Scalar and tensor perturbations in loop quantum cosmology: High-order
  corrections
  Loop quantum cosmology (LQC) provides promising resolutions to the
trans-Planckian issue and initial singularity arising in the inflationary
models of general relativity. In general, due to different quantization
approaches, LQC involves two types of quantum corrections, the holonomy and
inverse-volume, to both of the cosmological background evolution and
perturbations. In this paper, using {\em the third-order uniform asymptotic
approximations}, we derive explicitly the observational quantities of the
slow-roll inflation in the framework of LQC with these quantum corrections. We
calculate the power spectra, spectral indices, and running of the spectral
indices for both scalar and tensor perturbations, whereby the tensor-to-scalar
ratio is obtained. We expand all the observables at the time when the
inflationary mode crosses the Hubble horizon. As the upper error bounds for the
uniform asymptotic approximation at the third-order are $\lesssim 0.15\%$,
these results represent the most accurate results obtained so far in the
literature. It is also shown that with the inverse-volume corrections, both
scalar and tensor spectra exhibit a deviation from the usual shape at large
scales. Then, using the Planck, BAO and SN data we obtain new constraints on
quantum gravitational effects from LQC corrections, and find that such effects
could be within the detection of the forthcoming experiments.
