Dynamical formation of the GW190814 merger
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Dynamical formation of the GW190814 merger
Manuel Arca Sedda Astronomisches Rechen-Institut, Zentrum f¨ur Astronomie der Universit¨at Heidelberg, M¨onchhofstr. 12-14, D-69120 Heidelberg, Germany (Received; Revised; Accepted)
Submitted to ApJLABSTRACTWe investigate the possible dynamical origin of GW190814, a gravitational wave (GW) source dis-covered by the LIGO-Virgo-Kagra collaboration (LVC) associated with a merger between a stellarblack hole (BH) with mass 23 . (cid:12) and a compact object, either a BH or a neutron star (NS), withmass 2 .
59 M (cid:12) . Using a database of 240,000 N -body simulations modelling the formation of NS-BHmergers via dynamical encounters in dense clusters, we find that systems like GW190814 are likely toform in young, metal-rich clusters. Our model suggests that a little excess ( ∼ − . − (cid:12) in the compact remnants’ mass spectrum leads to a detection rate fordynamically formed “GW190814 -like” mergers of Γ GW190814 (cid:39) − − Gpc − , i.e. within the ob-servational constraints set by the GW190814 discovery, Γ LVC ∼ −
23 yr − Gpc − . Additionally, ourmodel suggests that ∼ . − .
8% of dynamical NS-BH mergers are compatible with GW190426 152155,the only confirmed NS-BH merger detected by the LVC. We show that the relative amount of lightand heavy NS-BH mergers can provide clues about the environments in which they developed.
Keywords: black holes - neutron stars - star clusters - gravitational waves INTRODUCTIONThe LIGO-Virgo collaboration (LVC) detected re-cently GW190814, a merger between a BH with mass M BH = 23 . +1 . − . M (cid:12) and a mysterious compact ob-ject with mass M CO = 2 . +0 . − . M (cid:12) (The LIGO Sci-entific Collaboration et al. 2020). The properties ofGW190814 challenge our understanding of compact bi-naries: i) the secondary mass falls in the “lower massgap”, a range of masses (2 . − (cid:12) ) characterised bythe observational absence of stellar remnants (Bailynet al. 1998; ¨Ozel et al. 2012), ii) the mass ratio is small, q = 0 . +0 . − . , iii) and the inferred merger rate is fairlylarge, Γ LVC = 7 +16 − yr − Gpc − . The unusual mass ofGW190814 secondary suggests that this merger involvedeither the heaviest NS or the lightest BH known in acompact binary system. Although the BH hypotesisseems to be the favoured one (The LIGO Scientific Col-laboration et al. 2020), the existence of NS with massesup to 3 M (cid:12) (e.g. Freire et al. 2008; Tsokaros et al. 2020) Corresponding author: Manuel Arca [email protected] or, more in general, the absence of a lower mass gap(e.g. Wyrzykowski & Mandel 2020; Zevin et al. 2020)cannot be completely ruled out. Whether the secondaryis a NS or a BH, matching all GW190814 features – low-mass companion, low mass-ratio, and large merger rate –poses a challenge to astrophysical theories. Populationsynthesis models for isolated binaries predict mass ra-tios q > . M BH ∼
50 M (cid:12) (Yang et al. 2019),but comparable mass ratios, i.e. q = 0 . − . ∼
4% of AGN-assisted mergers can haveone of the binary components in the lower mass-gap(Yang et al. 2020). Other explanations include the ac-cretion of material expelled during the NS formation re-mained bound to the binary due to the large mass of theprimary (Safarzadeh & Loeb 2020), through the devel-opment of a hierarchical merger involving two NSs and a r X i v : . [ a s t r o - ph . H E ] F e b Arca Sedda, Manuel a BH (Lu et al. 2021) or, more in general, in hierarchicaltriples assembled either in the field or in dense clusters(e.g. Liu & Lai 2020). Alternatively, GW190814-likesystems may hint at a supernova (SN) mechanism act-ing on longer timescales than previously thought, thusenabling the proto-compact remnant to accrete enoughmass before undergoing explosion (Zevin et al. 2020).However, even in such case the formation of mergerswith a secondary mass and mass ratio compatible withGW190814 is almost impossible in the isolated binaryscenario, regardless of the SN explosion mechanism as-sumed (Zevin et al. 2020). Another potential forma-tion channel for GW190814 is via dynamical encountersin a star cluster. The dynamical formation of massivebinaries (e.g. BH-BH) is efficient in globular clusters(GCs), where compact remnants undergo dozens of in-teractions before either merging inside the cluster, orgetting ejected and merge afterward (Rodriguez et al.2016; Askar et al. 2017; Rodriguez et al. 2018). How-ever, binary BHs in GCs tend to have high mass ratios( q > .
5, Rodriguez et al. 2016), while the formation ofNS-BH binaries is suppressed owing to BHs that quenchmass segregation of lighter objects. Therefore, the in-ferred NS-BH merger rate for GCs in the local Universeis rather low, Γ
NSBH = 10 − − − (Clausen et al. 2013;Arca Sedda 2020; Ye et al. 2020). Young and open clus-ters (YCs) might be suitable formation sites for NS-BHmergers (Ziosi et al. 2014; Rastello et al. 2020). Thelarge number of YCs expected to lurk in galaxies (upto 10 in Milky Way, e.g. Piskunov et al. 2006) canboost the overall NS-BH merger rate, especially if theycontain a high fraction of primordial binaries (Rastelloet al. 2020). In a recent work, we explored the NS-BHdynamical formation channel through a suite of 240,000 N -body simulations (Arca Sedda 2020) tailored to re-produce scatterings in star clusters with velocity dis-persion σ = 5 −
100 km s − , thus covering the rangefrom YCs to nuclear clusters (NCs). We predict thatsome NS-BH mergers display distictive features, namelya chirp mass > (cid:12) , a BH heavier than M BH >
10 M (cid:12) ,and the absence of an electromagnetic (EM) counter-part even if the BH is highly spinning. In this Letter,we exploit this database to quantify the likelihood fordynamically formed GW190814-like sources. DYNAMICAL NS-BH MERGER RATES ANDTHE FORMATION OF GW190814-LIKESOURCESOur simulations (Arca Sedda 2020) model binary-single hyperbolic encounters involving two compact ob-jects and a normal star through
ARGdf (Arca-Sedda& Capuzzo-Dolcetta 2019), an improved version of the M NS M ST M BH Configuration:
BHSTNS
Configuration:
NSSTBH M BH M ST M NS Figure 1.
