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

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 Hořava-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 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.

Power spectra and spectral indices of k-inflation: High-order corrections

“…[19,22], by introducing a dimensionless variable y ¼ −kη, let us first write the equation of the mode function in the form, d 2 μ k ðyÞ dy 2 ¼ ½λ 2ĝ ðyÞ þ qðyÞμ k ðyÞ: ð3:1Þ…”

“…Consequently, fðξÞ must be chosen so that f ð1Þ ðξÞ has zeros and singularities of the same type as that ofĝðyÞ. As shown in [18,19], such a requirement plays an essential role in determining the approximate solutions. In terms of U and ξ, Eq.…”

“…However, to match with the accuracy of the current and forthcoming observations, as pointed out in [19], consideration of the high-order corrections in the uniform asymptotic approximation is highly demanded. In this paper, with the general expressions of power spectra and spectral indices we obtained in [19], we calculate explicitly the power spectra and spectral indices of scalar and tensor perturbations of the k-inflation up to the third order in the uniform asymptotic approximation.…”

“…[16,17], and then was extended by us [18] to the more general case where the dispersion relation is not necessarily linear, and in general has multi-and/or high-order turning points. Furthermore, by considering the more accurate high-order uniform asymptotic approximations, we have obtained the general expressions of the power spectra and spectral indices up to the third order in the uniform asymptotic approximation, at which the error bounds are ≲0.15% [19]. With these results we have also calculated explicitly the power spectra and spectral indices of scalar and tensor perturbations with a modified dispersion relation up to the second order in the slow-roll approximation.…”

“…In this paper, with the general expressions of power spectra and spectral indices we obtained in [19], we calculate explicitly the power spectra and spectral indices of scalar and tensor perturbations of the k-inflation up to the third order in the uniform asymptotic approximation. These expressions represent a significant improvement of the previous results obtained by other methods.…”