We study, using Mean Curvature Flow methods, 3+1 dimensional cosmologies with a positive cosmological constant, matter satisfying the dominant and the strong energy conditions, and with spatial slices that can be foliated by 2-dimensional surfaces that are the closed orbits of a symmetry group. If these surfaces have non-positive Euler characteristic (or in the case of 2-spheres, if the initial 2-spheres are large enough) and also if the initial spatial slice is expanding everywhere, then we prove that asymptotically the spacetime becomes physically indistinguishable from de Sitter space on arbitrarily large regions of spacetime. This holds true notwithstanding the presence of initial arbitrarily-large density fluctuations.
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We thank the anonymous referees whose invaluable comments had contributed tremendously to this paper. We thank Matt Kleban, Shamit Kachru, Brett Kotschwar, Jonathan Luk, Richard Schoen for discussions. OH has been partially supported by a Koret Foundation early career scholar award. LS is partially supported by Simons Foundation Origins of the Universe program (Modern Inflationary Cosmology collaboration) and LS by NSF award 1720397 . AV is partially supported by NSF awards DMS-1664683 and DMS-1953987 . PC would like to thank the Stanford Institute for Theoretical Physics for hospitality and support during part of this work. LS and AV would like to thank the International Center for Theoretical Physics for hospitality and support during part of this work.
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