⚛️ general relativity

Testing $f(Q)$ Gravity with DESI DR2 and Strong-Lensing Time Delays

This paper constrains two $f(Q)$ gravity models using a comprehensive dataset including DESI DR2 and strong-lensing time delays, finding that the square-root exponential model outperforms the standard $\Lambda$ CDM model in explaining the Universe's late-time accelerated expansion without a cosmological constant.

Original authors: Darshan Kumar (HNAS), Saibal Ray, Fengge Zhang, Nisha Rani, Praveen Kumar Dhankar, Jie Zheng

Published 2026-08-31

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CC BY 4.0

Original authors: Darshan Kumar (HNAS), Saibal Ray, Fengge Zhang, Nisha Rani, Praveen Kumar Dhankar, Jie Zheng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, the most successful story we have about how the universe works has been built on a simple, if mysterious, premise. Albert Einstein's theory of general relativity describes gravity not as a force pulling objects together, but as a curvature in the fabric of space and time, much like a heavy ball sinking into a trampoline. This theory has passed every test thrown at it, from the bending of starlight to the existence of black holes. However, in the late 1990s, astronomers discovered something that this story could not explain on its own: the universe is not just expanding, but that expansion is speeding up. To make the math work, the standard model of cosmology had to invent a new ingredient called dark energy, often represented by a cosmological constant, to push the universe apart. Yet, this solution comes with its own headaches, including deep puzzles about why this push exists at all and why it seems to have the exact strength it does today.

Because of these lingering questions, scientists have begun to look for alternatives. Instead of adding a mysterious new ingredient to the cosmic soup, they have asked if the recipe for gravity itself might be slightly different than Einstein wrote it. One promising avenue involves a geometric description of gravity that swaps the familiar idea of curved space for a concept called non-metricity. In this view, gravity arises from how the rules of measuring distance and time change from point to point, rather than from the bending of space. This approach, known as symmetric teleparallel gravity, offers a fresh way to write the equations of the universe. By tweaking these equations, researchers can create models that might explain the accelerating expansion without needing to invent a cosmological constant.

In a recent study, a team of researchers put two specific versions of this new gravity theory to the test against the most precise cosmic measurements available today. They focused on two mathematical forms: one that follows a simple power-law pattern and another that uses a square-root exponential shape. To see which, if either, matched reality, they gathered a massive collection of data from the real universe. This included measurements of how fast the universe is expanding at different times, gathered by watching the aging of ancient galaxies; the rhythmic patterns of sound waves frozen in the distribution of galaxies; the time delays of light from distant quasars bent by massive galaxies; and the brightness of thousands of exploding stars known as Type Ia supernovae. These supernovae act as cosmic mile markers, allowing astronomers to measure distances across billions of light-years with high precision.

The researchers used powerful statistical tools to see how well their new gravity models fit this mountain of data compared to the standard model. They found that the first model, the power-law version, performed almost exactly like the standard theory. It could describe the universe's history just as well, but it did not offer a clear advantage that would justify its extra complexity. The second model, however, told a different story. When the researchers combined data from the cosmic expansion history, the galaxy sound waves, the time-delayed light, and the supernova catalogs known as Pantheon+ and Union 3.0, the square-root exponential model provided a significantly better fit than the standard theory. It matched the observations more closely, and this improvement was strong enough to outweigh the penalty for adding an extra parameter to the equation.

This result suggests that the universe might be accelerating not because of a mysterious constant pushing it, but because the fundamental rules of gravity have a slightly different shape than Einstein proposed. The square-root exponential model emerged as a particularly strong candidate, offering a more accurate description of the cosmic expansion for several of the data combinations they tested. However, the story is not entirely settled. When the team used a different set of supernova data, known as DES Y5, the standard model regained the upper hand, indicating that the answer may depend on which specific observations are weighed most heavily. Despite this nuance, the study demonstrates that a modified version of gravity can successfully explain the accelerating universe without requiring a cosmological constant.

The researchers also looked at a key moment in cosmic history: the transition point when the universe stopped slowing down under its own gravity and began to speed up. Both of their new models predicted that this shift happened at a cosmic redshift of roughly 0.66 to 0.69, a value that aligns perfectly with what we see in the standard model and with other independent observations. This agreement is crucial; it means that even if the underlying theory of gravity is different, the timeline of the universe's evolution remains consistent with what we observe. The study concludes that while the standard model remains a robust description of the cosmos, the square-root exponential form of modified gravity offers a compelling and statistically superior alternative for explaining the late-time acceleration of the universe. It stands as a viable path forward, suggesting that the secret to the universe's expansion might lie not in a hidden energy, but in a deeper understanding of the geometry of space itself.

