The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant

Igor Rodnianski, Jared Speck

Research output: Contribution to journalArticle

29 Scopus citations

Abstract

In this article, we study small perturbations of the family of Friedmann-Lemaǐtre-Robertson-Walker cosmological background solutions to the coupled Euler-Einstein system with a positive cosmological constant in 1 + 3 spacetime dimensions. The background solutions model an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing exponentially accelerated expansion. Our nonlinear analysis shows that under the equation of state p = c2s ρ, 0 < cs < √1/3, the background metric + fluid solutions are globally future-stable under small irrotational perturbations of their initial data. In particular, we prove that the perturbed spacetime solutions, which have the topological structure [0, ∞) × T3, are future causally geodesi-cally complete. Our analysis is based on a combination of energy estimates and pointwise decay estimates for quasilinear wave equations featuring dissipative inhomogeneous terms. Our main new contribution is showing that when 0 < cs < √1/3, exponential spacetime expansion is strong enough to suppress the formation of fluid shocks. This contrasts against a well-known result of Christodoulou, who showed that in Minkowski spacetime, the corresponding constant-state irrotational fluid solutions are unstable.

Original languageEnglish (US)
Pages (from-to)2369-2462
Number of pages94
JournalJournal of the European Mathematical Society
Volume15
Issue number6
DOIs
StatePublished - Dec 2 2013
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Cosmological constant
  • Energy dissipation
  • Expanding spacetime
  • Geodesically complete
  • Global existence
  • Irrotational fluid
  • Relativistic fluid
  • Wave coordinates

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