Zel'dovich approximation and General Relativity

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 ΛCDM 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.

5 pages, corrected a typo, accepted for publication in MNRAS Letters

References in corpus (8)

• Transients from Initial Conditions in Cosmological Simulations
• Resumming Cosmological Perturbations via the Lagrangian Picture: One-loop Results in Real Space and in Redshift Space
• Lagrangian perturbations and the matter bispectrum I: fourth-order model for non-linear clustering
• Lagrangian theory of structure formation in relativistic cosmology I: Lagrangian framework and definition of a nonperturbative approximation
• Quantifying distortions of the Lagrangian dark-matter mesh in cosmology
• Gradient expansion approach to nonlinear superhorizon perturbations
• Lagrangian perturbations and the matter bispectrum II: the resummed one-loop correction to the matter bispectrum
• 3+1 Approach to the Long Wavelength Iteration Scheme

Cited by in corpus (24)

• The Phenomenology of Right Handed Neutrinos
• Average expansion rate and light propagation in a cosmological Tardis spacetime
• General relativistic corrections to $N$-body simulations and the Zel'dovich approximation
• Lagrangian theory of structure formation in relativistic cosmology II: average properties of a generic evolution model
• First-order Cosmological Perturbations Engendered by Point-like Masses
• Non-Gaussian initial conditions in $\Lambda$CDM: Newtonian, relativistic and primordial contributions
• General relativistic weak-field limit and Newtonian N-body simulations
• How smooth are particle trajectories in a $\Lambda$CDM Universe?
• Einstein's signature in cosmological large-scale structure
• Lagrangian theory of structure formation in relativistic cosmology III: gravitoelectric perturbation and solution schemes at any order
• Relativistic Lagrangian displacement field and tensor perturbations
• Frame dragging and Eulerian frames in General Relativity
• Initial conditions for cold dark matter particles and General Relativity
• Lagrangian theory of structure formation in relativistic cosmology. IV. Lagrangian approach to gravitational waves
• General Relativistic Cosmological N-body Simulations I: time integration
• Second-order cosmological perturbation theory and initial conditions for $N$-body simulations
• On the accuracy of N-body simulations at very large scales
• A Relativistic view on large scale N-body simulations
• Astrophysical Bose-Einstein Condensates and Superradiance
• The adhesion model as a field theory for cosmological clustering
• Lagrangian theory of structure formation in relativistic cosmology. V. Irrotational fluids
• Neutrinos in N-body simulations
• Relativistic perturbations in $\Lambda$CDM: Eulerian & Lagrangian approaches
• Relativistic and non-Gaussianity contributions to the one-loop power spectrum