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’ ${)}^2$≠1 or if [${\mathit{m}}_{\mathrm{Pl}}^2$/(4π)]‖V’ ’ ’/V’‖≳1, where V(φ) is the inflaton potential and ${\mathit{m}}_{\mathrm{Pl}}$ 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.