Reheating constraints on modified single-field Natural Inflation models

In this paper, we discuss three modified single-field natural inflation models in detail, including Special generalized Natural Inflation model(SNI), Extended Natural Inflation model(ENI) and Natural Inflation inspired model(NII). We derive the analytical expression of the tensor-to-scalar ratio $r$ and the spectral index $n_s$ for those models. Then the reheating temperature $T_{re}$ and reheating duration $N_{re}$ are analytically derived. Moreover, considering the CMB constraints, the feasible space of the SNI model in $(n_s, r)$ plane is almost covered by that of the NII, which means the NII is more general than the SNI. In addition, there is no overlapping space between the ENI and the other two models in $(n_s, r)$ plane, which indicates that the ENI and the other two models exclude each other, and more accurate experiments can verify them. Furthermore, the reheating brings tighter constraints to the inflation models, but they still work for a different reheating universe. Considering the constraints of $n_s$, $r$, $N_k$ and choosing $T_{re}$ near the electroweak energy scale, one can find that the decay constants of the three models have no overlapping area and the effective equations of state $\omega_{re}$ should be within $\frac{1}{4}\lesssim \omega_{re} \lesssim \frac{4}{5}$ for the three models.