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We examine the variational and conformal structures of higher order theories of gravity which are derived from a metric-connection Lagrangian that is an arbitrary function of the curvature invariants. We show that the constrained first order formalism when applied to these theories may lead consistently to a new method of reduction of order of the associated field equations. We show that the similarity of the field equations which are derived from appropriate actions via this formalism to those produced by Hilbert varying purely metric Lagrangians is not merely formal but is implied by the diffeomorphism covariant property of the associated Lagrangians. We prove that the conformal equivalence theorem of these theories with general relativity plus a scalar field, holds in the extended framework of Weyl geometry with the same forms of field and self-interacting potential but, in addition, there is a new 'source term' which plays the role of a stress. We point out how these results may be further exploited and address a number of new issues that arise from this analysis.

We develop a Hamiltonian formulation of the Bianchi type I spacetime in conformal gravity - the theory described by the Lagrangian , which involves the quadratic curvature invariant constructed from the Weyl tensor, in a four-dimensional spacetime. We derive the explicit forms of the super-Hamiltonian and of the constraint expressing the conformal invariance of the theory and we write down the system of canonical equations. To seek out exact solutions of this system we add extra constraints on the canonical variables and we go through a global involution algorithm which eventually leads to the closure of the constraint algebra. The Painlevé approach provides us with a proof of non-integrability, as a consequence of the presence of movable logarithms in the general solution of the problem. We extract all possible particular solutions that may be written in closed analytical form. This enables us to demonstrate that the global involution algorithm has brought forth the complete list of exact solutions that may be written in closed analytical form. We discuss the conformal relationship, or absence thereof, of our solutions with Einstein spaces.

In a recent series of papers Endean examines the properties of spatially homogeneous and isotropic (FLRW) cosmological models filled with dust in the "conformally flat spacetime presentation of cosmology" (CFS cosmology). This author claims it is possible to resolve a certain number of the difficulties the standard model exhibits when confronted to observations, if the theoretical predictions are obtained in the special framework of CFS cosmology. As a by-product of his analysis Endean claims that no initial (big-bang) nor final (big-crunch) singularities occur in the closed FLRW model. In this paper we show up the fallacious arguments leading to Endean's conclusions and we consistently reject his CFS cosmology.Subject headings: cosmology : theory -large-scale structure of universe 1 Fonds pour la formationà la recherche dans l'industrie et dans l'agriculture

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 L = R + β c R 2 in four spacetime dimensions. 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 + β c R 2 or conformal L = α c C αβµν C αβµν 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 quadratic theory L = β c R 2 , we solve them 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.