Quantification of discreteness effects in cosmologicalN-body simulations: Initial conditions

Abstract

The relation between the results of cosmological N-body simulations, and the continuum theoretical models they simulate, is currently not understood in a way which allows a quantification of N dependent effects. In this first of a series of papers on this issue, we consider the quantification of such effects in the initial conditions of such simulations. A general formalism developed in [1] allows us to write down an exact expression for the power spectrum of the point distributions generated by the standard algorithm for generating such initial conditions. Expanded perturbatively in the amplitude of the input (i.e. theoretical, continuum) power spectrum, we obtain at linear order the input power spectrum, plus two terms which arise from discreteness and contribute at large wavenumbers. For cosmological type power spectra, one obtains as expected, the input spectrum for wavenumbers k smaller than that characteristic of the discreteness. The comparison of real space correlation properties is more subtle because the discreteness corrections are not as strongly localised in real space. For cosmological type spectra the theoretical mass variance in spheres and two point correlation function are well approximated above a finite distance. For typical initial amplitudes this distance is a few times the inter-particle distance, but it diverges as this amplitude (or, equivalently, the initial red-shift of the cosmological simulation) goes to zero, at fixed particle density. We discuss briefly the physical significance of these discreteness terms in the initial conditions, in particular with respect to the definition of the continuum limit of N-body simulations.

Quantification of discreteness effects in cosmologicalN-body simulations. II. Evolution up to shell crossing

“…We have chosen the value a = 5 because it is of the order of the typical one at which shell crossing occurs in N -body simulations, i.e., the initial amplitude in these simulations is typically such that this is the case 5 . As we have already noted, the discrete evolution and the fluid one are described by different exponents, so this figure of D disp (k, a) will change as we change a.…”

“…One of our results here is that this non-interchangeability of these limits also manifests itself in the perturbative approximation to the evolution we use. As explained briefly in the conclusion of [5], this behavior can be understood as a manifestation of the fact that the Vlasov limit of an N -body system with long-range interactions is valid for sufficiently short times [6,7]. More specifically, this means that taking z init → ∞, at fixed particle density, the evolution of the particle system diverges from that of the Vlasov-Poisson limit.…”

“…In the first paper [5] we have considered only the initial conditions (IC) for such simulations, studying in detail the correlation properties of the point distributions generated with the standard algorithm used to produce them. We have identified the limit in which the continuum IC are recovered, and quantified precisely the corrections to this limit which appear at finite particle number.…”

“…mass variance in spheres) become dominated by those of the underlying (unperturbed) particle distribution, and thus differ completely from those of the theoretical IC. As the analysis of [5] is limited to the IC, no conclusion can be drawn about the relevance of these behaviours in the evolved simulations. One of our results here is that this non-interchangeability of these limits also manifests itself in the perturbative approximation to the evolution we use.…”

“…In this paper we pursue the study of [5] of discreteness effects, considering the early times dynamics of these simulations. By "early times" we mean the regime of validity of a perturbative treatment of the evolution of the full N -body problem which has been reported in detail in [12,13].…”