinely works when all are run through the same analysis pipeline with the same CMB, BAO, and supernova data. It claims that only models with a localized, non-radiative energy injection around matter–radiation equality – the early dark energy family – pass both a frequentist and a Bayesian bar: they shift the inferred expansion rate to about 70 km/s/Mpc and reduce the residual tension from more than 5 sigma to around 2.5–3.6 sigma, while improving the joint fit over the standard model. Varying the electron mass at recombination helps only halfway, and neither extra radiation nor late-time modifications improve over the standard model. The paper's punchline is that the mechanism matters: the fix has to shrink the sound horizon, not simply tweak what happens later.

Core claim

The central claim is that, in a common statistical framework applied to the baseline CMB+BAO+SN dataset, the four early-energy models (axion-like early dark energy, cold NEDE, early modified gravity, and its Rock'n'Roll limit) are the only contenders that both reduce the residual calibration tension to the 2.5–3.6σ level and achieve strong joint-fit support over ΛCDM (−ΔAIC > 10 and ln BF > 3). A localized non-radiative contribution near matter–radiation equality is thus identified as the most effective mechanism, because it shrinks the sound horizon while leaving the detailed high-multipole CMB spectra and the low-redshift distance relation intact. The paper does not claim any model fully r

What carries the argument

The central mechanism is a transient, non-radiative energy component peaking near matter–radiation equality – a short-lived scalar field that briefly accelerates expansion before recombination and shrinks the sound horizon. This allows a higher H0 from early-universe data without disturbing the well-measured CMB damping tail and the BAO/SN distance relation. The comparison is carried by two complementary statistics: frequentist (AIC and ΔDMAP) and Bayesian (log-evidence and Δshift), with thresholds −ΔAIC > 10 and ln BF > 3.

Load-bearing premise

The verdict depends on the prior ranges chosen for each model's extra parameters (taken from the original proposals, not derived here) and on the inclusion of the ACT high-multipole data; the paper itself notes that removing ACT relaxes radiation-model constraints back to 3–3.5σ.

What would settle it

Re-run the baseline comparison without the ACT high-multipole dataset (or with a different high-resolution CMB experiment in its place). If an extra-radiation or late-time model then crosses the AIC/Bayes thresholds and pushes its residual tension below 3σ, the claim that a localized non-radiative injection is the most effective mechanism would be refuted.

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

If this is right

  • Successful models must reduce the sound horizon while preserving the high-multipole CMB spectra and the low-redshift distances fixed jointly by BAO and supernovae.
  • Radiation-only additions are limited by damping-tail and phase-shift signatures; post-recombination changes have too little freedom once BAO and SN are included.
  • Modified recombination (varying electron mass) can shift the relevant scales and qualifies, but with a larger residual tension (~4σ).
  • The neutrino-mass sum can be varied freely without significantly affecting the tension results, and its constraints are nearly as tight in most extended models as in ΛCDM.
  • Both statistical frameworks agree on the overall hierarchy, even though the internal ordering within the early-energy group depends on whether best-fit or posterior-volume measures are used.

Where Pith is reading between the lines

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

  • The paper's group-stage framing suggests a competition with winners, but the honest reading is that a class of mechanisms (early, non-radiative injections) is favoured; which specific model wins depends on the statistic.
  • The stated sensitivity of radiation models to the ACT dataset implies that the eliminated or relegated status of the radiation group is not robust to future high-multipole CMB data; a rerun with different high-resolution data could reshuffle the lower ranks.
  • A natural testable extension is to feed the same baseline into multi-field or otherwise extended early-energy models, which the paper hints can relax the tight constraints on the single-field case.
  • If the early-energy mechanism is correct, its signatures should appear in complementary probes such as cosmic birefringence, fifth-force constraints, or accelerator searches of the associated particles – a thread the paper leaves for later 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

3 major / 4 minor

Summary. This Letter presents a systematic comparison of fourteen cosmological models proposed to address the Hubble tension, all analyzed with a common CMB+BAO+SN dataset combination (A) and an optional local supernova-calibration prior B. The authors use two frequentist statistics, ΔDMAP and ΔAIC, and two Bayesian statistics, the log-Bayes factor and Δshift, to evaluate each model's ability to reduce the residual calibration tension and to improve the global fit relative to flat ΛCDM. The central claim is that all four Group E models (EDE, Cold NEDE, EMG, RnR) cross both selection thresholds, shift the inferred H0 to approximately 70 km/s/Mpc, and reduce the residual tension to about 2.5–3.6σ, making a localized non-radiative contribution near matter–radiation equality the most effective mechanism in the baseline comparison. Varying electron mass is the only non-Group-E model that qualifies; radiation and late-time models do not. Full model definitions, priors, and robustness tests are deferred to an in-preparation companion paper (Paper II).

