Termination of Superradiance from a Binary Companion

We start with the perturbed Hamiltonian $H=H_{0}+V_{*}$ of the boson. Its matrix element is given by

$$\langle\psi_{n^{\prime}l^{\prime}m^{\prime}}|H|\psi_{nlm}\rangle=\omega_{nlm}\delta_{n^{\prime}n}\delta_{l^{\prime}l}\delta_{m^{\prime}m}+\langle\psi_{n^{\prime}l^{\prime}m^{\prime}}|V_{*}|\psi_{nlm}\rangle~{}.$$

(16)

The diagonal terms are dominated by the eigenvalues of the free Hamiltonian while the off-diagonal terms are led by the gravitational level mixings due to the companion. The level mixings couple two states together whenever the selection rule allows. In particular, the superradiant states can be coupled to the absorptive states, bringing negative contributions to the imaginary part of the frequencies. Most dangerous of all absorptive states are the spherically symmetric $|\psi_{n00}\rangle$ states, since they decay the fastest. Hence we expect the leading order correction to the superradiant rate of $|\psi_{nlm}\rangle$ to come from its mixing with $|\psi_{n00}\rangle$ (if the selection rule allows). Therefore, we will focus on a two-state subspace $\{|1\rangle,|2\rangle\}$, where $|1\rangle$ denotes a superradiant state $|\psi_{nlm}\rangle$ and $|2\rangle$ denotes a highly absorptive state such as $|\psi_{n00}\rangle$. In matrix form, the perturbed Hamiltonian reads

$\displaystyle H=\left(\begin{matrix}\omega_{1}+V_{11}&V_{12}\\ V_{21}&\omega_{2}+V_{22}\end{matrix}\right)\equiv\left(\begin{matrix}\bar{E}_{1}+i\Gamma_{1}&\eta^{*}\\ \eta&\bar{E}_{2}+i\Gamma_{2}\end{matrix}\right)~{},$

(17)

where we have denoted $\eta\equiv V_{21}$, $\bar{\omega}_{i}\equiv\omega_{i}+V_{ii}$ and $\bar{E}_{i}\equiv E_{i}+V_{ii}$, $i=1,2$. Notice that both $\bar{E}_{i}(t)$ and $\eta(t)$ implicitly depend on time through the binary orbital motion. Thus in the adiabatic limit $|\frac{\dot{\eta}}{\bar{E}_{i}^{2}}|,|\frac{\dot{\bar{E}}_{i}}{\bar{E}_{i}^{2}}|\ll 1$, the state of the cloud $c_{i}(t)\equiv\langle\psi_{i}|\psi(t)\rangle$ can be solved using the WKB approximation,

$$c_{i}(t)=C_{i+}e^{-i\int\lambda_{+}dt}+C_{i-}e^{-i\int\lambda_{-}dt}~{},~{}i=1,2~{},$$

(18)

where $\lambda_{\pm}$ are the two instantaneous eigenvalues of the perturbed Hamiltonian (17),

	$\displaystyle\lambda_{\pm}\equiv$	$\displaystyle\frac{\bar{\omega}_{1}+\bar{\omega}_{2}}{2}\pm\sqrt{|\eta|^{2}+\left(\frac{\bar{\omega}_{1}-\bar{\omega}_{2}}{2}\right)^{2}}$
	$\displaystyle\simeq$	$\displaystyle\left\{\begin{array}[]{lr}\bar{E}_{1}+\frac{|\eta|^{2}}{\bar{E}_{1}-\bar{E}_{2}}+i\left[\Gamma_{1}-\frac{\Gamma_{1}-\Gamma_{2}}{(\bar{E}_{1}-\bar{E}_{2})^{2}}|\eta|^{2}\right]\text{, }+&\\ \\ \bar{E}_{2}+\frac{|\eta|^{2}}{\bar{E}_{2}-\bar{E}_{1}}+i\left[\Gamma_{2}-\frac{\Gamma_{2}-\Gamma_{1}}{(\bar{E}_{1}-\bar{E}_{2})^{2}}|\eta|^{2}\right]\text{, }-&\end{array}\right.$		(22)

