REVIEWARTICLE

Hierarchical mergers of stellar-mass black holes and their gravitational-wave signatures

1, 2 Davide Gerosa ∗, Maya Fishbach

We review theoretical findings, astrophysical modeling, and current gravitational-wave evidence of hierarchical stellar-mass black-hole mergers. While most of the compact binary mergers detected by LIGO and Virgo are expected to consist of first- generation black holes formed from the collapse of stars, others might instead be of second (or higher) generation, containing the remnants of previous black-hole mergers. Such a subpopulation of hierarchically assembled black holes presents distinctive gravitational-wave signatures, namely higher masses, possibly within the pair-instability mass gap, and dimensionless spins clustered at the characteristic value of 0.7. In order to produce hierarchical mergers, astrophysical environments need to ∼ overcome the relativistic recoils imparted to black-hole merger remnants, a condition which prefers hosts with escape speeds & 100 km/s. Promising locations for efficient production of hierarchical mergers include nuclear star clusters and accretion disks surrounding active galactic nuclei, though environments that are less efficient at retaining merger products such as globular clusters may still contribute significantly to the detectable population of repeated mergers. While GW190521 is the single most promising hierarchical-merger candidate to date, constraints coming from large population analyses are becoming increasingly more powerful.

1 Introduction Investigating the occurrence of hierarchical mergers consti- tutes an orthogonal, but at the same time complementary, direc- Gravitational-wave (GW) observations are revolutionizing the tion to the usual “field vs dynamics” formation-channel debate. field of astronomy and our understanding of compact objects. The key question is the following: The prototypical GW sources are merging binaries composed of What if, instead of being the direct product of stellar collapse, black holes (BHs) and neutron stars (NSs). At the time of writ- some of the BHs we observed with LIGO and Virgo are rem- 1–7 ing, more than 50 of these events have been detected by the nants of previous BH mergers? Advanced LIGO8 and Virgo9 detector network. Exploring the Universe using GWs relies on a deep interplay between astro- This line of research is often phrased as the reconstruction of the physics and relativity. Here we review one of the topics where BH “generation”, contrasting first-generation (1g) objects pro- the dialogue between these two disciplines is most evident: the duced by stellar collapse to second- (2g) or higher- generation hierarchical assembly of multiple generations of stellar-mass BH BHs involving remnants of previous mergers. In essence, a BH mergers. merger is a process that converts two parents into a single BH remnant and the emitted GWs. Crucially, the properties of post- Altough relativistic dissipation of energy and angular mo- merger BHs (masses, spins, and proper velocities) are set by the mentum via GWs is the process that ultimately drives the merg- relativistic dynamics of the merger process, not by their astro- ers we observe with LIGO and Virgo, this mechanism alone is physical environments. At the same time, however, assembling not sufficient to explain the occurrence of coalescing compact repeated mergers requires an astrophysical environment that ef- binaries. Quasi-circular BH binaries of 10M at separations ficiently retains post-merger remnants, such that they remain larger than 10R take more than a Hubble∼ time to merge available to merge again. By relying largely on relativity, hier- under GW radiation∼ reaction alone. One thus needs some ad- archical mergers introduce a clean feature in the astrophysical ditional astrophysical mechanisms acting at larger separations. modeling of GW events. While it is well understood that individual BHs and NSs form arXiv:2105.03439v2 [astro-ph.HE] 11 Aug 2021 This reasoning opens the door to a number of intertwined from the collapse of massive stars,10 conceiving plausible as- questions. Does hierarchical assembly of BHs happen at all at trophysical scenarios that can pair them in binaries while ex- the masses targeted by LIGO and Virgo? If yes, what is the frac- plaining the rates and properties of observed GW mergers still tion of hierarchically-formed events in the observable popula- presents many open issues (see Refs.11, 12 for recent reviews). tion of merging compact binaries? What does this constraint tell Leading models of compact-binary formation include the iso- us about the branching ratios between the different BH binary lated evolution of massive binary stars in galactic fields13 via formation channels? either common-envelope14, stable mass transfer,15 or chemical This review is restricted to hierarchical mergers of stellar mixing,16, 17 and dynamical assembly aided by either a tertiary mass and their GW signatures. Among the LIGO/Virgo events, companion,18, 19 multiple exchanges in dense clusters,20 or gas- only a small fraction of systems are expected to be of hierar- assisted migration.21 chical origin, with first-generation mergers comprising the bulk of the population. On the other hand, for supermassive BHs 1School of Physics and Astronomy & Institute for Gravitational Wave Astronomy, University of Birmingham, Birmingham, B15 2TT, UK. hosted at the center of galaxies, hierarchical assembly is believed 2Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) & Department 5 9 of Physics and Astronomy, Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA. to be the norm. The properties of BHs of 10 10 M are ∼ − 22 [email protected] strongly correlated with those of their host environments and ∗