Significance. If the results are correct, this is a valuable community benchmark: it places fourteen often-disparate models in a single likelihood framework with current CMB, BAO, and supernova data, and it cross-checks frequentist and Bayesian selection statistics. The paper is refreshingly explicit about its limitations: it notes that no model fully resolves the tension, that the Bayes factor depends on prior volume, and that conclusions change when ACT data are removed. The use of public pipelines and emulators, and the explicit comparison to the earlier H0 Olympics, make the work reproducible in principle. However, the central Bayesian claim rests on a single marginal crossing (Cold NEDE ln BF = 3.03 versus the 3.0 threshold) and on model priors that are not specified in this Letter. The qualitative hierarchy may well be robust, but the specific claim that 'all four Group E models cross both selection thresholds' is more fragile than the presentation suggests.

major comments (3)
  1. [Table I and Section 'We use parallel frequentist and Bayesian criteria'] Cold NEDE has ln BF = 3.03 against the stated threshold ln BF > 3, a margin of 0.03. No numerical uncertainty or stability analysis is reported for any of the evidence values in Table I. The paper itself warns that ln BF 'must be interpreted with the adopted priors in mind,' but it does not report the numerical accuracy of the nested-sampling estimates, nor does it test the sensitivity of the Cold NEDE crossing to reasonable changes in the inherited priors or to the evidence estimator. Since the crossing of the Bayesian threshold is a load-bearing part of the claim that 'all four Group E models' qualify, this needs a quantitative robustness check or an explicit statement of the evidence uncertainty.
  2. [Paragraph beginning 'In every case, the reference cosmology...'] All model definitions, parameter priors, and implementation details are deferred to the companion Paper II [1], which is 'in preparation.' The Bayesian evidence integrates over the prior volume, so the ranking and qualification status depend on the priors. The reader cannot verify that the priors are representative or unbiased, especially for models whose parent studies were themselves motivated by the Hubble tension. This is a correctness-risk issue rather than a presentation issue. The Letter should either include the prior ranges in an appendix, or the companion paper must be available and referenced before the central Bayesian comparison can be fully assessed.
  3. [Section 'The baseline competition gives a concise empirical target...'] The paper claims that 'a localized non-radiative contribution near matter–radiation equality is the most effective mechanism' based on the Group E results. This is a model-selection statement, but all four Group E models leave a residual tension of 2.5–3.6σ, and the paper admits they are not favored over ΛCDM without the local H0 prior. The conclusion is therefore an empirical ranking of phenomenological proxies, not a detection of the physical mechanism. This is acknowledged in the text, but the abstract and title may overstate the case. I would recommend softening the causal language or making the proxy status more prominent.
minor comments (4)
  1. [Table I caption] The caption says 'all 13 contenders' but the table lists 14 rows. The agnostic-reionization row has dashes for model-comparison metrics, so the caption should say '14 contenders, 13 of which have defined model-comparison statistics' or similar.
  2. [Figure 1] The left panel shows H0 intervals from A alone, while the middle and right panels show statistics from A+B. The caption is clear, but the reader must reconcile the different data treatments across the three panels. A sentence in the caption explicitly stating that only the left panel uses A alone would be helpful.
  3. [Equation (2)] The definition of ΔDMAP as a square root of a chi-square difference is reasonable, but the label 'non-Gaussian tension' is not defined precisely. It would help to state explicitly that this reduces to the usual number of sigma in the Gaussian limit and to note the interpretation when the profile is non-Gaussian.
  4. [References] The reference list is extensive, but Ref. [17] is cited in the introduction for the DESI dynamical dark energy preference and again in the text; consider consolidating. Also, several arXiv entries have incomplete bibliographic details (e.g., Refs. [14,15] with DOI placeholders); these should be completed during production.

