A prototype effective-one-body model for non-precessing spinning inspiral-merger-ringdown waveforms

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    a  r   X   i  v  :   1   2   0   2 .   0   7   9   0  v   2   [  g  r  -  q  c   ]   2   1   A  u  g   2   0   1   2 Prototype effective-one-body model for nonprecessing spinninginspiral-merger-ringdown waveforms Andrea Taracchini, 1 Yi Pan, 1 Alessandra Buonanno, 1,2 Enrico Barausse, 3,1, ∗ MichaelBoyle, 4 Tony Chu, 5 Geoffrey Lovelace, 4 Harald P. Pfeiffer, 5 and Mark A. Scheel 6 1 Maryland Center for Fundamental Physics & Joint Space-Science Institute,Department of Physics, University of Maryland, College Park, MD 20742, USA 2  Radcliffe Institute for Advanced Study, Harvard University, 8 Garden St., Cambridge, MA 02138, USA 3  Department of Physics, University of Guelph, Guelph, ON N1G 2W1, Canada  4 Center for Radiophysics and Space Research, Cornell University, Ithaca, NY, 14853, USA 5  Canadian Institute for Theoretical Astrophysics, 60 St. George Street,University of Toronto, Toronto, ON M5S 3H8, Canada  6  Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, CA 91125, USA (Dated: August 22, 2012)This paper presents a tunable effective-one-body (EOB) model for black-hole (BH) binaries of arbitrary mass ratio and aligned spins. This new EOB model incorporates recent results of small-mass-ratio simulations based on Teukolsky’s perturbative formalism. The free parameters of themodel are calibrated to numerical-relativity simulations of nonspinning BH-BH systems of five dif-ferent mass ratios and to equal-mass non-precessing BH-BH systems with dimensionless BH spins χ i  ≃± 0 . 44. The present analysis focuses on the orbital dynamics of the resulting EOB model, andon the dominant ( ℓ , m )=(2,2) gravitational-wave mode. The calibrated EOB model can generateinspiral-merger-ringdown waveforms for non-precessing, spinning BH binaries with any mass ratioand with individual BH spins − 1 ≤ χ i   0 . 7. Extremizing only over time and phase shifts, the cali-brated EOB model has overlaps larger than 0.997 with each of the seven numerical-relativity wave-forms for total masses between 20 M  ⊙  and 200 M  ⊙ , using the Advanced LIGO noise curve. We com-pare the calibrated EOB model with two additional equal-mass highly spinning ( χ i  ≃− 0 . 95 , +0 . 97)numerical-relativity waveforms, which were not used during calibration. We find that the calibratedmodel has overlap larger than 0.995 with the simulation with nearly extremal  anti-aligned   spins.Extension of this model to black holes with  aligned   spins  χ i   0 . 7 requires improvements of our mod-eling of the plunge dynamics and inclusion of higher-order PN spin terms in the gravitational-wavemodes and radiation-reaction force. PACS numbers: 04.25.D-, 04.25.dg, 04.25.Nx, 04.30.-w I. INTRODUCTION Coalescing compact-object binary systems (binaries,for short) are among the most promising sources of gravitational waves (GWs) for detectors like the U.S.Laser Interferometer Gravitational-Wave Observatory(LIGO), the British-German GEO, and the French-Italian Virgo [1–3]. LIGO and Virgo are undergoing up- gradesto Advanced configurations[4], which will improvesensitivity by about a factor of 10. A detailed and accu-rate understanding of the GWs radiated as the bodies ina binary spiral towards each other is crucial not only forthe initial detection of such sources, but also for max-imizing the information that can be obtained from theGW signals once they are observed.The matched-filtering technique is the primary data-analysis tool used to extract the GW signals from thedetectors’ noise. It requires accurate waveform modelsof the expected GW signals. Analytical