Relating spectral indices to tensor and scalar amplitudes in inflation

Within an expansion in slow-roll inflation parameters, we derive the complete second-order expressions relating the ratio of tensor to scalar density perturbations and the spectral index of the scalar spectrum. We find that ``corrections'' to previously derived formulas can dominate if the tensor to scalar ratio is small. For instance, if VV''/ (V') ^21 or if m₏₋^2/ (4) '''/V'1, where V () is the inflaton potential and m₏₋ is the Planck mass, then the previously used simple relations between the indices and the tensor to scalar ratio fails. This failure occurs in particular for natural inflation, Coleman-Weinberg inflation, and ``chaotic'' inflation.