We investigate the main features of the flat Friedmann-Lemaître-Robertson-Walker cosmological models in the $f(T)$ modified gravity regime. In particular, a general approach to find out exact cosmological solutions in $f(T)$ gravity is discussed. Instead of taking into account phenomenological models, we consider as a selection criterion, the existence of Noether symmetries in the cosmological $f(T)$ pointlike Lagrangian. We find that only the $f(T)=f_0{T}^n$ model admits extra Noether symmetries. The existence of extra Noether integrals can be used in order to simplify the system of differential equations (equations of motion) as well as to determine the integrability of the $f(T)=f_0{T}^n$ cosmological model. Within this context, we can solve the problem analytically and thus we provide the evolution of the main cosmological functions such as the scale factor of the Universe, the Hubble expansion rate, the deceleration parameter, and the linear matter perturbations. We show that the $f(T)=f_0{T}^n$ cosmological model suffers from two basic problems. The first problem is related to the fact that the deceleration parameter is constant which means that it never changes sign, and therefore the Universe always accelerates or always decelerates depending on the value of $n$. Second, we find that the clustering growth rate remains always equal to unity implying that the recent growth data disfavor the $f(T)=f_0{T}^n$ gravity. Finally, we prove that the $f(T)=f_0{T}^n$ gravity can be cosmologically equivalent with the $f(R)=R^n$ gravity model and the time varying vacuum model $\Lambda (H)=3\gamma H^2$ (for $n^{-1}=1-\gamma$) because the above cosmological scenarios share exactly the same Hubble expansion, despite the fact that the three models have a different geometrical origin. Finally, some important differences with power-law $f(R)$ gravity are pointed out.

• Received 14 September 2013

DOI:https://doi.org/10.1103/PhysRevD.88.103526