Schematization of the dynamical encounter driv-ing the formation of a neutron star - black hole binary (NS-BH) in a dense stellar environment. We assume that ei-ther a roaming NS scatters over a BH-star (ST) binary (leftpanel, configuration BHSTNS) or vice-versa (right panel,NSSTBH).
ARCHAIN N -body code that implements Post-Newtonianformalism up to order 2.5 (Mikkola & Merritt 2008) andenables a high-accuracy treatment for close encounters(Mikkola & Tanikawa 1999). We adopt a mass func-tion for compact objects such that all remnants with afinal mass < (cid:12) are labelled as NSs, whilst the re-maining are labelled as BHs. The database, containingover 240,000 simulations, is dissected into two configu-rations (see Figure 1): either the binary contains a BHand a star (ST) and the third object is a NS (configura-tion BHSTNS), or viceversa (NSSTBH). Details aboutthe initial conditions and the mass distribution of theobjects involved in the scattering are discussed in Ap-pendix A and B.We identify N GW = 1193 mergers, i.e. P NS − BH (cid:39) .
5% of the whole sample. To characterise how N GW varies across different values of the velocity dispersion( σ ), we define an individual merger rate (Γ ind ), namelythe number of mergers per unit time per cluster, asΓ ind = p GW N bin d R/ d t, (1)i.e. as the product between the fraction of NS-BH merg-ers ( p GW ) which is measured directly from our simula-tions, the average number of binaries containing eithera NS or a BH that at a given time coexist in the cluster( N bin ), and the rate of binary-single interactions thatlead to the formation of a NS-BH binary (d R/ d t ). Wefind that there is a tight relation between Γ ind and σ ,well described by a power-lawΓ ind (cid:39) kN bin (cid:18) σσ c (cid:19) δ . (2)In the equation above, σ c = 5 km s − , and k and δ ,whose values are summarized in Table 1, are best-fit ynamical formation of GW190814 merger N bin parameter is highly uncertain, as it dependson the mass of the cluster, the retention fraction of bothNSs and BHs, and the cluster relaxation time. To con-strain this quantity we resort to the suite of GCs MonteCarlo models named the MOCCA Survey Database I(Askar et al. 2017). In MOCCA we find on average N bin ∼ M < M (cid:12) and N bin = 2 − M andhalf-mass radius r h are linked to σ via (Arca Sedda 2020)Log ( GM/r h ) = (1 . ± .
03) + 2Log σ, (3)thus we can uniquely infer the individual merger rate fora cluster with given mass and half-mass radius throughits velocity dispersion. Table 2 lists Γ ind estimates as-suming typical values for σ in YCs, GCs, and NCs. Ata redshift z (cid:46)
1, the merger rate associated with a givencluster population can be roughly calculated as (ArcaSedda 2020; Ye et al. 2020):Γ NS − BH = Γ ind ρ MWEG N c , (4)where Γ ind is calculated through Eq. 2, ρ MWEG =0 . − is the local density of Milky Way equiva-lent galaxies (Abadie et al. 2010) and N c is the totalnumber of clusters in the galaxy. A typical NC hasΓ NC < . − , i.e. 1-3 orders of magnitude largerthan other cluster types. However, the contribution ofNCs to the population of NS-BH is likely rather low,as they are outnumbered by GCs (around 200 in theMilky Way Harris 2010) and YCs (up to 10 Piskunovet al. 2006). Assuming around 200 GCs and 1 NCsfor all MW-like galaxies in the local Universe impliesa merger rate Γ GC = (0 . − . × N bin yr − Gpc − and Γ NC = (3 × − − × − ) × N bin yr − Gpc − .To explain the LVC inferred rate, the number of binarieswith a compact object lurking in typical GCs and NCsshould thus be N bin ∼ − . However, N bin is morelikely to be (cid:46)
10 in GCs (e.g. Morscher et al. 2015; Kre-mer et al. 2020, see also Appendix A) and ∼ − inNCs (e.g. Arca Sedda et al. 2020), thus suggesting thatthese environments are unlikely to be main contributorto the population of NS-BH mergers. Extending our configuration Z k δ [Gyr − ]BHSTNS 0.0002 (5 . ± . × − . ± . . ± . × − . ± . . ± . × − . ± . . ± . × − . ± . Table 1.
Best fitting parameters for clusters individualmerger rate. Col. 1: scattering configuration. Col. 2: metal-licity. Col. 3-4: parameters of the fitting function for theindividual merger rate in Equation 2. calculations to YCs, instead, yields to:Γ YC = (cid:18) ρ MWEG . − (cid:19) (cid:18) N c (cid:19) N bin × (5) × . − . − Gpc − , σ = 0 . − , . − . − Gpc − , σ = 1 . − , . − . − Gpc − , σ = 3 . − , with lower limits corresponding to Z = 0 .
02 and theNSSTBH configuration. According to Eq. 3, the σ val-ues adopted above correspond to a cluster mass M YC =(280 − , − , (cid:12) assuming r h = 1 pc. Notethat such an optimistic rate is obtained assuming thatall MW-like galaxies in the local Universe have a numberof YCs ∼ , each of which contains at least N bin = 1binary that undergoes the type of scattering exploredhere.This rate falls within the LVC measurements (TheLIGO Scientific Collaboration et al. 2020) and is in re-markably good agreement with simulations of compactYCs (Rastello et al. 2020). Assuming instead similarnumber densities for YCs and GCs, ρ GC = 2 .
31 Mpc − ,leads to Γ YC = (1 . × − − .
06) yr − Gpc − , inagreement with recent results from Fragione & Banerjee(2020) (for more details see Appendix E). The agree-ment between our models and N -body simulations, inspite of the different assumptions adopted, suggests thatthe dynamical formation of NS-BH binaries is drivenmostly by stellar dynamics and is less affected by stellarevolution and post-Newtonian corrections. Our simpli-fied approach enables us to produce a catalogue of ∼ NS-BH mergers, which can be used to constrain theiroverall properties, and to access the NCs mass range,for which full direct simulations are prohibitive. DYNAMICAL FORMATION OF GW190814 ANDGW190426 152155Among all our models we find 11 mergers with aBH with mass 20 < M BH / M (cid:12) <
25 and a NSwith mass M NS > (cid:12) . One interesting example is Arca Sedda, Manuel Γ ind [Gyr − ]NSSTBH BHSSTNScluster M r h σ N bin Z Z type [ M (cid:12) ] [pc] [km s − ] 0.0002 0.02 0.0002 0.02YCs 3 × . . × − . × − . × − . × − YCs 3 × . . × − . × − . × − . × − YCs 3 × . . × − . × − . × − . × − GCs 10 . . × − . × − . × − . × − NCs 8 × . × − . × − . × − . × − Table 2.
