Detecting quantum gravitational effects of loop quantum cosmology in the early universe?

We derive the primordial power spectra and spectral indexes of the density fluctuations and gravitational waves in the framework of loop quantum cosmology (LQC) with holonomy and inverse-volume corrections by using the uniform asymptotic approximation method to its third order, at which the upper error bounds are $\lesssim 0.15\%$ and accurate enough for the current and forthcoming cosmological observations. Then, using the Planck, BAO, and supernova data, we obtain the tightest constraints on quantum gravitational effects from LQC corrections and find that such effects could be well within the detection of the current and forthcoming cosmological observations.