Black Hole Mergers From Globular Clusters Observable by LISA I: Eccentric Sources Originating From Relativistic N -body Dynamics
aa r X i v : . [ a s t r o - ph . H E ] A ug MNRAS , 1–6 (2018) Preprint 23rd August 2018 Compiled using MNRAS L A TEX style file v3.0
Black Hole Mergers From Globular Clusters Observable by LISA I:Eccentric Sources Originating From Relativistic N -body Dynamics Johan Samsing ,⋆ , Daniel J. D’Orazio Department of Astrophysical Sciences, Princeton University, Peyton Hall, 4 Ivy Lane, Princeton, NJ 08544, USA Department of Astronomy, Harvard University, 60 Garden Street Cambridge, MA 01238, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We show that nearly half of all binary black hole (BBH) mergers dynamically assembledin globular clusters have measurable eccentricities ( e > .
01) in the LISA band (10 − Hz),when General Relativistic corrections are properly included in the N -body evolution. If onlyNewtonian gravity is included, the derived fraction of eccentric LISA sources is significantlylower, which explains why recent studies all have greatly underestimated this fraction. Ourfindings have major implications for how to observationally distinguish between BBH form-ation channels using eccentricity with LISA, which is one of the key science goals of themission. We illustrate that the relatively large population of eccentric LISA sources reportedhere originates from BBHs that merge between hardening binary-single interactions insidetheir globular cluster. These results indicate a bright future for using LISA to probe the originof BBH mergers. Key words: gravitation – gravitational waves – stars: black holes – stars: kinematics anddynamics – globular clusters: general
Gravitational waves (GWs) from merging binary black holes(BBHs) have been observed (Abbott et al. 2016b,c,a, 2017a,b);however with the sparse sample collected to far, it is notclear where and how these BBHs formed in our Universe.From a theoretical perspective, several formation channels havebeen suggested including isolated field binaries (Dominik et al.2012, 2013, 2015; Belczynski et al. 2016b,a), dense stellarclusters (Portegies Zwart & McMillan 2000; Banerjee et al. 2010;Tanikawa 2013; Bae et al. 2014; Rodriguez et al. 2015, 2016a,b,b;Askar et al. 2017; Park et al. 2017), single-single GW captures ofprimordial BHs (Bird et al. 2016; Cholis et al. 2016; Sasaki et al.2016; Carr et al. 2016), active galactic nuclei discs (Bartos et al.2017; Stone et al. 2017; McKernan et al. 2017), galactic nuclei(O’Leary et al. 2009; Hong & Lee 2015; VanLandingham et al.2016; Antonini & Rasio 2016; Hoang et al. 2017), and very massivestellar mergers (Loeb 2016; Woosley 2016; Janiuk et al. 2017;D’Orazio & Loeb 2017). Although these proposed pathways seemto give rise to similar merger rates and observables, recent workinterestingly suggests that careful measurements of the BBH orbitaleccentricity and relative spins might be the key to disentanglingthem. For example, BBH mergers forming as a result of field binaryevolution are likely to have correlated spin orientations, except if athird object is bound and the three objects form a hierarchical triple ⋆ [email protected]; [email protected] ( e.g. , Liu & Lai 2017; Antonini et al. 2017), whereas BBH mergersforming in clusters are expected to have randomized orientationsdue to frequent exchanges ( e.g. , Rodriguez et al. 2016c). Regardingeccentricity, it was recently shown by Samsing (2017) that ≈ > .
