Quantifying distortions of the Lagrangian dark-matter mesh in cosmology

We examine the Lagrangian divergence of the displacement field, arguably a more natural object than the density in a Lagrangian description of a cosmological large-scale structure. This quantity, which we denote psi, quantifies the stretching and distortion of the initially homogeneous lattice of dark-matter particles in the universe. psi encodes similar information as the density, but the correspondence has subtleties. It corresponds better to the log-density A than the overdensity delta. A Gaussian distribution in psi produces a distribution in A with slight skewness; in delta, we find that in many cases the skewness is further increased by 3. A local spherical-collapse-based (SC) fit found by Bernardeau gives a formula for psi's particle-by-particle behaviour that works quite well, better than applying the Lagrangian perturbation theory (LPT) at first or second (2LPT) order. In 2LPT, there is a roughly parabolic relation between initial and final psi that can give overdensities in deep voids, so low-redshift, high-resolution 2LPT realizations should be used with caution. The SC fit excels at predicting psi until streams cross; then, for particles forming haloes, psi plummets as in a waterfall to -3. This gives a new method for producing N-particle realizations. Compared to LPT realizations, such SC realizations give reduced stream-crossing, and better visual and 1-point-probability density function (PDF) correspondence to the results of full gravity. LPT, on the other hand, predicts large-scale flows and the large-scale power-spectrum amplitude better, unless an empirical correction is added to the SC formula.