Parametrizing the power spectrum: beyond the truncated Taylor expansion

Abstract

The power spectrum is traditionally parametrized by a truncated Taylor series: ln P(kappa) = ln P(*) + (n(*) - 1) ln(kappa/kappa(*)) + 1/2n'(*) ln2(kappa/kappa(*)). It is reasonable to truncate the Taylor series if |n'(*) ln(kappa/kappa(*))| << |n(*) - 1|, but it is not if |n'(*) ln(kappa/kappa(*))| greater than or similar to |n(*) - 1|. We argue that there is no good theoretical reason to prefer |n'(*)| << |n(*) - 1|, and show that current observations are consistent with |n'(*) ln(kappa/kappa(*))| similar to |n(*) - 1| even for |ln(kappa/kappa(*))| similar to 1. Thus, there are regions of parameter space, which are both theoretically and observationally relevant, for which the traditional truncated Taylor series parametrization is inconsistent, and hence it can lead to incorrect parameter estimations. Motivated by this, we propose a simple extension of the traditional parametrization, which uses no extra parameters, but that, unlike the traditional approach, covers well motivated inflationary spectra with |n'(*)| similar to |n(*) - 1|. Our parametrization therefore covers not only standard slow-roll inflation models but also a much wider class of inflation models. We use this parametrization to perform a likelihood analysis for the cosmological parameters.