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 eﬀectiveonebody model for nonprecessing spinninginspiralmergerringdown 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 SpaceScience 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 35017, California Institute of Technology, Pasadena, CA 91125, USA
(Dated: August 22, 2012)This paper presents a tunable eﬀectiveonebody (EOB) model for blackhole (BH) binaries of arbitrary mass ratio and aligned spins. This new EOB model incorporates recent results of smallmassratio simulations based on Teukolsky’s perturbative formalism. The free parameters of themodel are calibrated to numericalrelativity simulations of nonspinning BHBH systems of ﬁve different mass ratios and to equalmass nonprecessing BHBH 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) gravitationalwave mode. The calibrated EOB model can generateinspiralmergerringdown waveforms for nonprecessing, spinning BH binaries with any mass ratioand with individual BH spins
−
1
≤
χ
i
0
.
7. Extremizing only over time and phase shifts, the calibrated EOB model has overlaps larger than 0.997 with each of the seven numericalrelativity waveforms for total masses between 20
M
⊙
and 200
M
⊙
, using the Advanced LIGO noise curve. We compare the calibrated EOB model with two additional equalmass highly spinning (
χ
i
≃−
0
.
95
,
+0
.
97)numericalrelativity waveforms, which were not used during calibration. We ﬁnd that the calibratedmodel has overlap larger than 0.995 with the simulation with nearly extremal
antialigned
spins.Extension of this model to black holes with
aligned
spins
χ
i
0
.
7 requires improvements of our modeling of the plunge dynamics and inclusion of higherorder PN spin terms in the gravitationalwavemodes and radiationreaction force.
PACS numbers: 04.25.D, 04.25.dg, 04.25.Nx, 04.30.w
I. INTRODUCTION
Coalescing compactobject binary systems (binaries,for short) are among the most promising sources of gravitational waves (GWs) for detectors like the U.S.Laser Interferometer GravitationalWave Observatory(LIGO), the BritishGerman GEO, and the FrenchItalian 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 accurate 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 maximizing the information that can be obtained from theGW signals once they are observed.The matchedﬁltering technique is the primary dataanalysis tool used to extract the GW signals from thedetectors’ noise. It requires accurate waveform modelsof the expected GW signals. Analytical templates basedon the postNewtonian (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 binary evolution. In 1999 a new approach to the twobodydynamics of compact objects, the socalled eﬀectiveonebody (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, blackholeperturbation theory, and, more recently, the gravitationalselfforce formalism. It does not, however, use the PNresults in their srcinal Taylorexpanded 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 blackhole perturbation theory and the closelimit approximation, Refs. [10,13] computed preliminary plunge, merger, and ringdown signals of nonspinning and spinning blackholebinaries. After breakthroughs in numerical relativity(NR) [14–16], the EOB inspiralmergerringdown wave
forms were improved by calibrating the model to progressively 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 binaries of comparable masses [27] and the smallmassratio
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 highmass and intermediatemass blackhole binariesin LIGO/Virgo data [29, 30] used the inspiralmerger
ringdown templates generated by the EOB model calibrated in Ref. [19], as well as the phenomenological templates proposed in Ref. [31].
Stellarmass black holes are expected to carry spins,which signiﬁcantly increases the dimension of the binary parameter space. The ﬁrst EOB Hamiltonian withleadingorder (1.5PN) spinorbit and (2PN) spinspincouplings was developed in Ref. [12]. Then, Ref. [13]
worked out the radiationreaction force in the EOB equations of motion in the presence of spins and computedinspiralmergerringdown waveforms for generic spinningbinaries, capturing their main features, including thesocalled “hang up”. Later, Ref. [32] incorporated the
nexttoleadingorder (2.5PN) spinorbit couplings in theEOB Hamiltonian. By construction, in the testparticlelimit the Hamiltonian of Ref. [32] does not reduce to theHamiltonian of a spinning test particle in the Kerr spacetime. Moreover, the Hamiltonian of Ref. [32] rewritesthe EOB radial potential using Pad´e summation, causing 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 towards improving the spin EOB Hamiltonian. Ref. [33]worked out the Hamiltonian for a spinning testparticlein a generic spacetime, which was used in Ref. [34] toderive a spin EOB Hamiltonian having the correct testparticle 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 consequence, the EOB Hamiltonian of Ref. [34] has desirablestrongﬁeld circularorbit features, such as the existenceof an innermoststable circular orbit (ISCO), a photoncircular orbit (or lightring), and a maximum in the orbital frequency during the plunge. Still preserving theseproperties, the spin EOB Hamiltonian of Ref. [34] wasrecently extended to include the nexttonexttoleadingorder (3.5PN) spinorbit couplings in Ref. [35]. The
EOB Hamiltonian of Ref. [32] was also recently extendedthrough 3.5PN order in the spinorbit sector in Ref. [36].
