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 $$1 + 3$$ 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 – $$f(T)= \mu T_{0}(T/T_{0})^{n}$$ f ( T ) = μ T 0 ( T / T 0 ) n and the more generalised case, where $$f(T)= T+ \mu T_{0} (T/T_{0})^{n}$$ f ( T ) = T + μ 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 $$\Lambda $$ Λ CDM behaviour even for small deviation 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.