Scattering Amplitudes and Conservative Binary Dynamics at {\cal O}(G^4)
Zvi Bern, Julio Parra-Martinez, Radu Roiban, Michael S. Ruf, Chia-Hsien Shen, Mikhail P. Solon, Mao Zeng
aa r X i v : . [ h e p - t h ] J a n CALT-TH-2021-004, FR-PHENO-2021-03, OUTP-21-03P
Scattering Amplitudes and Conservative Binary Dynamics at O ( G ) Zvi Bern, Julio Parra-Martinez, Radu Roiban, Michael S. Ruf, Chia-Hsien Shen, Mikhail P. Solon, and Mao Zeng Mani L. Bhaumik Institute for Theoretical Physics,University of California at Los Angeles, Los Angeles, CA 90095, USA Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park, Pz 16802, USA Physikalisches Institut, Albert-Ludwigs-Universit¨at Freiburg,Hermann-Herder-Strasse 3, 79104 Freiburg, Germany Department of Physics 0319, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA Rudolf Peierls Centre for Theoretical Physics, University of Oxford,Parks Road, Oxford OX1 3PU, United Kingdom
Using scattering amplitudes, we obtain the potential contributions to conservative binary dy-namics in general relativity at fourth post-Minkowskian order, O ( G ). As in previous lower-ordercalculations, we harness powerful tools from the modern scattering amplitudes program includinggeneralized unitarity, the double copy, and advanced multiloop integration methods, in combinationwith effective field theory. The classical amplitude involves polylogarithms with up to transcenden-tal weight two and elliptic integrals. We derive the radial action directly from the amplitude, anddetermine the corresponding Hamiltonian in isotropic gauge. Our results are in agreement withknown overlapping terms up to sixth post-Newtonian order, and with the probe limit. We alsodetermine the post-Minkowskian energy loss from radiation emission at O ( G ) via its relation tothe tail effect. Introduction.
The emergence of gravitational-wave sci-ence [1] has dramatically underscored the scientific valueof observing the universe through an entirely new lens,and will continue to fundamentally transform key areas inastronomy, cosmology, and particle physics. This calls forinvigorating the theoretical framework necessary for in-terpreting signals at current and future detectors [2], andhas thus galvanized new work in this direction. This in-cludes a new program [3–5] for understanding the natureof gravitational-wave sources based on modern tools fromscattering amplitudes and effective field theory (EFT).The connection of scattering amplitudes to general rel-ativity corrections to Newton’s potential has long beenknown [6–8]. Starting from foundational ideas from EFTapplied to gravitational-wave physics [9], this new efforthas integrated modern methods, including generalizedunitarity [10], double-copy relations between gauge andgravity theories [11, 12], EFT [3, 8], and advanced multi-loop integration [13–15]. These ideas culminated with ad-vancing the state of the art by obtaining the O ( G ) con-servative Hamiltonian for spinless compact binaries [4, 5],whose various aspects have now been confirmed in mul-tiple studies [16–20]. In this paper, we extend previousmethods and take the next step to obtain the contribu-tions to conservative binary dynamics at O ( G ).We focus here on the inspiral phase of binary dy-namics. Traditionally, this is approached using effec-tive one-body [21], numerical relativity [22], gravita-tional self-force [23, 24], and perturbation theory inthe post-Newtonian (PN) [25, 26], post-Minkowskian(PM) [27, 28], and non-relativistic general relativity(NRGR) [9, 29] frameworks. The PM approach has re-cently risen in prominence due to increased analytic con-trol [3–5, 28, 30–34]. We work in this context given the natural fit with relativistic amplitudes.The new amplitudes-based approach has, of course,benefited immensely from traditional methods, both inguidance and for confirming calculations. In turn, ithas revealed connections between scattering amplitudes,classical observables, and gravitational self-force, in-spiring new methods for obtaining perturbative correc-tions [19, 20, 34–37].In this paper, we use modern amplitudes methods toderive the scattering amplitude in the classical limit fortwo massive scalars interacting via potential gravitons at O ( G ) and all orders in velocity. We also obtain the cor-responding conservative two-body Hamiltonian and ra-dial action, and determine the energy loss due to gravi-ton emission at O ( G ) through its relation [26, 37, 38]to the O ( G ) tail effect [39, 40]. This order in perturba-tion theory presents new challenges from the tail effect,which manifests an infrared (IR) divergence due to theoverlap between the momentum regions [41] of potentialand radiation gravitons [42]. We also encounter a classof elliptic integrals, which complicates the analysis.Scattering amplitudes are independent of gauge or co-ordinate choices, while EFT exposes universality in phys-ical systems. These features greatly help identify emer-gent structures that can enhance our understanding ofbasic phenomena and lead to new tools that will furtherthe cycle of innovation. Here we present a remarkablysimple gauge-invariant relation between the conservativescattering amplitude and the radial action based on a re-organization of the amplitude into classical and iterationpieces, distinct from that of Refs. [3–5, 34]. It is wellknown that the radial action is also gauge-invariant andencodes the dynamics of both bound and unbound orbits(see e.g. Refs. [34, 37, 43]). Classical Dynamics from Scattering Amplitudes.
