Cosmology as Geodesic Motion

TL;DR

This paper demonstrates that flat cosmologies with scalar fields and arbitrary potentials can be represented as geodesic motion in an augmented target space, providing a geometric framework for analyzing cosmological solutions and their late-time behavior.

Contribution

It introduces a novel geometric formalism linking cosmological evolution to geodesics in an augmented space, applicable to various potentials and curvature cases.

Findings

Flat cosmologies correspond to geodesics in an augmented target space.

Accelerating cosmologies are associated with timelike geodesics within an acceleration subcone.

Explicit analysis of exponential potential models and their late-time behavior.

Abstract

For gravity coupled to N scalar fields with arbitrary potential V, it is shown that all flat (homogeneous and isotropic) cosmologies correspond to geodesics in an (N+1)-dimensional `augmented' target space of Lorentzian signature (1,N), timelike if V>0, null if V=0 and spacelike if V<0. Accelerating cosmologies correspond to timelike geodesics that lie within an `acceleration subcone' of the `lightcone'. Non-flat (k=-1,+1) cosmologies are shown to evolve as projections of geodesic motion in a space of dimension (N+2), of signature (1,N+1) for k=-1 and signature (2,N) for k=+1. This formalism is illustrated by cosmological solutions of models with an exponential potential, which are comprehensively analysed; the late-time behviour for other potentials of current interest is deduced by comparison.