Hamiltonian formalism of the Bianchi's models

Lately the Cosmic Background Radiation (CMB) data have resulted in anomalies or deviations with respect to the standard model of cosmology, which has led several cosmologists to consider alternative models to the standard model (homogeneous and isotropic), such as the Bianchi models, which are homogeneous but anisotropic. Based on these motivations to consider alternative models, we propose to study, in the present work, the algebraic classification of the Bianchi models and each of the Bianchi space-times, applying the ADM formalism of general relativity in its Hamiltonian version and the groups G3 . The dynamic equations are shown with the help of the Hamiltonian density H and the Poisson parentheses, in other words, the equations of motion are presented for each of the Bianchi space-times. Some theoretical consequences of these equations are discussed when we take the limit Ω → -∞ and the fixed parameters β + and β - , consequently, we find that the dependent part of the gravitational potential from the Hamiltonian Density tends to zero and from the equations of motion we find the constant of motion, pΩ = pβ+ = pβ- = constant .