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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 includesnonlinear effects in both Eulerian and Lagrangian frameworks, also correspondsto the Burgers equation of hydrodynamics. Focusing on the one-dimensional casewith power-law initial conditions, we obtain exact results for Eulerian andLagrangian propagators. We find that Eulerian propagators can be expressed interms of the one-point velocity probability distribution and show a strongdecay at late times and high wavenumbers, interpreted as a "sweeping effect"but not a genuine damping of small-scale structures. By contrast, Lagrangianpropagators can be written in terms of the shock mass function -- which wouldcorrespond to the halo mass function in cosmology -- and saturate to a constantvalue at late times. Moreover, they show a power-law dependence on scale orwavenumber which depends on the initial power-spectrum index and is directlyrelated to the low-mass tail of the shock mass function. These results stronglysuggest that Lagrangian propagators are much more sensitive probes of nonlinearstructures in the underlying density field and of relaxation processes thantheir Eulerian counterparts.

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