JournalsjemsVol. 15, No. 6pp. 2369–2462

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

  • Igor Rodnianski

    Massachusetts Institute of Technology, Cambridge, USA
  • Jared Speck

    Massachusetts Institute of Technology, Cambridge, USA
The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant cover
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Abstract

In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the coupled Euler-Einstein system with a positive cosmological constant in 1+31 + 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=c2ρ,0<c2<1/3p = c^2 \rho, 0 < c^2 < 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,)XT3[0,\infty) X T^3, are future causally geodesically 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<c2<1/30 < c^2 < 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.

Cite this article

Igor Rodnianski, Jared Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant. J. Eur. Math. Soc. 15 (2013), no. 6, pp. 2369–2462

DOI 10.4171/JEMS/424