1 are thus expected to grow through mergers as their host galax- instability BH mass threshold between 40M and 55M ies merge.23 Supermassive-BH evolution has been comprehen- (up to 90M is one allows for variations∼ at the 3-σ level).∼ Sim- sively reviewed elsewhere.24–29 Hierarchical mergers of stellar- ilarly, Renzo∼ et al.42 found a threshold of 48M , robust against mass BHs, which are the focus of this review, have important the treatment of convection in the stellar evolutionary models. synergies with the astrophysics of supermassive BHs, the former Additional pieces of the puzzle include potential correlations representing potential seeds for the latter. of the location of the pair-instability gap with the BH spin43, 44 This review is structured as follows. In Sec.2, we illustrate which might the impact edge of the gap by 15%, uncertainties ∼ how hierarchical mergers provide characteristic GW signatures in current stellar-wind prescriptions,45, 46 as well as dredge-up that are independent of the specific environment in which they episode during the helium-burning phase47 which can also push form. In Sec.3, we present the most promising astrophysical lo- the lower edge of the mass gap all the way to 90M . GW ob- ∼ cations that can produce hierarchical mergers, which notably in- servations from the first two observing runs of Advanced LIGO clude star clusters and disks of active galactic nuclei (AGN). We and Virgo provided observational evidence for a dearth of BHs stress that this paper is not designed to provide an exhaustive heavier than 45M ,48–51 widely thought to be a consequence ∼ review of GW formation channels, but only to highlight their rel- pair instabilities in supernovae. This observation offered the evance to the hierarchical-merger problem. In Sec.4, we review first empirical constraints on the pair-instability gap as well as the interpretations of individual GW events as repeated mergers. a chance to test the underlying stellar and nuclear astrophysics. In Sec.5, we summarize ongoing efforts to statistically extract The pair-instability process immediately translates into a subpopluations of hierarchical mergers from the current catalog promising signature of hierarchical mergers in GW observations: of LIGO/Virgo detections. Finally, we conclude and illustrate if BHs with 50M . m . 120M cannot be produced by future prospects in Sec.6. stars, they might well be the remnants of previous BH merg- ers. Possible caveats to this statement include envelope reten- 52–54 2 Gravitational-wave signatures tion in low-metallicity population III stars, stellar mergers prior to BH formation,37, 55–57 evolution in detached binaries,44 as 58 59 Hierarchical mergers have unique GW signatures that make well as accretion in either molecular clouds, minihalos, dense 60, 61 62 them distinguishable from BHs resulting from stellar collapse. clusters, or isolated binaries. Higher-generation BHs present, on average, both larger masses and larger spins. 2.2 Spins: a characteristic value

2.1 Masses: populating the pair-instability mass gap Spins of hierarchically-formed BHs also present a distinctive fea- Assembling multiple generations of BH mergers has the obvi- ture. As two BHs merge, the sum of their component spins Si ous effect of producing GW events with larger masses. The en- and the orbital angular momentum at plunge L is converted into the spin of the final BH: S L + S + S . For the sim- ergy emitted in GW following a BH merger is . 5% of the to- fin ' 1 2 tal mass of the merging binary.30, 31 BHs of higher generation in- pler case of an equal-mass, non-spinning merger, the remnant BH has a dimensionless Kerr parameter S/m2 χ 0.69.30, 63 herit 95% the combined mass of their parents and are therefore ≡ ∼ heavier∼ than both of them. Somewhat surprisingly, this remains true in a statistical sense 64–67 This point becomes particularly relevant in the context of the for generic BH mergers. This property of relativistic dynam- so-called “pair-instability mass gap” (sometimes also referred to ics can be explained heuristically using the so-called “orbital 68 as “upper mass gap”, in contrast with the putative “lower mass hang-up” effect. When BH spins are co-aligned with the or- gap” between BHs and NSs). At the onset of stellar collapse, suf- bital angular momentum, the binary inspirals for longer before ficiently large helium cores (in the mass range 30 130 M ) merger. Their delayed plunge decreases the magnitude of L, but reach central temperatures larger than 109K at∼ densities− below spins and angular momentum add constructively into the spin 6 3 of the remnant, S L + S + S . On the other hand, bina- 10 g/cm , triggering efficient electron-positron production. As fin ' 1 2 radiation-pressure support drops, the adiabatic index in the core ries with anti-aligned spins plunge from afar: the orbital angu- falls below 4/3, causing a contraction of the core. This, in turn, lar momentum at plunge is larger, but it is partially cancelled by the component spins, S L S S . In practice, these two ignites explosive burning of carbon and oxygen, producing an fin ' − 1 − 2 amount of energy that is comparable to the binding energy of the effects counterbalance each other and produce a distribution of star. Depending on the mass of the collapsing core, the star can remnant spins that is strongly peaked at the characteristic value 32 χ 0.7. be either completely (“pair-instability supernova”) or partially ' (“pulsational pair-instability supernova”)33 disrupted. This cre- Such BH spin values are thought to be larger than those that ates the lower edge of the pair-instability gap, an upper limit on can be produced by stellar collapse. Recent models of core- the BH mass that can be produced following stellar collapse.34–37 envelope interactions in massive stars point towards an efficient 69 In stars with even more massive helium cores (& 130M ), some transfer of angular momentum out of the collapsing core, re- sulting in BHs with χ 10 2. This finding is partially supported of the energy from the core contraction goes into photodisente- ∼ − gration, preventing complete stellar disruption. If such massive by current GW data, which points to a sizable population of BHs stars exist, this would result in a population of BHs “above the with relatively small spins49–51, 70 (but see also Refs.71, 72). Tidal gap” with masses & 120M .38, 39 interactions can also play a relevant role: for binaries formed in

Farmer et al.40, 41 found that pair instability creates a mass gap isolation, tides have the net effect of increasing the spin mag- in the BH mass spectrum starting at m 45M . This thresh- nitudes and aligning the spin directions with the binary orbital 73–78 old was found to be a relatively solid prediction' given the cur- angular momentum. rent understanding of stellar evolution. One of the largest un- If, overall, stars produce BHs which are slowly rotating, the certainties in the stellar evolution models is given by the un- high-spin region of the parameter space around χ 0.7 becomes known 12C(α, γ)16O reaction rate, which may shift the pair- exclusive to hierarchical mergers.65, 66, 79 Baibhav et' al.80 referred

