Gravitational quantum effects on power spectra and spectral indices with higher-order corrections

TL;DR

This paper develops high-order analytical approximations for the power spectra and spectral indices of cosmological perturbations considering gravitational quantum effects, applicable to various quantum gravity models, with error bounds below 0.15%.

Contribution

It extends previous first-order methods to third-order accuracy for cases with single or zero turning points in the dispersion relation, providing precise formulas for quantum-corrected spectra.

Findings

Power spectra and spectral indices calculated up to third-order with <0.15% error.

Formulas validated against Green function method, showing consistency.

Application to slow-roll inflation with nonlinear dispersion relation.

Abstract

The uniform asymptotic approximation method provides a powerful, systematically-improved, and error-controlled approach to construct accurate analytical approximate solutions of mode functions of perturbations of the Friedmann-Robertson-Walker universe, designed especially for the cases where the relativistic linear dispersion relation is modified after gravitational quantum effects are taken into account. These include models from string/M-Theory, loop quantum cosmology and Ho\v{r}ava-Lifshitz quantum gravity. In this paper, we extend our previous studies of the first-order approximations to high orders for the cases where the modified dispersion relation (linear or nonlinear) has only one-turning point (or zero). We obtain the general expressions for the power spectra and spectral indices of both scalar and tensor perturbations up to the third-order, at which the error bounds are $\lesssim 0.15\%$. As an application of these formulas, we calculate the power spectra and spectral indices in the slow-roll inflation with a nonlinear power-law dispersion relation. To check the consistency of our formulas, we further restrict ourselves to the relativistic case, and calculate the corresponding power spectra, spectral indices and runnings to the second-order. Then, we compare our results with the ones obtained by the Green function method, and show explicitly that the results obtained by these two different methods are consistent within the allowed errors.