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Beyond isotropic & homogeneous loop quantum cosmology: theory and predictions

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In this talk I will discuss several recent advances in loop quantum cosmology and its extension to inhomogeneous models. I will focus on spherically symmetric spacetimes and Gowdy cosmologies with local rotational symmetry in vacuum. I will discuss how to implement a quantum Hamiltonian evolution on these quantizations. Then, I will focus on how we can extract predictions from those quantum geometries, and finally analyze a concrete example: cosmological perturbations on Bianchi I spacetimes in LQC.