Neutron Star-Black Hole Mergers from Gravitational Wave Captures
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Neutron Star-Black Hole Mergers from Gravitational Wave Captures
Bao-Minh Hoang,
1, 2
Smadar Naoz,
1, 2 and Kyle Kremer
3, 4, 5, 6 Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095 Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208, USA Center for Interdisciplinary Exploration & Research in Astrophysics (CIERA), Northwestern University, Evanston, IL 60208, USA The Observatories of the Carnegie Institution for Science, Pasadena, CA 91101, USA TAPIR, California Institute of Technology, Pasadena, CA 91125, USA
ABSTRACTLIGO’s third observing run (O3) has reported several neutron star-black hole (NSBH) merger candi-dates. From a theoretical point of view, NSBH mergers have received less attention in the communitythan either binary black holes (BBHs), or binary neutron stars (BNSs). Here we examine single-single(sin-sin) gravitational wave (GW) captures in different types of star clusters— galactic nuclei (GN),globular clusters (GC), and young stellar clusters (YSC)— and compare the merger rates from thischannel to other proposed merger channels in the literature. There are currently large uncertaintiesassociated with every merger channel, making a definitive conclusion about the origin of NSBH mergersimpossible. However, keeping these uncertainties in mind, we find that sin-sin GW capture is unlikelyto significantly contribute to the overall NSBH merger rate. In general, it appears that isolated binaryevolution in the field or in clusters, and dynamically interacting binaries in triple configurations, mayresult in a higher merger rate. INTRODUCTIONThe recent gravitational wave (GW) detections ofmerging BBHs and BNSs (Abbott et al. 2017a) (Abbottet al. 2016a,b, 2017b,c,d; The LIGO Scientific Collabo-ration & The Virgo Collaboration 2018) by LIGO/Virgohave ushered in a golden age of GW astrophysics. Thefirst (O1) and second (O2) observing runs of the ad-vanced LIGO/Virgo detector network have yielded in-ferred estimates for the merger rate of BBHs (9 . −
101 Gpc − yr − ) and BNSs (110 − − yr − ) (Ab-bott et al. 2019). A very diverse range of mechanismsand astrophysical environments have been invoked toexplain these mergers, such as: dynamical interactionsin globular clusters (e.g. Portegies Zwart & McMillan2000; Wen 2003; O’Leary et al. 2006; Antonini et al.2014; Rodriguez et al. 2016; O’Leary et al. 2016; Kre-mer et al. 2020b) and galactic nuclei (e.g. O’Leary et al.2009; Kocsis & Levin 2012; Antonini & Perets 2012;Ramirez-Ruiz et al. 2015; Hoang et al. 2018; Fern´andez& Kobayashi 2019), active galactic nuclei (e.g. McKer-nan et al. 2012; Bartos et al. 2017; Stone et al. 2017;Secunda et al. 2019), isolated binary evolution in the Corresponding author: Bao-Minh [email protected] field (e.g. Mandel & de Mink 2016; de Mink & Man-del 2016; Belczynski et al. 2016; Marchant et al. 2016),Population III stars (e.g. Kinugawa et al. 2014, 2016;Hartwig et al. 2016; Inayoshi et al. 2016; Dvorkin et al.2016), and primordial black holes (e.g. Bird et al. 2016;Clesse & Garc´ıa-Bellido 2017; Sasaki et al. 2016).The non-detection of any neutron star-black hole(NSBH) mergers during O1 and O2 puts a 90% con-fidence interval upper limit on the NSBH merger rate of0 −
610 Gpc − yr − . However, there are currently severalcandidates for NSBH mergers in O3 and it is likely thatthere will be a confirmed detection within the decade(see The LIGO Scientific collaboration (2020)). NSBHsare much less well studied than either BBHs or BNSs.However, over the years there have been a number ofstudies estimating the rate of NSBH mergers from anumber of channels. In this work we explore a relativelyunexplored channel for producing NSBH mergers — ec-centric GW captures in dense clusters. GW capturesare well-studied as a promising formation mechanismfor BBHs, particularly in GN. NSBH captures are lessstraightforward than BBH captures due to the potentialpresence of tidal effects. Furthermore, NSs and BHsinteract with each other less often than they do withmembers of their own species due to mass segregationin clusters. We explore these complications in three a r X i v : . [ a s t r o - ph . H E ] S e p Hoang, Naoz & Kremer different types of cluster in this work: GNs, GCs, andYSCs. We then compare our results to results fromother channels, consider the limits of the different chan-nels, and finally discuss likely origins for a future NSBHmerger detection in LIGO. We begin by summarizingthe various channels that have been proposed to explainNSBH mergers below:1.
Mergers in the Field : The most well studied chan-nel for merging NSBH is binary evolution in thefield. Early studies using population synthesishave resulted in a large range of merger rates,0 . − . − in aLIGO, due to many uncer-tainties in binary evolution models (e.g., Sipior &Sigurdsson 2002; Pfahl et al. 2005; Belczynski et al.2007, 2010; O’Shaughnessy et al. 2010). Belczyn-ski et al. (2011) studied X-ray source Cyg X-1—a likely NSBH progenitor— and found an “empir-ical” rate of NSBH mergers based on the evolu-tion of this system: (2 − × − yr − gal − (in-ferred volumetric rate ∼ . − .
