Nonlinear Diffusion Models for Gravitational Wave Turbulence

TL;DR

This paper introduces nonlinear diffusion models in spectral space to describe gravitational wave turbulence, demonstrating their ability to conserve key quantities and reproduce known power law solutions, with insights into primordial universe dynamics.

Contribution

The paper develops and analyzes second- and fourth-order nonlinear diffusion models that capture energy and wave action cascades in gravitational wave turbulence, including self-similar solutions with front propagation.

Findings

Models conserve energy and wave action.

Reproduce power law solutions for cascades.

Identify anomalous scaling and front propagation law.

Abstract

A fourth-order and a second-order nonlinear diffusion models in spectral space are proposed to describe gravitational wave turbulence in the approximation of strongly local interactions. We show analytically that the model equations satisfy the conservation of energy and wave action, and reproduce the power law solutions previously derived from the kinetic equations with a direct cascade of energy and an explosive inverse cascade of wave action. In the latter case, we show numerically by computing the second-order diffusion model that the non-stationary regime exhibits an anomalous scaling which is understood as a self-similar solution of the second kind with a front propagation following the law $k_f \sim (t_*-t)^{3.296}$, with $t<t_*$. These results are relevant to better understand the dynamics of the primordial universe where potent sources of gravitational waves may produce space-time turbulence.