  We investigate the cosmological perturbations in f(T) gravity. Examining the
pure gravitational perturbations in the scalar sector using a diagonal
vierbien, we extract the corresponding dispersion relation, which provides a
constraint on the f(T) ansatzes that lead to a theory free of instabilities.
Additionally, upon inclusion of the matter perturbations, we derive the fully
perturbed equations of motion, and we study the growth of matter overdensities.
We show that f(T) gravity with f(T) constant coincides with General Relativity,
both at the background as well as at the first-order perturbation level.
Applying our formalism to the power-law model we find that on large subhorizon
scales (O(100 Mpc) or larger), the evolution of matter overdensity will differ
from LCDM cosmology. Finally, examining the linear perturbations of the vector
and tensor sectors, we find that (for the standard choice of vierbein) f(T)
gravity is free of massive gravitons.
