High-order Primordial Perturbations with Quantum Gravitational Effects

TL;DR

This paper systematically analyzes high-order primordial perturbations influenced by quantum gravitational effects, deriving explicit power spectra expressions, and exploring observational implications such as modifications to the tensor-to-scalar ratio.

Contribution

It introduces a method using uniform asymptotic approximations to analytically solve for high-order perturbations with quantum gravity effects, including multiple-turning points, with high accuracy.

Findings

Quantum gravitational effects can significantly alter the tensor-to-scalar ratio.

Analytical solutions with less than 0.15% error are obtained for high-order perturbations.

Quantum effects may help reconcile inflationary models with observational data.

Abstract

In this paper, we provide a systematic investigation of high-order primordial perturbations with nonlinear dispersion relations due to quantum gravitational effects in the framework of {\em uniform asymptotic approximations}. Because of these effects, the equation of motion of the mode function in general has multiple-turning points. After obtaining analytically approximated solutions to any order in different regions, associated with different types of turning points, we match them to the third one. To this order the errors are less than $0.15\%$. General expressions of the power spectra of the primordial tensor and scalar perturbations are derived explicitly. We also investigate effects of back-reactions of the quantum gravitational corrections, and make sure that inflation lasts long enough in order to solve the underlying problems, such as flatness, horizon and monopole. Then, we study various features of the spectra that are observationally relevant. In particular, under a moderate assumption about the energy scale of the underlying theory of quantum gravity, we have shown that the quantum gravitational effects may alter significantly the ratio between the tensor and scalar power spectra, thereby providing a natural mechanism to alleviate the tension between observations and certain inflationary models, including the one with a quadratic potential.