Individual merger rate for different cluster types. Column 1-4: cluster type, mass, half-mass radius, and typicalvelocity dispersion. Col. 5: average number of binaries containing a NS or a BH. Col. 6-7: individual merger rate for NSSTBHconfiguration and different metallicities. Col. 8-9: individual merger rate for BHSTNS configuration and different metallicities. (Piskunov et al. 2007; Soubiran et al. 2018; Jackson et al. 2020). (Harris 2010). (Feldmeier et al. 2014). − . − . . . x [AU] − . − . − . − . . . . y [ AU ] M BH = 23 . (cid:12) M ST = 0 . (cid:12) a BHST = 0 .
04 AU e BHST = 0 . M NS = 2 .
77 M (cid:12)
BHSTNS15ZH S M BH = 23 . (cid:12) M NS = 2 .
77 M (cid:12) a NSBH = 0 . e NSBH = 0 . t GW = 1 . Figure 2.
Formation of a GW190814 prototype in one of ourmodels ( Z = 0 . σ = 15 km s − ). During the scattering theNS swaps with the star (ST), which is ejected away, leadingto the formation of a highly eccentric binary with mergingtime t GW (cid:46) . BHSTNS15ZH-S5561 (configuration BHSTNS, σ = 15km s − , metallicity Z = 0 . M BH =23 . (cid:12) and M NS = 2 .
77 M (cid:12) , i.e. within ∼ .
4% and ∼
7% from the GW190814 measured values. As shownin Figure 2, at formation BHSTNS15ZH-S5561 has asemimajor axis a = 0 . e = 0 . t GW = 1 . ∼ (cid:12) at solar metallicity (seeAppendix B). To identify other NS-BH mergers similar toGW190814, we use the BH mass M BH and the mass ra-tio q . Figure 3 compares these quantities for GW190814,our dynamical mergers, and isolated mergers (adaptedfrom Giacobbo & Mapelli 2018) for metal-poor ( Z =0 . Z = 0 .
02) stellar progenitors. Itmust be noted that Giacobbo & Mapelli (2018) adoptsa rapid
SN explosion scheme (see Fryer et al. 2012),whereas our model is based on a delayed
SN scheme forthe calculation of compact remnants’ masses. Nonethe-less, updated models of isolated binary evolution ac-counting for both rapid and delayed SN, which haveshown a broad agreement with Giacobbo & Mapelli(2018) models, suggest that the amount of mergers withproperties similar to GW190814 is limited to < . − P LVC = 4 . − . P LVC =8 . − .
3% for solar metallicity models. From Equa-tion 6 and Table 2 we can thus derive a rate for mergerssimilar to GW190814 in YCs as:Γ
LVC = P LVC Γ YC = (6)= (0 . − . N bin yr − Gpc − Z/ Z (cid:12) = 0 . , (0 . − . N bin yr − Gpc − Z/ Z (cid:12) = 1 , with the lower(upper) limits corresponding to the case σ = 1(3) km s − , i.e. the typical value of velocity dis-persion for MW young and open clusters(Soubiran et al.2018; Kuhn et al. 2019; Jackson et al. 2020), and to con-figuration NSSTBH(BHSTNS).Using the same procedure, we also seek for mergerssimilar to GW190426 152155, a NS-BH merger detected ynamical formation of GW190814 merger M BH [M (cid:12) ]10 − q [ m a ss r a t i o ] GW190814GW190426Dyn. (poor)Dyn. (rich) Iso. (poor)Iso. (rich) − − Figure 3.
Mass ratio q versus BH mass M BH for NS-BH mergers in our metal-poor (green dots) and metal-rich(purple dots) models, compared to the measured values forGW190814 and GW190426 152155 (white stars). We in-clude the combined distribution for metal-poor (red squares)and metal-rich (yellow squares) binaries derived from Gia-cobbo & Mapelli (2018) (model CC α <
30% from the observed M BH and q .In the side histograms, the dotted lines identify the mea-sured mass and mass ratio for both GW sources, while theshaded(empty) areas encompass the measured 90% crediblelevel for GW190814(GW190426 152155). during the O3 LVC observation run, characterised by M BH = 5 . +4 . − . M (cid:12) and M NS = 1 . +0 . − . M (cid:12) . We findmergers with primary mass and mass ratio within 30%of the measured value for GW190426 152155 in 15% ofour models regardless the progenitor metallicity, thusindicating that dynamical mergers can produce a sub-stantial fraction of systems with a relatively low mass.Note that the interval of M BH and q values assumedin this case falls well within the observed 90% credibleinterval level.The analysis above does not account for potential ob-servation biases that can affect GW detectors. For in-stance, the volume within which LIGO can detect agiven class of sources depends on several parameters,like the source mass and mass ratio, the distance, the skylocation, or the mutual inclination of the spins. For bi-naries with a total mass M + M = (10 − (cid:12) , Fish-bach & Holz (2017) showed that this volume scales withthe primary mass following a power-law V T ∝ M . anddecreases at decreasing the mass ratio q . Using Figure 1 in Fishbach & Holz (2017), we extract V T and q at afixed primary mass value M = (10 , , , ,
50) M (cid:12) ,finding that such relation is well described by a power-law
V T ∝ q β , with β ∼ . − .
6. In the following,we adopt a slope β = 0 .
5, which is the value associ-ated with a primary mass M ∼
20 M (cid:12) . As we showin Appendix D, setting q = 0 . q = 0 .
6, i.e. thevalues typical of systems with a primary M <
35 M (cid:12) ,leads our estimated merger rate to vary by less than10%. To mimic the selection effect connected with thebinary primary and mass ratio, we augment our pop-ulation of NS-BH mergers to 10,000 by sampling themfrom the combined M BH − M NS distribution and, fromthe augmented sample, we extract 1,000 NS-BH mergersweighing the probability to select a given mass and massratio with the selection functions f M = k BH M . (with k BH a normalization constant), and f q = k q q . (seealso Arca Sedda & Benacquista 2019; Arca Sedda et al.2020). The resulting volume weighted mass distributionfor BHs and NSs, and the M BH − q plane are shown inFigure 4. The percentage of mergers falling inside thelimiting values adopted for GW190814 remains limitedto P LVC = 4 . ± .
