T. Baertschiger

Power-law correlation and discreteness in cosmological N-body simulations

We analyse with simple real-space statistics the Virgo consortiums cosmological N-body simulations. Significant clustering rapidly develops well below the initial mean interparticle separation \Lambda_i, where the gravitational force on a particle is dominated by that with its nearest neighbours. A power-law behaviour in the two point correlation function emerges, which in the subsequent evolution is continuously amplified and shifted to larger scales, in a roughly self-similar manner. We conclude that the density fluctuations at the smallest scales due to the particle-like nature of the distribution being evolved are thus essential in the development of these correlations, and not solely, as usually supposed, the very small continuous (fluid-like) fluctuations at scales larger than \Lambda_i >.

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

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 give rise to Poisson noise, which would then dominate the non-linear small-scale 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.

Force distribution in a randomly perturbed lattice of identical particles with 1/r2 pair interaction.

We study the statistics of the force felt by a particle in the class of a spatially correlated distribution of identical pointlike particles, interacting via a 1/r2 pair force (i.e., gravitational or Coulomb), and obtained by randomly perturbing an infinite perfect lattice. We specify the conditions under which the force on a particle is a well-defined stochastic quantity. We then study the small displacements approximation, giving both the limitations of its validity and, when it is valid, an expression for the force variance. The method introduced by Chandrasekhar to find the force probability density function for the homogeneous Poisson particle distribution is extended to shuffled lattices of particles. In this way, we can derive an approximate expression for the probability distribution of the force over the full range of perturbations of the lattice, i.e., from very small (compared to the lattice spacing) to very large where the Poisson limit is recovered. We show in particular the qualitative change in the large-force tail of the force distribution between these two limits. Excellent accuracy of our analytic results is found on detailed comparison with results from numerical simulations. These results provide basic statistical information about the fluctuations of the interactions (i) of the masses in self-gravitating systems like those encountered in the context of cosmological N -body simulations, and (ii) of the charges in the ordered phase of the one-component plasma, the so-called Coulomb or Wigner crystal.

Universality of power law correlations in gravitational clustering

We present an analysis of different sets of gravitational N-body simulations, all describing the dynamics of discrete particles with a small initial velocity dispersion. They encompass very different initial particle configurations, different numerical algorithms for the computation of the force, with or without the space expansion of cosmological models. Despite these differences we find in all cases that the non-linear clustering which results is essentially the same, with a well-defined simple power law behaviour in the two-point correlations in the range from a few times the lower cut-off in the gravitational force to the scale at which fluctuations are of order one. We argue, presenting quantitative evidence, that this apparently universal behaviour can be understood by the domination of the small-scale contribution to the gravitational force, coming initially from nearest-neighbor particles.

Reply to the Comment by A. Domínguez and A. Knebe on “On the problem of initial conditions in cosmological N-body simulations”

We reply to some comments in astro-ph/0309381 concerning the problem of setting-up initial conditions in cosmological N-body simulations

Initial Conditions, Discreteness and Non-Linear Structure Formation in Cosmology

In this lecture we address three different but related aspects of the initial continuous fluctuation field in standard cosmological models. Firstly we discuss the properties of the so-called Harrison-Zeldovich like spectra. This power spectrum is a fundamental feature of all current standard cosmological models. In a simple classification of all stationary stochastic processes into three categories, we highlight with the name “super- homogeneous” the properties of the class to which models like this, with P(0) = 0, belong. In statistical physics language they are well described as glass-like. Secondly, the initial continuous density field with such small amplitude correlated Gaussian fluctuations must be discretised in order to set up the initial particle distribution used in gravitational N-body simulations. We discuss the main issues related to the effects of discretisation, particularly concerning the effect of particle induced fluctuations on the statistical properties of the initial conditions and on the dynamical evolution of gravitational clustering.