Hamiltonian Cosmology

TL;DR

This paper presents a Hamiltonian formulation of cosmology where the scale factor is treated as a dynamical variable, providing an energy conservation perspective and new insights into cosmological red-shift and energy exchange.

Contribution

It introduces a Hamiltonian approach to cosmology that interprets the scale factor as a dynamical coordinate, offering a novel energy conservation viewpoint and intuitive understanding of cosmological phenomena.

Findings

Hamiltonian formulation reproduces Friedmann equation

Energy densities are formally conserved quantities

Cosmological red-shift explained as energy loss in EM fields

Abstract

The cosmological scale factor $a(t)$ of the flat-space Robertson-Walker geometry is examined from a Hamiltonian perspective wherein $a(t)$ is interpreted as an independent dynamical coordinate and the curvature density $\sqrt {- g(a)} R({a,\dot a,\ddot a})$ is regarded as an action density in Minkowski spacetime. The resulting Hamiltonian for $a(t)$ is just the first Friedmann equation of the traditional approach (i.e. the Robertson-Walker cosmology of General Relativity), as might be expected. The utility of this approach however stems from the fact that each of the terms matter, radiation, and vacuum, and including the kinetic / gravitational field term, are formally energy densities, and the equation as a whole becomes a formal statement of energy conservation. An advantage of this approach is that it facilitates an intuitive understanding of energy balance and exchange on the cosmological scale that is otherwise absent in the traditional presentation. Each coordinate system has its own internally consistent explanation for how energy balance is achieved. For example, in the spacetime with line element $ds^2 = dt^2 - a^2(t) d{\bf{x}}^2$, cosmological red-shift emerges as due to a post-recombination interaction between the scalar field $a(t)$ and the EM fields in which the latter loose energy as if propagating through a homogeneous lossy medium, with the energy lost to the scale factor helping drive the cosmological expansion.