Relativistic Lagrangian displacement field and tensor perturbations

TL;DR

This paper develops and compares three methods to extract the relativistic displacement field from Lagrangian solutions in cosmology, revealing the importance of gravitational wave solutions and tensor perturbations for accurate relativistic modeling.

Contribution

It introduces three techniques for deriving the relativistic displacement field from Lagrangian solutions, including a non-perturbative approach, and applies these to a ΛCDM universe with primordial non-Gaussianity.

Findings

Relativistic displacement field derived up to second order in ΛCDM.

Gravitational wave solutions are essential to avoid spurious tensor artifacts.

Tensor perturbations are excited by non-linear frame-dragging effects.

Abstract

We investigate the purely spatial Lagrangian coordinate transformation from the Lagrangian to the basic Eulerian frame. We demonstrate three techniques for extracting the relativistic displacement field from a given solution in the Lagrangian frame. These techniques are (a) from defining a local set of Eulerian coordinates embedded into the Lagrangian frame; (b) from performing a specific gauge transformation; and (c) from a fully non-perturbative approach based on the ADM split. The latter approach shows that this decomposition is not tied to a specific perturbative formulation for the solution of the Einstein equations. Rather, it can be defined at the level of the non-perturbative coordinate change from the Lagrangian to the Eulerian description. Studying such different techniques is useful because it allows us to compare and develop further the various approximation techniques available in the Lagrangian formulation. We find that one has to solve the gravitational wave equation in the relativistic analysis, otherwise the corresponding Newtonian limit will necessarily contain spurious non-propagating tensor artefacts at second order in the Eulerian frame. We also derive the magnetic part of the Weyl tensor in the Lagrangian frame, and find that it is not only excited by gravitational waves but also by tensor perturbations which are induced through the non-linear frame-dragging. We apply our findings to calculate for the first time the relativistic displacement field, up to second order, for a $\Lambda$CDM Universe in the presence of a local primordial non-Gaussian component. Finally, we also comment on recent claims about whether mass conservation in the Lagrangian frame is violated.