Binary black hole coalescence in the large-mass-ratio limit: the hyperboloidal layer method and waveforms at null infinity

Abstract

We compute and analyze the gravitational waveform emitted to future null
infinity by a system of two black holes in the large mass ratio limit. We
consider the transition from the quasi-adiabatic inspiral to plunge, merger,
and ringdown. The relative dynamics is driven by a leading order in the mass
ratio, 5PN-resummed, effective-one-body (EOB), analytic radiation reaction. To
compute the waveforms we solve the Regge-Wheeler-Zerilli equations in the
time-domain on a spacelike foliation which coincides with the standard
Schwarzschild foliation in the region including the motion of the small black
hole, and is globally hyperboloidal, allowing us to include future null
infinity in the computational domain by compactification. This method is called
the hyperboloidal layer method, and is discussed here for the first time in a
study of the gravitational radiation emitted by black hole binaries. We
consider binaries characterized by five mass ratios, $\nu=10^{-2,-3,-4,-5,-6}$,
that are primary targets of space-based or third-generation gravitational wave
detectors. We show significative phase differences between finite-radius and
null-infinity waveforms. We test, in our context, the reliability of the
extrapolation procedure routinely applied to numerical relativity waveforms. We
present an updated calculation of the gravitational recoil imparted to the
merger remnant by the gravitational wave emission. As a self consistency test
of the method, we show an excellent fractional agreement (even during the
plunge) between the 5PN EOB-resummed mechanical angular momentum loss and the
gravitational wave angular momentum flux computed at null infinity. New results
concerning the radiation emitted from unstable circular orbits are also
presented.

Full text

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arXiv:1107.5402v1 [gr-qc] 27 Jul 2011
Binary black hole coalescence in the large-mass-ratio limit:
the hyperboloidal layer method and waveforms at null infinity
Sebastiano Bernuzzi,1Alessandro Nagar,2and Anıl Zengino˘ glu3
1Theoretical Physics Institute, University of Jena, 07743 Jena, Germany
2Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France
3Theoretical Astrophysics, California Institute of Technology, Pasadena, California, USA
We compute and analyze the gravitational waveform emitted to future null infinity by a system
of two black holes in the large mass ratio limit. We consider the transition from the quasi-adiabatic
inspiral to plunge, merger, and ringdown. The relative dynamics is driven by a leading order in
the mass ratio, 5PN-resummed, effective-one-body (EOB), analytic radiation reaction. To compute
the waveforms we solve the Regge-Wheeler-Zerilli equations in the time-domain on a spacelike foli-
ation which coincides with the standard Schwarzschild foliation in the region including the motion
of the small black hole, and is globally hyperboloidal, allowing us to include future null infinity
in the computational domain by compactification. This method is called the hyperboloidal layer
method, and is discussed here for the first time in a study of the gravitational radiation emitted by
black hole binaries. We consider binaries characterized by five mass ratios, ν = 10−2,−3,−4,−5,−6,
that are primary targets of space-based or third-generation gravitational wave detectors. We show
significative phase differences between finite-radius and null-infinity waveforms. We test, in our
context, the reliability of the extrapolation procedure routinely applied to numerical relativity wave-
forms. We present an updated calculation of the final and maximum gravitational recoil imparted
to the merger remnant by the gravitational wave emission, vend
vmax
fractional agreement (even during the plunge) between the 5PN EOB-resummed mechanical angular
momentum loss and the gravitational wave angular momentum flux computed at null infinity. New
results concerning the radiation emitted from unstable circular orbits are also presented. The high
accuracy waveforms computed here could be considered for the construction of template banks or
for calibrating analytic models such as the effective-one-body model.
kick/(cν2) = 0.04474 ± 0.00007 and
kick/(cν2) = 0.05248 ± 0.00008. As a self consistency test of the method, we show an excellent
PACS numbers: 04.30.Db, 04.25.Nx, 95.30.Sf, 97.60.Lf
I.INTRODUCTION
Compact binaries with large mass ratios are pri-
mary targets for space-based detectors of gravitational
waves (GWs), like the Laser Interferometer Space An-
tenna (LISA) [1, 2] (or the similar ESA-led mission),
and for third-generation ground-based detectors, like the
planned Einstein Telescope [3]. For example, the quasi-
adiabatic inspiral of extreme-mass-ratio (EMR) binaries,
i.e. of mass ratio ν ∼ 10−6, is interesting for LISA
(see e.g. [4]), while the merger of intermediate-mass-ratio
(IMR) binaries, ν ∼ 10−2− 10−3, is in the band of sen-
sitivity of the Einstein Telescope [5].
modelling of such sources is a difficult task since nei-
ther numerical relativity (NR) simulations (due to their
computational cost [6, 7]), nor standard post-Newtonian
(PN) techniques [8] (due to the strong-field, high-velocity
regime) can be applied.
The theoretical
Black-hole perturbation theory is instead the natural
tool to model large mass ratio binaries [9–19]. The rel-
ative dynamics of the binary is described by the mo-
tion of a particle (representing the small black hole) in a
fixed background, black-hole spacetime (representing the
central, supermassive black hole). The dynamics of the
particle is driven away from geodesic motion by the ac-
tion of radiation reaction through a long, quasi-adiabatic
inspiral phase up to the nonadiabatic plunge into the
black hole. For what concerns nonconservative (dissipa-
tive) effects only, they can be modeled either numerically,
for example in the adiabatic approximation, (e.g.
in [9, 20, 21] and references therein) or analytically, using
PN-resummed results (` a la effective-one-body), going in
fact beyond the adiabatic approximation [10, 18, 22, 23].
Gravitational self-force calculations [24–29] can provide
corrections to the particle conservative and nonconser-
vative dynamics at next-to-leading/higher order in the
mass (away from geodesic motion), although the field is
not ready yet for waveform production. Finally, a very
promising (semi)-analytical approach to describe the bi-
nary dynamics and to produce waveform template banks
(for any mass ratio, including EMR and IMR binaries) is
the effective-one-body (EOB) model [30–39]. The EOB
approach is intrinsically nonadiabatic and it is designed
to take into account both conservative and nonconser-
vative back-reaction effects, but requires the calibration
of some flexibility parameters to account for (yet uncal-
culated) higher-order effects in the dynamics and wave-
forms [22, 40–46].
The most important output of these studies is the
GW signal which encodes the gauge-invariant informa-
tion about the source as it should be seen by detectors.
Gravitational waves are rigorously and unambiguously
defined only at null infinity. Numerical computations,
however, are confined to finite grids. A theoretical prob-
lem is thus to model and to compute the waveforms at
as

