Gravitational Evolution of a Perturbed Lattice and its Fluid Limit

Abstract

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...

Gravitational dynamics of an infinite shuffled lattice of particles

“…We have noted that the lagging of the evolution behind the asymptotic behavior in this regime can be ascribed to effects of discreteness (i.e. non fluid effects) slowing down the evolution of fluctuations at scales comparable to the inter-particle distance which have been quantified in [40,44]. We have seen also that the form of the correlation function emerges already at the very early times when the first non-linear correlations develop due to two-body correlations which develop under the effect of nearest neighbor interactions.…”

“…A very good approximation to the evolution of a perturbed lattice is provided by a perturbative treatment described in [40,44]. The force acting on particles is written as an expansion in the relative displacement of particles, in a manner completely analogous to a standard technique used in solid state physics to treat perturbations to crystals.…”

Gravitational dynamics of an infinite shuffled lattice: Early time evolution and universality of nonlinear correlations

“…In this limit, however, we have developed in [7,8] an analytical perturbative approach, which at linear order gives a very good approximation to the dynamical evolution. The treatment involves simply a Taylor expansion of the force between particles in their relative displacements from their initial lattice positions R, and thus breaks down when the latter become equal to the initial separation of the particles.…”

“…In the case of "shuffled lattice" (SL) initial conditions, which we consider in this work, such an approximation is not generically good: when the typical displacement of a particle is small compared to the lattice spacing, the high degree of symmetry gives that the force on a typical particle is the sum of comparable contributions from many particles. In this regime, however, we can describe the evolution very well by a simple perturbative approximation, which has been developed fully in [7,8]. This latter approximation breaks down, roughly, when particles start to approach one another, which is precisely when one expects a NN approximation for the force may become appropriate.…”

“…analytical approximation we have introduced and studied in [7,8]; in the second phase the particles evolve under a force coming solely from their nearest neighbor (NN).…”