The effect of accretion on scalar superradiant instability

The Kerr spacetime metric with mass $M$ and angular momentum $J$ is expressed in the Boyer-Lindquist coordinates as Boyer:1966qh

	$\displaystyle ds^{2}=$	$\displaystyle\left(1-\frac{2Mr}{\Sigma}\right)dt^{2}+\frac{4aMr}{\Sigma}\sin^{2}\theta dtd\varphi-\frac{\Sigma}{\Delta}dr^{2}$		(1)
		$\displaystyle-\Sigma d\theta^{2}-\left[\left(r^{2}+a^{2}\right)\sin^{2}\theta+2\frac{Mr}{\Sigma}a^{2}\sin^{4}\theta\right]d\varphi^{2},$

where

	$\displaystyle a$	$\displaystyle\equiv J/M,$		(2a)
	$\displaystyle\Delta$	$\displaystyle\equiv r^{2}-2Mr+a^{2},$		(2b)
	$\displaystyle\Sigma$	$\displaystyle\equiv r^{2}+a^{2}\cos^{2}\theta.$		(2c)

The inner horizon $r_{-}$ and the outer horizon $r_{+}$ are respectively defined as

$$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}.$$

(3)

The variable $a$ is the angular momentum of unit BH mass. Throughout this paper, we use the dimensionless variable $a_{*}\equiv a/M$ and refer to it as the BH spin for brevity.

In this work, we consider a real scalar field surrounding a Kerr BH. Due to the small energy density of the condensate, its backreaction to the metric and self-interaction of the scalar field can be safely ignored Brito:2014wla . They can be included systematically as perturbations if high precision is required, which is beyond the scope of this work. Below we study a free real scalar field on the Kerr metric

$$\left(\nabla^{\rho}\nabla_{\rho}+\mu^{2}\right)\Phi=0,$$

(4)

where $\mu$ is the scalar mass. Generally, the energy eigenvalues are complex numbers depending on three indices, which are the overtone number $n$, azimuthal number $l$, and magnetic number $m$. The principal number $\bar{n}=n+l+1$ is also widely used instead of $n$. For eigenstate with indices $\{n,l,m\}$, we use $\omega_{nlm}$ and $\Gamma_{nlm}$ to denote the real and imaginary parts of the eigenvalue, respectively. For bounded scalar fields, the $\omega_{nlm}$ can be expressed as a series of $\alpha=M\mu$ Baumann:2019eav

$\displaystyle\omega_{nlm}$

$\displaystyle=\mu\Big{(}1-\frac{\alpha^{2}}{2\bar{n}^{2}}-\frac{\alpha^{4}}{8\bar{n}^{2}}+\frac{f_{\bar{n}l}}{\bar{n}^{3}}\alpha^{4}+\frac{h_{l}a_{*}m}{\bar{n}^{3}}\alpha^{5}+\cdots\Big{)},$

(5)

with

	$\displaystyle f_{\bar{n}l}$	$\displaystyle\equiv-\frac{6}{2l+1}+\frac{2}{\bar{n}},$		(6)
	$\displaystyle h_{l}$	$\displaystyle\equiv\frac{16}{2l(2l+1)(2l+2)}.$		(7)

The imaginary part $\Gamma_{nlm}$, which is also called the superradiance rate, has been calculated at the leading order in Ref. Detweiler:1980uk and has been recently improved to next-to-leading order (NLO) in Ref. Bao:2022hew . Both the expressions have compact forms while the latter is much more accurate compared with the numerical calculation in Ref. Dolan:2007mj . In this work, we use the NLO expression, which gives¹¹1This expression is rewritten from the result in Ref. Bao:2022hew .