Circularity Check

0 steps flagged

No circular derivation: the paper reports model fits and comparison statistics; no claimed prediction is defined in terms of its own output, and the main result is externally benchmarked against the data.

full rationale

This is a model-comparison paper, not a derivation: each of the fourteen contenders is fitted to the same CMB+BAO+SN likelihoods, and the reported quantities (−ΔAIC, ln BF, ΔDMAP, ΔM_B^shift) are summaries of the fit quality. No equation in the paper defines a predicted quantity in terms of a fitted parameter in a way that would make the outcome tautological. The central claim that Group E models reduce the tension is supported by profile-likelihood and posterior intervals for H0 (Fig. 1, Table I), which are direct outputs of the data analysis rather than assumed results. The main caveats are prior sensitivity and reliance on companion material: the text states that 'each model is analyzed with the priors adopted for its physical parameters in Paper II [1], following the parent studies cited above,' and Paper II is cited as 'in preparation (2026)'. This makes the Bayesian evidence values, especially Cold NEDE's ln BF = 3.03 against a threshold of ln BF > 3, dependent on priors that are not fully described in this Letter. The paper itself acknowledges that 'the evidence averages over the prior volume, its numerical value must be interpreted with the adopted priors in mind.' These are robustness concerns, not instances of circularity: the priors come from earlier model-specific studies and are not defined in terms of the present paper's outputs. The self-citations to the authors' own EDE, NEDE, EMG, WZDR, and H0 Olympics work are normal references for model implementations and are not load-bearing in a circular sense, because the fits are evaluated against external data and the ranking is a data-driven outcome. The absence of explicit numerical error bars on ln BF and the near-threshold value for Cold NEDE are correctly classified as statistical fragility, not circular reasoning. I therefore find no significant circularity, assigning score 1 only to reflect the modest unverifiable reliance on unpublished Paper II priors.

Axiom & Free-Parameter Ledger

15 free parameters · 5 axioms · 0 invented entities

The paper's central ranking rests on choices it inherits rather than derives: fourteen model parameterizations fitted to the same data, priors from parent studies (often by the same authors), author-chosen selection thresholds, and a fixed local calibration. No new physical entity is introduced.

free parameters (15)
  • EDE parameters (f_EDE, log10 z_c, θ_i) = not reported in Letter
    Fitted to CMB+BAO+SN; drive the EDE H0 shift to ~70 km/s/Mpc and the 2.5σ residual tension claim.
  • Cold NEDE parameters (f_NEDE, z_*, etc.) = not reported in Letter
    Fitted model parameters for the cold NEDE early-dark-energy scenario.
  • EMG parameters = not reported in Letter
    Early modified gravity model parameters fitted to the baseline data.
  • RnR parameters = not reported in Letter
    Rock 'n' Roll minimally coupled early-dark-energy parameters fitted to the baseline data.
  • Varying-electron-mass amplitude (δm_e/m_e) = not reported in Letter
    Fitted to CMB+BAO+SN; produces an intermediate H0 shift and relaxes the neutrino-mass bound.
  • 4-parameter modified recombination parameters = not reported in Letter
    Flexible ionization-history parameters fitted to the data; improve χ² but penalized by AIC/evidence.
  • ΔN_eff (free-streaming radiation) = not reported in Letter
    Additional relativistic degrees of freedom fitted to the data.
  • SIDR self-interaction parameters = not reported in Letter
    Self-interacting dark radiation model parameters fitted to the data.
  • WZDR parameters = not reported in Letter
    Wess–Zumino dark radiation parameters fitted to the data; best performer in Group R.
  • DRMD parameters = not reported in Letter
    Dark-radiation–matter decoupling parameters fitted to the data.
  • Thawing gravity parameters = not reported in Letter
    Thawing gravity model parameters fitted to the data.
  • ΛsCDM parameters = not reported in Letter
    Sign-switching cosmological constant parameters fitted to the data.
  • iDM–DE interaction parameters = not reported in Letter
    Interacting dark matter–dark energy parameters fitted to the data.
  • Agnostic reionization parameters = not reported in Letter
    Low-multipole polarization treatment parameters; no AIC/evidence defined for this altered dataset.
  • Summed neutrino mass Σmν = free in all models
    Marginalized in every model; roughly ΛCDM-tight except in the varying-me model, where it relaxes by a factor of two.
axioms (5)
  • standard math The stated statistical criteria (ΔAIC>10, ln BF>3, ΔDMAP, ΔM_B^shift) are valid measures of tension and model preference.
    Eqs. (2)–(4) and the surrounding text; thresholds are author-chosen benchmarks rather than externally fixed.
  • domain assumption The public likelihoods and data products (Planck PR3/PR4, SRoll2, ACT DR6, SPT-3G, DESI DR2, Pantheon+) accurately represent the underlying measurements.
    The 'uncalibrated baseline' paragraph lists the datasets; no validation or systematics audit is provided in this Letter.
  • domain assumption Model priors inherited from parent studies are appropriate and not tuned to make contenders win or lose.
    'Each model is analyzed with the priors adopted for its physical parameters in Paper II [1], following the parent studies cited above.' Many parent studies share authors with this paper.
  • domain assumption The local calibration M_B = −19.253 ± 0.027 from Riess et al. (2022) is the correct local distance anchor.
    Eq. (1) adopts this prior; the abstract quotes a newer H0DN joint local value (73.50 ± 0.81) that is not used in the baseline.
  • domain assumption Flat ΛCDM with a free summed neutrino mass is the appropriate reference cosmology.
    Text: 'In every case, the reference cosmology is flat ΛCDM with a free summed neutrino mass'; all improvements are measured against this baseline.