2 to this peculiarity of hierarchical mergers as the “spin gap”, in 3 Astrophysical formation channels analogy with the mass gap described above. Understanding the formation channels of merging BH binaries 2.3 A simple realization is now one of the most pressing quests in high-energy astro- physics. Among the many proposed scenarios, two main classes The left panels of Fig.1 show a simple implementation of these of models can be identified. In the first class, merging BHs orig- ideas. We start from a population of 1g BHs modeled after cur- inate from isolated binary stars. Alternatively, BH mergers can rent LIGO/Virgo detections50 (thus implicitly assuming, for con- be aided by interactions with external bodies, such as other com- creteness in this example, that all current events are of first gen- pact objects or large gaseous structures. As schematically sum- eration). In particular, we consider their “power-law + peak” marized in Fig.2, hierarchical mergers are a prerogative of this mass model and “default” spin model, which were found to re- second class of models. turn the highest Bayesian evidence among various phenomeno- logical options. We extract binaries from the resulting pos- 3.1 The role of the escape speed terior population distribution and estimate the properties of their merger remnants using fits to numerical-relativity simu- The escape speed of the environment has a crucial role in the 89 lations by Varma et al.83, 84 The fits are evaluated at the refer- assembly of hierarchical mergers. Second-generation BH bina- ence GW frequency of 20 Hz. Because the LIGO/Virgo popu- ries can be produced only if the remnants of previous mergers lation model only captures masses, spin magnitudes and polar are efficiently retained within their astrophysical host. spin angles, we distribute the azimuthal spin angles uniformly, As two BH merge, asymmetric dissipation of linear momen- which is equivalent to the prior used in the underlying single- tum via GWs imparts a recoil (or “kick”) to the post-merger event analyses. About 0.1% of the samples have mass ratios remnant. These merger recoils range from 0 (for the case of 90 smaller than 1:6, which is outside the validity range of the rem- highly symmetric configurations) to 5000 km/s (for the case ∼ 91–93 nant numerical-relativity fits.83 For those few cases, we imple- of equal-mass systems with large, misaligned spins). For suf- 94–96 ment expressions for final mass,85 spin,86 and recoil87 which ex- ficiently broad populations, BH merger kicks are of O(100) plicitly include the test-particle limit. See Doctor et al.88 for a km/s. For instance, using the 1g population extracted from cur- study dedicated to the distribution of post-merger remnants ex- rent LIGO/Virgo data as in Sec. 2.3, one finds that 99%, 85%, and 3% of the resulting 2g BHs receive a kick∼ greater∼ than trapolated from the current LIGO/Virgo detections. ∼ In this data-driven 1g population, only 2% of the BH 30km/s, 100 km/s, and 1000 km/s, respectively. masses (including primary and secondary components)∼ are > In order to produce hierarchical mergers, the escape speed 45M . This fraction goes up to 20% if one considers the 2g of the host needs to be larger than the typical kick imparted to merger remnants, indicating that∼ the hierarchical mergers are BH remnants. For context, the escape speed of a typical globular cluster is 10 100 km/s, that of the Milky Way is 600 km/s, indeed an efficient way to populate the mass gap. The 1g spin ∼ − ∼ distribution extracted from current LIGO/Virgo data shows a while large elliptical galaxies can reach escape speeds of 2000 81, 82 ∼ broad preference for χ . 0.5 (although some effective spin pa- km/s. If all dynamical formation channels were to produce rameters are better measured).50 The spin magnitudes of 2g BHs, BH mergers at approximately the same rate, it would be natural however, are highly concentrated close to χ 0.7. to expect that hierarchical mergers are more likely to originate ∼ In general, repeated mergers are expected to present a strong from environments with larger escape speeds. For the forma- mass-spin correlation and cluster in the high-m high-χ region tion channels highlighted below, current BH-binary merger-rate 3 1 of the parameter space. This is a distinctive population feature predictions range from 4 60 Gpc− yr− (at z = 0) for globu- − that might aid the identification of hierarchical mergers from lar clusters,97, 98 1 Gpc 3yr 1 for nuclear star clusters,99, 100 and ∼ − − LIGO/Virgo data (cf. Sec.5) 0.002 60 Gpc 3yr 1 for AGN disks.101, 102 − − − It is worth noting that orbital eccentricities also affect merger 2.4 Other observables recoils. For eccentricities . 0.3, BH kicks can be enhanced by up to 25%,103–105 making it more difficult to retain the remnants The mass ratio of merging BHs might also provide precious in- of mergers formed in dynamical environments, a fraction of sights, especially if different generations of mergers are com- which is predicted to be eccentric.18, 19, 106–109 The consequences bined. Very simply, one can expect that a mixed binary will of eccentricity-enhanced kicks on the rate of hierarchical merg- present a mass ratio more unequal compared to other events ers is unexplored. where both components come from the same generation. For in- stance, the mass ratio of a 1g+2g event might be somewhat close 3.2 Star clusters to 1/2. The redshift distributions of merging BH binaries could also Repeated mergers in stellar cluster have long been investigated contain information about their generation. Some delay will be as potential formation pathways of intermediate-mass110–123 and present between the formation of first- and second-generation supermassive124–128 BHs. In the LIGO context, an early report BHs which implies that the redshift of the hierarchical merger of potential hierarchical mergers was made by O’Leary et al.,129 event should be lower than those of the previous generation.65 who claimed that such events could constitute up to 10% of Specific astrophysical environments can, however, overpopulate the GW mergers from dense clusters (see e.g. their Fig.∼ 2). Ro- the low-redshift region for 1g BHs. For instance, this might be driguez et al.130, 131 later presented a wide range of Monte Carlo the case for binaries that are ejected from clusters and take a long integrations specifically targeting repeated mergers in globulars. time to merge under gravitational radiation reaction, compared They identified a very strong dependence on the birth spins of to binaries that merge relatively quickly inside the cluster. One BHs: the contribution of hierarchical mergers to the BH merger also needs to consider that the higher masses involved in hier- rate drops from 10% if BHs are born non-spinning to . 1% for archical mergers imply that larger redshifts are accessible to the moderate spins χ∼ 0.5. This is because higher-spinning BHs are detectors. subject to larger merger∼ recoils and are thus more easily ejected.

