BGV theorem, Geodesic deviation, and Quantum fluctuations

TL;DR

This paper derives a simple expression for the BGV Hubble parameter, linking it to geodesic deviation and quantum fluctuations, and analyzes its behavior in various spacetimes to understand curvature effects.

Contribution

It provides a new, straightforward formula for the BGV Hubble parameter and analyzes its fluctuations due to curvature perturbations in different spacetime geometries.

Findings

Derived an exact expression for the BGV Hubble parameter in terms of geodesic deviation.

Computed the time dependence of the Hubble parameter and deviation magnitude in simple spacetimes.

Characterized the rms fluctuations of these quantities caused by curvature tensor fluctuations.

Abstract

I point out a simple expression for the "Hubble" parameter $\mathscr{H}$, defined by Borde, Guth and Vilenkin (BGV) in their proof of past incompleteness of inflationary spacetimes. I show that the parameter $\mathscr{H}$ which an observer $O$ with four-velocity $\bf v$ will associate with a congruence $\bf u$ is equal to the fractional rate of change of the magnitude $\xi$ of the Jacobi field associated with $\bf u$, measured along the points of intersection of $O$ with $\bf u$, with its direction determined by $\bf v$. I then analyse the time dependence of $\mathscr{H}$ and $\xi$ using the geodesic deviation equation, computing these exactly for some simple spacetimes, and perturbatively for spacetimes close to maximally symmetric ones. The perturbative solutions are used to characterise the rms fluctuations in these quantities arising due to possible fluctuations in the curvature tensor.