Dynamic Signatures of Black Hole Binaries with Superradiant Clouds

Home , Dynamical friction, Superradiance

Jun Zhang1, ∗ and Huan Yang2, 3, † 1Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2AZ, U.K. 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 3University of Guelph, Guelph, Ontario N2L 3G1, Canada Superradiant clouds may develop around a rotating black hole, if there is a bosonic ﬁeld with Compton wavelength comparable to the size of the black hole. In this paper, we investigate the eﬀects of the cloud on the orbits of nearby compact objects. In particular, we consider the dynamical friction and the backreaction due to level mixing. Under these interactions, the probability of a black hole dynamically capturing other compact objects, such as stellar mass black holes and neutron stars, is generally enhanced with the presence of the cloud. For extreme mass ratio inspirals and binary stellar mass binary black holes, the cloud-induced orbital modulation may be detected by observing the gravitational waveform using space borne gravitational wave detectors, such as LISA. Interestingly within certain range of boson Compton wavelength, the enhanced capture rate of stellar mass black holes could accelerate hierarchical mergers, with higher-generation merger product being more massive than the mass threshold predicted by supernova pair instability. These observational signatures provide promising ways of searching light bosons with gravitational waves.

I. INTRODUCTION [21–24]. The growth of the field can be very efficient if the Compton wavelength of the field is comparable to the The existence of light bosons have been motivated size of a rotating BH. More precisely, the growth rate of from various theoretical considerations. For example, a scalar field scales as [21] QCD axions are proposed as an extension of the stan- 4`+5 Γn`m ∝ (mΩH − ωn`m) α for α 1, (1) dard model which naturally solves the strong CP problem in particle physics [1,2]. It is also shown that light where α ≡ GMµ/~c is the ratio between the Compton bosons can form a Bose-Einstein condensate in galax- wavelength and the size of the BH, ΩH is the angular ies and could be a good candidate for dark matter [3– velocity of the BH, and ωn`m is the eigenfrequency of 9]. Moreover, light bosons are predicted in string theory a particular mode denoted by the “quantum” numbers [10, 11]. Scalar degrees of freedom could arise as prod- {n, `, m}. An eigenmode grows if mΩH > ωn`m, as neg- ucts of compactifying extra dimensions [12, 13], the mass ative energy flux falls into the black hole horizon. As a of which could be in a wide range with a lower bound result, the BH spins down. Eventually the mode instabil- possibly down to the Hubble scale. Therefore, the detec- ity saturates when mΩH ≈ ωn`m, with a long-live cloud tion of light bosons not only provides a smoking gun to rotating around the BH [25, 26]. new physics, but also has profound indications on string The superradiant cloud around a BH could lead to theory. The coupling between these light bosons and nor- many interesting observational effects. For instance, the mal matter is model dependent, but is weak in general, cloud can emit monochromic GWs [27–29], which may which makes the detection of such light bosons a difficult be observed from a single newly-formed BH right after task. Nevertheless, thanks to the equivalence principle, binary BH merger [30], or through coherent stacking a these light bosons at least couple to matter gravitation- set of events [31]. It may also be detected in all-sky ally, which provides a universal way for searching. searches [32, 33] or as stochastic background by GW de- Depending on their mass and coupling to the stan- tectors such as LIGO and LISA [34, 35]. The evolution dard model particles, light bosons could lead to different of the saturated cloud also has been studied in binary arXiv:1907.13582v3 [gr-qc] 24 Feb 2020 observable effects, including [10, 11, 14–20]. In particu- systems [36, 37]. If a superradiant cloud forms around a lar, the detection of gravitational waves (GWs) made by supermassive BH, it may affect the dynamics of compact LIGO and Virgo opened up an new observational win- objects orbiting around the BH, and leave fingerprints on dow. Together with space borne GW detectors, such as the waveform of GWs emitted by such objects, providing LISA (Laser Interferometer Space Antenna), GW obser- another approach to search for light bosons with GWs vations provide promising new ways of searching for light [38–40]. bosons. In fact, it has been suggested for a long time that Previous studies on cloud-induced orbital modulation a massive bosonic field near a rotating black hole (BH) usually assume fixed cloud density profile based on the may grow exponentially by extracting angular momen- mode wave function, so that the gravitational effects of tum from the BH, a process known as supperradiance the cloud may be obtained through multipole expansions [38, 39]. This assumption however neglects the cloud density perturbation generated by the gravitational interaction between the cloud and the orbiting object, as ∗Electronic address: [email protected] demonstrated in [36, 37, 40]. In particular, it has been †Electronic address: [email protected] shown in [40] that, gravitational tidal interaction could 2 deform the cloud in a way that, the backreaction lead cloud usually get dynamically excited after one pericen- to angular momentum transfer from the cloud to the or- tre passage, which leads to a change in the cloud energy biting object. For extreme-mass-ratio inspirals, such an- even if mode decay is neglected. In this case, the bo- gular momentum transfer can be sufficiently strong to son particles redistribute between different levels, and compensate the angular momentum loss caused by GW the energy change of the cloud should be proportional emission, if the object’s motion resonantly induces level to the energy gain/loss while shifting between different mixing of the cloud. As a result, the orbit stops decay- modes, i.e., ∆E ∼ α2µ∆c2, where ∆c is the change of ing and floats at a certain radius, emitting monochromic the amplitude of subdominant modes. Based on energy GWs for a long time until the cloud is depleted. Floating conservation, the orbital energy of the object should also orbits would not be possible if the cloud back reaction is change by the same amount, but with a negative sign. neglected. We shall refer to this process as dynamical level mixing. Given the equation of motion of the cloud in (20), In this work, we further investigate gravitational inter- we expect ∆c to be proportional to the strength of tidal actions between superradiant clouds and their surround- coupling. The overall effect of dynamical level mixing on ing compact objects, focusing on two effects: dynamical the orbits is proportional to the mass ratio between the friction and backreaction due to level mixing.1 When a orbiting object and the BH. It is sub-dominating in extra- compact object passes through a medium, the medium mass-ratio systems, but could have interesting impact on usually forms overdensity trail behind the object which comparable-mass binaries. exerts gravitational drag on the object, i.e., dynamical friction [41]. While dynamical friction generically occurs This paper is organized as follow. In Sec.II and for a compact object traveling through extended distribu- Sec.III, we will study the effects of dynamical friction and tion of matter, the magnitude of the drag force depends cloud level mixing respectively. In Sec.IV, we will focus on the properties of the component matter, such as the on extreme-mass-ratio-inspirals (EMRIs), which are im- sound speed of the sounding medium. In the case of a portant sources for LISA-like GW detectors. We will in- scalar field, the effects of dynamical friction depends on vestigate the effects of a superradiant cloud on the EMRI the mass of the field [9]. We will show that within some waveform as well as on the EMRI rate. In Sec.V, we will mass range the energy loss caused by dynamical friction discuss the effects of a superradiant cloud on stellar mass can be much larger than that caused by GW radiation, BH coalescence, and on the formation rate of stellar mass with the orbits significantly altered. BH binaries, which are the important sources for both space-based and ground-based GW detectors. Sec.VI is On the other hand, level mixing is a specific phe- devoted for discussion. nomenon associated with superradiant cloud [36, 37]. In almost all examples, we consider a scalar field of Generally speaking, a saturated cloud should be dom- mass µ developing a superradiance cloud around a BH of inated by a particular eigenmode, i.e. the mode that mass M. We use the subscription ∗ for quantities asso- grows fastest, while all modes evolve independently at ciated with the orbiting object, which could be a stellar the linear level. In the presence of an orbiting object, mass BH or a neutron star. For example, M∗ is the mass eigenmodes of cloud that evolve independently become of the orbiting object, and {r∗, θ∗, φ∗} are the position coupled under the tidal potential of the orbiting object, of the object in spherical coordinates. We sometimes re- and the wave function of the cloud starts oscillating be- fer the compact objects as stars, e.g., in Sec.IV. But tween different modes. If the object is in a quasi-circular one should keep in mind that they could be both stars orbit, the backreaction of level mixing “almost” vanishes and BHs. Unless specified, we assume natural units with after averaging over one orbital period. The extra sub- G = c = ~ = 1. tlety comes from the fact that the cloud will lose energy to the BH when a decay mode is excited during the Rabi oscillation, i.e. modes with mΩ < ω . Because of H n`m II. DYNAMICAL FRICTION this dissipation, backreaction on the orbits does not van- ish exactly, but is proportional to the decay rate of the decay modes [40]. Despite of the small decay rate, the As a compact object moves through the cloud, an over- backreaction on the orbits can still be significant, if the density trail forms behind it, which exerts a gravitational mass of the cloud is much larger than that of the orbit- drag on the object, i.e. dynamical friction. The friction ing object and the Rabi oscillation is resonantly excited. force can be written as [9, 42] We shall refer this process as resonant level mixing, as 2 2 studied in [40]. In a separate scenario, i.e., with highly 4πG M∗ ρ FDF = CΛ. (2) eccentric and hyperbolic orbits, different modes of the v2 In our case, v is the velocity of the object relative to the wave function of the cloud Ψ, ρ is the density of the 1 If the object is a stellar mass BH, its motion is also affected cloud, and CΛ = CΛ (ξ, krΛ) with ξ ≡ GM∗µ/~v, k ≡ by cloud accretion. However, as we will see later, the effect of µv/~, and rΛ representing the smaller quantity between accretion is negligible comparing to dynamical friction. for simplicity the size of the orbit and the size of the 3 √ cloud. In particular, it is calculated in [9] that where Jm = GMa is the angular momentum of the circular orbit and is also the maximal angular momentum πξ 2 Z 2krΛ e |Γ (1 − iξ)| 2 for a given . The energy loss during one orbital period CΛ ≡ dz |F(iξ, 1, iz)| 2ξ 0 T is z z Z T × − 2 − log , (3) FDF 3 2 krΛ 2krΛ ∆DF = vdt = qα xp IDF [xp, e, n,ˆ q] , (12) 0 M∗ where Γ is the gamma function and F is a confluent hy- pergeometric function. Following [9], the velocity of the wheren ˆ is the direction of the orbit angular momentum, and we have defined q ≡ M∗/M, α ≡ GMµ and field is defined as 2 xp ≡ α rp/GM with rp being the periapsis of the orbit. v ≡ ~ ∇Θ, (4) IDF [xp, e, n,ˆ q] is an integral given by µ Z 2π 4πC 1 − e cos z where Θ is the phase of the wave function, i.e., Ψ ∝ Λ 2 IDF ≡ dz 2 Rn` xp e−i(ωt−Θ). For fuzzy dark matter discussed in [9], the 0 (1 − e) 1 − e wave function has no angular dependence and the veloc- (1 − e cos z)3/2 × Y 2 (θ , φ ), (13) ity of the fluid vanishes. In the case of a superradiant 1/2 `m ∗ ∗ cloud, the wave function is rotating with a velocity along (1 + e cos z) the φ-direction, where Y`m is the spherical harmonic function and Rn` is m the radial function of the cloud (see App.A for the ex- v = ~ φ.ˆ (5) rµ sin θ plicit expression). We also defined xΛ ≡ krΛ, depending on xp, Considering a circular orbit, we have √ 3/2 r x (1−e cos z) , x ≤ x , |v| GM  p (1+e cos z)1/2 p 97 xΛ = 3/2 (14) ∼ 2 , (6) x97 (1−e cos z) √ 1/2 , xp > x97, v∗ rα  xp (1+e cos z) which means the relative velocity is dominated by the or- where x represents the size of the cloud and is chosen bital velocity if r > α−2GM. As most of the cloud mass 97 so that more than 97% of the cloud mass is within r < is in the region outside α−2GM, we will approximate the x α−2GM. In Fig.1, we show x2 I [x , e, nˆ] with e = velocity relative the wave function with the orbital ve- 97 p DF p 0 and 1, assuming the orbit lies in the equator. In the locity for simple. If we consider a stellar mass object following, we will neglect then ˆ dependence for simple. orbiting around a supermassive BH, we have We may expect the result would be changed by an O(1) M α factor after averaged overn ˆ. ξ = ∗ 1, (7) M v As a dissipation effect, dynamical friction accelerates the orbital decay. It would be intuitive to compare the in which case, energy loss caused by dynamical friction to that caused sin 2kr by GW radiation, the power of which averaged over one C (kr) = Cin(2kr) + − 1 + O(ξ) (8) Λ 2kr orbital period is [43] R z −7/2 with Cin(z) ≡ (1 − cos t)dt/t. 64π rp 0 P = ηf(e) , (15) We assume the orbits are Keplerian at the leading or- GW 5 GM der, in which case we neglect the relativistic corrections 2 as well as the gravitational effects of the cloud. For el- with η ≡ M∗M/(M∗ + M) being the dimensionless re- liptical orbits, we have duced mass and 2 1 + (73/24)e2 + (37/96)e4 a(1 − e ) f(e) = . (16) r = = a (1 − e cos z) , (9) 7/2 ∗ 1 + e cos ν (1 + e) r GM z − e sin z = (t − t ), (10) Together with Eq. (12), we find a3 0 −4 PDF 5 α 11/2 where a is the semi-major axis of the orbit, e is the eccen- = xp IDF [xp, e] , (17) tricity, and z is the eccentricity anomaly. It is convenient PGW 64π f(e) to define the specific energy and the specific angular mo- which means for orbits with radius comparable to the mentum of the orbit (here after energy and angular mo- size of the cloud, the energy loss could be enhanced by mentum): roughly a factor of α−4 if α 1. GM 1 GM p If the compact object is a stellar mass BH, it will = − v2 = ,J = J 1 − e2, (11) r 2 2a m continuously absorb the cloud, resulting in a force of 4

