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.
