$f(T)$ Gravity: Background Dependence and Propagating Degrees of Freedom

Abstract: The standard cosmological model, rooted in General Relativity (GR), has achieved remarkable success, yet it still faces unresolved issues like the nature of dark matter, dark energy, and the Hubble tension. These challenges might imply the need for alternative gravitational theories. Teleparallel gravity offers a compelling framework by reformulating the gravitational interaction using torsion, rather than curvature, as its fundamental geometrical property. This paper delves into $f(T)$ gravity, an extension of the Teleparallel Equivalent of General Relativity (TEGR), which introduces non-linear modifications of the torsion scalar $T$. We focus on the role of spacetime-dependent Lorentz transformations in the vierbein formalism, examining their impact on both background solutions and perturbation dynamics. Special attention is given to the homogeneous and isotropic FLRW spacetime, as well as the anisotropic Bianchi I spacetime. Furthermore, the analysis of the propagating degrees of freedom on these spacetimes is performed. While it is well established that TEGR reproduces the same results as GR, the propagating degrees of freedom in its non-linear extension, $f(T)$ gravity, is still debated in the literature. In this work, we find that only two fields propagate in the gravity sector, independently of the background spacetime considered, either FLRW or Bianchi I. Although not definitive, this paper provides fresh insights into the issue of the propagating degrees of freedom in $f(T)$ gravity, opening the door to intriguing new directions for further investigation.