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null infinity, as seen by a far-away idealized observer.
This problem is prominent especially in NR simula-
tions. When an asymptotically Cauchy foliation of the
spacetime is employed, the waveforms are typically ex-
tracted on coordinate spheres at finite distances from
the source. To compute waveforms at null infinity post-
simulation techniques are applied. Extrapolation to infi-
nite extraction radius [47–51] proved to be sufficiently ro-
bust and accurate, though somehow delicate due to ambi-
guities introduced by the gauge dynamics and the choice
of a fiducial background.An unambiguous procedure
based on the Cauchy-characteristic extraction (CCE)
method [52–54] has recently been implemented [55–57]
to extract waveforms from binary black hole mergers
of comparable masses.Although the set up of initial
data for the characteristic evolution is intricate [58],
the method successfully provides waveforms from binary
black hole mergers at null infinity and permits to cross-
check the standard extrapolation procedure.
An alternative approach that does not require post-
processing is to employ spacelike surfaces that approach
null infinity. Such surfaces are called hyperboloidal be-
cause their asymptotic behavior resembles that of stan-
dard hyperboloids in Minkowski spacetime [59]. Hyper-
boloidal foliations have already been considered in the
early days of numerical relativity and were expected to be
suitable for studying gravitational radiation [60–63]. The
hyperboloidal initial value problem for the Einstein equa-
tions has been analyzed by Friedrich [59, 64]. His con-
formally regular field equations have been implemented
numerically in certain test cases (for reviews see [65, 66]).
More recently, alternative hyperboloidal formulations
have been suggested [67–69] that do not exhibit explicit
conformal regularity. The only successful numerical im-
plementation of such a formalism is by Rinne in axisym-
metry [70]. It is an outstanding question whether this
or a similar hyperboloidal approach will lead to generic
numerical simulations of black hole spacetimes.
While the numerical properties of the hyperboloidal
method for Einstein equations is only poorly understood
in the general case, the situation is much clearer in per-
turbation theory where the background is given. There,
the best numerical gauge is to fix the coordinate loca-
tion of null infinity (scri), as first discussed by Frauen-
diener in the context of conformally regular field equa-
tions [71]. Moncrief presented the first explicit construc-
tion of a hyperboloidal scri-fixing gauge for Minkowski
spacetime [72] (for numerical implementations see [73–
75]). The application of the method in black hole space-
times proved to be difficult [76–80], until the general
construction of suitable hyperboloidal scri-fixing coor-
dinates on asymptotically flat spacetimes has been pre-
sented [81]. Since then, hyperboloidal scri-fixing coordi-
nates have been employed in a rich variety of problems
concerning black hole spacetimes [22, 82–91].
In particular, hyperboloidal compactification has been
applied to solve in time-domain the homogeneous Regge-
Wheeler-Zerilli (RWZ) equations [92–96] for metric per-
turbations of a Schwarzschild black hole [86]. This work
showed the efficiency of hyperboloidal compactification
as applied to the RWZ equations and discovered that the
asymptotic formula relating the curvature perturbation
ψ4 to the gravitational strain is invalid for the polyno-
mially decaying solution even at large distances used for
standard waveform extraction, thereby emphasizing the
importance of including null infinity in numerical studies
of gravitational radiation.
The solution of the inhomogeneous RWZ equations on
a hyperboloidal slicing of the Schwarzschild spacetime is
discussed in this paper for the first time. The presence
of a compactly supported matter source, such as a point-
particle [11, 12] or a test-fluid [18, 97–99], implies modi-
fications. It may be desirable to use standard techniques
in a compact domain including the central black hole
and the matter dynamics. The hyperboloidal method
shall then be restricted to the asymptotic domain only,
so that standard coordinates for matter dynamics can
be employed. Such a restricted hyperboloidal compacti-
fication provides the idealized waveform at null infinity,
avoids outer boundary conditions, and increases the ef-
ficiency of the numerical computation without changing
the coordinate description of matter dynamics.
A convenient technique to achieve this, called the hy-
perboloidal layer method, has been introduced in [100]. A
hyperboloidal layer is a compact radial shell in which the
spacelike foliation approaches null infinity and the radial
coordinate is compactifying. By properly attaching such
a layer to a standard computational domain, one makes
sure that outgoing waves are transported to null infin-
ity and no outer boundary conditions are needed. An
intuitive prescription for the construction of a suitable
hyperboloidal layer, that we describe in Sec. IIIB, is to
require that the spherically outgoing null surfaces have
the same representation in the layer coordinates as in
the interior coordinates. Because the hyperboloidal layer
is practically attached to an existing computational do-
main, only minimal modifications to current numerical
infrastructures are needed for its implementation.
In this paper we apply the hyperboloidal layer method
to improve the quality of recently computed RWZ wave-
forms emitted by the coalescence of (circularized) black-
hole binaries in the test-particle limit [10] (hereafter Pa-
per I) (see also Refs. [18, 22, 23]). The central new re-
sult of this paper is the computation of highly accurate
gravitational waveforms at future null infinity (I+) with
an efficient and robust method. As in Paper I, the rel-
ative motion of the binary is driven by 5PN-accurate,
EOB-resummed [39, 101] analytical radiation reaction
and we focus on the transition from quasi-adiabatic inspi-
ral to plunge, merger, and ringdown. To span the range
between IMR and EMR, we consider five mass ratios,
ν ≡ µ/M = 10−2,−3,−4,−5,−6, where M is the mass of
the central Schwarzschild black hole, and µ is the mass
of the small compact object approximated as a point par-
ticle. We estimate the differences between waveforms ex-
tracted at I+and waveforms extracted at finite radii,

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and we provide an updated estimate of the gravitational
recoil previously computed from finite-radius waveforms
in Refs. [9, 10]. The availability of I+waveforms also
allows us to assess, in a well controllable setup, the ac-
curacy of the extrapolation procedure that is routinely
applied to NR waveforms.
The new multipolar waveform extracted at I+pre-
sented here has already been used in Ref. [22] (here-
after Paper II) to obtain several results that are valu-
able for currently ongoing EOB/NR comparisons: (i)
finite-distance effects are significant even at compara-
tively large extraction radii (r ∼ 1000M); (ii) the agree-
ment between the EOB-resummed analytical multipo-
lar waveform [39, 101] and the RWZ waveform improves
when the latter is extracted at I+; (iii) the tuning of
next-to-quasi-circular corrections to the phase and am-
plitude of the EOB-resummed (multipolar) waveform im-
proves its agreement with the RWZ waveform during the
late-plunge and merger phase (See also Ref. [40] for a
similar tuning procedure applied to several black-hole bi-
naries with comparable mass ratios.)
The paper is organized as follows. In Sec. II we briefly
recall the model for the relative dynamics of the bi-
nary. The construction of the hyperboloidal layer in
Schwarzschild spacetime is carried out in Sec. III. We
discuss the RWZ equations with and without the hyper-
boloidal layer in Sec. IV. Details of the numerical imple-
mentation are presented in Sec. V. Physical results are
collected in Sec. VI, which consists of the following parts.
First, we assess the accuracy of our implementation in the
case of stable circular orbits, and present new results for
unstable circular orbits. We then focus on the gravita-
tional waveforms emitted during the transition from the
quasi-circular inspiral through plunge, merger, and ring-
down, and we quantify the differences with finite-radius
extraction. We discuss the performance of standard tech-
niques to extrapolate the finite-radius waveform to infi-
nite extraction radius. Concluding remarks are presented
in Sec. VI. In Appendix A we present convergence tests of
the code. In Appendix B we summarize the relations be-
tween the RWZ master functions and asymptotic observ-
ables. We mainly use geometrized units with G = c = 1.
II. RELATIVE DYNAMICS
The relative dynamics of the binary is computed as in
Paper I and II; here we review a few elements that are
relevant to our study.
The binary dynamics has a conservative part (Hamil-
tonian) and a dissipative part (radiation-reaction force).
The conservative part is described by the ν → 0 limit
of the EOB Hamiltonian (the Hamiltonian of a parti-
cle in Schwarzschild spacetime) with the following, di-
mensionless variables: the relative separation r = R/M,
the orbital phase ϕ, the orbital angular momentum
pϕ= Pϕ/(µM), and the orbital linear momentum pr∗=
Pr∗/µ, canonically conjugate to the tortoise radial coordi-
nate separation r∗= r+2ln(r/2−1). The Schwarzschild
metric in standard coordinates (t,r) reads
g = −Adt2+ A−1dr2+ r2dσ2,
where dσ2is the standard metric on the unit sphere and
A ≡ 1 − 2/r. The Schwarzschild Hamiltonian per unit
(µ) mass is
(1)
ˆH =
?
A
?
1 +p2
ϕ
r2
?
+ p2
r∗.(2)
The expression for the analytically resummed mechanical
angular momentum loss (our radiation-reaction force),
ˆFϕ, is accurate at first order in the mass ratio, Ø(ν),
and is computed from the 5PN-accurate EOB-resummed
waveform of Refs. [10, 23, 39, 101]. Following [10, 18, 23,
102], we use
ˆ Fϕ≡ −32
5νΩ5r4ˆf(vϕ),(3)
where Ω = dϕ/dt is the orbital frequency, vϕ = rΩ is
the azimuthal velocity, andˆf = Fℓmax/FNewt
the Newton-normalized (ν = 0) energy flux up to mul-
tipolar order ℓmax, analytically resummed according to
Ref. [23, 39]. The resummation procedure is based on a
certain multiplicative decomposition of the circularized
multipolar gravitational waveform. More precisely, for
circular orbits, the energy flux is written as
22
denotes
Fℓmax=
ℓmax
?
1
8π
ℓ=2
ℓ
?
ℓmax
?
m=1
Fℓm
=
ℓ=2
ℓ
?
m=1
(mΩ)2|rhℓm|2. (4)
Above, hℓmis the factorized waveform of [39],
hℓm(x) = h(N,ǫ)
ℓm
(x)ˆS(ǫ)(x)Tℓm(x)eiδℓm(x)(ρℓm(x))ℓ, (5)
where h(N,ǫ)
given by Eq. (4) of [39], ǫ = 0 (or 1) for ℓ+m even (odd).
The remaining terms are defined as follows:ˆS(ǫ)is the
(specific) source, Eqs. (15-16) of [39]; Tℓmis the tail factor
that resums an infinite number of leading logarithms due
to tail effects, Eq. (19) of [39]; δℓm is a residual phase
correction, Eqs. (20-28) of [39]; and ρℓm is the residual
amplitude correction, that we keep up to 5PN fractional
accuracy [101], although their knowledge (and that of the
δℓm’s) has been recently increased up to 14PN fractional
order [103].
Note that the argument in the multipoles of Eq. (5)
(and therefore in Eq. (3)) is x ≡ v2
is preferable to xcirc
≡ Ω2/3due to the violation
of the circular Kepler’s constraint during the plunge
phase [23, 102]. The sum in Eq. (4) is truncated at
ℓmax= 8 included, and the system is initialized (in the
ℓm
(x) represents the Newtonian contribution
ϕ= (rΩ)2, that

