Modified spacetime geometry offers an alternative path to cosmic acceleration

A new statistical analysis shows that modifying gravitational geometry can match astronomical observations of the expanding universe without invoking a cosmological constant.

Astronomers studying the distant reaches of the cosmos face a puzzle that has resisted resolution for nearly three decades: the expansion of the universe is speeding up over time.¹² In everyday physics, matter attracts other matter through gravity, which would lead one to expect the mutual gravitational pull of billions of galaxies to act as a brake on cosmic expansion. Yet deep astronomical observations across vast spans of space reveal that galaxies are flying apart from one another at an ever-accelerating pace. To make standard gravitational equations match this observation, physicists have traditionally added an invisible, uniform energy component known as dark energy or the cosmological constant, which accounts for the missing repulsive push across the universe.¹

That conventional picture, known as the Lambda-Cold Dark Matter cosmological framework, has successfully matched many observations of the cosmos. But it brings serious theoretical headaches. The energy density required for this cosmological constant is vastly smaller than the vacuum energy density predicted by quantum field theory, creating an immense theoretical discrepancy.¹ In addition, measurements of the cosmic expansion rate taken from the early universe disagree with direct measurements taken in the local universe.¹³ Rather than inventing an unobserved dark energy fluid to generate cosmic repulsion, an alternative line of physical reasoning asks whether the geometric laws governing gravity itself need adjustment over cosmic distances.¹

How can spacetime geometry explain accelerating cosmic expansion?

Spacetime geometry can generate cosmic acceleration when the mathematical connection describing how vectors move across space permits lengths to stretch, a property known in differential geometry as non-metricity.¹² In Albert Einstein's general theory of relativity, gravity is understood as the curvature of spacetime, while length preservation along parallel paths is taken as an underlying axiom.¹² But general affine spacetime geometry allows three distinct geometric manifestations: curvature, torsion, and non-metricity.¹ In a theoretical framework called symmetric teleparallel gravity, curvature and torsion are both set to zero, leaving non-metricity to carry the entire gravitational interaction.¹

The physical causal chain operates through straightforward geometric steps across expanding space. In standard spacetime, when a physical ruler travels along a trajectory, its measured length remains fixed even as its orientation might change. Under non-metricity, the covariant derivative of the metric tensor does not vanish, meaning the intrinsic measure of length itself varies as it traverses the gravitational field.¹² When this non-metricity is generalized from a simple linear term to a non-linear mathematical function, labelled f(Q), the extra geometric terms enter the cosmic expansion equations.¹ As the universe expands and matter dilutes, these non-linear geometric terms become dominant, creating an effective acceleration in cosmic expansion without demanding an actual cosmological constant.¹

Darshan, an author of the new research at the Henan Academy of Sciences, explained the intuitive physical reasoning underlying this mechanism in response to questions from Primary. "One way to picture this is to think of spacetime as something that is continuously stretching as the Universe expands," Darshan said, explaining that "the way this stretching responds can depend on the expansion itself."^contributed As matter dilutes across cosmic history, non-linear geometric terms become more prominent in determining the overall expansion rate. "In this sense, the effect can look like a repulsive component such as dark energy, but there is no additional energy component or physical force pushing the galaxies apart," Darshan told Primary.^contributed "The accelerated expansion comes from the changing dynamics of spacetime geometry itself. So what we interpret as the effect of dark energy in the standard picture can, in this case, emerge from the geometry of gravity."^contributed

To test whether such a geometric modification accurately reflects the real universe, cosmologists compare its predictions against multiple independent probes of cosmic history. Passive, ancient galaxies called cosmic chronometers serve as clocks to measure the expansion rate across different epochs.¹ Sound waves frozen into the distribution of matter from the early universe, known as baryon acoustic oscillations, act as standard cosmic rulers.¹ Exploding stars called Type Ia supernovae provide standard candles of known brightness to map cosmological luminosity distances.¹ Finally, light rays from distant variable sources bent around intervening galaxies create strong gravitational lensing time delays, measuring absolute geometric distances across billions of light years.³

What do the latest cosmological datasets reveal about f(Q) gravity?

A statistical analysis combining recent cosmological surveys finds that a specific square-root exponential form of f(Q) gravity provides a superior statistical fit to observational data compared to the standard Lambda-Cold Dark Matter model.¹⁵ In a research preprint posted to the arXiv server, Darshan Kumar and Fengge Zhang of the Henan Academy of Sciences, alongside Saibal Ray of GLA University, Nisha Rani of the University of Delhi, Praveen Kumar Dhankar of Symbiosis International University, and Jie Zheng of the Qingdao University of Science and Technology, investigated two separate f(Q) formulations against an extensive collection of cosmological measurements.¹

The authors reported that the square-root exponential model achieves a lower minimum chi-square goodness-of-fit value and better information criteria scores than the standard model when combining cosmic chronometers, strong lensing time delays, and baryon acoustic oscillations from Data Release 2 of the Dark Energy Spectroscopic Instrument with the Pantheon Plus or Union 3.0 supernova compilations.¹ The researchers also examined a normalized power-law formulation of f(Q) gravity, finding that it remains statistically competitive with standard cosmology across supernova-inclusive datasets, though statistical information criteria penalizing extra model parameters do not favor its additional degree of freedom.¹

