Primeval symmetries

TL;DR

This paper investigates how matter and radiation influence the symmetries of Robertson-Walker universes by analyzing Killing equations, revealing conditions under which maximal or reduced symmetries occur, and compares coordinate systems for clearer interpretation.

Contribution

It provides a detailed analysis of Killing symmetries in Robertson-Walker coordinates and relates symmetry breaking to the product of scale factor and Hubble function derivative.

Findings

Maximal symmetry when U is constant

Symmetry reduces to six parameters with matter/radiation

Comparison with static coordinates clarifies physical interpretation

Abstract

A detailed examination of the Killing equations in Robertson-Walker coordinates shows how the addition of matter and/or radiation to a de Sitter Universe breaks the symmetry generated by four of its Killing fields. The product U = (a^2)(dH/dt) of the squared scale parameter by the time-derivative of the Hubble function encapsulates the relationship between the two cases: the symmetry is maximal when U is a constant, and reduces to the six-parameter symmetry of a generic Friedmann-Robertson-Walker model when it is not. As the fields physical interpretation is not clear in these coordinates, comparison is made with the Killing fields in static coordinates, whose interpretation is made clearer by their direct relationship to the Poincare group generators via Wigner-Inonu contractions.