A Built-in Inflation in the $f(T)$-Cosmology

In the present work we derive an exact solution of an isotropic and homogeneous Universe governed by $f(T)$ gravity. We show how the torsion contribution to the FRW cosmology can provide a \textit{unique} origin for both early and late acceleration phases of the Universe. The three models ($k=0, \pm 1$) show a \textit{built-in} inflationary behavior at some early Universe time; they restore suitable conditions for the hot big bang nucleosynthesis to begin. Unlike the standard cosmology, we show that even if the Universe initially started with positive or negative sectional curvatures, the curvature density parameter enforces evolution to a flat Universe. The solution constrains the torsion scalar $T$ to be a constant function at all time $t$, for the three models. This eliminates the need for the dark energy (DE). Moreover, when the continuity equation is assumed for the torsion fluid, we show that the flat and closed Universe models \textit{violate} the conservation principle, while the open one does not. The evolution of the effective equation of state (EoS) of the torsion fluid implies a peculiar trace from a quintessence-like DE to a phantom-like one crossing a matter and radiation EoS in between; then it asymptotically approaches a de Sitter fate.