templates basedon the post-Newtonian (PN) approximation [5–8] of the Einstein field equations developed over the past thirty ∗ CITA National Fellow years accurately describe the inspiraling stage of the bi-nary evolution. In 1999 a new approach to the two-bodydynamics of compact objects, the so-called effective-one-body (EOB) approach, was proposed with the goal of extending the analytical templates throughout the laststages of inspiral, plunge, merger, and ringdown. TheEOB approach uses the results of PN theory, black-holeperturbation theory, and, more recently, the gravitationalself-force formalism. It does not, however, use the PNresults in their srcinal Taylor-expanded form (i.e., aspolynomials in  v/c ), but in a resummed form.The EOB formalism was first proposed in Refs. [9, 10] and subsequently improved in Refs. [11–13]. Using physical intuition and results from black-hole perturba-tion theory and the close-limit approximation, Refs. [10,13] computed preliminary plunge, merger, and ring-down signals of nonspinning and spinning black-holebinaries. After breakthroughs in numerical relativity(NR) [14–16], the EOB inspiral-merger-ringdown wave- forms were improved by calibrating the model to pro-gressively more accurate NR simulations, spanning largerregions of the parameter space [17–27]. More recently, an EOB model for the dominant (2 , 2) mode and foursubdominant modes was built for nonspinning bina-ries of comparable masses [27] and the small-mass-ratio  2limit [28]. These results, at the interface between nu- merical and analytical relativity, have already had animpact in LIGO and Virgo searches. The first searchesof high-mass and intermediate-mass black-hole binariesin LIGO/Virgo data [29, 30] used the inspiral-merger- ringdown templates generated by the EOB model cali-brated in Ref. [19], as well as the phenomenological tem-plates proposed in Ref. [31]. Stellar-mass black holes are expected to carry spins,which significantly increases the dimension of the bi-nary parameter space. The first EOB Hamiltonian withleading-order (1.5PN) spin-orbit and (2PN) spin-spincouplings was developed in Ref. [12]. Then, Ref. [13] worked out the radiation-reaction force in the EOB equa-tions of motion in the presence of spins and computedinspiral-merger-ringdown waveforms for generic spinningbinaries, capturing their main features, including theso-called “hang up”. Later, Ref. [32] incorporated the next-to-leading-order (2.5PN) spin-orbit couplings in theEOB Hamiltonian. By construction, in the test-particlelimit the Hamiltonian of Ref. [32] does not reduce to theHamiltonian of a spinning test particle in the Kerr space-time. Moreover, the Hamiltonian of Ref. [32] rewritesthe EOB radial potential using Pad´e summation, caus-ing spurious poles in some regions of parameter space.Nevertheless, the Hamiltonian of Ref. [32] was adopted in Ref. [25] to demonstrate the possibility of calibrating the EOB model for spinning binaries.Since then, substantial progress has been made to-wards improving the spin EOB Hamiltonian. Ref. [33]worked out the Hamiltonian for a spinning test-particlein a generic spacetime, which was used in Ref. [34] toderive a spin EOB Hamiltonian having the correct test-particle limit. Furthermore, Ref. [34] rewrote the EOBradial potential in a way that guarantees the absence of poles without employing the Pad´e summation. As a con-sequence, the EOB Hamiltonian of Ref. [34] has desirablestrong-field circular-orbit features, such as the existenceof an innermost-stable circular orbit (ISCO), a photoncircular orbit (or light-ring), and a maximum in the or-bital frequency during the plunge. Still preserving theseproperties, the spin EOB Hamiltonian of