The reconstruction of the early universe amounts to recovering the tiny density fluctuations of the early universe (shortly after the "big bang") from the current observation of the matter distribution in the universe. Following Zeldovich, Peebles and, more recently Frisch and collaboratoirs, we use a newtonian gravitational model with time dependent coefficients taking into accont general relativity effects. Due to the (remarkable) convexity of the corresponding action, the reconstruction problem apparently reduces to a straightforward convex minimization problem. Unfortunately, this approach completely ignores the mass concentration effects due to gravitational instabilities.
In this lecture, we show a way of modifying the action in order to take concentrations into account. This is obtained through a (questionable) modification of the gravitation model,
by substituting the fully nonlinear Monge-Amp`ere equation for the linear Poisson equation. (This is a reasonable approximation in the sense that it makes exact some approximate solutions advocated by Zeldovich for the original gravitational model.) Then the action can be written as a perfect square 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.