  We study the evolution of a homogeneous and isotropic spacetime whose spatial
sections have three-torus topology, coupled to a massless scalar field with
small scalar perturbations within loop quantum cosmology. We consider a
proposal for the effective dynamics based on a previous hybrid quantization
completed by us. Consequently, we introduce a convenient gauge fixing and adopt
reduced canonical variables adapted to that hybrid quantum description.
Besides, we keep backreaction contributions on the background coming from terms
quadratic in the perturbations in the action of the system. We carry out a
numerical analysis assuming that the inhomogeneities were in a massless vacuum
state at distant past (where the initial data are set). At distant future, we
observe a statistical amplification of the modes amplitude in the infrared
region, as well as a phase synchronization arising from quantum gravity
phenomena. A description of the perturbations in terms of the Mukhanov-Sasaki
gauge invariants provides the same qualitative results. Finally, we analyze
some consequences of the backreaction in our effective description.
