High-order Primordial Perturbations with Quantum Gravitational Effects
  In this paper, we provide a systematic investigation of high-order primordial
perturbations with nonlinear dispersion relations due to quantum gravitational
effects in the framework of {\em uniform asymptotic approximations}. Because of
these effects, the equation of motion of the mode function in general has
multiple-turning points. After obtaining analytically approximated solutions to
any order in different regions, associated with different types of turning
points, we match them to the third one. To this order the errors are less than
$0.15\%$. General expressions of the power spectra of the primordial tensor and
scalar perturbations are derived explicitly. We also investigate effects of
back-reactions of the quantum gravitational corrections, and make sure that
inflation lasts long enough in order to solve the underlying problems, such as
flatness, horizon and monopole. Then, we study various features of the spectra
that are observationally relevant. In particular, under a moderate assumption
about the energy scale of the underlying theory of quantum gravity, we have
shown that the quantum gravitational effects may alter significantly the ratio
between the tensor and scalar power spectra, thereby providing a natural
mechanism to alleviate the tension between observations and certain
inflationary models, including the one with a quadratic potential.
