Trajectories of point particles in cosmology and the Zel'dovich approximation

TL;DR

This paper uses a Green's function approach to compare classical particle trajectories in cosmology with Zel'dovich approximation trajectories, proposing an iterative scheme to improve the approximation and analyzing the gravitational potential effects.

Contribution

It introduces an iterative method to refine Zel'dovich trajectories using Green's functions, bridging the gap between approximate and exact particle paths in cosmological models.

Findings

Effective gravitational potential vanishes initially due to the continuity equation.

The iterative scheme produces trajectories that interpolate between Zel'dovich and exact paths.

Late-time potential acting on improved trajectories is significantly smaller than on exact trajectories.

Abstract

Using a Green's function approach, we compare the trajectories of classical Hamiltonian point particles in an expanding space-time to the effectively inertial trajectories in the Zel'dovich approximation. It is shown that the effective gravitational potential accelerating the particles relative to the Zel'dovich trajectories vanishes exactly initially as a consequence of the continuity equation, and acts only during a short, early period. The Green's function approach suggests an iterative scheme for improving the Zel'dovich trajectories, which can be analytically solved. We construct these trajectories explicitly and show how they interpolate between the Zel'dovich and the exact trajectories. The effective gravitational potential acting on the improved trajectories is substantially smaller at late times than the potential acting on the exact trajectories. The results may be useful for Lagrangian perturbation theory and for numerical simulations.