Inflationary spectra with inverse-volume corrections in loop quantum cosmology and their observational constraints from Planck 2015 data

We first derive the primordial power spectra, spectral indices and runnings of both scalar and tensor perturbations of a flat inflationary universe to the second-order approximations of the slow-roll parameters, in the framework of loop quantum cosmology with the inverse-volume quantum corrections. This represents an extension of our previous work in which the parameter $\sigma$ was assumed to be an integer, where $\sigma$ characterizes the quantum corrections and in general can take any of values from the range $\sigma \in (0, 6]$. Restricting to the first-order approximations of the slow-roll parameters, we find corrections to the results obtained previously in the literature, and point out the causes for such errors. To our best knowledge, these represent the most accurate calculations of scalar and tensor perturbations given so far in the literature. Then, fitting the perturbations to the recently released data by Planck (2015), we obtain the most severe constraints for various values of $\sigma$. Using these constraints as our referring point, we discuss whether these quantum gravitational corrections can lead to measurable signatures in the future cosmological observations. We show that, depending on the value of $\sigma$, the scale-dependent contributions to the relativistic inflationary spectra due to the inverse-volume corrections could be well within the range of the detectability of the forthcoming generations of experiments, such as the Stage IV experiments.