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0 2040 6080 100
r?
20
40
60
80
t
FIG. 1: Level sets of the hyperboloidal time τ as defined by
Eqs. (8), (15) and (29) with respect to standard Schwarzschild
coordinates {t,r∗}. The dashed line at r∗ = R∗ = 50 depicts
the location of the interface between the inner domain and
the hyperboloidal layer.
strong-field region 6 < r ≤ 7 ) with post-circular initial
data [18, 30], which yields negligible initial eccentricity.
The dynamics is then computed by solving Eqs. (1)-(7)
of Paper I.
III. A HYPERBOLOIDAL FOLIATION OF
SCHWARZSCHILD SPACETIME
In this Section we discuss the hyperboloidal layer ap-
proach in Schwarzschild spacetime.
hyperboloidal foliation by gluing together a truncated
Cauchy surface, which covers the strong-field region of
the particle motion, and a hyperboloidal surface [81].
Because a hyperboloidal surface is spacelike by construc-
tion, and because Cauchy surfaces are also spacelike, one
can choose a global hyperboloidal foliation to agree with
Cauchy surfaces in a compact inner domain that includes
the motion of the particle and the central black hole. This
choice allows us to employ standard coordinates near the
central black hole. The outer, asymptotic, domain is in-
cluded in the hyperboloidal layer.
We construct a
A. General properties
A hyperboloidal layer is defined as a compact radial
shell in which the spacelike foliation approaches null in-
finity and the radial coordinate is compactified. We de-
termine the coordinates by requiring that outgoing null
surfaces have the same representation in the layer coordi-
nates as in the inner domain coordinates. We connect the
coordinates used in the compact inner domain (Cauchy
region) with the coordinates used in the outer domain
(hyperboloidal layer) at an interface.
We depict such a foliation with respect to standard
coordinates {t,r∗} in Fig. 1. The level sets of the new
time function, τ(t,r∗), agree with the level sets of the
??
??
?0
?
??
??
??
??
singularity
FIG. 2: Penrose diagram of Schwarzschild spacetime depict-
ing the causal properties of the foliation plotted partially in
Fig. 1. The dashed line indicates the interface to the hy-
perboloidal layer.The time surfaces agree with standard
Schwarzschild time surfaces to the left of the interface. The
diagram also shows that the foliation stays spacelike every-
where, including the asymptotic domain near null infinity.
standard Schwarzschild time, t, for r∗≤ R∗= 50. The
dashed line indicates the timelike surface, referred to as
the interface (r∗ = R∗ = 50), at which we smoothly
modify the spacelike surfaces to approach outgoing null
rays asymptotically.
The spacelike surfaces partially depicted in Fig. 1 ap-
proach outgoing null rays, but never become null surfaces
themselves. The asymptotic causal structure can not be
clearly depicted in Fig. 1. A better visualization of the
causal structure is the Penrose diagram in Fig. 2. The
interface (still represented by a dashed line) is depicted
close to the black hole for visualization, but the causal
structure is accurate in this diagram. We see that the
hyperboloidal foliation agrees with standard t surfaces
near the black hole. Beyond the interface, the surfaces
smoothly approach future null infinity in a spacelike man-
ner. Although the surfaces look like they are becoming
null in Fig. 1, the Penrose diagram in Fig. 2 clearly shows
that the surfaces are spacelike everywhere. This causal
behavior allows us to solve a usual initial-boundary value
problem, while extracting gravitational waveforms at fu-
ture null infinity.
The hyperboloidal foliation that we employ is not only
suitable for wave extraction, it also provides a solution
to the outer boundary problem. Instead of truncating
the simulation domain at a finite but large distance,
we employ a compactifying coordinate with respect to
which null infinity is at a finite coordinate location. It
is well known that compactification leads to loss of res-
olution near the outer boundary when Cauchy foliations
are used [104]. We do not run into this problem because
we need to resolve only a finite number of oscillations on
an infinite domain along hyperboloidal foliations, as op-

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5
?40
?20204060
Ρ
?1.0
?0.5
0.5
1.0
Characteristic speed
FIG. 3: Characteristic speeds on the numerical grid as given
in Eq. (3). The dashed line denotes the location of the inter-
face between the inner domain and the hyperboloidal layer.
The outgoing speed has the same value in the inner domain as
in the layer, whereas the incoming speed smoothy approaches
zero in the layer.
posed to an infinite number of oscillations along Cauchy
foliations [100].
A good illustration that compactification solves the
outer boundary problem is given by a depiction of char-
acteristic speeds on the numerical grid (Fig. 3).
outgoing speed of characteristics is nonvanishing finite
at future null infinity. The incoming speed, on the other
hand, vanishes because future null infinity is itself an
incoming null surface (Fig. 2). No outer boundary con-
ditions are needed because there are no incoming char-
acteristics from the outer boundary.
Note that the compactifying coordinate is conceptually
independent from the hyperboloidal foliation. We can
choose any compactifying coordinate along the spacelike
surfaces of our foliation compatible with scri-fixing. The
choice of hyperboloidal foliation and compactification to-
gether determines the structure of characteristics on the
numerical grid. The choices for Fig. 3 ensure that the
outgoing characteristic speed is unity in the layer. In the
next Section we discuss how to achieve this.
The
B. Explicit construction of the hyperboloidal layer
There are different ways to construct a hyperboloidal
layer. One we find most lucid is to consider the expression
of outgoing null rays in local coordinates. In standard
waveform extraction methods, the solution is computed
along t surfaces and the waveform is plotted along the
outgoing null surfaces t − r∗. Naturally, we would like
to keep the expression of outgoing null rays invariant in
our formulation. We would also like to keep the time
direction invariant, so that ringdown frequencies or decay
rates that we compute are physical. Our requirements for
a suitable hyperboloidal layer are as follows:
1. The exterior timelike Killing vector field in local
coordinates is kept invariant in the layer.
2. The outgoing null rays in local coordinates is kept
invariant in the layer.
3. The local coordinates in the layer agree with the
standard {t,r∗} coordinates at the interface.
Now we formalize these requirements. The first re-
quirement gives a relation between the new time coordi-
nate τ and the standard time coordinate t. The require-
ment that the Killing field is kept invariant translates into
∂t= ∂τ. This condition is fulfilled by a transformation
of the form
τ = t − h(r∗) ,(6)
where the function h(r∗) is called the height function.
The height function can only depend on spatial coordi-
nates to leave the timelike Killing field invariant. We let
the height function depend only on the tortoise coordi-
nate because our problem is spherically symmetric.
Under the transformation (6) the Schwarzschild met-
ric (1) becomes
g = A?−dτ2− 2Hdτdr∗+?1 − H2?dr2
where H ≡ dh/dr∗is called the boost function. For ex-
ample, ingoing Eddington-Finkelstein coordinates are ob-
tained with H = −2/r. Similarly, Painlev´ e-Gullstrand
coordinates are obtained with H = −?2/r. The con-
event horizon instead of intersecting at the bifurcation
sphere and are therefore suitable for excision. Note that
both choices give H = −1 at the horizon [105].
We require an analogous behavior in the asymptotic
domain, in the sense that the resulting surfaces should
foliate future null infinity instead of intersecting at spa-
tial infinity. The analogy with excision indicates that one
needs to satisfy H = 1 at infinity. The choice of a suit-
able boost function follows from the second item in our
list. We require that the outgoing null rays in local coor-
dinates is kept invariant. Denoting the layer coordinates
with {τ,ρ}, we require
∗
?+ r2dσ2, (7)
stant time hypersurfaces in these coordinates foliate the
t − r∗= τ − ρ,(8)
where ρ is a yet unspecified compactifying coordinate. By
combining Eqs. (6) and (8) we get for the height function
h(r∗) = r∗−ρ(r∗). Taking the derivative of this equation
with respect to r∗, we obtain the following relation be-
tween the boost function H and the Jacobian dρ(r∗)/dr∗
of the spatial compactification
dρ
dr∗
= 1 − H .(9)
The Jacobian of any compactification vanishes at the do-
main boundary, so we have H = 1 at null infinity.
The condition (8) has two important consequences.
First, the outgoing characteristic speed, which is +1 in
the inner domain, remains +1 also across the hyper-
boloidal layer. Second, the incoming characteristic speed,
which is −1 in the inner domain, smoothly decreases in

