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

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.

21 pages, content matches published version in PRD, discussion on singularities added, some formulas added, some rewritten and some corrected

References in corpus (7)

• Accelerated expansion from structure formation
• Adhesive Gravitational Clustering
• Lagrangian perturbations and the matter bispectrum I: fourth-order model for non-linear clustering
• On the curvature of the present-day Universe
• Cosmological models from a geometric point of view
• On the Newtonian Limit of the Weyl Tensor
• An exact Lagrangian integral for the Newtonian gravitational field strength

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