Ref. [34] wasrecently extended to include the next-to-next-to-leading-order (3.5PN) spin-orbit couplings in Ref. [35]. The EOB Hamiltonian of Ref. [32] was also recently extendedthrough 3.5PN order in the spin-orbit sector in Ref. [36]. In the non-conservative sector of the EOB model, theradiation-reaction force in the EOB equations of mo-tion is built from the GW energy flux, which, in turn,is computed from a decomposition of the waveform intospherical harmonic ( ℓ,m ) modes. These modes, insteadof being used in their Taylor-expanded form, are re-summed (or factorized). This factorization was srci-nally proposed in Refs. [37, 38] for nonspinning black- hole binaries, and was then extended to include spin ef-fects in Ref. [39] and higher-order PN spinless terms inRefs. [40, 41]. In the test-particle limit, the factorized waveforms are known at very high PN order—for exam-ple their sum generates the GW energy flux for nonspin-ning binaries through 14PN [41] order and to 4PN order in terms involving the black-hole spins. However, in thecomparable-mass case the GW modes are known only ata much lower PN order. Despite the fact that the GW en-ergy flux in the comparable-mass case is known through3.5PN [42, 43] and 3PN [44] order in the nonspinning and spin-orbit sectors, and 2PN order in the spin-spin sector,the GW modes have been computed only through 1.5PNorder for spin-orbit couplings and 2PN order for spin-spincouplings [39, 45]. Currently, this lack of information in the GW modes is the main limitation of our spin EOBmodel, and, as we will see, it affects the performance of the model for prograde orbits and large spin values.In this paper, we build upon the past success in an-alytically modeling inspiral-merger-ringdown waveformsthrough the EOB formalism, and develop a prototypeEOB model for non-precessing spinning black-hole bina-ries that covers a large region of the parameter space andcan be used for detection purposes and future calibra-tions. More specifically, we adopt the EOB Hamiltonianderived in Refs. [34, 35], the GW energy flux and factor- ized waveforms derived in Refs. [38, 39], and calibrate the EOB (2,2) dominant mode to seven NR waveforms: fivenonspinning waveforms with mass ratios 1 , 1 / 2 , 1 / 3 , 1 / 4and 1 / 6 [27] and two equal-mass non-precessing spinning waveforms of spin magnitudes 0 . 44 [46]. We combine theabove results with recent small-mass-ratio results pro-duced by the Teukolsky equation [28] to build a proto- type EOB model for inspiral-merger-ringdownwaveformsfor non-precessing spinning black-hole binaries with anymass ratio and individual black-hole spins − 1 ≤ χ i   0 . 7.For  χ i   0 . 7, although the EOB dynamics can be evolveduntil the end of the plunge, the EOB (2,2) mode peakstoo early in the evolution, where the motion is still qua-sicircular. As a consequence, we cannot correct the EOB(2,2) mode to agree with the NR (2,2) mode peak usingnon-quasicircular amplitude coefficients. This limitation,which also affects the small-mass-ratio limit results [28], is caused by the poor knowledge of PN spin effects in theGW modes and makes the prototype EOB waveformsunreliable for  χ i   0 . 