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Testing Extended Exponential Infrared Teleparallel Gravity with CMB, DESI, and PantheonPlus

This study constrains an extended exponential infrared $f(T)$ teleparallel gravity model using combined CMB, DESI, and PantheonPlus datasets, finding that while the model successfully recovers standard early-universe behavior and exhibits late-time dark energy dynamics, it remains statistically indistinguishable from $\Lambda$ CDM with CMB data alone and shows only a weak preference for $\Lambda$ CDM when full datasets including SH0ES are incorporated.

Original authors: Mahmoud Hashim, Jackson Levi Said, Waleed El Hanafy, Eleonora Di Valentino

Published 2026-10-30

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CC BY 4.0

Original authors: Mahmoud Hashim, Jackson Levi Said, Waleed El Hanafy, Eleonora Di Valentino

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 slowly drifting apart, but speeding up in its journey through space. For decades, the standard explanation for this acceleration has been a mysterious force called dark energy, often imagined as a constant push inherent to space itself. This idea, known as the cosmological constant, fits most of our observations well, yet it leaves scientists with nagging theoretical problems and a stubborn mismatch in measurements. When astronomers look at the ancient light left over from the Big Bang, they calculate a certain speed for the universe's expansion today. But when they measure the expansion by looking at nearby exploding stars and other cosmic markers, they get a faster speed. This disagreement, known as the Hubble tension, suggests that our current understanding of gravity or the universe's contents might be incomplete.

To address this, a team of researchers has explored a different way of thinking about gravity. Instead of assuming the universe is curved like a bent sheet, as described by Einstein's general relativity, they investigated a framework where gravity arises from a twisting property of space called torsion. In this view, the acceleration of the universe isn't caused by a mysterious fluid, but by the geometry of spacetime itself changing over time. The researchers tested a specific, extended version of this theory, which introduces a new adjustable knob to control how this twisting gravity behaves in the recent history of the cosmos. By comparing their model against a vast collection of modern astronomical data, they sought to see if this geometric twist could resolve the tension in expansion rates without breaking the rest of our cosmic story.

The team, led by cosmologists from institutions in Egypt, Malta, and the United Kingdom, focused on a model called Extended Exponential Infrared teleparallel gravity. In this framework, the force of gravity is described not by the curvature of space, but by a torsion scalar, a measure of how much spacetime twists. The researchers proposed that this torsion acts like an effective dark energy, evolving smoothly over billions of years. They introduced an extra parameter, a number that dictates how the strength of this torsion changes as the universe ages. Their goal was to see if this single addition could allow the model to match the faster expansion rates seen in local measurements while still agreeing with the precise patterns of the ancient cosmic microwave background radiation.

To test this, the researchers combined the most comprehensive datasets available. They used observations of the cosmic microwave background, the afterglow of the Big Bang, from satellites and ground-based telescopes. They also incorporated data on the distribution of galaxies from the Dark Energy Spectroscopic Instrument and measurements of distant supernovae from the PantheonPlus catalog. These observations act as a cosmic ruler and clock, allowing scientists to trace how the universe has expanded from its earliest moments to the present day. The team ran complex statistical analyses to see how well their twisted-gravity model fit these numbers compared to the standard model of cosmology.

The results revealed a nuanced picture. When looking only at the ancient light from the early universe, the new model performed just as well as the standard theory, producing nearly identical predictions for the patterns seen in the cosmic background. In fact, the model allowed for a slightly higher expansion rate than the standard model, bringing it closer to the faster speeds measured by local astronomers. The researchers found that the best fit for their model required a specific value for their new parameter, placing it between 0.5 and 1. This value suggests that the effective dark energy in their model started out behaving like a standard accelerating force and then transitioned into a more extreme, "phantom-like" state as the universe aged.