Wefocus on conservative two-body dynamics for spinlesscompact objects, described by the four-point amplitude, M ( q ), of gravitationally interacting minimally-coupledmassive scalars. The two incoming particles of momenta p , p have masses m , m , and we define σ ≡ p · p m m inmostly minus signature. We work in the center-of-mass(COM) frame where the momentum transfer q µ = (0 , q )is purely spatial. Following Refs. [44, 45], we decom-pose p , p into components orthogonal and along q , i.e., p = ¯ p − q/ p = ¯ p + q/ p i · q = 0.As described in Refs. [3–5], major simplifications areobtained by taking the classical limit early at the level ofthe integrand. This is achieved by an expansion in largeangular momentum J ≫ ~ . We implement this by rescal-ing q, ℓ → λq, λℓ , where ℓ is any graviton momentum, andthen expanding in small λ .The classical limit therefore identifies the soft region,defined by the loop momentum scaling ℓ µ = ( ω, ℓ ) ∼ ( λ, λ ), as encoding classical dynamics. In the spirit ofEFT [9], we simplify the analysis, especially in the pres-ence of the tail effect at O ( G ), by focusing on the po-tential and (ultrasoft) radiation subregions defined by thescalings ∼ ( vλ, λ ) and ∼ ( vλ, vλ ), respectively. Here andbelow we use v to denote the typical velocity of the binaryconstituents, corresponding to the small velocity that de-fines the PN expansion.In the present work, we focus on the conservative partdescribed by the potential contribution, and do not in-clude radiation. This is sufficient for completely spec-ifying the conservative dynamics through O ( G ) [4, 5].However, at O ( G ), radiative effects contribute to con-servative dynamics via the tail effect [39]. Since the po-tential and radiation contributions overlap, this intro-duces scheme-dependence and IR divergence [42]. We useconventional dimensional regularization, where the am-plitudes, including graviton polarizations, are uniformlycontinued into D = 4 − ǫ dimensions. Amplitude-Action Relation.
Conservative binary dynam-ics is fully encoded in the four-point amplitude M ( q ),truncated to the classical order. There exists anotherscalar gauge invariant function which encodes the samedynamics, namely the radial action, which is definedas the integral of the radial momentum p r along thescattering trajectory, I r ( J ) ≡ R p r dr , with appropriateregularization of the long-distance contribution. Herewe present a simple relation between these two gauge-invariant quantities, exposed through the EFT intro-duced in Ref. [3].In the classical limit, the amplitude at O ( G n ) containsa classical contribution that scales as λ n − and iterationcontributions that scale as λ n − , λ n − , · · · , λ − . The lat-ter correspond to iterations of lower-order amplitudes,are IR divergent, and cancel in physical observables. Al-though the full amplitude is invariant, the choice of polestructure of the iterations is not unique and the classicalpart is modified accordingly. Previously this was cho-sen to align with the matter energy poles in the EFT for
21 3421 3412 34 12 34 21 3421 3421 34 21 34
FIG. 1. Generalized unitarity cuts encoding potential-regioncontributions to binary dynamics. Ovals represent tree am-plitudes while exposed lines depict on-shell states. Thin andthick lines denote gravitons and massive scalars, respectively. direct cancellation without explicit evaluation [3]. Thischoice also revealed a connection between the classicalamplitude and the local COM momentum in isotropicgauge, first observed in [4, 5] and later proven in [30, 34].In the present analysis, we instead expand the mat-ter poles about the momentum component along ˆ z , thedirection of the spatial component of ¯ p . Inspired bythe eikonal approximation [46], this prescription revealsa gauge-invariant “amplitude-action relation” i M ( q ) = Z J (cid:16) e iI r ( J ) − (cid:17) , (1)between the amplitude M ( q ) and the radial action I r ( J ).The classical part of the amplitude then corresponds tothe term linear in I r ( J ), given by˜ I r ( q ) = Z J I r ( J ) ≡ E | p | Z µ − ǫ d D − b e i q · b I r ( J ) , (2)where p is the spatial momentum, E is the total energy, | b | = J/ | p | is the impact