|211〉q≪1 |322〉q≪1 1.5 1.5 e=0 e=0

e=1 e=1 1.0 1.0 DF DF I I 2 2 p p x x

0.5 0.5

0.0 0.0 5 10 15 20 5 10 15 20 25 30

xp xp 2.0 1.5 |211〉q=1 |322〉q=1 e=0 e=0 1.5 e=1 1.0 e=1 DF DF I I

2 1.0 2 p p x x

0.5 0.5

0.0 0.0 0 5 10 15 20 5 10 15 20 25 30

xp xp

2 FIG. 1: The value of xp IDF [xp, e, nˆ] with e = 0 (blue) and 1 (orange, dotted), assuming an orbit lies in the equator. xp ≡ 2 α rp/GM with rp being the periapsis of the orbit. The left panels assume the cloud is saturated at |211i mode, while the right panels assume the cloud is saturated at |322i mode. The upper panels assume q = M∗/M 1, and the lower panels assume q = 1.

2 FAb = σAbρv , where σAb is the absorption cross section the object averaged over one orbit is effectively zero if and v is the relative velocity between the BH and the the modes do not decay. However, it is no longer true for cloud. The absorption cross section of a massive scalar hyperbolic orbits or highly eccentric orbits, in which case field with M∗µ 1 has been calculated in [44–46], the backreaction could be significant. When the object approaches the BH, the dominated mode of the cloud will 2 2 2 32π G M∗ α mix with the other modes. After one passage, the wave σAb ' . (18) v2 function of the cloud will be redistributed to each mode, Together with Eq. (2), we find that resulting in a change in the cloud’s energy, and hence in the orbital energy as well. In this section, we will in- F 8παv2 vestigate the backreaction of level mixing on orbits with Ab ' 1, (19) FDF CΛ eccentricity e ∼ 1. We will see that counter-rotating orbits always deposit energy to the cloud after one passage, which means that the force caused by absorption is neg- while a co-rotating orbit may gain energy from the cloud ligible comparing to dynamical friction. if the periapsis is close to the radius of the cloud, and will lose energy to the cloud if the periapsis is relatively far from the cloud. We will first discuss the gravitational III. LEVEL MIXING effects of the orbiting object on the cloud, and then estimate the energy change of orbits after one passage. A In the presence of a compact object, the gravitational similar calculation for tidal excitation of star oscillations potential of the object will introduce couplings between for arbitrary eccentricity orbits can be found in [47–49]. the modes of the cloud, which mix modes with different energy levels. The effects of level mixing on the superra- Throughout this paper we assume that the cloud size diant cloud have been studied in [36, 37], assuming the is much larger than the BH size, so that we can adopt a object is in a quasi-circular orbit or an elliptical orbit. In Newtonian approximation for the cloud. For full treat- particular, in [40] we have shown that a perturbed cloud ment in the relativistic setting, one can read [40] based backreacts on the orbit as well. In the adiabatic limit, the on techniques developed in [50–52]. Let us write the wave P backreaction on the quasi-circular orbits is proportional function of the cloud as |ψi = i ci(t) |ψii, where sub- to the decay rate of the decay modes, because the en- script i is a shorthand for n`m, and |ψii is the wave ergy/angular momentum transfer between the cloud and function of the eigenmode denoted by i. Initially, the 5 cloud is dominated by the saturated mode, say the mode usually have |1 − e| 1. Under this parametrization, it with i = s, and hence we have cs ' 1, while all the other would be convenient to solve ci in terms of z instead of ci ' 0. In the presence of an object, we have t. In this case, Eqs. (20) become

