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Helmholtz Decomposition of the Lagrangian Displacement

Lagrangian displacement field $Ψ$ is the central object in Lagrangian perturbation theory (LPT). LPT is very successful at high redshifts, but it performs poorly at low redshifts due to severe shell crossing. To understand and quantify the effects of shell crossing, we extract $Ψ$ from N-body simulation and decompose it into scalar and vector parts. We find that at late time the power spectrum of the scalar part agrees with 1-loop results from LPT at large scales, while the power in small scales is much suppressed due to shell crossing. At z=0, the power spectrum of $Ψ$ is 10% lower than the 1-loop results at k = 0.1 h/Mpc. Shell crossing also generates the vector contribution in $Ψ$, although its effect is subdominant in comparison with the power suppression in the scalar part. At z=0, the vector part contributes 10% to the total power spectrum of $Ψ$ at k = 1 h/Mpc, while only 1% is expected from the vector contribution in LPT. We also examine the standard LPT recipes and some of its variants. In one of the variants, we include a power suppression factor in the displacement potential to take into account the power suppression in small scales after shell crossing. However, these simple phenomenological approaches are found to yield limited improvement compared to the standard LPT after the onset of shell crossing.