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?40
?200 204060
Ρ
0
10
20
30
40
50
Τ
FIG. 4: The structure of the characteristics on the numerical
grid. Compare Figs. 1, 2, and 3.
the layer to reach zero at future null infinity. This is eas-
ily seen by writing the Schwarzschild metric (7) using the
compactifying coordinate ρ as1
g = A
?
−dτ2−
2H
1 − Hdτdρ +1 + H
1 − Hdρ2
?
+ r(ρ)2dσ2.
(10)
The outgoing (c+) and incoming (c−) characteristic
speeds of spherically symmetric null surfaces read
c+= 1,c−= −1 − H
1 + H.
(11)
Note that c−= 0 at the outer boundary of the ρ-domain,
where H = 1 by (9). The speeds are plotted in Fig. 3
for a particular choice of spatial compactification that we
describe in Sec. IIIC.
Let us now discuss the third condition in our list,
namely the requirement that the new coordinates {τ,ρ}
agree with standard coordinates {t,r∗} at the interface
between the inner domain and the hyperboloidal layer.
This condition can be fulfilled by a suitable choice of the
compactifying coordinate. The spatial compactification
r∗(ρ) shall have the following differentiability properties
along the interface at r∗= R∗
r∗(R∗) = R∗,
dr∗
dρ
ρ=R∗
drk
∗
dρk
ρ=R∗
(12)
????
= 1,(13)
????
= 0, k > 1.(14)
These relations imply that the coordinates r∗and ρ, and
therefore t and τ, agree along the interface to kth order.
1Note that the metric in Eq. (10) is singular at the boundary be-
cause the Jacobian of any compactification is singular. This sin-
gularity can be rescaled away with a conformal factor, but such
a rescaling is not necessary for our purposes because the RWZ
equation in hyperboloidal compactification is regular without an
explicit conformal rescaling of the background [86].
We give an explicit choice for the compactification r∗(ρ)
in Sec. IIIC.
For completeness, we finally depict in Fig. 4 the global
structure of the characteristics propagating along the nu-
merical grid {τ,ρ} in a suitable hyperboloidal compact-
ification. The outgoing characteristics are straight lines
with 45 degrees to the ρ-axis, just as for the {t,r∗} co-
ordinates. In agreement with Eq. (11) and Fig. 3, there
are no incoming characteristics from the outer boundary.
A point where our approach can be further improved
is indicated in Fig. 4. We truncate the infinite computa-
tional domain in r∗ to the left arbitrarily at r∗= −50.
As a result, there are incoming modes from the inner
boundary that need to be set by artificial boundary con-
ditions. This procedure can contaminate the interior so-
lution and make the calculation of GWs absorbed by
the black hole inaccurate (for example, to reduce con-
tamination Ref. [106] uses a very large value of the ex-
traction radius, rextr
∗
= −1500M, for computing the ab-
sorbed fluxes). In addition, the efficiency of the numeri-
cal computation is reduced by the coordinates used near
the black hole. We accept these disadvantages because
we want to describe the dynamics of the test-mass using
{t,r∗} coordinates, exactly as in Paper I.
A way to avoid the inner timelike boundary near the
black-hole horizon is to work in horizon-penetrating co-
ordinates in combination with excision. Then one needs
to transform the RWZ equations, their sources, as well
as the relative dynamics of the binary, that we used in
Paper I, to horizon-penetrating coordinates (coordinate-
independent expressions for the RWZ equations and
sources are explicitly given in Ref. [92, 96]).
horizon-penetrating, hyperboloidal foliation is the clean-
est option to compute accurately both the asymptotic
and the absorbed waves. Alternatively, one can construct
such coordinates also by attaching an internal layer to
the truncated {t,r∗} domain so that the event horizon,
r∗ = −∞, is compactified. Because our main focus in
this study is on the asymptotic waveform, we use the hy-
perboloidal layer only in the exterior asymptotic domain.
Using a
C.Spatial compactification
We present the form of the compactifying coordinate
that we use in our numerical calculations. We transform
r∗by introducing a compactifying coordinate ρ via
r∗=
ρ
Ω(ρ),(15)
where, Ω(ρ) is a suitable function of ρ (not to be confused
with the orbital frequency in Sec. II). The function Ω(ρ)
has similar properties as the conformal factor in the con-
formal compactification of asymptotically flat spacetimes
proposed by Penrose [107, 108]. For the regularity of the
transformation in the interior we require that Ω has a
definite sign, say, Ω > 0 for all ρ < S, where S denotes
the coordinate location of null infinity, and therefore the

Page 7

7
zero set of Ω. To map the infinite domain R∗≤ r∗< +∞
to the finite domain R∗≤ ρ ≤ S we require
Ω(S) = 0,Ω′(S) ?= 0.(16)
where Ω′≡ dΩ/dρ.
In addition, we also require that our coordinates agree
with standard coordinates in an inner domain. Therefore
we set Ω = 1 for all ρ ≤ R∗, where R∗denotes the loca-
tion of the interface. The transition to the layer at this
interface needs to be sufficiently smooth for a stable nu-
merical implementation. We require in accordance with
Eqs. (13)-(14)
dkΩ
dρk
???
ρ=R∗= 0withk ≥ 1.(17)
The maximum value of k for which the above property
is satisfied determines the differentiability of the layer.
By differentiating Eq. (15), we get with Eq. (9)
H(ρ) = 1 −
Ω2
Ω − ρΩ′. (18)
The form of the compactifying coordinate (15) is con-
venient because it allows us to control the hyperboloidal
foliation by a suitable function Ω(ρ) via Eq. (18).
also makes the connection to the definition of asymptotic
flatness within the Penrose conformal compactification
picture clear. However, we emphasize that, in our spe-
cific case, we can also use a more general transformation
than (15), which fulfills the conditions of a coordinate
compactification.
It
IV.THE RWZ EQUATIONS
In this Section we discuss the RWZ equations as im-
plemented numerically. For the relations of the RWZ
master function with the asymptotic observable quanti-
ties see Appendix B.
A. The RWZ equations in the interior
In the interior domain the RWZ equations with a point-
particle source are written as in Paper I. Given the dy-
namics of the particle, one solves the following two de-
coupled partial differential equations for each multipole
(ℓ,m) of even (e) or odd (o) type2
∂2
tΨ(e/o)
ℓm
− ∂2
r∗Ψ(e/o)
ℓm
+ V(e/o)
ℓ
Ψ(e/o)
ℓm
= S(e/o)
ℓm
, (19)
2In our case, these correspond respectively to multipoles with ℓ+
m = even and ℓ + m = odd.
with source terms S(e/o)
the phase-space variables (r∗,p∗). The sources have the
structure
ℓm
that are explicit functions of
S(e/o)
ℓm
= G(e/o)
+ F(e/o)
ℓm
ℓm(r,t)δ(r∗− r∗(t))
(r,t)∂r∗δ(r∗− r∗(t)) ,(20)
where r∗(t) is here indicating the particle radial coordi-
nate. The explicit expressions for the sources are given in
Eqs. (20)-(21) of [18], to which we address the reader for
further technical details. In our approach the distribu-
tional δ-function is approximated by a narrow Gaussian
of finite width σ ≪ M (see Sec. VB).
B. The RWZ equations in the hyperboloidal layer
As explained in Sec. III there are three essential steps
to the construction of the hyperboloidal layer:
1. Introduce a new time coordinate τ, Eq. (6), that
preserves the stationarity of the background,
∂t= ∂τ
⇒τ = t − h.(21)
2. Fix the time coordinate such that the expression of
the outgoing null rays is invariant in the layer,
t − r∗= τ − ρ⇒H = 1 −dρ
dr∗
.(22)
3. Choose a suitable compactifying coordinate ρ so
that the coordinates in the layer agree with the
coordinates near the black hole, satisfying the con-
ditions (12)-(14).
The whole prescription results in is a simple coordinate
transformation, {t,r∗} → {τ,ρ}, that satisfies the above
properties. The derivative operators in standard coordi-
nates transform as
∂t= ∂τ,∂r∗= −H ∂τ+ (1 − H)∂ρ.(23)
Applying this transformation on Eq. (19) (dropping all
multipolar indices)
(∂2
t− ∂2
r∗+ V )Ψ = S,(24)
we get for the wave operator in the new coordinates
∂2
+(1 − H)?−2H∂τ∂ρ+ (1 − H)∂2
We can take out a (1 − H) term from the operator. We
need to be careful with the lower order terms in (24). The
source term is compactly supported in a neighborhood
of the particle in the interior domain and therefore is
not a concern. The potential, however, is nonvanishing
in the wave zone.Its fall-off behavior is essential for
the applicability of the hyperboloidal method [86]. The
t− ∂2
r∗= −(1 − H2)∂2
τ+
ρ− (∂ρH)(∂τ+ ∂ρ)?.