dci X dci i = hψi| V∗ |ψji cj, (20) i = A c (24) dt dz ij j j with where V∗ is the tidal perturbation generated by the ap- q proaching object. See App.B for more details. The inner 3 X 4π Aij = iq 2xp IΩIij (x∗) Y`∗m∗ (θ∗, 0) product is defined as an integral weighted by wave func- `∗ + 1 tions. `∗,m∗ h q i 3 3 It is important to note a few issues regarding × exp i ωij 2xp z + z /3 ∓ m∗φ∗ , (25) hψi| V∗ |ψji. First of all, the monopole and the dipole pieces of the tidal potential do not contribute to mode 2 where ωij = (ωi − ωj) /µα , and the − sign in front of coupling if the object is far away from the cloud [36]. m∗φ∗ is associated with co-rotating orbits, while the + It is because the monopole does not lead to a shift in sign is for counter-rotating orbits. For `∗ ≥ 2, we have the energy level, and the dipole is fictitious by virtue of Z x∗ `∗ the equivalence principle (see the appendix in [36] for ex- 2 x 2 Iij = dx x Rn ` (x)Rn ` (x) plicit calculations). Second, hψi| V∗ |ψji is Hermitian, x`∗+1 i i j j P 2 0 ∗ which implies i |ci| = 1. Moreover, the tidal cou- Z ∞ `∗ 2 x∗ pling is proportional to the perturber’s mass, so that + dx x Rn ` (x)Rn ` (x), (26) `∗+1 i i j j x x hψi| V∗ |ψji ∝ q. The conservation of wave function im- ∗ plies that the energy change of the cloud is of order and for `∗ = 1 and 0, we have 2 3 2 2 α Mc 2 α q M Z ∞ x3 ∆ ∼ ∆ij ∆ |ci| ∼ 2 |qcs∆t| ∝ 2 , (21) 2n 2n Iij = dx x∗ 1 − 3 Rniì (x)Rnj `j (x) (27) x∗ x∗ 2 which is suppressed by the factor q . Here ∆ij ∼ 2 2 and α Mc/2n is the energy difference between mode i and ∞ mode j, and Mc ' αM [34] is the mass of the cloud. Z I = dx xR (x)R (x) (28) Third, in computing Eq. (20) (or equivalently Eq. (25)), ij niì nj `j x∗ `∗ should be summed over all `∗ that subject to the se- lection rules |`j − ì| ≤ `∗ ≤ `j + ì [36]. The coupling respectively. Given the initial conditions cs = 1 while strength hψi| V∗ |ψji, however, is typically suppressed by other ci = 0 as z → −∞, we can solve ci at z → +∞ `∗+1 1/r∗ when the object is outside of the cloud. There- numerically. The energy change of the cloud after one fore, Eq. (20) is dominated by the modes near the satu- passage is given by rated mode in the ` space. Similarly, although the energy difference between levels of different n is roughly of the Mc X + 2 − 2 ∆ = ωi c − c , (29) 2 µ i i same order, i.e. ∆ωij ∼ α µ, so that all these modes i should be excited during the pericentre passage, the coupling strength decreases if the difference between n of two where the superscriptions + and − denote z approaches modes increases. Therefore, it is sufficient in practice to +∞ and −∞ respectively. only consider modes that are near the saturated mode In a simple scenario, we consider parabolic orbits that for different n’s. lie in the equator, in which case θ∗ = π/2 and φ∗ = In the following, we approximate hyperbolic orbits and 2 arctan z. Therefore the orbit is determined given the highly eccentric orbits with parabolical orbits, parameter xp, i.e. the dimensionless periapsis. We assume a mass ratio of q = 1 and a cloud initially satu- 2 ν r∗ = rp(1 + z ), z = tan , (22) rated at |211i or |322i, and solve Eqs. (24) numerically 2 taking into account all the modes with n ≤ 7. The en- s 3 ergy change of the cloud is plotted in Fig.2. Inclusion 2rp 1 3 t = z + z , (23) of higher n modes is computational expensive. On the G (M + M∗) 3 other hand, we are more interested in the range of xp where ν is the true anomaly. This is a good approxima- that are capable to capture the object. Including modes tion, as the orbits that are of interest in a capture process with n > 7 does not change this range very much as the energy change is exponentially suppressed at large xp, and is not necessary for the discussion in this paper. The net change of the cloud energy depends on the 2 Here we have neglected the mode decay caused by the non-zero modes that are efficiently excited. In other words, the but tiny imaginary eigenfrequency. cloud will lose energy if the excitation is dominated by 6

0.030 mode and a higher energy mode, we may expect the lower |211〉 energy modes will get eﬃciently excited when the typi- 0.025 co-rotating cal orbital frequency is larger or equivalently when xp is 0.020 counter-rotating small. * As shown in the upper panel of Fig.2, if the cloud 0.015 n≤7 Δ E / M initially saturates at mode |211i, a co-rotating orbit can

- 3 4 α n≤6 only make it couple to higher energy modes, therefore 0.010 n≤5 the object can only lose energy to the cloud. If the cloud 0.005 initially saturates at |322i mode, a co-rotating orbit can n≤4 couple it to lower energy modes, such as modes with 0.000 n = 1. As xp decreases, the excitation is dominated by 10 15 20 25 30 35 40 the n = 1 modes, which explains the valley in the lower xp panel of Fig.2. Here we assume modes with same n have 0.010 the same energy. In principle, they have different energy |322〉 0.008 due to the higher order corrections to the BH potential. co-rotating However, the energy difference between modes with same 0.006 2 counter-rotating n is further suppressed by a factor of α , which is negligi- * ble. Depending on the periapsis of the orbit, the modes 0.004 Δ E / M of the cloud get exited to different extent, resulting a - 3

α 0.002 diﬀerent amount of exchanging in total energy.

0.000

-0.002 IV. EFFECTS ON EMRIS 10 20 30 40 50 xp In this section, we discuss the observational effects of a superradiant cloud on EMRIs. For quasi-circular or- FIG. 2: The energy change of the cloud after one passage. We −2 bits, the observational effects of level mixing has been consider parabolic orbits with periapsis rp = xp α GM and studied in [40] (also see App.C for further discussion). θ = π/2, and choose the mass ratio q = 1. The upper panel ∗ For hyperbolic and highly eccentric orbits, which are of shows the case with a cloud initially saturates at mode |211i, while the lower panel shows the case with a cloud initially most interest in the estimation of EMRI rate, the effects saturates at mode |322i. We show both co-rotating orbits (in of level mixing is suppressed by the mass ratio q to be blue) and counter-rotating orbits (in orange) considering all negligible. Therefore, we will mainly focus on dynamical modes with n ≤ 7 (solid lines). We also show the results with friction in the following discussion. different number of modes taken into account (n ≤ 4, 5, 6 with dashed, dash-dotted, and dotted lines). We can see that the result converges as n increases. We mark I = 0.001 defined in Eq. (51) with grey horizontal lines in the plots. A. Inspiral waveform

As shown in Sec.II, dynamical friction accelerates orbit decay and introduces phase shift in the EMRI wave- modes with energy lower than the initially saturated form. In fact, dynamical friction can be much more effi- mode, and vice versa. On the other hand, according to cient in draining orbital energy as compared to GW ra- Eq. (20), a mode j responds to the tidal force efficiently if diation, when the orbiting object immerses in the cloud. (ωs − ωj)/(ms − mj) is comparable to the typical orbital However, LISA usually starts seeing EMRIs when the 3 frequency. In order to take into account both co-rotating pericentre distance is at least within 20 GM away from and counter-rotating orbits, we say the orbital frequency the BH, while the density peak of the cloud is much fur- is positive if the orbit is co-rotating and is negative if the ther sway. Therefore, it is unlikely to observe the effect orbit is counter-rotating. Therefore, a counter-rotating of dynamical friction in the waveform, unless the central orbit always rises the energy of the cloud, as there is no 5 BH is less massive than 10 M . lower energy level with mj > ms. For co-rotating orbits, For example, let consider a circular obit lying in the the cloud can lose energy, if the tidal force excites the equator, and compare the energy loss power caused by lower energy modes with mj < ms. As the energy dif- dynamical friction PDF to that caused by GW radiation ference between the saturated mode and a lower energy mode is usually larger than that between the saturated

4 Mode |211i can couple to lower energy modes with n = 1 only through the monopole and dipole of V∗, which are eﬀectively zero 3 Recall that hψi| V∗ |ψj i 6= 0 only if m∗ = mi − mj [36]. when the star is far away from the cloud. 7

B. EMRI rate 9 10 |211〉 α=0.1 Besides the waveform study for individual sources, an- 106 α=0.01 other important message we may receive from a popu-

GW 1000 lation of future EMRI observations is the EMRI merger / P rate, which encodes information of galactic-centre stellar DF P 1 distributions [53]. The theoretical EMRI rate can be estimated by virtue of loss cone dynamics, which will be 0.001 brieﬂy summarized as follows (also see [54] for a review on loss cone dynamics). 10-6 Assuming spherical symmetry, orbits around the cen- 10-8 10-6 10-4 10-2 tral BH can be speciﬁed by two parameters, i.e., energy GMΩ and angular momentum J. Taking into account gravitational encounters, the distribution of the orbits around a 109 |322〉 supermassive BH can be obtained by solving the Fokker- α=0.1 106 Planck equation. Given the orbit distribution, the rate of α=0.01 the stars5 falling into the BH can be found by calculating

