Improved WKB analysis of cosmological perturbations

TL;DR

This paper develops improved WKB approximation methods for analyzing cosmological perturbations across all scales, including near the turning point, and demonstrates their effectiveness through power-law inflation benchmarks.

Contribution

It introduces advanced WKB techniques using Green's function and adiabatic expansions to accurately compute cosmological perturbation spectra beyond the lowest order.

Findings

Next-to-leading order adiabatic expansion accurately predicts power spectrum amplitudes.

Green's function method shows limited improvement over leading order.

Methods are applicable to general cases beyond power-law inflation.

Abstract

Improved Wentzel-Kramers-Brillouin (WKB)-type approximations are presented in order to study cosmological perturbations beyond the lowest order. Our methods are based on functions which approximate the true perturbation modes over the complete range of the independent (Langer) variable, from sub-horizon to super-horizon scales, and include the region near the turning point. We employ both a perturbative Green's function technique and an adiabatic (or ``semiclassical'') expansion (for a linear turning point) in order to compute higher order corrections. Improved general expressions for the WKB scalar and tensor power spectra are derived for both techniques. We test our methods on the benchmark of power-law inflation, which allows comparison with exact expressions for the perturbations, and find that the next-to-leading order adiabatic expansion yields the amplitude of the power spectra with excellent accuracy, whereas the next-to-leading order with the perturbative Green's function method does not improve the leading order result significantly. However, in more general cases, either or both methods may be useful.