A split result in the model comparison
An analysis of late-time cosmic expansion found that a square-root exponential f(Q) model was favored over flat ΛCDM in several data combinations, but that preference changed with the Type Ia supernova catalogue. It was favored for the baseline, Pantheon+ and Union 3.0 combinations after the comparison accounted for the model’s extra parameter. DES Y5 produced the opposite result under the study’s information criteria. A second, normalized power-law f(Q) model stayed close to flat ΛCDM and was not clearly favored.
The study is an arXiv preprint that asks whether normalized power-law and square-root exponential f(Q) models can describe late-time expansion and perform as well as or better than flat ΛCDM. The researchers used Markov chain Monte Carlo, a sampling method for estimating model parameters, and fitted H0, Ωm0 and either n for the power-law form or m for the square-root exponential form. The joint analysis treated the datasets as statistically independent and added their separate chi-square contributions.
The evidence came in different combinations
The underlying inputs were uneven in size. The analysis used 32 Cosmic Chronometer Hubble-parameter points and seven H0LiCOW strong-lensing systems. Pantheon+ supplied 1,701 spectroscopically confirmed Type Ia supernovae, with 1,590 analyzed after points below redshift 0.01 were excluded. Union 3.0 contained 2,087 supernovae, including 1,363 points common with Pantheon+. DES Y5 reported 1,635 photometrically classified supernovae between redshifts 0.0596 and 1.12, plus 194 low-redshift supernovae below 0.1. The reported totals for the baseline, Pantheon+, Union 3.0 and DES Y5 joint combinations were 52, 1,642, 74 and 1,687 data points.
What the fits found
In the baseline normalized power-law fit, the reported 68% central estimates were H0 = 68.9139, Ωm0 = 0.2916 and n = 0.0460. The study says that adding Type Ia supernova data tightened the H0 and Ωm0 constraints and shifted n toward positive values. For the square-root exponential model, the corresponding baseline estimates were H0 = 67.9802, Ωm0 = 0.3020 and m = 4.0212. In the Pantheon+ combination, Ωm0 was 0.3166. The reported 68% intervals were asymmetric, particularly for H0 in the square-root exponential fits.
On the model-comparison scores, the normalized power-law model stayed close to flat ΛCDM. The baseline minimum-chi-square difference, calculated as fI minus ΛCDM, was +0.56; the differences became negative when supernova data were included. AIC and BIC were also used as fit scores with a penalty for the extra parameter: AIC differences for the supernova combinations were less than two in magnitude, while BIC differences were positive for all four combinations. Taken together, the analysis did not clearly favor fI over flat ΛCDM.
The square-root exponential model produced more favorable comparisons for several combinations. In the baseline combination, its differences from ΛCDM were -7.6076 for minimum chi-square, -5.6076 for AIC and -3.6563 for BIC. With Pantheon+, the corresponding figures were -12.5354, -10.5354 and -5.1317. For Union 3.0, the differences were -12.4079, -10.4079 and -8.1038. DES Y5 did not show the same pattern: the minimum-chi-square difference was -1.3218, but AIC was +0.6782 and BIC was +6.1089, leaving flat ΛCDM favored by the information criteria.
A similar transition, an unsettled choice
Both models gave nearly the same estimate for when the expansion shifted from deceleration to acceleration. The study defines that transition as the point at which the deceleration parameter reaches zero. In the baseline fit, the transition redshift was 0.6849 for the power-law model and 0.6886 for the square-root exponential model. Across all four combinations, both models placed it between 0.66 and 0.69, with asymmetric 68% confidence intervals.
The central caveat is that the model preference was not universal. Pantheon+ and Union 3.0 remained favorable to fII after the parameter penalty, but DES Y5 favored flat ΛCDM by the information criteria. The analysis therefore points to dataset dependence, not a settled replacement for the standard model. The paper says larger and more homogeneous observations will be needed to test whether the fII preference persists.
The analysis is limited to background expansion history. It does not test cosmological perturbations, structure formation or other observational probes beyond the background. The DESI analysis also uses a baryon-density prior centered at 0.02218 with a spread of 0.00055, together with a Gaussian prior on the sound horizon rd whose numerical value is not reported.
The manuscript remains a preprint, and its front matter says it is to be submitted to Eur.Phys.J.C. The paper states that all observational data used are publicly available and that the code will be made available on reasonable request.
Paper data and sources
Original title: Testing $f(Q)$ Gravity with DESI DR2 and Strong-Lensing Time Delays
Authors: Darshan Kumar, Saibal Ray, Fengge Zhang et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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