⚛️ general relativity

Observational constraints on scalar-nonmetricity cosmology: the role of the connection

This paper systematically constrains three distinct connection branches of scalar-nonmetricity cosmology using modern observational data, finding that while all can reproduce cosmic acceleration, Connection I is statistically favored over the standard $\Lambda$ CDM model according to the Akaike information criterion.

Original authors: Ghulam Murtaza, Avik De, Emmanuel N. Saridakis

Published 2026-10-20

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CC BY 4.0

Original authors: Ghulam Murtaza, Avik De, Emmanuel N. Saridakis

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

The universe is expanding, and not just expanding, but accelerating. This discovery, made in the late 1990s, shattered the long-held belief that gravity would eventually slow the cosmos down. To explain this cosmic speed-up, scientists have proposed a mysterious force called dark energy, often modeled as a cosmological constant—a steady, unchanging energy density filling space. This standard model, known as Lambda Cold Dark Matter, works remarkably well, but it leaves many physicists uneasy. It treats gravity as the curvature of spacetime, a concept introduced by Einstein, but it does not explain why the universe is accelerating or what dark energy actually is.

In recent years, a different way of looking at gravity has gained traction. Instead of focusing on curvature, some theorists describe gravity through a property called nonmetricity. Imagine spacetime not as a flexible fabric that bends, but as a grid where the rules for measuring distance and time change from point to point. In this framework, gravity is encoded in how these measuring rules shift. A natural extension of this idea involves adding a scalar field—a single value that varies across space and time—to the mix. This creates a theory where gravity and this new field interact in complex ways. A unique feature of this approach is that even in a perfectly smooth, flat universe, the mathematical "connection" that links different points in space is not fixed. There are three distinct, valid ways to set up this connection, and until now, it was unclear if these different mathematical choices would lead to different, observable realities.

A team of researchers set out to answer this question by putting these three possibilities to the test against real-world data. They focused on a specific version of this theory where the scalar field has a "steep" potential, meaning its energy changes rapidly, and where it interacts with the nonmetricity field in a specific, power-law fashion. The researchers took the equations governing the three different connection branches and ran them against the most comprehensive astronomical datasets available today. These included observations of Type Ia supernovae, which act as cosmic mile markers to measure expansion; baryon acoustic oscillations, which are frozen sound waves from the early universe acting as a standard ruler; and compressed data from the cosmic microwave background, the afterglow of the Big Bang.

The results revealed that while all three connection branches can successfully reproduce the history of our universe—transitioning from a matter-dominated past to the current era of accelerated expansion—they do not all behave the same way when faced with the data. The first connection branch proved to be highly sensitive to which supernova data was used. When the team swapped one major supernova catalog for another, the inferred values for the universe's expansion rate and the current strength of dark energy shifted noticeably. This branch showed a great deal of flexibility, able to mold itself to different datasets, but this flexibility came at a cost.

In contrast, the second connection branch displayed a remarkable stability. No matter which combination of data the researchers used, the reconstructed history of the universe remained almost identical. The values for the expansion rate and the density of matter stayed tightly clustered, suggesting that this specific mathematical setup is far more rigid and less dependent on the specific details of the observational data. The third branch fell somewhere in between, showing a stable expansion rate but allowing for more variation in the estimated amount of matter in the universe.

The most striking finding emerged when the team compared how well each model fit the data compared to the standard cosmological model. Using a specific statistical tool that balances how well a model fits the data against how many extra parameters it requires, the researchers found that the standard model usually wins. However, there was one exception. When the team used a specific, newer set of supernova data combined with other measurements, the first connection branch provided a significantly better fit to the observations than the standard model. In this specific case, the improvement in the fit was large enough to justify the extra complexity of the new theory. The other two branches did not show this same advantage; they fit the data no better than the standard model, and their added complexity made them statistically less favorable.

This study demonstrates that the choice of mathematical connection in this theory is not just a technical detail hidden in the equations. It has real, observable consequences. The fact that one branch can adapt to fit new data better than the standard model, while the others remain rigid, suggests that the geometry of gravity itself might be testable. The researchers conclude that while the standard model remains the most economical description for most datasets, the freedom inherent in these alternative theories allows for scenarios that could be favored by future, more precise observations. The work opens a path to distinguishing between different geometric realizations of gravity, moving beyond abstract mathematics to see which version of the universe's structure actually matches the sky we observe.

Technical Summary: Observational Constraints on Scalar-Nonmetricity Cosmology: The Role of the Connection

Problem Statement
Scalar-nonmetricity gravity offers a natural extension of symmetric teleparallel gravity (STG), where gravitational interaction is encoded in nonmetricity ( $Q$ ) rather than curvature or torsion. A distinctive feature of STG is that, even for a spatially flat Friedmann-Robertson-Walker (FRW) geometry, cosmological symmetry does not uniquely determine the affine connection. Instead, three inequivalent connection branches exist. While previous studies have explored the dynamical properties of scalar-nonmetricity models with steep potentials, a systematic observational comparison of these three distinct connection branches has been lacking. The central question addressed is whether this geometrical freedom leads to observable consequences that can distinguish the branches or favor one over the standard $\Lambda$ CDM paradigm.