014 Gpc − yr − ),which is much smaller than the previously esti-mated rates from population synthesis. A morerecent binary population synthesis study by Do-minik et al. (2015) estimated NSBH merger ratesin aLIGO to be 0 . − . − for a standard bi-nary evolution model, up to 3 . − . − for anoptimistic common envelope evolution model, anddown to 0 . − . − for a pessimistic modelwith high BH kicks (inferred volumetric rate acrossall models ∼ . −
14 Gpc − yr − ). More recently,Kruckow et al. (2018) found an optimistic upperlimit to NSBH mergers via isolated binary evolu-tion in the field of up to ∼
53 Gpc − yr − .2. Mergers in Globular Clusters (GC) : NSs have beenwell-observed in GCs dating back to the 1970s asboth X-ray (e.g. Clark 1975; Heinke et al. 2005)and radio sources (e.g. Lyne et al. 1987; Ransom2008). Over the past decade, a growing amount ofevidence has suggested GCs also retain BH pop-ulations (e.g. Strader et al. 2012; Giesers et al.2019). Thus the question of NSBH formation inGCs arises naturally. Several studies have sug-gested that the rate of NSBH mergers is muchlower in GCs than in the field through variouslines of reasoning (Phinney 1991; Grindlay et al.2006; Sadowski et al. 2008). A few authors havesince calculated the rates of NSBH mergers inGC through dynamical processes. For example,Clausen et al. (2013) studied binary-single (bin-sin) interactions in a static cluster potential, andfound a merger rate of ∼ . − .
17 Gpc − yr − . More recently, Ye et al. (2020) studied dynamicallyformed NSBHs in GCs using the code CMC (Clus-ter Monte Carlo, see Kremer et al. (2020b) for de-tails), and found a rate of 0 . − .
06 Gpc − yr − .Arca Sedda (2020) studied hyperbolic bin-sin in-teractions in dense clusters, and found a NSBHmerger rate of 3 . × − − .
25 Gpc − yr − in GCs.The upper limits of these GC rates are indeedonly comparable to the lower limits of the theo-retical field rates, supporting the idea that fieldchannels dominate over GC channels for NSBHmergers. Note that all three of the aforemen-tioned studies focused on binary-mediated inter-actions, i.e. binary-single (bin-sin), and binary-binary (bin-bin) interactions, and did not include single-single (sin-sin) encounters, which will bethe focus of this paper.3. Mergers in Galactic Nuclei (GN) : The extentof previous studies for NSBH mergers in GNare similarly limited. For example, Arca Sedda(2020) studied bin-sin mergers in GN and founda rate of ∼ × − − . × − Gpc − yr − .O’Leary et al. (2009) focused on mergers ofBBHs in GN resulting from sin-sin GW captures(e.g., Quinlan & Shapiro 1987; Lee 1993), usingcompact object densities resulting from Fokker-Planck simulations. They estimated that therate of NSBH mergers from this channel willbe roughly 10 − yr − gal − for a GN around a4 × M (cid:12) SMBH, or 1% of the BBH rate. Tsang(2013) also estimated the rate of NSBH mergersin GN, but using density profiles of an isothermalsphere instead of density profiles from a Fokker-Planck simulation, and found a merger rate of ∼ × − − × − yr − gal − for a GN sur-rounding a 4 × M (cid:12) SMBH (calculated fromtheir Eq. A9).We note that there is a great deal of subtletyand uncertainty concerning the conversion of aper galaxy merger rate for fixed SMBH mass toa volumetric/expected detection rate for GW cap-tures in GN. The most straightforward way is tosimply multiply the per galaxy rate by a galaxynumber density in the universe to find the vol-umetric rate; and then multiply the volumetricrate by the volume observable by LIGO to findan expected detection rate. However, as O’Learyet al. (2009) noted, there may be significant vari-ance in central cusp densities between differentGNs with the same SMBH mass . Since the rate ofGW captures scales as density squared, this vari-ance may lead to an enhancement of the afore-mentioned volumetric/expected detection rate bya factor of ξ . The true value of ξ is highly un-certain. Whereas O’Leary et al. (2009) and Koc-sis & Levin (2012) estimate ξ to be (cid:38)
30, Tsang(2013) found that ξ is at most ∼
14 under veryoptimistic assumptions. O’Leary et al. (2009) andTsang (2013) found volumetric (expected LIGOdetection) rates of ∼ .
07 Gpc − yr − ( ∼ − )and ∼ . − . − yr − ( ∼ . −
20 yr − ), re-spectively, using their different values of ξ . With-out enhancement from ξ , these rates will decreaseto roughly ∼ .
002 Gpc − yr − ( ∼ .
03 yr − ) and ∼ . − .