The approximation is made under the limit $|\bar{E}_{1,2}|\gg|\eta|,|\Gamma_{1,2}|$, which is always valid given the superradiance context. Thus we see that $\lambda_{+}$ ($\lambda_{-}$) represents the corrected frequency of the quasi-stationary state $|1\rangle$ ($|2\rangle$). In particular, their imaginary parts are modified. Since we have assumed $\Gamma_{1}\geqslant 0$ and $\Gamma_{2}<0$, the effective superradiance rate is now suppressed:

$$\tilde{\Gamma}_{1}\equiv\Gamma_{1}+\Delta\Gamma_{1},~{}\Delta\Gamma_{1}\simeq-\frac{\Gamma_{1}-\Gamma_{2}}{(\bar{E}_{1}-\bar{E}_{2})^{2}}|\eta(R_{*})|^{2}<0~{}.$$

(23)

This suppression term in the effective superradiance rate mainly depends on the distance $R_{*}$ between the binary. If the absorptive state $|2\rangle$ happens to be a highly dangerous state such as $|\psi_{n00}\rangle$, the corresponding suppression can be significant if $R_{*}$ is not large. For instance, consider the case $n=3,l=m=2$. The fine-structure splitting is

$$E_{322}-E_{300}\simeq 0.04~{}\text{s}^{-1}\left(\frac{\alpha}{0.1}\right)^{5}\left(\frac{M}{10M_{\odot}}\right)^{-1}~{}.$$

(24)

The superradiance/absorption rates at maximal black hole spin ($\tilde{a}=1$) are

	$\displaystyle\Gamma_{322}$	$\displaystyle\simeq 3\times 10^{-13}~{}\text{s}^{-1}\left(\frac{\alpha}{0.1}\right)^{13}\left(\frac{M}{10M_{\odot}}\right)^{-1}~{},$		(25)
	$\displaystyle\Gamma_{300}$	$\displaystyle\simeq-3\times 10^{-3}~{}\text{s}^{-1}\left(\frac{\alpha}{0.1}\right)^{5}\left(\frac{M}{10M_{\odot}}\right)^{-1}~{}.$		(26)

Then at a binary separation $R_{*}=10^{5}M$, the size of the level mixing is

$$|\eta(R_{*})|\simeq 2\times 10^{-7}~{}\text{s}^{-1}\left(\frac{\alpha}{0.1}\right)^{-3}\frac{q}{0.2}\left(\frac{M}{10M_{\odot}}\right)^{-1}~{}.$$

(27)

This results in a correction to the superradiance rate

	$\displaystyle\Delta\Gamma_{322}\simeq$	$\displaystyle-7\times 10^{3}\frac{q^{2}}{\alpha^{10}}\frac{M^{5}}{R_{*}^{6}}$
	$\displaystyle\simeq$	$\displaystyle-0.6\times 10^{-13}~{}\text{s}^{-1}\left(\frac{\alpha}{0.1}\right)^{-10}\left(\frac{q}{0.2}\right)^{2}\left(\frac{M}{10M_{\odot}}\right)^{-1}~{},$		(28)

which already gives $\tilde{\Gamma}_{322}\sim\frac{4}{5}\Gamma_{322}$, $i.e.$, a reduction of $20\%$. As one expects from the quadrupole tidal perturbation, $\Delta\Gamma_{322}$ grows as $R_{*}^{-6}$ as the binary separation decreases. Therefore, inside a critical distance $R_{*}<R_{*,c}$ defined from the condition

$$\tilde{\Gamma}_{nlm}(R_{*,c})=\Gamma_{nlm}+\Delta\Gamma_{nlm}(R_{*,c})\equiv 0~{},$$

(29)

the effective superradiance rate becomes negative, and the mode $|\psi_{nlm}\rangle$ can no longer grow even at maximal black hole spin. For the $|\psi_{322}\rangle$ example, the critical distance is easily read out from (25) and (28):

$$R_{*,c}(322)\simeq 10^{6}~{}\text{km}\left(\frac{\alpha}{0.1}\right)^{-23/6}\left(\frac{q}{0.2}\right)^{1/3}\frac{M}{10M_{\odot}}~{}.$$

(30)

Before moving on to further discussions, we would like to make a few comments on suppressed superradiance and the critical distance.

