Inflationary cosmology with nonlinear dispersion relations

TL;DR

This paper develops an analytical method to accurately solve inflationary perturbations with nonlinear dispersion relations, accounting for quantum effects, and applies it to compute modified power spectra in slow-roll inflation.

Contribution

It introduces the uniform asymptotic approximation technique for linear perturbations with nonlinear dispersion relations, providing explicit error bounds and demonstrating high accuracy.

Findings

Quantum effects modify the amplitude of the power spectra.

Power spectrum indices remain unchanged despite quantum modifications.

Analytical solutions closely match exact evolution even at first order.

Abstract

We present a technique, {\em the uniform asymptotic approximation}, to construct accurate analytical solutions of the linear perturbations of inflation after quantum effects of the early universe are taken into account, for which the dispersion relations generically become nonlinear. We construct explicitly the error bounds associated with the approximations and then study them in detail. With the understanding of the errors and the proper choice of the Liouville transformations of the differential equations of the perturbations, we show that the analytical solutions describe the exact evolution of the linear perturbations extremely well even only in the first-order approximations. As an application of the approximate analytical solutions, we calculate the power spectra and indices of scalar and tensor perturbations in the slow-roll inflation, and find that the amplitudes of the power spectra get modified due to the quantum effects, while the power spectrum indices remain the same as in the linear case.