Reconstruction of the early Universe as a convex optimization problem
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 threedimensional 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 MongeAmpèreKantorovich 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 Nbody 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 MongeAmpère equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. A selfcontained discussion of relevant notions and techniques is provided.
 Publication:

Monthly Notices of the Royal Astronomical Society
 Pub Date:
 December 2003
 DOI:
 10.1046/j.13652966.2003.07106.x
 arXiv:
 arXiv:astroph/0304214
 Bibcode:
 2003MNRAS.346..501B
 Keywords:

 hydrodynamics;
 cosmology: theory;
 early Universe;
 largescale structure of Universe;
 Astrophysics;
 Condensed Matter;
 Mathematical Physics;
 Mathematics  Optimization and Control;
 Nonlinear Sciences  Pattern Formation and Solitons
 EPrint:
 26 pages, 14 figures