1. $\bullet$

   First, it is interesting to see the interplay of UV and IR effects here. The imaginary part of the eigenvalues of the free Hamiltonian $H_{0}$ can be understood as an UV effect, which is relevant near the black hole horizon scale $r_{UV}\sim M$. Yet its correction involves IR effects which has nothing to do with the black hole geometry. These co-rotating bosons produced via superradiant scattering leak to an IR scale $r_{IR}\sim(\mu\alpha)^{-1}$, and are turned into non-/counter-rotating bosons by the tidal perturbation of the binary companion, which then get absorbed by the black hole again in the UV (see again FIG. 1).

2. $\bullet$

   Second, if the binary separation is shorter than the critical distance $R_{*,c}(nlm)$, there can be two scenarios. If the cloud has not been formed, yet the black hole spin appears to be capable of superradiance ($\Gamma_{nlm}>0$), superradiance will be shut off. No $|\psi_{nlm}\rangle$-cloud will be produced, and black hole spin cannot drop below the saturation value for modes with magnetic quantum number $m$. This is because, unlike Hawking radiation, superradiance is not spontaneous and can only amplify a given cloud state. But if such a cloud state is easily depleted, there is no seed particles left for amplification, hence no extraction of black hole spin. On the other hand, if the cloud has already been formed, it will decay slowly at the rate $|\tilde{\Gamma}_{nlm}|$. Like GCP resonances, this process generates backreaction to the binary orbit and can be observed. More discussions on this possibility will be elaborated in Sect. IV.2.

3. $\bullet$

   Third, the superradiance termination we are dealing with here is similar but not identical to resonant depletion scenarios considered in the literature [16, 25, 26]. Specifically, we need no requirement on the orbital frequency or orbit direction (co-rotating/counter-rotating). In fact, we will mostly stay away from the GCP resonance bands and analyze the viability of GCP resonances considering superradiance termination in Sect. IV.1.

The above calculations are performed under the assumption of adiabaticity. However, when the orbital frequency is high, the naive WKB approximation (18) breaks down and we need a better treatment of the perturbed Hamiltonian $H$. For simplicity, we will specialize to the case of equatorial circular orbits. The tidal moments thus reduce to

$$\mathcal{E}_{l_{*}m_{*}}=\frac{4\pi}{2l_{*}+1}Y_{l_{*}m_{*}}(\pi/2,\varphi_{*}(t))=e_{l_{*}m_{*}}e^{im_{*}\varphi_{*}(t)}~{},$$

(31)

where

$$e_{l_{*}m_{*}}\equiv\sqrt{\frac{4\pi}{2l_{*}+1}\frac{(l_{*}-m_{*})!}{(l_{*}+m_{*})!}}P_{l_{*}}^{m_{*}}(0)$$

(32)

and $P_{l_{*}}^{m_{*}}(x)$ is the associated Legendre polynomial. This $e^{im_{*}\varphi_{*}(t)}$ time dependence is inherited by the off-diagonal term $\eta(t)$. Therefore, we follow the standard procedure and perform a time-dependent unitary transformation to go into the co-rotating frame [32, 17, 22],

	$\displaystyle H_{D}=U(t)^{\dagger}(H(t)-i\partial_{t})U(t),$
	$\displaystyle\text{with }U(t)\equiv e^{-i\varphi_{*}(t)L_{z}}~{},$		(33)

where $L_{z}$ is the cloud angular momentum operator along the spin axis. In component form, the Hamiltonian in the co-rotating frame reads

$$H_{D}=\left(\begin{matrix}\bar{E}_{1}+i\Gamma_{1}-m_{1}\dot{\varphi}_{*}&|\eta|\\ |\eta|&\bar{E}_{2}+i\Gamma_{2}-m_{2}\dot{\varphi}_{*}\end{matrix}\right)~{}.$$

