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Transients from initial conditions based on Lagrangian perturbation theory in $N$-body simulations II: the effect of the transverse mode

We study the initial conditions for cosmological $N$-body simulations for precision cosmology. In general, Zel'dovich approximation has been applied for the initial conditions of $N$-body simulations for a long time. These initial conditions provide incorrect higher-order growth. These error caused by setting up the initial conditions by perturbation theory is called transients. We investigated the impact of transient on non-Gaussianity of density field by performing cosmological $N$-body simulations with initial conditions based on first-, second-, and third-order Lagrangian perturbation theory in previous paper. In this paper, we evaluates the effect of the transverse mode in the third-order Lagrangian perturbation theory for several statistical quantities such as power spectrum, non-Gaussianty, and multifractal dimensions. Then we clarified that the effect of the transverse mode in the third-order Lagrangian perturbation theory is quite small.