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

Abstract

The simplest $\alpha$-attractor $T$ model is given by the potential $V=V_0{\tanh }^2(\lambda \varphi /M_{pl})$. However its generalization to the class of models of the type $V=V_0{\tanh }^p(\lambda \varphi /M_{pl})$ 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 $V=V_0(1-{\text{sech}}^p(\lambda \varphi /M_{pl}))$, 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 ${\varphi }^2$ monomial.

Comments: Published Phys. Rev. D