Effect of accretion on scalar superradiant instability

TL;DR

This work investigates how gas accretion affects the evolution of a Kerr black hole surrounded by a condensate formed via scalar superradiance, across multiple bound-state modes. By solving the coupled evolution equations and employing analytical small-$\alpha$ expansions, it reveals that accretion can dramatically accelerate condensate growth and GW emission, while shortening the overall signal duration. A universal Regge-trajectory pattern emerges: the system alternates between trajectories dictated by accretion and those defined by dominant $\{0,l,l\}$ modes, with transitions determined by analytic conditions on $\alpha=M\mu$ and mode-specific SR rates. The findings imply that accretion broadens the parameter space where GWs from ultra-light scalars could be detectable (e.g., by LISA) and provides practical formulas for phenomenology, while highlighting limitations like neglect of backreaction, scalar self-interactions, and possible overtone mixing in higher-$l$ sectors.

Abstract

Superradiance can lead to the formation of a black hole (BH) condensate system. We thoroughly investigate the accretion effect on the evolution of this system, and the gravitational wave signals it emits in the presence of multiple superradiance modes. Assuming the multiplication of the BH mass and scalar mass as a small number, we obtain the analytical approximations of all important quantities, which can be directly applied to phenomenological studies. In addition, we confirm that accretion could significantly enhance the gravitational wave (GW) emission and reduce its duration, and show that the GW beat signature is similarly modified.

Figures (8)

• Figure 1: The evolution of BH spin $a_*$ as a function of the mass ratio $M/M_0$ without the scalar field. Two initial BH spins $a_{*0}=0$ and $a_{*0}=0.7$ are considered. The red dashed curves are solved from Eq. \ref{['eq:das_dlnM_with_photon']}. The black curves further assumes the absence of the photon captured by the BH, which gives Eq. \ref{['eq:as_ini_not_0']}
• Figure 2: The time evolution of scalar condensate masses (panel a), GW emission fluxes (panel b), BH spins (panel c), and BH mass (panel d). Initial parameters are $a_{*0}=0.7$, $M_0=10^3M_\odot$, ${M}_\mathrm{s,0}^{(011)}=10^{-5}M_0$ and $M_0\mu=0.01$. Different colors represent different accretion rates indicated by the legends. The vertical lines are the superradiant timescale $\tau_\text{SR}$ and the GW emission timescale $\tau_\text{GW}$.
• Figure 3: The time evolution of the scalar condensate mass (panel a), the GW emission (panel b), the BH spin (panel c), and the BH mass (panel d). The black and orange dotted curves are the evolution with and without the GW emission, respectively. The green curve in panel (c) is calculated with Eq. \ref{['eq:a_critc']}. The blue line in panel (d) is the exponential increase of the BH mass given in Eq. \ref{['eq:BH_mass_efold']}.
• Figure 4: The time evolution of scalar condensate masses (panel a), GW emission fluxes (panel b), BH spins (panel c), and BH masses (panel d). Initial parameters are $M_0=10^3M_\odot$, ${M}_\mathrm{s,0}^{(011)}=10^{-5}M_0$, and $M_0\mu=0.01$. The solid and dashed curves are for initial BH spins $a_{*0}=0.7$ and $a_{*0}=0$, respectively. Different colors represent different accretion rates, indicated by the legends.
• Figure 5: The evolution of the BH spin as a function of the BH mass. The initial parameters for panels (a) and (b) are identical to those used in Fig. \ref{['fig:vs_fEdd']} and Fig. \ref{['fig:vs_fEdd_2']}, respectively. The $\{0,1,1\}$ Regge trajectory defined by Eq. \ref{['eq:a_critc']} is represented as the upper boundary of the shaded region.