Eulerian and Lagrangian propagators for the adhesion model (Burgers dynamics)

TL;DR

This paper computes Eulerian and Lagrangian propagators in the adhesion model, revealing distinct decay and saturation behaviors, and highlighting the Lagrangian propagators as more sensitive probes of nonlinear cosmic structures.

Contribution

It provides exact results for propagators in the one-dimensional adhesion model, linking them to velocity distributions and shock mass functions, and compares their effectiveness in probing nonlinearities.

Findings

Eulerian propagators decay strongly at late times and high wavenumbers.

Lagrangian propagators saturate to a constant at late times.

Lagrangian propagators depend on the initial power spectrum and shock mass function.

Abstract

Motivated by theoretical studies of gravitational clustering in the Universe, we compute propagators (response functions) in the adhesion model. This model, which is able to reproduce the skeleton of the cosmic web and includes nonlinear effects in both Eulerian and Lagrangian frameworks, also corresponds to the Burgers equation of hydrodynamics. Focusing on the one-dimensional case with power-law initial conditions, we obtain exact results for Eulerian and Lagrangian propagators. We find that Eulerian propagators can be expressed in terms of the one-point velocity probability distribution and show a strong decay at late times and high wavenumbers, interpreted as a "sweeping effect" but not a genuine damping of small-scale structures. By contrast, Lagrangian propagators can be written in terms of the shock mass function -- which would correspond to the halo mass function in cosmology -- and saturate to a constant value at late times. Moreover, they show a power-law dependence on scale or wavenumber which depends on the initial power-spectrum index and is directly related to the low-mass tail of the shock mass function. These results strongly suggest that Lagrangian propagators are much more sensitive probes of nonlinear structures in the underlying density field and of relaxation processes than their Eulerian counterparts.