GW 1000 the ﬂux of the stars into the loss cone, a regime deﬁned / P

DF in the two-parameter phase space and in which stars fall P 1 into the supermassive BH in one orbital time. In particular, orbits with higher magnitude of energy have short 0.001 periods,6 and stars in such orbits hardly penetrate beyond the loss-cone boundary before they are consumed 10-6 10-8 10-6 10-4 10-2 by the BH. This defines the empty-loss-cone, or the dif- GMΩ fusive regime of the phase space. On the other hand, it is possible for a star in a low energy orbit to diffuse across FIG. 3: The ratio between energy loss rate caused by dynam- the loss cone by gravitational encounters during a single ical friction and that caused by GW radiation. In the plot, orbital period. This defines the full-loss-cone, or pinhole we consider a cloud saturated at |211i mode (upper panel) regime of the phase space. It has been shown that both or |322i mode (lower panel). The horizontal axis shows the regimes contribute to the flux of stars falling into the BH. orbital frequency, scaled by the mass of the black hole. The However, a star falling into the BH does not necessarily blue/orange dashed line shows the orbital frequency corre- lead to an EMRI. To secure a successful EMRI, the star 2 −2 sponding to a radius of n α GM, which locates the density should avoid scattering during the orbital decay. That peak of the cloud. The shaded region shows the LISA fre- 5 means t0 < tJ , where t0 is the lifetime of the orbit, and quency band (0.1Hz - 0.001Hz), assuming a 10 M BH. The the t is the angular momentum relaxation time. The gray vertical line denotes the innermost stable circular orbit. J The observation band shifts toward the left for smaller BHs. criterion t0 < tJ indicates that the full loss cone regime barely contributes to the EMRIs. Stars in such regime are scattered multiple times each orbit. The probability of the same star falling into the BH in the corresponding [43] orbit lifetime, even in the presence of a large number of gravitational encounters, is effectively zero. The criterion 32 2 2 4 6 t0 < tJ defines a critical radius ac (or equivalently, a PGW ' − GM q r∗Ω , (30) 5 critical energy c), such that stars starting inspiral from an orbits with a < ac are able to fall into the central BH where we have used η ' q for M M∗. We plot the with high probability. Therefore, the inspiral rate can be dependence of PDF/PGW on the orbital frequency Ω in estimated as

Fig.3. Note that the ratio PDF/PGW does not depend Z ac on q. For a LISA-like GW detector, the effects of dy- ΓEMRI ' da F(a), (31) −5 namical frication may be detected if PDF/PGW ≥ 10 0 in the observation band, as the number of cycles staying where F(a) is the flux of stars into the loss cone per in band could be order 105. Taking M = 105M as a radius interval. Contributions to EMRIs are mainly from benchmark, we find that this requires α > 0.1. However, in such case the cloud might be depleted by the tidal perturbation of the earlier falling object as suggested in [36, 37] or the supperradiant spin-down, unless the BH 5 Here we refer all compact objects as stars for simplicity, but keep angular momentum is fed by accretion efficiently. The in mind that they can be neutron stars or stellar mass BHs. requirement of α > 0.1 can be relaxed for clouds around 6 Note that in our notation of energy, orbits of smaller semi-major intermediate mass BHs. axis have higher energy. 8 √ p 2 the diffusive regime, the orbit distribution in which can Here Jlc = 2rlc( − GM/rlc) ' 2GMrlc is the loss- be well approximated by the steady state distribution cone angular momentum corresponding to a loss cone obtained from the one-dimensional (angular momentum- radius rlc, which is about a few GM. As we mentioned J dependent) Fokker-Planck equation. In this case the star does not change too much for large a, and therefore J ∼ 6 flux into the loss cone is Jlc. As a benchmark, we take p = 0 and M = 10 M . In GW N (a) this case, we have ζGW ∼ 0.016 and ac = ζGWrh rh. F(a) = , (32) That means orbits leading to successful EMRIs usually ln (J /J ) t (a) m lc r have radius much smaller than the radius of influence. where√ N(a) is the number of stars within a, Jm(a) = Integrating Eq. (31) up to the critical radius gives GMa is the maximal (circular orbit) angular momen- GW 3/2−2p tum for a specific energy , and tr(a) is the angular mo- Nh ac ΓEMRI ∼ GW . (39) mentum relaxation time at radius a. In the following, th log [Jm(ac )/Jlc] rh we will first estimate the inspiral rate considering only GW radiation. We will mostly follow the estimation in Now let us discuss how dynamical friction affects the [55]. After that we will discuss how the inspiral rate is inspiral rate. We will show that dynamical friction does changed in the presence of a superradiant cloud. not affect the orbit with rp ' rlc. For α 1, the Without the cloud, the energy dissipation is caused cloud is expected to be far from the BH. The orbits by GW radiation (we neglect other dissipation effects for with rp ' rlc ∼ 10GM are affected by the cloud only simplicity), and the lifetime of the orbit is if e ' 1. Nevertheless, for these orbits the energy √ loss is dominated by GW radiation, as one can check Z ∞ d 2π GMa ∆ /∆ ∼ 0.06 α7I 1. On the other hand, tGW = ' , (33) DF GW DF 0 dynamical friction dominates GW dissipation if the peri- 0 d/dt ∆GW centre distance is comparable to the size of the cloud: where ∆GW is the energy loss caused by GW radiation −2 −2 rp ∼ α GM. In this case, orbits with rp ∼ α GM are in one orbital period. GWs also carry away angular mo- DF GW first circularized by the cloud in a timescale t0 t0 mentum. Generally, the changes of J during inspiral is and then continuously inspiraling into the MBH due to dominated by two-body scattering, and ∆JGW can be GW radiation. For a given orbit with semi-major axis a, neglected until a becomes very small. Given the stellar we have velocity dispersion of the host bulge σ, we can define the √ 2 radius of influence of the supermassive BH rh ≡ GM/σ . DF 2π GMa t0 ' . (40) It is convenient to refer the relaxation time to the relax- ∆DF ation time at the BH radius of influence, We still need to impose the condition that rp < −2 T (rh) (J/J )−10/(3−4p)aGW according to Eq. (37). That is the th = Ap q , (34) cl c Nh log Λ1 orbital life time is shorter than the angular momentum relaxation time, otherwise the orbit cannot complete the where T (rh) is the orbital period corresponding to inspiral stage even after the circularization. Here we keep rh, Nh is the number of stars within rh, and Λ1 = 1/4 the J-dependence in Eq. (36) as J is no longer J as in q−1 (2GM/r ) . In terms of t , the relaxation time lc h h the case of GW dissipation. By doing this, we take into at any radius a is given by account the contribution from orbits with small eccen- p a tricity. To summarize, in the presence of the cloud, an tr(a) = th , (35) rh orbit with radius a could complete inspiral if it can be circularized down to a radius of α−2GM in a timescale and the angular momentum relaxation time is DF −10/(3−4p) GW t0 tJ , and if rp < (J/Jlc) ac . Assuming 2 p J a xp ' 1, we can find that the former condition defines a tJ = th. (36) critical radius Jm(a) rh −10 2 3−2p 3−2p Simulations indicate that 0 ≤ p ≤ 0.25 [56–59], which DF J ∆DF GW ac = ac (41) means tr is roughly independent of radius a. In partic- Jlc ∆GW ular, we have Ap ' 0.2 for p = 0 [60]. Therefore, the 2 10 3−2p criterion for a successful EMRI becomes 3 × 2 IDF GW ' α ac . (42) 3 −p 85π GW 2 3 t0 a 2 −p = ζGW (J) < 1, (37) 6 DF 2/3 GW tJ rh For M = 10 M and p = 0, we have ac ' 0.6α ac , GW 2 which is usually smaller than ac . Therefore, the EMRI where we have used the fact that rp/GM = 8(J/Jlc) for rate is still decided by GW dissipation, and dynamical orbits with e ' 1, and have defined friction caused by the superradiant cloud should not al- 2 " √ !# 3−2p ter the EMRI rate significantly. This result is expected, J −5 85 GMt ζ (J) ≡ h . (38) because usually the EMRI rate is not determined by cap- GW 3/2 Jlc 10 3 × 2 rh ture mechanism, but by how efficient the loss cone gets 9