05 Gpc − yr − (0 . − . − ), respec-tively. Due to the high uncertainty in the value of ξ , in this work we calculate and adopt GW capturerates without ξ as our fiducial rates, which yieldsconservative estimations. However, we will discussthe implications for LIGO should ξ be significant.Aside from GW captures, GN can be the site ofbinary mergers induced by interactions with theSMBH (e.g., Antonini & Perets 2012; Naoz 2016;Stephan et al. 2016, 2019; Hoang et al. 2018),via the Eccentric Kozai-Lidov Mechanism (EKLKozai 1962; Lidov 1962; Naoz 2016). RecentlyLu & Naoz (2019) suggested that supernova na-tal kicks can tend to shrink the post supernovaseparation. Moreover they showed that these sys-tems are more likely to stay in a triple configura-tion near an SMBH. On the other had, a super-nova kick rarely keeps stellar-mass tertiary. Thus,torques from the SMBH can lead to enhancementof NSBH mergers, compared too field binaries.Subsequently, Stephan et al. (2019) studied stel-lar binary evolution in GN with EKL includingself-consistent post-main sequence stellar evolu-tion and found that LIGO may detect NSBHmergers from this mechanism at a rate of 2 − − (inferred volumetric rate: 0 . − .
33 Gpc − yr − ).Fragione et al. (2019a) performed a study of com-pact binary mergers in GN induced by EKL, con-sidering different binary parameter distributionsand SMBH masses, and found a NSBH mergerrate of 0 . − . − yr − . Comparing theseEKL rates to the sin-sin GW capture rates fromO’Leary et al. (2009) and Tsang (2013), we seethat if ξ is small (i.e. the variance in GN density islow), mergers induced by EKL will dominate overmergers from GW captures in GN. Conversely, if ξ is significant, then mergers from GW captures willbe either comparable to or dominate over mergersfrom EKL. Recently McKernan et al. (2020) studied compactobject binary mergers in AGN disks, the gas inwhich has previously been shown to potentiallyaccelerate binary mergers (e.g. McKernan et al.2012; Bartos et al. 2017; Stone et al. 2017; Se-cunda et al. 2019). They found that this channelcan potentially produce NSBH mergers at rates of f AGN , BBH (10 − − yr − , where f AGN , BBH is the fraction of BBH mergers observed by LIGOthat come from the AGN channel. They expectthat 10-20% of NSBH mergers from this channelwill have electromagnetic counterparts, which canhelp disentangle this channel from others.4.
Mergers in Young Stellar Clusters (YSC) : Moststars, including massive stars that are BH and NSprogenitors, are born in YSCs (Carpenter 2000;Lada & Lada 2003; Porras et al. 2003). Theirhigher density relative to the galactic field meansthat compact binaries can form from dynamicalinteractions similar to those in GCs, as well asfrom stellar binary evolution. As a result, a num-ber of studies have explored YSCs as a possi-ble birthplace for BBHs (e.g. Portegies Zwart &McMillan 2002; Banerjee et al. 2010; Kouwen-hoven et al. 2010; Goswami et al. 2014; Ziosi et al.2014; Mapelli 2016; Di Carlo et al. 2019; Baner-jee 2017, 2018; Fujii et al. 2017; Kumamoto et al.2019; Rastello et al. 2019), with promising results.Recently, Rastello et al. (2020) studied the forma-tion of NSBHs in YSCs from redshifts 0-15. Theyfound that YSCs can produce NSBHs that mergein the local universe (redshift < .
1) at a rate of ∼
28 Gpc − yr − , through a combination of bi-nary evolution and dynamical interactions. Mostof these NSBHs will be ejected from YSCs beforethey merge, and so will ostensibly be “field” bi-naries when they are detected by LIGO. However,NSBHs that formed in YSCs have a different massspectrum from those that formed in true isola-tion in the field, and may be differentiated in thisway. The rate found in Rastello et al. (2020) islikely an optimistic estimation of the YSC mergerrate, due to the following reasons. Rastello et al.(2020) assume a NS natal kick distribution witha root mean square of 15 km / s, whereas obser-vational studies of pulsar proper motions in theliterature show that a majority of NSs likely re-ceive very large natal kicks ( ∼ −
500 km / s)at birth (e.g. Hansen & Phinney 1997; Lorimeret al. 1997; Cordes & Chernoff 1998; Fryer et al.1999; Hobbs et al. 2004, 2005; Beniamini & Piran2016). High velocity natal kicks tend to disrupt Hoang, Naoz & Kremer binaries and may significantly reduce the rate ofNSBH formation from binary evolution. In addi-tion, the high stellar densities and fractal initialconditions adopted in Rastello et al. (2020) maynot be representative of all YSCs, and thereforemay overestimate the influence of dynamis. Forcomparison, lower density models found in anotherrecent work, Fragione & Banerjee (2020), resultedin an upper limit of 3 × − Gpc − yr − for theNSBH merger rate from binary evolution and dy-namical exchanges in YSCs. Note that while theanalysis in Rastello et al. (2020) and Fragione &Banerjee (2020) included dynamical binary inter-actions and exchanges, they also did not includesin-sin GW capture. We will give an order of mag-nitude upper limit estimation of the rate of NSBHmergers due to sin-sin GW captures in YSCs inthis work.5. Mergers in Triples:
