Singularities in FLRW spacetimes

Huibert Het Lam,Tomislav Prokopec

doi:10.1016/j.physletb.2017.10.070

Abstract

We point out that past-incompleteness of geodesics in FLRW spacetimes does not necessarily imply that these spacetimes start from a singularity. Namely, if a test particle that follows such a trajectory has a non-vanishing velocity, its energy was super-Planckian at some time in the past if it kept following that geodesic. That indicates a breakdown of the particle's description, which is why we should not consider those trajectories for the definition of an initial singularity. When one only considers test particles that do not have this breakdown of their trajectory, it turns out that the only singular FLRW spacetimes are the ones that have a scale parameter that vanishes at some initial time.

Highlights

Hubble’s law, the observed abundance of elements, the cosmic background radiation and the large scale structure formation in the universe are strong evidence that the universe expanded from an initial very high dense state to how we observe it
That definition is based on a test particle that has that geodesic as trajectory
This means that the particle stops being a test particle and it does not matter that its trajectory is past-incomplete

Summary

INTRODUCTION

Hubble’s law, the observed abundance of elements, the cosmic background radiation and the large scale structure formation in the universe are strong evidence that the universe expanded from an initial very high dense state to how we observe it now. When we want to use these theorems to say something about an initial singularity in an FLRW spacetime, we need a metric that has a scale parameter a that becomes zero at some time in the past. Another theorem that proves that a geodesic is past-incomplete was published in [5] and is applicable to spacetimes that have a(t) > 0 for all t. It says that when the average Hubble parameter H = a /a along a non-spacelike geodesic, Hav, satisfies Hav > 0, the geodesic must be pastincomplete. Examples are for instance given by spacetimes with H > 0 and a → a0 > 0 for t → −∞ (in this case we will have that H → 0 as t → −∞)

ENERGY OF TEST PARTICLES

CONCLUSION