⚛️ general relativity

A cosmological model with logarithmic f(T) gravity and H(z) quadratic expansion

This paper investigates a logarithmic f(T) modified teleparallel gravity model, demonstrating through Pantheon+SH0ES and cosmic chronometer data that its quadratic H(z) expansion yields an accelerated universe with a deceleration parameter of $q = -0.435 \pm 0.028$ and effective energy dynamics resembling quintessence and phantom models.

Original authors: Adriel O. Aquino, J. E. G. Silva

Published 2026-08-18

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CC BY 4.0

Original authors: Adriel O. Aquino, J. E. G. Silva

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, a fact established over a century ago, but the rate at which it expands has become a source of deep confusion for modern astronomers. For decades, the standard model of cosmology has relied on a mysterious force called dark energy to explain why this expansion is speeding up rather than slowing down. In this standard view, dark energy acts like a cosmological constant, a fixed energy density that pushes space apart. However, measurements of the universe's expansion rate have begun to disagree with one another. When scientists look back at the early universe using the afterglow of the Big Bang, they calculate a slower expansion rate than when they look at nearby exploding stars in the recent universe. This discrepancy, known as the Hubble tension, suggests that our current understanding of gravity and cosmic evolution might be incomplete. To resolve this, physicists are exploring whether the rules of gravity themselves need to be rewritten, rather than just adding new invisible ingredients to the cosmic recipe.

In a recent study, researchers A. O. Aquino and J. E. G. Silva investigated a specific alternative to Einstein's theory of gravity called teleparallel gravity. Instead of describing gravity as the curvature of space and time, this approach treats gravity as a twisting force, or torsion, acting on a flat background. The authors proposed a modified version of this theory where the mathematical description of gravity includes a logarithmic correction term. This modification is designed to behave like the standard cosm constant in some limits but offers new possibilities for how the universe evolves. To test if this idea holds water, the team did not rely on theoretical speculation alone. They built a mathematical model for how the universe's expansion rate changes over time and then compared that model against a vast collection of real-world astronomical data.

The researchers combined three distinct types of observational evidence to put their theory to the test. First, they used data from cosmic chronometers, which are essentially aging galaxies that act as cosmic clocks, allowing scientists to measure the expansion rate at different points in history. Second, they analyzed light from thousands of Type Ia supernovae, which serve as standard candles to measure vast distances across the universe. Finally, they incorporated measurements of baryon acoustic oscillations from the Dark Energy Spectroscopic Instrument, which act as a standard ruler to map the large-scale structure of the cosmos. By fitting their modified gravity model to this combined dataset, the team was able to determine the most likely values for the parameters governing their theory.

The results showed that the modified teleparallel model successfully describes a universe that is currently accelerating. The best-fit parameters indicated a present-day expansion rate that sits comfortably within the range of recent local measurements. More importantly, the study calculated a deceleration parameter of roughly -0.42. In cosmological terms, a negative value confirms that the expansion is speeding up, and the specific number found here aligns well with what we observe in the late-time universe. The model also revealed that the effective energy driving this expansion behaves dynamically. Depending on the specific values chosen for the model's constants, this effective energy could act like quintessence, a slowly evolving field, or like phantom energy, a more extreme form that would eventually tear the universe apart. In both scenarios, the model naturally produces the observed acceleration without needing to invent a new fluid, suggesting the effect comes purely from the geometry of gravity itself.

To ensure the model was not just a mathematical curiosity but a physically viable description of reality, the authors performed a stability analysis. They examined how small fluctuations in the density of matter and the expansion rate would behave over time. If a theory is unstable, tiny ripples in the early universe would grow uncontrollably, leading to a chaotic cosmos that bears no resemblance to the one we see today. The study found that in their model, these fluctuations decay smoothly over time, fading away as the universe expands. This indicates that the cosmological background described by the theory is dynamically stable and robust against the kinds of disturbances that occur in the real universe.

The study concludes that this specific modification to teleparallel gravity offers a compelling alternative to the standard cosmological model. It reproduces the observed accelerated expansion, fits the available data from supernovae, galaxy ages, and cosmic structures, and remains stable under scrutiny. While the model does not yet solve the Hubble tension completely, as the derived expansion rate still aligns with local measurements rather than the early universe values, it demonstrates that changing the fundamental laws of gravity can naturally account for the dark energy phenomenon. The work suggests that the mysterious force driving the universe apart may not be a new substance at all, but rather a consequence of how gravity operates on the largest scales.

Technical Summary: Testing a Cosmological Teleparallel Modified f(T) Model with Parametric H(z)

Problem Statement
The standard $\Lambda$ CDM cosmological model, while successful, faces significant theoretical and observational challenges. Theoretically, the nature of the cosmological constant ( $\Lambda$ ) as the source of dark energy remains unexplained, and the model relies on the introduction of an exotic fluid with negative pressure. Observationally, the "Hubble tension" persists, characterized by a discrepancy between the Hubble constant ( $H_0$ ) derived from early-universe Cosmic Microwave Background (CMB) data ( $67.4 \pm 0.5$ km s $^{-1}$ Mpc $^{-1}$ ) and late-universe standard candle measurements ( $73.04 \pm 1.04$ km s $^{-1}$ Mpc $^{-1}$ ). Furthermore, the $\Lambda$ CDM model is based on General Relativity (GR), where gravity is described by spacetime curvature ( $R \neq 0$ ). Alternative formulations, such as Teleparallel Equivalent of General Relativity (TEGR), describe gravity via torsion ( $T \neq 0$ ) in a flat spacetime ( $R=0$ ). Modified gravity theories, specifically $f(T)$ gravity, offer a potential geometric explanation for cosmic acceleration without invoking exotic fluids, potentially alleviating the issues associated with the cosmological constant.

