Maximal Fermi charts and geometry of inflationary universes

TL;DR

This paper analyzes the geometry of inflationary universes using Fermi coordinates, deriving formulas, bounds, and comparing properties of inflationary and non-inflationary cosmologies, especially regarding horizons and velocities.

Contribution

It provides exact formulas for Fermi coordinates in Robertson-Walker spacetimes and compares geometric and velocity properties between inflationary and non-inflationary models.

Findings

Fermi coordinate charts cover all events within the cosmological horizon or globally.

Proper radii diverge in non-inflationary cosmologies as time increases.

Fermi velocities are subluminal in inflationary models, but can be superluminal in non-inflationary models.

Abstract

A proof is given that the maximal Fermi coordinate chart for any comoving observer in a broad class of Robertson-Walker spacetimes consists of all events within the cosmological event horizon, if there is one, or is otherwise global. Exact formulas for the metric coefficients in Fermi coordinates are derived. Sharp universal upper bounds for the proper radii of leaves of the foliation by Fermi spaceslices are found, i.e., for the proper radii of the spatial universe at fixed times of the comoving observer. It is proved that the radius at proper time $\tau$ diverges to infinity for non inflationary cosmologies as $\tau\to\infty$, but not necessarily for cosmologies with periods of inflation. It is shown that any spacelike geodesic orthogonal to the worldline of a comoving observer has finite proper length and terminates within the cosmological event horizon (if there is one) at the big bang. Geometric properties of inflationary versus non inflationary cosmologies are compared, and opposite inequalities for the inflationary and non inflationary cases, analogous to Hubble's law, are obtained for the Fermi relative velocities of comoving test particles. It is proved that the Fermi relative velocities of radially moving test particles are necessarily subluminal for inflationary cosmologies in contrast to non inflationary models, where superluminal relative Fermi velocities necessarily exist.