parameter in the COM frame,and µ is the renormalization scale. As will be shownelsewhere [47], terms higher order in I r ( J ) in the relation(1) have the following structure under our prescription Z J ( iI r ( J )) n n ! = i Z ℓ ˜ I r ( ℓ ) . . . ˜ I r ( ℓ n ) Z . . . Z n − , (3) Z j = − E | p | (cid:0) ( ℓ + ℓ + · · · + ℓ j ) · ˆ z + i (cid:1) , where R ℓ ≡ R Q ni =1 d D − ℓ i (2 π ) D − (2 π ) D − δ ( P nj =1 ℓ j − q ) andwe only keep the leading classical expansion in the nu-merator of Eq. (3). Crucially, we manifest the pole struc-ture in Z j when computing the amplitude such that theclassical part can be isolated and iterations can be safelydropped without explicitly evaluating them, following thepath of Ref. [3]. With our prescription, we avoid trackingsuch terms, which is necessary in standard eikonal expo-nentiation [46]. The relation (1) can be established moregenerally, albeit care is needed for iteration terms [47].This amplitude-action relation then dictates the it-eration structure in the amplitude when expanded in
21 34 21 34 21 34
FIG. 2. Sample diagrams at O ( G ). From left to right: acontribution in the probe limit, a nonplanar diagram thatcontains iteration terms, and a diagram that contains contri-butions related to the tail effect. G . To illustrate, Eqs. (1) and (3) with the expansions M ( q ) = P n M n ( q ) and ˜ I r ( q ) = P n ˜ I r,n ( q ) yield M ( q ) = ˜ I r, ( q ) , M ( q ) = ˜ I r, ( q ) + Z ℓ ˜ I r, ˜ I r, Z , M ( q ) = ˜ I r, ( q ) + Z ℓ ˜ I r, Z Z + Z ℓ ˜ I r, ˜ I r, Z , (4)where the sum over permutations of distinct ˜ I r,n isimplicit; for instance, ˜ I r, ˜ I r, ≡ ˜ I r, ( ℓ ) ˜ I r, ( ℓ ) +˜ I r, ( ℓ ) ˜ I r, ( ℓ ) while ˜ I r, ≡ ˜ I r, ( ℓ ) ˜ I r, ( ℓ ) ˜ I r, ( ℓ ). Aswe can see, the classical part of the amplitude with thispole choice is directly the radial action, in contrast toRefs. [3–5]. We have explicitly verified Eqs. (1) and (3)through O ( G ) by comparing the amplitude calculationin EFT to the radial action from classical mechanics [47]. Constructing the Integrand.
The calculation of the ampli-tude through O ( G ) begins with the construction of theclassical limit of the three-loop integrand. We use gen-eralized unitarity, as described in Ref. [5], which buildsgravitational loop integrands directly from on-shell treeamplitudes of scalars and gravitons. The maximal-cutversion [48] of generalized unitarity adopted here orga-nizes the cuts hierarchically.As explained in Ref. [5], potential contributions comefrom cuts with at least one matter line per loop and withno gravitons beginning and ending on the same matterline. We drop contributions not of this type, such as self-energy loops and matter contact diagrams. The eightdistinct contributing generalized cuts are shown in Fig. 1,with the others given by simple relabelings.We use the D -dimensional tree-level BCJ doublecopy [12] to obtain the gravitational tree amplitudes fromcorresponding gauge-theory ones. The dilaton and anti-symmetric tensor which naturally appear as intermedi-ate states are straightforwardly eliminated by includinggraviton physical-state projectors on the cut legs. Conve-niently, the reference light-cone momentum in their def-inition cancels automatically, by organizing the tree am-plitudes so that they obey generalized Ward identities,following Ref. [49]. A similar strategy was used earlierfor simpler one- and two-loop calculations [4, 5, 33].The resulting integrand is then organized in terms of51 distinct cubic-vertex Feynman-like diagrams, of whichthree are shown in Fig. 2. All remaining diagrams are ob-tained by relabeling the external momenta. The integrals are then further reduced to a basis using integration byparts [13] implemented through FIRE6 [15], and graphsymmetries. We isolate iteration integrals, which cancelin the EFT matching prior to explicit integration.Throughout, we work with rescaled variables u i =¯ p i / | ¯ p i | , following Ref. [45], so that the integrals dependon the single variable y = u · u , enormously simplify-ing the analysis. This also factors out the mass depen-dence, clearly exposing the overlap between PM gravityand gravitational self-force [35, 36, 50, 51]. Evaluating the Integrals.
