[Paper Review] Turbulence model of the cosmic structure

José Gaite|arXiv (Cornell University)|Feb 14, 2012

Cosmology and Gravitation Theories1 references3 citations

TL;DR

This paper proposes a non-perturbative, Kolmogorov-based turbulence framework applied to the stochastic adhesion model of cosmic structure formation, using scaling laws to derive the matter density two-point correlation function. It finds the correlation exponent γ ∈ (1, 1.33), consistent with simulations and galaxy surveys, but suggests the adhesion model underestimates energy dissipation from filaments and nodes due to its neglect of gravitational chaos and relativistic effects.

ABSTRACT

The Kolmogorov approach to turbulence is applied to the Burgers turbulence in the stochastic adhesion model of large-scale structure formation. As the perturbative approach to this model is unreliable, here is proposed a new, non-perturbative approach, based on a suitable formulation of Kolmogorov's scaling laws. This approach suggests that the power-law exponent of the matter density two-point correlation function is in the range 1--1.33, but it also suggests that the adhesion model neglects important aspects of the gravitational dynamics.

Motivation & Objective

• To address the limitations of perturbative approaches in modeling nonlinear gravitational clustering in the cosmic web.
• To apply Kolmogorov’s scaling laws—typically used in fluid turbulence—to the stochastic adhesion model of large-scale structure formation.
• To derive the matter density two-point correlation function from velocity field scaling, avoiding reliance on linear approximations.
• To assess whether the adhesion model adequately captures energy dissipation and scaling behavior in gravitational clustering, especially at small scales.
• To identify the role of intermittency and small-scale dynamics in shaping the cosmic web’s multifractal structure.

Proposed method

• Formulates a non-perturbative approach based on Kolmogorov’s scaling laws adapted to Burgers turbulence in the adhesion model.
• Uses the velocity field’s scaling behavior to derive the density field via the exact solution of the Burgers equation in the zero-viscosity limit.
• Applies the relation ρ(x) = ρ₀ det[δij − ∂iuj(x)] to express density in terms of velocity gradients, enabling derivation of correlation functions.
• Introduces a stochastic closure approach to handle intermittency and derive the two-point correlation function’s scaling exponent.
• Analyzes the correlation function’s singular components, discarding the Poisson term and focusing on the dominant c(r) contribution from velocity correlations.
• Identifies the homogeneity scale L(t) = t^{1/(1−h)}L(1) as analogous to the integral scale in Navier-Stokes turbulence, defining the inertial range.

Experimental results

Research questions

• RQ1 Can Kolmogorov’s scaling laws be meaningfully applied to the stochastic adhesion model of cosmic structure formation?
• RQ2 What is the predicted value of the matter density two-point correlation function’s exponent γ in the non-perturbative regime of Burgers turbulence?
• RQ3 Why does the adhesion model fail to fully account for energy dissipation in the formation of filaments and nodes in the cosmic web?
• RQ4 How does intermittency in the velocity field affect the scaling of the density correlation function?
• RQ5 What role do small-scale dissipative processes—potentially involving black holes—play in gravitational clustering, and how are they missed by the adhesion model?

Key findings

• The power-law exponent γ of the matter density two-point correlation function is predicted to lie in the range 1 < γ < 1.33, with the upper bound γ = 4/3 ≈ 1.33 arising from the Kolmogorov scaling ζ(2) = 2/3.
• The correlation function is dominated by sheet-like structures (2D singularities), consistent with N-body simulations showing that most mass resides in sheets.
• The adhesion model underestimates energy dissipation from filaments and nodes (1D and 0D singularities), which involve stronger gravitational singularities and infinite energy dissipation in Newtonian gravity.
• The vanishing of the c(r) term for ρ > 5/2 implies that higher-order singularities do not contribute to the dominant scaling, but for ρ < 5/2, c(r) would lead to γ > 2, which is inconsistent with observations.
• The model suggests that the Kolmogorov scale may be physically linked to the formation of supermassive black holes, implying that relativistic effects are essential for a complete description of small-scale dissipation.
• The analysis indicates that the adhesion model’s lack of chaotic dynamics and energy re-injection limits its ability to reproduce the full scaling behavior seen in N-body simulations and galaxy surveys.