Optimal transport by omni-potential flow and cosmological reconstruction

 

 

One of the simplest models used in studying the dynamics of large-scale structure in cosmology, known as the Zeldovich approximation, is equivalent to the three-dimensional inviscid Burgers equation for potential flow. For smooth initial data and sufficiently short times it has the property that the mapping of the positions of fluid particles at any time t1 to their positions at any time t2 >= t1 is the gradient of a convex potential, a property we call omni-potentiality. We show that, in both two and three dimensions, there exist flows with this property, that are not straightforward generalizations of Zeldovich flows. How general are such flows? In two dimensions, for sufficiently short times, there are omni-potential flows with arbitrary smooth initial velocity. Mappings with a convex potential are known to be associated with the quadratic-cost optimal transport problem: Implications for the problem of reconstructing the dynamical history of the Universe from the knowledge of the present mass distribution are discussed. The talk is based on the joint work reported in the paper Frisch U., Podvigina O., Villone B., Zheligovsky V. "Optimal transport by omni-potential flow and cosmological reconstruction". J. Math. Phys. 53 (2012) 033703, http://arxiv.org/abs/1111.2516