Guide for the Hubble Tension: Mid-Time Solutions and Generalized PAge Parameterizations

Da-Chun Qiang; Hao Wei; Jing-Yi Jia

REVIEW

Guide for the Hubble Tension: Mid-Time Solutions and Generalized PAge Parameterizations

Jing-Yi Jia, Da-Chun Qiang, Hao Wei

Cosmology and Nongalactic Astrophysics · Submitted 28 Aug 2026

Pith's verdict: Based solely on the abstract, this paper proposes a plausible (and interesting) direction for the Hubble tension—mid-time physics via a generalized PAge parameterization—but without the full text, we cannot evaluate whether the claimed resolution is a fitting artifact or a real constraint.

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arXiv ID2608.28260 (version 1)

Pith ID2YAVH3K2, Pith's permanent record of this paper and its review

CitationDa-Chun Qiang, Hao Wei, Jing-Yi Jia. "Guide for the Hubble Tension: Mid-Time Solutions and Generalized PAge Parameterizations." arXiv:2608.28260 (2026). https://pith.science/paper/2YAVH3K2

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Editor's note

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Jia, Qiang, and Wei argue that the original PAge parameterization has flaws and introduce generalized versions (GPAge) that extend to CMB redshifts and improve low-z accuracy. Using these, they claim the Hubble tension can be alleviated (or resolved) and that models with mid-time new physics are overwhelmingly preferred over ΛCDM. This is an incremental but sensible extension of the PAge framework, and the focus on the mid-epoch as the key is a fresh angle in a debate that has mostly been split between early- and late-time fixes. That alone makes the paper worth a look.

That said, the abstract alone gives me no way to assess the core concern: the GPAge parameterization likely adds free parameters fitted to the same data that define the tension. The claimed preference over ΛCDM could come from that extra flexibility rather than from genuine physical content. The authors mention "all the information criteria," which is good, but without seeing the penalty terms, the number of parameters, and a robustness test (e.g., simulated data with ΛCDM), the claim remains unsubstantiated. The abstract also doesn't show the tension-reduction numbers—does it go from 5σ to 2σ or to 1σ? That matters.

I'd also want to see whether the GPAge forms are physically motivated or purely phenomenological. The abstract says "new physics in the mid-time might be the key," which is a strong claim; without a concrete model or a physical mechanism, it's just a curve-fit.

For whom is this paper? Any cosmologist following the Hubble tension will want to read the full paper once it's out, but with guarded expectations. For peer review: yes, a serious editor should send it out, because the idea is timely and the abstract shows a clear attempt to address recognized flaws in PAge. But I'd expect the referees to demand validation against overfitting, a clear statement of the effective model degrees of freedom, and preferably a forecast distinguishing mid-time physics from early-time alternatives.

Referee report

Minor objections (3)

[Abstract] The term 'mid-time' is introduced without a definition; it should be clarified whether this refers to a specific redshift range (e.g., recombination to reionization) or to an intermediate epoch between early and late universes.
[Abstract] The phrase 'overwhelmingly preferred over ΛCDM in terms of all the information criteria' requires reporting of the actual numerical values (e.g., ΔAIC, ΔBIC) even in the abstract to support the strength of the claim.
[Abstract] The abstract states that 'some flaws' exist in the original PAge parameterization but does not specify them; a brief indication of the nature of these flaws would help assess the motivation.

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The paper

The paper claims

By generalizing the PAge parameterization to include CMB data, the authors find that new physics in the mid-universe epoch—not the early or late universe alone—may resolve the Hubble tension, and that data overwhelmingly prefer such…

The reading

The paper argues that the Hubble tension (the mismatch between early-universe and late-universe measurements of the Hubble constant) can be alleviated by modifying the expansion history in the mid-time epoch, roughly between recombination and dark-energy domination. The authors first identify flaws in the standard PAge parameterization, which had previously been used to rule out late-time modifications. They then construct generalized PAge (GPAge) parameterizations that remain accurate at high redshifts (allowing inclusion of CMB data) and are also more precise at low redshifts. Fitting these GPAge forms to real data (late-time probes plus CMB) yields expansion histories that deviate significantly from ΛCDM and that are strongly preferred by information criteria. The central claim is that the tension can be resolved without invoking early-universe physics, provided the mid-time expansion is sufficiently different from ΛCDM.