$\displaystyle\begin{split}&\Gamma_{nlm}=\\ &\hskip 18.49411pt-\omega_{1}\left(4\kappa\sqrt{M^{2}-a^{2}}\right)^{2l^{\prime}+1}\frac{\Gamma(n+2l^{\prime}+2)}{n!}\frac{\sinh(2\pi p)}{2\pi}\\ &\times\frac{\left|\Gamma\left(l^{\prime}+1-ip+\sqrt{q-p^{2}}\right)\Gamma\left(l^{\prime}+1+ip+\sqrt{q-p^{2}}\right)\right|^{2}}{\left[\Gamma(2l^{\prime}+1)\Gamma(2l^{\prime}+2)\right]^{2}},\end{split}$

(8)

where $l^{\prime}\equiv l+\epsilon$, $p\equiv Mr_{+}(\omega_{nlm}-m\Omega_{\text{H}})/\sqrt{M^{2}-a^{2}}$, $\kappa\equiv\sqrt{\mu^{2}-\omega_{0}^{2}}$, $\Omega_{\mathrm{H}}\equiv a/(2Mr_{+})$, and

	$\displaystyle\epsilon$	$\displaystyle\equiv-\frac{8\alpha^{2}}{2l+1},$		(9a)
	$\displaystyle\begin{split}q&\equiv\frac{8Mr_{+}\omega_{0}(r_{+}\omega_{0}-mM\Omega_{\text{H}})}{r_{+}-r_{-}}\\ &\hskip 56.9055pt-\mu^{2}(r_{+}^{2}+a^{2})+4M^{2}(\mu^{2}-3\omega^{2}_{0}),\end{split}$			(9b)
	$\displaystyle\omega_{0}$	$\displaystyle\equiv\mu\sqrt{1-\frac{2\alpha^{2}}{\bar{n}^{2}+4\alpha^{2}+\bar{n}\sqrt{\bar{n}^{2}+8\alpha^{2}}}},$		(9c)
	$\displaystyle\omega_{1}$	$\displaystyle\equiv\frac{\mu^{2}-\omega_{0}^{2}}{\bar{n}\omega_{0}(1+4M^{2}(\omega_{0}^{2}-\mu^{2})/\bar{n})}.$		(9d)

Eq. (8) indicates that the superradiance condition $\Gamma_{nlm}>0$ is equivalent to $\omega<m\Omega_{\text{H}}$, from which one could derive the critical BH spin for each mode labeled by $\{n,l,m\}$

$\displaystyle a^{(nlm)}_{\mathrm{*c}}=\frac{4mM\omega_{nlm}}{m^{2}+\left(2M\omega_{nlm}\right)^{2}}\text{, for $M\omega_{nlm}\leq\frac{m}{2}$}.$

(10)

The superradiance of $\{n,l,m\}$ mode occurs only when the BH spin $a_{*}$ is greater than $a^{(nlm)}_{\mathrm{*c}}$.

The superradiance rate could be very different for different $\{n,l,m\}$ modes. In general, the most important modes are those with $l=m$ which is the smallest integer greater than $\omega_{nlm}/\Omega_{H}$. Among these modes, the superradiance rate is usually smaller with a larger value of $n$. This hierarchy is strict for $l=1$ and 2. For modes with $l\geq 3$, mode with larger $n$ may have a larger superradiance rate at some ranges of $M\mu$ and $a_{*}$, which is named “overtone mixing” in the literature Siemonsen:2019ebd . In this work, we first focus on the case in which modes with larger $n$ have smaller superradiance rates, obtaining analytical approximations of all important quantities. Then in the last section, we argue that these approximations also work in the presence of “overtone mixing”.

The evolution of the BH-condensate system includes the exchange of energy and angular momentum between the BH and the scalar field. The latter emits GW while rotating around the host BH. With accretion, the BH also absorbs energy from the accretion disk. In a more realistic scenario, the photons radiated by the accretion disk are important when the BH spin is very close to identity Thorne:1974ve . The evolution equations with all these effects are

