Hybrid quantization: From Bianchi I to the Gowdy model

We complete the quantization of the vacuum Bianchi I model within the framework of loop quantum cosmology adopting a new improved dynamics scheme put forward recently. In addition, we revisit the hybrid quantization of the Gowdy $T^3$ cosmologies with linear polarization using that scheme, proving with rigor some steps that remained unconcluded. The family of Gowdy $T^3$ cosmologies is an inhomogeneous model whose subset of homogeneous solutions is given precisely by the vacuum Bianchi I model. Our hybrid approach combines the new loop quantum cosmology description of this homogeneous sector with a Fock quantization of the inhomogeneities. Both in the Bianchi I model and in the Gowdy model the Hamiltonian constraint provides an evolution equation with respect to the volume of the Bianchi I universe, which is a discrete variable with a strictly positive minimum. We show that, in vacuo, this evolution is well defined inasmuch as the associated initial value problem is well posed: physical solutions are completely determined by the data on the initial section of constant Bianchi I volume. This fact allows us first to carry out to completion the quantization of the vacuum Bianchi I model which had not yet been achieved and then to confirm the feasibility of the hybrid procedure when the homogeneous sector is quantized with the new improved dynamics scheme.