[Paper Review] Cosmic density and velocity fields in Lagrangian perturbation theory

Mikel Susperregi, Thomas Buchert|arXiv (Cornell University)|Apr 25, 1995

Cosmology and Gravitation Theories3 citations

TL;DR

This paper develops a consistent, Lagrangian perturbation framework to relate cosmic density and peculiar velocity fields up to second order, using exact solutions of the Lagrange-Newton system. By deriving the density from the Jacobian of the Lagrangian displacement without truncating the continuity equation, it resolves inconsistencies present in prior Eulerian-based reconstruction schemes, yielding a self-consistent, accurate description of large-scale structure formation.

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.

Motivation & Objective

• To develop a consistent, higher-order perturbation framework for modeling cosmic density and velocity fields in the context of large-scale structure formation.
• To address the inconsistency between the continuity equation and Euler's equation in traditional reconstruction schemes that arise from improper handling of Lagrangian and Eulerian coordinates.
• To generalize previous results, such as those from the Zel'dovich approximation, by extending the formalism to second order in a fully Lagrangian framework.
• To eliminate the need for truncating the Jacobian expansion in the continuity equation, which leads to incompatibilities in earlier approaches.
• To provide an exact, self-consistent method for computing the density contrast from velocity fields using the Lagrangian displacement field.

Proposed method

• The method employs the second-order solutions of the Lagrange-Newton system derived by Buchert & Ehlers to describe the evolution of cosmic structures.
• It uses the Jacobian of the Lagrangian displacement field to compute the density contrast exactly, avoiding approximations in the continuity equation.
• The approach ensures that the derived density and velocity fields satisfy both the continuity equation and Euler's equation simultaneously by construction.
• The formalism is applied to two specific solutions with generic initial conditions to demonstrate consistency and accuracy.
• It avoids the common inconsistency in Eulerian-based schemes by working entirely in the Lagrangian picture, where the density is a direct functional of the displacement field.
• The method generalizes the Zel'dovich approximation by including second-order corrections in the perturbation series while preserving exact consistency.

Experimental results

Research questions

• RQ1 How can a consistent second-order relation between cosmic density and peculiar velocity fields be derived without truncating the Jacobian expansion in the continuity equation?
• RQ2 Why do traditional reconstruction schemes based on Eulerian coordinates produce inconsistencies between the continuity and Euler equations?
• RQ3 Can a fully Lagrangian perturbation approach yield a self-consistent description of density and velocity fields that avoids the approximations of the Zel'dovich approximation?
• RQ4 What is the role of the Lagrangian displacement field in enabling an exact calculation of the density contrast without ad hoc truncations?
• RQ5 How does the proposed method improve upon prior schemes in modeling the formation of large-scale structure?

Key findings

• The method provides a consistent, exact calculation of the density contrast from the Jacobian of the Lagrangian displacement field, eliminating inconsistencies from truncated expansions.
• The derived density and velocity fields satisfy both the continuity equation and Euler's equation simultaneously, ensuring dynamical consistency.
• The formalism generalizes the Zel'dovich approximation by including second-order corrections while preserving exact consistency in the Lagrangian framework.
• The approach avoids the need for ad hoc truncations in the Jacobian expansion, which were the root cause of incompatibilities in earlier reconstruction schemes.
• The method is validated on two generic initial condition solutions, demonstrating its robustness and accuracy in modeling cosmic density and velocity fields.