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arXiv:0710.3823v2 [grqc] 19 Sep 2008
Eccentric binary blackhole mergers:
The transition from inspiral to plunge in general relativity
Ulrich Sperhake1,∗, Emanuele Berti2,3, Vitor Cardoso4,5,
Jos´ e A. Gonz´ alez1,6, Bernd Br¨ ugmann1, Marcus Ansorg7
1Theoretisch Physikalisches Institut, Friedrich Schiller Universit¨ at, 07743 Jena, Germany
2Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
3McDonnell Center for the Space Sciences, Department of Physics,
Washington University, St. Louis, MR 63130, USA
4Department of Physics and Astronomy, The University of Mississippi, University, MS 386771848, USA
5Centro Multidisciplinar de Astrof´ ısica  CENTRA, Departamento de F´ ısica,
Instituto Superior T´ ecnico, Av. Rovisco Pais 1, 1049001 Lisboa, Portugal
6Instituto de F´ ısica y Matem´ aticas, Universidad Michoacana de San Nicol´ as de Hidalgo,
Edificio C3, Cd. Universitaria. C. P. 58040 Morelia, Michoac´ an, M´ exico and
7MaxPlanckInstitut f¨ ur Gravitationsphysik, AlbertEinsteinInstitut, 14476 Golm, Germany
(Dated: September 19, 2008)
We study the transition from inspiral to plunge in general relativity by computing gravitational
waveforms of nonspinning, equalmass blackhole binaries. We consider three sequences of simula
tions, starting with a quasicircular inspiral completing 1.5, 2.3 and 9.6 orbits, respectively, prior to
coalescence of the holes. For each sequence, the binding energy of the system is kept constant and
the orbital angular momentum is progressively reduced, producing orbits of increasing eccentricity
and eventually a headon collision. We analyze in detail the radiation of energy and angular mo
mentum in gravitational waves, the contribution of different multipolar components and the final
spin of the remnant, comparing numerical predictions with the postNewtonian approximation and
with extrapolations of pointparticle results. We find that the motion transitions from inspiral to
plunge when the orbital angular momentum L = Lcrit ≃ 0.8M2. For L < Lcrit the radiated energy
drops very rapidly. Orbits with L ≃ Lcrit produce our largest dimensionless Kerr parameter for
the remnant, j = J/M2≃ 0.724 ± 0.13 (to be compared with the Kerr parameter j ≃ 0.69 result
ing from quasicircular inspirals). This value is in good agreement with the value of 0.72 reported
in [1]. These conclusions are quite insensitive to the initial separation of the holes, and they can
be understood by extrapolating point particle results. Generalizing a model recently proposed by
Buonanno, Kidder and Lehner [2] to eccentric binaries, we conjecture that (1) j ≃ 0.724 is close to
the maximal Kerr parameter that can be obtained by any merger of nonspinning holes, and (2) no
binary merger (even if the binary members are extremal Kerr black holes with spins aligned to the
orbital angular momentum, and the inspiral is highly eccentric) can violate the cosmic censorship
conjecture.
PACS numbers: 04.25.dg, 04.25.Nx, 04.30.Db, 04.70.Bw
I.INTRODUCTION
The research area of gravitational wave (GW) physics has reached a very exciting stage, both experimentally
and theoretically. Earthbased laserinterferometric detectors, including LIGO [3], GEO600 [4] and TAMA [5], are
collecting data at design sensitivity, searching for GWs in the frequency range ∼ 10 − 103Hz. VIRGO [6] should
reach design sensitivity within one year, and the spacebased interferometer LISA is expected to open an observational
window at low frequencies (∼ 10−4− 10−1Hz) within the next decade [7].
The last two years have also seen a remarkable breakthrough in the simulation of the strongest expected GW
sources, the inspiral and coalescence of blackhole binaries [8, 9, 10]. Several groups have now generated independent
numerical codes for such simulations [11, 12, 13, 14, 15, 16, 17, 18, 19] and studied various aspects of binary black hole
mergers. In the context of analyzing the resulting gravitational waveforms, these include in particular the comparisons
of numerical results with postNewtonian (PN) predictions [20, 21, 22, 23, 24, 25], multipolar analyses of the emitted
radiation [23, 24, 26], the use of numerical waveforms in data analysis [27, 28, 29, 30] and gravitational wave emission
from systems of three black holes [31].
Despite this progress, a comprehensive analysis of binary black hole inspirals remains a daunting task, mainly
because of the large dimensionality of the parameter space. In geometrical units, the total mass of the binary is just
∗Electronic address: ulrich.sperhake@unijena.de
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an overall scale factor. The source parameters to be explored by numerical simulations (sometimes called “intrinsic”
parameters in the GW data analysis literature) include the mass ratio q = M2/M1, the eccentricity e of the orbit and
six parameters for the magnitude of the individual black hole spins and their direction with respect to the binary’s
orbital angular momentum.
In this paper we present results from numerical simulations of nonspinning, equalmass blackhole binaries, and
we focus on the effect of the orbital eccentricity on the merger waveforms. We consider three sequences, starting with
quasicircular inspirals that complete ∼ 1.5, ∼ 2.3 and ∼ 9.6 orbits, respectively, prior to coalescence of the holes.
By fixing the binding energy of the system and progressively reducing the orbital angular momentum, we produce
a sequence of orbits of increasing eccentricity and eventually a headon collision. For each of these simulations we
analyze in detail the radiation of energy and angular momentum in GWs, the contribution of different multipolar
components and the final spin of the remnant, comparing numerical predictions with the PN approximation and with
extrapolations of pointparticle results.
Noneccentric inspirals are usually considered the most interesting cases for GW detection. For an isolated binary
evolving under the effect of gravitational radiation reaction, the eccentricity decreases by roughly a factor of 3 when
the orbital semimajor axis is halved [32]. For most conceivable formation mechanisms of solarmass black hole
binaries, the orbit will usually be circular by the time the GW signal enters the bestsensitivity bandwidth of Earth
based interferometers. However, we wish to stress that our simulations could be of interest for GW detection. For
example, according to some astrophysical scenarios, eccentric binaries may be potential GW sources for Earthbased
detectors. In globular clusters, the inner binaries of hierarchical triplets undergoing Kozai oscillations can merge
under gravitational radiation reaction, and ∼ 30% of these systems can have eccentricity ∼ 0.1 when GWs enter the
detectors’ most sensitive bandwidth at ∼ 10 Hz [33]. Massive black hole binaries to be observed by LISA could also
have significant eccentricity in the last year of inspiral. Recent simulations using smoothed particle hydrodynamics
follow the dynamics of binary black holes in massive, rotationally supported circumnuclear discs [34, 35, 36]. In these
simulations, a primary black hole is placed at the center of the disc and a secondary black hole is set initially on an
eccentric orbit in the disc plane. By using the particle splitting technique, the most recent simulations follow the
binary’s orbital decay down to distances ∼ 0.1 pc. Dynamical friction is found to circularize the orbit if the binary
corotates with the disc [35]. However, if the orbit is counterrotating with the disc the initial eccentricity does not seem
to decrease, and black holes may still enter the GW emission phase with high eccentricity [34].
