  We study the hybrid quantization of the linearly polarized Gowdy $T^3$ model
with a massless scalar field with the same symmetries as the metric. For
simplicity, we quantize its restriction to the model with local rotational
symmetry. Using this hybrid approach, the homogeneous degrees of freedom of the
geometry are quantized \`a la loop, leading to the resolution of the
cosmological singularity. A Fock quantization is employed both for the matter
and the gravitational inhomogeneities. Owing to the inclusion of the massless
scalar field this system allows us to modelize flat Friedmann-Robertson-Walker
cosmologies filled with inhomogeneities propagating in one direction, providing
a perfect scenario to study the quantum back-reaction of the inhomogeneities on
the polymeric homogeneous and isotropic background.