	$\displaystyle\dot{M}$	$\displaystyle=\dot{M}_{\mathrm{acc}}\left(1+\frac{E_{\mathrm{rad}}^{\dagger}}{E_{\mathrm{ms}}^{\dagger}}\right)-\sum_{nlm}\dot{E}_{\mathrm{SR}}^{(nlm)},$		(11a)
	$\displaystyle\dot{J}$	$\displaystyle=\frac{\dot{M}_{\mathrm{acc}}}{E_{\mathrm{ms}}^{\dagger}}\left(J_{\mathrm{ms}}^{\dagger}+J_{\mathrm{rad}}^{\dagger}\right)-\sum_{nlm}\frac{m}{\omega_{nlm}}\dot{E}_{\mathrm{SR}}^{(nlm)},$		(11b)
	$\displaystyle\dot{M}_{\mathrm{s}}^{(nlm)}$	$\displaystyle=\dot{E}_{\mathrm{SR}}^{(nlm)}-\dot{E}_{\mathrm{GW}}^{(nlm)},$		(11c)
	$\displaystyle\dot{J}_{\mathrm{s}}^{(nlm)}$	$\displaystyle=\frac{m}{\omega_{nlm}}\left(\dot{E}_{\mathrm{SR}}^{(nlm)}-\dot{E}_{\mathrm{GW}}^{(nlm)}\right),$		(11d)

where $M_{s}^{(nlm)}$, $J_{s}^{(nlm)}$, $\dot{E}_{\mathrm{SR}}^{(nlm)}$ and $\dot{E}_{\mathrm{GW}}^{(nlm)}$ are the mass, angular momentum, superradiance rate and the GW emission rate of the $\{n,l,m\}$ mode, respectively. The $\dot{E}_{\mathrm{SR}}^{(nlm)}$ is related to the imaginary part of the eigenfrequency by $\dot{E}_{\mathrm{SR}}^{(nlm)}\equiv 2{M}_{\mathrm{s}}^{(nlm)}\Gamma_{nlm}$. The $\dot{M}_{\mathrm{acc}}$ is the mass absorption rate of the BH from the accretion disk. The $E_{\mathrm{ms}}^{\dagger}$ and $J_{\mathrm{ms}}^{\dagger}$ are the energy and angular momentum of unit mass in the last stable circular orbit, respectively. The accretion disk radiates photons, some of which are later captured by the BH. Its contribution to the BH mass and angular momentum are described by $E_{\mathrm{rad}}^{\dagger}$ and $J_{\mathrm{rad}}^{\dagger}$ for unit accreted mass, respectively, which are calculated in Refs. Page:1974he ; Thorne:1974ve . The expressions for $E_{\mathrm{ms}}^{\dagger}$, $J_{\mathrm{ms}}^{\dagger}$ and $\dot{E}_{\mathrm{GW}}^{(nlm)}$ are listed in App. A for convenience.

In obtaining Eqs. (11), we have made the following assumptions:

• •

  The backreactions of the condensate and the accretion disk on the metric, as well as the scalar self-interaction are ignored due to the low energy densities of the condensate and the disk Brito:2014wla .

• •

  The quasi-adiabatic approximation is employed, which assumes that the dynamical timescale of the BH is much shorter than the timescales of both the accretion and superradiant instability Brito:2014wla ; Brito:2017zvb .

• •

  The gravitational effect from other celestial bodies is neglected. The BH-condensate system with the accretion disk is effectively isolated.

These contributions can be added perturbatively which are beyond the scope of this work. In the rest of this section, we study the evolutions without either the superradiance or accretion. Then we move on to the case with all these effects from the next section.

If superradiance could happen without accretion, there are two widely separated scales: the superradiance time scale $\tau_{\text{SR}}$ and the GW emission time scale $\tau_{\text{GW}}$, with the latter much larger than the former. With accretion, there is a new time scale $\tau_{\text{acc}}$. Another time scale can be found by requiring the critical BH spin in Eq. (10) to be less than the limiting value 0.998, which gives a critical value of the BH mass

$\displaystyle M_{\text{de}}^{(nlm)}\approx\frac{0.47m}{\mu},$

(34)

where the subscript stands for “decaying”. Once the BH mass surpasses this critical value, the superradiant mode turns to a decaying mode. In this section, we consider the case with $M_{0}<M_{\text{de}}^{(011)}$. We refer to the time cost as the decaying time scale $\tau_{\text{de}}$. Using Eq. (19), one could estimate this decaying time scale as

$\displaystyle\tau_{\text{de}}=\tau_{\text{acc}}\log\frac{0.47m}{\alpha_{0}},$

(35)

where $\alpha_{0}=M_{0}\mu$. For the phenomenologically important cases, the value of $\alpha_{0}$ is roughly $0.01$ and the $\tau_{\text{de}}$ is in the same order as $\tau_{\text{acc}}$. Thus it is not considered as a separate time scale below.

