Hamiltonian formalism of Bianchi models with cosmological constant

Abstract

Cosmic Background Radiation (CMB) has resulted in anomalies or deviations from the standard cosmology. Consequently, we propose to study Bianchi spacetimes with cosmological constant by applying the ADM formailsm of general relativity in the Hamiltonian version. From the Lagrangian density and with the use of the Legendre transformation we can calculate the Hamiltonian density and the Poisson parentheses. Subsequently, we present the equations of motion for each of the Bianchi spacetimes. We can find that the dependent part of the gravitational potential of the Hamiltonian density tends to zero and from the equations of motion we find that canonical moments are constant. On the other hand, FLRW models can be generalized only to some Bianchi models. The Bianchi types I and VII are a generalization of the Euclidean FLRW model (k=0), the Bianchi IX for the spherical FLRW model (k=1) and the Bianchi types V and VII are for the hyperbolic FLRW model (k=-1). The rest of the Bianchi models do not contain the FLRW models as a particular case.