Advances in Theoretical and Mathematical Physics

Volume 21 (2017)

Number 4

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

Pages: 857 – 899

DOI: http://dx.doi.org/10.4310/ATMP.2017.v21.n4.a1

Authors

Artur Alho (Centro de Matemática, Universidade do Minho, Braga, Portugal; and Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Lisboa, Portugal)

Filipe C. Mena (Centro de Matemática, Universidade do Minho, Braga, Portugal)

Juan A. Valiente Kroon (School of Mathematical Sciences, Queen Mary, University of London, United Kingdom)

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–Lemaître–Robertson–Walker backgrounds, with scalar field potentials satisfying certain future asymptotic conditions, decay exponentially to zero, in synchronous time.

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AA and FM were supported by projects PTDC/MAT/108921/2008, CERN/FP/123609/2011 and PTDC/MAT-ANA/1275/2014 and by CMAT, Univ. Minho, through FEDER Funds COMPETE and FCT Project Est-OE/MAT/UI0013/2014. AA thanks the Relativity Group at the School of Mathematical Sciences, Queen Mary, University of London, for their warm hospitality while most of this work was done and FCT for grant SFRH/BD/48658/2008. JAVK was supported by an EPSRC Advanced Research Fellowship and by a project research grant from the Leverhulme Trust (F/07476/AI). JAVK thanks the Centre of Mathematics of the University of Minho for its hospitality.