4% for metal-poor clusters, owing tothe fact that heavier BHs have a larger probability tobe selected. For metal-rich environments and the BH-STNS configuration, instead, this probability increasesto P LVC = 22 ± GW190814 , V = (cid:18) ρ MWEG . − (cid:19) (cid:18) N c (cid:19) N bin × (7) × (0 . − .
6) yr − Gpc − σ = 0 . − , (0 . − .
0) yr − Gpc − σ = 1 . − , (0 . − .
8) yr − Gpc − σ = 3 . − , at redshift z < ρ MWEG = 0 . − , N c = 10 , and N bin = 1 as scaling values.The lower(upper) limit corresponds to configurationNSSTBH(BHSTNS). The same calculation for metal-poor clusters yields to a maximum value of 0 . − . − Gpc − , thus suggesting that a population ofmetal-rich YCs with σ = 1 − − , i.e. r h ∼ . − . M ∼ − M (cid:12) , represent the most suitedclass of environments to explain the origin of GW190814.In comparison, the isolated scenario predicts a mergerrate of < . − Gpc − , regardless of the assumptionson the SN mechanism (Zevin et al. 2020).In the case of GW190426 152155, our procedure leadsto P LVC = 1 . − . Z = 0 . . Arca Sedda, Manuel contribution of dynamical mergers to the population oflow-mass NS-BH mergers can be non negligible.The merger rates above represent optimistic estimatesthat rely on the assumption that YCs: 1) have aroundsolar metallicity, 2) have all the same velocity disper-sion, and 3) are ∼ in MW-like galaxies. The dis-covery of other mergers similar to GW190814, i.e. withchirp masses (cid:38) (cid:12) and mass ratio < . < , − , ≥
15] M (cid:12) , and a companion mass in therange [ < , − . , > .
5] M (cid:12) . As summarized inTable 3, we find that a dominant contribution to NS-BHmergers from metal-rich clusters would result in a 96%probability to detect a primary heavier than >
15 M (cid:12) ,whereas for metal poor clusters this probability is com-parable for light and heavy primary components.In the extreme case in which NS-BH mergers formdynamically and mostly in metal-rich clusters, the esti-mate above implies that 9 out of 10 detections of NS-BH mergers would involve a BH with M BH >
15 M (cid:12) ,and 2-3 among them will have a companion with mass > (cid:12) . Comparing these predictions with expecta-tions from other channels and actual detections can helpin unravelling the markers of different formation chan-nels in detected sources and shed a light on the role ofdynamics in determining the assembly of NS-BH merg-ers. SUMMARY AND CONCLUSIONSIn this Letter we exploited a suite of 240,000 N -body simulations of hyperbolic encounters in star clus-ters to investigate the dynamical formation of NS-BH mergers with properties similar to GW190814 andGW190426 152155. Our main results can be summa-rized as follows:1. We find that the NS-BH merger probability de-pends strongly on the star cluster velocity disper-sion, following a power law with slope δ ∼ . − . .
5% of our models lead to a NS-BH merger;2. We derive an individual merger rate, i.e. numberof mergers per time unit, for typical NCs (up to ∼ . < .
06 Gyr − ), andYCs ( < .
01) Gyr − ;3. Since YCs outnumber GCs and NCs by a factorup to 10 in typical galaxies, they might be themajor contributor to the population of dynamical NS-BH mergers. In the local Universe, we infera NS-BH merger rate for YCs of Γ YC = (0 . −
36) yr − Gpc − ;4. Among all simulations, we identify ∼ −
10% NS-BH mergers with a BH mass and mass ratio com-patible with GW190814, and ∼
15% with proper-ties similar to GW190426 152155. We exploit ourmodels to derive a “raw” merger rate for dynam-ically formed GW190814-like sources, i.e. withmass and mass-ratio within 30% the observed val-ues, of Γ ∼ (0 . − .
9) yr − Gpc − ;5. To place our models in the context of LVC de-tections, we assume that the probability to selecta NS-BH merger in our sample depends on theprimary mass ( ∝ M . ) and mass ratio ( ∝ q . )to mimic the potential selection effects to whichGW detectors might be subjected. This “vol-ume weighted” sample contains only ∼ . ± . ∼ ± . − . − Gpc − at low redshift, in the ballpark of LVC predictionsand up to 100 times larger than the estimates ob-tained from isolated binary stellar evolution mod-els;7. Using the same selection procedure, we find ∼ . − .
7% of mergers with properties comparableto GW190426 152155, with the lower limit corre-sponding to metal-poor clusters. This relativelylow occurrence disfavours, but does not rule out,a dynamical origin for GW190426 152155;8. We suggest that the mass spectra of compact rem-nants in NS-BH merger candidates can be usedto identify markers of different formation chan-nels. In the extreme case in which all mergersformed dynamically in metal-rich clusters, we pre-dict that 9 out of 10 mergers should involve a BHwith M BH >
15 M (cid:12) and at least 2 of them in-volve a companion with mass > (cid:12) . Comparingthese predictions with future detections can sheda light on the dynamical channel and provide newinsights on the mass spectrum of compact objectsin the lower mass-gap range.ACKNOWLEDGEMENTS ynamical formation of GW190814 merger Z M NS [ M (cid:12) ] M BH [ M (cid:12) ] ( M BH , M NS ) [ M (cid:12) ] < − . ≥ . < − ≥ > > .
8% 19 .
5% 2 .
7% 4 .
0% 40 .
9% 55 .
2% 13 . .
4% 12 .
3% 0 .
3% 1 .
3% 3 .
3% 95 .
5% 23 . Table 3.
Occurrence of NS-BH mergers in different mass ranges. Col. 1: metallicity. Col. 2-4: probability to have a NS masswithin a given mass range. Col. 5-7: the same as for columns 2-4, but for BHs. Col. 8: probability to have a merger with aprimary mass > (cid:12) and a companion mass > (cid:12) . MAS acknowledges financial support from the Alexan-der von Humboldt Foundation for the research program“The evolution of black holes from stellar to galacticscales”, the Volkswagen Foundation Trilateral Partner-ship through project No. I/97778 “Dynamical Mecha- nisms of Accretion in Galactic Nuclei”, and the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) – Project-ID 138713538 – SFB 881 “The MilkyWay System”. MAS is grateful to Martina Donnari forproviding useful comments to an earlier version of thismanuscript.APPENDIX A. INITIAL CONDITIONS I. BINARY-SINGLE INTERACTION RATESThe idea at the basis of our approach is that BH-NS binaries form via interaction of a free roaming single compactobject (a BH or NS) and another compact object (a NS or BH) paired with a star. Here, we consider “NSs” allcompact objects with a mass < (cid:12) and “BHs” otherwise. Since the two compact objects are heavier than the star,on average this configuration favours the ejection of the least massive component and the formation of a binary witha higher binding energy (Sigurdsson & Phinney 1993). We explore two different configurations: NSSTBH (NS-starbinary impacting over a single BH), and BHSTNS (BH-star binary impacting over a single NS), and we vary the stellarmetallicity to either Z = 0 . Z = 0 .
