Cosmic density and velocity fields in Lagrangian perturbation theory

TL;DR

This paper develops a purely Lagrangian second-order relation between cosmic density and velocity fields, providing a consistent and accurate method that improves upon previous approximations by avoiding truncation and coordinate inconsistencies.

Contribution

It introduces a novel Lagrangian approach to relate cosmic density and velocity fields at second order, ensuring consistency without truncating the Jacobian expansion.

Findings

Derives a second-order relation between density and velocity fields in Lagrangian perturbation theory.

Shows the approach automatically satisfies Euler's equation, ensuring physical consistency.

Provides a more accurate reconstruction scheme compared to previous methods.

Abstract

A first- and second-order relation between cosmic density and peculiar-velocity fields is presented. The calculation is purely Lagrangian and it is derived using the second-order solutions of the Lagrange-Newton system obtained by Buchert & Ehlers. The procedure is applied to two particular solutions given generic initial conditions. In this approach, the continuity equation yields a relation between the over-density and peculiar-velocity fields that automatically satisfies Euler's equation because the orbits are derived from the Lagrange-Newton system. This scheme generalizes some results obtained by Nusser et al. (1991) in the context of the Zel'dovich approximation. As opposed to several other reconstruction schemes, in this approach it is not necessary to truncate the expansion of the Jacobian given by the continuity equation in order to calculate a first- or second-order expression for the density field. In these previous schemes, the density contrast given by (a) the continuity equation and (b) Euler's equation are mutually incompatible. This inconsistency arises as a consequence of an improper handling of Lagrangian and Eulerian coordinates in the analysis. Here, we take into account the fact that an exact calculation of the density is feasible in the Lagrangian picture and therefore an accurate and consistent description is obtained.