pith-pipeline@v1.3.0-alltime-deepseek · 13991 in / 11536 out tokens · 128086 ms · 2026-08-02T05:37:02.599822+00:00 · methodology

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read the original abstract

The Hubble tension has reached a nominal significance above $7\sigma$, while new high-precision measurements of the cosmic microwave background (CMB) and baryon acoustic oscillations (BAO) sharpen the test of proposed solutions. Using a common framework, we compare fourteen representative alternatives to the standard $\Lambda$ Cold Dark Matter ($\Lambda$CDM) model in light of up-to-date CMB, BAO and supernovae data to gauge their ability to resolve the tension. The models span late-time modifications, modified recombination, and exotic pre-recombination expansion histories driven by additional radiation or a localized dark energy injection. We evaluate each proposal with complementary frequentist and Bayesian measures of the residual calibration tension and of the improvement in the joint fit. Both approaches identify the same broad hierarchy. Early dark energy and early modified gravity models perform best, shifting the $H_0$ inference without local measurement priors toward $70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ and reducing the residual discrepancy with SH0ES to approximately $2.5-3.6\sigma$, depending on the model and statistic, while receiving strong support over $\Lambda$CDM in the combined fit. Varying the electron mass at recombination yields an intermediate improvement, whereas the enhanced-radiation and late-time scenarios do not improve over $\Lambda$CDM. This Letter summarizes the group stage of the competition; in a companion paper (Paper II) we present the results of an exhaustive set of analyses and assess their robustness to variations in modeling assumptions and datasets.

Figures

Figures reproduced from arXiv: 2607.13282 by Angelo G. Ferrari, Fabio Finelli, Julien Lesgourgues, Luca Morelli, Markus R. Mosbech, Nils Sch\"oneberg, Ravi Kumar Sharma, Th\'eo Simon, Vivian Poulin.

Figure 1
Figure 1. Figure 1: FIG. 1. Combined baseline comparison. Left: frequentist profile-likelihood intervals (purple diamonds) and Bayesian marginalized [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

discussion (0)

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Forward citations

Cited by 2 Pith papers

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

  1. Hubble tension: the shape wall

    astro-ph.CO 2026-07 accept novelty 6.0

    Late-time modifications to the expansion history can raise H0 by at most about 2% (conservative) to 3.7% (permissive) if the CMB acoustic scale is fixed.

  2. What could an emerging Big Bang Nucleosynthesis discrepancy be hinting at?

    astro-ph.CO 2026-07 conditional novelty 5.0

    A transient expansion-rate boost during deuterium burning can reconcile the measured deuterium abundance with the high baryon density preferred by early dark energy models, without changing helium-4.

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