The value of $\tau_{\text{acc}}$ can be adjusted by changing the accretion rate $f_{\text{Edd}}$ as in Eq. (18). There are five scenarios with this new scale, i,e, ${\tau}_{\mathrm{acc}}\ll{\tau}_{\mathrm{SR}}^{(011)}$, ${\tau}_{\mathrm{acc}}\sim{\tau}_{\mathrm{SR}}^{(011)}$, ${\tau}_{\mathrm{SR}}^{(011)}\ll{\tau}_{\mathrm{acc}}\ll{\tau}_{\mathrm{GW}}^{(011)}$, ${\tau}_{\mathrm{acc}}\sim{\tau}_{\mathrm{GW}}^{(011)}$ and ${\tau}_{\mathrm{acc}}\gg{\tau}_{\mathrm{GW}}^{(011)}$. In Fig. 2, we numerically solve Eqs. (11) for these five scenarios. The initial BH mass and spin are chosen as $M_{0}=10^{3}M_{\odot}$ and $a_{*0}=0.7$, respectively. The initial mass coupling is set as $M_{0}\mu=0.01$, which then gives the scalar mass $\mu\approx 1.34\times 10^{-15}\,\mathrm{eV}$. The superradiance and the GW emission time scales without accretion are ${\tau}_{\mathrm{SR}}^{(011)}\approx 6.01\times 10^{9}\,\mathrm{yr}$ and ${\tau}_{\mathrm{GW}}^{(011)}\approx 1.03\times 10^{22}\,\mathrm{yr}$. Their values are plotted in Fig. 2 as vertical lines. The case without accretion ($f_{\text{Edd}}=0$) is plotted in dashed curves for comparison. One can see that the BH-condensate system evolution is dramatically affected by a large accretion rate. Specifically, both the superradiance and the GW emission proceed much faster than the case with $f_{\text{Edd}}=0$. By observing the curves in Fig. 2, we further categorize the above five scenarios into three groups, which are studied in the next three subsections respectively.

On the other hand, if the superradiance could not happen initially, neither the superradiance nor the GW emission time scales exist. The BH evolution is controlled by the accretion until the extraction of the angular momentum is faster than the inwards flow due to the accretion. This case is studied in Sec. III.5.

The case of ${\tau}_{\mathrm{acc}}\ll{\tau}_{\mathrm{SR}}^{(011)}$ corresponds to the curves with ${f}_{\mathrm{Edd}}=1$ in Fig. 2. They are re-plotted with more details in Fig. 3. The evolution was split into four distinct phases in Ref. Guo:2022mpr . Below we briefly review these phases:²²2Here we rename the phases of those in Ref. Guo:2022mpr .

• •

  Spin-up phase. This phase corresponds to the interval $0<t<t_{0}^{\prime}$, where $t_{0}^{\prime}$ represents the time at which the BH spin reaches its first local maximum. During this phase, the BH evolves following the process described in Sec. II.2.1, while the scalar condensate is too small in mass to be important.

• •

  Spin-down phase. This phase spans $t_{0}^{\prime}<t<t_{1}$, where $t_{1}$ marks the time at which the BH spin reaches its minimum. As the BH mass and the condensate mass increase with time, the angular momentum extraction rate grows faster than exponentially. At some point, the extraction surpasses the accretion, resulting in a decrease of $a_{*}$ until it approaches the $\{0,1,1\}$ Regge trajectory determined by Eq. (10). This Regge trajectory is plotted as the green curve in Fig. 3(c).

