Gravitational evolution of a perturbed lattice and its fluid limit
Résumé
We apply a simple linearization, well known in solid state physics, to approximate the evolution at early times of cosmological N-body simulations of gravity. In the limit that the initial perturbations, applied to an infinite perfect lattice, are at wavelengths much greater than the lattice spacing $l$ the evolution is exactly that of a pressureless self-gravitating fluid treated in the analagous (Lagrangian) linearization, with the Zeldovich approximation as a sub-class of asymptotic solutions. Our less restricted approximation allows one to trace the evolution of the discrete distribution until the time when particles approach one another (i.e. "shell crossing''). We calculate modifications of the fluid evolution, explicitly dependent on $l$ i.e. discreteness effects in the N body simulations. We note that these effects become increasingly important as the initial red-shift is increased at fixed $l$. The possible advantages of using a body centred cubic, rather than simple cubic, lattice are pointed out.