Local Nonlinear Approximations to the Growth of Cosmic Structures

Local nonlinear approximations to the growth of cosmic perturbations are developed, resulting in relations at a given epoch, between the peculiar velocity and gravity fields and their gradients. The continuity equation and the gravitational field are computed exactly at that epoch, and only the equation of motion is approximated. The second-order relation is derived for arbitrary geometry and cosmological parameters. Solutions are developed to fourth order for laminar spherical perturbations in an Einstein e Sitter universe, with exact mass conservation at all times, but the gain in accuracy for higher orders is modest. All orders become important when the specific peculiar kinetic energy becomes comparable to the peculiar gravitational potential ½υ2 ∼ φ, typically at relative density perturbations, δ ∼ 4. The general second-order relation, while implicit, is simple to solve. N-body simulations show that it provides moderate gains in accuracy over other local approximations. It can therefore be easily applied in the comparison of large-scale structures and velocities in the quasi-linear regime, δ ∼ 1-4, as well as in the reconstruction of the primordial perturbations from which they grew.