What carries the argument

The generalized PAge (GPAge) parameterization extends the original PAge parameterization, which uses a small number of parameters to describe the expansion history, to remain valid for z > z_CMB (allowing CMB data to be included) and to be more accurate at low redshifts. The key is that by relaxing the assumptions of the original PAge (which implicitly assumed a ΛCDM-like early expansion), the GPAge forms capture a broader range of possible expansion histories and reveal that data prefer a mid-time deviation from ΛCDM, contradicting earlier no-go arguments that had been based on the original PAge framework.

Extended reading notes

Core claim

The authors claim that the Hubble tension can be resolved (or significantly alleviated) when using generalized PAge parameterizations that allow the expansion history to differ from ΛCDM in the mid-time epoch (redshifts between the CMB last-scattering surface and the onset of late-time acceleration). The original PAge parameterization could not incorporate CMB data and had flaws that made it unsuitable for drawing no-go conclusions about late-time modifications; the corrected GPAge versions overcome these issues. The data (including CMB, BAO, SNe Ia, and local H0 measurements) overwhelmingly favor GPAge models over ΛCDM by all common information criteria, and these preferred models have expansion histories that differ substantially from ΛCDM at intermediate redshifts, pointing to new mid-time physics as the key.

Load-bearing premise

The GPAge parameterization is sufficiently flexible to represent the true cosmic expansion history over the full range of redshifts covered by the data, and the data-driven preference for non-ΛCDM shapes is not an artifact of overfitting or parameter degeneracy inherent in the parameterization.

Editorial extensions

If this is right

If correct, the Hubble tension does not require new physics in the early universe (before recombination) or only in the local late universe; rather, modifications at intermediate redshifts (e.g., changes in the dark energy equation of state or interactions between dark components) become plausible candidates.
Future surveys covering the mid-redshift range (e.g., DESI, Euclid, Roman) should detect a distinct expansion history that deviates from ΛCDM at a few percent level, which would be a direct test.
The no-go arguments against late-time modifications based on the original PAge parameterization need to be reconsidered, as they assumed a parameterization that is both inaccurate at high z and flawed in its low-z interpretation.
The GPAge parameterization itself becomes a useful model-independent tool for testing the expansion history, complementing other non-parameteric approaches.
The overwhelming preference for GPAge over ΛCDM by information criteria suggests that the standard cosmological model may be falsified by current data if the GPAge parameterization is indeed flexible enough to capture the true expansion history.

Reading between the lines

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

A natural extension: the mid-time deviation implied by GPAge fits may be connected to a period of modified gravity or a transition in the dark energy equation of state (e.g., from an early matter-like component to late-time acceleration). This could be tested with cosmic shear and galaxy clustering data.
The original PAge-based no-go arguments were widely cited; this paper implies they were premature. A re-analysis of earlier constraints using the corrected parameterization may find that late-time modifications are not ruled out after all, opening a wider range of model-building possibilities.
If the mid-time solution holds, the Hubble tension may be accompanied by other signature discrepancies in the growth of structure (e.g., the σ₈ tension), which would then require a single physical mechanism linking expansion and growth at intermediate redshifts.
The paper's finding that CMB data can be accommodated within mid-time modifications suggests that recombination physics need not be altered, simplifying the theoretical landscape.

Original abstract

In cosmology, the Hubble tension has become one of the most serious challenges in the last decade. Assuming that all the observational data are right, one of the ways out is to modify the standard $\Lambda$CDM cosmology. In the literature, the theoretical modifications are mainly made in the early and the late/local universes. But on both sides, some no-go arguments have been proposed recently. In particular, the general no-go arguments for the late-time modifications are mainly based on the PAge parameterization. In the present work, we note that there are some flaws in the (original) PAge parameterization, and then propose various generalized PAge (GPAge) parameterizations, which can be extended to $z>z_{\rm CMB}$ and hence the cosmic microwave background (CMB) can be taken into account, while they are also more accurate at low redshifts. With these GPAge parameterizations, we revisit the Hubble tension by using not only the late-time observations but also the observational data of CMB. We find that the Hubble tension could be alleviated (or even resolved) in GPAges significantly different from the standard $\Lambda$CDM cosmology, while they are overwhelmingly preferred over $\Lambda$CDM by the observational data in terms of all the information criteria. Surprisingly, we find that the new physics in the mid-time might be the key to resolve the Hubble tension.

