Initial condition problem for cosmological N-body simulations

For a long time, the Lagrangian linear perturbation (Zel'dovich approximation, ZA) is used for initial conditions for cosmological N-body simulations. Recently, Crocce et al. (2006) proposed the improvement of the initial conditions. They applied Lagrangian second-order perturbation (2LPT) for the initial condition. Then they calculated the Lagrangian linear perturbation and showed that 2LPT initial conditions improve the evolution of non-Gaussianity for the density fluctuation in the Universe. Here we have one question. Although 2LPT initial conditions improve the cosmological N-body simulation, is 2LPT initial conditions sufficient? To answer this question, in addition to the ZA and 2LPT, we also calculate the non-Gaussianity with the initial conditions based on third-order Lagrangian perturbation theory (3LPT). Then we show reasonable order of perturbation and redshift for the initial conditions.