• •

  Attractor phase. This phase extends from $t_{1}$ to $t_{3}$, where $t_{3}$ denotes the moment when the BH mass reaches $M_{\text{de}}^{(011)}$ given in Eq. (34). Throughout this phase, the BH evolves along the Regge trajectory of the $\{0,1,1\}$ mode, which acts as an attractor mathematically. Concurrently, as the condensate accumulates mass, its GW emission intensifies, and ${M}_{\mathrm{s}}^{(011)}/M$ reaches its maximum at $t_{2}$. Note that it does not coincide with the peak of the GW emission flux in panel (b), since the latter depends on both the mass ratio and the mass coupling, illustrated in Eq. (82).

• •

  Decaying phase. This phase lasts from $t_{3}$ until the condensate depletes. At $t_{3}$, the superradiance rate equals zero and the condensate becomes a decaying mode beyond $t_{3}$. Consequently, the scalar condensate quickly shrinks, returning its mass and angular momentum to the BH. For the single-mode example, $t_{3}$ is the time when the BH mass reaches its critical value $M_{\text{de}}^{(011)}$.

The important quantities in the numerical evolution can be estimated with simple analytic expressions at $\alpha\ll 1$ limit. To facilitate the analysis, we also solve the evolution equations without GW emission. The results are shown with orange dashed curves in Fig. 3. The exponential rise of the BH mass in Eq. (19) is also displayed as a thin blue line in Fig. 3 (d). For compactness, the quantities at a specific time are labeled with the corresponding subscripts, with subscript 0 reserved for the initial time, e.g. $\alpha_{0}\equiv M_{0}\mu$, $M_{1}\equiv M(t_{1})$ and $M_{\text{s,0}^{\prime}}\equiv M_{\text{s}}(t_{0}^{\prime})$.

The spin-up phase spans from $t=0$ to $t_{0}^{\prime}$, when the BH spin reaches the local maximum. In the case of ${\tau}_{\mathrm{acc}}\ll{\tau}_{\mathrm{SR}}^{(011)}$, this phase can be further divided into two segments, separated by the time $t_{0}^{\prime\prime}$ when the BH mass reaches $\sqrt{6}kM_{0}$. This time can be readily calculated

$\displaystyle t_{0}^{\prime\prime}=\tau_{\mathrm{acc}}\log\left(\sqrt{6}k\right).$

(36)

In the range from $t=0$ to $t_{0}^{\prime\prime}$, the BH mass grows according to Eq. (19) until $M=\sqrt{6}kM_{0}$, and the BH spin follows Eq. (13) up to $a_{*}\approx a_{*\lim}$. Ignoring GW emission and inserting Eq. (25) and Eq. (13) into Eq. (11c), one could calculate the condensate mass

$\displaystyle\begin{split}M^{(011)}_{\text{s}}=M^{(011)}_{\text{s0}}\exp\left\{{\alpha_{0}^{8}\mu\tau_{\text{acc}}}\left[g\left(\frac{M}{M_{0}}\right)-g\left(1\right)\right]\right\},\hskip 14.22636pt&\\ \text{for }1\leq\frac{M}{M_{0}}\leq{\sqrt{6}\,k},&\end{split}$

(37)

with

$\displaystyle\begin{split}g(x)\equiv&\ \frac{1}{36}\bigg{[}-\frac{2\alpha_{0}x^{9}}{3}+\frac{2}{7}\sqrt{6}kx^{7}\\ &+\frac{\sqrt{3}k}{35}\left(216k^{4}+36k^{2}x^{2}+5x^{4}\right)(9k^{2}-x^{2})^{3/2}\bigg{]}.\end{split}$

(38)

This estimate also applies to the case when $a_{*0}<a_{*\text{c}}^{(001)}$.

Subsequently, the BH spin remains at approximately $a_{*\lim}$ from $t_{0}^{\prime\prime}$ to $t_{0}^{\prime}$. As the condensate mass increases, the BH spin evolution is increasingly affected by superradiance, which results in $\dot{a}_{*}=0$ at $t_{0^{\prime}}$. In this time range, one could simplify $\Gamma_{011}$ in Eq. (25) by taking the leading order (LO) of the $\alpha$ and setting $a_{*}=1$. Then Eq. (11c) can be solved by ignoring the GW emission, arriving at

$\displaystyle\begin{split}M^{(011)}_{\text{s}}&=M_{\text{s,0}^{\prime\prime}}\exp\left[\frac{\mu\,\tau_{\text{acc}}}{192}\left(\alpha^{8}-\alpha_{0^{\prime\prime}}^{8}\right)\right],\end{split}$

(39)

where $\alpha_{0^{\prime\prime}}=\sqrt{6}k\alpha_{0}$ and $M_{\text{s,0}^{\prime\prime}}$ is determined by Eq. (37).