Our strategy for integra-tion combines the nonrelativistic method of Refs. [3–5] and the method of differential equations followingRef. [45]. The two methods complement each other tosystematically determine the analytic result to all or-ders in velocity, and have been applied to multiple ex-amples [17, 18, 52–54].The nonrelativistic method evaluates integrals as anexpansion in powers of v , where the leading term serves asthe boundary value for solving the differential equations.This also efficiently isolates the iterations in Eq. (3),which cancel directly in the EFT matching. The firststep is to perform energy loop integration by localizingto residues given by matter poles in the potential region.For a given integral I the result of energy integration is Z Y i =1 dω i π I ( ω , ω , ω ) = X i S i Res i I ( ω , ω , ω ) , (5)where the sum runs over triplets of matter poles on whichthe residues are evaluated. The symmetry factors S i aredetermined from the cuts in Fig. 1, building upon theprescription in Ref. [5]. The remaining three-dimensionalintegrals are then expanded in v , and reduced to masterintegrals via integration by parts [13] using FIRE6 [15].Iterations are identified by the pole structure in Eq. (3),and the final integrals are the same as in NRGR [9].Following Ref. [45], we use differential equations to an-alytically solve integrals, or to obtain solutions expandedin v to very high orders. The boundary conditions forthe univariate differential equations are imposed in theleading PN expansion. A new feature at O ( G ) is theappearance of elliptic integrals, which precludes a canon-ical basis [55]. Although this complicates the structure ofthe differential equations, we are able to solve part of thesystem in terms of classical polylogarithms up to weightthree, and verify with the velocity expansion. The con-tribution of the elliptic functions to the amplitude is de-termined by constructing an ansatz based on solutions tosimpler integrals in the same class and fixing its parame-ters using data up to O ( v ). We then verify the solutionup to O ( v ). Details will be presented in Ref. [47]. Amplitude.
Performing this calculation, we obtain thefollowing 4PM classical amplitude in the potential region: M ( q ) = G M ν | q | (cid:18) q ˜ µ (cid:19) − ǫ π (cid:20) M p4 + ν (cid:18) M t4 ǫ + M f4 (cid:19)(cid:21) + Z ℓ ˜ I r, Z Z Z + Z ℓ ˜ I r, ˜ I r, Z Z + Z ℓ ˜ I r, ˜ I r, Z + Z ℓ ˜ I r, Z , M p4 = − (cid:0) − σ + 33 σ (cid:1) σ − , M t4 = h + h log (cid:0) σ +12 (cid:1) + h arccosh( σ ) √ σ − , (6) M f4 = h + h log (cid:0) σ +12 (cid:1) + h arccosh( σ ) √ σ − h log( σ ) − h π h arccosh ( σ ) σ − h (cid:20) Li (cid:0) − σ (cid:1) + 12 log (cid:0) σ +12 (cid:1)(cid:21) + h (cid:20) Li (cid:0) − σ (cid:1) − π (cid:21) + h (cid:20) Li (cid:16) − σ σ (cid:17) − Li (cid:16) σ − σ +1 (cid:17) + π (cid:21) + h σ (2 σ − σ − / h Li (cid:16)q σ − σ +1 (cid:17) − Li (cid:16) − q σ − σ +1 (cid:17)i + 2 h √ σ − (cid:20) Li (cid:16) − σ − p σ − (cid:17) − Li (cid:16) − σ + p σ − (cid:17) + 5Li (cid:16)q σ − σ +1 (cid:17) − (cid:16) − q σ − σ +1 (cid:17) + 2 log (cid:0) σ +12 (cid:1) arccosh( σ ) (cid:21) + h K (cid:16) σ − σ +1 (cid:17) + h K (cid:16) σ − σ +1 (cid:17) E (cid:16) σ − σ +1 (cid:17) + h E (cid:16) σ − σ +1 (cid:17) , where M = m + m is the total mass, ν = m m /M isthe symmetric mass ratio, ˜ µ = 4 πµ e − γ E is the renor-malization scale in MS scheme. Li is the dilogarithm,and K and E are the complete elliptic integrals of the firstand second kind, respectively. The coefficient functions h i are collected in Table I. h = 1151 − σ + 3148 σ − σ + 339 σ − σ + 210 σ σ − h = 12 (cid:0) − σ + 150 σ − σ − σ (cid:1) h = σ (cid:0) − σ (cid:1) σ − (cid:0) − σ + 35 σ (cid:1) h = 1144 (cid:0) σ − σ ( −