However, the story changed when the researchers added the data from the nearby, recent universe. While the model could produce a higher expansion rate, it lacked the flexibility to fit all the new observations simultaneously. When the team combined the ancient cosmic background data with the local measurements of galaxies and supernovae, the standard model of cosmology still provided a better overall fit to the data. The new model, while successful at raising the expansion rate, was too rigid in its predictions for the recent history of the universe. It could not stretch enough to accommodate the full range of distances and speeds observed by modern telescopes.

Ultimately, the study concludes that while this twisted-gravity theory is a compelling alternative that preserves the successes of the standard model in the early universe, it does not fully solve the Hubble tension. The researchers found that the model is statistically indistinguishable from the standard theory when only early-universe data is considered, but the standard model remains slightly preferred when all available data is taken into account. The work demonstrates that modifying gravity can indeed shift the predicted expansion rate, but it also highlights the difficulty of finding a theory that is flexible enough to satisfy both the ancient and the modern universe at the same time. The search for a complete explanation of cosmic acceleration continues, with this study narrowing the path by showing exactly where this particular geometric idea succeeds and where it falls short.

Technical Summary: Testing Extended Exponential Infrared Teleparallel Gravity with CMB, DESI, and PantheonPlus

Problem Statement
The accelerated expansion of the Universe remains a central challenge in modern cosmology. While the standard $\Lambda$ CDM model successfully fits a wide range of observations, it relies on a cosmological constant ( $\Lambda$ ) that suffers from theoretical issues such as fine-tuning and coincidence problems. Furthermore, a significant tension exists between early-Universe determinations of the Hubble constant ( $H_0$ ) from Cosmic Microwave Background (CMB) data and late-Universe local measurements (e.g., SH0ES). Modified gravity theories, particularly those based on teleparallel geometry where gravity is described by torsion rather than curvature, offer a potential geometric alternative to dark energy. This paper investigates the Extended Exponential Infrared (EExpIR) $f(T)$ teleparallel gravity model, which introduces an additional parameter to govern late-time dynamics, aiming to determine if this geometric modification can alleviate the $H_0$ tension while remaining consistent with the full suite of cosmological data.

Methodology
The authors employ a comprehensive observational analysis combining background expansion history and linear perturbation evolution.

Theoretical Framework: The study focuses on the EExpIR model defined by $f(T) = T e^{\beta(T_0/T)^b}$ , where $T$ is the torsion scalar, $T_0 = 6H_0^2$ , and $b$ is a dimensionless parameter governing the infrared modification. The parameter $\beta$ is constrained by the present-day Friedmann equation via the Lambert-W function, leaving $b$ as the sole free parameter relative to $\Lambda$ CDM.
Phase-Space Analysis: Before confronting data, the authors perform a dynamical systems analysis to identify viable cosmological branches. They examine the Lambert-W solution branches and the evolution of the torsion equation of state ( $w_T$ ). This analysis excludes branches that fail to reproduce the standard matter-dominated (sCDM) era or lead to unphysical singularities, selecting the principal branch with $0.5 < b < 1$ as the physically viable regime. This range allows for a smooth transition from quintessence-like to phantom-like behavior.
Perturbation Theory: The study derives linear perturbation equations in the Newtonian gauge for the EExpIR model. Unlike General Relativity (GR), the tetrad formalism introduces an extra scalar degree of freedom ( $\zeta$ ) constrained by the antisymmetric field equations. This leads to a non-vanishing gravitational slip ( $\psi \neq \phi$ ) even in the absence of anisotropic stress, characterized by effective phenomenological functions ( $Q, \mu, \Sigma, \Xi$ ).
Observational Data & Statistical Inference: The model is constrained using the Cobaya package and a modified CLASS solver. The datasets include:
- CMB-SPA: A combined compilation of Planck 2018, ACT DR6, and SPT-3G data (temperature, polarization, and lensing).
- DESI: Baryon Acoustic Oscillation (BAO) measurements from DESI Data Release 2.
- PantheonPlus (PP): 1550 Type Ia supernovae.
- SH0ES: Distance-ladder calibration included in the PPS dataset.
  The analysis utilizes Bayesian inference (Metropolis–Hastings) to sample the posterior distributions of cosmological parameters, including $b$ . Model performance is evaluated using $\chi^2$ , the Akaike Information Criterion (AIC), and Bayesian evidence (Bayes factor) against the standard $\Lambda$ CDM model.

