Classical stability of a homogeneous, anisotropic inflating space-time

TL;DR

This paper investigates the classical stability of an anisotropic inflationary space-time influenced by a vector field, finding conditions under which the background remains stable or becomes unstable based on the vector field's kinetic terms.

Contribution

It introduces a model for anisotropic inflation driven by a vector field and analyzes its stability, highlighting the importance of the vector field's kinetic term form.

Findings

Linear perturbations do not grow for certain parameters.

Negative energy modes indicate instability when kinetic terms are not of the field strength tensor form.

Positive energy perturbations ensure stability when kinetic terms are of the field strength tensor form.

Abstract

We study the classical stability of an anisotropic space-time seeded by a spacelike, fixed norm, dynamical vector field in a vacuum-energy-dominated inflationary era. It serves as a model for breaking isotropy during the inflationary era. We find that, for a range of parameters, the linear differential equations for small perturbations about the background do not have a growing mode. We also examine the energy of fluctuations about this background in flat-space. If the kinetic terms for the vector field do not take the form of a field strength tensor squared then there is a negative energy mode and the background is unstable. For the case where the kinetic term is of the form of a field strength tensor squared we show that perturbations about the background have positive energy at lowest order.