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 $\lesssim 0.15\%$. 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.

revtex, 5 tables. Added one figure, and various typos were corrected

References in corpus (16)

• Quantum Gravity at a Lifshitz Point
• Cosmological Constraints from the SDSS Luminous Red Galaxies
• Chiral Primordial Gravitational Waves from a Lifshitz Point
• DBI and the Galileon reunited
• Measuring the matter density using baryon oscillations in the SDSS
• Loop quantum gravity corrections to gravitational wave dispersion
• Consistency of holonomy-corrected scalar, vector and tensor perturbations in Loop Quantum Cosmology
• Minimum Length Cutoff in Inflation and Uniqueness of the Action
• Gravitational Wave Constraints on DBI Inflation
• K-inflationary Power Spectra in the Uniform Approximation
• Inflationary observables in loop quantum cosmology
• Vector and tensor perturbations in Horava-Lifshitz cosmology
• Breaking the scale invariance of the primordial power spectrum in Horava-Lifshitz Cosmology
• Fine-Tuning in DBI Inflationary Mechanism
• Inflationary Spectra from Lorentz Violating Dissipative Models
• Perturbative evaluation of scalar two-point function in the cosmic microwave background power spectrum

Cited by in corpus (7)

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• Power spectra and spectral indices of $k$-inflation: high-order corrections
• Modified Lyth bound and implications of BICEP2 results
• Gravitational Particle Production and the Validity of Effective Descriptions in Loop Quantum Cosmology
• Inflationary Cosmology with Quantum Gravitational Effects and the Swampland Conjectures
• Effects of high-order operators in non-relativistic Lifshitz holography