45 + 207 σ − σ + 13349 σ − σ + 104753 σ − σ − σ − σ + 134745 σ + 83844 σ − σ + 13644 σ + 10800 σ (cid:1) h = 14( σ − (cid:0) − σ + 3407 σ − σ + 957 σ − σ + 341 σ + 100 σ (cid:1) h = 124( σ − (cid:0) σ − σ + 12915 σ + 18102 σ − σ − σ + 2973 σ + 5816 σ − σ (cid:1) h = 2 σ (cid:0) − − σ − σ + 75 σ (cid:1) σ − h = σ σ − (cid:0) − − σ + 672 σ + 402 σ − σ − σ − σ + 540 σ + 240 σ − σ (cid:1) h = 12 (cid:0) − σ + 351 σ − σ + 30 σ − σ (cid:1) h = 2 (cid:0)
27 + 90 σ + 35 σ (cid:1) h = 20 + 111 σ + 30 σ − σ h = 834 + 2095 σ + 1200 σ σ − h = − σ + 2660 σ + 1200 σ σ − h = 7 (cid:0)
169 + 380 σ (cid:1) σ − TABLE I. Functions specifying the amplitude in Eq. (6).
We emphasize that Eq. (6) uses dimensional regular-ization with D = 4 − ǫ and that ˜ I r, , ˜ I r, and ˜ I r, are ex-panded to the classical limit. The tail effect manifests asa 1 /ǫ IR divergence in the classical term, due to the over-lap between potential and radiation contributions [42].Including the latter would cancel this divergence and theassociated scheme dependence, replace ˜ µ with a physicalscale, and also add finite terms. Note that the schemedependence starts at 4PN and enters only through thecoefficient functions h , h , and h .The amplitude naturally exposes the simple depen-dence in the symmetric mass ratio ν , consistent withRef. [35]. The leading term M p4 agrees with the resultobtained using the Schwarzschild solution [56, 57]. Thenext-to-leading terms M t4 and M f4 overlap with first-order self-force [35].As for the O ( G ) case, the ultrarelativistic limit ofthe conservative result in Eq. (6) does not smoothlymatch onto the massless case. The amplitude has aleading power discontinuity of the form ∼ G p | q | ( m + m ) / ( m m ), consistent with dimensional analysis. Onecan expect this to cancel with radiative effects [54, 58].Given the relation in Eq. (1), it is straightforward toderive the radial action from the classical term in Eq. (6)via inverting Eq. (2) I r, ( J ) = − G M ν π p EJ (cid:18) µ e γ E J p (cid:19) ǫ × (cid:20) M p4 + ν (cid:18) M t4 ǫ + M f4 − M t4 (cid:19)(cid:21) , (7)which inherits the simple mass dependence from the am-plitude [19, 35]. Moreover, we have checked that I r, , I r, and I r, obtained from the iteration terms in Eq. (6) areconsistent with known results.The scattering angle is then given by χ = − ∂I r /∂J .We compare to the O ( G ) scattering angle obtained frompotential contributions to the Hamiltonian up to 5PN,given in Eqs. (21)–(26) of [59] and Eq. (5) of [60]. We findagreement including terms that depend on conventionaldimensional regularization, which first enter at 4PN. Wealso compare the regularization-scheme-independent π terms with the 6PN result in Eq. (8.4) of Ref. [20], andfind agreement. A full comparison would require radia-tion contributions, which we have not pursued here.As discussed in Refs. [26, 37, 38], the tail term M t4 isrelated to the energy loss ∆ E from radiation emission,and we thus identify∆ E = G M ν π p E J M t4 , (8)in the COM frame. We have compared this to the directcalculation of the energy loss in Ref. [61] using the for-malism of Ref. [32], finding agreement. Additional checks of ∆ E are discussed in Ref. [61]. We can also obtainother observables for bound orbits via analytic continu-ation [34]; details will be presented elsewhere [47]. Hamiltonian.