(34)

Notice that we have applied the selection rule (14) and set $m_{*}=m_{2}-m_{1}$. Now the fast oscillations in $H$ are eliminated and we are left with a co-rotating frame Hamiltonian $H_{D}$ that varies slowly in time only through $R_{*}(t)$. Going through the same procedure as before, we obtain the correction to the superradiance rate of $|1\rangle$:

$$\Delta\Gamma_{1}\simeq-\frac{\Gamma_{1}-\Gamma_{2}}{\left[\bar{E}_{1}-\bar{E}_{2}-(m_{1}-m_{2})\dot{\varphi}_{*}(R_{*})\right]^{2}}|\eta(R_{*})|^{2}~{}.$$

(35)

Clearly, in the adiabatic limit, $|\frac{\dot{\varphi}}{E_{i}}|\ll 1$, the above expression reduces to (23). The impact on superradiance becomes severe when the denominator vanishes. This corresponds to the resonant depletion scenario [16, 25, 26], where the cloud state $|1\rangle$ largely mixes into $|2\rangle$ and is quickly absorbed into the black hole. Since we focus more on the off-resonance scenario, the denominator will typically be dominated either by $\bar{E}_{1}-\bar{E}_{2}$ (the adiabatic case) or by $(m_{1}-m_{2})\dot{\varphi}$ (the diabatic case).

Since both the tidal perturbations and the instability rates are small compared to the energy levels, the two-state result (35) can be readily generalized to that of multiple states. To leading order in $|\eta_{ij}|$, the impact on superradiance is just a simple summation,

$$\Delta\Gamma_{1}\simeq-\sum_{i=n^{\prime}l^{\prime}m^{\prime}}\frac{\Gamma_{1}-\Gamma_{i}}{\left[\bar{E}_{1}-\bar{E}_{i}-(m_{1}-m_{i})\dot{\varphi}_{*}(R_{*})\right]^{2}}|\eta_{1i}(R_{*})|^{2}~{},$$

(36)

where $\eta_{1i}$ contains a sum of tidal moments $\mathcal{E}_{l_{*},m_{i}-m_{1}}$. The critical distance is again determined by requiring $\tilde{\Gamma}_{1}=\Gamma_{1}+\Delta\Gamma_{1}=0$. Due to the $R_{*}$ dependence in the denominator, there may be multiple solutions of $R_{*,c}$ when entering/exiting a resonance. We are more interested in the off-resonance solutions.

In FIG. 2, we have plotted the binary-separation-dependence of the effective superradiance rate $\tilde{\Gamma}_{nlm}(R_{*})$ for different states and black hole spins. It can be seen that the effective superradiance rate $\tilde{\Gamma}_{nlm}$ tends to the Detweiler value $\Gamma_{nlm}$ at large binary separations, since the tidal perturbations become negligible in this limit. As we decrease the binary separation, $\tilde{\Gamma}_{nlm}$ quickly drops to zero near a critical distance, where superradiance is terminated due to mixing into absorptive states. The sharp peaks and valleys are caused by GCP resonances, where the mixing effect is non-perturbatively large. Interestingly, when the black hole spin is high ($e.g.$, $\tilde{a}=1$), the high-$l$ states such as $|\psi_{433}\rangle$ and $|\psi_{544}\rangle$ receive positive enhancement to the superradiance rate first, before negative suppression terms take over and terminate superradiance as $R_{*}$ decreases. This is because of their mixings into low-$l$ states such as $|\psi_{411}\rangle$ and $|\psi_{522}\rangle$, which possess larger positive growth rates. Yet as the black hole spin drops below the threshold for superradiating these low-$l$ states, the superradiance rate $\tilde{\Gamma}_{nlm}$ for high-$l$ states can no longer benefit from the mixings, and thus monotonically drop to zero as $R_{*}$ decreases, as indicated by the dashed lines in FIG. 2.

