Helmholtz Decomposition of the Lagrangian Displacement
  Lagrangian displacement field $\Psi$ 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 $\Psi$ 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 $\Psi$ is 10% lower than the
1-loop results at k = 0.1 h/Mpc. Shell crossing also generates the vector
contribution in $\Psi$, 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 $\Psi$ 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.