Following the approach in Ref. [3], we canconstruct the two-body Hamiltonian in isotropic gauge H iso = E + E + ∞ X n =1 G n ( r ˜ µ e γ E ) nǫ r n c n ( p ) , (9)where r is the distance between bodies, and E i = p p + m i are the energies of the incoming particles.The O ( ǫ ) corrections are relevant when c n ( p ) has di-vergence, which first occurs at O ( G ) in isotropic gauge.The coefficients c n ( p ) are determined by matching asin Refs. [3–5], but using the new pole choice in Eq. (3).Upon accounting for this we find, c = M ν ξE (cid:20) M p4 + ν (cid:18) M t4 ǫ + M f4 − M t4 (cid:19)(cid:21) + D (cid:20) E ξ c (cid:21) + D (cid:20)(cid:18) E ξ p + Eξ (3 ξ − (cid:19) c − E ξ c c (cid:21) + (cid:18) D + 1 p (cid:19) (cid:20) Eξ (2 c c + c ) + (cid:18) ξ − E + 2 E ξ p + Eξ (3 ξ − p (cid:19) c + (cid:18) (1 − ξ ) − E ξ p (cid:19) c c (cid:21) , (10)where ξ = E E /E , and D = dd p denotes differentiationwith respect to p . The lower-order coefficients c , c ,and c can be found in Eq. (10) of Ref. [4]. The finalexplicit result for c is included in the ancillary file [62].Note that the iteration terms in Eq. (6) cancel in thismatching, providing another nontrivial check. Conclusions.
In this letter we applied and extended am-plitudes and EFT based methods to determine the poten-tial contribution to binary dynamics at O ( G ), offering afirst look at the PM tail effect. It would be interesting tostudy closed-orbit observables at O ( G ), as was done for O ( G ) [51], and via analytic continuation [34]. The inter-play with gravitational self-force [23, 24, 35, 36, 50, 63],the structure of long-distance logarithms at higher or-ders via renormalization group techniques [38, 40, 64],and the complete tail contribution in the PM frameworkalso deserve further study.Aside from the obvious application to gravitational-wave physics, our calculation elucidates emerging struc-tures and identifies new tools. The amplitude-action re-lation in Eq. (1) greatly clarifies the link between scat-tering amplitudes and classical mechanics. It is natu-ral to expect that this structure holds more generally. Obvious extensions within the PM framework includespin [50, 52, 65], tidal [53, 57, 66], and radiation [67, 68]effects. Most excitingly, the methods applied here arenot close to being exhausted. Acknowledgments.
We thank Samuel Abreu, Jo-hannes Bl¨umlein, Alessandra Buonanno, Clifford Che-ung, Thibault Damour, Lance Dixon, Enrico Herrmann,Andr´es Luna, Rafael Porto, Ira Rothstein, Jan Stein-hoff, Gabriele Veneziano and Justin Vines for helpfuldiscussions. Z.B. is supported by the U.S. Departmentof Energy (DOE) under Award Number DE-SC0009937.J.P.-M. is supported by the U.S. Department of Energy(DOE) under award number DE-SC0011632. R.R. is sup-ported by the U.S. Department of Energy (DOE) un-der grant no. DE-SC0013699. M.S.R.’s work is fundedby the German Research Foundation (DFG) within theResearch Training Group GRK 2044. C.-H.S. is sup-ported by the U.S. Department of Energy (DOE) underaward number DE-SC0009919. M.P.S. is supported bythe David Saxon Presidential Term Chair. M.Z.’s workis supported by the U.K. Royal Society through GrantURF \ R1 \ [1] B. P. Abbott et al. [LIGO Scientific and Virgo Collab-orations], Phys. Rev. Lett. , no. 6, 061102 (2016)[arXiv:1602.03837 [gr-qc]]; B. P. 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