Page 8

8
potential in the RWZ equation falls off as r−2both for
even and odd parity perturbations. Therefore we can
introduce the rescaled potential
¯V ≡ V/(1 − H),(25)
which has a regular limit at null infinity. To see this,
consider for example the odd-parity (Regge-Wheeler) po-
tential
V(o)=
1
r2
?
ℓ(ℓ + 1) −6
r
?
,(26)
we have with (18)
¯V(o)=
V(o)
1 − H=(Ω − ρΩ′)
ρ2
r
?
ℓ(ℓ + 1) −6Ω
ρr
?
, (27)
where ρr≡ Ωr. The rescaled Schwarzschild radius ρrhas
a nonvanishing limit at infinity because r and r∗coincide
asymptotically. As a result, we have ρr = ρ = S at
infinity. An analogue regular expression holds also for
the even-parity (Zerilli) potential.
Then we can write the RWZ equation in the layer as
−(1 + H)∂2
−(∂ρH)(∂τ+ ∂ρ)Ψ +¯V Ψ = 0.
From this form of the equation, it is immediately clear
that setting H = 0 recovers the standard RWZ equa-
tion (19). We also see that the equation is regular and
pure outflow at infinity (H = 1).
τΨ − 2H∂τ∂ρΨ + (1 − H)∂2
ρΨ
(28)
V.NUMERICS
The numerical technique employed in our code is a
standard combination of finite-difference approximation
for the spatial derivatives and Runge-Kutta methods for
time integration [10, 86]. In this section we briefly review
the method.
A.Numerical methods
Our code solves the RWZ equation in first-order-in-
time second-order-in-space form adopting the method of
lines and the Runge-Kutta 4th order scheme. The right
hand side is discretized in space on a uniform grid in
the coordinate ρ ∈ [ρmin,S]R∗, where R∗ denotes the
interface to the hyperboloidal layer and S the coordi-
nate location of I+. Finite differences are employed for
the derivatives. We use 4th order central stencils in the
bulk, lop-sided or sided 4th order stencils for the outer-
most points (ρ = ρmin and ρ = S). No boundary data
is prescribed at I+, whereas maximally dissipative 4th
order convergent outgoing boundary conditions [109] are
imposed at the inner boundary. Kreiss-Oliger type dis-
sipation is added to the RWZ equation. The particle
trajectory is updated using a 4th order Runge-Kutta in-
tegrator with adaptive time-step. The convergence of the
code is demonstrated in Appendix A.
In our numerical computations we set
Ω = 1 −
?ρ − R∗
S − R∗
?4
Θ(ρ − R∗), (29)
though various other choices are possible. The step func-
tion, Θ(ρ − R∗), indicates that compactification is per-
formed only for ρ > R∗. We choose the numerical domain
as [ρmin,S]R∗= [−50,70]50; Figs. 1, 3, and 4 refer to
these settings. For the production runs that we present
below, the ρ-domain is covered by 12001 points, that cor-
respond to gridspacing ∆ρ = 0.01.
B.Particle treatment
Following previous work [10, 18], the δ-function in the
RWZ source is represented by a narrow Gaussian of fi-
nite width ∆ρ < σ ≪ M. The hyperboloidal compact-
ification has an advantage also on the treatment of the
Dirac distribution via a smooth Gaussian because most of
the computational resources are used for the strong-field,
bulk region so that narrow Gaussians can be efficiently
resolved. For the production runs that we present below,
we use σ = 0.08M.
We inject zero initial data for the RWZ master func-
tions switching on the sources progressively in time3
following the prescription [110],
S ?→
S
exp[−a0(t − t0)] + 1, (30)
where typically a0 = 1/M and t0 = 40M.
served that this smooth switch-on significantly reduces
the (localized)“junk” radiation contained in the initial
data, without, obviously, eliminating it completely.
We ob-
VI. RESULTS
Let us briefly summarize our main results. In Sec. VIA
we focus on circular orbits to assess the performance of
our new numerical implementation.
gravitational energy flux emitted at null infinity by a
particle on stable circular orbits and compare it with
the semi-analytic data of Fujita et al. [113].
compute (and characterize) the GW energy flux emitted
by the particle on unstable circular orbits. In particu-
lar, we extract from the data the corresponding residual
amplitude corrections ρℓm introduced in Ref. [39]. We
focus then on the transition from quasi-circular inspiral
We compute the
We also
3This approach has been suggested to reduce the impact of Jost
solutions [110–112].

Page 9

9
0.13 0.1350.140.1450.150.1550.160.165
0.016
0.017
0.018
0.019
0.02
0.021
0.022
0.023
0.024
x
∆F/Fexact[%]
FIG. 5: Stable circular orbits: fractional difference between
the RWZ total energy flux computed with our code and ex-
tracted at I+(up to ℓ = 8) and the corresponding semi-
analytic data computed by Fujita et al. [113].
to plunge, merger and ringdown. In Sec. VIB we dis-
cuss the total gravitational waveform, including up to
ℓmax = 8 multipoles, extracted at I+. This waveform
is then compared in Sec. VIB1 to waveforms extracted
at finite radii. We estimate phase and amplitude dif-
ferences and test the standard extrapolation procedure
that is routinely applied to NR waveforms. In Sec. VIC
a self-consistency check of the treatment of the dynamics
is presented. Our prescription for the radiation reaction
is checked on consistency (even beyond the LSO cross-
ing) between the GW angular momentum flux extracted
at I+and the (5PN EOB-resummed) mechanical angu-
lar momentum loss Fϕ. In Sec. VID we compute the final
and maximum gravitational recoil of the final black-hole
in the ν → 0 limit, obtaining a more accurate estimate
than the ones given in Paper I.
A.Circular orbits
1.Accuracy: comparison with data by Fujita et al.
As a test of the accuracy of our new setup we compute
the gravitational wave energy and angular momentum
fluxes emitted by a particle on stable circular orbits. For
each orbital radius, r0(in units of M hereafter), we con-
sider the complete multipolar waveform (up to ℓmax= 8)
measured at I+and compute the fluxes summing to-
gether all multipoles via Eqs. (B2) and (B3). We con-
sider circular orbits belonging to both the stable branch
(r0 ≥ 6) and the unstable branch (3 < r0 < 6). The
computation of the GW fluxes from stable circular orbits
in Schwarzschild spacetime has been performed several
times in the past, with different integration techniques
(either in time domain or in frequency domain) and with
0.120.140.160.180.20.220.240.260.280.30.32
0
2
4
6
8
10
12
14
16
18
20
ˆF