Key Contributions

Branch Selection: The paper rigorously identifies the principal branch ( $0.5 < b < 1$ ) of the EExpIR model as the only one capable of reproducing the standard early-Universe evolution while allowing for late-time acceleration, justifying the prior range used in subsequent analyses.
Full Perturbation Analysis: Unlike previous studies focusing solely on background observables, this work incorporates the full linear perturbation equations, including the effects of the extra tetrad degree of freedom on the gravitational slip and CMB anisotropies.
Comprehensive Dataset Integration: The study combines the latest high-precision CMB data (Planck, ACT, SPT) with the newest late-time probes (DESI BAO and PantheonPlus) to test the model's viability across cosmic history.

Results

Parameter Constraints: Using the full dataset (CMB-SPA + PPS + DESI), the model constrains the new parameter to $b = 0.573^{+0.016}_{-0.060}$ at 68% CL. This value corresponds to a dark-energy sector that transitions from a quintessence-like phase to a phantom regime at a redshift of $z_{trans} \approx 1.29$ .
Hubble Constant: The model yields $H_0 = 70.67^{+0.31}_{-0.35}$ km s $^{-1}$ Mpc $^{-1}$ (with SH0ES calibration), which is higher than the corresponding $\Lambda$ CDM value derived from the same dataset and closer to local measurements.
CMB Fit: The EExpIR model provides a fit to the CMB-SPA data that is statistically indistinguishable from $\Lambda$ CDM ( $\Delta \chi^2 = -2.24$ ). The model predicts a mild enhancement in the TT power spectrum at low multipoles ( $\ell \lesssim 5$ ) due to modified late-time Integrated Sachs-Wolfe (ISW) effects and gravitational slip, but these deviations remain within observational uncertainties.
Model Comparison:
- CMB Only: The models are statistically indistinguishable.
- Full Dataset: When including late-time probes (DESI and PantheonPlus), the $\Lambda$ CDM model achieves a better fit to the combined data. The Bayes factor indicates a weak preference for $\Lambda$ CDM ( $\ln B_{ij} = -1.61$ ) when SH0ES is included, and a strong preference when SH0ES is excluded.
$H_0$ Tension: While the EExpIR model reduces the tension with local $H_0$ measurements compared to $\Lambda$ CDM (e.g., $2.34\sigma$ vs $5.13\sigma$ relative to SH0ES for CMB-only data), it does not fully resolve the discrepancy. The rigidity of the model's low-redshift expansion history prevents it from accommodating the combined distance measurements as effectively as $\Lambda$ CDM.

Significance and Claims
The authors conclude that the EExpIR $f(T)$ model represents a viable geometric alternative to dark energy that successfully mimics the standard cosmological evolution at early times while generating a dynamical dark-energy sector at late times. The model's ability to yield a higher $H_0$ while preserving the successful CMB phenomenology of $\Lambda$ CDM highlights the potential of late-time gravitational modifications.

However, the paper modestly claims that the model does not successfully resolve the Hubble tension when confronted with the full suite of modern cosmological data. The additional parameter $b$ provides a richer dark-energy evolution (quintessence-to-phantom transition) but lacks the necessary flexibility to simultaneously fit the CMB, BAO, and supernova data as well as the standard model. The study emphasizes that while the EExpIR model is statistically competitive with $\Lambda$ CDM in CMB-only analyses, the inclusion of low-redshift probes exposes the limitations of its specific expansion history. The authors assert that future high-precision measurements of large-scale structure and weak lensing will be crucial for further testing the torsional sector of such modified gravity theories.

Technical Summary: Testing Extended Exponential Infrared Teleparallel Gravity with CMB, DESI, and PantheonPlus

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