Black holes as particle detectors: evolution of superradiant instabilities

TL;DR

This work develops a quasi-adiabatic, fully relativistic treatment of Kerr black hole superradiance including gravitational-wave emission and gas accretion. It demonstrates that GW losses negligibly affect BH mass and spin evolution, while accretion plays a crucial role in entering and traversing the instability window; the scalar cloud can become a sizeable fraction of the BH mass yet remains very diffuse, leaving the geometry effectively Kerr. Monte Carlo simulations show the linearized analysis remains valid and reproduce the predicted depleted regions (Regge holes) in the BH parameter space, strengthening existing bounds on ultralight bosons. The results have important implications for using BH spin measurements and future gravitational-wave observations to constrain or detect ultralight fields, while highlighting the limited prospects for detectable Kerr deviations caused by the cloud in electromagnetic probes.

Abstract

Superradiant instabilities of spinning black holes can be used to impose strong constraints on ultralight bosons, thus turning black holes into effective particle detectors. However, very little is known about the development of the instability and whether its nonlinear time evolution accords to the linear intuition. For the first time, we attack this problem by studying the impact of gravitational-wave emission and gas accretion on the evolution of the instability. Our quasi-adiabatic, fully-relativistic analysis shows that: (i) gravitational-wave emission does not have a significant effect on the evolution of the black hole, (ii) accretion plays an important role and (iii) although the mass of the scalar cloud developed through superradiance can be a sizeable fraction of the black-hole mass, its energy-density is very low and backreaction is negligible. Thus, massive black holes are well described by the Kerr geometry even if they develop bosonic clouds through superradiance. Using Monte Carlo methods and very conservative assumptions, we provide strong support to the validity of the linearized analysis and to the bounds of previous studies.

Figures (5)

• Figure 1: Pictorial description of a bosonic cloud around a spinning BH in a realistic astrophysical environment. The BH loses energy $E_S$ and angular momentum $L_S$ through superradiant extraction of scalar waves and emission of GWs, while accreting gas from the disk, which transports energy $E_{\rm ACC}$ and angular momentum $L_{\rm ACC}$. Notice that accreting material is basically in free fall after it reaches the innermost stable circular orbit. The cloud is localized at a distance $\sim 1/M\mu^2>2M$.
• Figure 2: Evolution of the BH mass and spin and of the scalar cloud due to superradiance, accretion of gas and emission of GWs. The two sets of plots show two different cases. In Case I (left set) the initial BH mass $M_0=10^4 M_\odot$ and the initial BH spin $J_0/M_0^2=0.5$. The BH enters the instability region at about $t\sim 6{\rm Gyr}$, when its mass $M\sim10^7 M_\odot$ and its spin is quasi-extremal. The set of plots on the right shows Case II, in which $M_0=10^7M_\odot$ and $J_0/M_0^2=0.8$, and the evolution starts already in the instability region for this scalar mass $\mu=10^{-18}{\rm eV}$. For both cases, the left top panels show the dimensionless angular momentum $J/M^2$ and the critical superradiant threshold $a_{\rm crit}/M$ (cf. Eq. \ref{['acrit']}); the left bottom panels show the mass of the scalar cloud $M_S/M$ (note the logarithmic scale in the x-axis for Case II); and the right panels show the trajectory of the BH in the Regge plane Arvanitaki:2010sy during the evolution. The dashed blue line denotes the depleted region as estimated by the linearized analysis, i.e. it marks the threshold at which $\tau\sim\tau_{\rm ACC}$.
• Figure 3: The final BH mass and spin in the Regge plane for initial data consisting of $N=10^3$ BHs with initial mass and spin randomly distributed between $\log_{10}M_0\in[4,7.5]$ and $J_0/M_0^2\in[0.001,0.99]$. The BH parameters are then extracted at $t=t_F$, where $t_F$ is distributed on a Gaussian centered at $\bar{t}_{F}\sim 2\times 10^9{\rm yr}$ with width $\sigma=0.1\bar{t}_{F}$. We considered $\mu=10^{-18}{\rm eV}$. The dashed blue line is the prediction of the linearized analysis obtained by comparing the superradiant instability time scale with the accretion time scale, $\tau\approx\tau_{\rm Salpeter}/f_{\rm Edd}$, whereas the solid green line denotes the region defined through Eq. \ref{['region']}. Old BHs do not populate the region above the green threshold curve. The experimental points with error bars refer to the supermassive BHs listed in Brenneman:2011wz.
• Figure 4: The energy-density profile of the scalar cloud on the equatorial plane and at azimuthal angle $\varphi=0$ in units of the BH density, $1/M^2\sim 10^6 {\rm kg/m}^3$, at different time snapshots for the evolution shown in the right panel of Fig. \ref{['fig:evolution']}.
• Figure 5: Evolution with the same initial conditions as in the right set of Fig. \ref{['fig:evolution']} but turning off the emission of GWs as in the case of a complex scalar field. The left panels show the case of mass accretion at the rate $f_{\rm Edd}=0.1$, whereas the right panels show the case in which also accretion is turned off. For comparison, the scalar mass in the right bottom panel is also shown in the left bottom panel by a dashed black curve. When accretion is effective, the scalar cloud can become heavier.