Darshan explained why the two mathematical formulations behave differently when confronted with observational data. "The main difference is in how the two models modify the gravitational dynamics as the Universe evolves," Darshan told Primary.^contributed "The power law has a more restricted dependence on the geometric quantity, so its effect changes in a relatively fixed way with redshift. This makes it harder to accommodate the different behaviours seen across the observational datasets at the same time."^contributed By contrast, "the square root exponential form allows a more gradual change in the geometric modification as the Universe evolves, giving it more flexibility to follow the observed expansion and distance measurements over the full redshift range," Darshan said.^contributed

Both modified gravity models successfully reproduce the historical transition of the cosmos from an early decelerating phase into its present accelerating epoch.¹ The paper states that the computed transition redshifts, which mark the point in cosmic history where the universe switched from slowing down to speeding up, remain consistent across different observational dataset combinations and match empirical estimates of the cosmic acceleration epoch.¹ By providing a natural mechanism for this cosmic transition, the modified geometry accounts for late-time cosmic dynamics directly through the non-metricity scalar.¹

How did the researchers measure and test these geometric models?

The research team evaluated the geometric models by setting up cosmological equations within a spatially flat Friedmann-Lemaitre-Robertson-Walker universe and constraining their free parameters using Markov chain Monte Carlo sampling. In this cosmological setting, the non-metricity scalar simplifies to six times the square of the Hubble parameter.¹ By applying the boundary condition that the expansion rate at a redshift of zero equals the present-day Hubble constant, the authors expressed the background evolution for each model using the Hubble constant, the matter density parameter, and one additional dimensionless model parameter.

To confront the theoretical equations with reality, the authors assembled an observational baseline comprising four distinct types of cosmological data. For direct expansion rate measurements, the study used 32 Hubble parameter data points from cosmic chronometers, which determine the differential age of passively evolving galaxies where star formation has ceased.¹ For standard rulers, the analysis incorporated baryon acoustic oscillation measurements from the Dark Energy Spectroscopic Instrument Data Release 2, which captures acoustic density ripples mapped through millions of galaxies, quasars, and the high-redshift Lyman-alpha forest.¹⁶

For distance calibrations, the authors tested three separate Type Ia supernova compilations: the Pantheon Plus sample of 1,590 light curves spanning redshifts from 0.01 to 2.261, the Union 3.0 compilation of 2,087 supernovae analyzed with Bayesian hierarchical modeling, and the Dark Energy Survey five-year dataset containing 1,635 photometrically classified supernovae. To anchor absolute distance scales, the researchers integrated strong gravitational lensing time-delay observations from multiply imaged quasars, where differences in light travel times across gravitational potentials provide geometric measurements independent of the local cosmic distance ladder.

What are the limitations of these modified gravity findings?

These findings represent a statistical parameter estimation and model comparison rather than direct experimental proof of modified spacetime geometry. The study is based on numerical fits to cosmological datasets under the standard assumption of a homogeneous and isotropic spatially flat universe evaluated within the coincident coordinate gauge, where the affine connection vanishes globally.¹ If the large-scale universe possesses significant spatial curvature or breaks homogeneity, the underlying mathematical simplifications of the field equations would require substantial revision. Furthermore, the analysis tests background cosmic expansion history and does not fully incorporate the perturbation-level growth of large-scale structures, where modified gravity theories often produce subtle signatures distinct from general relativity. The research is currently posted as a preprint on arXiv and has not yet completed formal peer review.⁵

Darshan acknowledged these boundaries in discussions with Primary, clarifying that the background calculations do not capture how the geometry behaves under local perturbations. "The coincident gauge and spatially flat background provide a convenient description for studying the background expansion, but they do not by themselves tell us how the model behaves in more general situations," Darshan said.^contributed "In particular, the present analysis does not include cosmological perturbations, so we cannot yet make a strong statement about the stability of the model or its behaviour for the growth of cosmic structures. Extending the analysis to perturbations, including possible effects on structure growth and gravitational potentials, is therefore an important next step."^contributed

Extrapolating the geometric framework further back in cosmic history also remains an open theoretical boundary. Asked how far into the early universe the model can extend before clashing with primordial physics, Darshan emphasized that the current investigation focuses strictly on late cosmic times. "Our analysis is mainly focused on the late time Universe, so we have not extrapolated the model quantitatively all the way to the nucleosynthesis era," Darshan said.^contributed "The important point is that any viable modified gravity model must recover the standard cosmological behaviour at sufficiently high redshift, otherwise it could affect the expansion rate during Big Bang nucleosynthesis and change the predicted light element abundances. Therefore, extending our model to the early Universe and testing it against BBN and other early Universe constraints is an important next step. At this stage, we would not claim that the present analysis by itself establishes the validity of the model back to the nucleosynthesis epoch."^contributed

What do these results mean for the future of cosmology?

