New generalization of the simplest $\alpha$-attractor $T$ model

The simplest $\alpha$-attractor $T$ model is given by the potential $V=V_0\tanh^2(\lambda\phi/M_{pl})$. However its generalization to the class of models of the type $V = V_0 \tanh^p (\lambda \phi/M_{pl})$ is difficult to interpret as a model of inflation for most values of $p$. Keeping the basic model, we propose a new generalization, where the final potential is of the form $V = V_0 (1-\sech^p (\lambda \phi/M_{pl}))$, which does not present any of the problems that plague the original generalization, allowing a successful interpretation as a model of inflation for any value of $p$ and, at the same time, providing the potential with a region where reheating can occur for any $p$ (including odd and fractional values) without difficulty. In the cases $p = 1, 2, 4$ we obtain the solutions $r(n_s, N_{ke})$ where $r$ is the tensor-to-scalar ratio, $n_s$ the spectral index and $N_{ke}$ the number of $e$-folds during inflation. We also show how these solutions connect to the $\phi^2$ monomial.