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.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,004 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.54 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In cosmology, the Hubble tension has become one of the most serious challenges in the last decade. Assuming that all the observational data are right, one of the ways out is to modify the standard $\Lambda$CDM cosmology. In the literature, the theoretical modifications are mainly made in the early and the late/local universes. But on both sides, some no-go arguments have been proposed recently. In particular, the general no-go arguments for the late-time modifications are mainly based on the PAge parameterization. In the present work, we note that there are some flaws in the (original) PAge parameterization, and then propose various generalized PAge (GPAge) parameterizations, which can be extended to $z>z_{\rm CMB}$ and hence the cosmic microwave background (CMB) can be taken into account, while they are also more accurate at low redshifts. With these GPAge parameterizations, we revisit the Hubble tension by using not only the late-time observations but also the observational data of CMB. We find that the Hubble tension could be alleviated (or even resolved) in GPAges significantly different from the standard $\Lambda$CDM cosmology, while they are overwhelmingly preferred over $\Lambda$CDM by the observational data in terms of all the information criteria. Surprisingly, we find that the new physics in the mid-time might be the key to resolve the Hubble tension.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In cosmology, the Hubble tension has become one of the most serious challenges in the last decade.	background 1.00	0.04	0.01	asserted 1.00
s1	Assuming that all the observational data are right, one of the ways out is to modify the standard $\Lambda$CDM cosmology.	background 0.98	0.04	0.02	hedged possibility 0.80
s2	In the literature, the theoretical modifications are mainly made in the early and the late/local universes.	background 0.99	0.03	0.01	asserted 1.00
s3	But on both sides, some no-go arguments have been proposed recently.	background 0.82	0.04	0.02	asserted 0.99
s4	In particular, the general no-go arguments for the late-time modifications are mainly based on the PAge parameterization.	background 0.95	0.04	0.03	asserted 0.99
s5	In the present work, we note that there are some flaws in the (original) PAge parameterization, and then propose various generalized PAge (GPAge) parameterizations, which can be extended to $z>z_{\rm CMB}$ and hence the cosmic microwave background (CMB) can be taken into account, while they are also more accurate at low redshifts.	contribution 0.92	0.93	0.08	asserted 1.00
s6	With these GPAge parameterizations, we revisit the Hubble tension by using not only the late-time observations but also the observational data of CMB.	method 0.88	0.90	0.05	asserted 0.85
s7	We find that the Hubble tension could be alleviated (or even resolved) in GPAges significantly different from the standard $\Lambda$CDM cosmology, while they are overwhelmingly preferred over $\Lambda$CDM by the observational data in terms of all the information criteria.	result 1.00	0.96	0.55	hedged possibility 0.52
s8	Surprisingly, we find that the new physics in the mid-time might be the key to resolve the Hubble tension.	interpretation 0.80	0.94	0.77	asserted 0.79

What kind of paper the classifier reports this is: method or tool 0.82 · evidence from analysis of existing data 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Forward citations

Cited by 2 Pith papers

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

GPAge: a Global Parameterization based on cosmic Age for some unified early-time and late-time solutions to the Hubble tension
astro-ph.CO 2026-10 accept novelty 6.0 of 10

Varying early-time matter density even with flexible late-time expansion cannot resolve the Hubble tension, based on a global parametrization of cosmic age.
Cosmological anatomy of interacting dark energy
astro-ph.CO 2026-09 accept novelty 6.0 of 10

A controlled analysis reveals that the genuine cosmological signatures of a widely studied interacting dark energy model are much smaller than naive fixed-parameter comparisons suggest.

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 77 sentences without a confident label

Contests 3 papers

Contrary to the no-go arguments of [ref] , we find that all the GPAge parameterizations are overwhelmingly preferred over the standard $Λ$CDM model in terms of all the information criteria $lnB$, $ΔAIC$, $ΔBIC$.

No-go guide for the Hubble tension : Late-time solutions · Analysis · contest 0.88
Contrary to the no-go arguments of [ref] , we find that all the GPAge parameterizations are overwhelmingly preferred over the standard $Λ$CDM model in terms of all the information criteria $lnB$, $ΔAIC$, $ΔBIC$.

No-go guide for late-time solutions to the Hubble tension: Matter perturbations · Analysis · contest 0.88
Contrary to the no-go arguments of [ref] , we find that all the GPAge parameterizations are overwhelmingly preferred over the standard $Λ$CDM model in terms of all the information criteria $lnB$, $ΔAIC$, $ΔBIC$.