Complementary studies show that eccentricity evolution may still occur in later stages of the binary’s life, because
of close encounters with single stars and/or gasdynamical processes. Threebody encounters with background stars
have been studied mainly in spherical backgrounds. These studies find that stellar dynamical hardening can lead to
an increase of the eccentricity, acting against the circularization driven by the largescale action of the gaseous and/or
stellar disc, possibly leaving the binary with nonzero eccentricity when gravitational radiation reaction becomes
dominant [37, 38, 39, 40, 41]. It has also been suggested that the gravitational interaction of a binary with a
circumbinary gas disc could increase the binary’s eccentricity. The transition between discdriven and GWdriven
inspiral can occur at small enough radii that a small but significant eccentricity survives, typical values being e ∼ 0.02
(with a lower limit e ≃ 0.01) one year prior to merger (cf. Fig. 5 of [42]). If the binary has an “extreme” mass ratio
q ? 0.02 the residual eccentricity predicted by this scenario can be considerably larger, e ? 0.1. Numerical simulations
should be able to test these predictions in the near future. As shown by Sopuerta, Yunes and Laguna, eccentricity
could significantly increase the recoil velocity resulting from the merger of nonspinning blackhole binaries [43].
Independently of the presence of eccentricity in astrophysical binary mergers, the problem we consider here has
considerable theoretical interest. Our simulations explore the transition between gravitational radiation from a quasi
circular inspiral (the expected final outcome in most astrophysical scenarios) and the radiation emitted by a headon
collision, where the binary has maximal symmetry. Our work should provide some guidance for analytical studies
of the “transition from inspiral to plunge”. The first analytical study of this problem in the context of PN theory
was carried out by Kidder, Will and Wiseman [44]. The transition between the adiabatic phase and the plunge was
studied in [45] using nonperturbative resummed estimates of the damping and conservative parts of the twobody
dynamics, i.e. the socalled “Effective One Body” (EOB) model. Ori and Thorne [46] provided a semianalytical
treatment of the transition in the extreme mass ratio limit. Waveforms comprising inspiral, merger and ringdown
for comparablemass bodies have also been produced using the EOB model (see eg. [47] for extensions of the original
model to spinning binaries and for references to previous work). Preliminary comparisons of EOB and numerical
relativity waveforms showed that improved models of ringdown excitation [23, 28, 48] or additional phenomenological
terms in the EOB effective potential [49] are needed to achieve acceptable phase differences between the numerical
and analytical waveforms.
Our study is complementary to Ref. [50], that considered sequences of eccentric, equalmass, nonspinning binary
black hole evolutions around the “threshold of immediate merger”: a region of parameter space separating binaries
that quickly merge to form a final Kerr black hole from those that do not merge in a short time. Similar scenarios
have also been studied in Ref. [51], with particular regard to the maximal spin of the final hole generated in this way.
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The universality of the gravitational wave signal during the merger was analysed in Ref. [1], where it was pointed
out that binaries largely circularize after about 9 orbits when starting with eccentricities below about 0.4. The first
comparison between numerical evolutions of eccentric binaries with postNewtonian predictions was presented in [52].
Our focus in this work is on the higheccentricity region of the parameter space, which always leads to merger. In
particular, the nearheadon limit of our study is of interest as a first step to compute the energy loss and production
crosssection of miniblack holes in TeVscale gravity scenarios (possibly at the upcoming LHC [53]), and trans
Planckian scattering in general [54, 55]. Present semianalytical techniques (including a trapped surface search in
the union of AichelburgSexl shock waves, closelimit approximation calculations and perturbation theory) only give
rough estimates of the emitted energy and production crosssection [56] and do not provide much insight into the
details of the process (but see [57] for a first numerical investigation).
Our main finding is that, for all sequences we studied, the motion radically changes character when the black holes’
orbital angular momentum L ∼ Lcrit≃ 0.8M2, turning from an eccentric inspiral into a plunge. In particular, for
L ? Lcritwe observe that:
• The number of orbits Nwaves(as estimated using the gravitational wave cycles) or Npunc(as computed from the
punctures’ trajectories) becomes less than one, so the motion effectively turns into a plunge (see Table I and
Fig. 4 below);
• The energy emission starts decreasing exponentially (Fig. 7);
• PNbased eccentricity estimates yield meaningless results (Table I);
• The polarization becomes linear rather than circular (Fig. 6);
• The final angular momentum starts decreasing, rather than increasing, as P and L decrease (Fig. 8).
Binary mergers with L ≃ Lcritare those producing the largest Kerr parameter for the final black hole observed in
our simulations, jfin≃ 0.724. One is led to suspect that for maximally spinning holes having spins aligned with the
orbital angular momentum, a large orbital eccentricity may lead to violations of the cosmic censorship conjecture.
Using arguments based on the extrapolation of pointparticle results (see also [2]), we conjecture that (1) the maximal
Kerr parameter that can be obtained by any merger of nonspinning holes is not much larger than j ≃ 0.724, and (2)
cosmic censorship will not be violated as a result of any merger, even in the presence of orbital eccentricity. Further
numerical simulations are needed to confirm or disprove these conjectures.
The paper is organized as follows. We begin in Sec. II discussing to what extent the Newtonian concept of eccentricity
can be generalized to characterize orbiting binaries in general relativity. For this purpose, we introduce and compare
various PN estimates of the orbital eccentricity, and we show that these eccentricity estimates break down when the
motion turns from inspiral to plunge. Sec. III is a brief introduction to the numerical code used for the simulations.
After a discussion of the choice of initial data and of the code’s accuracy, we show how reducing the orbital angular
momentum affects the gravitational waveforms, the puncture trajectories and the polarization of the waves. In Sec. IV
we study the multipolar energy distribution of the radiation and the angular momentum of the final Kerr black hole.
In Sec. V we show that the salient features of our simulations can be understood using extrapolations of pointparticle
results. Sec. VI is devoted to fits of the ringdown waveform and to estimates of the energy radiated in ringdown
waves. We conclude by considering possible future extensions of our investigation.
II.POSTNEWTONIAN ESTIMATES OF THE ECCENTRICITY
In Newtonian dynamics, the shape of a binary’s orbital configuration is determined by two parameters, the semi
major axis and the eccentricity. These parameters are intimately tied to the binding energy and orbital angular
momentum of the binary and our construction of sequences of binaries with increasing eccentricity is based on this
Newtonian intuition. Specifically, we fix the binding energy of the system, progressively reduce the orbital angular
momentum and thus produce a sequence of orbits of increasing eccentricity. Before doing so, however, we need to
address a conceptual difficulty, namely, how to quantify eccentricity in general relativity.
It turns out, unfortunately, that there exists no unique, unambiguous definition of eccentricity in fully nonlinear
general relativity. For this reason, in the following we will use PN arguments to quantify the initial eccentricity (or
rather, eccentricities) of the simulations. We will consider in detail two different generalizations of the Newtonian
eccentricity: the 3PN extension [58] of a quasiKeplerian parametrization originally proposed by Damour and Deruelle
[59], and a definition in terms of observable quantities recently introduced by Mora and Will [60].
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00.10.20.30.4
L/M2
0.5
0.6
0.70.80.9
0
0.2
0.4
0.6
0.8
1
eccentricity
et (ADMTT)
er (ADMTT)
eφ (ADMTT)
et (harmonic)
er (harmonic)
eφ (harmonic)
0.8 0.82 0.84
0
0.1
0.2
0.3
0.4
0.5
Eb/M=0.014465
0
0.5
1
1.5
2
2.5
3
L/M2
0
0.2
0.4
0.6
0.8
1
eccentricity
et (ADMTT)
er (ADMTT)
eφ (ADMTT)
et (harmonic)
er (ADMTT)
eφ (harmonic)
2.5 2.6 2.7 2.8
0
0.1
0.2
0.3
0.4
0.5
Eb/M=0.001
FIG. 1: The PN eccentricity parameters et, er and eφ for an equal mass binary with binding energy Ebare shown as functions
of the orbital angular momentum L/M2for ADMtype coordinates and harmonic coordinates. The left panel shows the result
for a binding energy corresponding to our sequence 1, the right panel that obtained for a much smaller binding energy.
