Higher order corrections to primordial spectra from cosmological inflation

TL;DR

This work develops high-precision predictions for primordial scalar and tensor spectra from inflation without relying on slow-roll, by introducing horizon-flow functions and two non-slow-roll approximations: constant-horizon and growing-horizon. The authors derive scalar and tensor power spectra at third order, show no running of the spectral index within these schemes, and establish generalized consistency relations between r and n_T that extend beyond standard slow-roll. The approach provides accurate results for models with large slow-roll parameters, including chaotic monomial potentials and inflation near a maximum, thereby broadening the set of inflationary scenarios Compatible with current and future CMB data. These results offer practical guidance for interpreting observations and testing inflationary dynamics beyond the slow-roll paradigm.

Abstract

We calculate power spectra of cosmological perturbations at high accuracy for two classes of inflation models. We classify the models according to the behaviour of the Hubble distance during inflation. Our approximation works if the Hubble distance can be approximated either to be a constant or to grow linearly with cosmic time. Many popular inflationary models can be described in this way, e.g., chaotic inflation with a monomial potential, power-law inflation and inflation at a maximum. Our scheme of approximation does not rely on a slow-roll expansion. Thus we can make accurate predictions for some of the models with large slow-roll parameters.

Figures (1)

• Figure 1: Regions in the $\epsilon_1$-$|\epsilon_2|$ parameter space where the spectral amplitudes can be calculated with an accuracy better than $1\%$. In the dark shaded region the Stewart-Lyth (SL) approximation SL, as well as all other approximations are fine. Second-order corrections, as calculated by Stewart and Gong (SG) GS, extend that region to the light shaded region. The constant horizon approximation at order $n$ (ch$n$), and the growing horizon approximation at order $n$ (gh$n$), do well below the thick line. The rays indicate where the corresponding higher order corrections are necessary. The thick line itself is the condition $\epsilon_1 |\epsilon_2| < (A/100\%)/\Delta N$, with $\Delta N =10$ and $A = 1\%$. As is easily seen the ch2 and gh2 regions are included within the SG region. The ch$n$ and gh$n$ regions with $n > 2$ allow us to go beyond the SG approximation.