3/5 1/5 repopulated, where the later is still determined by grav- where M ≡ (MM∗) / (M + M∗) is the chirp mass, itational scattering even in the presence of the superra- and P is the dissipation power of the orbital energy. Us- diant cloud. ing P = PGW as a benchmark, we can write However, a superradiant cloud can affect the eccen- df 96 tricity distribution of EMRIs. During orbital decay, the = π8/3M5/3f 11/3 (1 + γ) , (48) evolution of the eccentricity e in terms of the semi-major dt 5 axis a is given by where γ is the fractional energy loss power caused by √ √ additional dissipation mechanisms other than the GW de 1 − e2 GM dL dE −1 1 − e2 = − + , (43) radiation, for example γDF = PDF/PGW for dynamical da e 2a5/2 dt dt 2ea friction. The lifetime of the orbit can be estimated by Z dL dE t 40 −11/3 8πMdf where dt and dt are the angular momentum and = (8πMf) . (49) the energy loss rate averaged over one orbital period. If M 3 1 + γ the orbital decay is dominated by GW radiation, we have In terms of observation, we are interested in the timescale de 1 1 − e2 208e2 + 37e4 that a GW signal sweeps over the observation band, given = . (44) the initial frequency f (or equivalently the initial sepa- da 2 ea 96 + 292e2 + 37e4 i ration of the binary). This timescale can be obtained by If the orbital decay is dominated by a friction F , we have integrating Eq. (49) from fi to fmax with fmax being the higher boundary of the observation band. In Fig.4, we dL 1 Z r3/2F show the timescales taking into account dynamical fric- = √ dθ (45) tion caused by the cloud. Note that the cloud could be dt 2π a 2a − r depleted due to the level mixing happened earlier when the BHs were in a lower frequency orbit. In this case, the and mass of the cloud could be suppressed at most by a factor r −4 dE 1 Z r2F 2GM GM of e for all α < 0.05 [36, 37]. Nonetheless, the cloud = √ − dθ, (46) is still able to accelerate the orbit decay significantly. In dt 2π 2 2 r a a 1 − e fact, we find that in most cases considered here the GW signals sweep through the observation band within one with r = a(1 − e2)/(1 + e cos θ), which generally results year, if 0.012 < α < 0.04 assuming no depletion of the in a different evolution of eccentricity as it is dominated cloud, or 0.016 < α < 0.036 assuming a depleted cloud. by GW radiation. The detection of continuous signals is sensitive to the accumulated phase, which can be estimated by [65]

V. EFFECTS ON STELLAR MASS BINARY Z ff 10 −8/3 8πMdf BLACK HOLE COALESCENCE Φ = (8πMf) . (50) fi 3 1 + γ

Stellar mass BH binary is one of the main sources Here fi and ff are the initial and final frequency of the for future spaced-based and ground-based GW detectors, observation. In Fig.5, we calculated the phase difference and some of them will be detected by both types of detec- between the case with and without dynamical friction. tors to allow multiband analysis [61–64]. In this section, We assume one year integration time, and a binary of we would like to investigate the effects of a superradiant two 30M BHs spiraling in a circular orbit. We can see cloud on inspiral wavefom of stellar mass BH binaries, as that the phase difference caused by the cloud can be as well as on the binary formation rate. large as 104 when the dynamical friction is much stronger than the GW radiation reaction. Order O(1) phase shift is likely detectable by LISA parameter estimation proce- A. Inspiral wavefom dures. In principle, level mixing also affects the inspiral waveform. However, for stellar mass binaries, the effects of Let us consider a binary of two 30M BHs, inspiraling in a circular orbit. The separation of the binary that level mixing is either subdominated comparing to dy- corresponds to LISA’s most sensitive observation band namical friction or happening outside the observation 7 (10−3Hz - 0.1 Hz) is from 2 × 104GM to 103GM. As a band. Depending on the orbit frequency, level mixing result, LISA may observe rapid orbit decay if the cloud can be dominated by Bohr resonance or hyperfine res- radius is within such range, for example when α ∼ 0.01. onance [36, 37]. If the transient resonance timescale is At leading order, the chirp of GWs emitted by the orbit is

7 df 3 −1 Note that this is not true for EMRIs, in which case eﬀects of level = − M P, (47) −2 dt π mixing is enhanced by η with η being the small mass ratio. 10

-2.0 4 2 104 No cloud 1000 α= 0.005 0 -2.2 100 α= 0.01 10

t / year α= 0.015 1 -2.4 i f

0.10 10 Log 0.01 -2.6 0.001 0.005 0.010 0.050 0.100

fi /Hz

-2.8 104 No cloud 1000 α= 0.005 100 -3.0 3 1 α= 0.01 0.04 0.05 0.06 0.07 0.08 0.09 0.10 10 α t / year α= 0.015 1 -2.0 4 2

0.10 0

0.01 -2.2

0.001 0.005 0.010 0.050 0.100

fi /Hz -2.4 i FIG. 4: The timescales that the GW signal sweeps over the f observation band of a LISA-like GW detector, i.e. 0.001Hz − 10 0.1Hz. We assumed a binary of two 30M BHs, and show Log -2.6 the timescales with different initial frequency fi, and different α (solid lines). We also show the case with no cloud (gray dashed lines) for comparison. In the upper panel, we assume no depletion of the cloud. In the lower panel, we assume the -2.8 mass of the cloud is suppressed by a factor of e−4 due to a previous depletion [36, 37]. 3 -3.0 1 0.04 0.05 0.06 0.07 0.08 0.09 0.10 longer than the observation time, one can use similar α argument as in the dynamical friction case to estimate the phase modulation due to a continuous tidal force by FIG. 5: The phase difference ∆Φ between the case with and mode mixing. On the other hand, if the transient reso- without dynamical friction, assuming a binary of two 30M nance phase is much shorter than the observation time, BHs and one year integration time. In the gray region, the the resonance essentially introduces a “kick” in the or- signal will sweep the observation band within one year tak- bital energy (see similar discussion in [66–68]) that affects ing into account the dynamical friction. The contours are the accumulated phase later on. labeled by log10 ∆Φ. In the lower panel, we assume the mass −4 Bohr resonance usually happens when the orbit size of the cloud is suppressed by a factor of e due to a previous depletion [36, 37]. is comparable to that of the cloud, in which case the energy loss is usually dominated by dynamical friction. In particular, the resonance condition is that the orbital frequency Ω ' ∆ω/∆m ∼ µα2/∆m, which corresponds to an orbital separation of 21/3∆m2/3α−2GM. 4`−1 In this case, PLM/PDF ∼ α 1. Comparing to Bohr resonance, hyperfine resonance could happen further away from the cloud, i.e. with a separation of 1/3 2/3 −4 2 ∆m α GM. In this case, we have PLM/PDF ∼ However, the corresponding frequency is usually outside 4`+3 1/3 2/3 −2 α IDF(x ∼ ∆2 m α ). As IDF decreases expo- LISA’s observation band for such a small α. For these nentially as x increases, we may expect that for small α, reasons, we will not consider the effects of level mixing level mixing will eventually dominate dynamical friction. on the waveform. 11

B. Binary formation rate the merger timescale. The lifetime of a highly eccentric (e ' 1) orbit (due to GW emission) is given by [43] Binary BHs may be born out of star binaries, or 3 a4 through dynamical multibody interactions in dense stel- 0 2 7/2 t0 ' 3 (1 − e0) , (56) lar environments, e.g., galactic nuclei where the num- 85 Mtotη 10 −3 ber density can exceed 10 pc [69] in the inner region while the typical timescale for an encounter to disrupt and globular clusters where the number density is about the binary is [69] 105pc−3 [70]. In the LIGO O3 run new binary BH merger events are roughly detected in a weekly basis, where for 1 v3 t ' ' , (57) third-generation detectors the event rate could be once e 12πa2n v 12πM 2 n per a few minutes [71]. With more detections it will 0 BH tot BH be interesting to understand different possible formation where nBH is the number density of the BHs, and v is the channels. In the following, we will show that, in a cer- typical velocity of the binary. If the energy dissipation tain range of velocity, the presence of superradiant clouds is dominated by dynamical friction or level mixing, we leads to a larger impact parameter for gravitational cap- have ture and hence a higher dynamical formation rate for a M η binaries. It is particularly interesting if one or both BHs 0 ' tot α−3I−1, (58) in the binary is(are) already product of previous merg- Mtot 2M ers, because hierarchical mergers may be able to produce and BHs heavier than the upper bound predicted by supernova pair instability [72–77]. 2 4M 1 − e0 ' αIxp. (59) It would be convenient to write the loss of the orbit Mtotη energy after one pericenter passage as Therefore, we have 3 ∆ = α I, (51) 7/2 −3 t0 −3 −1/2 xp v = 2 × 10 I −1 where we approximately take I > 0.01 for 1 < xp < 10 te 10 100kms −17/2 3 due to dynamical friction (see Fig.1), and I > 0.001 for α M nBH 10 < x < 20 due to level mixing (see Fig.2). The × 8 −3 , (60) p 0.1 30M 10 pc angular momentum change caused by the cloud satisfies ∆J J and will be neglected in the following discussion. which means t0 te if the BH number density is Let us consider two BHs of comparable masses aiming not extreme large. For example, we can find that for −1 at each other with an impact parameter b and relative v = 100kms , in order for t0 ∼ te, the number den- 10 −3 velocity v∞. In this case, the closest distance is sity of BHs must be nBH ≥ 2 × 10 pc for α = 0.1 or 8 −3 nBH ≥ 8×10 pc for α = 0.05. As long as the BH num- −1 s 2 ! ber density is below the corresponding value, the rate of 1 Mtot Mtot rp = 2 + 4 4 + 2 2 (52) merger is roughly the rate of binary formation. b b v∞ b v∞ For simplicity, we simplify the mass distribution of BHs 2 2 2 4 b v∞ b v∞ by assuming all BHs have the same mass, in which case ' 1 − 2 . (53) the binary formation rate can be estimated as 2Mtot 4Mtot Z 2 2 2 2 These two BHs will form a binary if they release enough ΓBF = nBH 4π bmaxv∞r dr, (61) 2 energy during one passage, i.e. ∆ > v∞/2. For ∆ v2 /2, the semi-major axis of new formed binary can be ∞ where bmax is the maximum impact parameter for two estimated by BHs with relative velocity v∞ to be bound. For instance, β 2 in the galactic nuclei, we have nBH ∝ r , where r is the Mtotη p a ' , (54) distance to the galaxy centre, and v∞ ∼ GMSMBH/r 0 2∆M BH with MSMBH being the mass of the supermassive BH in the centre of the galaxy. For globular clusters, v is and the eccentricity is ∞ given by the typical velocity dispersion. If we are only s interested in binaries that lead to successful mergers, the 2 2 s ∆MBHb v∞ ∆MBHrp integral (61) should have a lower cutoff at r such that e0 ' 1 − 2 3 ' 1 − 4 2 , (55) Mtotη Mtotη t0 ' te. This cutoff may be relevant for binary formation in galactic nuclei, where the BH number density in the 2 2 where we have used that rp ' b v∞/2Mtot. As ∆ 1, inner region could be too large for new formed binaries to we have e0 ' 1. In addition, a successful merger requires survive. On the other hand, it may not be a problem in that the newly formed binary should not be disrupted by globular clusters where the BH number density is usually further gravitational encounter with a third body within much lower. 12

