[Paper Review] Lagrangian Perturbation Approach to the Formation of Large-scale Structure

Thomas Buchert|arXiv (Cornell University)|Sep 1, 1995

Aquatic and Environmental Studies1 references15 citations

TL;DR

This paper presents a Lagrangian perturbation approach to modeling large-scale structure formation in cosmology, reformulating Newtonian fluid dynamics using fluid element trajectories as the sole dynamical variable. It demonstrates that the method captures nonlinear evolution—including shell-crossing and hierarchical structure formation—while identifying limitations such as violation of mass equivalence post-shell-crossing and the need for high-frequency mode filtering to maintain accuracy beyond collapse.

ABSTRACT

The present lecture notes address three columns on which the Lagrangian perturbation approach to cosmological dynamics is based: 1. the formulation of a Lagrangian theory of self--gravitating flows in which the dynamics is described in terms of a single field variable; 2. the procedure, how to obtain the dynamics of Eulerian fields from the Lagrangian picture, and 3. a precise definition of a Newtonian cosmology framework in which Lagrangian perturbation solutions can be studied. While the first is a discussion of the basic equations obtained by transforming the Eulerian evolution and field equations to the Lagrangian picture, the second exemplifies how the Lagrangian theory determines the evolution of Eulerian fields including kinematical variables like expansion, vorticity, as well as the shear and tidal tensors. The third column is based on a specification of initial and boundary conditions, and in particular on the identification of the average flow of an inhomogeneous cosmology with a ``Hubble--flow''. Here, we also look at the limits of the Lagrangian perturbation approach as inferred from comparisons with N--body simulations and illustrate some striking properties of the solutions.

Motivation & Objective

• To establish a rigorous Lagrangian framework for self-gravitating cosmological fluids, treating fluid element trajectories as the fundamental dynamical variable.
• To derive Eulerian field quantities (velocity, density, shear, vorticity) from the Lagrangian trajectory field, enabling direct comparison with observations and simulations.
• To define a Newtonian cosmological framework where the average flow corresponds to a Hubble-like expansion, ensuring consistency with cosmological principles.
• To identify the theoretical and practical limits of the Lagrangian perturbation approach, especially regarding shell-crossing and mode filtering.
• To compare the method's performance with N-body simulations, highlighting its strengths in early nonlinear evolution and weaknesses in post-collapse dynamics.

Proposed method

• Formulates Newtonian cosmology using the Lagrangian coordinate system, where fluid elements are labeled by initial positions $\vec{X}$, and their motion is described by the trajectory field $\vec{f}(\vec{X}, t)$.
• Transforms the Eulerian equations (Euler equation, continuity equation, Poisson equation) into Lagrangian form, enabling exact integration of velocity and density fields along trajectories.
• Derives kinematical variables (expansion, shear, vorticity, tidal tensors) from the deformation tensor of the trajectory field, capturing nonlinear evolution without explicit field equations.
• Applies periodic boundary conditions to ensure uniqueness of solutions and to maintain a homogeneous background Hubble flow.
• Uses second-order Lagrangian perturbation theory to model higher-order structures such as 'second-generation' pancakes, filaments, and clusters.
• Imposes a cutoff on high-frequency modes in the initial power spectrum to avoid unphysical behavior after shell-crossing, effectively setting a lower mass scale of applicability (~$10^{13} M_\odot$).

Experimental results

Research questions

• RQ1 How can the dynamics of self-gravitating fluids in cosmology be reformulated in a Lagrangian framework using a single field variable?
• RQ2 To what extent can Lagrangian perturbation theory accurately reproduce the evolution of Eulerian fields such as velocity, density, and shear beyond linear order?
• RQ3 What are the physical and mathematical limits of the Lagrangian perturbation approach, particularly after shell-crossing occurs?
• RQ4 Why does the method fail to conserve Einstein’s equivalence principle of inertial and gravitational mass in the post-shell-crossing regime?
• RQ5 How does the initial power spectrum’s slope (e.g., $n = -3$) affect the onset of nonlinear collapse and the validity of the Lagrangian approximation?

Key findings

• The first-order Lagrangian perturbation solution reduces exactly to the Zel’dovich approximation when initial peculiar velocity and acceleration are parallel.
• The method inherently includes nonlinear terms, even in the first-order solution, due to the nonlinear dependence of Eulerian fields on the trajectory field.
• Shell-crossing marks the temporal limit of validity for standard Lagrangian schemes, as displacements of all orders become comparable in magnitude.
• Post-shell-crossing evolution is unstable in the standard formulation due to violation of mass equivalence, where individual fluid elements are accelerated by their own gravitational fields rather than the total field.
• For models with high small-scale power (e.g., standard CDM), the method requires filtering of high-frequency modes to prevent unphysical behavior; a spectral index of $n = -3$ represents a critical boundary between pancake-like and hierarchical structure formation.
• Comparisons with N-body simulations (e.g., BSI model) show good agreement before shell-crossing, but performance degrades afterward unless high-frequency components are truncated, with the effective lower limit of applicability corresponding to galaxy group mass scales ($\sim 10^{13} M_\odot$).