Non-linear Approximations to Gravitational Instability: a Comparison in the Quasi-linear Regime

We compare different nonlinear approximations to gravitational clustering in the weakly nonlinear regime, using as a comparative statistic the evolution of non-Gaussianity which can be characterised by a set of numbers Sp describing connected moments of the density field at the lowest order in 〈δ2〉: 〈δ〉c ≃ Sn〈δ〉. Generalizing earlier work by Bernardeau (1992) we develop an ansatz to evaluate all Sp in a given approximation by means of a generating function which can be shown to satisfy the equations of motion of a homogeneous spherical density enhancement in that approximation. On the basis of the values of we show that approximations formulated in Lagrangian space (such as the Zeldovich approximation and its extensions) are considerably more accurate than those formulated in Eulerian space such as the Frozen Flow and Linear Potential approximations. In particular we find that the nth order Lagrangian perturbation approximation correctly reproduces the first n + 1 parameters Sn. We also evaluate the density probability distribution function for the different approximations in the quasi-linear regime and compare our results with an exact analytic treatment in the case of the Zeldovich approximation. Subject Headings: cosmology: theory – large scale structure of Universe