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

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.