Direct Evidence of a Dual Cascade in Gravitational Wave Turbulence

We present the first direct numerical simulation of gravitational wave turbulence. General relativity equations are solved numerically in a periodic box with a diagonal metric tensor depending on two space coordinates only, $g_{ij}\equiv g_{ii}(x,y,t){\delta }_{ij}$, and with an additional small-scale dissipative term. We limit ourselves to weak gravitational waves and to a freely decaying turbulence. We find that an initial metric excitation at intermediate wave number leads to a dual cascade of energy and wave action. When the direct energy cascade reaches the dissipative scales, a transition is observed in the temporal evolution of energy from a plateau to a power-law decay, while the inverse cascade front continues to propagate toward low wave numbers. The wave number and frequency-wave-number spectra are found to be compatible with the theory of weak wave turbulence and the characteristic timescale of the dual cascade is that expected for four-wave resonant interactions. The simulation reveals that an initially weak gravitational wave turbulence tends to become strong as the inverse cascade of wave action progresses with a selective amplification of the fluctuations $g_{11}$ and $g_{22}$.

Figure 1

Time evolution (rms values) of the basic fields $\tilde{\alpha }$, $\tilde{\beta }$, and $\lambda$ (inset).

Figure 2

Time evolution of the normalized energy. Inset: the same variation in log-log reveals a decay close to $t^{-1/4}$. The colored vertical lines at the top correspond to the times chosen to plot the spectra in Fig. 3.

Figure 3

Time evolution of $k^{2/3}{N}_{1d}(k)$ (solid lines) with the initial spectrum in black dashed line. Corresponding times are given in Fig. 2. The same spectrum, renormalized in scale and amplitude, is shown for an inviscid simulation at a resolution of 1024 points (blue dashed line). The 1D spectra $2\pi k{\int }_0^{2\pi }|{\tilde{\alpha }}_{\mathbf{k}}{|}^{2}d\varphi$ and $2\pi k{\int }_0^{2\pi }|{\tilde{\beta }}_{\mathbf{k}}{|}^{2}d\varphi$ are also shown (dotted lines) at the final time of the simulation (at a resolution of 512 points) and compared with the power law $k^{-3.5}$.

Figure 4

$N(\mathbf{k})$ around the final time of the simulation (mean over 4 times). The center of the domain of initial excitation is indicated by the symbol “o”.

Figure 5

$\omega \text{−}k$ spectrum of wave action. The dotted line corresponds to the dispersion relation of a GW.

Figure 6

Metric components $g_{00}$ (blue), $g_{11}$ (orange), $g_{22}$ (green), and $g_{33}$ (red) at the final time of the simulation. $g_{00}$ and $g_{33}$ have been vertically shifted by $+1.9$ and $+0.15$, respectively.