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 in [24]), and the related Zeldovich approximate model [39]. 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 [27]. 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 [24, 17]. We refer to these papers for the detailed physical background of this important problem in cosmology. Here is a simplified mathematical formulation. We consider (for simplicity) a smooth closed bounded 3D domain D ⊂ R and denote the space variable by x ∈ D. We are given two times t1 > t0 > 0, two probability measures ρ0(dx), ρ1(dx) on D. We look for a time-dependent family of probability measures ρ(t, dx) on D (depending continuously on the time variable t, with respect to the weak convergence of measures), interpolating ρ0 and ρ1, at t = t0 and t = t1 respectively, and minimizing the following action