Hamiltonian Formulation of Bianchi Cosmological Models in Quadratic Theories of Gravity

TL;DR

This paper applies Hamiltonian formalism to quadratic gravity theories to analyze classical Bianchi cosmological models, providing explicit formulations and solutions for certain models, and distinguishing between different quadratic gravity variants.

Contribution

It introduces a canonical transformation to differentiate quadratic gravity variants and derives explicit Hamiltonian formulations for Bianchi models, including analytical solutions for Bianchi I.

Findings

Explicit super-Hamiltonian and Hamiltonian densities for Bianchi I and IX models.

Analytical solutions obtained for Bianchi I model in pure R-squared gravity.

Reduction of equations to coupled second-order forms for Bianchi IX model.

Abstract

We use Boulware's Hamiltonian formalism of quadratic gravity theories in order to analyze the classical behaviour of Bianchi cosmological models for a Lagrangian density containing quadratic terms in the curvature. For this purpose we define a canonical transformation which leads to a clear distinction between two main variants of the quadratic theory, namely the R-squared or conformal Lagrangian densities. In this paper we restrict the study to the first variant. For Bianchi type I and IX models we give the explicit forms of the super-Hamiltonian constraint, of the ADM Hamiltonian density and of the corresponding canonical equations. In the case of a pure R-squared theory we solve these equations analytically for Bianchi I model. For Bianchi type IX model, we reduce the first-order equations of the Hamiltonian system to three coupled second-order equations for the true physical degrees of freedom. This discussion is extended to isotropic FLRW models.