Numerical (RWZ)
Analytical (EOB-resummed)
0.120.140.16 0.18 0.20.22 0.24 0.260.280.30.32
0
0.05
0.1
x
∆ˆF/ˆFRWZ
FIG. 6: Newton-normalized total gravitational wave energy
flux summed up to ℓ = 8.The analytical (5PN-accurate,
EOB-resummed) flux is compared with the numerical points,
that include also unstable circular orbits. The vertical dashed
line indicates the LSO location at x = 1/6.
increasing level of accuracy [106, 113–117]. Currently,
the method that yields the most accurate results is the
one developed by Fujita et al. [113], which allows for the
computation of emitted fluxes with a relative error of or-
der 10−14. We checked the accuracy of our numerical
setup (finite differencing with a hyperboloidal layer and
wave extraction at I+) by considering a small sample
of stable orbits, with radii in the range 6 ≤ r0≤ 7.9456
and spaced by ∆r0 = 0.1 for 6 ≤ r0 ≤ 7.
multipolar information for r0 = 7.9456 (both energy
and angular momentum fluxes) is listed in Table II in
Appendix A, so to facilitate the comparison with pub-
lished data [106, 114].In addition, a direct compari-
son with the data kindly given to us by Ryuichi Fu-
jita and computed as in Ref. [113], that we consider
“exact”, reveals that our finite-differencing, time-domain
computation is rather accurate: The relative difference
∆Fℓm/FExact
ℓm
= (FRWZ
ℓm
− FExact
is below 0.8 % in almost every multipolar channel (see
Appendix A for more detailed information). Summing
together all multipoles, we find that the total energy flux,
dominated by the modes with smaller values of ℓ and
with m = ℓ, agrees with the exact data within 0.02 %.
In Fig. 5 we show the relative difference between total
fluxes, ∆F/FExact= (FRWZ− FExact)/FExact(summed
up to ℓmax= 8), versus x = 1/r0.
The full
ℓm
)/FExact
ℓm
in energy flux
2.Total energy flux, unstable orbits and the “exact”
multipolar amplitudes ρℓm
Now that we have assessed the accuracy of our finite-
difference, time-domain code, we calculate the GW en-

Page 10

10
0.15 0.20.25 0.3
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
x
ρ2m


ℓ = 2,m = 2
ℓ = 2,m = 1
0.15 0.2 0.250.3
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
x
ρ3m


ℓ = 3,m = 3
ℓ = 3,m = 2
ℓ = 3,m = 1
0.120.140.160.180.20.220.240.260.280.30.32
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
x
ρ4m


ℓ = 4,m = 4
ℓ = 4,m = 3
ℓ = 4,m = 2
ℓ = 4,m = 1
0.120.14 0.160.180.20.22 0.240.260.280.30.32
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
x
ρ5m


ℓ = 5, m = 5
ℓ = 5, m = 4
ℓ = 5, m = 3
ℓ = 5, m = 2
ℓ = 5, m = 1
0.15 0.20.250.3
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
x
ρ6m


ℓ = 6,m = 6
ℓ = 6,m = 5
ℓ = 6,m = 4
ℓ = 6,m = 3
ℓ = 6,m = 2
0.15 0.20.250.3
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
x
ρ7m


ℓ = 7,m = 7
ℓ = 7,m = 6
ℓ = 7,m = 5
ℓ = 7,m = 4
ℓ = 7,m = 3
FIG. 7: The “exact” functions ρℓm extracted from the numerical fluxes for 1/7.9456 ≤ x ≤ 1/3.1. The vertical dashed line
indicates the LSO location, x = 1/6.
ergy flux for unstable circular orbits, i.e.
radii in the range 3 < r0 < 6. This computation has
received a rather poor attention in the literature. To our
knowledge, the only computation along unstable orbits
was performed in Ref. [39] for the ℓ = m = 2 flux and
with less good accuracy than what we are able to do
here. In [39] it was pointed out that the knowledge of
orbits with
the emitted flux also below the LSO might be helpful to
improve the resummation of the residual amplitude cor-
rections ρℓmthat enter the factorized (EOB-resummed)
multipolar waveform introduced there.
We compute the multipolar fluxes for a sample of un-
stable circular orbits with 3.1 ≤ r0 < 6, spaced by
∆r0= 0.1. Figure 6 shows in the top panel (as a solid

Page 11

11
line with circles) the two branches together, both for sta-
ble and unstable orbits, of the Newton-normalized total
energy flux,ˆF = Fℓm/FN
up to ℓmax = 8. The vertical dashed line indicates the
location of the LSO at x = 1/6.
It is interesting to ask how reliable is the 5PN-accurate
EOB-resummed analytical representation of the flux over
the sequence of unstable orbits. We recall that Ref. [39]
introduced a specific factorization and resummation of
the PN waveform such that the related analytical flux
was found to agree very well with the numerical one (see
Fig. 1 (d) of [39]). For this reason the top panel of Fig. 6
additionally shows the energy flux constructed analyti-
cally from the resummed circularized multipolar wave-
form of [39] that includes all the 5PN-accurate terms
computed in [101]. The relative difference between fluxes
is plotted in the bottom panel. The figure indicates a re-
markable agreement between the analytical and numer-
ical fluxes also for circular orbits below the LSO, with a
relative difference that is almost always below 5%. Note
that the difference becomes as large as 10% only for the
last 6-7 orbits, which are very close to the light ring
(x = 1/3). It is, however, remarkable that the analyt-
ical expression for the flux, based on suitably resummed
5PN-accurate (only) results remains rather reliable in a
region were the velocity of the orbiting particle is about
half the speed of light. It will be interesting in the future
to perform such a comparison with the 14PN-accurate
expression of the waveform recently computed analyti-
cally by Fujita [103].
In the spirit of the factorized form of the multipolar
waveform entering the analytical flux, Eqs. (4)-(5), the
most important information one wants to extract from
the numerical data is the behavior of the residual ampli-
tudes ρExact
ℓm
(x) also along unstable orbits. These quan-
tities are the real unknowns of the problem, since all
other factors, i.e. the source S(ǫ)(x) and the tail factor
Tℓm(x), are known analytically. In this respect, the com-
plete knowledge of the ρExact
ℓm
field information that is only partially available via their
PN expansion The computation of ρExact
for the first time in Ref. [39]. It was restricted mainly
to stable orbits, with multipoles up to ℓmax = 6, and
was based on the numerical data computed by Emanuele
Berti [116, 117]. In addition, as mentioned above, a small
sample of unstable orbits were also considered to explore
the behavior of ρExact
22
toward the light ring.
The exact ρExact
ℓm
are obtained from the partial fluxes
FExact
ℓm
as
22, summed over all multipoles
’s brings in the full strong-
ℓm
was performed
ρExact,(ǫ)
ℓm
(x) =



?
FExact
ℓm
|Tℓm|ˆS(ǫ)
/FNewton
ℓm



1/ℓ
(31)
where the source S(ǫ)is either the energy (for even-parity
multipoles, ǫ = 0), or the Newton-normalized angular
momentum (for odd-parity multipoles, ǫ = 1) along cir-
0500100015002000 2500300035004000
−1
−0.5
0
0.5
1
Rh+/(Mν)
4180 4200 4220424042604280
(τ-S)/M
4300432043404360 43804400
−1
−0.5
0
0.5
1
Rh+/(Mν)


ℓmax= 4 (m ?= 0)
ℓmax= 6 (m ?= 0)
ℓmax= 8 (m = 0)
tLR
FIG. 8: (Color online). The Rh+/(Mν) polarization (from
Eq. (B1)) of the gravitational waveform for ν = 10−3. The
top panel shows the complete wave train (∼ 40 orbits up to
merger). The bottom panel focuses around the merger time
and illustrates the impact of subdominant multipoles. The
vertical dashed line indicates the light-ring crossing time by
the point-particle.
cular orbits, i.e.
ˆS(0)(x) =
1 − 2x
√1 − 3x
1
√1 − 3x.
(32)
ˆS(1)(x) =
(33)
The square modulus of the tail factor Tℓmreads [23, 39]
|Tℓm|2=
1
(ℓ!)2
4πˆˆk
1 − e−4πˆˆk
ℓ?
s=1
?
s2+
?
2ˆˆk
?2?
(34)
whereˆˆk = mx3/2.
The result of the computation is presented in Fig. 7
including multipoles up to ℓmax= 7. The figure clearly
shows that, for some multipoles, the quasi-linear behavior
of the ρℓm(x) above the LSO (explained in detail in [39])
is replaced by a more complicated shape below the LSO,
where high-order corrections seem relevant. The figure
completes below the LSO the data of Fig. 3 of [39], where
only stable orbits were considered. Indeed, in the stable
branch, the curves presented here perfectly overlap with
those of [39].
We postpone to future work the analytical understand-
ing of the behavior of the various ρExact
On the basis of the analytical information already con-
tained in Fig. 5 of Ref. [39], it seems unlikely that the
current 5PN-accurate analytical knowledge of the ρℓm(x)
functions can by itself explain the structure of the ρExact
close to the light-ring.It will be interesting to see
whether this structure can be fully accounted for by the
14PN-accurate results of Ref. [103].
lm
when x → 1/3.
lm

