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

Reconstructing the density fluctuations in the early Universe that evolved into the distribution of galaxies we see today is a challenge to modern cosmology. 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 proposed, 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 way, 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.

[1]  Y. Zel’dovich.  Gravitational instability: An Approximate theory for large density perturbations , 1969 .

[2]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[3]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[4]  A. I. Saichev,et al.  Probability distribution and spectra of potential hydrodynamic turbulence , 1984 .

[5]  Michel Balinski,et al.  A competitive (dual) simplex method for the assignment problem , 1986, Math. Program..

[6]  Phillip James Edwin Peebles,et al.  Tracing galaxy orbits back in time , 1989 .

[7]  Reconstructing Primordial Density Fluctuations. , 1989 .

[8]  Sergei F. Shandarin,et al.  The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium , 1989 .

[9]  Edmund Bertschinger,et al.  Recovering the full velocity and density fields from large-scale redshift-distance samples , 1989 .

[10]  F. Bouchet,et al.  Precollapse Scale Invariance in Gravitational Instability , 1991 .

[11]  Avishai Dekel,et al.  Tracing large-scale fluctuations back in time , 1992 .

[12]  D. Weinberg,et al.  Reconstructing primordial density fluctuations – I. Method , 1992 .

[13]  T. Buchert,et al.  Lagrangian theory of gravitational instability of Friedman-Lemaitre cosmologies and the 'Zel'dovich approximation' , 1992 .

[14]  Non-Linear Approximations to Gravitational Instability: A Comparison in the Quasi-Linear Regime , 1994, astro-ph/9402065.

[15]  F. Pearce,et al.  Hydra: An Adaptive--Mesh Implementation of PPPM--SPH , 1994 .

[16]  Lagrangian dynamics in non-flat universes and non-linear gravitational evolution , 1994, astro-ph/9406016.

[17]  R. Croft,et al.  Reconstruction of cosmological density and velocity fields in the Lagrangian Zel'dovich approximation , 1996, astro-ph/9602100.

[18]  W. Gangbo,et al.  The geometry of optimal transportation , 1996 .

[19]  Vijay K. Narayanan,et al.  Recovering the Primordial Density Fluctuations: A Comparison of Methods , 1998, astro-ph/9806255.

[20]  The Optimal Time-Continuous Mass Transport Problem and its Augmented Lagrangian Numerical Resolution , 1998 .

[21]  D. Culler,et al.  Comparison of methods , 2000 .

[22]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[23]  W. Saunders,et al.  Reconstructing the IRAS Point Source Catalog Redshift Survey with a generalized PIZA , 2000 .

[24]  A. Szalay,et al.  Large-scale structure: entering the precision era , 2000 .

[25]  On the least action principle in cosmology , 1999, astro-ph/9908167.

[26]  D. Spergel,et al.  Using Perturbative Least Action to Recover Cosmological Initial Conditions , 1999, astro-ph/9912408.

[27]  Michel Hénon,et al.  A mechanical model for the transportation problem , 2002, ArXiv.

[28]  E. Branchini,et al.  Peculiar velocity reconstruction with the fast action method: tests on mock redshift surveys , 2001, astro-ph/0110618.