Natural inflation with natural number of e-foldings

We examine natural inflation without the use of the standard slow-roll approximation by considering the number of physical e-folds $\ln (a_e{H}_e\mathrm{/}aH)$. We show that $\tilde{H}=aH\propto \cos (A\varphi \mathrm{/}2{)}^{2\mathrm{/}{A}^2}\sin (A\varphi \mathrm{/}2)$ produces a natural inflationary scenario. This model may be solved exactly, showing that the slow-roll approximation overestimates the tensor-to-scalar ratio by about $13-19\mathrm{\%}$ for $n_s\approx 0.96$ and $50-60$ $e$-folds.