The critical distance can be solved numerically from (29) in the case of multiple states beyond adiabaticity. As mentioned before, we focus on systems with an orbital frequency away from the GCP resonance bands, because such a configuration occupies the most proportion of the binary lifetime and are more typical statistically. Henceforth, it will be convenient to perform a “Wick” rotation and replace the denominator in (36) by

	$\displaystyle\frac{1}{\left[\bar{E}_{1}-\bar{E}_{i}-(m_{1}-m_{i})\dot{\varphi}_{*}(R_{*})\right]^{2}}$
	$\displaystyle\to\frac{1}{(\bar{E}_{1}-\bar{E}_{i})^{2}+[(m_{1}-m_{i})\dot{\varphi}_{*}(R_{*})]^{2}}~{}.$		(37)

This modification removes the resonance poles, but it keeps the off-resonance physics mostly unaltered. Since the tidal-perturbation-theory calculations are under the assumption of $R_{*}>r_{n}$, the resulting critical distance must be greater than the cloud radius,

$$R_{*,c}(nlm)>r_{n}=n^{2}r_{1}\text{ for consistency}~{}.$$

(38)

Under this constraint, we solve (29) for the critical distance and plot its dependence on $\alpha$ and $q$ in FIG. 3. Because we have chosen a maximal black hole spin, the resulting critical distance is the minimal requirement for cloud stability. This means superradiance of a given state $|\psi_{nlm}\rangle$ will be terminated completely below $R_{*}<R_{*,c}(nlm)$, and an existing cloud will decay quickly at a rate comparable to its original growth rate. From FIG. 3, we see that for fixed $\alpha$ and $M$ (hence fixed boson mass), the critical distance decreases with a smaller mass ratio. This is in agreement with intuition, since a lighter (heavier) binary companion is expected to have weaker (stronger) gravitational perturbations on the cloud dynamics, resulting in a broader (narrower) safe region. Cloud states with higher $l$ appear to be less stable in the presence of a binary companion, as their critical distances are larger. At last, we stress that as an off-resonance quantity, the critical distance solved in this way does not depend on the orbit orientation, which matters only for the triggering of a GCP resonance.

First, the termination of superradiance poses a constraint on observable GCP transitions. This is natural since GCP is based on resonant cloud transitions. If there is no cloud when the binary enters the resonance band, there is certainly no cloud transition, and no observable signal. Therefore, roughly speaking, in order to have successful GCP transitions, we must require that the binary separation at resonance to be greater than the critical distance of the initial cloud state²²2Notice that both sides of (39) depend on $\alpha$ and $q$. Thus this is effectively a constraint on the relative sizes of three mass parameters (namely, $\mu,M,M_{*}$) in the system.,

$$R_{*,r}(nlm\to n^{\prime}l^{\prime}m^{\prime})>R_{*,c}(nlm)~{}.$$

(39)

Otherwise, the cloud either cannot form, or would have been depleted via $\tilde{\Gamma}_{nlm}<0$ long before entering the resonance band.

Scanning through the parameter space, we plot the “safe” region for various GCP transitions in FIG. 4. It is clear from the plot that superradiance termination poses a tight bound on the parameters of the binary system. Bohr transitions with floating orbits are most severely constrained, because they happen at relatively high orbital frequencies. This means the binary separation at resonance is short, and the cloud is vulnerable to absorption. Hyperfine transitions, on the contrary, are influenced the least. In particular, we note that the $|\psi_{211}\rangle\to|\psi_{21~{}-1}\rangle$ and $|\psi_{322}\rangle\to|\psi_{300}\rangle$ transitions are safe to occur throughout the parameter space. This is because they happen at low orbital frequencies, where the overlap between the superradiant mode and absorptive modes is still small. Overall, except the hyperfine ones and Bohr transitions starting from the state $|\psi_{211}\rangle$, most transitions are forbidden for mass ratio $q\gtrsim 10^{-3}$. They are allowed only below a certain mass ratio, corresponding to a small mass of the binary companion.

