Quantum Cosmology with Many Fluids and the Choice of Cosmological Time

In this work, we propose the quantization of a cosmological model describing the primordial universe filled with five barotropic fluids, namely radiation, dust, vacuum, cosmic strings, and domain walls. We intend to identify which fluid is best suited to provide phenomenologically the temporal variable in accordance with the observable universe. Through the Galerkin spectral method and the finite difference method in the Crank-Nicolson scheme (vacuum case), the quantum cosmological solutions are obtained and compared. We, also, compare the quantum cosmological solutions with the corresponding classical ones. The vacuum case is especially interesting because it provides a tunneling transition mechanism from the quantum to the classical phase and the possibility of calculating quantum tunneling probabilities.