Without superradiant cloud, BHs can form binaries BHs. The estimation can also be improved by consid- by GW dissipation, which defines a maximum impact ering a more accurate velocity distribution instead of p parameter [69] v∞ ∼ GMSMBH/r. A detailed modeling is beyond the scope of this paper, and will be left for future works. 340π 1/7 b = η1/7M v−9/7. (62) GW 3 tot ∞ VI. DISCUSSION Similarly, we can also find the maximum impact parameter if the dissipation is caused by the cloud Superradiant cloud may develop around a rotating (as- s trophysical) BH, if the Compton wavelength of the boson 2α−2x M p tot field is comparable to the size of the BH. In this paper, bcloud ' 2 . (63) v∞M we investigate the effects of the superradiant cloud of a scalar field on the orbits of nearby compact objects, fo- As the dissipation caused by cloud becomes significant cusing on dynamical friction and backreaction of cloud only when the periapsis is comparable to the size of the −1 level mixing. We compute the dynamical friction caused cloud, we can choose xp ∼ 10 for v∞ < 500kms with by the cloud, see Eq. (12). Depending on the mass of α ranged from 0.01 to 0.1. We see that the maximum the scalar field, the dynamical friction dissipation can impact parameter becomes larger in the presence of su- DF GW be larger than the dissipation caused by GW radiation. perradiant clouds. For example, we have bmax = 10 bmax −1 Therefore the dynamical friction dissipation is able to for α = 0.01 and v∞ = 100kms . accelerate orbit decay, leaving fingerprints in the GW Beside the formation dynamics, binary formation rate waveform. For EMRIs around super-massive BHs, it is also depends on the environments, such as the BH dis- not very likely to detect the effects of dynamical friction tribution in the galactic nuclei. Therefore, we shall in the EMRI waveform, as a LISA-like detector usually compare the formation rate in the presence of cloud see inspirals with separations much less than the size of with that associated with GW radiation, by computing cloud GW the cloud and the effects of dynamical friction is not max- γ ≡ ΓBF /ΓBF . In galactic nuclei, integrating Eq. (61) imized in the observation band. The effects are much with Eq. (62) and Eq. (63), gives more significant on stellar mass BH inspirals. For example, with 0.016 < α < 0.036, GW signals from a binary −2 2/7 xp α MSMBH of two 30M BHs can sweep over the whole LISA band γGN ' 56 Cβ 6 , (64) 10 0.01 10 M within one year, which usually takes 104 year in the ab- sence of the cloud. For α > 0.036, the cloud could also where introduce a large phase difference that is up to 104 assum- R xmax 2β+ 5 ing one year observation. However, for EMRIs around cloud dx x 2 xmin Cβ = (65) intermedia mass BHs, there could be interesting GW sig- R xmax 2β+ 39 GW dx x 14 nals that can be observed by LISA [40]. In addition to xmin dynamical friction, when a compact object approaches is an order one coefficient depends on the slope of the BH the cloud, it could dynamically excite the eigenmodes of number density β. For instance, we have Cβ ' 0.7 for the cloud. In most cases, the object will lose energy to β = −2 [78], and Cβ ' 1.6 for β = −7/4 [79]. For binary the cloud due to the tidal interaction. We calculate the formation in globular clusters, both number density nBH energy loss during one passage by considering a parabolic and v∞ are smaller than that in galactic nuclei, therefore orbit, see Fig.2. we have In both cases, we expect that the dissipation caused by b2 the cloud will facilitate the cloudy BH capturing other γ = cloud objects, such as BHs and neutron stars. We estimate GC b2 GW the effects of the superradiant cloud on the EMRI rate, x α −2 v 4/7 and find the EMRI rate does not change very much in ' 213 p ∞ . (66) 10 0.01 60kms−1 the presence of the cloud. It is because the EMRI rate is mainly determined by repopulation mechanism of the Eq. (64) and Eq. (66) indicate that, in the presence of loss cone. We also investigate the formation rate of stellar superradiant cloud, the binary formation rate will be sig- mass BH binaries. Assuming a single mass distribution of nificantly enhanced if α < 0.1. BHs, we find the formation rate is enhanced in the pres- In the above estimation, we have assumed that all BHs ence of superradiant cloud, if α < 0.1. All of these effects have the same mass. In the realistic case, we may expect provide promising ways of searching for light bosons. BH mass ranges from ∼ 5M to ∼ 50M . Given the A large binary formation rate could lead to interesting mass of the scalar field, BHs of different mass should observational signatures. One possible consequence is hi- be surrounded with cloud of different α. According erarchical mergers, i.e. mergers with one BH being the to Eq. (64) and Eq. (66), we may expect that small remnant of a previous merger. If there is a scalar field of BHs are more like to form binaries comparing to large mass 10−11eV, we would have cloud developing around 13 stellar mass BHs. The growth time of superradiant cloud the action of ψ is with α 1 can be estimated as [21] 1 R 4 √ h ∗ a 2 00 ∗ S = − d x −g ∂aψ ∂ ψ + µ (g + 1)ψ ψ τ ' 24(a/M)−1α−(4`+5) GM/c3 , (67) 2µ 0a ∗ ∗ i +iµg (ψ ∂aψ − ψ∂aψ ) . (A3) where a is the spin of the BH. For 30M BHs, the 8 growth time for ` = 1 mode is about 10 years if −1 2 α = 0.01 or equivalently if there is a scalar field of mass Keeping only terms up to first order in r and α , we µ = 4.5 × 10−11eV. Obviously in order to be astrophys- can obtain the effective action, ically relevant, according to Eq. (67) α cannot be much Z 3 ∗ 1 ∗ α ∗ smaller than 0.01, or the growth timescale will be longer S2 = dt d r ψ ∂tψ − ∇ψ ∇ψ + ψ ψ , (A4) 2µ r than the Hubble time. With α ≥ 0.01 the cloud may develop around the remnant BHs of previous mergers and which leads to the Schrödingerequation enhance the formation rate of binaries with higher generation BHs. ∂ 1 α i ψ(t, r) = − ∇2 − ψ(t, r) . (A5) On the other hand, such rate enhancement mechanism ∂t 2µ r may play a role in the growth of supermassive BHs, e.g. QSO SDSS 1148+5251 found at z = 6.43 and with mass The stationary eigenmode of cloud is denoted by |n`mi 9 ∼ 10 M [80]. It remains an open question that how with the wave function do the supermassive BHs grow out of their much lighter −i(ω−µ)t seeds. Gas accretion may not be able to support suffi- ψn`m = e Rn` (x) Y`m(θ, φ), (A6) ciently fast mass growth [81], so that BH merger is ex- 2 pected to play a role. The enhanced capture rate with where x = α r/GM, Y`m is the spherical harmonic func- scalar cloud may allow fast growth of the host BH across tion, and one order of magnitude. " 3 #1/2 ` 2 (n − ` − 1)! − x 2x R (x) = e n n` n 2n(n + 1)! n Acknowledgments 2`+1 2x ×Ln−`+1 (A7) We thank Sam Dolan for helping with the accretion n cross section of a BH in scalar cloud. J.Z. is supported by with L2`+1 [x] being the generalized Laguerre polyno- European Union’s Horizon 2020 Research Council grant n−`+1 mial of degree n − ` − 1. 724659 MassiveCosmo ERC-2016-COG. H.Y. acknowl- edges support from the Natural Sciences and Engineering Research Council of Canada, and in part by the Perime- Appendix B: Tidal perturbation ter Institute for Theoretical Physics. Research at Perime- ter Institute is supported by the Government of Canada through the Department of Innovation, Science and Eco- In this appendix, we show the explicit expression of nomic Development Canada, and by the Province of On- V∗. In the frame centred at the BH, the Hamiltonian of tario through the Ministry of Research and Innovation. the system is p2 µM Htot = − Appendix A: Supperradiance cloud in 2µ r non-relativistic limit 2 p∗ M∗M M∗µ M∗µ + − − + 3 r · R∗ , 2M∗ R∗ |R∗ − r| R∗ In this appendix, we summarize the non-relativistic de- (B1) scription of a superradiant cloud. The Lagrangian of a free complex scalar field Ψ is where p = µr˙, p∗ = M∗R˙ ∗ with r and R∗ being the ab ∗ 2 ∗ positions of the cloud and the star relative to the BH. L = −g ∂aΨ ∂bΨ − µ Ψ Ψ, (A1) For r < R∗, we have where g is the metric of Kerr BH, assuming the back- ab 1 1 r cos ∆θ X X 4π reaction of the field’s stress-energy on the metric is neg- = + 2 + |R∗ − r| R∗ R∗ 2`∗ + 1 ligible. One can make an ansatz `∗≥2 |m∗|≤`∗

