Scalar Field in the Bianchi I: Noncommutative Classical and Quantum Cosmology

Using the ADM formalism in the minisuperspace, we obtain the commutative and noncommutative exact classical solutions and exact wave function to the Wheeler-DeWitt equation with an arbitrary factor ordering, for the anisotropic Bianchi type I cosmological model, coupled to a scalar field, cosmological term and barotropic perfect fluid. We introduce noncommutative scale factors, considering that all minisuperspace variables q i do not commute, so the symplectic structure was modified. In the classical regime, it is shown that the anisotropic parameter β ±nc and the field φ, for some value in the λ eff cosmological term and noncommutative θ parameter, present a dynamical isotropization up to a critical cosmic time t c ; after this time, the effects of isotropization in the noncommutative minisuperspace seems to disappear. In the quantum regimen, the probability density presents a new structure that corresponds to the value of the noncommutativity parameter.