An Eulerian perturbation approach to large-scale structures: extending the adhesion approximation

A description of the dynamics of a collisionless, self-gravitating fluid is developed and applied to follow the development of large-scale structures in the Universe. Such a description takes on one of the assumptions of the adhesion approximation (AA) model, i.e., the introduction of an artificial viscosity ν in the Euler equation, but extends it to deeper non-linear stages, where the extrapolation of the linear relation ψ=-φ between the velocity and the gravitational potentials – at the basis of both the Zel'dovich and the adhesion models – is no longer valid. This is achieved by expanding the relation between φ and ψ, in general not explicitly computable, in powers of the small viscosity ν. In this case, the evolution of the velocity potential is described by a diffusion-like equation for the ‘expotential’ field ξ=exp(-ψ/2ν). Such an equation includes a source term V(ξ), which expresses the relation between φ and ψ as a series expansion in powers of ν. Such a term is related to the onset of the non-linear evolution of the velocity potential and grows from zero (the limit corresponding to the AA) with increasing time. For terms in V(ξ) up to order O(ν) (the only ones that can be expressed in a fully Eulerian form), the diffusion equation is solved using the path-integral approach. The AA is then recovered as a ‘free-particle’ theory [corresponding to V(ξ)=0], where the dynamics is determined by the initial value of ξ. Our inclusion of the lowest order term in V(ξ) substantially changes the dynamics, so that the velocity potential at a given time in a given Eulerian position depends on the values taken at all previous times in all other coordinates. This is expected in the non-linear regime, where perturbations no longer evolve independently, but ‘feel’ the changes of the surrounding density field. The path-integral solution is computed numerically through an algorithm based on Monte Carlo realizations of random walks in the Eulerian space. In particular, the solution ξ at any cosmic time is obtained upon averaging the value of the potential V(ξ) at previous times in the Eulerian locations reached by the random walks. The solution is applied to the cosmological evolution of a cold dark matter density field, and the results are compared to the outcomes of an N-body simulation with the same initial condition. The velocity field in the presented extended adhesion (EA) description is obtained numerically using the random walks algorithm described above. For the case of a null potential V(ξ)=0, this constitutes a novel implementation of the AA which is free from the numerical errors affecting finite-difference solution schemes for partial differential equation and is faster than the Gaussian convolution algorithm adopted by Weinberg & Gunn. When the first-order term of the potential V(ξ) is included, the proposed extension of the adhesion approach provides a better description of small-scale, deeply non-linear regions, as is quantitatively shown by the computation of some statistical indicators. At larger scales, the satisfactory description of the large-scale texture and of the voids given by the canonical AA is preserved in the extended model.