The cosmology of a fundamental scalar

TL;DR

This paper explores a geometric scalar field emerging from conformally flat cosmologies, linking it to the sine-Gordon equation and a new perspective on the universe's stability and mass scale.

Contribution

It introduces a stable family of perturbed conformally flat metrics with an associated scalar field obeying the sine-Gordon equation, connecting geometric and physical scalar fields.

Findings

Standard cosmologies are parametrically unstable within a larger conformally flat family.

A scalar field obeying the sine-Gordon equation naturally arises in the stable metric family.

The constants in the theory must satisfy  > m^2/4 for a non-singular universe.

Abstract

We observe that the standard homogeneous cosmologies, those of Minkowski, de Sitter, and anti-de Sitter, which form the matrix for the Robertson--Walker scale factor, live naturally as isolated points inside a larger family of conformally flat metrics obtained by allowing a tensor containing the information of conformal symmetry breaking to be more general. So the standard cosmological metrics are parametrically unstable in this sense, and therefore unphysical. When we pass to the stable family of perturbed metrics, we immediately encounter a scalar field, which drives the conformal expansion of the universe and which automatically obeys the non-linear sine-Gordon equation. The Lagrangian for the sine-Gordon equation is a cosine potential agreeing to the fourth order with the potential used in the approach to the generation of mass in gauge theories. Accordingly we identify our geometric scalar field---actually of the type of an abelian gauge field---with the recently discovered scalar field. There are two constants in the theory: the first, named $m$, is positive and defines a mass scale for the universe; the second, named $\Lambda$, is the cosmological constant. For the space-time to be everywhere non-singular, equivalently for the (strict) dominant energy condition to hold, these constants must obey the inequality $\Lambda > m^2/4$.