In this work we study the cosmology of the general f(T) gravity theory. We express the modified Einstein equations using covariant quantities, and derive the gauge-invariant perturbation equations in covariant form. We consider a specific choice of f(T), designed to explain the observed late-time accelerating cosmic expansion without including an exotic dark energy component. Our numerical solution shows that the extra degree of freedom of such f(T) gravity models generally decays as one goes to smaller scales, and consequently its effects on scales such as galaxies and galaxies clusters are small. But on large scales, this degree of freedom can produce large deviations from the standard ΛCDM scenario, leading to severe constraints on the f(T) gravity models as an explanation to the cosmic acceleration.
Li, B., Sotiriou, T., & Barrow, J. (2011). Large-scale structure in f(T) gravity. Physical Review D, 83, Article 104017. https://doi.org/10.1103/physrevd.83.104017