  We apply the full theory of Loop Quantum Gravity (LQG) to cosmology and
present a top-down derivation of gauge-invariant cosmological perturbation
theory from quantum gravity. The derivation employs the reduced phase space
formulation of LQG and the new discrete path integral formulation defined in
arXiv:1910.03763. We demonstrate that in the semiclassical approximation and
continuum limit, the result coincides with the existing formulation of
gauge-invariant cosmological perturbation theory in e.g. arXiv:0711.0117. Time
evolution of cosmological perturbations is computed numerically from the new
cosmological perturbation theory of LQG, and various power spectrums are
studied for scalar mode and tensor mode perturbations. Comparing these power
spectrums with predictions from the classical theory demonstrate corrections in
the ultra-long wavelength regime. These corrections are results from the
lattice discretization in LQG. In addition, tensor mode perturbations at late
time demonstrate the emergence of spin-2 gravitons as low energy excitations
from LQG. The graviton has a modified dispersion relation and reduces to the
standard graviton in the long wavelength limit.