7. Two NR waveforms with nearlyextremal spin magnitudes [47, 48] became available to us when we were finishing calibration of the spin EOBmodel. We use them to examine the limitations of thespin prototype EOB model, and extract from them usefulinformation for future work.The paper is organized as follows. In Sec. II, we de-scribe the spin EOB model used in this work, its dy-namics, waveforms, and adjustable parameters. Sec-tion IIIA discusses the alignment procedure used to com-pare EOB and NR waveforms at low frequency, and thestatistics used to quantify the differences between thewaveforms. We then calibrate the EOB model to theNR waveforms in Sec. IIIB. In Sec. IV, we combine the results of Sec. IIIA with those of Ref. [28] to build a prototype EOB model that interpolates between the cal-ibrated EOB waveforms and extends them to a larger  3region of the parameter space. We also investigate howthis prototype EOB model performs with respect to twoNR waveforms with nearly extremal spin, which were notused in the calibration. Finally, Sec. V summarizes ourmain conclusions. In Appendix A we explicitly write thefactorized waveforms used in this work, including spineffects. II. EFFECTIVE-ONE-BODY DYNAMICS ANDWAVEFORMS IN THE PRESENCE OF SPINEFFECTS In this section, we define the spin EOB model that wewill later calibrate using NR waveforms. Henceforth, weuse geometric units  G  =  c  = 1.In the spin EOB model [12, 32, 34–36] the dynamics of  two black holes of masses  m 1  and  m 2  and spins  S  1  and S  2  is mapped into the dynamics of an effective particleof mass  µ  =  m 1 m 2 / ( m 1  +  m 2 ) and spin  S  ∗  moving ina deformed Kerr metric with mass  M   =  m 1  +  m 2  andspin  S  Kerr . The position and momentum vectors of theeffective particle are described by  R   and  P  , respectively.Here, for convenience, we use the reduced variables r  ≡  R  M  ,  p ≡  P  µ .  (1)Since we will restrict the discussion to spins alignedor anti-aligned with the orbital angular momentum, wedefine the (dimensionless) spin variables  χ i  as  S  i  ≡ χ i m 2 i  ˆL , where  ˆL  is the unit vector along the direc-tion of the orbital angular momentum. We also write S  Kerr  ≡ χ Kerr M  2 ˆL . A. The effective-one-body dynamics In this paper we adopt the spin EOB Hamiltonian pro-posed in Refs. [33–35]. The real (or EOB) Hamiltonian is related to the effective Hamiltonian  H  eff   through therelation H  real  ≡ µ  ˆ H  real  =  M    1 + 2 ν   H  eff  µ  − 1  − M ,  (2)where  H  eff   describes the conservative dynamics of an ef-fective spinning particle of mass  µ  and spin  S  ∗ movingin a deformed Kerr spacetime of mass  M   and spin  S  Kerr .The symmetric mass ratio  ν   =  µ/M   acts as the deforma-tion parameter. Through 3.5PN order in the spin-orbitcoupling, the mapping between the effective and real spinvariables reads [34, 35] S  Kerr  =  S  1  + S  2 ,  (3a) S  ∗ =  m 2 m 1 S  1  +  m 1 m 2 S  2  + ∆ (1) σ ∗  + ∆ (2) σ ∗  ,  (3b)where ∆ (1) σ ∗  and  ∆ (2) σ ∗  are the 2.5PN and 3.5PN spin-orbitterms given explicitly in Eqs. (51) and (52) of Ref. [35]. They depend on the dynamical variables r and  p , the spinvariables S  i , and on several gauge parameters. These pa-rameters are present because of the large class of canon-ical transformations that can map between the real andeffective descriptions. Their physical effects would cancelout if the PN dynamics were known at arbitrarily highorders; since this is clearly not the case, the gauge param-eters can have a noticeable effect [35] and may in princi- ple be used as spin EOB adjustable parameters. In thispaper however, we set all gauge parameters to zero andintroduce a spin EOB adjustable parameter at 4.5PN or-der in the spin-orbit sector by adding the following termto Eq. (3b) ∆ (3) σ ∗  =  d SO ν r 3  m 2 m 1 S  1  +  m 1 m 2 S  2   .  (4)Here  d SO  is the spin-orbit EOB adjustable parameter.The effective Hamiltonian reads [34] H  eff  µ  =  β  i  p i  + α   1 + γ  ij  p i  p j  + Q 4 (  p ) +  H  SO µ  +  H  SS µ −  12 Mr 5 ( r 2 δ  ij − 3 r i r j ) S  ∗ i  S  ∗ j  , (5)where the first two terms are the Hamiltonian of a non-spinning test particle in the deformed Kerr spacetime, α ,  β  i and  γ  ij are the lapse, shift and 3-dimensionalmetric of the effective geometry and  Q 4 (  p ) is a non-geodesic term quartic in the linear momentum intro-duced in Ref. [49]. The quantities  H  SO  and  H  SS  inEq. (5) contain respectively spin-orbit and spin-spin cou-plings that are  linear   in the effective particle’s spin  S  ∗ ,while the term  − 1 / (2 Mr 5 )( r 2 δ  ij −  3 r i r j ) S  ∗ i  S  ∗ j  is theleading-order coupling of the particle’s spin to itself,with  δ  ij being the Kronecker delta. More explicitly, us-ing Ref. [34] we can obtain  H  SO  and  H  SS  by insertingEqs. (5.31), (5.32), Eqs. (5.47a)–(5.47h), and Eqs. (5.48)–(5.52) into Eqs. (4.18) and (4.19);  α ,  β  i and  γ  ij are givenby inserting Eqs. (5.36a)–(5.36e), Eqs. (5.38)–(5.40) andEqs. (5.71)–(5.76) into Eqs. (5.44)–(5.46). We will eluci-date our choice of the quartic term  Q 4 (  p ) at the end of this section, when introducing the tortoise variables.Following Ref. [25], we introduce another spin EOBadjustable parameter in the spin-spin sector. Thus, weadd to Eq. (5) the following 3PN term d SS ν r 4  m 2 m 1 S  1  +  m 1 m 2 S  2  · ( S  1  + S  2 ) ,  (6)with  d SS  the spin-spin EOB adjustable parameter. Forwhat concerns the nonspinning EOB sector, we adoptthe following choice for the EOB potentials ∆ t  and ∆ r entering  α ,  β  i  and  γ  ij  (see Eq. (5.36) in Ref. [34]). The  4potential ∆ t  is given through 3PN order by∆ t ( u ) = 1 u 2  ∆ u ( u ) ,  (7a)∆ u ( u ) =  A ( u ) + χ 2Kerr u 2 ,  (7b) A ( u ) = 1 − 2 u + 2 ν u 3 + ν   943  −  4132 π 2   u 4 , (7c)where  u  ≡  1 /r . Reference [34] suggested rewriting the quantity ∆ u ( u ) as∆ u ( u ) = ¯∆ u ( u )  1+ ν   ∆ 0  + ν   log  1 + ∆ 1 u + ∆ 2 u 2 +∆ 3 u 3 + ∆ 4 u 4   ,  (8)where ∆ i  with  i  = 1 , 2 , 3 , 4 are explicitly given inEqs. (5.77)–(5.81) of Ref. [34], and¯∆ u ( u ) = χ 2Kerr  u −  1 r EOB+   u −  1 r EOB −   ,  (9a) r EOB ±  =  1 ±   1 − χ 2Kerr   (1 − K ν  ) .  (9b)Here,  r EOB ±  are radii reducing to those of the Kerr eventand Cauchy horizons when the EOB adjustable param-eter  K   goes to zero. The logarithm in Eq. (8) was in-troduced in Ref. [34] to quench the divergence of thepowers of   u  at small radii. Its presence also allows theexistence of an ISCO, a photon circular orbit (or light-ring), and a maximum in the orbital frequency duringthe plunge. The reason for modeling ∆ u ( u ) with Eq. (8)instead of using the Pad´e summation of ∆ u ( u ), as pro-posed in Ref. [32], is threefold. First, we did not want to use the Pad´e summation of ∆ u ( u ) because Ref. [25]found that for certain regions of the parameter spacespurious poles can appear. Second, although we couldhave applied the Pad´e summation only to  A ( u ) and usedthe Pad´e potential  A ( u ) calibrated to nonspinning wave-forms in Ref. [27], we want to take advantage of the goodproperties of the potential (8) during the late inspiral, asfound in Ref. [34]. Third, we find it useful to develop avariant of the EOB potential so that in the future we cantest how two different EOB potentials (both calibratedto NR waveforms at high frequency) compare at low fre-quency.Furthermore, for the potential ∆ r  at 3PN order enter-ing the EOB metric components (5.36) in Ref. [34], wechoose∆ r ( u ) = ∆ t ( u ) D − 1 ( u ) ,  (10a) D − 1 ( u ) = 1 + log[1 + 6 ν u 2 + 2(26 − 3 ν  ) ν u 3 ] . (10b)Once expanded in PN orders, the EOB Hamiltonian(2) with the effective Hamiltonian defined in Eq. (5) and the spin mapping defined in Eqs. (3a) and (3b), reproduces all known PN orders—at 3PN, 3.5PN and2PN order in the nonspinning, spin-orbit and spin-spinsectors, respectively—except for the spin-spin terms at3PN and 4PN order, which have been recently com-puted in Refs. [50–57]. Furthermore, in the test-particle limit