Methodology
The authors investigate a specific modified teleparallel gravity model where the gravitational Lagrangian is extended from the TEGR torsion scalar $T$ to a function $f(T)$ . The specific model proposed is:
$f(T) = -2\Lambda (\log_\gamma (1 - \lambda T))^n$
where $\Lambda$ , $n$ , and $\lambda$ are constants. This formulation is designed to recover a cosmological constant term when $n=0$ and provide power-law corrections for small $\lambda$ .

To test this model against observational data without assuming a specific cosmological solution a priori, the authors adopt a model-independent approach by parameterizing the Hubble expansion rate $H(z)$ as a quadratic function of redshift:
$H(z) = H_0(1 + \alpha z + \beta z^2)$
Here, $\alpha$ and $\beta$ act as free parameters linked to the constants in the $f(T)$ function.

The methodology involves a joint statistical analysis using the maximum likelihood method to constrain the parameters $H_0$ , $\alpha$ , and $\beta$ . The analysis utilizes three distinct observational datasets:

Cosmic Chronometers (CC): 31 model-independent measurements of $H(z)$ derived from the differential age of passively evolving galaxies ( $0.07 \leq z \leq 1.965$ ).
Type Ia Supernovae (Pantheon+): 1701 observations from the Pantheon+SH0ES dataset ( $0.01 < z < 2.3$ ), utilizing the distance modulus and full covariance matrix.
Baryon Acoustic Oscillations (BAO): Data from the Dark Energy Spectroscopic Instrument (DESI) Release 1 (DR1), including 1D isotropic ( $D_V/r_d$ ) and 2D anisotropic ( $D_M/r_d$ , $D_H/r_d$ ) measurements across 7 tracers ( $0.1 < z < 2.45$ ).

The total likelihood is constructed as the product of the individual likelihoods from these datasets. Following the parameter constraints, the authors derive the effective energy density ( $\rho_g$ ), effective pressure ( $p_g$ ), and the effective equation-of-state parameter ( $\omega_g$ ) for the modified gravity sector. Finally, a linear scalar perturbation analysis is performed to assess the dynamical stability of the cosmological background against homogeneous fluctuations.

Key Contributions and Results

Parameter Constraints: The joint fit yields best-fit values of $H_0 = 72.819^{+0.227}_{-0.228}$ , $\alpha = 0.583^{+0.033}_{-0.031}$ , and $\beta = 0.206^{+0.016}_{-0.017}$ . The sound horizon parameter is constrained to $r_d = 136.832^{+1.111}_{-1.134}$ .
Cosmic Acceleration: The model successfully reproduces a late-time accelerated expansion. The deceleration parameter at the present epoch is calculated as $q_0 = -0.417^{+0.021}_{-0.022}$ , indicating acceleration consistent with observations, though slightly less negative than the standard $\Lambda$ CDM prediction ( $q_0 \approx -0.531$ ) for the same datasets.
Effective Fluid Behavior: The analysis of the effective equation of state $\omega_g$ $ω_{g}$ reveals that the model naturally accommodates both quintessence-like ( $-1 < \omega_g < -1/3$ $- 1 < ω_{g} < - 1/3$ ) and phantom-like ( $\omega_g < -1$ $ω_{g} < - 1$ ) behaviors depending on the sign of the parameter $n$ $n$ .
- For $n > 0$ , the model exhibits quintessence behavior.
- For $n < 0$ , the model exhibits phantom behavior.
- In the limits of $z \to \infty$ and $z \to 0$ , the solution tends toward a cosmological constant behavior ( $\omega_g \to -1$ ).
Dynamical Stability: The linear scalar perturbation analysis demonstrates that the cosmological background is dynamically stable. The authors derive an exact analytical solution showing that both matter density perturbations ( $\delta_m$ ) and Hubble parameter perturbations ( $\delta$ ) decay monotonically over time, scaling directly with the background Hubble parameter $H(t)$ . This stability holds provided the regularity condition $f_T(T) + 2Tf_{TT}(T) - 1 \neq 0$ is maintained.

Significance and Claims
The paper claims that the proposed $f(T)$ model offers a viable geometric alternative to the standard $\Lambda$ CDM model for explaining late-time cosmic acceleration. By modifying the gravitational sector rather than introducing an exotic fluid, the model naturally generates an effective energy density and pressure that drive acceleration. The authors emphasize that the model's ability to transition between quintessence and phantom regimes based on the parameter $n$ provides flexibility in fitting cosmological data without violating the energy conditions in the effective fluid interpretation.

Furthermore, the study validates the model's physical viability by confirming its dynamical stability against linear fluctuations, a crucial requirement often challenging in higher-order modified gravity theories (such as $f(R)$ ). The authors conclude that teleparallel modified gravity models represent a promising alternative for addressing the open problems of cosmic expansion, offering a framework that is consistent with current observational data from cosmic chronometers, supernovae, and BAO.

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