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

Abstract

k-inflation represents the most general single-field inflation, in which the perturbations usually obey an equation of motion with a time-dependent sound speed. In this paper, we study the observational predictions of the k-inflation by using the high-order uniform asymptotic approximation method. We calculate explicitly the slow-roll expressions of the power spectra, spectral indices, and running of the spectral indices for both the scalar and tensor perturbations. These expressions are all written in terms of the Hubble and sound speed flow parameters. It is shown that the previous results obtained by using the firstorder uniform asymptotic approximation have been significantly improved by the high-order corrections of the uniform asymptotic approximations. Furthermore, we also check our results by comparing them with the ones obtained by other approximation methods, including the Green's function method, WKB approximation, and improved WKB approximation, and find the relative errors.

Scalar and tensor perturbations in loop quantum cosmology: high-order corrections

“…(A.27). As we discussed in [28], in order to do so, first we need to expand ν(η) and c s (η) around the turning point y 0 ofĝ(y) (i.e.,ĝ(ȳ 0 ) = 0 withȳ 0 = −kη 0 ), then perform the integral of ĝ(y) in Eq. (A.26), and calculate the error control function H (+∞).…”

“…(2.8)) perturbations in the form (A.1) by introducing a new variable y = −kη. Then, we find thatĝ(y) and q(y) must be chosen as [27,28],…”

“…More specifically, we consider the slow-roll inflation with the quantum gravitational corrections, but ignore the pre-inflation dynamics. By using the general expressions of power spectra, spectral indices, and running of spectral indices obtained in [29], we calculate explicitly these quantities, up to the third-order approximations in terms of the uniform asymptotic approximation parameter, for which the upper error bounds are 0.15%. All the inflationary observables are expressed in terms of slow-roll parameters and parameters representing quantum corrections explicitly.…”

“…Although a lot of effort has already been devoted to the inflationary models of LQC with both holonomy and inverse-volume quantum corrections, very accurate calculations of inflationary observables in LQC are still absent, and with the arrival of the era of precision cosmology, such calculations are highly demanded. Recently, we have developed a powerful method, the uniform asymptotic approximation method [27][28][29][30], to make precise observational predictions from inflation models, after quantum gravitational effects are taken into account. We note here that such method was first applied to inflationary cosmology in [31], and then we have developed it to more general mult-turning points cases and more precise higher order approximations [27][28][29][30].…”

“…Recently, we have developed a powerful method, the uniform asymptotic approximation method [27][28][29][30], to make precise observational predictions from inflation models, after quantum gravitational effects are taken into account. We note here that such method was first applied to inflationary cosmology in [31], and then we have developed it to more general mult-turning points cases and more precise higher order approximations [27][28][29][30]. The main purpose of the present paper is to use this powerful method to derive the inflationary observables in LQC with holonomy and inverse-volume quantum corrections with high accuracy.…”