Demonstrating that modified non-metricity can match or outperform the standard cosmological model opens a promising avenue for resolving fundamental cosmological paradoxes without invoking hypothetical energy fields. If cosmic acceleration emerges from the geometric structure of spacetime rather than an added cosmological constant, the enormous theoretical fine-tuning problem associated with vacuum energy calculations disappears. A geometric origin for acceleration also changes how astrophysicists interpret discrepancies between early-universe sound horizon calibrations and late-universe distance measurements.

Future astronomical surveys will provide the data necessary to confirm or falsify these modified gravity models against standard cosmological predictions. Asked what measurements could rule out the square-root exponential model, Darshan pointed to joint constraints on cosmic expansion and the formation of cosmic structure. "I think the most important test would be a precise measurement of the expansion history of the Universe together with the growth of cosmic structures," Darshan told Primary.^contributed "Our model gives a different late time evolution compared with the cosmological constant, so if future surveys measure the expansion and structure growth with enough precision and find them consistent with the \Lambda CDM predictions, while excluding the parameter range of our model, then the model would be ruled out."^contributed

Rather than relying on any single survey, Darshan stressed that independent tests will decide the question. "The combination of future BAO, supernova and large scale structure measurements would be much more powerful," Darshan said.^contributed "If these independent observations consistently show no deviation from the \Lambda CDM prediction, then there would be no need for the modified gravity explanation and the cosmological constant would be the simpler model."^contributed

This piece was prepared from the arXiv preprint and public records together with answers from Darshan to five questions from the Primary news team, completed September 2026.

References

This article is based on 15 sources, with 5 statements from 1 contributor, listed in the order they are cited.

1. 1 DK Darshan Kumar (HNAS), Saibal Ray, Fengge Zhang, Nisha Rani, Praveen Kumar Dhankar, Jie Zheng announcement · 1 Sep 2026 Testing $f(Q)$ Gravity with DESI DR2 and Strong-Lensing Time Delays See the source
2. 2 A ar5iv.labs.arxiv.org Cosmological Dynamics on a Novel f ( Q ) Gravity Model with Recent DESI DR2 Observation See the source
3. 3 A arxiv.org TDCOSMO 2025: Cosmological constraints from strong lensing time delays See the source
4. 4 D Darshan added 5 Sep 2026 Contributor commentary — Darshan 5 statements added to this article
5. 5 A arxiv.org Testing $f(Q)$ Gravity with DESI DR2 and Strong-Lensing Time Delays See the source
6. 6 D desi.lbl.gov DESI DR2 Results: March 19 Guide See the source
7. 7 A arxiv.deeppaper.ai Arxiv Paper Search Engine - Find AI Research Papers | Arxiv - DeepPaper See the source
8. 8 A arxiv.org Constraint on Symmetric Teleparallel Gravity with Different Dark energy Parametrizations from DESI DR2 BAO Data See the source
9. 9 A arxiv.org A cosmographic analysis using DESI- DR2 and strong lensing: II. Distance Ratio measurements See the source
10. 10 A arxiv.org Computer Science Aug 2026 See the source
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12. 12 A arxiv.org A cosmographic analysis using DESI- DR2 and strong lensing: I. Time- Delay measurements See the source
13. 13 D desi.lbl.gov DESI DR2 cosmology chains and data products released See the source
14. 14 N nature.com f(Q) gravity as a possible resolution of the H0 and S8 tensions with DESI DR2 - Scientific Reports See the source
15. 15 V validate.perfdrive.com Radware Bot Manager Captcha See the source
16. 16 V validate.perfdrive.com Radware Bot Manager Captcha See the source

Article history

1. 5 statements 5 Sep 2026, 07:21
   What was added

   "The coincident gauge and spatially flat background provide a convenient description for studying the background expansion, but they do not by themselves tell us how the model behaves in more general situations," Darshan said.

   On the record as reference 4 go to the sentence ↑
   What was added

   "Our analysis is mainly focused on the late time Universe, so we have not extrapolated the model quantitatively all the way to the nucleosynthesis era," Darshan said.

   On the record as reference 4 go to the sentence ↑
   What was added

   "The main difference is in how the two models modify the gravitational dynamics as the Universe evolves," Darshan told Primary.

   On the record as reference 4 go to the sentence ↑
   What was added

   "I think the most important test would be a precise measurement of the expansion history of the Universe together with the growth of cosmic structures," Darshan told Primary.

   On the record as reference 4 go to the sentence ↑
   What was added

   As the universe expands and matter dilutes, these non-linear geometric terms become dominant, creating an effective acceleration in cosmic expansion without demanding an actual cosmological constant.

   On the record as reference 4 go to the sentence ↑
   D Darshan · Contributor An author of the new research at the Henan Academy of Sciences.
2. Published 6 Sep 2026, 18:18
   Assembled by the Primary desk from 15 sources · 42 cited sentences