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Analysis of the adhesion model and the reconstruction problem in cosmology

Jian-Guo Liu, Robert L. Pego

TL;DR

<3-5 sentence high-level summary>The paper analyzes the adhesion model for cosmological mass transport, rigorously deriving the zero-viscosity limit and proving the existence of a limiting sticky Lagrangian flow that is uniquely characterized by a differential inclusion. It shows that the mass distribution ρ_t is the pushforward of Lebesgue measure by the limiting flow X_t, and it decomposes into absolutely continuous and singular parts, with the absolutely continuous part necessarily matching the Monge-Ampère measure κ_t while singular parts can differ in certain merging scenarios. A key contribution is the identification of conditions under which reconstruction via Monge-Ampère/OT agrees with the true Lagrangian map, and a detailed 2D three-sector example demonstrates when a monotone reconstruction fails to be exact a.e. This work clarifies the limits of Monge-Ampère-based reconstruction in the presence of singular mass concentrations and special merging configurations, with implications for understanding the cosmic web and mass-sheet dynamics in the adhesion framework.

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-Ampère 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-Ampère 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.

Analysis of the adhesion model and the reconstruction problem in cosmology

TL;DR

<3-5 sentence high-level summary>The paper analyzes the adhesion model for cosmological mass transport, rigorously deriving the zero-viscosity limit and proving the existence of a limiting sticky Lagrangian flow that is uniquely characterized by a differential inclusion. It shows that the mass distribution ρ_t is the pushforward of Lebesgue measure by the limiting flow X_t, and it decomposes into absolutely continuous and singular parts, with the absolutely continuous part necessarily matching the Monge-Ampère measure κ_t while singular parts can differ in certain merging scenarios. A key contribution is the identification of conditions under which reconstruction via Monge-Ampère/OT agrees with the true Lagrangian map, and a detailed 2D three-sector example demonstrates when a monotone reconstruction fails to be exact a.e. This work clarifies the limits of Monge-Ampère-based reconstruction in the presence of singular mass concentrations and special merging configurations, with implications for understanding the cosmic web and mass-sheet dynamics in the adhesion framework.

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-Ampère 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-Ampère 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.

Paper Structure

This paper contains 42 sections, 49 theorems, 255 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Present results
  4. Related literature
  5. Discussion
  6. Plan of the paper
  7. Lagrangian flow in the adhesion model
  8. Symmetries
  9. Semi-concavity and stability estimates for Lagrangian flows
  10. Zero-viscosity limit and differential inclusion
  11. Mass flow for the adhesion model
  12. Zero-viscosity limit for mass flow
  13. Lebesgue decomposition of the mass distribution
  14. Unique backward images a.e.
  15. Continuity equation
  16. ...and 27 more sections

Key Result

Lemma 2.1

Assume that $\varphi:\mathbb{R}^d\to\mathbb{R}$ is subquadratic at infinity and is $\lambda$-concave for some $\lambda\ge0$. Then with $\lambda_t = \lambda/(1+\lambda t)$, for any $t>0$ we have:

Figures (3)

  • Figure 1: Transport map for 3-sector velocity. Left panel: Lagrangian plane. Right panel: Eulerian plane. Red arrows indicate components of triple-point velocity $v_\star$ along mass filaments.
  • Figure 2: Flow examples. Insets indicate $v_i$ on a circle with center $v_\star=0$.
  • Figure 3: Pre-images $X_s^{-1}(x)$ (black), $X_t^{-1}(x)$ (blue) for $x=sv_{12}\in R_{12}$, $t>s$. Velocities $v_j$ are as in Fig. \ref{['f:stationary']}(a).

Theorems & Definitions (109)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • Lemma 2.6
  • ...and 99 more