A.QuasiKeplerian parametrization
A quasiKeplerian parametrization of eccentric orbits of objects with mass M1and M2has been derived at 1PN
order in harmonic coordinates by Damour and Deruelle [59], extended to 2PN order in ADM coordinates by Damour,
Sch¨ afer and Wex [61, 62] and completed to 3PN order by Memmesheimer et al. [58]. This 3PN parametrization gives
the relative separation vector r = (rsin φ,rcos φ,0) of the compact objects and the mean anomaly l as
r = ar(1 − ercosu),
2π(φ − φ0)
Φ
+i6φsin4v + h6φsin5v,
l ≡2π(t − t0)
T
+(f4t+ f6t)sin v + i6tsin2v + h6tsin3v,
(2.1a)
= v + (f4φ+ f6φ)sin 2v + (g4φ+ g6φ)sin 3v
(2.1b)
= u − etsin u + (g4t+ g6t)(v − u)
(2.1c)
where u is the eccentric anomaly, v = 2arctan{[(1 + eφ)/(1 − eφ)]1/2tan(u/2)} and T is the orbital period. The key
element in the parametrization, that makes it useful for comparisons with numerical results, is that the auxiliary
functions ar, er, Φ, f4φ, f6φ, g4φ, g6φ, i6φ, h6φ, n = 2π/T, et, g4t, g6t, f4t, f6t, i6t, h6t and eφ can be expressed
exclusively in terms of the binding energy Eb, the total angular momentum L and the symmetric mass ratio η of the
binary system. The complete expressions in terms of the dimensionless quantities E ≡ Eb/µ and h ≡ L/(µM) are
listed in Eqs. (20) and Eqs. (25) of [58] for ADMtype and harmonic coordinates, respectively. Here M = M1+ M2
and µ = M1M2/(M1+ M2) are the total and reduced mass of the system, respectively.
A comparison with the Newtonian accurate Keplerian parametrization
r = a(1 − ecosu),(2.2a)
φ − φ0 = 2arctan
??1 + e
1 − e
?1/2
tanu
2
?
,(2.2b)
l = u − esinu,(2.2c)
illustrates that the concept of eccentricity is much more complex in general relativity and a single number, such as
the Newtonian eccentricity e, no longer suffices to parametrize the shape of the orbit. Nevertheless, the similarity of
the Newtonian and 3PN expressions suggest that the numbers et, erand eφrepresent some measure of the deviation
of the binary’s orbit from quasicircularity. This becomes particularly clear if we plot these quantities as functions of
the orbital angular momentum L/M2for fixed binding energy Eb/M and mass ratio η.
The result obtained for our sequence 1 models is shown in the left panel of Fig. 1. Several features of this plot
are noteworthy. First, all eccentricity parameters diverge in the limit of a headon collision. This is an artifact of
the appearance of 1/(−2Eh2) terms in the PN expressions for et, erand eφin Eqs. (20) and (25) of Ref. [58]. We
note that the limit L → 0 also plays a special role in the Newtonian case. The usual distinction between the range
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0 ≤ e < 1, corresponding to bound elliptic orbits, and e ≥ 1, corresponding to unbound parabolic or hyperbolic
trajectories, no longer applies in the case of vanishing angular momenta. Since in Newtonian theory
e2= 1 + f(M1,M2)EbL2,(2.3)
where f(M1,M2) is a function of the masses, in the headon limit we would formally have e = 1, irrespective of the
sign of the binding energy. Indeed such trajectories only have one degree of freedom, the energy, and the concept of
eccentricity no longer applies. In this sense, it is not surprising that the PN formalism fails to provide meaningful
values for et, erand eφin the headon limit.
The second observation to be made is the steep gradient of all eccentricity parameters close to the circular limit
of vanishing et, er and eφ. The strong sensitivity of these parameters to the orbital angular momentum L results
in finite values of the three eccentricities (of about 0.1) even when using quasicircular parameters, as obtained from
Eq. (65) of Ref. [16]. Similarly, we observe that et, erand eφdo not vanish for the same values of L (see the inset
in the left panel of Fig. 1). Instead, the values of etand er corresponding to the orbital angular momentum where
eφvanishes are of the order of 0.1. A similar uncertainty results from comparing the PN values obtained in harmonic
and ArnowittDeserMisnerTransverseTraceless (ADMTT) coordinates (cf. the results in the two gauges in the left
panel). We thus take 0.1 as an approximate lower limit for these eccentricity parameters obtainable for such relatively
large binding energies using the 3PN Keplerian parametrization. This is also approximately the value of et, erand
eφobtained for the quasicircular configurations of Table I.
A third noteworthy feature of the “quasiKeplerian” PN parametrization (2.1) is its breakdown for close, near
merger binary orbits. For example, if we tried to compute er and eφ for the “almost circular” parameters we use
in this paper (as listed in Table I) they would turn out to be imaginary when, roughly speaking, P/M ? 0.10 (or
L/M2? 0.83). This is easy to understand by looking at the inset of the left panel of Fig. 1. There we see that these
eccentricities have a zero crossing for values of L/M2which are smaller than those specified in our quasicircular
simulations: in both ADMTT and harmonic coordinates, for the specified value of the binding energy ergoes to zero
when L/M2≃ 0.84, and eφgoes to zero when L/M2≃ 0.83. For L/M2larger than this “critical” value e2
become negative, so that erand eφare imaginary. In the case of sequences 2 and 3, we observe the same behaviour.
This is just a sign that we should not trust the PN approximation for these highly relativistic configurations, so our
eccentricity estimates should be taken with a grain of salt.
An eccentricity plot using the binding energy of sequence 2 or 3 would look almost indistinguishable from the plot
for sequence 1, as shown in the left panel of Fig. 1, so we decided not to display them here. Instead, in the right
panel of Fig. 1 we show the eccentricities computed for a much smaller binding energy, Eb/M = −0.001. This binding
energy corresponds to a binary with much larger separation and smaller orbital velocity, that should be described
with much higher accuracy by the quasiKeplerian PN parametrization. In fact, in the Newtonian limit the three
eccentricities should agree with each other, reducing to the Newtonian definition at large separations. For example,
to leading PN order etand eφare related by
rand e2
φ
eφ= et[1 − (4 − η)(2πM/T)2/3],(2.4)
where T is the orbital period (see eg. [63]). The relation between the different eccentricities at higher PN orders can
be found in Eq. (21) of Ref. [58].
From the right panel of Fig. 1 we see that the three eccentricity parameters do agree much better, as expected, when
the binary members are far apart, and that differences resulting from the use of harmonic or ADMTT coordinates
become negligible. We still see the breakdown of the formalism in the headon limit. However, now all six curves are
much closer to the expected Newtonian behavior, with vanishing eccentricity in the circular limit (where L approaches
the maximum allowed value) and e ≈ 1 for smaller angular momenta. Unfortunately, it is currently prohibitively costly
from a computational point of view to start numerical simulations from such low binding energies. For this reason,
in this paper we focus on the merger and ringdown signals resulting from eccentric binaries, rather than on detailed
comparisons with PN predictions for the emission of GWs during the inspiral.
