No abstract

We study, using numerical simulations, the dynamical evolution of self-gravitating point particles in static euclidean space, starting from a simple class of infinite "shuffled lattice" initial conditions. These are obtained by applying independently to each particle on an infinite perfect lattice a small random displacement, and are characterized by a power spectrum (structure factor) of density fluctuations which is quadratic in the wave number k, at small k. For a specified form of the probability distribution function of the "shuffling" applied to each particle, and zero initial velocities, these initial configurations are characterized by a single relevant parameter: the variance δ 2 of the "shuffling" normalized in units of the lattice spacing ℓ. The clustering, which develops in time starting from scales around ℓ, is qualitatively very similar to that seen in cosmological simulations, which begin from lattices with applied correlated displacements and incorporate an expanding spatial background. From very soon after the formation of the first non-linear structures, a spatio-temporal scaling relation describes well the evolution of the two-point correlations. At larger times the dynamics of these correlations converges to what is termed "self-similar" evolution in cosmology, in which the time dependence in the scaling relation is specified entirely by that of the linearized fluid theory. Comparing simulations with different δ, different resolution, but identical large scale fluctuations, we are able to identify and study features of the dynamics of the system in the transient phase leading to this behavior. In this phase, the discrete nature of the system explicitly plays an essential role.

We apply a simple linearisation, used standardly in solid state physics, to give an approximation describing the evolution under its self-gravity of an infinite perfect lattice perturbed from its equilibrium. In the limit that the initial perturbations are restricted to wavelengths much larger than the lattice spacing, the evolution corresponds exactly to that derived from an analagous linearisation of the Lagrangian formulation of the dynamics of a pressureless self-gravitating fluid, with the Zeldovich approximation as a sub-class of asymptotic solutions. Our less restricted approximation allows one to trace the evolution of the fully discrete distribution until the time when particles approach one another (i.e. "shell crossing"), with modifications of the fluid limit explicitly depending on the lattice spacing. We note that the simple cubic lattice presents both oscillating modes and modes which grow faster than in the fluid limit. In current cosmological theories the physics of structure formation in the universe reduces, over a large range of scales, to understanding the evolution of clustering under Newtonian gravity, with only a simple modification of the dynamical equations due to the expansion of the Universe. The primary instrument for solving this problem is numerical N -body simulation (NBS, see e.g.[1]). These simulations are most usually started from configurations which are simple cubic (sc) lattices perturbed in a manner prescribed by a theoretical cosmological model. In this letter we observe that, up to a change in sign in the force, the initial configuration is identical to the Coulomb lattice (or Wigner crystal) in solid state physics (see e.g.[2]), and we exploit this analogy to develop an approximation to the evolution of these simulations. We show that one obtains, for long wavelength perturbations, the evolution predicted by an analagous fluid description of the self-gravitating system, and in particular, as a special case, the Zeldovich approximation [3]. Further we can study precisely the deviations from this fluid-like behaviour at shorter wavelengths arising from the discrete nature of the system. This analysis should be a useful step towards a precise quantitative understanding, which is currently lacking, of the role of discreteness in cosmological NBS (see e.g. [4,5,6]). One simple conclusion, for example, is that a body centred cubic (bcc) lattice may be a better choice of discretisation, as its spectrum has only growing modes with exponents bounded above by that of fluid linear theory.The equation of motion of particles moving under their mutual self-gravity is [1]Here dots denote derivatives with respect to time t, x i is the comoving position of the ith particle, of mass m i , related to the physical coordinate by r i = a(t)x i , where a(t) is the scale factor of the background cosmology with Hubble constant H(t) =ȧ a . We treat a system of N point particles, of equal mass m, initially placed on a Bravais lattice, with periodic boundary conditions. Perturbations from the Coulom...

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.

Abstract. -Cosmological N-body simulations aim to calculate the non-linear gravitational growth of structures via particle dynamics. A crucial problem concerns the setting-up of the initial particle distribution, as standard theories of galaxy formation predict the properties of the initial continuous density field with small amplitude correlated Gaussian fluctuations. The discretisation of such a field is a complex issue and particle fluctuations are especially relevant at small scales where non-linear dynamics firstly takes place. In general, most of the procedures which may discretise a continuous field, gives rise to Poisson noise, which would then dominate the non-linear small-scales dynamics due to nearest-neighbours interactions. In order to avoid such a noise, and to consider the dynamics as due only to large scale (smooth) fluctuations, an ad-hoc method (lattice or glassy system plus correlated displacements) has been introduced and used in cosmological simulations. We show that such a method gives rise to a particle distribution which does not have any of the correlation properties of the theoretical continuous density field. This is because discreteness effects, different from Poisson noise but nevertheless very important, determine particle fluctuations at any scale, making it completely different from the original continuous field. We conclude that discreteness effects play a central role in the non-linear evolution of N-body simulations.The purpose of cosmological N-body simulations is to calculate the non-linear growth of structures in the universe by following individual particles trajectories under the action of their mutual gravity [1,2,3]. These particles are not galaxies but are meant to represent some sorts of collisionless clouds of elementary dark matter particles. In order to make them move, one must calculate the force acting on each of them due to all the others. In general one may find several algorithms which speed up the N 2 sum necessary to compute the force on each particle. In this paper we study simulations from the Virgo project [3] which are done