Covariant density and velocity perturbations of the quasi-Newtonian cosmological model in $f(T)$ gravity.

We investigate classes of shear-free cosmological dust models with irrotational fluid flows within the framework of $f(T)$ gravity. In particular, we use the $1 + 3$ covariant formalism and present the covariant linearised evolution and constraint equations describing such models. We then derive the integrability conditions describing a consistent evolution of the linearised field equations of these quasi-Newtonian universes in the $f(T)$ gravitational theory. Finally, we derive the evolution equations for the density and velocity perturbations of the quasi-Newtonian universe. We explore the behaviour of the matter density contrast for two models - $f(T)= \mu T_{0}(T/T_{0})^{n}$ and the more generalised case, where $f(T)= T+ \mu T_{0} (T/T_{0})^{n}$, with and without the application of the quasi-static approximation. Our numerical solutions show that these $f(T)$ theories can be suitable alternatives to study the background dynamics, whereas the growth of energy density fluctuations change dramatically from the expected $\Lambda$CDM behaviour even for small deviations away from the general relativistic limits of the underlying $f(T)$ theory. Moreover, applying the so-called quasi-static approximation yields exact-solution results that are orders of magnitude different from the numerically integrated solutions of the full system, suggesting that these approximations are not applicable here.