Stellar multiplicity studieshave shown that ∼
15% of massive stars — pro-genitors of BHs and NSs — have at least two stel-lar companions (e.g. Raghavan et al. 2010; Sanaet al. 2013; Dunstall et al. 2015; Jim´enez-Estebanet al. 2019). Several studies have explored theformation of BBH mergers in stellar triples andquadruples (e.g. Antonini et al. 2017; Silsbee &Tremaine 2017; Fragione & Kocsis 2019; Liu &Lai 2019). Recently, Fragione & Loeb (2019a)and Fragione & Loeb (2019b) studied NSBH merg-ers in field triples and found merger rates of ∼ . × − −
22 Gpc − yr − , where the wide rangecomes from uncertainties in the metallicity of theprogenitor population, and the magnitude of BHand NS natal kick velocities.The paper is organized as follows: We begin with de-scribing the basic equations that govern sin-sin GW cap-ture in Section 2. We then calculate the sin-sin NSBHmerger rate in GCs, GNs, and YSCs in Section 3. Fi-nally, we offer our discussions about the most probablyNSBH merger channels in Section 4. SINGLE-SINGLE GRAVITATIONAL WAVECAPTURESTwo compact objects undergoing a close encountercan emit enough gravitational wave energy to become abound binary. Because these encounters are relativistic,and the velocity dispersion of galactic nuclei and otherclusters are much less than the speed of light, they are al-most always nearly parabolic (Quinlan & Shapiro 1987;Lee 1993). We consider these approximately parabolicencounters and subsequent gravitational wave captures of stellar-mass black holes of mass m BH and neutronstars of mass m NS , total mass M tot = m BH + m NS , asymmetric mass ratio η = m BH m NS / (( m BH + m NS ) ), arelative velocity of v rel , and an impact parameter of b .The energy that is emitted in GWs in such an encounteris (Peters & Mathews 1963; Turner 1977):∆ E GW = − πG / √ c η M / r / (1)where c is the speed of light, G is the gravitational con-stant, and r p is the distance of closest approach of theencounter: r p = (cid:32)(cid:115) b + G M b v + GM tot b v (cid:33) − . (2)If | ∆ E GW | > M tot ηv (the kinetic energy of the en-counter), a bound NSBH binary is formed (e.g., Lee1993). This criterion implies a maximum impact param-eter b max to form a bound binary (e.g., O’Leary et al.2009; Gond´an et al. 2018a): b max = (cid:18) πη (cid:19) / GM tot c (cid:16) v rel c (cid:17) − / . (3)There is also a minimum impact parameter b min to forma bound binaries, as encounters with b < b min will resultin a direct collision rather than a bound binary. Theseencounters may result in an electromagnetic event butwill likely not result in any strong GW signals. Thiscollisional impact parameter is defined as (Gond´an et al.2018a), b min = 4 GM tot c (cid:16) v rel c (cid:17) − . (4)The total GW capture cross section is thus: σ ( m BH , m NS , v rel ) = π ( b − b ) . (5)We note that during these encounters, energy is alsolost due to tidal oscillations in the neutron star excitedby the black hole (e.g., Press & Teukolsky 1977), andcontributes to σ . To check whether we should includethis effect in our calculations or whether it can be safelyneglected, we approximate the tidal energy dissipatedduring a parabolic encounter according to the formalismpresented in Press & Teukolsky (1977):∆ E T = (cid:16) Gm R NS (cid:17)(cid:16) m BH m NS (cid:17) (cid:88) l =2 , ,... (cid:16) R NS R min (cid:17) l +2 T l , (6)where R NS is the radius of the neutron star, R min is theperiastron of the approach, and T l are dimensionless val-ues associated with each spherical harmonic l (see Press& Teukolsky (1977) for calculation of T l ). We only con-sider the quadrupole mode ( l = 2), which dominatesover the other modes (Press & Teukolsky 1977). We ap-proximate the NS as a polytropic star of index n = 0 . T l . Notethat since there is a minimum impact parameter, thereis minimum possible value of R min . For a parabolic en-counter the relationship between the impact parameterand the periastron distance is: R min ( b ) = b v GM tot . (7)Thus, combining Equations (4) and (7), we find the min-imum possible R min to be: R min ( b min ) = 8 GM tot c . (8)Encounters with R min < R min ( b min ) will result in a di-rect collision between the BH and NS instead of a boundbinary. In Figure 1 we plot ∆ E T / ∆ E GW — the ratio ofenergy lost to tidal oscillations to the energy lost to grav-itational waves— as a function of R min , for a 5 M (cid:12) BHand a 1 . (cid:12) NS. We have also marked the region where R min < R min ( b min ). We see that in the region of interestwhere R min > R min ( b min ), i.e., where bound binaries canform, ∆ E T / ∆ E GW < − , an extremely small value.We