Reconstruction of the early Universe as a convex optimization problem |
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Authors: | Y Brenier U Frisch M Hénon G Loeper S Matarrese R Mohayaee A Sobolevski |
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Institution: | CNRS, UMR 6621, Universitéde Nice-Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France;CNRS, UMR 6529, Observatoire de la Côte d'Azur, BP 4229, 06304 Nice Cedex 4, France;Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA;Dipartimento di Fisica 'G. Galilei' and INFN, Sezione di Padova, via Marzolo 8, 35131-Padova, Italy;Department of Physics, M. V. Lomonossov Moscow University, Leninskie Gory, 119992 Moscow, Russia |
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Abstract: | We show that the deterministic past history of the Universe can be uniquely reconstructed from knowledge of the present mass density field, the latter being inferred from the three-dimensional distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales – a few Mpc – where multistreaming becomes significant. Above 6 h ?1 Mpc we propose and implement an effective Monge–Ampère–Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation, a problem which goes back to Monge. After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than O ( N 3) time complexity and reasonable CPU time requirements. Testing against N -body cosmological simulations gives over 60 per cent of exactly reconstructed points. We apply several interrelated tools from optimization theory that were not used in cosmological reconstruction before, such as the Monge–Ampère equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. A self-contained discussion of relevant notions and techniques is provided. |
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Keywords: | hydrodynamics cosmology: theory early Universe large-scale structure of Universe |
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