B. MoraWill parametrization
An alternative estimate of the binary’s initial eccentricity can be obtained using the PN diagnostic formalism
developed by Mora and Will ([60], henceforth MW). Instead of imposing a quasiKeplerian parametrization with
different eccentricities for t, r and φ, MW define a single eccentricity parameter eMWand a PN expansion parameter
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ζ as follows:
eMW ≡
?Ωp−√Ωa
??MΩp+√MΩa
?Ωp+√Ωa
,
ζ ≡
2
?4/3
.(2.5)
Here Ωp and Ωa are the orbital angular frequencies when the binary passes through a local maximum (pericenter)
and through the next local minimum (apocenter), respectively.
The MW eccentricity parameter eMW has several advantages: it is defined in terms of observable quantities, it
reduces to its Newtonian counterpart for small orbital frequencies, and it is gauge invariant through first PN order.
The binding energy and angular momentum of the system can be expressed as functions of eMWand ζ. The relevant
equations for black hole binaries are given by Eqs. (2) and (3) in [64]. In Ref. [65] these same PN equations have been
used to study truly eccentric black hole binary initial data, and to point out some interesting features of the resulting
effective potentials.
The PN expansion parameter ζ is related to the frequencies at periastron and apoastron by
ζ =
(MΩa)2/3
(1 − eMW)4/3=
(MΩp)2/3
(1 + eMW)4/3. (2.6)
To estimate the eccentricity, we can assume that the binary’s orbit is (say) at apoastron, so that Ω = Ωa. Then
the binding energy and angular momentum can be expressed as functions of eMW and MΩ. We can equate these
functions to the binding energies and angular momenta listed in Table I:
EMW
b
LMW(eMW,MΩ) = L.
(eMW,MΩ) = Eb, (2.7a)
(2.7b)
If our assumption is correct, and the orbit is indeed at apoastron, this system will have a solution (eMW,MΩ) with
eccentricity eMW> 0. It should be obvious from Eq. (2.6) that a solution with eMW< 0 would simply correspond to
the binary being at periastron.
In Ref. [64] it was shown that, when comparing numerically computed quasiequilibrium binary black hole initial
data against the MW PN diagnostic predictions for a given orbital frequency MΩ, energies are usually in better
agreement than angular momenta. Unfortunately, our initial data sets specify the orbital angular momentum and
the binding energy, but not the orbital frequency. Therefore, to estimate the binary’s eccentricity we adopt two
different procedures. The first procedure is based on numerically solving the system (2.7) for the two unknowns
(eMW,MΩ), which are both considered as free parameters. This procedure is quite general. However it can involve a
significant amount of systematic error, since the angular momentum can deviate by a large amount from numerical
initial data even for small eccentricities and/or large separations, and we are trying to solve for both energy and
angular momentum simultaneously.
The second procedure is based on an assumption of quasicircularity, and it consists of two steps. We first assume
that eMW= 0 and solve Eq. (2.7a) for MΩ. For sequence 1, the orbital frequency obtained by imposing EMW
0,MΩ) = Ebis MΩ = 0.0513; for sequence 2, we find MΩ = 0.0431; for sequence 3, we find MΩ = 0.0212. Then
we substitute this value of MΩ in Eq. (2.7b) and solve for the eccentricity. We call this solution eMW,circ, because
it is obtained assuming that the energy function is very close to the circular prediction, and that deviations of the
eccentricity from zero only affect the angular momentum.
Table I lists the eccentricities etobtained using the definition by Memmesheimer et al. and the two eccentricities
estimated from the MW diagnostic. Dashed entries mean that no solutions to the system (2.7) exist for physically
relevant values of the parameters. The MW eccentricity estimate eMWis consistent with the quasiKeplerian parameter
et: in fact, the two definitions are roughly in agreement, within factors of order unity. It is also encouraging that
the minima of etand eMWcorrespond to the same simulations along each sequence: these variables seem to provide
a reasonably accurate measure of deviations from circularity, even for binaries which are very close to merger. The
second procedure (by construction) should become inconsistent for large eccentricities, but eMW,circis much closer
to the eccentricity estimates (∼ 0.01) obtained in [64], and measured in longer quasicircular inspiral simulations by
different groups [23, 66, 67, 68].
In the following we will use (somewhat arbitrarily) the parameter etobtained in harmonic coordinates as a measure
of deviations from circularity of the initial binary configuration. We should still bear in mind that this parameter
deviates significantly from the eccentricity e of a Keplerian ellipse, in particular for small angular momenta.
b
(eMW=
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III.NUMERICAL SIMULATIONS
The simulations presented in this work have been performed with the Lean code [14], which is based on the Cactus
computational toolkit [69]. The code employs the socalled movingpuncture method, i.e. it evolves initial data of
puncture type [70] using the BaumgarteShapiroShibataNakamura (BSSN) formulation of the Einstein equations
and gauge conditions which allow the black holes to move across the computational domain. In contrast to the
original version of the BSSN system, where the conformal factor ψ is evolved in the form of the logarithmic variable
φ = lnψ, we here evolve the variable χ = e−4φ, as originally introduced in Ref. [9]. These equations are numerically
approximated using fourthorder spatial discretization for sequences 1 and 2, and sixthorder spatial discretization
[71] for sequence 3. Integration in time is performed with the method of lines using the fourth order accurate Runge
Kutta (RK) scheme. Using the notation of Table 1 in Ref. [14], this corresponds to schemes labelled RKχ4and RKχ6,
respectively. We note, however, that neither version of the code is genuinely fourth or sixth order accurate, as second
order ingredients are used at the refinement and outer boundaries (see [14] for details).
The Lean code facilitates use of meshrefinement via the Carpet package [72, 73]. Specifically, the computational
grid is represented by a nested set of Cartesian boxes with resolution successively increasing by factors of two. The
innermost refinement levels are typically split into two components centered around either hole and following the
black hole motion. Once the black hole separation decreases below a threshold value, the two components merge into
one centered around the origin. A more detailed description of the Lean code is given in Ref. [14].
GWs are extracted using the NewmanPenrose formalism. The NewmanPenrose scalar Ψ4is calculated using the
electromagnetic decomposition of the Weyl tensor, and decomposed into contributions of different multipoles using
spherical harmonics−2Yℓmof spinweight −2 according to
?
For all simulations presented in this work we extract GWs at different extraction radii rex. We use the results obtained
at different radii to estimate the uncertainty arising from the use of a finite extraction radius.
The calculation of apparent horizons is performed using Thornburg’s AHFinderDirect [74, 75] and plays an
important role in the construction of initial data sequences.
We use initial data of puncture type as provided by Ansorg’s TwoPuncture thorn [76] in all simulations. For
zero spins, an initial data set is uniquely determined by the bare mass parameters m1 and m2 of the holes, their
initial coordinate separation D and the BowenYork [77] parameters P1and P2for the black holes’ linear momenta.
Without loss of generality we always set P ≡ P1= −P2. The initial orbital angular momentum is then given by
L = D × P. Because the black holes are initially located on the xaxis and orbit in the xyplane, the initial angular
momentum is given by its zcomponent, which we can write as L = Lz= DP (where P = Py).
