Linear perturbative theory of the discrete cosmologicalN-body problem

Abstract

We present a perturbative treatment of the evolution under their mutual self-gravity of particles displaced off an infinite perfect lattice, both for a static space and for a homogeneously expanding space as in cosmological N-body simulations. The treatment, analogous to that of perturbations to a crystal in solid state physics, can be seen as a discrete (i.e. particle) generalization of the perturbative solution in the Lagrangian formalism of a self-gravitating fluid. Working to linear order, we show explicitly that this fluid evolution is recovered in the limit that the initial perturbations are restricted to modes of wavelength much larger than the lattice spacing. The full spectrum of eigenvalues of the simple cubic lattice contains both oscillatory modes and unstable modes which grow slightly faster than in the fluid limit. A detailed comparison of our perturbative treatment, at linear order, with full numerical simulations is presented, for two very different classes of initial perturbation spectra. We find that the range of validity is similar to that of the perturbative fluid approximation (i.e. up to close to "shell-crossing"), but that the accuracy in tracing the evolution is superior. The formalism provides a powerful tool to systematically calculate discreteness effects at early times in cosmological N-body simulations.

Particle linear theory on a self-gravitating perturbed cubic Bravais lattice

“…In this section, we present a summary of the general method we have developed in [10,11] to calculate the evolution of self-gravitating particles perturbed off a perfect lattice.…”

“…The expression of the dynamical matrix for a generic interaction potential vðrÞ is [10] D ðR Þ 0Þ ¼ @ @ wðRÞ; (5a)…”

“…[10]) that the term k ¼ 0 is not included in the sum (11b) due to the presence of the neutralizing background (or the space expansion in the cosmological context). An explicit expression of the dynamical matrix is given in Appendix A.…”

“…In [9] and subsequently [10][11][12] we have started to develop a program to precisely fill this gap. We have developed a framework which allows us to calculate the evolution-in the linear regime-of a system of self-interacting particles.…”

“…In the first section we give a summary of the PLT. Further details can be found in [10]. In the following section, we explicitly give the details of the PLT for sc, bcc, and fcc lattices.…”