have verified (not shown to avoid clutter), that forlarger BH masses, ∆ E T / ∆ E GW is even smaller. This isconsistent with previous studies about NS-NS captures(e.g., Gold et al. 2012; Chirenti et al. 2017). Thus, wecan safely neglect tides in our calculation of the capturecross section. EVENT RATESThe rate of NSBH binary formation for a single clusteris: Γ cl = (cid:90) d r πr n BH ( r ) n NS ( r ) × (cid:90) d m BH F BH ( m BH ) (cid:90) d m NS F NS ( m NS ) × (cid:90) d v rel ψ m BH ,m NS ( r, v rel ) σv rel , (9)where n BH ( r ) and n NS ( r ) are the number densities of ablack holes and neutron stars, respectively; F BH ( m BH )and F NS ( m NS ) are the mass probability distribu-tions for black holes and neutron stars, respectively; ψ m BH ,m NS ( r, v rel ) is the distribution of the relative ve-locity between m BH and m NS at r.For GCs, we approximate the BH and NS populationsof each cluster as following a Maxwellian velocity distri-bution. Thus, the BH and NS populations have velocity Figure 1. Ratio of energy lost to tides to energy lostto GWs ( ∆ E T / ∆ E GW ), as a function of encounter pe-riastron ( R min ) . Plotted for parabolic encounters betweena 5 M (cid:12) BH and a 1 . (cid:12) NS. The region highlighted blue de-notes encounters with impact parameter less than b min (givenby Eq. (4))– these encounters will result in a direct collisionbetween the BH and NS. The region of interest for us isthe region to the right of the blue line, where a NSBH canform. In this region, ∆ E T / ∆ E GW < − , an extremelysmall value. We note that encounters between a NS and aBH of mass greater than 5 M (cid:12) will result in even smaller val-ues of ∆ E T / ∆ E GW . Thus, we conclude that we can ignoretidal effects in our calculations. dispersions of v d , BH and v d , NS , respectively. We thencalculate the average relative velocity between BHs andNSs in a cluster, < v rel > = (cid:113) (8 /π )( v , BH + v , NS ), anduse this constant value in Equation (9). Note that in thiscalculation we have neglected any mass or r dependencein v rel for simplicity. Thus we have: (cid:90) GC d v rel ψ m BH ,m NS ( r, v rel ) σv rel ≈ σ < v rel > . (10)For GNs, O’Leary et al. (2009) showed that the lastintegral in Equation (9) is only weakly dependent on therelative velocity distribution, and is well approximatedby: (cid:90) GN d v rel ψ m BH ,m NS ( r, v rel ) σv rel ≈ σv c ( r ) , (11)where v c ( r ) = (cid:112) Gm SMBH /r is the circular velocity at r and σ is evaluated at v rel = v c ( r ).We can then calculate the nominal volumetric NSBHmerger rate due to sin-sin GW capture:Γ NSBH = n cl Γ cl , (12)where n cl is the density of clusters (either GN or GC, weuse a slightly different calculation for YSCs, see section Hoang, Naoz & Kremer
Figure 2. Cumulative merger rate per cluster Γ cl ( yr − ) as a function of r . Note that for GN cases (red), r denotes the distance from the SMBH; whereas for GC cases(blue), r denotes the distance away from GC center. TheGC lines do not extend below r ∼ . cl is the rate per cluster.Note that even though the capture rate is not technicallythe same as the merger rate, the vast majority of binariesthat form due to GW capture tend to be very tight,eccentric, and merge very quickly after capture. Thus,the capture rate is an extremely good approximation ofthe merger rate (e.g. O’Leary et al. 2009; Gond´an et al.2018a). 3.1. Globular Clusters (GC)
We calculate the capture rates for a single simulatedcluster that is representative of a typical Milky Waycluster (initial cluster mass 4 × M (cid:12) , final mass2 × M (cid:12) , core radius ∼ . Z (cid:12) ), taken from the latest CMC catalogue (Kre-mer et al. 2020b). We compare the contribution fromyounger clusters versus older clusters by considering thissimulated cluster at 1 Gyr and 10 Gyr. To performthe integral in Equation (9), we numerically calculatethe densities n BH ( r ) and n NS ( r ), and the mass distri-butions F BH ( m BH ) and F NS ( m NS ) from the simulationdata (note this particular simulation contains 527 BHsand 825 NSs at t = 1 Gyr and contains 38 BHs and 778 NSs at t = 10 Gyr). As previously mentioned in thissection, we adopt Maxwellian velocity distribution forthe BH and NS populations. We fit the BH and NS ve-locities with respect to cluster center to a Maxwelliandistribution in order to find their velocity dispersions,and from those velocity dispersions calculate the aver-age relative velocity < v rel > (see Equation (10)). InFigure 2, we show the cumulative merger rate of a clus-ter as a function of distance, r , from the center for theyoung (blue, solid line) and old (blue, dashed line) clus-ter.We find a total merger rate per GC of Γ cl , GC ∼ × − yr − (2 × − yr − ) for the 1 Gyr (10 Gyr)GC. In the younger 1 Gyr cluster, the BH populationcontains many higher mass BHs (mass range 5 −