1) when entering the observablerange of the ‘Laser Interferometer Gravitational-Wave Observatory’(LIGO). As argued by Samsing (2017), this population originatesfrom GW capture mergers forming in chaotic three-body interac-tions ( e.g. , Gültekin et al. 2006; Samsing et al. 2014) during clas-sical hardening, which explains why all recent Newtonian N -bodystudies have failed in resolving the correct fraction. In fact, it wasanalytically derived in Samsing (2017) that a Newtonian N -bodycode will always result in a rate of eccentric mergers that is ≈ e.g. , © 2018 The Authors Samsing, D’Orazio
Blanchet 2014), have previously been shown to play a crucial role inresolving eccentric LIGO populations ( e.g. , Samsing et al. 2018b);however, the possible effects related to LISA have not yet been prop-erly studied. Our motivation is to explore what can be learned aboutwhere and how BBH mergers form in our Universe from a LISAmission; we identify possible observable differences between dif-ferent BBH populations formed in GCs compared to those formedin the field. Motivated by previous studies, we focus in this paperon the eccentricity distribution. We note that the recent work byBreivik et al. (2016) did indeed look into this; however, the dataused for that study did not include GR effects, which we in thispaper show are extremely important.Using a semi-analytical approach, we find that ≈ > .
01) in the LISA band(10 − Hz) compared to the results reported by Breivik et al. (2016),when GR effects are included. This leads to the exciting conclu-sion that about 40% of all GC BBH mergers are expected to havea measurable eccentricity in the LISA band, whereas a field BBHpopulation in comparison will have ≈ between their hardening binary-single interactions inside their GC ( e.g. , Rodriguez et al. 2017). Thispopulation was not included in the recent study by Samsing et al.(2018a), which focused solely on the BBH mergers forming during the binary-single interactions. These BBH mergers where shown toelude the LISA band, and joint observations with LIGO are there-fore necessary to tell their GC origin. The fact that BBH mergers canbe jointly observed by LISA and LIGO was recently pointed out bySesana (2016) and Seto (2016). Discussions on BBH merger chan-nels and eccentricity distributions relevant for LISA were presentedin Nishizawa et al. (2016) and (Nishizawa et al. 2017). However,we note again that all previous studies have greatly underestimatedthe fraction of eccentric LISA sources from GCs, mainly due tothe omitted GR effects in the data set derived in Rodriguez et al.(2016a). It would be interesting to see how the results presented inthis paper affect those previous studies.The paper is organized as follows. In Section 2 we describethe approach we use for modeling the dynamical evolution of BBHsinside GCs and their path towards merger, when GR effects areincluded in the problem. Our main results are discussed in Section3, where we show for the first time that, with the inclusion of GReffects, nearly half of all BBH mergers forming in GCs are expectedbe eccentric in LISA. We conclude our study in Section 4. In this section we describe the new approach we use in this paper forestimating the distribution of GW frequency, f GW , and eccentricity, e , of BBH mergers forming in globular clusters (GCs). Using this,we explore the possible observable differences between differentBBH populations forming in GCs and those forming in the field foran instrument similar to LISA, and the role of GR in that modeling.As described in the Introduction, in the recent work by Breivik et al.(2016) it was claimed that ≈
10% of the GC mergers will havean eccentricity > .
01 at 10 − Hz, compared to ≈
0% for thefield population. However, the simulations used is Breivik et al.(2016) did not treat the relativistic evolution of BBHs inside the GCcorrectly, which essentially prevented BBHs to merge inside theirGCs (see Rodriguez et al. (2017) for a description). To improve ontheir study we combine in the sections below a simple Monte Carlo(MC) method with the analytical framework from Samsing (2017)
Figure 1.
The graphics in the three columns above illustrate the threedifferent dynamical pathways for merging BBHs to form, each of whichresult in a different type of GW merger. The horizontal steps from top tobottom illustrate the stepwise decrease in the BBH’s SMA due to harden-ing binary-single interactions, which progresses as δ a HB , δ a HB , ..., untila merger or an ejection takes place. The illustration complements the de-scription of our model from Section 2. In short, our model assumes theBBH in question starts with an SMA = a HB , after which it hardens throughbinary-single interactions, each of which leads to a decrease in its SMAfrom a to δ a . This hardening continues until the SMA reaches a ej , belowwhich the BBH will be ejected from the GC through three-body recoil. If theBBH merges outside the GC within a Hubble time, we label it an ‘ejectedmerger’ (left column). The ejected merger progenitors form via interactionsinvolving Newtonian gravity alone; however, when GR effects are included,the BBH can also merge inside the cluster, before ejection takes place (e.g.Samsing 2017; Rodriguez et al. 2017). This can happen either between or during its hardening interactions, outcomes we refer to as a ‘2-body merger’(middle column) and a ‘3-body merger’ (right column), respectively. Allprevious studies on the eccentricity distribution of LISA sources have onlyconsidered the ‘ejected mergers’; however, as we show in this paper, the‘2-body mergers’ clearly dominate the eccentric population observable byLISA ( e > .
