Relating spectral indices to tensor and scalar amplitudes in inflation.

Abstract

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'${)}^{2}$\ensuremath{\ne}1 or if [${\mathit{m}}_{\mathrm{Pl}}^{2}$/(4\ensuremath{\pi})]\ensuremath{\Vert}V'''/V'\ensuremath{\Vert}\ensuremath{\gtrsim}1, where V(\ensuremath{\varphi}) 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.