REVIEW
Revisiting the Hubble tension with an inverse power-law early dark energy
Cosmology and Nongalactic Astrophysics · Submitted 27 Aug 2026 · 45 references
Pith's verdict: New EDE potential shape but posterior violates the paper's own fine-tuning bound, undercutting the H0 result.
T0 · AI review
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Referee report
The paper
The paper claims
A new early dark energy model with an inverse power-law potential raises the inferred Hubble constant to about 70.2 km/s/Mpc, reducing the tension with local measurements.
The reading
What carries the argument
The potential V(ϕ) = V₀ (1 + (ϕ/f)²)^{-n} with n>0. The scalar field is frozen near ϕ=0 at early times, then rolls down the inverse power-law tail when Hubble friction drops, providing a brief epoch of dark energy that peaks near matter–radiation equality and subsequently dilutes. This modifies the angular sound horizon and damping scale, which raises the H₀ inferred from CMB data while preserving the fit to anisotropies. The mapping from model parameters (V₀, f) to phenomenological parameters (f_EDE, z_c) is handled by a shooting algorithm.
Extended reading notes
Core claim
For the inverse power-law EDE model with index n=10, fitting CMB+BAO+SN+H0DN data gives a posterior mean H₀ = 70.20⁺⁰·⁶⁰₋₀·₄₇ km/s/Mpc (best-fit 70.56), compared to ΛCDM’s 68.62. The EDE fraction peaks at f_EDE ≈ 0.079 at redshift z_c ≈ 3.5, and the model is preferred over ΛCDM with ΔAIC = −11.1. The Hubble tension, measured by Q_DMAP, drops from 6.25σ (ΛCDM) to 4.79σ (EDE). The model also predicts an increased S₈, worsening the S₈ tension, and leaves detectable imprints in CMB and matter power spectra that future data can test.
Load-bearing premise
The initial scalar field value Θᵢ must be ≤ 10⁻³ for the EDE to decay sufficiently by late times; without this fine-tuning the residual dark energy at z≈0 would spoil the fit to data.
Editorial extensions
If this is right
- The Hubble constant inferred from early-universe data is pushed to about 70.2 km/s/Mpc, reducing the discrepancy with the local SH0ES and H0DN measurements but not eliminating it entirely — the tension remains at ∼4.8σ.
- The model worsens the S₈ tension, as the increased dark matter density and spectral tilt raise the predicted matter clustering amplitude relative to ΛCDM.
- Future CMB data (e.g., from ACT, SPT-3G, or CMB-S4) can test the predicted deviations in the TT, TE, and EE power spectra, which currently lie within cosmic variance.
- Large-scale structure surveys (e.g., DESI, Euclid) can probe the ∼10% deviations in the matter power spectrum at k ∼ 1 h/Mpc predicted by the model.
- If the model is confirmed, it would imply that a new scalar degree of freedom with an inverse power-law potential operated near matter–radiation equality, motivating further theoretical work on ultraviolet completions.
Reading between the lines
- The inverse power-law shape resembles potentials that arise from non-perturbative effects in string theory or from certain D-term potentials; if the model holds, it could point toward a specific class of ultraviolet theories.
- The required fine-tuning of Θᵢ ≤ 10⁻³ suggests that the model is not fully natural; a dynamical mechanism that sets this small initial value (e.g., a phase transition or a separate inflationary stage) would be needed.
- Combining the model with weak-lensing data from upcoming surveys (e.g., Rubin Observatory) could sharply test the predicted high S₈, potentially discriminating it from other EDE models that have different S₈ predictions.
- The model’s preference over ΛCDM depends on including the H0DN prior; without it, the EDE fraction is unconstrained and the model is disfavored. This implies that the evidence for EDE is contingent on the local H₀ measurement used, and a shift in that prior could alter the conclusions.