02 and the cluster velocity dispersion to σ = 5 − − − − − − , thus covering the range of values going from young massive clusters (YCs), to globular clusters (GCs) andnuclear clusters (NCs).In such stellar ensembles, the interaction rate between a binary with component mass M , , semimajor axis a , andeccentricity e , and a single object with mass M can be written asd R/ d t = ˙ R = N bin nσ Σ , (A1)being N bin the average number of binaries coexisting in the cluster that contain either an NS or a BH, n the densityof scattering stars, σ the environment velocity dispersion, and Σ the binary cross sectionΣ = πa (1 − e ) (cid:20) G ( M + M + M ) σ a (1 − e ) (cid:21) . (A2)The density of a population of N CO (either BHs or NSs) with mean mass (cid:104) M CO (cid:105) inhabiting a cluster with mass M and half-mass radius R h can be estimated as n CO ∼ N CO / ( R ). For BHs, this quantity can be inferred for instancefrom recent studies on BH retention fraction and consequent formation of a tight BH subsystem (Breen & Heggie 2013;Morscher et al. 2015; Arca Sedda et al. 2018). In this work, we assume that the segregated population of BHs hasa density comparable to the overall density of the cluster, n BH ∼ M/ ( (cid:104) M BH (cid:105) R h ), as suggested in Arca Sedda et al.(2018). For NSs instead, we consider the fact that mass-segregation is prevented by the presence of BHs in the clustercentre, and that their total mass is around 0.01 times the cluster mass. Thus, we adopt n NS = 0 . n GC as an upperlimit on the average density of NSs. The number of binaries in the cluster is a crucial parameter. To bracket thisquantity, we take advantage of the MOCCA Survey Database I (Askar et al. 2017), a suite of around 2,000 MonteCarlo models of star clusters with initial masses in the range M = 2 . × − . × M (cid:12) , thus covering the massrange of massive YCs and GCs.Using the MOCCA models, we calculate the average number of binaries containing a BH(NS) in cluster at differenttimes (0 . − − − −
12 Gyr respectively) and in different mass bins, as shown in Figure 5.The figure highlights a clear, although non-trivial, dependence between N bin and the cluster mass. Clusters lighterthan M < M (cid:12) are characterised by N bin ∼ Arca Sedda, Manuel d N / d M M NS [M (cid:12) ]0 . . . . . . Z = 0 . Z = 0 . M BH [M (cid:12) ]BH Z = 0 . Z = 0 . Volume Weighted
20 40 60 M BH [M (cid:12) ] − q [ m a ss r a t i o ] GW190814 volume weighted
Dynamical (metal-poor)Dynamical (metal-rich) M chirp [M (cid:12) ]0 . . . . . n ( M c h i r p ) Z = 0 . (cid:12) Z =Z (cid:12) Figure 4.
Top: volume weighted mass spectrum for merging NSs (left, blue) and BHs (right, red) for a metallicity Z = 0 . Z = 0 .
02 (filled steps). The vertical lines mark the value measured for GW190814 and correspondinguncertainties. Central: as in Figure 3, but here the mass ratio and primary mass distribution are weighted with two selectionfunctions to mimic the dependencies affecting the detector accessible volume. Bottom: mergers chirp mass for metal-rich (filledred steps with dashed edge) and metal-poor (filled grey steps with straight edge) clusters. ynamical formation of GW190814 merger M GC [M (cid:12) ] N b i n BHs - 12GyrNSs - 12Gyr10 BHs - 6GyrNSs - 6Gyr10 BHs - 3GyrNSs - 3Gyr10 BHs - 1GyrNSs - 1Gyr10 BHs - 0.5GyrNSs - 0.5Gyr
Figure 5.
Average number of BH and NS in binaries as a function of the cluster mass and for different times, from top tobottom the time considered is 12 − − − − . between N bin ∼ − M > M (cid:12) instead, the number of BH in binaries varies between N bin = 1 −
10, almost regardless of the time, whereas the number of NS in binaries tends to be smaller N bin = 1 − N bin as a scaling value.We note that this assumption is compatible with results of other models (see e.g. Morscher et al. 2015), even the mostrecent one that implements updated stellar evolution for binary and single stars, new prescriptions for SN explosionmechanisms and recoil kicks, and post-Newtonian formalism for compact object interactions (e.g. Kremer et al. 2020),thus suggesting that the number of binaries left in the cluster at late evolutionary stages is likely the result of dynamics,rather than other mechanisms that, on the other hand, can affect the cluster structure.Under the set of assumptions above, adopting a cluster velocity dispersion of σ = 5 km s − , and using the medianvalues of the masses of the binary and third object and the binary semimajor axis and eccentricity, we estimate aninteraction rate of ˙ R ∼ − − for configuration NSSTBH and ˙ R ∼ −
400 Gyr − for configuration BHSTNS.To check the reliability of our calculations, we compare our results with MOCCA models as described in ourcompanion paper (Arca Sedda 2020), finding a range of values fully compatible with our theoretical estimate.0 Arca Sedda, Manuel M C O [ M (cid:12) ] NS BH Z = 1 Z (cid:12) Ze20SM17Be02This work0 25 50 75 100 M ZAMS [M (cid:12) ]10 M C O [ M (cid:12) ] NS BH Z = 0 .
01 Z (cid:12)
Ze20SM17Be02This work M NS [M (cid:12) ]10 − − n ( M N S ) Z = 1 Z (cid:12) Ze20 SM17 Be02 . . . . M NS [M (cid:12) ]10 − − n ( M N S ) Z = 0 .
01 Z (cid:12)
Ze20 SM17 Be02
Figure 6.
Left panel: final mass for compact objects as a function of the ZAMS mass adopted in this work (straight thin redline) in comparison with Ze20 (dashed thick black line), SM17 (dashed thick light green line), and Be02 (dashed thick lightblue line) models. The light grey area encompass the supposed lower mass-gap, whereas the dark grey area labels the mass-gapfeatured in our models. Right panel: NS mass spectrum adopted in this work (thick red line) compared with Ze20 (black steps),SM17 (dashed green steps), and Be02 (filled blue steps).B.