Before entering the critical distance $R_{*,c}$, the cloud may have already grown up. Then the negative effective superradiance rate within $R_{*,c}$ will gradually deplete the cloud. The loss of angular-momentum-carrying particles backreacts on the orbit and produces an effective torque on the binary companion, leading to floating/sinking orbits in a manner similar to GCP transitions. The difference here is that such orbital backreaction does not occur at a fixed resonance band.

The total angular momentum of the cloud is

$$S_{c}(t)=S_{c,0}\left(m_{1}|c_{1}(t)|^{2}+\sum_{i\neq 1}m_{i}|c_{i}(t)|^{2}\right)~{},$$

(40)

where $S_{c,0}$ is the cloud angular momentum at the saturation of $|\psi_{1}\rangle$, and $0\leqslant|c_{1}|^{2}\leqslant 1$ is the percentage of cloud occupation. It effectively describes the boson particle number, $i.e.$, the larger $|c_{1}|^{2}$ is, there are more particles in the cloud. Since except the superradiant mode $|\psi_{1}\rangle$ with $m_{1}>0$, most other modes are usually unoccupied ($c_{i}\simeq 0$), we can approximate $S_{c}(t)\simeq S_{c,0}m_{1}|c_{1}(t)|^{2}$. In the case of a planar circular orbit, considering backreaction of the cloud evolution yields an orbital period derivative

$$\dot{P}=(\dot{P})_{\text{GR}}+(\dot{P})_{\text{C}}~{},$$

(41)

where $(\dot{P})_{\text{GR}}$ represents the usual orbital decay in General Relativity (GR),

$$(\dot{P})_{\text{GR}}=-\frac{96}{5}(2\pi)^{8/3}\frac{q}{(1+q)^{1/3}}M^{5/3}P^{-5/3}~{},$$

(42)

and $(\dot{P})_{\text{C}}$ is the extra contribution due to cloud backreaction [17, 19],

$$(\dot{P})_{\text{C}}=-3(2\pi)^{1/3}(1+q)^{-2/3}\frac{S_{c,0}m_{1}}{M^{2}}\frac{d|c_{1}(t)|^{2}}{dt}M^{1/3}P^{2/3}~{}.$$

(43)

The cloud evolution would have been unitary but for the absorption, which means $c_{1}(t)\propto e^{-\tilde{\Gamma}_{1}t}$ and

$$\frac{d|c_{1}(t)|^{2}}{dt}=-2\tilde{\Gamma}_{1}(R_{*})|c_{1}(t)|^{2}~{}.$$

(44)

For instance, using Kepler’s law to rewrite $R_{*}$ in terms of $P$, we can obtain the fractional correction to the period derivative for $|\psi_{322}\rangle$,

		$\displaystyle\frac{(\dot{P})_{\text{C}}}{(\dot{P})_{\text{GR}}}$
	$\displaystyle\simeq$	$\displaystyle-15|c_{322}|^{2}\left(\frac{\alpha}{0.1}\right)^{-9}\frac{q}{(1+q)^{7/3}}\left(\frac{M}{10M_{\odot}}\right)^{5/3}\left(\frac{P}{1\text{ hr}}\right)^{-5/3}.$		(45)

The minus sign shows that this is a floating orbit. Thus the correction to the orbital period derivative can be significant for certain parameter choices³³3Since the cloud has been decaying for some time after entering the critical distance, we generally expect $|c_{1}|^{2}$ to be a small number. In principle, assuming natural evolution, one can solve the whole history and determine $|c_{1}|^{2}$. Here, for simplicity, we will treat it as a free parameter.. Such corrections should be detectable via multi-messengers such as gravitational waves and pulsar timing [33, 34]. For instance, consider the gravitational atom in a pulsar-black hole binary, the periastron time shift is calculated by

$$\Delta_{P}\equiv t-P(0)\int_{0}^{t}\frac{dt^{\prime}}{P(t^{\prime})}\approx\frac{1}{2}\frac{\dot{P}}{P}t^{2}~{},$$