In the nonconservative sector of the EOB model, theradiationreaction force in the EOB equations of motion 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 Taylorexpanded form, are resummed (or factorized). This factorization was srcinally proposed in Refs. [37, 38] for nonspinning black
hole binaries, and was then extended to include spin effects in Ref. [39] and higherorder PN spinless terms inRefs. [40, 41]. In the testparticle limit, the factorized
waveforms are known at very high PN order—for example their sum generates the GW energy ﬂux for nonspinning binaries through 14PN [41] order and to 4PN order
in terms involving the blackhole spins. However, in thecomparablemass case the GW modes are known only ata much lower PN order. Despite the fact that the GW energy ﬂux in the comparablemass case is known through3.5PN [42, 43] and 3PN [44] order in the nonspinning and
spinorbit sectors, and 2PN order in the spinspin sector,the GW modes have been computed only through 1.5PNorder for spinorbit couplings and 2PN order for spinspincouplings [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 analytically modeling inspiralmergerringdown waveformsthrough the EOB formalism, and develop a prototypeEOB model for nonprecessing spinning blackhole binaries that covers a large region of the parameter space andcan be used for detection purposes and future calibrations. 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 equalmass nonprecessing spinning
waveforms of spin magnitudes 0
.
44 [46]. We combine theabove results with recent smallmassratio results produced by the Teukolsky equation [28] to build a proto
type EOB model for inspiralmergerringdownwaveformsfor nonprecessing spinning blackhole binaries with anymass ratio and individual blackhole 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 quasicircular. As a consequence, we cannot correct the EOB(2,2) mode to agree with the NR (2,2) mode peak usingnonquasicircular amplitude coeﬃcients. This limitation,which also aﬀects the smallmassratio 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 describe the spin EOB model used in this work, its dynamics, waveforms, and adjustable parameters. Section IIIA discusses the alignment procedure used to compare 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 calibrated 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. EFFECTIVEONEBODY 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 antialigned 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 direction of the orbital angular momentum. We also write
S
Kerr
≡
χ
Kerr
M
2
ˆL
.
A. The eﬀectiveonebody dynamics
In this paper we adopt the spin EOB Hamiltonian proposed 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 effective 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 deformation parameter. Through 3.5PN order in the spinorbitcoupling, 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 spinorbitterms 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 parameters are present because of the large class of canonical 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 parameters 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 order in the spinorbit 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 spinorbit 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 nonspinning test particle in the deformed Kerr spacetime,
α
,
β
i
and
γ
ij
are the lapse, shift and 3dimensionalmetric of the eﬀective geometry and
Q
4
(
p
) is a nongeodesic term quartic in the linear momentum introduced in Ref. [49]. The quantities
H
SO
and
H
SS
inEq. (5) contain respectively spinorbit and spinspin couplings 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 theleadingorder coupling of the particle’s spin to itself,with
δ
ij
being the Kronecker delta. More explicitly, using 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 elucidate 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 spinspin 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 spinspin 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 parameter
K
goes to zero. The logarithm in Eq. (8) was introduced 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 lightring), 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 proposed 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 waveforms 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 frequency.Furthermore, for the potential ∆
r
at 3PN order entering 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, spinorbit and spinspinsectors, respectively—except for the spinspin terms at3PN and 4PN order, which have been recently computed in Refs. [50–57]. Furthermore, in the testparticle
limit the real Hamiltonian contains the correct spinorbitcouplings linear in the testparticle spin, at
all
PN orders [33, 34].
Let ˆ
t
≡
t/M
. In terms of the reduced Hamiltonianˆ
H
real
, the EOB Hamilton equations are given in dimensionless 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 nonconservative force that accounts for radiationreaction 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 frequency 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 eliminates the dominant behavior of
h
ℓm
. We sum over positive
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 nonprecessing binaries, 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 momentum of the EOB tortoise radial coordinate
r
∗
, one canimprove the numerical stability of the EOB equationsof motion. This happens because
p
r
diverges when approaching
r
EOB+
while
p
r
∗
does not. In this paper we follow 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 nonspinning case [27, 37, 58], we neglect contributions higher
1
The overdot 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 evolution.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 eﬀectiveonebody waveforms
Following Refs. [24–27, 37] we write the inspiralplunge
modes as
h
inspplunge
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 nonquasicircular (NQC)corrections that model deviations from quasicircular motion, which is assumed when deriving the
h
F
ℓm
. The factorized 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 comparablemass case is knownthrough 3PN order in the spinorbit sector [44], the spin
orbit couplings in the factorized (or PNexpanded) 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 testparticle 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 waveforms. However, since the mapping between the Kerrspin parameter in the testparticle limit and the blackhole spins in the comparablemass case is not yet unambiguously 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 testparticlelimit couplings in the factorized waveforms computed in Ref. [39]. We have checked before performing
any calibration that EOB models with or without testparticle 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 potential
A
(
r
). As discussed in the previous section, forthe EOB model adopted in this paper, the EOB nonspinning conservative dynamics depend so far only on theadjustable parameter
K
. We introduce a second EOB ad justable parameter in the nonconservative nonspinningEOB sector by adding a 4PN order nonspinning 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 dynamics. 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 mergerringdown waveform is built as a linear superposition of the quasinormal modes (QNMs) of the ﬁnal Kerr black hole [10, 17, 19, 21, 23, 24, 59], as
h
mergerRD22
(
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 complex 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 reproducing 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 simulations of smallspin blackhole binaries with mildly unequal masses. As a consequence, the formula is not veryaccurate for the largespin, unequalmass binaries considered in this paper. However, other formulas availablein the literature are either very accurate but only validfor equalmass binaries [63], or have not been yet extensively 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.