Cosmological perturbations in $f(T)$ 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 $f(T)$ Ansätze 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 ($\mathcal{O}(100\mathrm{Mpc})$ or larger), the evolution of matter overdensity will differ from $\Lambda \mathrm{CDM}$ 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.

Figure 1

The evolution of the matter overdensity $\delta$ as a function of the redshift $z$, on a scale of $k=0.1h{\mathrm{Mpc}}^{-1}$, for three choices of $n$, for the power-law model given by (45).

Figure 2

The evolution of the matter overdensity $\delta$ as a function of the redshift $z$, on a scale of $k=0.01h{\mathrm{Mpc}}^{-1}$, for three choices of $n$, for the power-law model given by (45).

Figure 3

The evolution of the matter overdensity $\delta$ as a function of the redshift $z$, on a scale of $k=0.001h{\mathrm{Mpc}}^{-1}$, for three choices of $n$, for the power-law model given by (45).

Figure 4

The evolution of the matter overdensity $\delta$ as a function of the redshift $z$, for a fixed $n$ for the power-law model given by (45), for three different scales.