3 0.12 1g 0.10 2g 0.08 )

m 0.06 ( p 0.04

0.02

0.00 a. b. 1.0 0.0025

0.8 0.0020

0.6 0.0015 ) v χ ( 0.4 p 0.0010

0.2 0.0005

0.0 0.0000 0 25 50 75 100 0 2 4 6 8 10 12 0 500 1000 1500 m [M ] p(χ) v [km/s]

Figure 1 Masses, spins, and recoil velocities of first- and second-generation BHs. The corner plot on the right (panel a.) shows BH masses m and spins χ. The| histogram on the left (panel b.) shows the corresponding kick velocities v. Blue scatter points and histograms indicate a population of 1g BHs extracted from current LIGO/Virgo population fits.50 Orange scatter points and histograms indicate the corresponding distribution of their merger remnants, which might form 2g GW events. Black dotted lines indicate typical values of astrophysical relevance: (i) the edge of the pair-instability mass gap40 m = 45M , (ii) the remnant spin of equal-mass, non-spinning BH mergers63 χ = 0.69, and (iii) an approximate upper limit to the escape speed of globular clusters 81, 82 v = 100 km/s.

The simplified model by Gerosa and Berti89 also finds very small Nevertheless, it is important to point our that globular clus- fractions of hierarchical mergers from globulars if spinning BHs ters could also host a sizable population of second-generation are considered. mergers if BH spins at birth turn out to be small, which is in line with some of the current predictions.69 Furthermore, glob- Semi-analytical treatments based on simulated stellar ular clusters were on average 5 times more massive at birth populations132–134 suggest that the fraction of repeated mergers compared to the present time,∼143 which increases their escape in nuclear star cluster is 1 order of magnitude larger than speeds by a factor of √5 > 2. These are crucial details be- that of globulars and 3 orders∼ of magnitude larger than that cause globulars are thought∼ to be extremely efficient factories of of young star clusters.∼ Similarly, energy arguments that relate GW events.144–151 the hardening rate of BH binaries to the global properties of the clusters135 indicate that the occurrence of repeated merg- The cluster metallicity might play an important role in the ers presents a steep increase in systems with escape speeds formation of hierarchical GW events, with a preference for metal-poor environments133 (but see Ref.152 for opposite claims). & 300 km/s and mass densities & 105 M /pc3. Monte Carlo simulations136 and further analytical modeling 80, 137–139 produce Additionally, a notable boost in the rate of hierarchical stellar- mass BH mergers in clusters could be provided by Kozai-Lidov qualitatively similar results: populating the upper mass gap via 138 in-cluster GW mergers seems possible, but requires sufficiently oscillation induced by a massive perturber. We also note that massive environments. hierarchical mergers involving NSs have also been explored as a potential formation channel of GW events with one of more Overall, these findings point towards galactic components in the lower mass gap (3M . M . 5M ), both in nuclei99, 100, 140, 141 as the most likely cluster environments to clusters153 and few-body configurations in the field.154 –157 host repeated mergers. The key binary formation mechanism is different for nuclear star cluster that do or do not host a 3.3 AGN disks central supermassive BH.142 In the former case, short relaxation time can result in the formation of a steep density cusp of Gaseous AGN disks are also promising environments for stellar-mass BHs around the central supermassive BH, which the production of BH binaries merging in the LIGO/Virgo facilitates mergers by GW captures.140 On the other hand, band.21, 101, 102, 158–161 In this scenario, stellar-mass BHs are embed- nuclear star clusters without a supermassive BH are akin to ded in accretion disks surrounding supermassive BHs, and their heavier globulars where hardening is driven by three-body evolution is driven by angular-momentum transfer via viscous encounters.100 interactions —a process that is analogous to that of planetary

4 Triples and few-body Young star clusters confgurations Globular clusters Isolated binary evolution (common envelope)

Hierarchical First-generation mergers Isolated binary evolution mergers (stable mass transfer) Nuclear star clusters

Isolated binary evolution (chemically homogeneous) Primordial AGN disks black holes (migration traps)

