Hamiltonian Approach to the Dynamics of Expanding Homogeneous Universes in the Brans-Dicke Cosmology

Open Access
Publisher Website
Google Scholar
Abstract
In view of a grave importance of the problem of initial singularity in theoretical cosmology, the dynamical behavior of expanding homogeneous universes (without rotation) in the Brans-Dicke cosmology is studied by means of extending suitably the canonical formalism due to Arnowitt, Deser and Misner. It is shown that, even if the inertial (scalar) mode characteristic in their theory of gravitation is omitted, our Hamiltonian is somewhat different from Ryan's Hamiltonian in relativistic cosmology. This is due to the difference in the manipulation of the source (consisting of matter and radiation assumed as a perfect fluid whose total density and pressure are represented by ρ and p, respectively) Lagrangian, and it seems that our manipulation is superior to Ryan's. In spite of this, so far as 0 ≤p ≤ρ/3, there exists an extremely early stage at which the source term in our Hamiltonian becomes negligible in such a way that it is reduced to a generalized version of Misner's Hamiltonian derivable from Ryan's under the same approximation. If ρ/3 ≪ p ≤ρ, however, such a stage cannot exist because of some peculiar role of the inertial mode interacting with matter and radiation. Accordingly, the dynamical behavior of gravitational and inertial modes at the extremely early stage of both the Bianchi-type IX universe with p = ρ/3 and the Bianchi-type I universe with p = ρ is analyzed in detail. The dynamical behavior is described as the three-dimensional motion of a world point in the presence of either the tri-angular potential walls (found by Misner) with gravitational origin or another potential field with inertial origin. It is shown in the former case that the inertial mode plays a significant role to modify Misner's bounce law for the collision of the world point with the potential walls, but is incompetent to eliminate the initial singularity of infinite density. On the other hand, in the latter case, the initial singularity may be formally removed under some condition which, however, contradicts with the requirement due to Brans-Dicke that the coupling constant ω must be larger than about 6.

This publication has 0 references indexed in Scilit: