Reconstructing inflation in scalar-torsion f(T, ϕ) gravity

It is investigated the reconstruction during the slow-roll inflation in the most general class of scalar-torsion theories whose Lagrangian density is an arbitrary function f(T, ϕ) of the torsion scalar T of teleparallel gravity and the inflaton ϕ. For the class of theories with Lagrangian density f(T,ϕ)=-Mpl2T/2-G(T)F(ϕ)-V(ϕ), with G(T) ∼ T^s+1 and the power s as constant, we consider a reconstruction scheme for determining both the non-minimal coupling function F(ϕ) and the scalar potential V(ϕ) through the parametrization (or attractor) of the scalar spectral index n_s(N) and the tensor-to-scalar ratio r(N) as functions of the number of e- folds N. As specific examples, we analyze the attractors n_s- 1 ∝ 1 / N and r∝ 1 / N, as well as the case r∝ 1 / N(N+ γ) with γ a dimensionless constant. In this sense and depending on the attractors considered, we obtain different expressions for the function F(ϕ) and the potential V(ϕ) , as also the constraints on the parameters present in our model and its reconstruction.