Original abstract
We propose a new early dark energy with a potential that has an inverse power-law asymptotic tail to alleviate the Hubble tension. Fitting this model to the datasets of CMB+BAO+SN+H0DN, one obtains a 68$\%$ confidence-level value of $H_0\sim 70.20^{+0.60}_{-0.47}$ km s$^{-1}$Mpc$^{-1}$ and a best-fit value of $H_0\sim 70.56$ km s$^{-1}$Mpc$^{-1}$ for index $n=10$. We show that this model can be tested by future data of CMB and matter power spectra. Finally, we briefly discuss fine-tuning problems related to this model.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We propose a new early dark energy with a potential that has an inverse power-law asymptotic tail to alleviate the Hubble tension. Fitting this model to the datasets of CMB+BAO+SN+H0DN, one obtains a 68$\%$ confidence-level value of $H_0\sim 70.20^{+0.60}_{-0.47}$ km s$^{-1}$Mpc$^{-1}$ and a best-fit value of $H_0\sim 70.56$ km s$^{-1}$Mpc$^{-1}$ for index $n=10$. We show that this model can be tested by future data of CMB and matter power spectra. Finally, we briefly discuss fine-tuning problems related to this model.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose a new early dark energy with a potential that has an inverse power-law asymptotic tail to alleviate the Hubble tension. | contribution 0.99 | 0.87 | 0.34 | asserted 0.84 |
| s1 | Fitting this model to the datasets of CMB+BAO+SN+H0DN, one obtains a 68$\%$ confidence-level value of $H_0\sim 70.20^{+0.60}_{-0.47}$ km s$^{-1}$Mpc$^{-1}$ and a best-fit value of $H_0\sim 70.56$ km s$^{-1}$Mpc$^{-1}$ for index $n=10$. | result 1.00 | 0.94 | 0.26 | asserted 1.00 |
| s2 | We show that this model can be tested by future data of CMB and matter power spectra. | future work 0.86 | 0.77 | 0.11 | asserted 0.89 |
| s3 | Finally, we briefly discuss fine-tuning problems related to this model. | limitation 0.97 | 0.63 | 0.03 | asserted 0.68 |
What was examined, and what was not
- arXiv listing abstract v2
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- P abstract
- S sections
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References
How the literature uses it
What this paper does with the arXiv papers it cites
What it cites · 23 sentences without a confident label
Uses the method or tool of 8 papers
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In this section we carry out the MCMC analysis on the EDE model with two representative values of $n={8,10}$ in terms of implementing the background and linear perturbation equations in Sec.[ref] into the Boltzmann solver [ref] [<https://github.com/lesgourg/class_public>] and [ref] [<https://github.com/mwt5345/class_ede>] interfaced with [ref] .
The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes · MCMC analysis · 1.00
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In this section we carry out the MCMC analysis on the EDE model with two representative values of $n={8,10}$ in terms of implementing the background and linear perturbation equations in Sec.[ref] into the Boltzmann solver [ref] [<https://github.com/lesgourg/class_public>] and [ref] [<https://github.com/mwt5345/class_ede>] interfaced with [ref] .
Early Dark Energy Does Not Restore Cosmological Concordance · MCMC analysis · 1.00
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In this section we carry out the MCMC analysis on the EDE model with two representative values of $n={8,10}$ in terms of implementing the background and linear perturbation equations in Sec.[ref] into the Boltzmann solver [ref] [<https://github.com/lesgourg/class_public>] and [ref] [<https://github.com/mwt5345/class_ede>] interfaced with [ref] .
Cobaya: Code for Bayesian Analysis of hierarchical physical models · MCMC analysis · 1.00
Uses data or values from 11 papers
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We fit the EDE model to the following datasets: * CMB: We use the Planck 2018 low-\(ℓ\) temperature and polarization likelihood [ref] , the high-\(ℓ\) temperature, polarization spectra, and the TT, TE, and EE measurements from the NPIPE CamSpec data release [ref] , and finally the Planck 2018 CMB lensing likelihood [ref] . * BAO: We adopt the BAO measurements from the DESI DR2 data release [ref] , which provide both isotropic and anisotropic distance constraints over the redshift interval covered by the survey. * SN: We include Type Ia supernova distance measurements from the Pantheon+ [ref] compilation. * H0DN: We adopt a Gaussian prior on the Hubble constant, $H_0 = 73.50 ±0.81$ km/s/Mpc, based on the baseline results from the $H0DN$ Collaboration [ref] .
