Structure formation in $f(T)$ gravity and a solution for $H_0$ tension

TL;DR

This work formulates scalar perturbations in $f(T)$ teleparallel gravity and derives their impact on the CMB, introducing a power-law model $f(T)=T+oldsymbol{α}(-T)^n$ to confront Planck CMB and BAO data plus a local $H_0$ measurement. It shows that the perturbation dynamics imply a scale- and time-dependent effective gravitational coupling $G_{ m eff}=G/f_T$ and a gravitational slip, with CMB observations favoring extremely small departures from $Λ$CDM (roughly $n\, ext{of order}\,10^{-3}$). Crucially, the $f(T)$ framework can shift the inferred $H_0$ to values compatible with local measurements, thereby alleviating the $H_0$ tension, although it does not fully resolve the $oldsymbol{σ_8}$ tension. The results indicate that $f(T)$ gravity remains a viable modified gravity candidate consistent with current CMB data, meriting further exploration of tensor modes, nonlocal formulations, and broader datasets.

Abstract

We investigate the evolution of scalar perturbations in $f(T)$ teleparallel gravity and its effects on the cosmic microwave background (CMB) anisotropy. The $f(T)$ gravity generalizes the teleparallel gravity which is formulated on the Weitzenböck spacetime, characterized by the vanishing curvature tensor (absolute parallelism) and the non-vanishing torsion tensor. For the first time, we derive the observational constraints on the modified teleparallel gravity using the CMB temperature power spectrum from Planck's estimation, in addition to data from baryonic acoustic oscillations (BAO) and local Hubble constant measurements. We find that a small deviation of the $f(T)$ gravity model from the $Λ$CDM cosmology is slightly favored. Besides that, the $f(T)$ gravity model does not show tension on the Hubble constant that prevails in the $Λ$CDM cosmology. It is clear that $f(T)$ gravity is also consistent with the CMB observations, and undoubtedly it can serve as a viable candidate amongst other modified gravity theories.

Figures (5)

• Figure 1: Left painel: Evolution of the gravitational slip as a function of the scale factor $a$ to $k = 10^{-3}/{\rm Mpc}$ for the flat $\Lambda$CDM cosmology and $f(T)$ gravity for some values of $n$ from the model eq.(\ref{['modf1']}). Right panel: Evolution of the quantity $\phi + \psi$ at late time.
• Figure 2: Left panel: the CMB TT power spectrum, $D^{TT}_l =l(l+ 1)C_l^{TT}/ 2 \pi\mu K^2$, for the flat $\Lambda$CDM cosmology and $f(T)$ gravity for some values of $n$ from the model eq.(\ref{['modf1']}). Right panel: Relative deviation of CMB TT power spectrum from the base line Planck 2015 $\Lambda$CDM model in comparison with $f(T)$ gravity for $n = 0.1$ (black line), $n = 0.01$ (blue line), and $n = 0.001$ (green line).
• Figure 3: The same as in figure \ref{['cmb_TT']}, but for the CMB EE power spectrum, $D^{EE}_l =l(l+ 1)C^{EE}_l/2 \pi\mu K^2$.
• Figure 4: Parametric space in the plane $H_0$ - $\sigma_8$, where the regions in red (blue) show the constraints for $\Lambda$CDM model from CMB + BAO (CMB + BAO + $H_0$), respectively. The regions in black (green) show the constraints for $f(T)$ gravity from CMB + BAO (CMB + BAO + $H_0$), respectively. The vertical gray band corresponds to $H_0=73.24\pm 1.74$ km s${}^{-1}$ Mpc${}^{-1}$.
• Figure 5: Parametric space in the plane $n$ - $\Omega_{F0}$, where the regions in black (green) show the constraints from CMB + BAO (CMB + BAO + $H_0$), respectively.