Relating spectral indices to tensor and scalar amplitudes in inflation.

Within an expansion in slow-roll in ation parameters, we derive secondorder expressions relating the ratio of tensor to scalar density perturbations and the spectral index of the scalar spectrum. We nd that \corrections" to previously derived formulae can dominate if the tensor to scalar ratio is small. For instance, if V V 00=(V ) 6= 1 or if mP l=(4 ) jV 000=V j > 1, where V ( ) is the in aton potential and mP l is the Plank mass, then the previously used simple relations between the indices and the tensor to scalar ratio fails. This failure occurs in particular for natural in ation, Coleman-Weinberg in ation, and \chaotic" in ation.