the real Hamiltonian contains the correct spin-orbitcouplings linear in the test-particle spin, at  all   PN or-ders [33, 34]. Let ˆ t  ≡  t/M  . In terms of the reduced Hamiltonianˆ H  real , the EOB Hamilton equations are given in dimen-sionless form by [25] d r d ˆ t =  { r ,  ˆ H  real } =  ∂   ˆ H  real ∂   p  ,  (11a) d  p d ˆ t =  {  p ,  ˆ H  real } + ˆ F   = − ∂   ˆ H  real ∂  r  + ˆ F  ,  (11b)where ˆ F   denotes the non-conservative force that ac-counts for radiation-reaction effects. Following Ref. [13], we use  1 ˆ F   = 1 ν  ˆΩ | r ×  p | dE dt  p ,  (12)where ˆΩ  ≡  M  | r ×  ˙ r | /r 2 is the dimensionless orbital fre-quency and  dE/dt  is the GW energy flux for quasicircularorbits obtained by summing over the modes ( ℓ,m ) as dE dt  =ˆΩ 2 8 π 8  ℓ =2 ℓ  m =0 m 2  R M h ℓm  2 .  (13)Here  R  is the distance to the source, and simply elimi-nates the dominant behavior of   h ℓm . We sum over pos-itive  m  modes only since  | h ℓ,m |  =  | h ℓ, − m | . Expressionsfor the modes  h ℓm  are given in the next section. In thispaper, we restrict the calibration to non-precessing bi-naries, and thus we omit the Hamilton equations of thespin variables.It was demonstrated in previous work [37, 58] that by replacing the radial component of the linear momentum  p r  ≡  (  p · r ) /r  with  p r ∗ , which is the conjugate momen-tum of the EOB tortoise radial coordinate  r ∗ , one canimprove the numerical stability of the EOB equationsof motion. This happens because  p r  diverges when ap-proaching  r EOB+  while  p r ∗  does not. In this paper we fol-low the definition of the EOB tortoise radial coordinatein Appendix A of Ref. [25]. 2 However, when applying thetortoise coordinate transformation to the quartic term inEq. (5), we get [25] Q 4 (  p ) ∝  p 4 r ∗ r 2 D 2 ∆ 4 t ( r 2 + χ 2Kerr ) 4 ,  (14)which clearly diverges at  r  =  r EOB+  . As in the nonspin-ning case [27, 37, 58], we neglect contributions higher 1 The over-dot stands for  d/dt . 2 Note that all the formulas in Appendix A of Ref. [25] are writtenin physical dynamical variables, namely  R   and  P  , while here weuse reduced variables  r  and  p .  5than 3PN order and rewrite Eq. (14) as Q 4 (  p ) ∝  p 4 r ∗ r 2  ( r 2 + χ 2Kerr ) 4 ,  (15)which is well behaved throughout the EOB orbital evo-lution.Lastly, we integrate the EOB Hamilton equations. Inorder to get rid of any residual eccentricity when the EOBorbital frequency is close to the initial frequency of theNR run, we start the EOB evolution at large separation,say 50 M  , and use the quasispherical initial conditionsdeveloped in Ref. [13]. We stop the integration when the orbital frequency Ω reaches a maximum. B. The effective-one-body waveforms Following Refs. [24–27, 37] we write the inspiral-plunge modes as h insp-plunge ℓm  =  h F ℓm N  ℓm ,  (16)where the  h F ℓm  are the factorized modes developed inRefs. [37–39], and the  N  ℓm  are non-quasicircular (NQC)corrections that model deviations from quasicircular mo-tion, which is assumed when deriving the  h F ℓm . The fac-torized modes read h F ℓm  =  h ( N,ǫ ) ℓm ˆ S  ( ǫ )eff  T  ℓm e iδ ℓm ( ρ ℓm ) ℓ ,  (17)where  ǫ  is the parity of the waveform. All the factorsentering the  h F ℓm  can be explicitly found in Appendix A.We emphasize here again that despite the fact that theGW energy flux in the comparable-mass case is knownthrough 3PN order in the spin-orbit sector [44], the spin- orbit couplings in the factorized (or PN-expanded) modeshave been computed only through 1.5PN order [39, 45]. This limitation will degrade the performances of our spinEOB model for prograde orbits and large spin values, asalreadyobservedin the test-particle limit in Refs. [28, 39]. To improve the