HAMILTONIAN-FORMULATION OF BIANCHI COSMOLOGICAL MODELS IN QUADRATIC THEORIES OF GRAVITY

We use Boulware's Hamiltonian formalism of quadratic gravity theories in order to analyse the classical behaviour of Bianchi cosmological mo dels for a Lagrangian density L = R + beta(c)R(2) in four spacetime di mensions. For this purpose we define a canonical transformation which leads to a clear distinction between two main variants of the general quadratic theory, i.e. for L = R + beta(c)R(2) or conformal L = alpha( c)C(alpha beta mu nu)C(alpha beta mu nu) Lagrangian densities. in this paper we restrict the study to the first variant. For the Bianchi-typ e I and IX models, we give the explicit forms of the super-Hamiltonian constraint, of the ADM Hamiltonian density and of the corresponding c anonical equations. In the case of a pure quadratic theory L = beta(c) R(2), we solve them analytically for the Bianchi I model. For the Bian chi-type IX model, we reduce the first-order equations of the Hamilton ian system to three coupled second-order equations for the true physic al degrees of freedom. This discussion is extended to isotropic FLRW m odels.