The spin-down phase starts at $t_{0}^{\prime}$ and ends at $t_{1}$, when the BH spin reaches the local minimum. In this time range, the accretion with photon capture first keeps the BH spin close to $a_{*\text{lim}}$ while both the BH mass and the scalar condensate mass increase with time. Eventually, close to $t_{1}$, the condensate is dense enough to quickly extract the energy and angular momentum from the BH, causing a rapid drop of the BH spin to $a_{*\text{c}}^{(011)}$. The GW is still irrelevant in this time range, as shown in Fig. 3 (a).

The evolution of the BH spin $a_{*}$ can be derived from Eqs. (11a) and (11b)

$\displaystyle\dot{a}_{*}=\frac{1}{\tau_{\text{acc}}}\left(\frac{J_{\mathrm{ms}}^{\dagger}}{ME_{\mathrm{ms}}^{\dagger}}-2a_{*}\right)-\frac{2M_{\text{s}}}{M}\left(\frac{1}{M\omega_{011}}-2a_{*}\right)\Gamma_{011},$

(40)

where the photon capture is ignored. From $t_{0}^{\prime\prime}$ to $t_{0}^{\prime}$, both terms on the right side are small, resulting in an almost constant $a_{*}$. From $t_{0}^{\prime}$ to $t_{1}$, the accretion term is still small, while the superradiance term grows quickly, leading to the fast drop of the BH spin. Thus one could ignore the accretion term in the time range between $t_{0}^{\prime\prime}$ and $t_{1}$. The BH mass $M$ can be estimated with Eq. (19) and the condensate mass continues to evolve according to the form given by Eq. (39). Then, using $\Gamma_{011}$ at the LO of the $\alpha$ and setting $a_{*}=1$, one rewrites Eq. (40) as

$\displaystyle\begin{split}\dot{a}_{*}=-&\frac{M_{\mathrm{s,0^{\prime\prime}}}^{(011)}}{24}M^{6}\mu^{8}\exp\left[\frac{\tau_{\text{acc}}}{192}(M^{8}-M^{8}_{0^{\prime\prime}})\mu^{9}\right].\end{split}$

(41)

Integrating both sides of Eq. (41) from $t_{0^{\prime\prime}}$, one obtains the following solution

$\displaystyle\begin{split}a_{*}(\alpha)&=a_{*0^{\prime\prime}}-\frac{1}{192}M^{(011)}_{\mathrm{s0^{\prime\prime}}}\tau_{\mathrm{acc}}\mu^{2}e^{-\frac{\alpha_{0^{\prime\prime}}^{8}\tau_{\mathrm{acc}}\mu}{192}}\\ &\,\times\left[\alpha_{0^{\prime\prime}}^{6}E_{\frac{1}{4}}\left(\frac{-\alpha_{0^{\prime\prime}}^{8}\tau_{\mathrm{acc}}\mu}{192}\right)-\alpha^{6}E_{\frac{1}{4}}\left(\frac{-\alpha^{8}\tau_{\mathrm{acc}}\mu}{192}\right)\right],\end{split}$

(42)

where $E_{\frac{1}{4}}(x)$ denotes the exponential integral function. At $t_{1}$, the BH spin decreases to a value close to $a_{\mathrm{*c}}^{(011)}\sim 4\alpha_{1}\ll a_{*\lim}$. Thus, one can approximate $a_{*1}-a_{*0^{\prime\prime}}$ by $-a_{*\lim}$, and perform the following expansion

