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