Lagrangian theory of structure formation in relativistic cosmology I: Lagrangian framework and definition of a nonperturbative approximation

TL;DR

This paper develops a Lagrangian framework in general relativity for structure formation, introducing a nonperturbative approximation analogous to Zel'dovich's, useful for modeling inhomogeneities and initial conditions in relativistic cosmology.

Contribution

It presents the first relativistic Lagrangian approach and a nonperturbative approximation for structure formation, extending Zel'dovich's method to general relativity.

Findings

Derived a relativistic analogue of Zel'dovich's approximation.

Compared the new approximation with existing literature.

Addressed the singularity problem in the model.

Abstract

In this first paper we present a Lagrangian framework for the description of structure formation in general relativity, restricting attention to irrotational dust matter. As an application we present a self-contained derivation of a general-relativistic analogue of Zel'dovich's approximation for the description of structure formation in cosmology, and compare it with previous suggestions in the literature. This approximation is then investigated: paraphrasing the derivation in the Newtonian framework we provide general-relativistic analogues of the basic system of equations for a single dynamical field variable and recall the first-order perturbation solution of these equations. We then define a general-relativistic analogue of Zel'dovich's approximation and investigate its implications by functionally evaluating relevant variables, and we address the singularity problem. We so obtain a possibly powerful model that, although constructed through extrapolation of a perturbative solution, can be used to put into practice nonperturbatively, e.g. problems of structure formation, backreaction problems, nonlinear properties of gravitational radiation, and light-propagation in realistic inhomogeneous universe models. With this model we also provide the key-building blocks for initializing a fully relativistic numerical simulation.