40 M (cid:12) ).Through mass segregation, this BH population formsa dense subsystem in the cluster’s center that subse-quently generates significant energy through “BH burn-ing” (the cumulative effect of dynamical binary forma-tion, hardening, and ejections; for review, see Kremeret al. 2020a). This process influences the large-scalestructural properties of the host cluster, in particular,delaying the onset of cluster core collapse by preventingthe migration of lower-mass stars, including NSs, to thecluster’s center (e.g., Merritt et al. 2004; Mackey et al.2008; Breen & Heggie 2013; Peuten et al. 2016; Wanget al. 2016; Arca Sedda et al. 2018; Kremer et al. 2019;Ye et al. 2020). As a result, there is only density overlapbetween BHs and NSs in the outer regions of the clusterwhere densities are low (i.e., there are no NSs in the in-ner most regions where the BHs dominate). This resultsin an extremely low rate of capture.However, as the cluster ages, the total number of re-tained BHs decreases as a result of the dynamics withinthe BH subsystem (see, e.g., Morscher et al. 2015). Ad-ditionally, because the highest mass BHs are dynami-cally ejected first, as the cluster ages, the BH mass dis-tribution becomes increasingly dominated by lower-massBHs (5 −
15 M (cid:12) ). As a consequence of these effects, theenergy generated through the BH burning process (andtherefore effect of the BHs on the clusters radial profile)becomes less significant as the cluster ages. Thus, theNS population is able to infiltrate the inner regions ofthe cluster more effectively, resulting in more overlap be-tween the BH and NS in the r ∼ . − n cl = 2 . − for the density of GCs in theuniverse (e.g., Portegies Zwart & McMillan 2000), wefind volumetric rates of Γ NSBH , GC ∼ − Gpc − yr − (7 × − Gpc − yr − ) for the 1 Gyr (10 Gyr) case. Thisis at least one order of magnitude below every other pro-posed channels, see Figure 3. Thus, we conclude thatsingle-single GW capture is not a major contributor tothe NSBH merger rate in GCs.3.2. Galactic Nuclei (GN)
It has been shown (e.g., Bahcall & Wolf 1977) thata spherically symmetric multi-mass stellar populationorbiting a SMBH within its radius of influence will re-lax into a power-law number density profile of the form n ∝ r − α , where α varies with mass. The most mas-sive members of the cluster will tend to segregate tothe center of the cluster. This problem has been stud-ied by various authors using Fokker-Planck formalismwith various assumptions about the GN environment(e.g., Bahcall & Wolf 1977; Freitag et al. 2006; Hopman& Alexander 2006; Alexander & Hopman 2009; Keshetet al. 2009; O’Leary et al. 2009; Aharon & Perets 2016).For this work we calculate GW capture rates in GN us-ing BH and NS density profiles from four different sce-narios studied by Aharon & Perets (2016) (henceforthAP16). AP16 studied a cluster surrounding a SMBH ofmass 4 × M (cid:12) , composed of a two-mass populationsof BH (10 M (cid:12) & 30 M (cid:12) ), NS (1 . (cid:12) ), white dwarfs(0 . (cid:12) ), and main-sequence stars (1 M (cid:12) ). We digi-tally analyzed their Figures 1 and 2 to obtain BH andNS density estimates from the four different GN evolu-tion models they considered: Model 1 — cluster evolvedfrom pre-existing cusp with compact object (CO) forma-tion in outer cluster regions, does not include 30 M (cid:12) BHpopulation; Model 2 — similar to Model 1, but includesthe 30 M (cid:12)
BH population; Model 3 — built-up clusterwith in situ star formation in the outer cluster regions;Model 4 — cluster evolved from pre-existing cusp withCO formation in the inner cluster regions. See AP16 formore details about the assumptions that went into thecalculation of these density profiles. We approximatethe relative velocity of an encounter at distance r withthe circular velocity at distance r , as shown in Equation11. We show the cumulative merger rate per GN as afunction of r for these four scenarios in Figure 2.We find a total merger rate per GN ranging fromΓ cl , GN ∼ × − − × − yr − for our four GNevolution scenarios. The density of GN in the universeis a very uncertain value, but is often assumed in theliterature to be in the range of ∼ . − .
04 Mpc − (e.g., Conselice et al. 2005; Tsang 2013). However, con-sidering a wide range of SMBH masses it may be as highas ∼ . − (Aller & Richstone 2002; O’Leary et al.2009). Thus, we adopt n cl = 0 . − . − for thecosmic density of GN. We then find a total volumetric rate of NSBH mergers in GN due to sin-sin GW capturesof Γ NSBH , GN ∼ . − .
06 Gpc − yr − As explained in the introduction (also see O’Learyet al. (2009) and Tsang (2013)), this rate maybe en-hanced by a factor of ξ , which accounts for the variancein central cusp densities between different GNs. Thevalue of ξ is highly uncertain, but it maybe as high as afew tens (O’Leary et al. 2009). Thus, our nominal rateof Γ NSBH , GN ∼ . − .
