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A reconstruction of the initial conditions of the Universe by optimal mass transportation

Abstract

Reconstructing the density fluctuations in the early Universe that evolved into the distribution of galaxies we see today is a challenge to modern cosmology1. An accurate reconstruction would allow us to test cosmological models by simulating the evolution starting from the reconstructed primordial state and comparing it to observations. Several reconstruction techniques have been proposed2,3,4,5,6,7,8,9, but they all suffer from lack of uniqueness because the velocities needed to produce a unique reconstruction usually are not known. Here we show that reconstruction can be reduced to a well-determined problem of optimization, and present a specific algorithm that provides excellent agreement when tested against data from N-body simulations. By applying our algorithm to the redshift surveys now under way10, we will be able to recover reliably the properties of the primeval fluctuation field of the local Universe, and to determine accurately the peculiar velocities (deviations from the Hubble expansion) and the true positions of many more galaxies than is feasible by any other method.

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Acknowledgements

Special thanks are due to E. Branchini (observational and conceptual aspects), Y. Brenier (mathematical aspects) and M. Hénon (algorithmic aspects and the handing of spatial periodicity and of scatter plots). We also thank J. Bec, H. Frisch, B. Gladman, L. Moscardini, A. Noullez, C. Porciani, M. Rees, E. Spiegel, A. Starobinsky and P. Thomas for comments. This work was supported by the BQR program of Observatoire de la Côte d'Azur, by the TMR program of the European Union (U.F., R.M.), by MIUR (S.M.), by the French Ministry of Education, the McDonnel Foundation, the Russian RFBR and INTAS (A.S.).

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Correspondence to Uriel Frisch.

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The authors declare that they have no competing financial interests.

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Figure 1: N-body simulation output (present epoch) used for testing our reconstruction method.
Figure 2: Tests of MAK reconstructions of the lagrangian positions, using the data shown in Fig. 1.
Figure 3: Reconstruction test in redshift space with the same data as for the real-space reconstruction tested in the upper left histogram of Fig. 2.

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