NONLINEAR APPROXIMATIONS TO GRAVITATIONAL-INSTABILITY - A COMPARISON IN 2ND-ORDER PERTURBATION-THEORY

Nonlinear approximation methods such as the Zel'dovich approximation a nd, more recently, the frozen flow and linear potential approximations are sometimes used to simulate nonlinear gravitational instability in the expanding universe. We investigate the relative accuracy of these approximations by comparing them with the exact solution using second -order perturbation theory. We evaluate the density and velocity field s in these approximations to second order and also determine the skewn ess parameter S3 = [delta3]/[[delta(1)]2]2 for each of the approximati ons, again in second order. We find that S3 = 4, 3, 3.4 for the Zel'do vich approximation, the frozen flow, and the linear potential approxim ations, respectively, as compared to S3 = 34/7 for the exact solution. Our results show that, of all the approximations considered, the Zel' dovich approximation is the most accurate in describing the weakly non linear effects of gravity. Moreover, the Zel'dovich approximation is m uch closer to the exact results for matter and velocity distributions than the other approximations, if the slope of the power spectrum of d ensity perturbations is -3 < n less-than-or-equal-to -1.