A modified least action principle allowing mass concentrations for the early universe reconstruction problem

We address the early universe reconstruction (EUR) problem (as considered by Frisch and coauthors), and the related Zeldovich approximate model. By substituting the fully nonlinear Monge-Ampere equation for the linear Poisson equation to model gravitation, we introduce a modified mathematical model, 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. A fully discrete algorithm is introduced for the EUR problem in one space dimension