On the $\alpha$-attractor T-models

We carry out a fully analytical study of the phenomenology of $\alpha$-attractor T-models defined by the potential $V = V_0\tanh^ p\left(\lambda \phi/M_{pl}\right)$. We obtain expressions for the number of e-folds during inflation $N_{ke}$ in terms of the scalar spectral index $n_s$ and independently in terms of the tensor-to-scalar ratio $r$. From these expressions we obtain exact solutions for both $n_s$ and $r$ in terms of $N_{ke}$ along with their expansions for large $N_{ke}$, in full agreement with known expressions. Eliminating the parameter $\lambda$ from the model in terms of $n_s$ and $r$ we can obtain exact solutions for $r$ in terms of $n_s$ and $N_{ke}$ which allows us to reproduce, in particular, numerical solutions presented by the Planck Collaboration for the monomial potentials. We explicitly show how these solutions are contained in the solutions for the $\alpha$-attractors and are also the end points of these. Finally, by also eliminating the global scale $V_0$ in terms of the observables $n_s$ and $r$ we show how in the appropriate limit the $\alpha$-attractor potential exactly reduces to the monomials potential. We also briefly show that for $\alpha$-attractor E-models, which generalize the Starobinsky potential in the Einstein frame, a similar transition occurs.