Lagrangian theory of structure formation in relativistic cosmology III: gravitoelectric perturbation and solution schemes at any order

TL;DR

This paper develops a relativistic Lagrangian perturbation framework that generalizes Newtonian solutions to any order, focusing on gravitoelectric perturbations in cosmology.

Contribution

It introduces construction rules to derive relativistic solutions from Newtonian ones, providing complete perturbation and solution schemes at all orders.

Findings

Generalized hierarchy of solutions for relativistic perturbations

Construction rules linking Newtonian and relativistic solutions

Framework applicable at any perturbation order

Abstract

The relativistic generalization of the Newtonian Lagrangian perturbation theory is investigated. In previous works, the first-order trace solutions that are generated by the spatially projected gravitoelectric part of the Weyl tensor were given together with extensions and applications for accessing the nonperturbative regime. We furnish here construction rules to obtain from Newtonian solutions the gravitoelectric class of relativistic solutions, for which we give the complete perturbation and solution schemes at any order of the perturbations. By construction, these schemes generalize the complete hierarchy of solutions of the Newtonian Lagrangian perturbation theory.