  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.

High Energy Physics - Phenomenology