(46)

where we have Taylor expanded the orbital period to the linear order. In order to observe the extra periastron time shift caused by the backreaction of cloud absorption, we must require the deviation from the GR result to be larger than the timing error of pulse counting,

$$|\Delta_{P}-(\Delta_{P})_{\text{GR}}|>\sigma_{\Delta_{P}}~{}.$$

(47)

For a pulsar with rotation period $\tau$ and pulse width $w\lesssim\tau$, the timing error $\sigma_{\Delta_{P}}$ can be roughly estimated as [19]

$$\sigma_{\Delta_{P}}\sim\frac{1}{\sqrt{t/1\text{day}}}\frac{w}{t_{\text{obs}}/P}~{},$$

(48)

where $t_{obs}$ is the duration of a continuous observation window every day. In FIG. 5, we plot the timing accuracy for a pulsar of width $w=0.01$s and total observation time $T_{\text{obs}}$. We see that with enough observational time ($e.g.$, $T_{\text{obs}}\sim$ 1 decade), much parameter region can be uncovered, even when the cloud is extremely dilute ($e.g.$, $|c_{1}|^{2}\sim 10^{-6}$).

As mentioned before, boson clouds in isolated black holes only deplete by emitting gravitational waves whose frequency is peaked around the boson rest mass $\mu$. Thus the null detection of such near-monochromatic gravitational waves poses constraints on the mass of the ultralight boson, under certain assumptions on the initial black hole spin, cloud ages and the distance from earth. For instance, the LIGO search for continuous gravitational waves from the Milky Way center has reported constraints for the boson mass rage $10^{-13}$eV$<\mu<10^{-12}$eV [14] (see also [12, 35, 13, 15]). However, it is likely that spinning black holes are not isolated in the galactic center, where the density of objects is high. Therefore, in such a high-density and highly dynamical region, black hole superradiance may be influenced by the presence of a binary companion or other objects, possibly even suppressed to the extent of terminated cloud growth. Henceforth, the gravitational wave null detection bound could be relaxed. We have also seen in Sect. III.2 that states with higher $l$ are more severely affected by the tidal perturbation of nearby objects. Thus we expect the bound should be state-dependent, with cloud states of higher $l$ being constrained less.

Another boson mass bound comes from measuring the mass-spin distribution of black holes (the so-called Regge plane). Assuming efficient superradiance growth of the cloud, the black hole angular momentum will quickly be extracted, and its spin parameter is cut off at a value less than 1. This effect manifests itself as a gap on the black hole Regge plane, which can be tested statistically with LIGO and LISA [8, 9, 10]. Considering superradiance termination, such constraints are also subjected to relaxation if the black hole is in a binary. As mentioned in Sect. III.2, if the superradiance of a certain state is shut off by a close companion, the black hole spin cannot further decrease by producing that state. If, in addition, this superradiant state is the one with the lowest $l$ that the black hole spin is capable of producing, all other states will be shut off. For example, if $R_{*}<R_{*,c}(322)$, and $\tilde{a}<\frac{4\alpha}{1+4\alpha^{2}}$, both $|\psi_{322}\rangle$ and $|\psi_{211}\rangle$ cannot grow, and neither can the higher-$l$ modes (since they have even larger $R_{*,c}$). Then the black hole spin will not be cut off at the saturation value $\frac{\alpha}{1+\alpha^{2}}$ of the $|\psi_{322}\rangle$ state. Such considerations must be taken into account in the statistical analysis of the black hole Regge plane.

Other bounds come from, for example, testing the birefringence of photons [36] and the stellar kinematics [37] near supermassive black holes, with the help of powerful telescopes such as EHT [38, 39] and the Keck Observatory [40]. These bounds also assume the presence of cloud, and are subjected to a similar relaxation as mentioned here, since supermassive black holes are known to be surrounded by numerous stars. A detailed analysis of the impact of superradiance termination on various boson mass bounds is beyond the scoop of this paper, but it no doubt deserves further investigations in future works.