Constraining $\alpha$-attractor models from reheating

We eliminate the parameters originally present in models of inflation of the $\alpha$-attractor type in favor of the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$. We then write expressions for the number of $e$-folds during reheating $N_ {re}$. By imposing reasonable conditions on $N_{re}$ we can restrict $n_s$ and $r$ and in turn, we use these constraints in order to find bounds for cosmological quantities of interest such as the number of $e$-folds during inflation and the radiation dominated eras, as well as for the reheating temperature and the running index. The minimum condition that $N_ {re}$ must satisfy is $N_ {re}\geq 0$ which we use to constrain the cosmological quantities mentioned above. In particular, we find that the tensor-to-scalar ratio $r$ (and as a consequence the energy scale of inflation) is bounded from below. We provide figures illustrating the behavior of these quantities as functions of $r$ for several values of $n_s$ and tables containing the bounds so obtained.