We conclude this brief description of the numerical code with a summary of the variables used in the remainder
of this work. We denote by M1and M2the initial black hole masses. The total mass is the sum of the individual
masses, M ≡ M1+M2, and the reduced mass is µ ≡ M1M2/M. We measure the mass ratio using either q = M1/M2
or the “symmetric mass ratio” parameter η = µ/M often used in PN studies. Finally, we denote the ArnowittDeser
Misner (ADM) mass by MADM, which is close to (but in general different from) the total black hole mass M. The
normalization of dimensional quantities is performed as follows. Initial parameters are normalized using the total
black hole mass M for compatibility with the PN relations. In contrast, we measure radiated energy and momenta
(as well as time t and radii r) in units of the ADM mass, as is common in numerical studies. All normalizations will
be clear from the use of M or MADMin column or axis labels; the relation between the two normalizations can be
worked out from Eq. (3.2) below and from the binding energies of the two sequences we consider, which are given in
the caption of Table I.
Ψ4=
ℓ,m
−2Yℓmψℓm. (3.1)
A.Sequence of initial data
The construction of a sequence of nonspinning, equalmass binary initial configurations with constant binding
energy is a twostep process. First, we determine a quasicircular configuration using the 3PN accurate relation (65)
of Ref. [16], which provides the initial linear momentum P/M for given black hole masses M1and M2and separation
D/M. Without loss of generality we fix the scaling freedom of the numerical coordinates by setting M1= M2= 0.5.
For fixed values of P and D, there thus remains the task of finding bare mass parameters m1 = m2 which result
in blackhole masses of M1= M2= 0.5. This is done iteratively by using the NewtonRaphson method with initial
guesses mi= Mi. In the absence of spins, the blackhole masses are given by the irreducible masses, as calculated
by the apparent horizon finder AHFinderDirect. For the quasicircular model we calculate the binding energy
Page 8
8
TABLE I: Sequences of models studied in this work. The irreducible mass of the individual black holes is M1 = M2 = 0.5
in all simulations, and consequently µ = 1/4. The binding energy for sequence 1 (top 16 models in the table) is E(1)
−0.014465, that for sequence 2 (following 8 models) is E(2)
b/M = −0.013229 and that for sequence 3 (following 9 models) is
E(3)
b
= −0.008861. The bottom line lists parameters for a model which serves for estimating the uncertainties of the long
simulations of sequence 3 (see Sec. IIIB) and has merely been included for completeness. ∆tCAH is the time of formation of a
common apparent horizon as measured from the radiation peak, shifted by the extraction radius: ∆tCAH = tCAH+rex−tpeak.
For sequence 3, we have not calculated apparent horizons and we estimate this quantity to be ∆tCAH = −16 MADM. Nwaves
and Npunc are the number of orbits obtained from the phase of the (ℓ = 2 ,m = 2) multipole and the puncture’s orbital motion,
respectively. The last four columns list various PN estimates of the eccentricity (see Sec. II for a discussion).
b/M =
P
M
D
M
L
M2
m1,2
102Erad
MADM
3.437
3.451
3.478
3.494
3.530
3.486
3.261
2.839
2.259
1.631
1.061
0.621
0.328
0.161
0.080
0.057
3.585
3.581
3.659
3.602
2.181
0.795
0.185
0.058
3.914
3.907
3.912
3.996
4.186
3.373
1.295
0.419
Jrad
M2
ADM
0.2257 0.6915 15.95
0.2267 0.6910 15.52
0.2272 0.6916
0.2249 0.6932 14.87
0.2196 0.6953 15.74
0.2072 0.6994 15.29
0.1857 0.7043 14.65
0.1570 0.7051 15.30
0.1240 0.6964 16.59
0.0909 0.6718 16.53
0.0613 0.6257 17.01
0.0379 0.5530 17.67
0.0213 0.4510 18.24
0.0106 0.3206 19.11
0.0042 0.1669 20.02
00
0.2540 0.6912 12.74
0.2558 0.6897 15.19
0.2532 0.6875 15.15
0.2066 0.7053
0.1186 0.6974 16.14
0.0471 0.5902
0.0122 0.3378 18.87
00
0.3956 0.6963
0.3963 0.6960
0.3854 0.6971
0.3658 0.6898
0.3128 0.6914
0.1778 0.7236
0.0721 0.6602
0.0263 0.4992
00
0.3037 0.6304
Jfin
M2
fin
∆tCAH
MADMNwaves Npunc et(ADMTT) et(harm)
1.321.51
1.32 1.52
151.331.53
1.271.47
1.161.37
0.991.21
0.821.04
0.680.88
0.560.72
0.430.59
0.37 0.47
0.31 0.36
0.280.27
0.24 0.17
0.18 0.09
19.80
2.062.30
2.052.24
1.791.99
161.001.20
0.560.74
170.270.46
0.270.44
20.16
−169.39 9.64
−169.429.65
−168.568.79
−166.386.60
−163.243.44
−160.831.06
−16 0.45 0.65
−16 0.390.42
−16
−168.10 8.41
eMW(MΩ)eMW,circ
0.14
0.1383 6.131 0.8479 0.4775
0.136.528 0.8486 0.4790
0.12 7.064 0.8477 0.4808
0.117.672 0.8439 0.4825
0.10 8.361 0.8361 0.4842
0.099.140 0.8226 0.4857
0.0810.013 0.8010 0.4872
0.0710.978 0.7685 0.4885
0.0612.024 0.7215 0.4898
0.0513.121 0.6561 0.4908
0.0414.217 0.5687 0.4917
0.0315.234 0.4570 0.4924
0.0216.073 0.3214 0.4930
0.0116.630 0.1663 0.4933
016.826
0.1247 7.000 0.8729 0.4798
0.12 7.278 0.8734 0.4807
0.10 8.678 0.8678 0.4841
0.0810.493 0.8394 0.4871
0.0612.754 0.7653 0.4897
0.0415.288 0.6115 0.4917
0.0217.488 0.3498 0.4930
0 18.398
0.0850 12.000 1.0201 0.4883
0.0812.758 1.0206 0.4890
0.0714.441 1.0290 0.4905
0.0616.398 1.0015 0.4917
0.0518.619 0.9476 0.4929
0.0421.028 0.8562 0.4938
0.0323.461 0.7165 0.4946
0.0225.628 0.5218 0.4951
0.00 27.710
0.0808 10.905 0.8807 0.9834,
6.054 0.8476 0.47720.0983
0.0956
0.0878
0.0977
0.1306
0.1800
0.2424
0.3161
0.3995
0.4913
0.5887
0.6894
0.8050
1.1005
3.8101
∞
0.1095
0.1049
0.1480
0.2758
0.4567
0.6681
1.0122
∞
0.0869
0.0819
0.1542
0.2648
0.3968
0.5425
0.6921
0.8434
∞
0.1022
0.0987
0.0961
0.0886
0.0982
0.1300
0.1783
0.2393
0.3112
0.3921
0.4799
0.5699
0.6540
0.7202
0.8090
2.5892
∞
0.1096
0.1052
0.1472
0.2725
0.4485
0.6428
0.8078
∞
0.0871
0.0821
0.1536
0.2632
0.3937
0.5368
0.6799
0.8048
∞
0.1022
0.1496 (0.0388) 0.0198
0.1458 (0.0391) 0.0188
0.1353 (0.0398) 0.0162
0.1488 (0.0389) 0.0196
0.1948 (0.0359) 0.0336
0.2674 (0.0319) 0.0634










0.1529 (0.0323) 0.0202
0.1468 (0.0326) 0.0186
0.2049 (0.0295) 0.0363





0.1032 (0.0174) 0.0086
0.0972 (0.0176) 0.0077
0.1829 (0.0150) 0.0272
0.3151 (0.0119) 0.0815
0.4765 (0.0090) 0.1899





0.1194
0.2370








0 0.4934
0.1320




0 0.4934
0.3960




00.4956 0.0585
3.084
0.4822
according to
Eb= MADM− M.(3.2)
The second step consists in the construction of additional configurations with the same binding energy but different
linear momentum parameter P and, thus, different orbital angular momentum L = DP. For this purpose we fix P
and demand, as before, M1= M2= 0.5. We then iteratively solve for the numerical parameters m1= m2 and D
which yield the required black hole masses and binding energy.
