The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant
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Speck JEMS 4th revision.pdf
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Author(s)
Rodnianski, Igor
Speck, Jared R.
Date Issued
2013
Journal
Journal of the European Mathematical Society
Publisher
European Mathematical Society Publishing House
Citation
Rodnianski, Igor, and 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, no. 6 (2013): 2369–2462.
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Author's final manuscript
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+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=c[superscript 2]ρ,0 < c[superscript 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,∞)XT[superscript 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 < c[superscript 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.4171/JEMS/424
