Tensor to scalar ratio of perturbation amplitudes and inflaton dynamics

Abstract

For the inflaton perturbations it is shown that the evolution of the difference between the spectral indices can be translated into information on the scale dependence of the tensor to scalar amplitudes ratio, $r$, and how the scalar field potential can be derived from that information. Examples are given where $r$ converges to a constant value during inflation but dynamics are rather different from the power--law model. Cases are presented where a constant $r$ is not characteristic of the inflationary dynamics though the resulting perturbation spectra are consistent with the CMB and LSS data. The inflaton potential corresponding to $r$ given by a n--th order polynomial of the e--folds number is derived in quadratures expressions. Since the observable difference between the spectral indices evaluated at a pivot scale yields information about the linear term of that polynomial, the first order case is explicitly written down. The solutions show features beyond the exponential form corresponding to power--law inflation and can be matched with current observational data.