On Shear-Free perturbations of FLRW Universes

TL;DR

This paper proves that in linearized barotropic perfect fluid cosmologies, shear-free matter must be either expansion-free or rotation-free, except for a non-physical equation of state, highlighting differences from Newtonian cosmology.

Contribution

It extends the shear-free perturbation result to barotropic perfect fluids in linearized FLRW universes, excluding a non-physical equation of state.

Findings

Shear-free matter must be either expansion-free or rotation-free in linearized FLRW models.

The result holds for all physically realistic barotropic fluids.

Newtonian cosmology does not replicate this shear-free condition behavior.

Abstract

A surprising exact result for the Einstein Field Equations is that if pressure-free matter is moving in a shear-free way, then it must be either expansion-free or rotation-free. It has been suggested this result is also true for any barotropic perfect fluid, but a proof has remained elusive. We consider the case of barotropic perfect fluid solutions linearized about a Robertson-Walker geometry, and prove that the result remains true except for the case of a specific highly non-linear equation of state. We argue that this equation of state is non-physical, and hence the result is true in the linearized case for all physically realistic barotropic perfect fluids. This result, which is not true in Newtonian cosmology, demonstrates that the linearized solutions, believed to result in standard local Newtonian theory, do not always give the usual behaviour of Newtonian solutions.