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We develop approximation methods in the hybrid quantization of the Gowdymodel with linear polarization and a massless scalar field, for the case ofthree-torus spatial topology. The loop quantization of the homogeneousgravitational sector of the Gowdy model (according to the improved dynamicsprescription) and the presence of inhomogeneities lead to a very complicatedHamiltonian constraint. Therefore, the extraction of physical results calls forthe introduction of well justified approximations. We first show how toapproximate the homogeneous part of the Hamiltonian constraint, correspondingto Bianchi I geometries, as if it described a Friedmann-Robertson-Walker (FRW)model corrected with anisotropies. This approximation is valid in thehigh-energy sector of the FRW geometry (concerning its contribution to theconstraint) and for anisotropy profiles that are sufficiently smooth. Inaddition, for certain families of states associated to regimes of physicalinterest, with negligible effects of the anisotropies and smallinhomogeneities, one can approximate the Hamiltonian constraint of theinhomogeneous system by that of an FRW geometry with a relatively simple mattercontent, and then obtain its solutions.

Approximation methods in Loop Quantum Cosmology: From Gowdy cosmologies to inhomogeneous models in Friedmann-Robertson-Walker geometries