01 at 10 − Hz). In comparison, the ‘3-body mergers’ dominatethe eccentric population observable by LIGO ( e > . to estimate what the actual BBH eccentricity distribution is expectedto be in the LISA band, taking into account that BBHs can formboth during and between hardening binary-single interactions ( e.g. ,Rodriguez et al. 2017). Although our approach is highly simplified,we do clearly find that GR effects play a central role in such a study.Figure 1 schematically illustrates our dynamical model describedbelow. We assume that the dynamical history of a BBH in a GC from itsformation to final merger follows the idealized model described inSamsing (2017), in which it first forms dynamically at the hard-binary (HB) limit ( e.g. , Heggie 1975; Aarseth & Heggie 1976;
MNRAS000
MNRAS000 , 1–6 (2018) ccentric Black Hole Mergers in LISA Hut & Bahcall 1983), after which it hardens through equal massthree-body interactions. Each interaction leads to a fixed decreasein its semi-major axis (SMA) from a to δ a , where the average valueof δ is 7 / δ for ourmodeling. The BBH will harden in this way until it either mergesinside the GC, or its three-body recoil velocity exceeds the escapevelocity of the GC, v esc , after which it escapes. In this model, suchan ‘ejection’ can only happen if the SMA of the BBH is below thefollowing characteristic value (Samsing 2017), a ej ≈ (cid:18) δ − (cid:19) Gm v , (1)where m is the mass of one of the three interacting (assumed equalmass) BHs. The mergers that are normally considered, using New-tonian prescriptions, are the BBHs that will merge outside the GC,i.e the subset of the ejected BBHs that has a GW lifetime that isless than the Hubble time, t H . However, when GR effects are in-cluded, a BBH can also merge inside the GC in at least two differentways ( e.g. , Samsing 2017; Rodriguez et al. 2017): The first way is between its hardening binary-single interactions – a merger type wewill refer to in short as a ‘2-body merger’ (2b). A BBH will undergosuch a merger if its GW lifetime is shorter than the time it takes forthe next interaction to occur. The second way is during its hardeningbinary-single interactions – a merger type we will refer to in shortas a ‘3-body merger’ (3b). Such a merger occurs if two of the threeinteracting BHs undergo a two-body GW capture merger during thechaotic evolution of the three-body system (Gültekin et al. 2006;Samsing et al. 2014).These three different types of mergers (ejected merger, 2-bodymerger, and 3-body merger) arise, as described, from different mech-anisms that each have their own characteristic time scale (Hubbletime, binary-single encounter time, three-body orbital time), whichexplains why they give rise to different distributions in GW fre-quency and eccentricity, as will be shown in Section 3. Belowwe describe how we construct these distributions from our simplemodel. We start by considering two BHs each with mass m , in a binary withSMA equal to their HB value given by ( e.g. , Hut & Bahcall 1983), a HB ≈ Gm v , (2)where v dis is the velocity dispersion of the interacting BHs. Asdescribed in Section 2.1 and shown in Figure 1, we assume that thedynamical evolution of this BBH is governed by isolated binary-single interactions that lead to a stepwise decrease in its SMA asfollows, δ a HB , δ a HB , δ a HB , ..., δ n a HB , ..., until δ N ej a HB ≈ a ej ,where n is the n ’th binary-single interaction, and N ej is the numberof interactions it takes to bring the BBH to its ejection value.For deriving the BBH merger fractions, GW frequencies, andeccentricity distributions, we perform the following calculations ateach interaction step n starting from n =
0, until the BBH eitherundergoes a merger or escapes the GC: We first estimate if the BBHwill undergo a 2-body merger, i.e. merge before the next encounter.For this estimation, we start by calculating the time between suc-cessive binary-single interactions, t bs , which can be approximatedby ≈ /( n s σ bs v dis ) , where n s is the number density of single BHs,and σ bs is the cross section for a binary-single interaction at step n ( e.g. , Samsing et al. 2018b). We then derive the GW-inspiral life-time of the BBH assuming its eccentricity is =
