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

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. In the first part of this paper, we discuss the essential question of how the relevant parameters introduced by this discretization should be extrapolated in convergence studies if the goal is to recover the Vlasov-Poisson limit. In the second part of the paper, we study numerically, and with analytical methods developed recently by us, the central issue of how finite particle density affects the precision of results above the force-smoothing scale. In particular, we focus on the precision of results for the power spectrum at wavenumbers around and above the Nyquist wavenumber, in simulations in which the force resolution is taken to be 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 lower bound on the real discreteness error. With the guidance of our analytical results, which match extremely well this measured dispersion into the weakly non-linear regime, and of further controlled tests for dependences on the relevant discreteness parameters, we establish with confidence that the measured dispersion is not contaminated either by finite box size effects or by subtle numerical effects. Our results notably show that, at wavenumbers 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 the order of 1 per cent, and larger above this scale. Our main conclusion is that the only way this error may be reduced below these levels at these physical 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. © 2009 RAS.