Figure 2 Interplay between the occurrence of hierarchical mergers and (some of) the proposed formation channels of merging compact binaries. While all| channels naturally produce 1g BHs (blue) either from stellar collapse or cosmological perturbations, only a subset of them can efficiently assemble hierarchical mergers (orange). Solid (dashed) lines indicate formation channels that are efficient (inefficient) at producing hierarchical mergers, though their overall contribution to the observed population will also depend on the relative merger rates from the different channels. migration in protoplanetary disks.162 Bellovary et al.163 pointed This argument has recently been made explicit by McKernan out that semi-realistic disk models164, 165 predict the existence of et al.168 Crucially, the Bardeen-Petterson effect allows for both “migration traps.” These are specific locations in the disk where co- and counter-alignment.176 This is because the disk consti- the viscous torque changes signs: disk perturbers at radii larger tute an essentially axisymmetric environment. Each of the two than the trap migrate inwards, while those at smaller radii mi- spins of a BH binary can thus be either co- or counter-aligned grate outwards. Trapped migration is an ideal mechanism to with the orbital angular momentum which implies the pres- assemble multiple generation of BH mergers: stellar-mass BHs ence of four, roughly equally populated subpopulations (“up- either formed by disk fragmentation or captured by the gravi- up”, “up-down”, “down-up”, and “down-down”). Notably, tational pull of the AGN tend to migrate toward the very same three of these will still be close to their aligned configuration as locations and thus meet each other. Orbital velocities at the mi- they enter the LIGO band, while “up-down” binaries, in which gration traps are of (104) km/s,163 which makes these systems the more (less) massive BH is co- (counter-) aligned, are ex- largely insensitive toO BH merger kicks of 100 km/s. pected to precess away because of a dynamical instability.177–180 ∼ Counter-alignment thus distinguishes the AGN scenario from Hierarchical mergers in AGN disks have been explored, both isolated binaries (where only co-alignment is predicted) 166, 167 with focus on both intermediate mass and stellar-mass and dynamical assembly in clusters (where spin orientations 102, 160, 168, 169 events detectable by LIGO/Virgo. Assuming that (i) are randomized by frequent encounters), constituting a poten- the number of BHs dragged into a migration trap within the tial smoking-gun signature of disk-driven mergers. AGN lifetime follows a Poisson distribution and (ii) once a new Because of their gaseous environment, GW events from BH reaches the trap it mergers immediately with the BH that BHs in AGN disks could present electromagnetic counterparts. 169 is already there, Yang et al. found that the fraction of higher- Super-Eddington accretion,21 as well as re-adjustment of the Hill generation mergers from AGN disks is & 50%. Combining sphere due to GW mass loss and recoil181, 182 are expected to pro- 102 N-body simulations and analytical arguments, Tagawa et al. duce detectable flares. Suggestive associations with fast radio reports fractions between 20% and 45%, although in their bursts (FRBs) have also been proposed.183 models migration traps are∼ less relevant∼ to the overall merger rate. For BHs formed in AGN disks, repeated mergers are a very 3.4 Exotica likely outcome, perhaps even the most likely. An entertaining possibility is that BH mergers detectable by Unique signatures of repeated mergers formed in AGN disks LIGO/Virgo have a cosmological origin and constitute (a include not only large masses (Sec. 2.1) and large spin magni- fraction of) the Universe’s dark matter184 see e.g. Ref.185 for tudes (Sec. 2.2), but also preferential spin alignment. This pro- detection-rate estimates and Refs.186, 187 for model selection). cess is commonly refereed to as the Bardeen-Petterson effect170 The impact of second-generation mergers on the predicted and has long been explored for the case of supermassive BH population of such primordial BHs was explored in Refs.188–190. binaries.171–174 Some gas from the AGN disc will form a smaller De Luca et al.190 found that the overall contribution of re- circumbinary disk surrounding the two stellar-mass BHs which, peated mergers to the primordial BH population is . 0.5% 3 16/37 × in turn, feeds individual disks around each of the two compo- ( fPBH/10− ) , where fPBH is the fraction of primordial BH in nents. General-relativistic frame dragging excites warps in these dark matter. This estimate depends very mildly on the assumed “minidisks”175 which tend to co/counter-align the BH spins mass spectrum: it thus seems unlikely that the merger history with the disk’s angular momentum. Assuming that the binary’s of primordial BHs could play a relevant role in the LIGO/Virgo orbital plane and the AGN disk are coplanar, the occurrence of context (but see e.g. Ref.191 for a different result). the Bardeen-Petterson effect implies some degree of preferential Bianchi et al.192 considered an interpretation of GW obser- alignment between the spins and the binary’s angular momen- vations in quantum gravity, arguing that BHs are in micro- tum, with potential GW signatures. canonical equilibrium (i.e. at fixed energy and particle num-

