The Hamiltonian formalism for scalar fields coupled to gravity in a cosmological background

Alex E. Bernardini, O. Bertolami

Published 2012-12-03, updated 2013-07-08Version 2

A novel routine to investigate the scalar fields in a cosmological context is discussed in the framework of the Hamiltonian formalism. Starting from the Einstein-Hilbert action coupled to a Lagrangian density that contains two components - one corresponding to a scalar field Lagrangian, ${\mathcal{L}}_{\phi}$, and another that depends on the scale parameter, ${\mathcal{L}}_{a}$ - one can identify a generalized Hamiltonian density from which first-order dynamical equations can be obtained. This set up corresponds to the dynamics of Friedmann-Robertson-Walker models in the presence of homogeneous fields embedded into a generalized cosmological background fluid in a system that evolves all together isentropically. Once the generalized Hamiltonian density is properly defined, the constraints on the gravity-matter-field system are straightforwardly obtained through the first-order Hamilton equations. The procedure is illustrated for three examples of cosmological interest for studies of the dark sector: real scalar fields, tachyonic fields and generalized Born-Infeld tachyonic fields. The inclusion of some isentropic fluid component into the Friedmann equation allows for identifying an exact correspondence between the dark sector underlying scalar field and an ordinary real scalar field dynamics. As a final issue, the Hamiltonian formulation is used to set the first-order dynamical equations through which one obtains the exact analytical description of the cosmological evolution of a generalized Chaplygin gas (GCG) with dustlike matter, radiation or curvature contributions. Model stability in terms of the square of the sound velocity, $c_{s}^{2}$, cosmic acceleration, $q$, and conditions for inflation are discussed.