$\displaystyle\begin{split}E_{\frac{1}{4}}\left(\frac{-\alpha_{1}^{8}\tau_{\mathrm{acc}}\mu}{192}\right)&=\\ e^{\frac{\alpha_{1}^{8}\tau_{\mathrm{acc}}\mu}{192}}&\left\{-\frac{192}{\alpha_{1}^{8}\tau_{\mathrm{acc}}\mu}+\mathcal{O}\left[\left(\frac{1}{\alpha_{1}^{8}\tau_{\mathrm{acc}}\mu}\right)^{2}\right]\right\}.\end{split}$

(43)

Finally, we arrive at an estimate for $\alpha_{1}$

$\displaystyle\alpha_{1}=\left\{\frac{-48}{\tau_{\text{acc}}\mu}W_{-1}\left[-\left(\frac{32\cdot 3^{3/4}\mu M_{\mathrm{s,0^{\prime\prime}}}^{(011)}\left(\tau_{\mathrm{acc}}\mu\right)^{1/4}}{\alpha_{0^{\prime\prime}}^{6}M_{\mathrm{s,0^{\prime\prime}}}^{(011)}\tau_{\mathrm{acc}}\mu^{2}E_{\frac{1}{4}}\left(\frac{-\alpha_{0^{\prime\prime}}^{8}\tau_{\mathrm{acc}}\mu}{192}\right)-192a_{*0^{\prime\prime}}e^{\frac{\alpha_{0^{\prime\prime}}^{8}\tau_{\mathrm{acc}}\mu}{192}}}\right)^{4}\right]\right\}^{1/8},$

(44)

where $W_{-1}(x)$ is the Lambert $W$ function. If $M_{\text{s},0^{\prime\prime}}^{(011)}\mu\ll\alpha_{0^{\prime\prime}}^{2}$, the term with the Exponential integral function is a small contribution and can be dropped. With the value of $\alpha_{1}$, the time $t_{1}$ can be obtained using Eq. (19) and the condensate mass is estimated with Eq. (39).

The attractor phase spans from $t_{1}$ to $t_{3}$, when the BH mass reaches the critical value $M_{\text{de}}^{(011)}$ defined in Eq. (34). In this time range, the BH follows the Regge trajectory closely. We further separate this phase into two ranges with $t_{2}$. The condensate mass to BH mass ratio first increases with time, reaches the maximum at $t_{2}$, and then drops gradually due to the GW emission.

From $t_{1}$ to $t_{2}$, the photon absorption from the accretion disk is unimportant since the BH spin is smaller than $0.97$. The GW radiation has a small effect on $M_{s}/M$ at $t_{2}$, which can be seen from the difference of the black and red dashed curves at $t_{2}$ in Fig. 3. Below we treat this effect as perturbation. At the zeroth order, the GW radiation is ignored and the BH spin evolves along the Regge trajectory given by Eq. (10). Inserting Eqs. (11a) and (11b) into the derivative of $a_{*\text{c}}^{(011)}$ with respect to $t$, one could then solve the superradiance flux order by order in $\alpha=M\mu$

$\displaystyle\begin{split}\dot{E}_{\mathrm{SR}}^{(011)}=\frac{\alpha M}{{\tau}_{\mathrm{acc}}}\Bigg{(}3\sqrt{\frac{3}{2}}-\frac{31\alpha}{2}+\frac{787\alpha^{2}}{8\sqrt{6}}-\frac{20305\alpha^{3}}{216}&\\ +\frac{1564549\alpha^{4}}{3456\sqrt{6}}+\mathcal{O}(\alpha^{5})&\Bigg{)}.\end{split}$

(45)

Ignoring GW emission and using Eq. (11a), one could integrate this equation directly

$\displaystyle\begin{split}{{M}_{\mathrm{s,Regge}}^{(011)}(\alpha)}\equiv\frac{\alpha^{2}}{\mu}\Bigg{(}\frac{3}{2}\sqrt{\frac{3}{2}}-\frac{2\alpha}{3}-\frac{473\alpha^{2}}{32\sqrt{6}}-\frac{223\alpha^{3}}{270}&\\ +\frac{999421\alpha^{4}}{20736\sqrt{6}}+\mathcal{O}(\alpha^{5})&\Bigg{)}.\end{split}$

