Reconstruction of the Early Universe, Zeldovich Approximation and Monge-ampère Gravitation

We address the early universe reconstruction (EUR) problem (as considered by Frisch and coauthors in [26]), and the related Zeldovich approximate model [45]. By substituting the fully nonlinear Monge-Ampère equation for the linear Poisson equation to model gravitation, we introduce a modified mathematical model (”Monge-Ampère gravitation/MAG”), for which the Zeldovich approximation becomes exact. The MAG model enjoys a least action principle in which we can input mass concentration effects in a canonical way, based on the theory of gradient flows with convex potentials and somewhat related to the concept of self-dual Lagrangians developped by Ghoussoub [29]. A fully discrete algorithm is introduced for the EUR problem in one space dimension. Introduction This paper addresses the early universe reconstruction (EUR) problem discussed by Frisch and coauthors in [26, 18], following Peebles’ seminal paper [38]. In these references, gravitation is not modelled according to the full Einstein equations, but rather to a semiNewtonian approximation, where classical Newtonian interactions just take place in an Einstein-de Sitter space, corresponding to a big bang scenario. In suitable coordinates, the model can be described as follows. Let us denote, for each gravitating body, its label by a and its position at time t by X(t, a) ∈ R. The density field ρ is defined by