The Einstein-Friedrich-nonlinear scalar field system and the stability of scalar field Cosmologies

TL;DR

This paper develops a hyperbolic system for scalar fields coupled to gravity and proves that small perturbations of certain cosmological models decay exponentially, indicating stability of these scalar field cosmologies.

Contribution

It introduces a new hyperbolic formulation for scalar field gravity systems and demonstrates exponential decay of perturbations in expanding cosmological backgrounds.

Findings

Small nonlinear perturbations decay exponentially.

Stability of scalar field cosmologies under certain conditions.

New hyperbolic system formulation for scalar fields and gravity.

Abstract

A frame representation is used to derive a first order quasi-linear symmetric hyperbolic system for a scalar field minimally coupled to gravity. This procedure is inspired by similar evolution equations introduced by Friedrich to study the Einstein-Euler system. The resulting evolution system is used to show that small nonlinear perturbations of expanding Friedman-Lema\^itre-Robertson-Walker backgrounds, with scalar field potentials satisfying certain future asymptotic conditions, decay exponentially to zero, in synchronous time.