Cosmological Viability of Exponential Infrared $f(T)$ Gravity

Technical Summary: Cosmological Viability of Exponential Infrared f(T) Gravity

Problem Statement
The standard $\Lambda$CDM cosmological model, while successful across a wide range of observations, faces significant theoretical and observational challenges. These include the unknown physical nature of dark matter and dark energy, the $H_0$ tension (a $\gtrsim 5\sigma$ discrepancy between early-Universe CMB inferences and local distance-ladder measurements), observational hints of evolving dark energy (e.g., from DESI DR2), the $A_{\text{lens}}$ anomaly, and recent indications of early structure formation (JWST). Modified gravity theories offer a potential resolution by generating cosmic acceleration as a geometric effect rather than through a fundamental dark-energy fluid. This paper investigates the cosmological viability of exponential infrared (IR) $f(T)$ teleparallel gravity, a specific modification where the torsion scalar $T$ is modified via an exponential function. The model is unique in that it reproduces General Relativity (GR) at early times while generating an effective dynamical dark-energy sector at late times, without introducing any additional free parameters beyond the six of spatially flat $\Lambda$CDM.

Methodology
The authors perform a comprehensive Bayesian parameter inference using the latest cosmological datasets. The analysis utilizes the Cobaya package for sampling and a modified version of the CLASS Boltzmann solver to compute theoretical predictions, including CMB power spectra, matter power spectra, and background expansion histories.

The study focuses on two distinct solution branches of the exponential IR $f(T)$ model, derived from the two real branches of the Lambert-W function ($W_0$ and $W_{-1}$):

1. Model I (Principal Branch, $W_0$): Corresponds to $\beta > 0$. This branch behaves as an effective phantom-like dark energy component at late times.
2. Model II (Secondary Branch, $W_{-1}$): Corresponds to $\beta < 0$. This branch features a transition where the effective torsional dark-energy density evolves from negative to positive values.

The parameter space is restricted to the standard six $\Lambda$CDM parameters: $\{\Omega_b h^2, \Omega_c h^2, H_0, \tau_{\text{reio}}, \ln(10^{10}A_s), n_s\}$. The model parameter $\beta$ is not free but is determined entirely by the present-day matter density $\Omega_m$ via the Lambert-W relation.

The analysis confronts these models with five dataset combinations:

• Late-Universe only: Pantheon+ (PP) + DESI DR2 BAO; and the SH0ES-calibrated Pantheon+ (PPS) + DESI.
• Early-Universe only: CMB-SPA (a combination of Planck 2018, ACT DR6, and SPT-3G D1 data, including lensing).
• Combined: CMB-SPA + PP + DESI and CMB-SPA + PPS + DESI.

The study explicitly includes linear cosmological perturbations, accounting for the gravitational slip and modified Poisson equations inherent to $f(T)$ gravity, to test the models against CMB anisotropies and structure growth.

Key Contributions and Results

• Model I (Phantom-like Branch):

  • Hubble Tension: When constrained by CMB data alone, Model I yields $H_0 = 71.97 \pm 0.45$ km s$^{-1}$ Mpc$^{-1}$. This reduces the tension with the Hubble Distance Network measurement ($H_0 \approx 73.50$) from $\sim 6.8\sigma$ (in $\Lambda$CDM) to $\sim 1.65\sigma$.
  • Stability: Unlike $\Lambda$CDM, Model I does not exhibit significant shifts in derived parameters ($\Omega_b h^2, A_s, n_s, \tau_{\text{reio}}$) when late-time data (DESI, Pantheon+) are added to CMB constraints. It provides a stable cosmological solution across different dataset combinations.
  • Goodness-of-Fit: Despite alleviating the $H_0$ tension, Model I remains statistically disfavoured compared to $\Lambda$CDM when all datasets are combined ($\Delta \chi^2 \approx 55.7$ for CMB+PP+DESI). The model becomes "overconstrained": the requirement to simultaneously match the CMB acoustic scale, the high local $H_0$, and the precise DESI/Supernova expansion history leaves no freedom to adjust the expansion history shape, leading to a degraded global fit.

• Model II (Negative-to-Positive Density Branch):

  • Ruling Out: This branch is decisively ruled out by current observations. While it theoretically allows for a large $H_0$ in CMB-only analyses ($H_0 \approx 83.3$ km s$^{-1}$ Mpc$^{-1}$), the inclusion of BAO and Supernova data forces the matter density $\Omega_m$ to lower values ($\approx 0.26$).
  • Perturbation Failure: The failure of Model II is driven by the interplay between background and perturbation constraints. The transition from negative to positive dark-energy density induces a large gravitational slip on super-horizon scales, significantly enhancing the late-time Integrated Sachs-Wolfe (ISW) effect. This leads to an excess of power in the low-$\ell$ CMB temperature spectrum that is incompatible with observations.
  • Parameter Degeneracies: To compensate for the tension between the CMB damping tail and the late-time expansion history, the model forces unphysical shifts in other parameters. Specifically, the optical depth $\tau_{\text{reio}}$ is driven to unphysical values ($\approx 0.02$) to satisfy the tight constraint on the combination $A_s e^{-2\tau_{\text{reio}}}$, resulting in a massive increase in $\chi^2$ ($\Delta \chi^2 \approx 631$ for CMB+PP+DESI).

Significance and Claims
The paper demonstrates that perturbation observables provide stringent and complementary tests of teleparallel gravity beyond the background expansion history. While both branches of the exponential IR $f(T)$ model can modify the late-time expansion to accommodate higher values of $H_0$, only the principal branch (Model I) remains compatible with the full suite of cosmological data, albeit with a statistical penalty.

The authors claim that the failure of Model II is not necessarily evidence against the concept of a negative-to-positive transition in effective torsional density itself, but rather an indication that the specific minimal realization (with no extra degrees of freedom) is insufficient to reconcile early- and late-Universe constraints. The study highlights that the lack of additional free parameters in the minimal model prevents it from simultaneously satisfying CMB damping tail constraints, reionization priors, and late-time distance measurements.

The paper concludes that while the exponential IR $f(T)$ model offers a geometric explanation for cosmic acceleration without new parameters, its viability is limited. The authors suggest that future investigations should focus on generalized versions of the model (e.g., introducing an additional parameter $b$ or $\alpha$) which may possess the necessary degrees of freedom to accommodate the complex interplay of current cosmological tensions.