Gravitational Many Body Problem


The standard model of the formation of large scale structure of the universe is based on the gravitational growth of small initial density fluctuations in a homogeneous and isotropic medium. In the current most popular model, Cold Dark Matter (CDM), the universe is thought to be constituted by non-visible (“dark matter”) particles that interacts only gravitationally. The fact that they are very cold permits to model this system with a collision-less Boltzmann equation and, for sufficiently large scales, pressure-less fluid equations. Then it is possible to solve in a perturbative way, for small density fluctuations, these fluid equations.

The most convenient way to formulate fluid theory is the Lagrangian approach, in which the trajectories of fluid elements are computed through integral curves. This method is known to describe better the evolution of fluctuations than the Eulerian theory, in which the dynamical variables are the density and velocity field. The Lagrangian formulation, that was originally introduced by Zeldovich for a particular case, provides a quite accurate description of the gravitational clustering even in the quasi-linear regime. Despite of the success of the Lagrangian theory in the linear and weakly non-linear regime, this treatment is inapplicable in the strong non-linear regime. Then, the most widely used tool to study gravitational clustering in the various regimes is by means of N-body simulations (NBS) which are based on the computation of particle gravitational dynamics in an expanding universe. In general one may consider an infinite periodic system, i.e. a finite system with periodic boundary condition. Despite the simplicity of the system, where dynamics is purely Newtonian at all but the smallest scales, the analytic understanding of this crucial problem is limited to the regime of very small fluctuations where a linear analysis can be performed. Moreover the cosmological density field must be discretized into “macro-particles” interacting gravitationally which are, for numerical reasons, tens of order of magnitudes heavier than the CDM particles. This procedure introduces discreteness at a much larger scale than the discreteness inherent to the CDM particles. By discreteness we mean statistical and dynamical effects which cannot be treated by the self-gravitating fluid approximation. The discreteness has different manifestations in th eevolution of the system. In this context we aim to consider the issue of the physical role of discrete fluctuations in the dynamics, which go beyond a description where particles play the role of collision-less fluid elements and the evolution can be understood in terms of a self-gravitating fluid.

In order to study the full gravitational many-body problem problem, we started by considering a very simple initial particle distribution represented by a slightly perturbed cubic lattice with zero initia velocities. A perfect cubic lattice is an unstable equilibrium configuration for gravitational dynamics, being the force on each particle equal to zero. A slightly perturbed lattice (see figure above left panel) represents instead a situation where the force on each particle is small, and linearly proportional to the average root mean square displacement of the particle from its lattice position. When the system is evolved for long times it creates complex non linear structures as shown in the right panel of the above figure. The full analytical understanding of the clustering dynamics is the target of our studies.

We have introduced a perturbative theory (analogous to the Lagrangian perturbative theory for a fluid) to study the gravitational evolution of perturbed lattices. We have observed 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, and we have exploited this analogy to develop an approximation to the evolution of the gravitational many-body problem. In this way we have found that the equation of motion of each particles can be found diagonalizing a dynamical matrix and this process is very low cost in computational resources because of the symmetry of the problem. The limit of zero inter-particles distance (or infinite number of particles) corresponds exactly to the Lagrangian fluid theory at any order. One of the most powerful features of this approach is that we can extract the linear behavior (and even any order) of a N-body system and compare it with the fluid limit to measure quantitatively the discreteness effects up to shell crossing, i.e. up to when two nearest particles collide.

Particularly we have shown that one obtains, for long wavelength perturbations, the evolution predicted by an analogous fluid description of the self-gravitating system, and in particular, as a special case, the Zeldovich approximation. Further we can study precisely the deviations from this fluid-like behavior at shorter wavelengths arising from the discrete nature of the system. This analysis should be a useful step toward a precise quantitative understanding, which is currently lacking, of the role of discreteness in cosmological NBS.

We found that there are qualitatively two kinds of effects introduced by the discretization: (i) an average slowing down of the growth of the modes relative to the theoretical fluid evolution, and (ii) a pronounced anisotropy in k-space. There are notably a small fraction of modes with growth exponents larger than in linear fluid theory which, for sufficiently large times, will always dominate the evolution. Moreover we found that there are modes sinusoidally oscillating in addition to the expected growing instabilities.

We concluded that these features of the simple cubic lattice discretization, usually adopted in cosmological NBS, can be circumvented by employing a body centered cubic lattice. The known stability of this configuration of the Coulomb lattice implies that the fluid exponent is in this case an upper bound for all modes and that there are no oscillating modes for the case of gravity. One simple conclusion is then that a body centered cubic lattice may be a better choice of discretization, as its spectrum has only growing modes with exponents bounded above by that of fluid linear theory. Another important conclusion which we have drawn is the following: for a given physical wavelength, the discrepancy between the fluid and full evolution grows, up to shell crossing, with time. Thus, for a given physical scale, discreteness effects increase when the starting time of the simulation is decreased. This implies that at least one of the conditions for keeping discreteness effects under control in an NBS will be a constraint on the initial time. A full analysis of discreteness effects in cosmological NBS and their role in the formation of large scale structures is now in progress.


Some References

T. Baertschiger and F. Sylos Labini “On the problem of initial conditions in cosmological N-body simulations” , Europhys.Lett. 57 , 322-329 (2002)

T. Baertschiger, M. Joyce & F. Sylos Labini , “Power-law correlation and discreteness in cosmological N-Body simulations” Astrophys.J.Lett. 581 , L63-L67 (2002)

T. Baertschiger & F. Sylos Labini “Reply to the comment on the paper `On the problem of initial conditions in cosmological N-body simulations”’ Europhysics Letters 63 , 633-634, (2003)

A. Gabrielli, P. A. Masucci & F. Sylos Labini “Gravitational force probability density in weakly correlated pointdistributions” , Phys.Rev.E 69 , 031110-031117 (2004)

F. Sylos Labini , T. Baertschiger & M. Joyce “Universality of power law correlations in gravitational clustering” , Europhys.Lett. 66 , 171-177, (2004)

T. Baertschiger, F. Sylos Labini “Growth of correlation in gravitational N-body simulations”, Phys.Rev.D, 69, 123001-1 123001-14 (2004)

M. Joyce, B. Marcos, A. Gabrielli, T. Baertschiger, F. Sylos Labini “Gravitational evolution of a perturbed lattice and its fluid limit” Physical Review Letters 95, 011304 (2005)