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We use a numerical code for accurate computation of the amplitude of linear density perturbations and gravitational waves generated by single-field inflation models to study the accuracy of existing analytic results based on the slow-roll approximation. We use our code to calculate the coefficient of an expansion about the exact analytic result for power-law inflation; this generates a fitting function which can be applied to all inflationary models to obtain extremely accurate results. In the appropriate limit our results confirm the Stewart-Lyth analytic second-order calculation, and we find that their results are very accurate for inflationary models favored by current observational constraints.
Grivell et al. (Sun,) studied this question.