Nonlinear diffusion models for gravitational wave turbulence

Highlights: • 4th and 2nd order nonlinear diffusion models in spectral space are derived. • The models describe gravitational wave turbulence for strongly local interactions. • Analytical properties of these equations are given (solution, flux sign, etc). • Numerical simulation of the 2nd order diffusion model is made. • An anomalous scaling is measured in the non-stationary regime. -- Abstract: A fourth-order and a second-order nonlinear diffusion model 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 $k_f\sim {(t_{\ast }-t)}^{3.296}$, with $t<t_{\ast }$. These results are relevant to better understand the dynamics of the primordial universe where potent sources of gravitational waves may produce space–time turbulence.