No-go guide for the Hubble tension: late-time or local-scale new physics · Analysis · contest 0.88

Uses the method or tool of 14 papers

Therefore, in [ref] , the so-called PAge parameterization proposed in [ref] was used instead, namely eq1 Ht=2/3+(H_0 t_0 -2/3(1+η)) (t/t_0)+2/3 η(t/t_0)^2 , which is in fact the Taylor expansion of $Ht$ with respect to $τ=t/t_0$ around $τ=0$ up to the second order, where $H$, $t$ and $t_0$ are the Hubble parameter, the cosmic time, and the current age of the universe, respectively.

Supernova Magnitude Evolution and PAge Approximation · Introduction · 0.82
Therefore, in [ref] , the so-called PAge parameterization proposed in [ref] was used instead, namely eq1 Ht=2/3+(H_0 t_0 -2/3(1+η)) (t/t_0)+2/3 η(t/t_0)^2 , which is in fact the Taylor expansion of $Ht$ with respect to $τ=t/t_0$ around $τ=0$ up to the second order, where $H$, $t$ and $t_0$ are the Hubble parameter, the cosmic time, and the current age of the universe, respectively.

Reaffirming the Cosmic Acceleration without Supernova and CMB · Introduction · 0.82
Therefore, in [ref] , the so-called PAge parameterization proposed in [ref] was used instead, namely eq1 Ht=2/3+(H_0 t_0 -2/3(1+η)) (t/t_0)+2/3 η(t/t_0)^2 , which is in fact the Taylor expansion of $Ht$ with respect to $τ=t/t_0$ around $τ=0$ up to the second order, where $H$, $t$ and $t_0$ are the Hubble parameter, the cosmic time, and the current age of the universe, respectively.

No-go guide for the Hubble tension : Late-time solutions · Introduction · 0.82

Uses data or values from 7 papers

Thus, we fix $H_0, EU=67.36 km/s/Mpc$ [ref] and $H_0, LU=73.04 km/s/Mpc$ [ref] in TGPAge-Fix, respectively.

Planck 2018 results. VI. Cosmological parameters · Two-segment generalized PAge (TGPAge) · 1.00
Thus, we fix $H_0, EU=67.36 km/s/Mpc$ [ref] and $H_0, LU=73.04 km/s/Mpc$ [ref] in TGPAge-Fix, respectively.

A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team · Two-segment generalized PAge (TGPAge) · 1.00
In this work, we use the 32 CC data sample given by Table 1 of [ref] in the redshift range of $0.07≤z≤1.965$, and the covariance matrix $C_CC, stat+sys$ including statistical and systematic uncertainties can be found from [ref] (see [ref] and Sec. 3.1 of [ref] for details).

Unveiling the Universe with Emerging Cosmological Probes · Observational data · 1.00

Compares with 1 paper

In fact, the reconstructed $Ω_m(z)$ given by Figs. $3-6$ of [ref] becomes $Ω_m>1$ at redshifts $z≳1.5-2$, in well consistent with our $Ω_m(z)$ of the GPAge parameterizations (n.b. the middle-right and the bottom-right panels of our Fig. [ref]).

Non-parametric Reconstruction of Growth Index via Gaussian Processes · Concluding remarks · 0.92

Cites as support 3 papers

However, as shown in [ref] , both the cosmographies with respect to $z$ and $y$ even up to the fifth order cannot faithfully approximate the model they claim to parameterize at redshift $z≳1$, even for the simplest $Λ$CDM model.

No-go guide for late-time solutions to the Hubble tension: Matter perturbations · Introduction · 0.96
However, as shown in [ref] , both the cosmographies with respect to $z$ and $y$ even up to the fifth order cannot faithfully approximate the model they claim to parameterize at redshift $z≳1$, even for the simplest $Λ$CDM model.

No-go guide for the Hubble tension : Late-time solutions · Introduction · 0.96
Noting that the $fσ_8$ data of the growth history has also been used in [ref] , the results of [ref] might be regarded as a support for the mid-time guide for the Hubble tension proposed in the present work.

Non-parametric Reconstruction of Growth Index via Gaussian Processes · Concluding remarks · 1.00

Cites as background 31 papers

In particular, the general no-go arguments for the late-time modifications in [ref] are mainly based on the PAge parameterization.

No-go guide for the Hubble tension : Late-time solutions · Concluding remarks · 0.83
In particular, the general no-go arguments for the late-time modifications in [ref] are mainly based on the PAge parameterization.

No-go guide for the Hubble tension: late-time or local-scale new physics · Concluding remarks · 0.83
In particular, the general no-go arguments for the late-time modifications in [ref] are mainly based on the PAge parameterization.

No-go guide for late-time solutions to the Hubble tension: Matter perturbations · Concluding remarks · 0.83

Discussion

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