06 Gpc − yr − is a conservativeone. 3.3. Young Stellar Clusters (YSC)
We perform an order of magnitude estimation adopt-ing models of massive YSCs from Banerjee (2017) andBanerjee (2018). We calculate the number density dis-tributions of BHs and NSs, n BH ( r ) and n NS ( r ), from thecumulative radial distributions obtained by digitally an-alyzing the left hand side plots of Figure 8 from Baner-jee (2018). These radial distributions are given for acluster with initial mass M cl ( t = 0) = 7 . × M (cid:12) and metallicity Z = 0 . Z (cid:12) , in snapshots at t = 100Myr, 1000 Myr, 5000 Myr, 7500 Myr, and 10 Gyr Weassume a cluster velocity dispersion of 3 km / s for v rel (e.g., Portegies Zwart et al. 2010), and single mass dis-tributions of BH and NS of 20 M (cid:12) and 1 . (cid:12) , respec-tively. We calculate the per cluster merger rate, Γ cl , YSC ,at different time snapshots using Equation 9. The percluster merger rate for our nominal YSC model rapidlydecreases as the cluster ages, going from Γ cl , YSC ∼ − yr − at t = 100 Myr to Γ cl , YSC ∼ − yr − for t > ∼ ∼
15 pc between 100 Myr and 10 Gyr, asshown in Figure 2 of Banerjee (2018). Cluster expan- The radial distributions are normalized with respect to thetotal number of bound BH and NS in the cluster, N BH , bound and N NS , bound , which are unfortunately given in neither Banerjee(2018) nor Banerjee (2018) for a M cl (0) = 7 . × M (cid:12) clus-ter. However, the time evolution of N BH , bound is given for fourother cluster masses in the range M cl (0) = (1 − × M (cid:12) in Figure 4 of Banerjee (2017). Based on numbers obtainedfrom Banerjee (2017), figure 4, we fit N BH , bound ( M cl ) withboth a linear and quadratic distribution to extrapolate a rangefor N BH , bound ( M cl (0) = 7 . × ) at the different time snap-shots. N NS , bound is given for a M cl (0) ≈ × M (cid:12) in Fig-ure 2 of Banerjee (2017), from which we can calculate the ratio N NS , bound /N BH , bound for M cl (0) ≈ × M (cid:12) . Assuming thatthis ratio is roughly constant with increasing cluster mass, we cancalculate N NS , bound ( M cl (0) = 7 . × ) from the extrapolated N BH , bound values. We can now unnormalize n BH ( r ) and n NS ( r )and estimate per cluster sin-sin capture rate. Hoang, Naoz & Kremer sion leads to a decrease in stellar density, which greatlylowers the frequency of dynamical captures.We then calculate the volumetric rate similarly toZiosi et al. (2014):Γ
NSBH , YSC ≈ Γ cl , YSC M cl (0) t eff ρ SF f SF , (13)where ρ SF = 1 . × − M (cid:12) Mpc − is the density ofstar formation at redshift 0 (adopted from Hopkins &Beacom 2006), and f SF = 0 . t eff = 100 Myr(even though a 7 . × M (cid:12) cluster can live up toabout 10 Gyr). Thus, we take Γ cl , YSC in Equation 13to be Γ cl , YSC ( t = 100 Myr) ≈ − yr − . We findΓ NSBH , YSC ≈ × − Gpc − yr − .We note that this is an upper-limit rate estimationfor the following reasons. First of all, our nominal YSCmodel, with a mass of 7 . × M (cid:12) , is much more mas-sive and contain more stellar and compact objects thanthe average YSC/open cluster (for comparison, the clus-ter models in Rastello et al. (2020) have masses rangingfrom 3 × − M (cid:12) ). Thus, by using this clustermodel as our fiducial model, we are overestimating theper cluster contribution for the average YSC. Secondly,our nominal YSC has a low metallicity of Z = 0 . Z (cid:12) .Since lower cluster metallicity increases the number ofcompact objects formed, our fiducial cluster metallicityis on the optimistic end of the spectrum.We see that our optimistic estimation for the sin-sinmerger rate in YSCs is still very low compared to themajority of other merger channels, as seen in Figure3. Thus, we can conclude that sin-sin GW capture inYSCs most likely do not contribute to the overall NSBHmerger rate. These relatively more massive young clusters are often re-ferred to in the literature as “young massive clusters” (YMCs),see Portegies Zwart et al. (2010) for a review.
Figure 3. Comparison of NSBH merger rates fromvarious channels.
We group merger channels into fourmajor categories: mergers taking place in galactic nu-clei (GN), globular clusters (GC), young stellar clusters(YSC), and the galactic field. Within the GN cate-gory we highlight three channels: EKL-assisted mergers(Stephan et al. 2019); binary-mediated interactions (bin-sin/bin) (Arca Sedda 2020); and sin-sin GW captures (thiswork). Within the GC category we show rates from binary-mediated interactions (bin-sin/bin) (Clausen et al. 2013; Yeet al. 2020; Arca Sedda 2020); and sin-sin GW captures (thiswork). Within the YSC category we show the rate fromsin-sin GW captures (this work). We denote this rate withan arrow to indicate that this is an upper-limit estimation.Within the Field category we show rates from isolated bin.evol. (iso. bin, Dominik et al. 2015; Kruckow et al. 2018);and field triples (Fragione & Loeb 2019a,b).4.