5 ber) and might undergo repeated mergers. In this scenario, the al.210 showed that a 1g+2g interpretation of GW190412 returns Bekenstein-Hawking entropy formula predicts minuscule spin similar Bayesian odds compared to a standard 1g+1g analysis 38 magnitudes (χ 10− for BHs heavier than a Planck mass), but favors an environment with escape speed vesc & 150 km/s, while the high-spin∼ region of the parameter space is populated thus excluding globular clusters. In those models, a large es- exclusively by merger remnants. cape speed is required to accommodate the measured values of the mass ratio and the effective spins. Rodriguez et al.211 inter- 4 Individual-event constraints preted the event as a 1g+3g merger in a super star cluster, which naturally explains the 1/3 mass ratio. GW190412 was also in- ∼ 212, 213 The first three observing runs of LIGO/Virgo saw a number terpreted in the context of disk-driven formation and as 214 of exceptional systems, including very massive BHs or signif- the end product of a quadrupole star system. However, one icantly asymmetric binaries. Several authors have speculated cannot exclude that this system belongs to the tail of the mass- 202 whether some of these systems may be products of hierarchical ratio distribution, rather than a different formation channel 50, 215 assembly. and, indeed, the more recent analyses of Refs. did not sin- gle GW190412 out as an outlier. Olejak et al.216 argues that the mass ratio of GW190412 can be predicted by isolated common- 4.1 GW170729 envelope evolution, where that 10% of isolated, 1g binary BH ∼ With a masses of 50M + 35M and mild evidence for spin systems might have q < 0.4. The isolated channel has a more precession, GW170729∼ is the heaviest system reported in the difficult time producing systems with significant spin precession GWTC-1 catalog from the first and second LIGO/Virgo observ- (but see Refs.75, 78, 217). ing runs.2 The possibility that GW170729 contains a 2g BH was approached by tuning the Bayesian prior,193 statistical compar- isons with the other GWTC-1 event,194–196 and more specifically 4.4 GW190521 in the context of BH formed in clusters132 and AGN disks.169 The heaviest binary BH system observed thus far is GW190521, These studies found only weak preference in favor of a hierar- with a total mass of 150 M .218 The primary mass of chical interpretation indicating that, overall, data are inconclu- GW190521 is confidently above∼ 65 M , placing it inside the pair- sive. Refs.49, 197 both argue that masses and spins of GW170729 instability mass gap and making GW190521 the most promis- are consistent with the other GWTC-1 events, suggesting that it ing hierarchical merger candidate to date. A repeated merger is not necessary to invoke a hierarchical origin to explain this origin is arguably considered to be the leading explanation of system. this event215, 219— a conclusion that is further strengthened by evidence of spin precession218 and/or residual eccentricity.220–222 4.2 GW170817A These features support a dynamical origin for GW190521, which 152, 223 The independent re-analysis of data from first two LIGO/Virgo is a prerequisite for hierarchical formation (see also Refs. observing runs by Zackay et al.198 reported the additional trig- for further modeling in this direction). The statistical mixture- 219 ger GW170817A, which is not present in GWTC-1 (not to be model analyses of Abbott et al. ended up favoring a 1g+1g confused with the more famous NS merger GW170817199 from model for GW190521, but this is because the underlying dis- 196 the same calendar day). If this event is of astrophysical origin tributions from Kimball et al. are specifically tuned to glob- (which is unclear198, 200) its total mass of 100M and effective ular clusters, which present escape speed lower than the typi- aligned spin parameter of χ 0.5 makes∼ it a promising can- cal merger kicks (cf. Sec.3). Subsequent work from the same eff 215 didate for a 2g merger. According∼ to the models of Gayathri group classified GW190521 as a 2g merger but boosted the et al.,201 a repeated merger in an AGN disk should be preferred cluster escape speed to & 100 km/s. Specifically in the context of 213 compared to an isolated origin. The event remains consistent AGN disks, Tagawa et al. finds that GW190521 is compatible with component masses below 50 M . with a 2g+2g (higher-g) BH merger in metal-poor (metal-rich) environments. In addition to the GW constraints, the electromagnetic flare 4.3 GW190412 ZTF19abanrhr from AGN J124942.3+344929 detected182 by the The first reported system with confidently unequal masses is Zwicky Transient Facility224 presents suggestive,225 but far from GW190412, with a mass ratio of q 0.3.202 This is in contrast conclusive,226, 227 association with GW190521. The interpretation ∼ to the 10 BH events from GWTC-1, which were all consistent put forward by Graham et al.182 relies on shocks generated by a with q = 1 and suggested a strong preference for equal mass recoiling BH in the disk surrounding the AGN. A post-merger mergers and a median mass ratio of 0.9.203 Most of the GWTC- remnant of mass m with kick velocity v will influence a sphere ∼ 2 events3 are also consistent with q = 1. GW190412 exhibits a of gas with radius r . Gm/v2. The BH leaves bound gas behind weak preference for spin precession caused by component BH after a time t r/v ( 30 days for ZTF19abanrhr) and enters an spins that are misaligned with the orbital angular momentum. unperturbed region∼ of∼ the disk. During this phase, the accretion While measurements of some effective spin quantities204–207 are rate becomes super-Eddington and the Bondi-Hoyle-Lyttleton solid, identifying the relative contribution of the two component luminosity reaches (1045) ergs/s, providing the necessary en- spins is more challenging. The LIGO/Virgo analysis202 found ergy to power theO observed transient.181, 182 Interestingly, they that the primary BH is mostly responsible for the spin signature predict that a similar flare should be detected after 1.6 yr as the in the data. Mandel and Fragos208 argued that, because of tidal kicked BH re-enters the disk. While it is not possible∼ to rule out a spin-up in isolated binaries, one should rather prefer a spinning chance coincidence for an AGN flare in the 4 Gpc3 90% local- secondary. Follow-up parameter-estimation studies209 indicate ization volume of GW190521, which likely contains∼ tens of thou- that the former interpretation should be preferred. sands of AGNs, it remains an exciting possibility. The presence Hierarchical mergers provide a viable scenario to explain or absence of future associations between GW events and AGN both the asymmetric masses and the spin properties. Gerosa et flares will reveal whether or not there is a physical connection.227