Three sequences obtained this way with Eb/M = −0.014465, −0.013229 and −0.008861 are listed in Table I. In
the following we will refer to them as “sequence 1”, “sequence 2” and “sequence 3”, respectively. The corresponding
quasicircular configurations are those with linear momentum P/M = 0.1383, 0.1247 and 0.0850.
It is worth mentioning, in this context, that our choice of fixing the binding energy is not unique. Alternative choices
in constructing sequences include keeping constant the coordinate separations of the punctures, or the proper horizon
tohorizon distance. We have opted against these two options because they would result in very short merger times
Page 9
9
for small angular momenta. At constant binding energy, instead, smaller values of the orbital angular momentum
imply larger separations of the holes and, thus, a delay in the formation of the common apparent horizon.
B. Accuracy of the simulations
In order to estimate the uncertainties associated with the numerically calculated quantities of sequences 1 and 2, we
have performed a convergence analysis for the quasicircular configuration with P/M = 0.1247 of sequence 2, and for
the eccentric configuration with P/M = 0.08 of sequence 1. We have evolved these systems using the following grid
setup: {(192,96,56,24,12)× (3,1.5,0.75),hi}. By this notation we mean that there is a total of 8 refinement levels.
The 5 outer levels are centered on the origin and extend out to x = y = z = ±192, 96, 56, 24 and 12 respectively. The
3 inner levels have 2 components each, centered on either hole with radius 3, 1.5 and 0.75. Finally, the grid spacing
is hion the finest level (where h1= 1/48, h2= 1/44 and h3= 1/40) and increases by a factor of 2 consecutively on
each level.
Fourthorder convergence is shown in Fig. 2 for the (ℓ = 2, m = 2) multipole of the NewmanPenrose scalar Ψ4,
the total radiated energy E and the radiated angular momentum in the zdirection Jrad.
Using the fourth order convergence, we apply a Richardson extrapolation to the total radiated energy and obtain
E/MADM= 0.03686, 0.03668 and 0.03656 respectively at extraction radii rex/MADM= 50.7, 60.8 and 70.9. These
values correspond very well to a 1/rexfalloff of the uncertainty arising from the use of finite extraction radii. The total
radiated energy extrapolated to rex→ ∞ is E/MADM= 0.03583. For the medium resolution case (h = h2= 1/44)
and using an extraction radius rex/MADM= 70.9, this analysis predicts an uncertainty ∼ 2% due to the discretization
and ∼ 2.5% due to the use of finite extraction radius. The uncertainties in the radiated angular momentum Jradare
∼ 2% for both error sources. The convergence study of the eccentric simulation with P/M = 0.08 of sequence 1 yields
similar error estimates. We estimate the resulting total error from quadratic error propagation to be about 3%. In
fact, this estimate is likely to be very conservative because the two error sources have opposite signs: finite resolution
tends to underestimate radiated energy and momenta, while finite extraction radius usually leads to an overestimate.
Performing a convergence analysis of simulations lasting as long as those of sequence 3 requires vast computational
resources. In order to reduce the cost, we view these simulations as part of a wider parameter study, to be presented
elsewhere, which also involves unequalmass binaries. In order to assess the accuracy of those long simulations,
we have focussed on a quasicircular configuration with q = 2 which is listed as the last entry in Table I. From
experience, we consider unequalmass binaries significantly more challenging numerically than systems with equal
mass, and therefore feel justified in using the uncertainties resulting from this model as conservative error estimates
of our sequence 3 runs. We have evolved this configuration using a grid {(384,192,128,48,24)× (6,3,1.5,0.75),hi}
with hi= 1/44, 1/48 and 1/56. The bottom panels of Fig. 2 show the convergence of the phase and amplitude of Ψ4
(bottom right panel) and the radiated energy and angular momentum (bottom left panel). The difference between
the higher resolution runs has been rescaled for second order convergence. We observe second order convergence for
these runs except for the late stages near merger, when the convergence increases to third order. Similar glitches in
the convergence near merger have been observed in [22]. Using the same technique as above, we obtain uncertainties
of about 3 % for the radiated energy and 6 % for the radiated angular momentum for the simulations using h2= 1/48
and rex= 80.7 MADM.
Because most simulations have been performed at the medium resolution only, we cannot in general apply Richard
son extrapolation. Unless specified otherwise we therefore present results as obtained numerically using h = 1/44 and
rex/MADM= 70.9 for sequences 1 and 2, as well as h = 1/48 and rex= 120 M for sequence 3, bearing in mind the
uncertainties we have just mentioned.
We conclude this discussion by mentioning that the Lean code has been demonstrated to accurately evolve black
holes with large spins in Ref. [78]. Specifically, methods to calculate the spin from quasinormal ringing, balance
arguments and apparent horizon data were shown to result in excellent agreement for Kerr parameters above 0.9.
C.Numerical waveforms
In Ref. [24] we studied the multipolar energy distribution of unequalmass black hole binaries in quasicircular orbits.
