Power spectra and spectral indices of $k$-inflation: high-order corrections

TL;DR

This paper improves the theoretical predictions of $k$-inflation's power spectra and spectral indices by applying high-order uniform asymptotic approximations, enhancing accuracy over previous first-order results and validating with multiple methods.

Contribution

It introduces high-order uniform asymptotic approximation calculations for $k$-inflation, providing more precise expressions for power spectra and spectral indices compared to earlier first-order methods.

Findings

High-order corrections significantly improve accuracy.

Results agree with other approximation methods.

Explicit expressions in terms of flow parameters.

Abstract

$k$-inflation represents the most general single-field inflation, in which the perturbations usually obey an equation of motion with a time-dependent sound speed. In this paper, we study the observational predictions of the $k$-inflation by using the high-order uniform asymptotic approximation method. We calculate explicitly the slow-roll expressions of the power spectra, spectral indices, and running of the spectral indices for both the scalar and tensor perturbations. These expressions are all written in terms of the Hubble and sound speed flow parameters. It is shown that the previous results obtained by using the first-order uniform asymptotic approximation have been significantly improved by the high-order corrections of the uniform asymptotic approximations. Furthermore, we also check our results by comparing them with the ones obtained by other approximation methods, including the Green's function method, WKB approximation, and improved WKB approximation, and find the relative errors.