Zel'dovich approximation and General Relativity

TL;DR

This paper derives the Zel'dovich approximation within a relativistic framework, showing it remains valid on large scales with modifications, and introduces relativistic corrections relevant for cosmological simulations.

Contribution

It demonstrates how the Zel'dovich approximation and second order displacement fields can be obtained from a general relativistic gradient expansion, extending its validity to horizon scales.

Findings

Zel'dovich approximation holds on horizon scales with small modifications.

Density perturbations follow a modified Helmholtz equation due to causality.

Relativistic corrections are significant near the horizon, aiding initial condition setup.

Abstract

We show how the Zel'dovich approximation and the second order displacement field of Lagrangian perturbation theory can be obtained from a general relativistic gradient expansion in \Lambda{}CDM cosmology. The displacement field arises as a result of a second order non-local coordinate transformation which brings the synchronous/comoving metric into a Newtonian form. We find that, with a small modification, the Zel'dovich approximation holds even on scales comparable to the horizon. The corresponding density perturbation is not related to the Newtonian potential via the usual Poisson equation but via a modified Helmholtz equation. This is a consequence of causality not present in the Newtonian theory. The second order displacement field receives relativistic corrections that are subdominant on short scales but are comparable to the second order Newtonian result on scales approaching the horizon. The corrections are easy to include when setting up initial conditions in large N-body simulations.