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 $\ln (a_eH_e/aH)$. We show that $\tilde{H} = aH \propto \cos(A \phi/2)^{2/A^2} \sin(A \phi/2)$ 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 $13-19 \%$ for $n_s \approx 0.96$ and $50-60$ $e$-folds.