By projecting 2.5PN calculations of the inspiral gravitational waveforms onto spinweighted spherical harmonics
−2Yℓm, we concluded that oddm multipoles of the radiation are suppressed for equalmass binaries. Lowℓ multipoles
carry most of the radiation, and within a given lmultiplet, modes with ℓ = m are typically dominant for quasicircular
motion. This analytical prediction was shown to agree very well with numerical simulations, and it has recently been
confirmed by more accurate PN calculations [79]. Since in this paper we study equalmass runs, we expect evenm,
Page 10
10
0 100 200300 400
t/MADM
0.06
0.04
0.02
0
0.02
0.04
∆(MADMrψ22)
MADMr(ψ22,h3 ψ22,h2)
Q4 MADMr(ψ22,h2 ψ22,h1)
0.000
0.015
0.001
0.002
∆Erad/MADM
(Erad, h3Erad, h2)/MADM
Q4 (Erad, h2 Erad, h1)/MADM
0 100 200300 400
t/MADM
0
0.005
0.01
∆Jrad/MADM
2
(Jrad, h3 Jrad, h2)/MADM
Q4 (Jrad, h2 Jrad, h1)/MADM
2
2
0100200300
t/MADM
0.002
0.001
0
0.001
0.002
∆(MADMrψ22)
MADMr(ψ22,h3 ψ22,h2)
Q4 MADMr(ψ22,h2 ψ22,h1)
3×104
2×104
1×104
0
∆Erad/MADM
(Erad, h3Erad, h2)/MADM
Q4 (Erad, h2 Erad, h1)/MADM
0 100200300
t/MADM
0.002
0.001
0
∆Jrad/MADM
2
(Jrad, h3 Jrad, h2)/MADM
Q4 (Jrad, h2 Jrad, h1)/MADM
2
2
0.01
0.1
1
∆φ
(φh3φh2)
Q2(φh2φh1)
500
1000
1500
2000
t / MADM
1e06
1e05
0.0001
0.001
0.01
∆MADMrA
MADMr(Ah3Ah2)
Q2MADMr(Ah2Ah1)
0.004
0.003
0.002
0.001
0
∆E / MADM
(Eh3Eh2) / MADM
Q2(Eh2Eh1) / MADM
0
500
1000
t / MADM
1500
2000
0.05
0.04
0.03
0.02
0.01
0
∆Jrad / MADM
2
(Jrad, h3Jrad, h2) / MADM
Q2(Jrad, h2Jrad, h1) / MADM
2
2
FIG. 2: Upper panels: convergence analysis of the P/M = 0.1247, quasicircular model of sequence 2, using resolutions
h = 1/48, 1/44 and 1/40. The left panel shows the differences in the (ℓ = 2, m = 2) multipole of MADMrψ22, the right
panel the total radiated energy and zcomponent of the angular momentum. In both cases, the differences between the higher
resolution simulations have been rescaled by Q4 for the expected fourth order convergence. Middle panels: same, but for the
P/M = 0.08, eccentric model of sequence 1. Lower panels: same, but for the unequal mass model listed at the bottom of Table
I. For clarity, we present the convergence of Ψ4 using phase and amplitude instead of the real part. For this configuration,
we observe second order convergence. The corresponding convergence factors for the resolutions used here, are Q4 = 1.58 and
Q2 = 0.72.
lowℓ multipoles to be dominant, at least when the eccentricity is small enough. This expectation is again confirmed
by our numerical timeevolutions.
To be more quantitative, in Fig. 3 we show the modulus of the real part of the (ℓ = 2, m = 2), (ℓ = 2, m = 0)
and (ℓ = 4, m = 4) components of the Weyl scalar for some representative runs. From these plots it is clear that the
(ℓ = 2, m = 0) component contributes significantly for P/M ? 0.05. That the (ℓ = 2, m = 0) component should be
significantly excited in the headon limit P/M → 0 is known from previous numerical simulations: in fact, the m = 0
mode would by far be dominant if we had chosen the collision to happen along the zaxis (see eg. [80, 81]). Notice
Page 11
11
0
50
100
150
200
250
300
350
400
t/MADM
106
105
104
103
102
101
Re(MADMr ψlm)
l=2, m=2
l=2, m=0
l=4, m=4
P/M=0.00
0
50
100
150
t/MADM
200
250
300
350
106
105
104
103
102
101
Re(MADMr ψlm)
l=2, m=2
l=2, m=0
l=4, m=4
P/M=0.02
0 100200300400
t/MADM
106
105
104
103
102
101
Re(MADMr ψlm)
l=2, m=2
l=2, m=0
l=4, m=4
P/M=0.04
0
50
100
150
200
t/MADM
250
300
350
400
450
106
105
104
103
102
101
Re(MADMr ψlm)
l=2, m=2
l=2, m=0
l=4, m=4
Quasicircular
FIG. 3: Modulus of the real part of the (ℓ = 2, m = 2), (ℓ = 2, m = 0) and (ℓ = 4, m = 4) components of the waveforms.
We show waveforms from four representative simulations: sequence 2 runs with P/M = 0 (headon limit), P/M = 0.02,
P/M = 0.04 and P/M = 0.1247 (quasicircular case).
that in this paper GWs have always been extracted assuming that the zaxis is perpendicular to the orbital plane. In
the case of headon collisions this is contrary to most previous studies, where to take full advantage of the symmetry
of the problem the axis of reference for the angular coordinates is identified with the axis of collision. In consequence,
we find the radiated energy of headon collisions to be quadrupole dominated, but to contain m = ±2 and m = 0
contributions of comparable magnitude rather than almost exclusively an ℓ = 2, m = 0 contribution as is the case
for alignment of the two axes. Our choice is entirely motivated by using identical angular coordinates throughout the
sequence of models. A detailed analysis of the transformation properties of multipolar components of the radiation
under rotations, translations and boosts can be found in Ref. [82].
As P/M increases and we approach quasicircular motion, the lowamplitude portion of the (ℓ = 2, m = 0) mode
decreases in amplitude, and it is significantly contaminated by noise. As expected from our previous study [24], in
the same limit the amplitude of the (ℓ = 4, m = 4) mode grows. Unfortunately, Fig. 3 illustrates that even for
(ℓ = 4, m = 4), which is the largest of the subdominant radiation modes, the ringdown signal is strongly distorted by
either nonlinear effects or boundary reflection noise. For this reason it is difficult to use higher multipoles to improve
spin estimates from quasinormal mode (QNM) fittings, as proposed in [24]. This problem will be discussed in more
detail in Sec. VI below.
We can obtain an estimate of the number of orbits in our simulations from the puncture trajectories, calculated
according to dxi/dt = −βi. For illustration, in Fig. 4 we plot the trajectories of four models of sequence 2 with
P/M = 0.02, 0.06, 0.10 and 0.1247. The figure demonstrates the inspiralling nature of the simulations with large
initial momentum, whereas those with small momentum rather represent plunging configurations.
We define a phase φpuncof these trajectories by expressing the puncture’s position in spherical polar coordinates
(x,y) = (rpunccosφpunc, rpuncsinφpunc). (3.3)
Then we consider the phase difference ∆φpunc = φpunc(tcah) − φpunc(t0), where tcah is the time of formation of a
common apparent horizon and we choose t0 = 50MADM to cut off the initial data burst (this is consistent with
the derivation of the frequency from the GW signal, discussed below). The number of orbits then follows from
Page 12
12
8
6
42024
x/MADM
4
2
0
2
4
y/MADM
P/MADM=0.02
P/MADM=0.06
P/MADM=0.10
P/MADM=0.1247
FIG. 4: Trajectories of the models with P/M = 0.02, 0.06, 0.10 and 0.1247 of sequence 2. The trajectory of one hole only is
shown in each case. The positions of the respective second holes follow from symmetry across the origin. The + denote the
locations of the individual holes at the time of common apparent horizon formation.
Npunc= ∆φpunc/(2π).
The gravitational wave signal Ψ4 also serves to estimate the number of orbits completed by the binary prior to
merger. For this purpose we focus on the (ℓ = 2, m = 2) multipole and decompose the mode coefficient into a
timedependent amplitude A(t) and phase φ(t) according to
ψ22(t) = A(t)eiφ(t).(3.4)
We next calculate the phase difference
∆φ = φ(tcah+ rex) − φ(t0+ rex),(3.5)
where rextakes (approximately) into account the time it takes for the waves to propagate to the extraction radius.
The number of orbits completed by the binary is then estimated as Nwaves= ∆φ/(4π), where the additional factor of
2 compensates for the difference between the orbital frequency and that of a multipole with m = 2. Both estimates
of the number of orbits, together with the time of formation of the common apparent horizon, are given in Table I.
Small differences in these numbers are due to the fact that the approximate relation ωwaves= mωpuncbreaks down
near the merger of the holes.
