Time-reparametrization invariance and Hamilton-Jacobi approach to the cosmological sigma-model

TL;DR

This paper reviews models with local time-reparametrization invariance, develops a covariant Hamilton-Jacobi formalism, and applies it to solve cosmological models including scalar fields within FLRW universes.

Contribution

It introduces a covariant, gauge-invariant Hamilton-Jacobi approach to cosmological sigma-models with time-reparametrization invariance, providing explicit solutions for non-linear O(N)-models.

Findings

Negative-energy contributions are essential for time-reparametrization symmetry.

A covariant Hamilton-Jacobi formalism is successfully developed and applied.

Explicit gauge-invariant solutions are obtained for non-linear sigma-model cosmologies.

Abstract

The construction of physical models with local time-reparametrization invariance is reviewed. Negative-energy contributions to the hamiltonian are shown to be crucial for the realization of this reparametrization symmetry. The covariant formulation of the dynamics is used to develop a time and gauge invariant Hamilton-Jacobi theory. This formalism is applied to solve for the cosmology of a homogeneous universe of the Friedmann-Lemaitre-Robertson-Walker type. After a discussion of empty universes, the FLRW theory is extended with homogeneous scalar fields generically described by a $\sg$-model on some scalar manifold. An explicit gauge-invariant solution is constructed for the non-linear O(N)-models.