0, denoted by t c ,using the prescriptions from Peters (1964). From these two derivedtime scales, we can then calculate what the minimum eccentricityof the BBH must be for it to undergo a GW merger before its nextencounter, denoted by e , which is the solution to the followingrelation t bs = t c ( − e ) / , assuming e ≫ e ≈ r(cid:16) − ( t bs / t c ) / (cid:17) . (3)To now determine if the BBH will actually undergo a 2-body mergerinside the GC at this interaction step n , we draw a value for theeccentricity of the BBH, e , assuming a thermal distribution P ( e ) = e (Heggie 1975). If the drawn eccentricity is ≥ e , the BBH willundergo a 2-body merger, and we record its orbital elements. If theBBH does not merge, i.e. if the drawn eccentricity is < e , we thenmove on to estimate if the BBH instead undergoes a merger duringits next binary-single interaction.For estimating the probability of a 3-body merger we use theframework first presented in Samsing et al. (2014), in which thebinary-single interaction is pictured as a series of states composedof a binary, referred to as an intermediate state (IMS) binary, anda bound single. As described in Samsing (2017), on average about N IMS ≈
20 IMS binaries will form per binary-single interaction,each with a SMA that is about the initial SMA of the target bin-ary and an eccentricity that is drawn from the thermal distribution P ( e ) = e . The probability for a 3-body merger to form during thethree-body interaction is equal to the probability for an IMS bin-ary to undergo a GW merger within the orbital time of the boundsingle. To calculate this probability, we first estimate the character-istic pericentre distance an IMS binary must have for it to undergoa GW capture merger during the interaction, a distance we denoteby r . Although this distance changes between each IMS in thethree-body interaction (Samsing et al. 2014), one finds that on aver-age r is about the distance for which the energy loss through GWemission integrated over one IMS binary orbit is similar to the ini-tial total energy of the three-body system. From this it follows that, r ≈ R m × ( a / R m ) / , where R m here denotes the Schwarzschildradius of a BH with mass m , and a is the SMA of the target binary,which in our step wise hardening series equals δ n a HB for step n (see Samsing 2017). The minimum eccentricity of an IMS BBHneeded to undergo a GW capture merger during the interaction isthen given by, e ≈ − r / a . (4)As for the 2-body mergers, we then draw a value for the IMS BBHeccentricity from the thermal distribution 2 e . We do this up to N IMS =
20 times for each interaction. If one of the drawn eccentri-cities is ≥ e a 3-body merger has formed and we save its orbitalelements.If neither a 2-body nor a 3-body merger has formed at theconsidered step n , we move on to the next SMA step in the hardeningseries, which after the last binary-single interaction is now δ n + a HB ,and redo the above calculations. If no merger has formed when theBBH SMA falls below a ej , we assume the BBH escapes the clusterwith a SMA = a ej . For this BBH we then calculate what its minimumeccentricity must be for it to merge within in a Hubble time, e Ht .We again draw from a thermal distribution in eccentricity, and if thevalue is > e Ht we label the BBH as an ejected BBH merger.For this paper we follow 10 such BBHs starting at their a HB from which we then derive BBH merger fractions, frequency and ec- MNRAS , 1–6 (2018)
Samsing, D’Orazio −5 −4 −3 −2 −1 0 1 2log f GW [Hz] n r e v e n t s GW freq ency f GW at formation ejected mergers (o tside cl ster)2-body mergers (inside cl ster)3-body mergers (inside cl ster) Figure 2.
Distribution of GW peak frequency f GW at formation for mergingBBHs forming dynamically in GCs. The distributions here are derived usingour simple BBH hardening model described in Section 2.1, which combinesan MC approach with the analytical framework presented in (Samsing 2017). Blue:
Distribution of BBHs that are ejected from the GC and merge withina Hubble time (ejected mergers).
