Towards Gravitational Wave Turbulence within the Hadad-Zakharov metric

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.

Figures (15)

• Figure 1: Temporal evolution of the conserved quantities $E$ (top) and $N$ (bottom). In both plots, the blue line ($E_\mathrm{n}$ and $N_\mathrm{n}$) represents the value normalised by its initial condition, while the red line shows the corresponding relative error ($E_\mathrm{err}$ and $N_\mathrm{err}$).
• Figure 2: Energy (left) and wave action (right) two-dimensional spectra at time $t = 80 \times 10^3~t_\mathrm {GW}$ in a logarithmic grid. The injection wavenumber normalises the axes, and a small white disk localises the initial excitation of the spectra.
• Figure 3: Energy (left) and wave action (right) one-dimensional spectra at times $t = 0$ (red) and $t = 80 \times 10^3~t_\mathrm {GW}$ (black) compensated by the Kolmogorov-Zakharov spectra $k^{0}$ and $k^{-2/3}$, respectively. An empirical observation of the spectrum is given by a red dashed line in the two panels. The injection wavenumber normalises the horizontal axis.
• Figure 4: Temporal evolution of the fronts (in units of $k_i$) for the direct (blue) and inverse (red) cascades. A linear regression is used to estimate their variations (solid lines).
• Figure 5: Time evolution of the mean (top) and the standard deviation (bottom) values of the Ricci and Krestchmann scalars.