Weakly turbulent saturation of the nonlinear scalar ergoregion instability

TL;DR

This work investigates the nonlinear saturation of the ergoregion instability on a horizonless spinning ultracompact spacetime by simulating a nonlinear scalar field with potential-type and derivative self-interactions on a spinning boson star background. The fastest-growing $m=1$ ergoregion mode triggers a weakly turbulent direct cascade that transfers energy from large scales to small-scale, highly trapped modes around the counter-rotating stable light ring, with nonlinear transfer times shorter than linear growth rates. The monopole mode acts as a pump driving higher azimuthal modes, and derivative self-interactions enhance the cascade efficiency, producing multiple saturation events and a long-lived turbulent state near the light ring. These results suggest turbulence could play a crucial role in fully gravitational ergoregion saturation and may imprint characteristic gravitational-wave signatures relevant to black hole mimickers and GW searches.

Abstract

We perform time-domain evolutions of the ergoregion instability on a horizonless spinning ultracompact spacetime in scalar theories with potential-type and derivative self-interactions mimicking the nonlinear structure of the Einstein equations. We find that the instability saturates by triggering a weakly turbulent direct cascade, which transfers energy from the most unstable and large-scale modes to small scales. The cascade's nonlinear timescales of each mode are orders of magnitude shorter than the corresponding linear e-folding times. Through this mechanism, the counter-rotating stable light ring is filled with a spectrum of higher-order azimuthal modes forming a ring-like shape. Thereby we demonstrate that turbulent processes are likely also important during the fully gravitational saturation of the instability, leaving imprints in the gravitational wave emission.

Figures (11)

• Figure 1: Evolution of the energies $E_{\ell\ell}$ for $\ell\geq 0$ from the linear into the nonlinear regime. We include only potential-type self-interactions (top), or only derivative self-interactions (bottom). Here $\langle\dots\rangle$ indicates a rolling time-average. All shown energies are negative, except $E_{00}>0$ (dashed black lines). The gray shaded regions indicate the periods, when the $\ell=m=1$ mode is subject to the linear ergoregion instability. Dotted vertial lines indicate the times shown in Fig. \ref{['fig:spectrum']}.
• Figure 2: The spectrum of linear energies $E_{\ell\ell}$ at various times during the evolution shown in Fig. \ref{['fig:mode_saturation']}. The top includes only potential-type self-interactions, while the bottom only derivative self-interactions.
• Figure 3: (top) The nonlinear energy transfer time of the potential-type self-interaction case, $\tau^{\rm pNL}_\ell$, and derivative self-interaction scenario, $\tau^{\rm dNL}_\ell$, compared to the energy e-folding timescales, $\tau^{\rm EI}_{m=\ell}$, of the linear ergoregion instability (obtained in Ref. inprep). (bottom) The evolution of the frequency $\omega_R=\mathcal{F}_{11}/\mathcal{J}_{11}$ and growth rate $\omega_I=-\mathcal{F}_{11}/(2E_{11})$ of the $\ell=m=1$ mode compared with their linear values, $\text{Re}(\omega_{1})$ and $\text{Im}(\omega_{1})$, respectively. Here we compare the impact of pure derivative ($\alpha<0$ and $\alpha>0$) to purely potential-type ($\kappa>0$) self-interactions.
• Figure 4: The evolution of the interaction energy $E_{\rm int}$, the sum of linear energies $E_{\rm lin}$, and the linear energy $E_{11}$, throughout saturation of the system, when considering only potential-type self-interactions.
• Figure 5: (left) The quantity $D_{6\ell\ell'\ell"}^{6\ell\ell'\ell"}$ plotted over the index space $(\ell,\ell',\ell")$ up to a maximum mode number of $\ell,\ell',\ell"\leq 20$. Color indicates its value. For clarity, when the coupling coefficients vanish, we remove the dots. While the value of $E_{6\ell\ell'\ell"}^{6\ell\ell'\ell"}$ differs slightly, they are qualitatively similar. (center and right) The coupling coefficients at fixed $\bar{m}=m=m'=m"=1$, i.e., $D^{1111}_{6\ell\ell'\ell"}$ (center) and $E^{1111}_{6\ell\ell'\ell"}$ (right). In the eikonal limit, these coefficients behave as $E^{1111}_{1\ell\ell\ell}\sim\ell^{1/2}$ (for odd-$\ell$ modes), whereas $D^{1111}_{1\ell\ell\ell}\sim\ell^{-3/2}$.