Green:
Distribution of BBHs that mergeinside the GC between their hardening binary-single interactions (2-bodymergers).
Red:
Distribution of BBHs that merge inside the GC during theirhardening binary-single interactions (3-body mergers). The relative contri-bution from each population depends on the masses of the interacting BHs,the density of single BHs in the GC core, and the escape velocity of theGC; however, all reasonable values lead to about half of all merging BBHsmerging inside the cluster (green/red), where about 5% of merging BBHsform in 3-body mergers. We emphasize that the 2-body and 3-body mergerpopulations only can be resolved with GR included in the N -body modeling. centricity distributions, from the above procedure by going througheach of the hardening steps. This allows us to investigate the roleof GR effects in what effectively corresponds to > The following results are derived using the method described above,applied to the scenario for which the interacting BHs are identical,with a mass 30 M ⊙ , and for which the population of GCs all have anescape velocity of 50 kms − ( e.g. , Harris 1996). We further assumethat the number density of single BHs, n s , in each GC core is 10 pc − . This number is highly uncertain; however, one finds that therelative number of 2-body mergers only scales weakly with densityas n − / , which follows from Samsing (2017). Finally, we note thatour chosen values robustly result in that ≈
50% of all BBH mergersare in the form of 2-body mergers, which is in agreement with therecent PN simulations presented in Rodriguez et al. (2017).The distributions of peak GW frequency, f GW , at the timeof formation of the BBHs that are merging through the three dif-ferent pathways considered in this work (ejected merger, 2-bodymerger, 3-body merger) are shown in Figure 2. For this we usedthe approximation f GW = π − q Gm / r , where r p is the pericentre −3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0log e n r e v e n t s Eccentricity e at f GW = 10 −2 Hz ejected mergers (outside cluster)2-body mergers (inside cluster) Figure 3.
Distribution of BBH orbital eccentricity e at 10 − Hz derived usingall the BBH mergers from the set presented in Figure 2 that have an initial f GW < − Hz. As the 3-body mergers peak at much higher frequencies,the considered set is completely dominated by the ejected (blue) and 2-body(green) mergers. As seen, the 2-body mergers dominate the fraction that willhave a resolvable eccentricity ( > .
01) in the LISA band (10 − Hz). Thispopulation will therefore play a key role in determining the origin of BBHmergers using a LISA-like instrument, as, e.g. , field BBHs are expected tobe circular to a much higher degree in LISA. distance at the time of formation of the merging BBH ( e.g. , Wen2003; Samsing 2017). The ejected mergers (blue) initially distrib-ute at relatively low f GW with a peak between 10 − − − Hz, andwill therefore drift through both LISA and LIGO. The possibilityfor joint observations have been suggested for such a population( e.g. , Sesana 2016; Seto 2016). The 3-body mergers (red) all havea much higher initial f GW with a peak between 10 − Hz, andwill therefore elude the LISA band and form directly in the pro-posed DECIGO (Kawamura et al. 2011; Isoyama et al. 2018)/TianQin (Luo et al. 2016) band before entering the LIGO band ( e.g. ,Chen & Amaro-Seoane 2017; Samsing et al. 2018a). We note herethat these two distributions are in full agreement with those foundin Samsing et al. (2018a), in which the distributions were resolvedusing full numerical 2.5 PN scatterings using data from the
MOCCA code (Giersz et al. 2013; Askar et al. 2017). This validates at leastthis part of our framework. The 2-body mergers (green) interest-ingly distribute between the ejected and the 3-body mergers, with apeak only slightly below the maximum sensitivity region of LISA.A proper understanding and modeling of this population is requiredfor using LISA to determine the origin of BBH mergers. As statedin Section 2, we note that this population has not been studied inthis context before. In Paper II of this series we investigate in detailthe GW signatures of these three dynamically formed populationsin the LISA band.The eccentricity distribution of the BBH mergers evaluated at10 − Hz, near the peak of the LISA sensitivity, is shown in Figure3. To derive this distribution, we use the evolution equations fromPeters (1964) to propagate the initial BBH eccentricity distribu-tion, with initial peak GW frequency f GW < − Hz, to the valueat f GW = − Hz. As seen, the 2-body mergers, i.e. the BBHsthat merge between encounters inside the GC, completely domin-
MNRAS000
MNRAS000 , 1–6 (2018) ccentric Black Hole Mergers in LISA −3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0log e0.00.20.40.60.81.0 c u m u l a t i v e n r e v e n t s Eccentricity e at f GW = 10 −2 H ejected mergersejected + 2-body mergers Figure 4.
