Analysis of the adhesion model and the reconstruction problem in cosmology

TL;DR

This paper analyzes the adhesion model in cosmology, focusing on mass flow and Lagrangian advection, revealing unique limiting behaviors and limitations in reconstructing inverse Lagrangian maps near singular structures.

Contribution

It provides a rigorous analysis of the zero-viscosity limit in the adhesion model, characterizing particle paths and their merging behavior using differential inclusions and Monge-Ampère measures.

Findings

Unique limiting Lagrangian semi-flow exists under mild conditions.

Mass measure's absolutely continuous part matches Monge-Ampère measure.

Reconstruction of inverse Lagrangian maps cannot be exact near merging singularities.

Abstract

In cosmology, a basic explanation of the observed concentration of mass in singular structures is provided by the Zeldovich approximation, which takes the form of free-streaming flow for perturbations of a uniform Einstein-de Sitter universe in co-moving coordinates. The adhesion model suppresses multi-streaming by introducing viscosity. We study mass flow in this model by analysis of Lagrangian advection in the zero-viscosity limit. Under mild conditions, we show that a unique limiting Lagrangian semi-flow exists. Limiting particle paths stick together after collision and are characterized uniquely by a differential inclusion. The absolutely continuous part of the mass measure agrees with that of a Monge-Amp\`ere measure arising by convexification of the free-streaming velocity potential. But the singular parts of these measures can differ when flows along singular structures merge, as shown by analysis of a 2D Riemann problem. The use of Monge-Amp\`ere measures and optimal transport theory for the reconstruction of inverse Lagrangian maps in cosmology was introduced in work of Brenier & Frisch et al. (Month. Not. Roy. Ast. Soc. 346, 2003). In a neighborhood of merging singular structures in our examples, however, we show that reconstruction yielding a monotone Lagrangian map cannot be exact a.e., even off of the singularities themselves.