Cosmological perturbations in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>gravity

We investigate the cosmological perturbations in $f(T)$ gravity. Examining the pure gravitational perturbations in the scalar sector using a diagonal vierbein, we extract the corresponding dispersion relation, which provides a constraint on the $f(T)$ Ans\"atze 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 ($\mathcal{O}(100\text{ }\text{ }\mathrm{Mpc})$ or larger), the evolution of matter overdensity will differ from $\ensuremath{\Lambda}\mathrm{CDM}$ 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.

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