Lagrangian theory of structure formation in relativistic cosmology. IV. Lagrangian approach to gravitational waves

TL;DR

This paper develops a relativistic Lagrangian perturbation framework for gravitational waves in cosmology, providing master equations and solution methods that connect to standard perturbation results and explore global properties of perturbations.

Contribution

It introduces a detailed first-order Lagrangian scheme for gravitational wave propagation in relativistic cosmology, including master equations and a connection to standard perturbation solutions.

Findings

Derived master equations for Lagrangian perturbations

Connected Lagrangian and Eulerian perturbation frameworks

Showed how to recover standard solutions from the Lagrangian approach

Abstract

The relativistic generalization of the Newtonian Lagrangian perturbation theory is investigated. In previous works, the perturbation and solution schemes that are generated by the spatially projected gravitoelectric part of the Weyl tensor were given to any order of the perturbations, together with extensions and applications for accessing the nonperturbative regime. We here discuss more in detail the general first-order scheme within the Cartan formalism including and concentrating on the gravitational wave propagation in matter. We provide master equations for all parts of Lagrangian-linearized perturbations propagating in the perturbed spacetime, and we outline the solution procedure that allows one to find general solutions. Particular emphasis is given to global properties of the Lagrangian perturbation fields by employing results of Hodge-de Rham theory. We here discuss how the Hodge decomposition relates to the standard scalar-vector-tensor decomposition. Finally, we demonstrate that we obtain the known linear perturbation solutions of the standard relativistic perturbation scheme by performing two steps: first, by restricting our solutions to perturbations that propagate on a flat unperturbed background spacetime and, second, by transforming to Eulerian background coordinates with truncation of nonlinear terms.