Page 12

12
B.Gravitational radiation from inspiral, plunge,
merger and ringdown
Now we discuss the properties of the gravitational
wave signal emitted by the five binaries with ν =
10−2,−3,−4,−5,−6. The initial relative separation is r0= 7
for ν = 10−2,−3,−4, r0= 6.3 for ν = 10−5and r0= 6.1
for ν = 10−6. These latter values are chosen so that
the evolution time is approximately equally long for
ν = 10−4,−5,−6(∼ 400 inspiral orbits, see Table III).
The relative dynamics is started using post-circular ini-
tial data as described in [18, 30], assuring a negligible ini-
tial amount of eccentricity. The system is then driven by
radiation reaction, Eq. (3), into a (long) quasi-adiabatic
inspiral, which is then smoothly followed by the nonadi-
abatic plunge phase, which terminates with the merger
of the two bodies and the final ringdown. The relative
dynamics and the multipolar structure of the waveforms
are qualitatively the same as described in Paper I and II.
Let us discuss the mass ratio ν = 10−3as case study.
We counted about 40 orbits up to merger4, defined as the
time at which the particle crosses the light-ring (r = 3).
Figure 8 (displayed also in Paper II and Ref. [9]) shows
the Rh+/(Mν) polarization, Eq. (B1), of the gravita-
tional waveform for this binary along the fiducial direc-
tion (θ,ϕ) = (π/4,0) for various multipolar approxima-
tion. The waveforms are displayed versus retarded time
at I+, τ − S. The most accurate waveform includes
the multipoles up to ℓmax= 8 (dash-dotted line). Sum-
ming up to ℓmax= 4 captures most of the behavior up to
the light ring crossing (tLR, vertical dashed line), while
the higher multipoles are more relevant during the late-
plunge phase and ringdown. Note also the importance of
the m = 0 modes during the ringdown.
1. Comparing waves extracted at I+and at finite radii
Access to the radiation at I+enables us to evaluate
finite distance effects in the waveform phase and ampli-
tude. We work again with mass ratio ν = 10−3only and
compare waves extracted at I+with those extracted
at three large, but finite, extraction radii rextr
(250, 500, 1000). Figure 9 displays the phase differences
∆φℓm ≡ φI+
amplitude difference ∆Aℓm/Aℓm≡ (AI+
(right panels) for the most relevant multipoles. On aver-
age, the phase differences accumulated between waves at
rextr
∗
/M = 250 and at I+is ∆φℓm∼ 0.125 − 0.25 rad,
∗
/M =
ℓm− φrextr
∗
ℓm
(left panels) and the fractional
ℓm− Arextr
∗
ℓm)/AI+
ℓm
4With a slight abuse of definition, we consider the number of “or-
bits” as the value of the orbital phase at the end of the dynamical
evolution divided by 2π. In doing so we are also including in the
computation the plunge phase, where the dynamics is nonadi-
abatic and cannot be approximated by a sequence of circular
orbits.
which decreases to ∆φℓm ∼ 0.05 rad when rextr
1000. The corresponding fractional variation of the am-
plitude is ∆Aℓm/Aℓm∼ 0.2% for rextr
drops down by roughly a factor of 10 for rextr
The phase differences shown in Fig. 9 are significant, in
that they are much larger than the numerical uncertainty
(δφ ∼ 10−6; see convergence results in Appendix A).
An interesting feature that is common to both the
phase difference and the fractional amplitude difference
is that their variation is rather small during the inspi-
ral, then decreases abruptly during the plunge (the LSO
crossing is at tLSO= u = 4076.1 for this binary) and the
smallest values are reached during the ringdown. The
multipolar behavior of Fig. 9 carries over to the total
gravitational waveform. Figure 10 shows the phase differ-
ence between the total polarization Rh+/(Mν) extracted
at I+and at finite radii. The phase difference amounts
to (on average) ∆φ ∼ 0.125 rad for rextr
∆φ ∼ 0.025 rad for rextr
ulation in the phase difference is not numerical noise, but
it is an actual physical feature due to the combination of
the (different) dephasings of the various multipoles.
We finally note that our ℓ = m = 2 EMR results
are consistent with the corresponding equal-mass re-
sults displayed in Fig. 10 of Ref. [118], where they com-
pare the extrapolated waveform to the one extracted at
rextr/M = 225. After applying both a time and a phase
shift to the finite-radius waveform, they found that the
accumulated phase difference to the extrapolated wave-
form is of order 0.2 rad, i.e. about two times our (average)
dephasing for the rextr
∗
/M = 250 waveform.
∗
/M =
∗
/M = 250, which
/M = 1000.
∗
∗
/M = 250 and
∗
/M = 1000. Note that the mod-
2.Extrapolating finite-radius waveforms to r → ∞
Now that we have shown that finite-radius effects are
significant, we use the data at I+to test, in a well con-
trollable setup, the standard extrapolation to r → ∞
routinely applied to NR finite-radius waveforms.
Indicating with r the radius at which radiation is mea-
sured in NR simulations, the waveforms are extrapolated
to r → ∞ by assuming an expansion in powers of 1/r
(see e.g. Refs. [47, 48, 50, 118]),
f(u,r) =
K
?
k=0
fk(u)
rk
,(35)
where f can be either the amplitude or the phase of the
gravitational waveform5. The extrapolation procedure
of NR data is affected by the fictitious identification of
a background (Schwarzschild or Kerr) in the numerically
generated spacetime and by subtleties in the definition
of the retarted time for each observer (see e.g. Sec. IIB
5In NR studies the extrapolation is usually applied to the curva-
ture waveform r ψ4.

Page 13

13
05001000 1500 200025003000 35004000
0
0.05
0.1
0.15
u/M
∆φ22[rad]


05001000150020002500 300035004000
0
0.05
0.1
0.15
0.2
0.25
∆φ21[rad]


0 500100015002000250030003500 4000
−0.002
0
0.002
0.004
0.006
0.008
u/M
∆A22/A22


0500 10001500 20002500 3000 35004000
−0.01
−0.005
0
0.005
0.01
0.015
∆A21/A21
rextr
∗
rextr
∗
rextr
∗
/M = 250
/M = 500
/M = 1000
0500 10001500 2000250030003500 4000
0
0.05
0.1
0.15
0.2
∆φ33[rad]


0 500100015002000 25003000 35004000
0
0.05
0.1
0.15
0.2
0.25
u/M
∆φ44[rad]


0 50010001500 20002500300035004000
−0.002
0
0.002
0.004
0.006
0.008
∆A33/A33
0 50010001500 20002500 300035004000
−0.002
0
0.002
0.004
0.006
0.008
u/M
∆A44/A44
FIG. 9: (Color online) Phase difference (left panels) and relative amplitude difference (right panels) between multipoles extracted
at I+and at finite radii. Extraction radii are rextr
∗
/M = (250, 500, 1000). Data refer to the ν = 10−3binary.
05001000 15002000 25003000 35004000
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
u/M
∆φ [rad]


rextr
∗
rextr
∗
rextr
∗
/M = 250
/M = 500
/M = 1000
FIG. 10:
Rh+/(Mν) total gravitational wave polarization at I+and
at finite radii. Data refer to the ν = 10−3binary.
(Color online) Phase difference between the
of Ref. [48] and Sec. IIIC of Ref. [118]). Thanks to the
aforementioned CCE procedure to compute the GW sig-
nal at I+, Ref. [55] was able to provide an independent
check of the extrapolation procedure. Reference [55] fo-
cused on the ℓ = m = 2 ψ4 waveform from an equal-
mass black-hole binary and considered data extracted
at r/M = (280,300,400,500,600,1000) as input for the
extrapolation procedure. Over the 1000M of evolution
from early inspiral to ringdown, Ref. [55] found a de-
phasing of 0.019 rad and a maximum fractional ampli-
tude difference of 1.08% between the extrapolated and
the I+waveforms.
Our setup permits the validation of the expansion in
Eq. (35) and a quantification of the extrapolation errors
in the absence of ambiguities related to the definition
of the extraction spheres and retarded times on a dy-
namical spacetime. The radius, r, is the areal radius of
the Schwarzschild background and the retarded time is
by construction u = τ − ρ. To produce a meaningful
comparison with the estimates of [55], we use waveforms
extracted at rextr
∗
/M = (250,500,750,1000) as input for
the extrapolation procedure, and we work again with the
ν = 10−3binary.
The phase and amplitude differences are plotted in
Fig. 11, where we show only ℓ = 2 multipoles for defi-
nitess (the picture does not change for other multipoles):
m = 1 (left panel) and m = 2 (right panel). Different

Page 14

14
0500 10001500 20002500 3000 35004000
−9
−8
−7
−6
−5
−4
−3
−2
log10(∆A21/A21)