6 If confirmed, this interpretation would indicate that the host of because of their unusual properties. However, it is generally dif- GW190521 is an environment that can efficiently retain kicked ficult to disentangle a genuine population outlier (for example, a remnants and thus host hierarchical mergers. binary BH system with a different origin) from a statistical fluc- Several alternative explanations for the occurrence of tuation in the tails of the population, especially in the presence GW190521 have also been put forward, ranging from criti- of large measurement uncertainties.197 Furthermore, inference cally assessing uncertainties in the lower edge of the mass based on a single event can be highly dependent on the choice gap,47, 53, 228, 229 Population III stars at very low-metallicity,53, 54, 230 of parameter-estimation prior, at least at the present signal-to- accretion onto either stellar-origin58, 59, 213 or primordial231, 232 noise ratios.71, 209, 238 By considering a single outlier, the measure- BHs, and stellar mergers.37, 55, 56 Additional speculations include ment uncertainties on its parameters, together with theoretical invoking beyond-standard-model physics233, 234 exotic compact uncertainties on the potential subpopulation (from which there objects,235, 236 and dark-matter annihilation.237 From a data- are 1 observations), mean that one can rarely discern its origin ≤ analysis perspective alone, there is also the concrete possibility with high confidence. On the other hand, combining informa- that the primary and secondary components of GW190521 are tion across multiple events might reveal whether there is indeed actually above and below the pair instability gap respectively, a subpopulation of BHs with the properties characteristic of hi- and no object is inside it.238, 239 erarchical assembly. As discussed in Sec.2, hierarchical mergers leave a distinct 4.5 GW190814 imprint on the spins of BHs. For individual events, BH spins are poorly measured, but, with (100) events, a potential subpopu- While GW190521 contains a BH that appears to sit in the pair- lation with χ 0.7 can be confidentlyO identified in the data.65, 66 instability mass gap, the event GW190814240 is notable because This method∼ relies on simultaneously inferring, or knowing a one of its components lies in the purported lower mass gap, be- priori, the spin distribution of first-generation BH binaries. If tween NSs and BHs.241, 242 The secondary mass of GW190814 natal BH spins are very small,69 it will take fewer observations at 2.6 M is probably too large to be a NS,240, 243 but ro- to reveal the imprint of hierarchical mergers on the spin distribu- ∼ tation might be an key player to discriminate between the tion. On the opposite extreme, if BHs are born with spins tightly two outcomes.244–247 It has been proposed that the empiri- clustered around χ 0.7, it will be difficult to infer the presence cal low mass gap, which was first observed by studying the of hierarchical mergers∼ from the spin distribution alone.80 masses of X-ray binaries, is caused by the supernova explosion In addition to imprinting spins, repeated mergers increase 248, 249 mechanism, although selection biases might not be fully the masses of the merging BHs. Therefore, identifying a positive 250, 251 under control. If a different origin is really behind the ex- correlation between spins and masses may point to the role of hi- istence of the low mass gap, the outlier mass of GW190814’s erarchical mergers.79 Interestingly, some formation models75, 256 secondary BH may indicate formation through a different path- predict a negative mass-spin correlation for 1g BHs which, if way compared to the other events. At roughly the same mass as confirmed, might enhance the distinguishability of the 2g sub- 199 the merger product of GW170817, the secondary GW190814 population. Using 47 events from GWTC-1 and GWTC-2, Ab- 153, 155 is suggestive of a hierarchical merger origin between NSs. bott et al.50 searched for evidence that the most massive BH However, NS mergers are predicted to be extremely rare in dy- mergers in the catalog follow a spin distribution that is distinct 252 namical environments, which challenges this interpretation. from the spin distribution of the lower-mass systems. This anal- In AGN disks, the interplay between repeated mergers and ac- ysis modeled the binary BH primary mass distribution as a mix- cretion has been suggested as a viable mechanism to produce ture of a power-law component with a high-mass Gaussian com- 213, 253 GW190814-like events. ponent, and allowed the distribution of spin magnitudes and spin tilts to vary between both components. While there are 5 Model selection for populations of events hints that the spin distribution within the high-mass Gaussian component favors larger spin magnitudes and larger spin tilts While individual events may hint at a hierarchical origin, ro- (approaching an isotropic tilt distribution), large uncertainties bust constraints on the presence of hierarchical mergers in a GW remain and there is no conclusive evidence that the spin distri- catalog requires a population analysis. We refer the reader to bution varies with mass. Additional phenomenological trends Refs.254, 255 for pedagogical introductions and Refs.49–51 for state- that can reveal the presence of hierarchical mergers include a se- of-the-art applications. ries of peaks257 or a “smoothed staircase” structure194 in the mass distribution, a forbidden region in the mass-spin plane,258, 259 and the flattening of the spin distribution at high masses.260 Mea- 5.1 Outliers and subpopulations suring the evolution of the mass and/or spin distribution with The goal of population studies is to infer the collective proper- redshift could also reveal subpopulations of higher generation ties —masses, spins, redshifts, eccentricities— from the detected mergers.261 set of binary BHs. If the observed population contains a subpop- Beside phenomenological models that search for trends in ulation of systems formed through hierarchical mergers, jointly the BH component mass, mass ratio and spin distribution, data analyzing all of the events in the GW catalog can reveal the pres- analysis strategies based on physical coagulation models have ence of this distinct set of events. Population analyses can pro- also been successfully employed.194, 196, 262, 263 These methods typ- vide powerful constraints on the astrophysical rate of hierarchi- ically parameterize the population of first-generation BHs with cal mergers and the natal masses and spins of BHs, even in the a phenomenological model and introduce additional parame- case where one cannot confidently identify which specific events ters calibrated on cluster simulations to model the coagulation in the catalog are of hierarchical origin. process and production of higher generation mergers. The re- The first signs of a new subpopulation may appear in the sulting fit simultaneously measures the properties of the first- form of outliers in the data. As summarized in Sec.4, a hier- generation BH subpopulation and the higher-generation sub- archical merger origin was explored for some individual events populations. Both Doctor et al.194 and Kimball et al.196 find that

