On the problem of initial conditions in cosmological N -body simulations

Abstract

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

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

“…Thus we will consider only the constraints at small k, i.e., the lower bound placed on the index n. Firstly we note that, using Eqs. (20) and (26), it is simple to show that d ij (r) is well defined only if…”

“…The exact ensemble average results given above, on the other hand, are easily calculated. The simplified and more explicit expressions for the relevant quantities are given in Appendix C. There is no intrinsic difference of importance between one and three dimensions for the questions we address 20 . We consider the case in which the pre-initial distribution is a lattice.…”

“…Their finding was, very surprisingly, that the measured and theoretical values of both the mass variance in spheres and the two point correlation function did not match. In [21] the same analysis was repeated by a different set of authors, and an error in the normalization in [20] of the theoretical variance was identified. Correcting for this error the authors concluded that the agreement between the measured and theoretical properties was good for the variance, while the authors of [20], in a reply [22], argued that the agreement was still very poor.…”

“…In [21] the same analysis was repeated by a different set of authors, and an error in the normalization in [20] of the theoretical variance was identified. Correcting for this error the authors concluded that the agreement between the measured and theoretical properties was good for the variance, while the authors of [20], in a reply [22], argued that the agreement was still very poor. For the two point correlation function the results of both sets of authors agreed, showing an estimated correlation function qualitatively and quantitatively different to the expected one.…”

“…The direct motivation for this work on IC in NBS came originally from some numerical studies of these issues by two groups [20,21,22,23], who have drawn quite different conclusions about the accuracy with which the IC produced by the canonically used algorithm in NBS represent the input IC 1 . The results in this paper, which are essentially analytic, clarify the issues underlying the discussion in and differences between these numerical studies.…”