Towards quantitative control on discreteness error in the non-linear regime of cosmological N body simulations

TL;DR

This paper investigates how discreteness errors in cosmological N-body simulations affect the accuracy of power spectrum measurements, providing bounds and methods to control these errors in the non-linear regime.

Contribution

It introduces analytical and numerical methods to quantify and bound discreteness errors, guiding how to extrapolate to the physical limit in cosmological simulations.

Findings

Discreteness error at the Nyquist wavenumber is about 1% for typical cosmological simulations.

Errors increase monotonically over time both below and above the Nyquist wavenumber.

Extrapolation to smaller interparticle distances is necessary for convergence to the physical limit.

Abstract

The effects of discreteness arising from the use of the N-body method on the accuracy of simulations of cosmological structure formation are not currently well understood. After a discussion of how the relevant discretisation parameters introduced should be extrapolated to recover the Vlasov-Poisson limit, we study numerically, and with analytical methods we have developed recently, the central issue of how finite particle density affects the precision of results. In particular we focus on the power spectrum at wavenumbers around and above the Nyquist wavenumber, in simulations in which the force resolution is taken smaller than the initial interparticle spacing. Using simulations of identical theoretical initial conditions sampled on four different "pre-initial" configurations (three different Bravais lattices, and a glass) we obtain a {\it lower bound} on the real discreteness error. With the guidance of our analytical results, we establish with confidence that the measured dispersion is not contaminated either by finite box size effects or by subtle numerical effects. Our results show notably that, at wavenumbers {\it below} the Nyquist wavenumber, the dispersion increases monotonically in time throughout the simulation, while the same is true above the Nyquist wavenumber once non-linearity sets in. For normalizations typical of cosmological simulations, we find lower bounds on errors at the Nyquist wavenumber of order of a percent, and larger above this scale. The only way this error may be reduced below these levels at these scales, and indeed convergence to the physical limit firmly established, is by extrapolation, at fixed values of the other relevant parameters, to the regime in which the mean comoving interparticle distance becomes less than the force smoothing scale.