Methodology
The authors construct a scalar-nonmetricity framework where a scalar field $\phi$ couples nonminimally to the nonmetricity scalar $Q$ via a function $f(\phi) = \beta \phi^m$ . The scalar field is governed by a steep potential $U(\phi) = U_0 \exp[\alpha (\phi/M_{Pl})^n]$ with $n > 1$ , and the kinetic sector is defined by a constant $h(\phi) = h_0$ .

Theoretical Formulation: The authors derive the background cosmological equations for all three allowed connection branches (labeled I, II, and III) in a spatially flat FRW metric.
- Connection I: The affine function $\gamma(t)$ drops out of the background equations, recovering the standard coincident gauge expression $Q = -6H^2$ .
- Connection II & III: The connection function $\gamma(t)$ (or its rescaled form) enters explicitly into the nonmetricity scalar and the field equations, introducing connection-dependent contributions to the cosmological evolution.
  The system is reformulated using dimensionless variables ( $x, y, \zeta, \eta$ ) suitable for numerical integration with respect to redshift $z$ .
Observational Analysis: The models are constrained using a Markov Chain Monte Carlo (MCMC) analysis implemented within the COBAYA framework. The datasets employed include:
- Type Ia Supernovae: Pantheon Plus and DES Year 5 (DES Y5) compilations.
- Baryon Acoustic Oscillations (BAO): DESI DR2 measurements.
- Cosmic Microwave Background (CMB): Compressed CMB likelihood (acoustic scale $\theta_*$ and physical densities $\omega_b, \omega_{bc}$ ).
  Three dataset combinations are tested: (Pantheon Plus + CMB + DESI DR2), (DES Y5 + CMB + DESI DR2), and (CMB + DESI DR2).
Statistical Comparison: The authors compare the performance of the three connection branches against the standard $\Lambda$ CDM model using the minimum chi-square ( $\chi^2_{min}$ ) and the Akaike Information Criterion (AIC) to penalize models with additional free parameters.

Key Contributions and Results
The study provides the first systematic observational confrontation of all three STG connection branches within a scalar-nonmetricity framework with a steep potential.

Viability: All three branches successfully reproduce the transition from a matter-dominated era to late-time accelerated expansion.
Connection-Dependent Sensitivity: The observational behavior of the models is strongly dependent on the choice of connection:
- Connection I: Exhibits the strongest sensitivity to the supernova compilation. Replacing Pantheon Plus with DES Y5 results in significant shifts in the inferred Hubble constant ( $H_0$ ) and the present effective equation of state ( $w_{eff,0}$ ).
- Connection II: Yields remarkably stable reconstructed histories. The three dataset combinations produce nearly coincident expansion histories and parameter constraints, showing weak dependence on the specific supernova sample used.
- Connection III: Displays an intermediate behavior. The value of $H_0$ is highly stable across datasets, but the reconstructed matter density fraction shows slightly more freedom compared to Connection II.
Statistical Performance (Model Comparison):
- For Pantheon Plus + CMB + DESI DR2, Connection I slightly improves $\chi^2_{min}$ over $\Lambda$ CDM ( $\Delta \chi^2_{min} = -1.18$ ), but the AIC favors $\Lambda$ CDM ( $\Delta AIC = 6.82$ ) due to the penalty for extra parameters. Connections II and III are disfavored.
- For DES Y5 + CMB + DESI DR2, Connection I achieves a substantial improvement in fit ( $\Delta \chi^2_{min} = -13.99$ ). This reduction is sufficient to overcome the AIC penalty, resulting in $\Delta AIC = -5.99$ , making Connection I the only model favored over $\Lambda$ CDM in this specific dataset combination.
- For CMB + DESI DR2 (supernova-independent), all three connections yield $\chi^2_{min}$ values very close to $\Lambda$ CDM, and the AIC again favors the more economical $\Lambda$ CDM model.

Significance and Claims
The paper claims that the choice of symmetric-teleparallel connection is not merely a technical gauge choice but has observable consequences at the background level. The analysis demonstrates that:

The same scalar-nonmetricity action and potential can lead to distinct posterior structures and statistical responses depending on the connection branch.
Connection I possesses significant phenomenological flexibility that allows it to fit specific supernova datasets (DES Y5) better than $\Lambda$ CDM, whereas Connections II and III are more rigid and dataset-independent.
The ability of one branch to exploit geometrical freedom to improve the fit to modern data, while others cannot, highlights that the affine connection in scalar-nonmetricity gravity is observationally distinguishable.

The authors conclude that while the additional freedom of scalar-nonmetricity gravity is not generically rewarded by current data, the specific case of Connection I with DES Y5 data suggests that the geometrical realization of nonmetricity gravity is physically relevant and distinguishable. Future work is suggested to include linear perturbations, full CMB likelihoods, and Bayesian model comparison to further test these distinctions.

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