INITIAL CONDITIONS II. MASS AND ORBITAL PROPERTIES OF BINARY-SINGLE SCATTERINGEXPERIMENTSThe binary-single scattering configuration explored in this work is characterised by the masses of the three objects(star, BH, and NS), the orbit of the initial binary, and the orbital properties of the single-binary interaction.We sample the zero age main sequence (ZAMS) mass of the three objects from a Salpeter mass function. Atmetallicity values Z = 0 . . M minBH > . (cid:12) evolve intoBHs (Belczynski et al. 2002; Spera & Mapelli 2017), whereas lighter stars with a mass above M minNS > .
5) M (cid:12) evolve into NSs.For stars, we select only masses in the range 0 . − (cid:12) . This choice is motivated by the fact that the timescaleassociated with the binary single scatterings modelled here is 10-100 times longer than the half-mass relaxation time ofthe host cluster (see Appendix C), thus generally longer than the evolutionary time for stars with a mass m > (cid:12) ,which is t age = 10 Gyr( m/ M (cid:12) ) − . (cid:46) − . < M st / M (cid:12) < . − N -body models. Given the fact thatthey constitute a little fraction of the stellar population, as stars with a mass < (cid:12) constitute over the 95 .
5% ofthe whole population, and that their stellar evolution timescale is close (or even longer) than the typical time for thescatterings studied here, we exclude 1 − (cid:12) stars from our models.For BHs, we adopt the mass spectrum described in Spera & Mapelli (2017) (hereafter SM17). In SM17 the authorsuse the SEVN tool. This tool implements prescriptions for single stellar evolution that include several SN explosionmechanisms and a treatment for pair instability and pulsational pair instability SNe, which naturally lead to a dearthof compact remnants with a mass in the range 65 −
120 M (cid:12) (the so-called upper mass gap). To derive the BH massfrom the ZAMS mass we exploit Table 1-3 in Spera & Mapelli (2017) , according to which the compact remnant massis calculated adopting the delayed SN explosion mechanism (Fryer et al. 2012).For NS, instead, we adopt the single star stellar evolution model from Belczynski et al. (2002) (hereafter Be02)implemented in the BSE package (Hurley et al. 2002). The maximum NS mass in our models reaches ∼ . (cid:12) for The SEVN code was note publicly available when this study be-gan. ynamical formation of GW190814 merger Z = 0 . . . − .
05) at solar metallicities having a mass M NS > . (cid:12) , thusenabling us to explore the lower mass-gap region.The combined use of SM17 prescriptions for BHs and Be02 for NSs leads our simulations to naturally exhibit a narrowlower mass-gap in the range (3 − .
2) M (cid:12) at solar metallicity and (2 . − .
1) M (cid:12) for metal-poor systems. Figure 6compares the relation between the ZAMS and the remnant masses and the compact remnants’ mass spectrum adoptedhere, in SM17, in Be02, and the new single stellar evolution models described in Zevin et al. (2020) (hereafter Ze20).We note that the mass spectrum adopted here is half-way between the case in which a wide lower mass-gap do existand that in which NSs and BHs are linked by a continuous mass spectrum, and can provide a description of how thedelayed SN mechanism or, more in general, an explosion mechanism that leads to a narrower mass-gap, can impactthe properties of dynamically formed NS-BH mergers. In fact, our simulations suggest that an excess of 3 −
5% ofcompact remnants with masses in the range 2 . − (cid:12) can lead to a dynamical merger rate compatible with LIGOexpectations.We note that the delayed SN model adopted here is only one among many possibilities, e.g. rapid SN mechanism(Fryer et al. 2012), electron-capture SN (Podsiadlowski et al. 2004). Nonetheless, none of the current models in theliterature is capable of capturing the complex phases of SN physics, especially in the case of core-collapse SN, whichcan be altered significantly by stellar rotation (Mapelli et al. 2020) and requires full 3D hydrodynamical simulationsto be fully unveiled (e.g. Burrows et al. 2019). From the theoretical point of view, recent single and binary stellarevolution population synthesis suggest that matching the GW190814 features requires that SN explosion proceeds ontimescale longer than typically assumed (Zevin et al. 2020). Moreover, observations of NSs and BHs detected throughtheir electromagnetic counterparts seem to be inconclusive about the existence of a lower mass-gap, suggesting thatit might be populated of both massive NSs (Freire et al. 2008) and light BHs (Giesers et al. 2018; Thompson et al.2019). This is also suggested by recent measurements based on microlensing events detected by the OGLE surveycoupled with GAIA DR2 data, which favor a continuous mass spectrum in the 2 − (cid:12) mass range, rather than alower mass-gap (Wyrzykowski & Mandel 2020).Regarding the orbital properties of the binary and the incoming object, as detailed in our companion paper (ArcaSedda 2020) we assume that the binary-single interaction is hyperbolic and in the regime of strong deflection. For thebinary, we adopt as minimum semimajor axis the maximum between 100 times the star’s Roche lobe and 1000 timesthe ISCO of the compact object in the binary, to avoid that the star is disrupted or swallowed before the scatteringtakes place. The maximum semimajor axis allowed is, instead, calculated as the minimum between the hard-binaryseparation (Heggie 1975) and the relation suggested by Rodriguez et al. (2016), who have shown that dynamicallyprocessed binaries have typical semimajor axis proportional to the binary reduced mass µ and the ratio betweenthe cluster mass and semimajor axis, namely a ∼ k d µM/R h . We adopt k d = 10, which produces semimajor axisdistribution in full agreement with binary-single scatterings of this kind found in MOCCA simulations with a σ = 5km s − (see Figure 10 in Arca Sedda 2020). C. CRITERIA FOR THE IDENTIFICATION OF MERGER CANDIDATES.To identify NS-BH merger candidates we refine the selection procedure described in Arca Sedda (2020) as follows.We first calculate the GW timescale t GW for all binaries, assuming (Peters 1964) t GW = 5256 c a f f ( e f ) G M M ( M + M ) , (C3)and f ( e ) = (1 − e ) / / e + (37 / e . (C4)We mark all NS-BH binaries with t GW <
14 Gyr as ”merger candidates”. To determine whether these candidatescan undergo merger in a cluster environment we need to infer the time at which the scattering takes place, i.e. theNS-BH binary formation time t f , and the timescale over which the binary can get disrupted, e.g. via further strongencounters or secular perturbations.In a real cluster, the NS-BH binary formation time t f depends on a number of factors: the mass segregation time-scale, the core-collapse process, the formation of binaries and multiples in the cluster core, the formation or not ofa BH subsystem. All these features are not captured by our three-body models, but are naturally accounted forin the MOCCA models. Therefore, we use a multi-stepped approach exploiting these high-resolution Monte Carlo2 Arca Sedda, Manuel models. First, we use the MOCCA database to reconstruct the logarithmic distribution of the ratio between t f andthe half-mass cluster relaxation time calculated at 12 Gyr for all NS-BH binaries formed in MOCCA models,Log τ ≡ Log( t f /t r ) , (C5)where the relaxation time is calculated as (Binney & Tremaine 2008) t r = 0 .
