Perturbation-theory informed integrators for cosmological simulations

TL;DR

This paper introduces LPT-informed time-stepping schemes for cosmological simulations that improve accuracy and efficiency, especially before shell-crossing, by matching particle trajectories to perturbation theory predictions.

Contribution

The authors develop new integrators based on Lagrangian perturbation theory that outperform existing methods like FastPM in fast, low-timestep simulations.

Findings

LPT-inspired integrators outperform FastPM in power spectrum accuracy.

Convergence is limited post-shell-crossing due to acceleration field regularity.

Symplecticity has minor impact in low-timestep, approximate simulations.

Abstract

Large-scale cosmological simulations are an indispensable tool for modern cosmology. To enable model-space exploration, fast and accurate predictions are critical. In this paper, we show that the performance of such simulations can be further improved with time-stepping schemes that use input from cosmological perturbation theory. Specifically, we introduce a class of time-stepping schemes derived by matching the particle trajectories in a single leapfrog/Verlet drift-kick-drift step to those predicted by Lagrangian perturbation theory (LPT). As a corollary, these schemes exactly yield the analytic Zel'dovich solution in 1D in the pre-shell-crossing regime (i.e. before particle trajectories cross). One representative of this class is the popular FastPM scheme by Feng et al. 2016, which we take as our baseline. We then construct more powerful LPT-inspired integrators and show that they outperform FastPM and standard integrators in fast simulations in two and three dimensions with $\mathcal{O}(1 - 100)$ timesteps, requiring less steps to accurately reproduce the power spectrum and bispectrum of the density field. Furthermore, we demonstrate analytically and numerically that, for any integrator, convergence is limited in the post-shell-crossing regime (to order 3/2 for planar wave collapse), owing to the lacking regularity of the acceleration field, which makes the use of high-order integrators in this regime futile. Also, we study the impact of the timestep spacing and of a decaying mode present in the initial conditions. Importantly, we find that symplecticity of the integrator plays a minor role for fast approximate simulations with a small number of timesteps.