0 500 1000150020002500 300035004000
−5.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
log10∆φ21
u/M
K = 1K = 2K = 3
0 50010001500 20002500 3000 35004000
−7
−6
−5
−4
−3
−2
log10(∆A22/A22)


0 5001000 150020002500 300035004000
−5.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
log10∆φ22
u/M
K = 1K = 2K = 3
FIG. 11: (Color online). Fractional amplitude difference (top panels) and dephasing (bottom panel) between I+and extrap-
olated waveforms. Note that we plot the log10. Multipoles are ℓ = m = 2 (left panels) and ℓ = 2, m = 1 (right panels).
Extraction radii are r∗/M ≃ (250,500,750,1000). Different lines refer to different polynomial order in the extrapolation i.e. K
in Eq. (35). The plot refers to the ν = 10−3binary.
lines in the plot correspond to different choices of the
maximum power K in the polynomial expansion (35).
The phase difference between the wave at I+and the ex-
trapolated one decreases uniformly in time: It is between
10−2and 10−3rad when a linear polynomial (K = 1)
in 1/r is assumed in Eq. (35) and it drops to between
10−4and 10−5when a cubic polynomial is used (K = 3).
In this analysis we considered only up to K = 3 be-
cause this value seems to give the best compromise be-
tween noise and accuracy when extrapolating NR wave-
forms [50, 118]. We remark, however, that in our setup
we are not limited in the choice of K. This is evident in
Fig. 12 where we use higher values of K and more extrac-
tion radii rextr
∗
/M = (250, 500, 750, 1000, 2000, 4000),
for the ℓ = m = 2 waveform. Both the phase and am-
plitude differences decrease monotonically with increas-
ing K, showing that more powers in the expansion (35)
lead to more accurate extrapolation. The simple extrap-
olation formula (35) proves robust and leads to reliable
waveforms.
C.Angular momentum loss
The main uncertainty in our approach lies, as discussed
above, on the accuracy of the analytically resummed ra-
diation reaction, Eq. (3). Several studies [10, 23] have
shown the consistency between the gravitationalwave an-
gular momentum flux computed from the RWZ waveform
(measured at a large, finite radius) and the mechanical
angular momentum loss −ˆFϕ obtained by suitably re-
summing (a la Pad´ e) the Taylor-expanded PN flux [23],
or via the multiplicative decomposition of the waveform
of [23, 39, 101], as performed in [10].
Ref. [10] pointed out a fractional difference between me-
chanical and GW angular momentum fluxes at the 10−3
level up to (and even below) the adiabatic LSO crossing.
In particular,
0500 1000150020002500 30003500 4000
−9
−8
−7
−6
−5
−4
log10∆A22/A22


K = 2K = 3K = 4K = 5
0 5001000 15002000
u/M
250030003500 4000
−7
−6.5
−6
−5.5
−5
−4.5
−4
−3.5
log10∆φ22
FIG. 12: (Color online). Residual of amplitude (top) and
phase (bottom) between the I+and the extrapolated ℓ =
m = 2 waveform. Note that we plot the log10. Extraction
radii are rextr
∗
/M = (250, 500, 750, 1000, 2000, 4000). Differ-
ent lines refer to different polynomial order in the extrapo-
lation i.e. different K in Eq. (35). Data refer to ν = 10−3
binary.
The common drawback of these studies is that the tar-
get “exact” flux is computed at a finite extraction radius
(typically r∗/M = 1000), whereas the analytical Fϕ is
computed (by construction) at I+. Because we can com-
pute the RWZ flux at I+, the comparison between the
instantaneous GW angular momentum flux˙JGW/ν2and
the mechanical angular momentum loss˙JM/ν2= −Fϕ/ν
is more meaningful, and can be calculated without the
ambiguity caused by a relative time-shift that one should
include when˙J/ν2is computed at a finite radius (it was
not included in [10] for simplicity).

Page 15

15
3.98 3.9944.014.02
t/M
4.034.04 4.054.06
x 104
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1


˙JM/ν2=-ˆ Fϕ/ν [mechanical]
˙JGW/ν2[GWs]
FIG. 13: (Color online). Late-time comparison between two
angular momentum losses for the binary with ν = 10−4. The
GW flux (˙JGW/ν2, solid line) computed from the RWZ wave-
form and extracted at I+(including up to ℓmax = 8 ra-
diation multipoles) is contrasted with the EOB-resummed,
analytical mechanical angular momentum loss −ˆ F/ν (dashed
line). The two vertical lines correspond (from left to right) to
the particle crossing respectively, the adiabatic LSO location
(r = 6, tLSO = 39974.40), and the light-ring location (r = 3,
tLR = 40388).
0.0550.06 0.0650.070.075
MΩ
0.080.0850.09 0.095
−0.01
−0.005
0
0.005
0.01
0.015
0.02
(˙JM-˙JGW)/˙JM


ν = 10−2
ν = 10−3
ν = 10−4
ν = 10−5
ν = 10−6
LSO crossing
FIG. 14: (Color online). Relative difference between the me-
chanical angular momentum loss and the GW energy flux for
the five mass ratios considered. The figure highlights how a
very small fractional difference is maintained also after the
LSO crossing.
We focus first on the ν = 10−4simulation.
Fig. 13 we compare the mechanical angular momentum
loss (changed sign, −ˆFϕ/ν, dashed line) to the instan-
taneous angular momentum flux (˙JGW/ν2, solid line)
extracted at I+and plotted versus the corresponding
retarded time τ − S. SinceˆFϕ is parametrized by the
mechanical time t, we use this as x-axis label. The two
vertical lines on the figure indicate (from left to right) the
In
particle crossing of the adiabatic LSO location (r/M = 6,
tLSO= 39974.40, dashed black line), which can be con-
sidered approximately as the end of the inspiral, and the
light-ring crossing (r/M = 3, tLR = 40388, dashed red
line). Consistently with the findings of Paper I (compare
Fig. 8 in Paper I, which used the flux at rextr
the figures confirm visually the good agreement between
the two fluxes also below the LSO crossing, and actu-
ally almost during the entire plunge phase. The rela-
tively large difference between the fluxes around the light-
ring crossing is due to the lack of next-to-quasi-circular
(NQC) corrections in the waveform amplitude as well
as of ringdown quasi-normal-modes, in the analytically
constructed ˙JM/ν2. Note that Paper II has explicitly
shown how these corrections can be effectively added to
the “bare” inspiral resummed multipolar waveform that
we use to compute radiation reaction to obtain a much
closer agreement between the waveform moduli in the
strong-field-fast-velocityregime. We work with NQC-free
radiation reaction because the late part of the dynamics
(and waveform) is practically unaffected by details of the
radiation reaction, as discussed in [23].
∗
/M = 1000),
The qualitative agreement seen in Fig. 13 is depicted
more accurately in Fig. 14.
the five mass ratios considered) the relative difference
(˙JM−˙JGW)/˙JM versus the orbital frequency MΩ. For
reference, the LSO crossing frequency, MΩLSO≈ 0.136,
is marked by a vertical dashed line (red online) in the
figure6. For ν = 10−3, the relative difference is initially
at 2.5 × 10−3and then it slowly increases to reach only
5 × 10−3at the LSO crossing. These (rather small) dif-
ferences are due to the limited PN knowledge (5PN) at
which the residual multipolar amplitudes ρℓm are im-
plemented in the radiation reaction. When considering
ν = 10−4, still starting at r0 = 7, the picture remains
practically unchanged (solid line in the figure), although
the difference is slightly larger at the LSO crossing and
during the plunge. The cases of ν = 10−5and ν = 10−6
(that start respectively at r0 = 6.3 and r0 = 6.1) are
practically superposed and one sees again a slight in-
crease of the difference around the LSO. This agreement
is a strong indication that the analytically resummed
radiation-reaction force is suitable to drive the dynam-
ics of a (circularized) EMRI, notably with ν = 10−6, an
interesting source for LISA7. In the future, it should be
explored how this agreement improves when the 14PN-
accurate corrections to the ρℓmfrom [103] are included
in the flux.
The figure displays (for
6Note that the other two apparent vertical lines are actually the
junk radiation corresponding to the beginning of the ν = 10−5
and ν = 10−6simulations.
7A similar conclusion was also reached in Refs. [41, 42], that actu-
ally pointed out that one should properly calibrate the Fϕ func-
tion to have an accurate representation of the EMRI dynamics.
Note however that here, contrarily to Refs. [41, 42], we include
in the discussion also the late inspiral and plunge regime.