WKB approximation for inflationary cosmological perturbations

TL;DR

This paper introduces a novel WKB-based method to predict inflationary power spectra that remains valid beyond the slow-roll regime. By applying a cosmological analogue of Langer’s transformation, the authors reduce the problem to a single quadrature and derive general expressions for scalar and tensor spectra, with spectral indices computed exactly in power-law inflation and with high precision in slow-roll cases. In power-law inflation, the WKB approach yields exact indices and amplitudes within a small error that vanishes as |β| grows, while in slow-roll it reproduces the correct spectral indices to leading order and achieves the expected sub-10% amplitude accuracy. The work provides a complementary, non-local method for forecasting inflationary perturbations and outlines clear paths for systematic improvements and extensions.

Abstract

A new method for predicting inflationary cosmological perturbations, based on the Wentzel-Kramers-Brillouin (WKB) approximation, is presented. A general expression for the WKB scalar and tensor power spectra is derived. The main advantage of the new scheme of approximation is that it is valid even if the slow-roll conditions are violated. The method is applied to power-law inflation, which allows a comparison with an exact result. It is demonstrated that the WKB approximation predicts the spectral indices exactly and the amplitude with an error lower than 10%, even in regimes far from scale-invariance. The new method of approximation is also applied to a situation where the slow-roll conditions hold. It is shown that the result obtained bears close resemblance with the standard slow-roll calculation. Finally, some possible improvements are briefly mentioned.

Figures (1)

• Figure 1: Evolution of the ratio $g(\beta)/f(\beta)$ with $\beta$. In the limit $\vert \beta \vert \rightarrow \infty$, the ratio tends towards $1$. This can only be guessed from the figure because the convergence rate is slow.