(46)

With this expression, the LO condensate mass at time $t_{2}$ is

$\displaystyle{{M}_{\mathrm{s,2}}^{(011),\text{LO}}}={M}_{\mathrm{s,1}}^{(011)}+{M}_{\mathrm{s,Regge}}^{(011)}(\alpha_{2})-{M}_{\mathrm{s,Regge}}^{(011)}(\alpha_{1}).$

(47)

To obtain a more precise value of ${M}_{\mathrm{s,2}}^{(011)}$, we next add the GW emission effect perturbatively. Assuming that the BH mass increases exponentially, the mass correction can be estimated by

$\displaystyle\delta{{M}_{\mathrm{s,2}}^{(011)}}$

$\displaystyle=-\int^{t_{2}}_{t_{1}}\dot{E}_{\mathrm{GW}}^{(011)}(t)dt\approx-5.24\times 10^{-3}{{\tau}_{\mathrm{acc}}}\alpha_{2}^{16},$

(48)

where Eqs. (19) and (46) have been used. Then the condensate mass at $t_{2}$ is

$\displaystyle\begin{split}{{M}_{\mathrm{s,2}}^{(011)}}&={{M}_{\mathrm{s,2}}^{(011),\text{LO}}}+\delta{{M}_{\mathrm{s,2}}^{(011)}}.\end{split}$

(49)

To determine the value of $\alpha_{2}$, we make use of $d\left({M}_{\mathrm{s}}^{(011)}/M\right)/dt=0$ at $t_{2}$, which gives

$\displaystyle\left(1+\frac{{M}_{\mathrm{s,Regge}}^{(011)}(\alpha_{2})}{M_{2}}\right)\dot{E}_{\mathrm{SR}}^{(011)}-\dot{E}_{\mathrm{GW}}^{(011)}=\frac{{M}_{\mathrm{s,Regge}}^{(011)}(\alpha_{2})}{{\tau}_{\mathrm{acc}}}.$

(50)

Here we have assumed the condensate obtains most of its mass in the attractor phase, i.e., $M_{\text{s}}^{(011)}(\alpha_{2})\approx M_{\text{s,Regge}}^{(011)}(\alpha_{2})$. Inserting the expressions in Eqs. (45), (46) and (82) and assuming $\alpha_{2}\gg\alpha_{1}$, one finds the perturbation is invalid unless $\mu\tau_{\text{acc}}\sim\alpha_{2}^{14}$. Then the LO value of $\alpha_{2}$ is

$\displaystyle\alpha_{2}\approx 1.25\times\sqrt[14]{\frac{1}{\mu{\tau}_{\mathrm{acc}}}}.$

(51)

The value of $t_{2}$ can be estimated with Eq. (11a). Ignoring the photon-capturing and using the expression in Eq. (45), one arrives at

$\displaystyle\dot{M}\approx\frac{M}{{\tau}_{\mathrm{acc}}}-\dot{E}_{\mathrm{SR}}^{(011)}\approx\frac{M}{{\tau}_{\mathrm{acc}}}-3\sqrt{\frac{3}{2}}\frac{M^{2}\mu}{{\tau}_{\mathrm{acc}}},$

(52)

at the LO. This differential equation can be integrated directly, which gives

$\displaystyle M\approx\frac{2M_{1}e^{(t-t_{1})/\tau}}{2+3\sqrt{6}M_{1}\mu\left[e^{(t-t_{1})/\tau}-1\right]}.$

(53)

Then $t_{2}$ could be obtained as

$\displaystyle t_{2}\approx t_{1}+{\tau}_{\mathrm{acc}}\log\left[\frac{\alpha_{2}(2-3\sqrt{6}\,\alpha_{1})}{\alpha_{1}(2-3\sqrt{6}\,\alpha_{2})}\right].$

(54)

With the initial parameters used in Fig. 3, the estimates of ${M}_{\mathrm{s,2}}^{(011)}/M_{2}$, $\alpha_{2}$ and $t_{2}$ are compared to the numerical results in Tab. 1.