`∗ 1 −iµt r ∗ Ψ(t, r) = √ ψ(t, r)e , (A2) × Y (θ∗, φ∗) Y` m (θ, φ) , (B2) `∗+1 `∗m∗ ∗ ∗ 2µ R∗ where ψ is a complex scalar ﬁeld which varies on a where ∆θ is the angle between r and R∗. The ﬁrst term timescale much longer than µ−1. Given Lagrangian (A1), on the r.h.s. of Eq. (B2) does not change the eigenstate 14 of the cloud, and the second term cancels with the last 105 <210|211> term in Eq. (B1). For R∗ < r, we have <21-1|211> 1000 211 322 1 1 r · R 4π < | > ∗ X X <210|322> = + 3 + |R∗ − r| r r 2`∗ + 1 10 <21-1|322> `∗≥2 |m∗|≤`∗ GW / P `∗ R LM 0.100

∗ ∗ P × Y` m (θ∗, φ∗) Y`∗m∗ (θ, φ) , (B3) r`∗+1 ∗ ∗ 0.001 in which case the monopole and dipole terms contribute. The inner product can be written as 10-5

10 7 4 1 X 4π 10- 10- 10- 10- hψ |V | ψ i = −M µ I I (r )Y (θ , 0) i ∗ j ∗ 2` + 1 Ω r ∗ `∗m∗ ∗ GMΩ `∗, m∗ 6 × exp [i(ω − ω )t ∓ im φ ] , (B4) 10 <210|211> i j ∗ ∗ <21-1|211> 104 where the upper sign in front of m φ corresponds to <211|322> ∗ ∗ <210|322> 100 co-rotating orbits, while the lower sign corresponds to <21-1|322> counter-rotating orbits. For ` ≥ 2, we have GW ∗ / P 1 LM Z ∞ `∗ P 2 r< Ir = dr r Rn ` (x) Rn ` (x) , (B5) 0.01 `∗+1 i i j j 0 r> 10-4 −2 where x ≡ α r/GM, r< is the smaller of r and r∗ and 10-6 r> is the lager of r and r∗. For `∗ = 1 and 0, we have 10-7 10-6 10-5 10-4 0.001 0.010 0.100 Z ∞ r3 GMΩ I = dx r 1 − R (x) R (x) (B6) r ∗ r3 ni`i nj `j r∗ ∗ FIG. 6: The ratio between angular momentum changing rate caused by level mixing and that caused by GW radiation. We and consider a circular orbit with the inclination i = π/4 (as some Z ∞ coupling strength is proportional to cos i or sin i), and show

Ir = dr rRniì (x) Rnj `j (x) (B7) the ratio for the dominant level mixing with |211i and |322i. r∗ α = 0.1 for the upper plot and α = 0.3 for the lower plot. The respectively. gray dashed line shows the orbital frequency corresponding to a radius 4α−2, which locates the density peak of the cloud. The shaded region shows the LISA frequency band (0.1Hz - 0.001Hz), assuming a 105M BH. The gray vertical line Appendix C: Effects of mode decay and floating denotes the innermost stable circular orbit. orbits

In this appendix, we will discuss the effects of mode where ∆ω and ∆m are the energy level and quantum decay. In particular, we will show that for EMRIs with number difference between the two modes. On the other quasi-circular orbits, dynamical friction dominates over hand, the angular momentum carried away by GW radi- the backreation of level mixing during most of the time, ation is while the floating orbits proposed in [40] are still possible at resonance frequency. dL 32 2 7/3 = − η M (GMΩ) . (C3) The tidal field of the star can mix the modes of the dt GW 5 cloud, which leads to angular momentum transfer between the star and the BH-cloud system. For a quasi- Fig.6 shows the relative magnitude of angular momen- circular orbit, the angular momentum transfer has been tum changing caused by level mixing and GW radiation. calculated in [40], Floating orbits occur when the angular momentum loss by GW radiation is compensated by that gained through 2 dL∗ Q Mc level mixing. When there is dynamical friction, the oc- = ∆mΓ 2 , (C1) dt 2ωR µ currence of floating orbits also requires energy gained from level mixing can balance the dissipation caused by where Γ is the decay rate of the mode that the saturated dynamic friction. The maximum power of energy gaining mode coupling to, Q is the coupling strength, ωR is the due to level mixing, comparing GW radiation is Rabi frequency defined as

PLM 5 10/3 −2 p q 2 ∼ ∆m η α ` , (C4) ω = Q2 + (∆ω − ∆mΩ) /4, (C2) R PGW max 64 15 where p` = 4` − 1 for Bohr resonance and p` = 4` − 8 termedia mass BH, the energy gain due to the coupling to for hyperfine resonance. On the other hand, the ratio a decaying mode may no longer support the orbital decay between the effects of GW is given by Eq. (17), comparing in the presence of dynamical friction. However, a floating to which we have orbit still exists if one considers the time dependence of the orbital frequency as discussed in [82]. In the pres- −2 PLM| PLM| 10 η max max 9 3 p`+4 ence of the dynamical friction, the change of the orbital > ∼ 10 ∆m α −5 . PDF PDF|max 10 frequency near the resonance frequency may be domi- (C5) nated by dynamical friction. Nevertheless, the Landau- Zener parameter defined in [82] is still much greater than For orbits around a super-massive BH, we usually have 1. Therefore, one can still expect long-lasting monochro- η 10−5, in which case the energy gain due to the cou- matic GW signals from floating orbits around intermedia pling to a decaying mode can still compensate the energy mass BHs as predicted in [40]. loss caused dynamical friction. For orbits around an in-