knowledge of spin effects in the GWmodes, Refs. [25, 39] added spin couplings in the test- particle limit through 4PN order in the factorized wave-forms. However, since the mapping between the Kerrspin parameter in the test-particle limit and the black-hole spins in the comparable-mass case is not yet unam-biguously determined [34, 35], and since we do not have many NR spinning waveforms at our disposal to test themapping, we decide not to include here the spinning test-particle-limit couplings in the factorized waveforms com-puted in Ref. [39]. We have checked before performing any calibration that EOB models with or without test-particle spin effects (with Kerr spin parameter χ Kerr ) givesimilar performances.In all the calibrations of the nonspinning EOB model,two EOB adjustable parameters were needed to calibratethe EOB Hamilton equations—for example Refs. [26, 27] used the 4PN and 5PN order coefficients in the EOB po-tential  A ( r ). As discussed in the previous section, forthe EOB model adopted in this paper, the EOB non-spinning conservative dynamics depend so far only on theadjustable parameter K  . We introduce a second EOB ad- justable parameter in the non-conservative non-spinningEOB sector by adding a 4PN order non-spinning termin  ρ 22  and denote the coefficient of this unknown PNterm by  ρ (4)22  [see Eq. (A8a)]. This adjustable parameterenters the EOB Hamilton equations through the energyflux defined in Eq. (13).As shown in Ref. [27], the NQC corrections of modes with ( ℓ,m )   = (2 , 2) have marginal effects on the dy-namics. Also, our goal in this work is to calibrate onlythe (2 , 2) mode, so in the following we set  N  ℓm  = 1 for( ℓ,m )  = (2 , 2). We have 3 N  22  =  1 +   p r ∗ r  ˆΩ  2  a h 22 1  + a h 22 2 r  + a h 22 3 r 3 / 2  + a h 22 4 r 2  + a h 22 5 r 5 / 2  × exp  i p r ∗ r  ˆΩ  b h 22 1  +  p 2 r ∗ b h 22 2  +  p 2 r ∗ r 1 / 2 b h 22 3  +  p 2 r ∗ r b h 22 4  , (18)where  a h 22 i  (with  i  = 1 ... 5) are the (real) NQC amplitudecoefficients and  b h 22 i  (with  i  = 1 ... 4) are the (real) NQCphase coefficients. We will explain in detail how thesecoefficients are determined at the end of this section.The EOB merger-ringdown waveform is built as a lin-ear superposition of the quasinormal modes (QNMs) of the final Kerr black hole [10, 17, 19, 21, 23, 24, 59], as h merger-RD22  ( t ) = N  − 1  n =0 A 22 n  e − iσ 22 n ( t − t 22match ) ,  (19)where  N   is the number of overtones,  A 22 n  is the complexamplitude of the  n -th overtone, and  σ 22 n  =  ω 22 n − i/τ  22 n is the complex frequency of this overtone with positive(real) frequency  ω 22 n  and decay time  τ  22 n . The com-plex QNM frequencies are known functions of the massand spin of the final Kerr black hole. Their numericalvalues can be found in Ref. [60]. The mass and spin of the final black hole,  M  f   and  a f  , can be computedthrough analytical phenomenological formulas reproduc-ing the NR predictions. Here, we adopt the formulasgiven in Eq. (8) of Ref. [61] and in Eqs. (1) and (3)of Ref. [62]. We notice that the formula for the final mass in Ref. [61] was obtained using numerical simula-tions of small-spin black-hole binaries with mildly un-equal masses. As a consequence, the formula is not veryaccurate for the large-spin, unequal-mass binaries con-sidered in this paper. However, other formulas availablein the literature are either very accurate but only validfor equal-mass binaries [63], or have not been yet exten-sively tested against NR simulations [64, 65]. Thus, for 3 Note that in Ref. [28] the  N  ℓm  were written in terms of physicaldynamical variables, rather than the reduced variables used here.
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