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

Abstract

In two recent articles a detailed study has been presented of the out of equilibrium dynamics of an infinite system of self-gravitating points initially located on a randomly perturbed lattice. In this article we extend the treatment of the early time phase during which strong non-linear correlations first develop, prior to the onset of "self-similar" scaling in the two point correlation function. We establish more directly, using appropriate modifications of the numerical integration, that the development of these correlations can be well described by an approximation of the evolution in two phases: a first perturbative phase in which particles' displacements are small compared to the lattice spacing, and a subsequent phase in which particles interact only with their nearest neighbor. For the range of initial amplitudes considered we show that the first phase can be well approximated as a transformation of the perturbed lattice configuration into a Poisson distribution at the relevant scales. This appears to explain the "universality" of the spatial dependence of the asymptotic non-linear clustering observed from both shuffled lattice and Poisson initial conditions.

Dynamics of finite and infinite self-gravitating systems with cold quasi-uniform initial conditions

“…In [36] we have examined the issue of the resultant errors, placing non-trivial lower bounds on them which show that they are certainly at a level relevant to the precision required in coming years for these simulations. In our study of the SL in [29] we have shown that the asymptotic form of the correlation function in this case is very similar to that which emerges at early times when the evolution is clearly far from the VP limit. This suggests that careful further study of the relation between the two regimes, < ε and > ε, even if numerically costly, will be necessary to resolve these issues, which are of both theoretical and practical importance in the theory of structure formation in the universe.…”

“…In the set of initial conditions that we have discussed, we have, as we discussed, only a single parameter, the normalized shuffling δ. It is straightforward to see that variation of this parameter does not give convergence to the VP limit (just as variation of the single parameter N in the finite case did not define such convergence): at any value of δ an analysis of the dynamics (see in particular [29]) shows that there is always a phase of the evolution at early times in which the forces on particles are determined predominantly by their nearest neighbors, which means that the mean-field VP limit is certainly not valid. Further for small values of δ one can show, using a perturbative treatment of the dynamics, that there are measurable deviations from the fluid linear theory limit at any wavenumber k [37].…”

“…As initial conditions for our study, which is reported in much greater detail in [27]- [29], we consider, for reasons that we will now explain, a class of 'shuffled lattice' (SL) distributions, defined as follows: particles initially on a perfect lattice, of lattice spacing (=mean inter-particle spacing), are displaced randomly ('shuffled') about their lattice site independently of all the others. A particle initially at the lattice site R is then at x(R) = R + u(R), where the random vectors u(R) are specified by p(u), the PDF for the displacement of a single particle.…”