7 the GWTC-1 catalog does not present evidence for hierarchical diately finds f = 1 f f /λ . For instance, if GW190521 B − A ' 2g B mergers. On the other hand, subsequent work by Kimball et were the only 2g merger in GWTC-1 and GWTC-2 ( f2g 1/50), 215 ∼ al. reports that the updated sample of GWTC-2 is likely to con- a single dynamical channel that predicts λB = 10% of repeated tain a subpopulation made of hierarchical merger products pro- mergers would indicate that 10 events should be dynamically ∼ vided that the cluster escape speed is & 100 km/s. Among the assembled while the remaining 40 evolved in isolation. events, that which is more likely to be a repeated mergers is, per- ∼ haps unsurprisingly, GW190521 (c.f. Sec. 4.4). This finding is at 6 Conclusions odds with an earlier analysis by Abbott et al.219 which relies on previous models196 and includes only the GWTC-1 events and In the context of LIGO/Virgo observations, the hierarchical BH GW190521. In the analysis of Kimball et al.,215 the model with merger scenario attracted a steep increase of interest in the last highest Bayes factor has v 300 km/s, which is relatively esc few years. We have attempted a concise review of both obser- far from the globular-cluster regime∼ their models are tuned to. vational constraints and modeling advances that underpin this For this parameter choice, the events which might be of hi- progress. erarchical nature beside GW190521 are GW190517, GW190519, LIGO and Virgo are sensitive to BHs with masses of 10 GW190602, GW190620, and GW190706 —a list that notably ex- 100M , similar to those of stars. Stars have been long∼ pre-− cludes GW190412 despite its mass ratio (cf. Sec. 4.3). Baxter dicted and observed to leave behind compact objects at the end et al.264 explicitly introduced a 2g component in their popula- of their lives: the working assumption when interpreting GW tion fit and found that it plays a marginal role. Veske et al.265 data, therefore, is that the observed BHs are produced by stellar attempted the more ambitious identification of pairs of events collapse. However, both individual GW observations as well as which may be directly related to each other. The search did not the statistical properties of the entire detected population seem yield a statistically significant association between event pairs, to indicate that stars might not be the progenitors of all LIGO the most likely relationship being GW190514 as the predecessor events. Other BHs might have a role to play: some of the events of GW190521. LIGO/Virgo observes might not be the direct end-product of Finally, another analysis strategy is to compare the data di- massive stars, but rather originate from previous BH mergers. rectly to a set of simulations, inferring unknown physical pa- Such repeated assembly is very reminiscent of supermassive rameters in the simulation rather than introducing any phe- 266–268 BHs formation pathways, as those objects have long been pre- nomenological parameters. This is currently challenging dicted to grow hierarchically, tracking the formation of structure due to the large number of theoretical uncertainties, including 69, 76, 77, 256 in our Universe. Current observations of stellar-mass compact but not limited to, the distribution of birth spins of BHs, objects by LIGO and Virgo might be giving us a glimpse of how which plays a key role in determining the retention fraction the very first few generations of hierarchical BHs are assembled. of BH merger products and therefore the expected rate of 2g Notably, hierarchical BH mergers present key features that mergers.89, 131 Additionally, the large number of proposed for- are set by the very fact that they are hierarchical, irrespectively mation channels makes it challenging to choose a particular of their precise astrophysical origin. First, hierarchical BHs can set of simulations to compare against the data. Some work in evade the mass cutoff imposed by the pair-instability supernova this direction includes the development of importance-sampling process and populate the predicted upper mass gap. Second, algorithms269 and machine-learning emulators266, 270 to speed up the spin of hierarchical BHs is almost entirely determined by the the evaluation of the population likelihood. A direct application relativistic emission of angular momentum at merger, a process of these ideas to the hierarchical-merger problem has, to be best which erases information of the spins inherited from the stellar of our knowledge, not yet been presented. progenitors. At the same time, BH remnants are imparted strong recoils at merger, which might eject them from their astrophys- 5.2 Measuring the mixing fraction ical host and prevent them from merging again. The key point is that all of these features (mass build up, remnant spin, and The confident identification of one or more hierarchical mergers merger recoil) are set by the relativistic dynamics of the merger in the GW catalog will be an important step towards inferring process. There is, therefore, the exciting possibility of identi- the contributions of different BH-binary formation channels to fying some of the LIGO/Virgo events as hierarchical, indepen- the total merger rate. Although this measurements is a natu- dently of the details of their specific formation pathway. That ral by-product of a complete population analysis which include said, such identifications then need to be explained within as- multiple channels, it is worth sketching out the key principle trophysical models able to accommodate repeated mergers in a here. consistent fashion. In particular, dynamical formation in nuclear Let us suppose for the sake of this argument that there are star clusters and gas-assisted migration in AGN disks appear to two channels at play, A and B, such that each of them is re- be the most promising candidates. For these reasons, one should sponsible for a given fraction of GW events in the observed cat- see the hierarchical assembly of BHs as an orthogonal but com- alog: fA + fB = 1. This can be be further separated into first- plementary strategy to constraining the formation channel(s) of generation and hierarchical mergers, i.e. fA,1g + fA,2g + fB,1g + merging compact binaries. fB,2g = 1. If a given theoretical model predicts that hierarchi- The LIGO, Virgo, and soon KAGRA interferometers are now cal mergers are possible with efficiencies λi = fi,2g/ fi (with i = on a solid path to deliver thousands of BH binary observations 271 A, B) and the fraction of hierarchical mergers f2g = fA,2g + fB,2g within a few years. A population of stellar-mass hierarchical can be identified experimentally from the data, one immediately BH mergers, if it is truly there in the Universe as tentatively sug- obtains the branching ratios f = (λ f )/(λ λ ) and gested by current data, is bound to emerge with increasing clar- A A − 2g A − B fB = (λB f2g)/(λB λA). The limit λA λB corresponds to ity. In the coming years, the study of hierarchical BH mergers of a scenario− where one of− the two pathways is much more likely to stellar origin and their GW signatures will be a continuous dia- produce repeated mergers, as in the case of isolated binary for- logue between theoretical modeling and observational findings, mation (A) and dynamical assembly (B). In this case, one imme- each field repeatedly prompting the other.

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Acknowledgements D.G. is supported by European Union’s H2020 ERC Starting Grant No. 945155–GWmining, Leverhulme Trust Grant No. RPG-2019-350, and Royal Society Grant No. RGS-R2-202004. M.F. is supported by NASA through the NASA Hubble Fel- lowship grant HST-HF2-51455.001-A awarded by the Space Telescope Science Institute. Computational work was per- formed on the University of Birmingham BlueBEAR cluster, the Athena cluster at HPC Midlands+ funded by EPSRC Grant No. EP/P020232/1, and the Maryland Advanced Research Comput- ing Center (MARCC).

Author Contributions This paper was written jointly by the two authors. D.G. pre- pared the figures. M.F. extracted distributions from LIGO/Virgo public data. Correspondence and requests for materials should be ad- dressed to D.G. The authors declare no competing interests.

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