[1] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 906, pp.1 (2015), 1501.06570. (1977). [24] S. R. Dolan, Phys. Rev. D76, 084001 (2007), 0705.2880. [2] S. Weinberg, Phys. Rev. Lett. 40, 223 (1978). [25] W. E. East and F. Pretorius, Physical review letters 119, [3] M. S. Turner, Phys. Rev. D 28, 1243 (1983). 041101 (2017). [4] W. H. Press, B. S. Ryden, and D. N. Spergel, Phys. Rev. [26] W. E. East, Physical Review D 96, 024004 (2017). Lett. 64, 1084 (1990). [27] H. Yoshino and H. Kodama, PTEP 2014, 043E02 (2014), [5] W. Hu, R. Barkana, and A. Gruzinov, Phys. Rev. Lett. 1312.2326. 85, 1158 (2000), astro-ph/0003365. [28] A. Arvanitaki, M. Baryakhtar, and X. Huang, Physical [6] P. J. E. Peebles, Astrophys. J. 534, L127 (2000), astro- Review D 91, 084011 (2015). ph/0002495. [29] M. Baryakhtar, R. Lasenby, and M. Teo, Physical Review [7] L. Amendola and R. Barbieri, Phys. Lett. B642, 192 D 96, 035019 (2017). (2006), hep-ph/0509257. [30] M. Isi, L. Sun, R. Brito, and A. Melatos, Physical Review [8] H.-Y. Schive, T. Chiueh, and T. Broadhurst, Nature D 99, 084042 (2019). Phys. 10, 496 (2014), 1406.6586. [31] H. Yang, K. Yagi, J. Blackman, L. Lehner, V. Paschalidis, [9] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Phys. F. Pretorius, and N. Yunes, Physical review letters 118, Rev. D95, 043541 (2017), 1610.08297. 161101 (2017). [10] D. J. E. Marsh, Phys. Rept. 643, 1 (2016), 1510.07633. [32] B. Goncharov and E. Thrane, Physical Review D 98, [11] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, 064018 (2018). and J. March-Russell, Physical Review D 81, 123530 [33] A. Pierce, K. Riles, and Y. Zhao, Physical review letters (2010). 121, 061102 (2018). [12] M. B. Green, J. H. Schwarz, and E. Witten, Super- [34] R. Brito, S. Ghosh, E. Barausse, E. Berti, V. Cardoso, string Theory, Cambridge Monographs on Mathematical I. Dvorkin, A. Klein, and P. Pani, Phys. Rev. D96, Physics (1987). 064050 (2017), 1706.06311. [13] P. Svrcek and E. Witten, JHEP 06, 051 (2006), hep- [35] R. Brito, S. Ghosh, E. Barausse, E. Berti, V. Cardoso, th/0605206. I. Dvorkin, A. Klein, and P. Pani, Phys. Rev. Lett. 119, [14] A. Arvanitaki and S. Dubovsky, Phys. Rev. D83, 044026 131101 (2017), 1706.05097. (2011), 1004.3558. [36] D. Baumann, H. S. Chia, and R. A. Porto, Phys. Rev. [15] D. Baumann, D. Green, and B. Wallisch, Phys. Rev. Lett. D99, 044001 (2019), 1804.03208. 117, 171301 (2016), 1604.08614. [37] E. Berti, R. Brito, C. F. B. Macedo, G. Raposo, and J. L. [16] J. G. Rosa and T. W. Kephart, Phys. Rev. Lett. 120, Rosa, Phys. Rev. D99, 104039 (2019), 1904.03131. 231102 (2018), 1709.06581. [38] M. C. Ferreira, C. F. B. Macedo, and V. Cardoso, Phys. [17] L. Sagunski, J. Zhang, M. C. Johnson, L. Lehner, Rev. D96, 083017 (2017), 1710.00830. M. Sakellariadou, S. L. Liebling, C. Palenzuela, and [39] O. A. Hannuksela, K. W. K. Wong, R. Brito, E. Berti, D. Neilsen, Phys. Rev. D97, 064016 (2018), 1709.06634. and T. G. F. Li, Nat. Astron. 3, 447 (2019), 1804.09659. [18] T. Fujita, R. Tazaki, and K. Toma, Phys. Rev. Lett. 122, [40] J. Zhang and H. Yang, Phys. Rev. D99, 064018 (2019), 191101 (2019), 1811.03525. 1808.02905. [19] J. Huang, M. C. Johnson, L. Sagunski, M. Sakellari- [41] E. C. Ostriker, Astrophys. J. 513, 252 (1999), astro- adou, and J. Zhang, Phys. Rev. D99, 063013 (2019), ph/9810324. 1807.02133. [42] S. Chandrasekhar, Astrophys. J. 97, 255 (1943). [20] T. Ikeda, R. Brito, and V. Cardoso, Phys. Rev. Lett. [43] P. C. Peters and J. Mathews, Phys. Rev. 131, 435 (1963). 122, 081101 (2019), 1811.04950. [44] C. L. Benone, E. S. de Oliveira, S. R. Dolan, and L. C. B. [21] S. L. Detweiler, Phys. Rev. D22, 2323 (1980). Crispino, Phys. Rev. D89, 104053 (2014), 1404.0687. [22] T. J. Zouros and D. M. Eardley, Annals of physics 118, [45] C. L. Benone, E. S. de Oliveira, S. R. Dolan, and L. C. B. 139 (1979). Crispino, Phys. Rev. D95, 044035 (2017), 1702.06591. [23] R. Brito, V. Cardoso, and P. Pani, Lect. Notes Phys. [46] W. G. Unruh, Phys. Rev. D14, 3251 (1976). 16

[47] W. Press and S. Teukolsky, The Astrophysical Journal (2017). 213, 183 (1977). [68] B. Bonga, H. Yang, and S. A. Hughes, arXiv preprint [48] H. Yang, W. E. East, V. Paschalidis, F. Pretorius, and arXiv:1905.00030 (2019). R. F. Mendes, Physical Review D 98, 044007 (2018). [69] R. M. O’Leary, B. Kocsis, and A. Loeb, Mon. Not. Roy. [49] H. Yang, arXiv preprint arXiv:1904.11089 (2019). Astron. Soc. 395, 2127 (2009), 0807.2638. [50] Z. Mark, H. Yang, A. Zimmerman, and Y. Chen, Physical [70] L. Barack et al. (LIGO), Class. Quant. Grav. 36, 143001 Review D 91, 044025 (2015). (2019), 1806.05195. [51] H. Yang, A. Zimmerman, and L. Lehner, Physical review [71] B. P. Abbott, R. Abbott, T. Abbott, M. Abernathy, letters 114, 081101 (2015). K. Ackley, C. Adams, P. Addesso, R. Adhikari, V. Adya, [52] H. Yang and F. Zhang, The Astrophysical Journal 817, C. Affeldt, et al., Classical and Quantum Gravity 34, 183 (2016). 044001 (2017). [53] C. P. Berry, S. A. Hughes, C. F. Sopuerta, A. J. [72] K. Belczynski, A. Heger, W. Gladysz, A. J. Ruiter, Chua, A. Heffernan, K. Holley-Bockelmann, D. P. Mi- S. Woosley, G. Wiktorowicz, H.-Y. Chen, T. Bulik, haylov, M. C. Miller, and A. Sesana, arXiv preprint R. O?Shaughnessy, D. E. Holz, et al., Astronomy & As- arXiv:1903.03686 (2019). trophysics 594, A97 (2016). [54] D. Merritt, Classical and Quantum Gravity 30, 244005 [73] M. Spera and M. Mapelli, Monthly Notices of the Royal (2013), 1307.3268. Astronomical Society 470, 4739 (2017). [55] C. Hopman and T. Alexander, Astrophys. J. 629, 362 [74] P. Marchant, M. Renzo, R. Farmer, K. M. Pappas, R. E. (2005), astro-ph/0503672. Taam, S. de Mink, and V. Kalogera, arXiv preprint [56] M. Freitag and W. Benz, Astron. Astrophys. 394, 345 arXiv:1810.13412 (2018). (2002), astro-ph/0204292. [75] S. Stevenson, M. Sampson, J. Powell, A. Vigna-Gómez, [57] H. Baumgardt, J. Makino, and T. Ebisuzaki, Astrophys. C. J. Neijssel, D. Szécsi,and I. Mandel, arXiv preprint J. 613, 1133 (2004), astro-ph/0406227. arXiv:1904.02821 (2019). [58] H. Baumgardt, J. Makino, and T. Ebisuzaki, Astrophys. [76] S. Woosley, The Astrophysical Journal 836, 244 (2017). J. 613, 1143 (2004), astro-ph/0406231. [77] D. Gerosa and E. Berti, arXiv preprint arXiv:1906.05295 [59] M. Preto, D. Merritt, and R. Spurzem, Astrophys. J. (2019). 613, L109 (2004), astro-ph/0406324. [78] C. Hopman and T. Alexander, Astrophys. J. 645, 1152 [60] T. Alexander and C. Hopman, Astrophys. J. 590, L29 (2006), astro-ph/0601161. (2003), astro-ph/0305062. [79] J. N. Bahcall and R. A. Wolf, Astrophys. J. 209, 214 [61] A. Sesana, Physical Review Letters 116, 231102 (2016). (1976). [62] S. Vitale, Physical review letters 117, 051102 (2016). [80] X. Fan, M. A. Strauss, D. P. Schneider, R. H. Becker, [63] K. W. Wong, E. D. Kovetz, C. Cutler, and E. Berti, R. L. White, Z. Haiman, M. Gregg, L. Pentericci, E. K. Physical review letters 121, 251102 (2018). Grebel, V. K. Narayanan, et al., The Astronomical Jour- [64] C. Cutler, E. Berti, K. Jani, E. D. Kovetz, L. Ran- nal 125, 1649 (2003). dall, S. Vitale, K. W. Wong, K. Holley-Bockelmann, [81] P. Natarajan, A. Ricarte, V. Baldassare, J. Bellovary, S. L. Larson, T. Littenberg, et al., arXiv preprint P. Bender, E. Berti, N. Cappelluti, A. Ferrara, J. Greene, arXiv:1903.04069 (2019). Z. Haiman, et al., arXiv preprint arXiv:1904.09326 [65] C. Cutler and E. E. Flanagan, Phys. Rev. D49, 2658 (2019). (1994), gr-qc/9402014. [82] D. Baumann, H. S. Chia, R. A. Porto, and J. Stout [66] E. E. Flanagan and T. Hinderer, Physical review letters (2019), 1912.04932. 109, 071102 (2012). [67] H. Yang and M. Casals, Physical Review D 96, 083015