  We study the evolution of quantum fluctuations of gravity around an
inflationary solution in renormalizable quantum gravity, in which the initial
scalar-fluctuation dominance is shown by the background-free nature expressed
by a special conformal invariance. Inflation ignites at the Planck scale and
continues until spacetime phase transition occurs at a dynamical scale about
$10^{17}$GeV. We can show that during inflation, the initially large
scale-invariant fluctuations reduce in amplitude to the appropriate magnitude
suggested by tiny CMB anisotropies. The goal of this research is to derive the
spectra of scalar fluctuations at the phase transition point, that is, the
primordial spectra. A system of nonlinear evolution equations for the
fluctuations is derived from the quantum gravity effective action. The running
coupling constant is then expressed by a time-dependent average following the
spirit of the mean field approximation. In this paper, we determine and examine
various nonlinear terms, not treated in previous studies such as the
exponential factor of the conformal mode. These contributions occur during the
early stage of inflation when the amplitude is still large. Moreover, in order
to verify their effects concretely, we numerically solve the evolution equation
by making a simplification to extract the most contributing parts of the terms
in comoving momentum space. The result indicates that they serve to maintain
the initial scale invariance over a wide range beyond the comoving Planck
scale. This is a challenge toward the derivation of the precise primordial
spectra, and we expect in the future that it will lead to the resolution of the
tensions that have arisen in cosmology.