65 Gyrlog(Λ) (cid:115) M c M (cid:12) (cid:18) r h (cid:19) / . (C6)This enables us to provide an estimate of t f for a given value of the cluster relaxation time, which is directly connectedwith the cluster velocity dispersion, mass, and half-mass radius (Binney & Tremaine 2008).The disruption of the NS-BH binary can be driven by either impulsive mechanisms, e.g. due to a strong encounterwith another compact object, or diffusive mechanisms, e.g. due to the effect of the continuous interactions withpassing-by stars or the mean field of the cluster.If the binary is soft, i.e. G ( M + M ) / (2 σ a ) > catastrophic regime,i.e. the binary is disrupted in a single interaction, and a diffusive regime, i.e. the binary is disrupted owing to thesecular effect impinged by interactions with cluster stars. A catastrophic interaction occurs if the impact parameterfalls below a maximum value (Binney & Tremaine 2008) b max = 1 . a (cid:18) GM p ( M + M ) σ a (cid:19) / , (C7)otherwise, the binary evolution is dominated by the secular, diffusive, mechanism.In the catastrophic regime, the binary disruption occurs over a timescale (Bahcall et al. 1985) t cat = 4 . (cid:18) k cat . (cid:19) (cid:18) (cid:12) pc − ρ (cid:19) (cid:18) M + M
20 M (cid:12) (cid:19) / (cid:18)
10 AU a (cid:19) / . (C8)In the diffusive regime, the binary can disrupt via high-speed encounters with perturbers of mass M p , a processcharacterised by a time scale t ds (Heggie 1975; Binney & Tremaine 2008) t ds (cid:39) . (cid:18) k d . (cid:19) (cid:16) σ
40 km s − (cid:17) (cid:18) M + M
20 M (cid:12) (cid:19) (cid:18)
20 M (cid:12) M p (cid:19) (cid:18)
10 pc − n p (cid:19) (cid:18)
10 AU a (cid:19) , (C9)where k d is a factor inferred from scattering experiments (Bahcall et al. 1985) and n p represents the perturbersnumber density. However, if these encounters are sufficiently rare, the binary can undergo disruption secularly, due tothe cumulative effect of all the weak interactions with cluster stars over. This process takes place over an evaporationtime t e (Binney & Tremaine 2008): t ev (cid:39) . (cid:16) σ
40 km s − (cid:17) (cid:18) ρ p
100 M (cid:12) pc (cid:19) (cid:16) a
10 AU (cid:17) (cid:18) logΛ6 . (cid:19) . (C10)If the binary is hard, instead, an interaction with a passing-by star tends, on average, to harden the binary further(Heggie 1975). As the binary hardens, the encounters become rarer but more violent, possibly resulting in the ejectionof the binary over a time t ej = (cid:18) k hd (cid:19) (cid:18) t r . (cid:19) , (C11)where t r is the cluster relaxation time and k hd is a parameter derived from scattering experiments (Goodman & Hut1993; Heggie & Hut 2003).All the timescales listed above depend on the cluster properties and the average values of scattering parameters.However, our three-body simulations are a) uniquely defined by the cluster velocity dispersion, which is degenerate inthe cluster mass and half-mass radius through Equation 3, and b) do not take into account the disruption, evaporation,or ejection processes. To partly solve the degeneracy and provide a more reliable description of the possible NS-BH ynamical formation of GW190814 merger • extract the τ value from the distribution reconstructed through MOCCA models; • extract the cluster half-mass radius r h from the observed distribution of Galactic GCs and local NCs; • combine r h and σ (which is fixed for each model) to calculate the cluster mass M c and the correspondingrelaxation time t r from Equation C6;for each i -th version of the same NS-BH merger candidate in a different environment, the formation time is thusuniquely defined as t f,i = τ i t r,i . This sampling is done for each value of the velocity dispersion explored here, i.e. σ = 5 , , , , ,
100 km s − .To characterise the disruption processes, instead, we extract an impact parameter b i from a linear distribution 2 bdb ,as expected from geometrical considerations, limited above by the cluster free mean path and below by 0.1 times thebinary semimajor axis, i.e. sufficiently hard to pose a threat to the binary survival . The mass of the perturber isextracted from a power-law mass function with slope α MF = − . M p = (0 . −
60) M (cid:12) . However,it must be noted that if the cluster core is dominated by heavy remnants, the mass function can be significantly steeper.For instance, if the cluster contains a BH subsystem, Arca Sedda et al. (2018) suggested that half of the mass insidethe subsystem is contributed from stars and the remaining in BHs. If we assume that stellar BHs have masses in therange (5 −
60) M (cid:12) and that the overall mass function in the subsystem is described by a power-law, it is possible toshow that inside the subsystem α MF ∼ −
1. We found that a steeper mass function decreases the number of mergercandidates by 12%.If the binary is soft, we check whether it is in the diffusive ( b i > b max ) or in the catastrophic regime, calculating thecorresponding disruption timescales.According to this statistical procedure, each NS-BH is characterised by 10,000 potential outcomes. We identify aspotential mergers those fullfilling one of the following conditions: • t GW + t f <
14 Gyr and t GW < t cat if the binary is soft and in the catastrophic regime, • t GW + t f <
14 Gyr and t GW < min( t ds , t ev ) if the binary is soft and in the diffusive regime, • the binary is hard.If one of the conditions is fullfilled in at least 5% of the interactions modelled for the same NS-BH candidate, we labelit as a merger. The number of times the merging binary has been identified as hard ( N hd ) or soft ( N sft ) determinesthe binary status, which is labelled as “hard” if N hd > N sft or “soft” otherwise. Similarly, we label the merger asin-cluster or ejected depending on the number of times that the binary has been identified as a candidate for ejectionbefore merging or not.We repeated the same procedure 10 times to verify the impact of randomization onto the calculation of the individualmerger rate (Equation 2) and we notice an overall variation of 2 −
4% in the values quoted in Table 4.In general, we find that the vast majority of NS-BH mergers in our models come from hard binaries ( > − σ = 15 −
35 km s − are ejected from the parent cluster before the mergertakes place. D. THE ROLE OF THE DETECTOR SENSITIVITYThe detection of a GW source depends intrinsically on several parameters, such as the distance at which the mergertook place, the direction of the wave hitting the detector, and the properties of the binary emitting GWs. Recently, We verified that decreasing further this limit does not impact theresults. Arca Sedda, Manuel configuration
Z σ f sft f hd , in f hd , ej [Z (cid:12) ] [km s − ] [%] [%] [%]NSSTBH 0.0002 5 0.0 100.0 0.00.0002 15 0.0 95.2 4.80.0002 20 0.0 90.3 9.70.0002 35 0.0 98.5 1.50.0002 50 1.7 98.3 0.00.0002 100 15.5 84.5 0.0BHSTNS 0.0002 5 0.0 100.0 0.00.0002 15 0.0 90.9 9.10.0002 20 0.0 95.5 4.60.0002 35 2.3 97.7 0.00.0002 50 7.3 91.0 1.80.0002 100 5.5 94.5 0.0NSSTBH 0.02 5 0.0 100.0 0.00.02 15 4.0 88.0 8.00.02 20 0.0 88.1 11.90.02 35 0.0 95.1 4.90.02 50 3.0 94.1 3.00.02 100 10.6 88.7 0.7BHSTNS 0.02 5 0.0 100.0 0.00.02 15 0.0 91.7 8.30.02 20 0.0 94.4 5.60.02 35 0.0 98.5 1.50.02 50 1.0 99.0 0.00.02 100 4.4 95.6 0.0 Table 4.