DISCUSSIONIn Figure 3, we compile and compare the predictedNSBH merger rates from the merger channels discussedin Section 1, as well as the results from this work. Wegroup these channels into four major categories: merg-ers taking place in GN, GC, YSC, and the galactic field.For the GN category we include the following chan-nels: EKL-assisted mergers (rate from Stephan et al.(2019)); binary mediated interactions (rate from ArcaSedda (2020)); and sin-sin GW captures (rate com-piled from O’Leary et al. (2009), Tsang (2013), and thiswork). For the GC category we include the followingchannels: binary mediated interactions (rate compiledfrom Clausen et al. (2013), Ye et al. (2020), and ArcaSedda (2020)); and sin-sin GW captures (rate from thiswork). For the YSC category we include sin-sin GWcaptures (rate from this work). For the field channel weused the rates from Dominik et al. (2015) and Kruckowet al. (2018).We see from Figure 3 that sin-sin GW captures ishighly unlikely to be the dominant mechanism for theproduction of NSBH mergers. Indeed, sin-sin GW cap-tures do not appear to contribute to the overall NSBHrate in any significant way. The most major caveat tothis statement concerns the sin-sin estimate in the GNcategory. As discussed in Section 1, the sin-sin ratesfor GN estimated in this work may be underestimatedby a “ ξ ” factor, which is due to the variance of stellardensities in the GN cusp. The actual value of ξ remainshighly uncertain — there are both pessimistic (Tsang2013) and optimistic (O’Leary et al. 2009) estimates of ξ in the literature. If we assume an optimistic value for ξ of a few tens, then the sin-sin GW capture rates inGN will be comparable to both the EKL and field rates.In the future, once LIGO has detected a statisticalpopulation of NSBH mergers, we will know whether thesin-sin merger rates have been systematically underesti-mated here by looking at the eccentricity distribution ofthese mergers. It has been shown that aLIGO can dis-tinguish eccentric stellar-mass compact binary mergersfrom circular ones for e (cid:38) . − .
081 at 10 Hz (Loweret al. 2018; Gond´an & Kocsis 2018). Broadly speak-ing, mergers from dynamical channels are expected tobe more eccentric in the LIGO band than mergers fromisolated binary evolution in the field, which are expectedto be predominantly circular (e.g., O’Leary et al. 2009;Cholis et al. 2016; Rodriguez et al. 2018b; Zevin et al.2019; Lower et al. 2018; Samsing 2018; Gond´an et al.2018b; Randall & Xianyu 2018). Amongst dynamicalmerger channels, some channels are predicted to yieldmore eccentric mergers than others. For example, Ro-driguez et al. (2018a) found that roughly 6% of BBHmergers from bin-sin interactions in GCs have e (cid:38) . . For NSBH mergers from bin-sin en-counters in GN, Arca Sedda (2020) found that none willhave eccentricity above the minimum detection thresh-old in the LIGO band, but that a large fraction have e > . Rodriguez et al. (2018a) distinguishes between bin-sin mergersthat take place after one or more bin-sin encounters, and merg-ers that take place during a bin-sin encounter due to significantGW emission at close passage. They have termed the latter “GWcapture” mergers. These mergers make up virtually all of theirmergers with e (cid:38) .
05 in the LIGO band. While the physicalmechanism underlying the GW emission in these “GW capture”mergers is the same physical mechanism underlying GW emissionin sin-sin GW captures, and despite of the similar name, we stressthat they are a completely distinct merger channel. For the pur-poses of this work, we group them with the other bin-sin mergersin GCs. For more information about these mergers, see Samsing(2018), Rodriguez et al. (2018a), and Rodriguez et al. (2018b).
BBH EKL mergers will have detectable eccentricity inthe LIGO band. However, sin-sin GW capture is by farand away the merger channel that results in the most ec-centric mergers in the LIGO band. Tak´atsy et al. (2019)studied BBH mergers in GN from both EKL and sin-sinGW captures, and found that ∼
75% of sin-sin GWcapture mergers will have e > . ∼
10% for EKL. Thus, if future NSBHmerger observations show a preponderance of eccentricmergers, then it is likely that we have underestimatedour sin-sin merger rates here, and most likely in the con-text of GN.The current nominal estimates shown in Figure 3 showfour channels that are possible dominant contributorsto the NSBH merger rate. These channels have over-lapping statistical uncertainties, and are isolated binaryevolution, triples in the field, binary-mediated interac-tions in GCs, and EKL in GN . There is a high prob-ability that the future observed NSBH merger popula-tion is a “blend” of two or more merger channels. Wemay be able to disentangle the contributions from dif-ferent channels by looking at merger distributions ineccentricity, mass, spin, etc., as different merger chan-nels produce different characteristic distributions in themerger parameter space (e.g., Rodriguez et al. 2018a;Tak´atsy et al. 2019; Arca Sedda 2020; Rastello et al.2020). However, different channels do sometimes over-lap in merger parameter space, so it may be very difficultto fully quantify the contribution of each merger channelto the observed distributions. We may also be able toclassify individual GW source (although this is proba-bly not be possible for a majority of GW mergers). Thiscan be accomplished with the detection of electromag-netic counterparts (e.g. Lee et al. 2010; Tsang 2013), orthrough the detection of imprints on the GW waveformpresent with some merger channels. For instance, forsome EKL-assisted mergers in GN, the gravitational pullof the SMBH on the merging binary can be detected inboth LIGO and LISA due to induced GW phase shifts(Inayoshi et al. 2017; Meiron et al. 2017), and eccen-tricity variations (Hoang et al. 2019; Randall & Xianyu2019; Emami & Loeb 2020; Deme et al. 2020; Guptaet al. 2020). ACKNOWLEDGEMENTSWe thank Bence Kocsis for helpful discussions.B.M.H. and S.N. acknowledge the partial support ofNASA grant No. 80NSSC19K0321 and No. 80NSSC20K0505.S.N. also thanks Howard and Astrid Preston for their Note that the degree of uncertainty in the different channelsvaries considerably. Hoang, Naoz & Kremer generous support. K.K. is supported by an NSF Astron- omy and Astrophysics Postdoctoral Fellowship underaward AST-2001751.REFERENCES
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