An arXiv preprint reports a sharp change in the Hubble-tension analysis when CMB distance information is included. All three generalized PAge, or GPAge, expansion models are reported as overwhelmingly preferred to standard flat ΛCDM by ln B, ΔAIC and ΔBIC. The authors regard TGPAge-Fix as the most reasonable preferred model and say it could alleviate or even resolve the tension.
That preference is not present in the late-time-only fits. Without CMB data, ΔBIC strongly favors ΛCDM over the GPAge models, while ln B and ΔAIC favor ΛCDM over some GPAges and are inconclusive for another. Once CMB information is added, the direction reverses, with all GPAge forms favored by all three comparison measures.
The comparison behind the result
The comparison sets flat ΛCDM against SGPAge, TGPAge-Fix and TGPAge-Free. The GPAge family is intended to extend beyond the CMB redshift and to be more accurate at low redshifts, giving researchers another way to describe the universe's expansion history. TGPAge-Fix holds the early-universe Hubble constant at 67.36 km/s/Mpc and the late-universe value at 73.04 km/s/Mpc.
TGPAge-Free is less decisive about the early-universe value. The paper says it does not constrain that Hubble constant well: its best fit slightly departs from the Planck-inferred value, while its late-universe value is consistent with SH0ES. The contrast matters because TGPAge-Fix fixes both early- and late-universe values in advance.
The analysis draws on 1,701 Pantheon+ light curves from 1,550 Type Ia supernovae and 77 Cepheid-calibrated host-galaxy distance moduli. The fits use several combinations of supernova, CMB, FAP, cosmic-chronometer and SH0ES data.
Parameter constraints came from Markov chain Monte Carlo, a method that explores many possible parameter combinations, using Cobaya and GetDist. The researchers estimated Bayesian evidence from those chains with MCEvidence and also used AIC and BIC for model comparison. The resulting numbers are scores for the competing fits.
For the combined dataset labelled SCFC, the TGPAge-Fix row reports an ln B of 26.16 relative to ΛCDM, a ΔAIC of -77.54 and a ΔBIC of -61.16. Under the paper's comparison convention, all three figures favor TGPAge-Fix. Those values provide the clearest numerical example of the paper's CMB-inclusive preference.
A middle-redshift signal
The physical discussion moves to the middle and high redshift range. The plotted GPAge histories are reported to depart from ΛCDM there. The derived effective ΩX is negative from approximately redshift 2 to 60 and can reach roughly -0.25 to -0.3 at its most negative. Because ΩX is an effective quantity within the parameterization, this describes the fitted expansion history rather than directly establishing a separate physical component with negative density.
The authors discuss modified gravity as a possible interpretation of this pattern. They also say that conclusion is not solid from expansion-history data alone because growth-history observations were not used. The result therefore points to a question for future tests, rather than identifying a mechanism.
Why the result is provisional
Another boundary is in the CMB input itself. The analysis uses CMB distance priors rather than the full Planck CMB data, leaving open whether the same model ranking would survive a full-CMB analysis. The paper's interpretation also depends on the phenomenological GPAge parameterization and its best-fit history.
For now, the result is best read as a strong statistical preference within a particular set of models and data. The authors present TGPAge-Fix as a possible way to alleviate or resolve the Hubble tension, not as a settled physical explanation. Full Planck data and growth-history observations are the key checks still needed to test how far the result extends.
Paper data and sources
Original title: Guide for the Hubble Tension: Mid-Time Solutions and Generalized PAge Parameterizations
Authors: Jing-Yi Jia, Da-Chun Qiang, Hao Wei
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text