Col. 1: configuration. Col. 2: cluster metallicity. Col. 3: cluster velocity dispersion. Col. 4-6: percentage of mergersfrom soft binaries, from hard binaries merging inside the cluster, from hard binaries merging outside the cluster.Z P LVC β = 0 . β = 0 . β = 0 . Table 5.
Col. 1: metallicity. Col. 2-4: percentage of models with GW190814-like mass and mass ratio assuming that theaccessible volume scales with the mass ratio as q β . Fishbach & Holz (2017) have shown that the volume (
V T ) accessible to the LIGO detector scales with the mass of theprimary through a power-law
V T ∝ M α , with α = 2 .
2, assuming 10 < M / M (cid:12) <
100 and at a fixed mass ratio, andincreases at increasing the mass ratio if the primary mass is kept fixed (e.g. cfr. with their Figure 1). In order to takeinto account this observational bias in our analysis, we extract the data from Fishbach & Holz (2017) Figure 1 . Weassociate to the extracted data a conservative error of 10% and reconstruct the dependence between the volume andthe mass ratio at fixed primary mass. As shown in Figure 7, we find, for primary masses in the range 10 −
50 M (cid:12) , thatthe volume-mass ratio relation is well represented by a power-law
V T ∝ q β with a slope in the range β = 0 . − . β = 0 . β = 0 . β = 0 . P LVC to find a GW190814-like mergerin our database once this ”volume weighting” procedure is taken into account. We find that varying the slope in the We use the data extraction tool https://apps.automeris.io/wpd/. ynamical formation of GW190814 merger . . . . . − − − S e n s i t i v e V o l u m e [ G p c − y r ] m = 10 M (cid:12) ; α = 0 . ± . m = 20 M (cid:12) ; α = 0 . ± . m = 25 M (cid:12) ; α = 0 . ± . m = 30 M (cid:12) ; α = 0 . ± . m = 50 M (cid:12) ; α = 0 . ± . Figure 7.
Sensitive redshifted spacetime volume,
V T , of the LIGO detectors in observation runs O1 and O2 as a function ofthe mass ratio and for different values of the primary mass. The dotted lines represent least square fits of the data. The datapoint are extracted from Fishbach & Holz (2017).
V T − q relation causes a maximum variation in P LVC of up to (cid:46) P LVC = (22 ± P LVC = (4 . ± . E. COMPARISON WITH SIMILAR WORKSIn this section we compare our inferred merger rate with recent results obtained through full N -body simulations ofYCs (Rastello et al. 2020; Fragione & Banerjee 2020).To perform the comparison, we make use of Equation 3 to derive the mass, half-mass radius, or velocity dispersionof host clusters. For the sake of comparison, Figure 8 shows the link between these three parameters and highlightstypical values for Galactic GCs, YCs, and the NC.Rastello et al. (2020) predict a global merger rate of 28 yr − Gpc − , but only ∼ −
55% of these mergers formdynamically, thus implying a dynamical merger rate of Γ (cid:39)
14 yr − Gpc − . The simulations presented by Rastelloet al. (2020) focus on star clusters with masses in the range M = (0 . − × M (cid:12) and half-mass radii R h = 0 . − . σ = 0 . − . − , according to our Equation 3.In this range of σ values, our inferred individual merger rate is roughly Γ ind ∼ (3 × − − × − ) Gyr − , dependingon the metallicity (the larger the metallicity the lower the rate) and configuration (BHSTNS produces more mergers).If we assume that Rastello et al. (2020) models represent the “normal” population of YCs in a MW-sized galaxy,we infer a merger rate of Γ YC ∼ . −
13 yr − Gpc − (see Equation 6 in the main paper), thus bracketing the valueinferred from direct models. Note that the assumption that the density of YCs in the local Universe is 2.31 Mpc − (e.g. similar to that of globular clusters Rodriguez et al. 2016; Ye et al. 2020; Fragione & Banerjee 2020) leads to amerger rate of Γ YC = (7 × − − . × − ) yr − Gpc − . In a more recent work, Fragione & Banerjee (2020) exploredthe output of 65 N -body simulations of clusters with masses M = 10 − M (cid:12) and half-mass radii R h = 1 − ρ YC ∼ .
31 Mpc − , the authorsderive an upper limit on the NS-BH merger rate of 3 × − yr − Gpc − . For these models, the velocity dispersioninferred from Equation 2 in the main paper is σ (cid:39) − . − . According to our calculations, the individualmerger rate for these types of clusters is Γ ind = (9 . × − − . − , depending on the configuration and themetallicity. This implies a local Universe merger rate of Γ = ρ Γ ind ∼ (2 . × − − .
13) yr − Gpc − , thus embracing6 Arca Sedda, Manuel M/ M (cid:12) − − L og r h / p c − σ [ k m s − ] Figure 8.
Surface map showing the cluster velocity dispersion as a function of mass and half-mass radius. From left to right,black lines identify clusters with a velocity dispersion of 0 . , , , , ,