The Einstein-Klein-Gordon-Friedrich system and the non-linear stability of scalar field Cosmologies

Artur Alho, Filipe C. Mena and Juan A. Valiente Kroon

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-perfect fluid system. The resulting evolution system is used to show that, for some classes of scalar field potentials, small non-linear perturbations of an expanding Klein-Gordon-Robertson-Walker background decay exponentially to zero or converge to constant values.