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The Stability of the Euler-Einstein system with a positive Cosmological Constant

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The Euler-Einstein system models the evolution of a dynamic spacetime containing a perfect fluid. In this talk, I will discuss the nonlinear stability of the Friedmann-Lemaˆıtre-Robertson-Walker family of background cosmological solutions to the Euler-Einstein system in 1 + 3 dimensions with a positive cosmological constant Λ. The background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. The main result is a proof that under the equation of state p = cs2 ρ, 0 < cs2 < 1/3, the background solutions are globally future-stable under small perturbations. In particular, the perturbed spacetimes, which have the topological structure [0, ∞) × T3 , are future causally geodesically complete. The results I will present are extensions of previous joint work with Igor Rodnianski, which covered the case of an irrotational fluid, and of work by Hans Ringstrom on the Einstein-non-linear-scalar-field system. Mathematically, the main result is a proof of small-data global existence for a modified version of the Euler-Einstein equations that are equivalent to the un-modified equations. The proof is based on the vectorfield method of Christodoulou and Klainerman.

It is of special interest to note that the behavior of the fluid in an exponentially expanding spacetime differs drastically from the case of flat spacetime. More specifically, Christodoulou has recently shown that on the Minkowski space background, data arbitrarily close to that of an initially uniform quiet fluid state can lead to solutions that form shocks. In view of this fact, we remark that the proof of our result can be used to show the following: exponentially expanding spacetime backgrounds can prevent the formation of shocks.

This talk is part of the Cosmology Lunch series.

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