CMB power spectra and cosmological parameters from Planck PR4 with CamSpec · Datasets · 1.00
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We fit the EDE model to the following datasets: * CMB: We use the Planck 2018 low-\(ℓ\) temperature and polarization likelihood [ref] , the high-\(ℓ\) temperature, polarization spectra, and the TT, TE, and EE measurements from the NPIPE CamSpec data release [ref] , and finally the Planck 2018 CMB lensing likelihood [ref] . * BAO: We adopt the BAO measurements from the DESI DR2 data release [ref] , which provide both isotropic and anisotropic distance constraints over the redshift interval covered by the survey. * SN: We include Type Ia supernova distance measurements from the Pantheon+ [ref] compilation. * H0DN: We adopt a Gaussian prior on the Hubble constant, $H_0 = 73.50 ±0.81$ km/s/Mpc, based on the baseline results from the $H0DN$ Collaboration [ref] .
Planck 2018 results. VIII. Gravitational lensing · Datasets · 1.00
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We fit the EDE model to the following datasets: * CMB: We use the Planck 2018 low-\(ℓ\) temperature and polarization likelihood [ref] , the high-\(ℓ\) temperature, polarization spectra, and the TT, TE, and EE measurements from the NPIPE CamSpec data release [ref] , and finally the Planck 2018 CMB lensing likelihood [ref] . * BAO: We adopt the BAO measurements from the DESI DR2 data release [ref] , which provide both isotropic and anisotropic distance constraints over the redshift interval covered by the survey. * SN: We include Type Ia supernova distance measurements from the Pantheon+ [ref] compilation. * H0DN: We adopt a Gaussian prior on the Hubble constant, $H_0 = 73.50 ±0.81$ km/s/Mpc, based on the baseline results from the $H0DN$ Collaboration [ref] .
Planck 2018 results. V. CMB power spectra and likelihoods · Datasets · 1.00
Cites as background 5 papers
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More specifically, this EDE can be understood as a fundamental scalar degree of freedom beyond the Standard Model.[We refer the reader to [ref] for alternative EDEs models that lack particle interpretations and are studied through effective fluid approximation.] In single-field realizations, there are several examples of EDE such as axion-like EDE [ref] , Rock `n' Roll EDE [ref] and $α$-attractor EDE [ref] , among which the axion-like EDE offers the largest value of $H_0$, of order $∼70-72$ km s$^-1$Mpc$^-1$ depending on the datasets used.
Early Dark Energy Can Resolve The Hubble Tension · Introduction · 0.91
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More specifically, this EDE can be understood as a fundamental scalar degree of freedom beyond the Standard Model.[We refer the reader to [ref] for alternative EDEs models that lack particle interpretations and are studied through effective fluid approximation.] In single-field realizations, there are several examples of EDE such as axion-like EDE [ref] , Rock `n' Roll EDE [ref] and $α$-attractor EDE [ref] , among which the axion-like EDE offers the largest value of $H_0$, of order $∼70-72$ km s$^-1$Mpc$^-1$ depending on the datasets used.
Rock 'n' Roll Solutions to the Hubble Tension · Introduction · 0.91
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More specifically, this EDE can be understood as a fundamental scalar degree of freedom beyond the Standard Model.[We refer the reader to [ref] for alternative EDEs models that lack particle interpretations and are studied through effective fluid approximation.] In single-field realizations, there are several examples of EDE such as axion-like EDE [ref] , Rock `n' Roll EDE [ref] and $α$-attractor EDE [ref] , among which the axion-like EDE offers the largest value of $H_0$, of order $∼70-72$ km s$^-1$Mpc$^-1$ depending on the datasets used.
Oscillating scalar fields and the Hubble tension: a resolution with novel signatures · Introduction · 0.91
Discussion
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