Cumulative distribution of the eccentricity distributions shown inFigure 3. When the 2-body mergers are included (black), ≈
40% of mergingBBHs will have an eccentricity > .
01 at 10 − Hz, near the peak sensitivityof the LISA band. We note that this fraction is about 4 times higher thanrecently reported by (Breivik et al. 2016), who effectively only consideredthe ejected population. A substantial fraction of eccentric BBH mergers aretherefore expected in LISA if the dynamical GC channel contributes to theBBH merger rate. This finding should be taken into account when optimizingscience cases and instrumental designs. ate the fraction of mergers that will have an eccentricity resolvableby LISA ( e > . ≈
10% will have an eccentricity > .
01 at 10 − Hz (blue); however, when the 2-body mergers are included ≈ > .
01 (black). Thisis an important correction, as some recent studies have argued thateccentric populations would hint for BBH mergers forming nearmassive BHs ( e.g. , Nishizawa et al. 2017). Our results show thatGCs can produce eccentric mergers in LISA as well, greatly motiv-ating further and more detailed studies on systems.From this we conclude that BBH mergers forming in GCsare expected to lead to a notable distribution of eccentric sources( > .
01) in the LISA band (10 − Hz), with a relative fraction that issignificantly higher than recently reported by Breivik et al. (2016).This not only shows the importance of a proper inclusion of GRterms in current N -body studies, but also the bright prospects ofobservationally distinguishing where and how BBH mergers formin our Universe with LISA. In Paper I of this series, we have explored the role of GR effectsin the dynamical evolution of BBHs inside GCs, and found thatthe population that merges through GW emission between theirhardening binary-single interactions, referred to as 2-body mergers,all appear with a notable eccentricity ( > .
01) in the LISA band(10 − Hz). Using a simple MC approach together with the ana-lytical framework presented in Samsing (2017), we find that withthe inclusion of these 2-body mergers, ≈
40% of all BBH mergersfrom GCs will be eccentric in LISA, which is ≈ e.g. ,Nishizawa et al. 2016; Breivik et al. 2016; Nishizawa et al. 2017;Samsing et al. 2018a). The reason is that different channels will havedifferent eccentricity distributions, e.g. isolated field binaries arebelieved to have almost circularized once entering LISA, whereasBBH mergers assembled near massive black holes have been shownto have a notable eccentricity in LISA ( e.g. , Nishizawa et al. 2017).Our results likewise indicate that the background of unresolvedsources observable by LISA, is likely to have a significant fraction ofeccentric sources. Including such a population will lead to changesin the expected background spectrum, which often is assumed tobe dominated by circular BBHs partly due to the Newtonian resultsderived in Rodriguez et al. (2016a), that we argue greatly underes-timates the true fraction of eccentric sources. In Paper II of thisseries, we explore the tracks of individually resolvable, eccentricBBHs through the LISA band as well as the effect of unresolvableeccentric systems on the gravitational wave background detectableby LISA, each a result of the GR effects discussed in this paper. ACKNOWLEDGEMENTS
It is a pleasure to thank M. Giersz, A. Askar, E. Kovetz, and M.Kamionkowski for helpful discussions. J.S. acknowledges supportfrom the Lyman Spitzer Fellowship. D.J.D. acknowledges finan-cial support from NASA through Einstein Postdoctoral Fellow-ship award number PF6-170151. D.J.D. also thanks Adrian Price-Whelan and Lauren Glattly for their hospitality during the concep-tion of this work.
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