Technical Summary: Testing f(Q) Gravity with DESI DR2 and Strong-Lensing Time Delays

Problem Statement
The late-time accelerated expansion of the Universe remains a fundamental challenge in modern cosmology. While the standard $\Lambda$ CDM model, attributing acceleration to a cosmological constant, successfully describes a wide range of observations, it faces unresolved theoretical issues, including the coincidence problem, the cosmological constant problem, and the Hubble tension. Consequently, there is a motivation to explore modified theories of gravity that can explain cosmic acceleration without invoking a cosmological constant. This work investigates symmetric teleparallel gravity, a geometric framework where gravity is characterized by non-metricity ( $Q$ ) rather than curvature or torsion. Specifically, the study tests two specific extensions of this framework, $f(Q)$ gravity, to determine if they offer a statistically superior or competitive description of the Universe's expansion history compared to the standard flat $\Lambda$ CDM model.

Methodology
The authors analyze two specific functional forms of $f(Q)$ gravity within a spatially flat FLRW background:

Normalized Power-law Model ( $f_I(Q)$ ): Defined as $f(Q) = Q + \alpha Q_0 (Q/Q_0)^n$ , where $n$ controls the deviation from linearity.
Square-root Exponential Model ( $f_{II}(Q)$ ): Defined as $f(Q) = Q + \alpha Q_0 [1 - \exp(-m\sqrt{Q/Q_0})]$ , where $m$ characterizes the exponential correction.

To constrain the free parameters of these models ( $H_0$ , $\Omega_{m0}$ , and the model-specific parameters $n$ or $m$ ), the authors employed Markov Chain Monte Carlo (MCMC) analyses using a comprehensive suite of observational datasets:

Cosmic Chronometers (CC): 32 direct measurements of the Hubble parameter $H(z)$ .
DESI DR2: Baryon Acoustic Oscillation (BAO) measurements from the Dark Energy Spectroscopic Instrument (Data Release 2).
Strong Gravitational Lensing (Time Delay): Observations from the H0LiCOW sample (seven lens systems) providing time-delay distances.
Type Ia Supernovae (SNIa): Three distinct compilations were used to test robustness: Pantheon+, Union 3.0, and DES Y5.

The statistical performance of the $f(Q)$ models was compared against the flat $\Lambda$ CDM model using the minimum chi-square ( $\chi^2_{min}$ ), the Akaike Information Criterion (AIC), and the Bayesian Information Criterion (BIC). These criteria were used to assess whether the improved fit of the modified gravity models justified the inclusion of an additional free parameter. The study also calculated the transition redshift ( $z_t$ ) from cosmic deceleration to acceleration for both models.

Key Contributions and Results

Parameter Constraints: The inclusion of SNIa data generally tightened constraints on $H_0$ and $\Omega_{m0}$ compared to using CC, DESI DR2, and Time-delay data alone. The posterior distributions revealed strong correlations between the modified-gravity parameters and the background expansion parameters.
Normalized Power-law Model ( $f_I(Q)$ ): This model provided a fit to the data comparable to $\Lambda$ CDM. While the minimum $\chi^2$ was slightly lower for SNIa-inclusive combinations, the AIC and BIC values remained positive (favoring $\Lambda$ CDM) or showed only marginal differences. This indicates that the additional parameter in the power-law model does not provide a statistically significant improvement over the standard model given current data.
Square-root Exponential Model ( $f_{II}(Q)$ ): This model demonstrated a substantially better statistical fit for three of the four dataset combinations (CC + DESI DR2 + Time-delay, + Pantheon+, and + Union 3.0).
- For the Pantheon+ and Union 3.0 combinations, the model yielded negative $\Delta$ AIC and $\Delta$ BIC values (e.g., $\Delta$ AIC $\approx -10.5$ for Pantheon+), indicating very strong to strong evidence in favor of $f_{II}(Q)$ over $\Lambda$ CDM, even after penalizing for the extra parameter.
- However, for the DES Y5 combination, the information criteria favored $\Lambda$ CDM, highlighting that the statistical preference depends on the specific supernova compilation used.
Transition Redshift: Both models predicted a transition redshift $z_t$ in the range of 0.66 to 0.69, which is consistent with observational estimates and the standard $\Lambda$ CDM prediction. This suggests that the improved statistical fit of the square-root exponential model does not come at the cost of an unrealistic expansion history.

Significance and Claims
The paper concludes that $f(Q)$ gravity is a viable alternative to $\Lambda$ CDM for explaining late-time cosmic acceleration without a cosmological constant. The authors claim that the square-root exponential form of $f(Q)$ gravity is particularly promising, as it provides a significantly better statistical fit to current observational data (specifically when combined with Pantheon+ and Union 3.0 SNIa data) than the standard model, a preference that persists even when accounting for model complexity.

The study emphasizes that the functional form of $f(Q)$ is critical; while the normalized power-law model remains competitive but not preferred, the square-root exponential model offers a distinct improvement. The authors note that their results are based on background expansion history and that a complete assessment would require future work on cosmological perturbations and structure formation. They also caution that the preference for the square-root exponential model is not universal across all supernova compilations (specifically noting the DES Y5 exception), suggesting that larger and more homogeneous datasets are needed to confirm these findings.

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