Towards Gravitational Wave Turbulence within the Hadad-Zakharov metric

TL;DR

This paper investigates gravitational wave turbulence within general relativity using the Hadad-Zakharov metric, combining analytical methods and GPU-based simulations to explore energy cascades and statistical properties.

Contribution

It introduces a new numerical code TIGER for simulating gravitational turbulence and analyzes the compatibility of Einstein equations in the weakly nonlinear regime.

Findings

Reproduction of dual energy and wave action cascades.

Observation of Kolmogorov-Zakharov spectrum in simulations.

Discovery of Gaussian statistics with intermittent structures in canonical variables.

Abstract

The theory of gravitational wave turbulence describes the long-term statistical behaviour of a set of weakly nonlinear interacting waves. In this paper, we aim to study aspects of gravitational turbulence within the framework of general relativity using the Hadad-Zakharov (HZ) metric. The latter is parameterised by four functions (the coefficients of a diagonal metric) that must satisfy seven non-trivial Einstein equations, six of which are independent. The issue of their mutual compatibility is therefore essential, yet it has so far been overlooked. In this work, we argue that these equations can be compatible in the weakly nonlinear regime under specific conditions. Our analytical investigation is complemented by direct numerical simulations performed with a new GPU-based code, TIGER. A comparative analysis of the evolution of the Ricci and Kretschmann scalars indicates that gravitational wave turbulence corresponds to the propagation of a genuine physical degree of freedom. These numerical findings, however, must be interpreted with caution, given the difficulty of satisfying all seven Einstein equations simultaneously with sufficient accuracy. On the other hand, our simulations reproduce well the expected properties of the wave turbulence regime, with the emergence of a dual cascade of energy and wave action, and for the latter the observation of the Kolmogorov-Zakharov spectrum. In addition, our analysis reveals that the canonical variables of the problem evolve towards a nearly Gaussian statistical distribution punctuated by intermittent coherent (spatially localised and long-living) structures. In contrast to the canonical variables, the structure functions of the gauge-invariant metric components exhibit monofractal behaviour, which is a classical property of wave turbulence.