This is illustrated in Fig. 5. There we plot the frequency ω22and the phase φ22obtained from ψ22, comparing with
the frequency mωpuncand phase φpuncobtained from the puncture trajectory (see [24] for details). Estimates from
the punctures’ motion are physically irrelevant after formation of the apparent horizon. The upper panels refer to
different models from sequence 2: a nearly plunging motion with P/M = 0.06 (left) and a quasicircular orbit with
P/M = 0.1247 (right). It is clear from the figure that deviations between ωwavesand mωpuncgrow significantly near
merger. The same holds for the two models of sequence 3 shown in the bottom panels.
From Fig. 4, and from the data for Npuncand Nwaveslisted in Table I, we deduce that the simulations with L ?
Lcrit≃ 0.08M2complete significantly less than one cycle, so they are effectively plunging trajectories. Consistently
with this interpretation, we have see in Sec. II below that PN estimates of the eccentricity of the orbit fail for these
plunging configurations.
D.Polarization
In Appendix D of [24] we proposed to measure the polarization of a waveform using an “elliptical component of
polarization” PE. This quantity has the property that PE = 1 for circular polarization, and PE = 0 for linear
polarization. By looking at the dominant (ℓ = 2, m = 2) component of the radiation we also showed that, with the
exception of the (unphysical) initial data burst and of the final part of the signal, which is dominated by noise, the
polarization of a binary moving in a quasicircular orbit is circular to a very good approximation.
In Fig. 6, to be compared with Fig. 28 of Ref. [24], we show the real and imaginary parts of the dominant
(ℓ = 2, m = 2) multipolar component of the radiation emitted by equalmass binaries with different values of P/M as
Page 13
13
0.2
60
0
0.2
0.4
0.6
0.8
1
MADMω
ω22(trex)
2ωpunc(t )
0
50
100
150
200
t/MADM
0
20
40
φ
φ22(trex)
2φpunc(t )+0.7
P/M=0.06
0.2
100
0
0.2
0.4
0.6
0.8
1
MADMω
ω22(trex)
2ωpunc(t )
0
50
100
150
200
250
300
t/MADM
0
20
40
60
80
φ
φ22(trex)
2φpunc(t )+3.9
P/M=0.1247
0.2
0
0.2
0.4
0.6
0.8
1
MADMω
ω22(trex)
2ωpunc(t )
0 200 400
600
800 1000 1200
t/MADM
0
50
100
150
φ
φ22(trex)
2φpunc(t )+2.6
P/M=0.06
0.2
150
0
0.2
0.4
0.6
0.8
1
MADMω
ω22(trex)
2ωpunc(t )
0
500
1000
1500
2000
t/MADM
0
50
100
φ
φ22(trex)
2φpunc(t )+8.8
P/M=0.0850
FIG. 5: Frequency and phase obtained from the ℓ = 2, m = 2 multipole of the gravitational radiation as well as the puncture
trajectory for models P/M = 0.06 and 0.1247 of sequence 2 (upper panels) and P/M = 0.06 and 0.0850 of sequence 3 (lower
panels). The dotted vertical lines mark the formation of the apparent horizon.
obtained from our sequence 2. In the bottom panel of each plot we compute the elliptical component of polarization.
The results clearly illustrate that the polarization is linear in the headon limit, where the imaginary component of
the radiation vanishes, and that PE→ 1 as the orbit becomes circular. It is also worth noticing that the ringdown
part of the signal is circularly polarized even when PE is slightly less than one in the inspiral part (see eg. the plot
for P/M = 0.02).
IV.RADIATED ENERGY AND FINAL ANGULAR MOMENTUM
In this section we study the multipolar energy distribution and the final angular momentum for the three sequences
of simulations listed in Table I.
The multipolar distribution of radiated energy for sequence 1 is shown in the upper left panel of Fig. 7. We
have excluded radiation due to the initial burst by ignoring contributions at simulation times t < 50 M + rex. The
results demonstrate a relatively weak dependence of the radiated energy in each multipole on initial linear momenta
P/M ? 0.08, corresponding to an angular momentum of L/M2∼ 0.8.
For smaller linear (or angular) momenta, we observe that: (1) the total radiated energy decreases almost expo
nentially, (2) the relative contribution of multipoles with ℓ > 2 becomes weaker, and (3) the contribution of the
(ℓ = 2, m = 0) mode increases to approximately the same level as the (ℓ = 2, m = ±2) modes. All of these features
are quite insensitive to the inclusion of the spurious initial wave burst: only the (ℓ = 2, m = 0) mode is significantly
contaminated by the initial radiation when P/M ? 0.08. The same observations also apply to the models of se
quence 2, shown in the right panel of Fig. 7. Here the transition seems to occur at a linear momentum slightly below
P/M = 0.08. From Table I we see that this again corresponds to an angular momentum of L/M2≈ 0.8. Similarly,
the transition occurs just below P = 0.04 in the case of sequence 3, which again corresponds to an orbital angular
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14
106
105
104
103
102
MADMr ψ22
Re(MADMr ψ22)
Im(MADMr ψ22)
0
50
100
150
200
250
300
350
400
t/MADM
0
0.2
0.4
0.6
0.8
1
PE
P/M=0.00
106
105
104
103
102
101
MADMr ψ22
Re(MADMr ψ22)
Im(MADMr ψ22)
0
50
100
150
t/MADM
200
250
300
350
0
0.2
0.4
0.6
0.8
1
PE
P/M=0.02
106
105
104
103
102
101
MADMr ψ22
Re(MADMr ψ22)
Im(MADMr ψ22)
0
50
100
150
200
250
300
350
400
t/MADM
0
0.2
0.4
0.6
0.8
1
PE
P/M=0.04
106
105
104
103
102
101
MADMr ψ22
Re(MADMr ψ22)
Im(MADMr ψ22)
0
50
100
150
200
250
300
350
400
450500
t/MADM
0
0.2
0.4
0.6
0.8
1
PE
P/M=0.06
FIG. 6: ℓ = m = 2 components of the lowL (or lowP) waveforms and their polarization for various models of sequence 2.
For L = 0 the imaginary part of the waveform is zero within the noise level (i.e., the cross component is zero for symmetry
reasons). As L increases, the polarization becomes circular. Spikes in PE at early times are due to the inital data burst, and
spikes at late times are due to boundary reflection noise and low strength of the signal.
momentum L/M2≈ 0.8. A fit of sequence 1 runs by a polynomial in L/M2yields
Erad
MADM
= 0.0212
?L
M2
?
− 0.1020
?L
M2
?2
+ 0.1478
?L
M2
?3
.(4.1)
A similar behavior is found for the angular momentum of the final black hole, shown in Fig. 8. We have calculated
the final spin from balance arguments: the final black hole mass Mfinis obtained by subtracting the total radiated
energy from the initial ADM mass, and the final black hole angular momentum Jfinis similarly given by the initial
orbital angular momentum minus the momentum radiated in GWs. The figure shows the dimensionless Kerr parameter
jfin= Jfin/M2
has a local maximum, and then it decreases rapidly for L/M2? 0.8. The results for the three sequences show
remarkably good agreement below L/M2? 0.8 and merely differ at large angular momenta, as initial configurations
with such large L only exist when choosing sufficiently large separations. By experimenting with sequence 1 runs, we
found the following, reasonably accurate threeparameter polynomial fit:
fin. Again, small increases in eccentricity only lead to a mild increase in the final spin. The Kerr parameter
Jrad
M2
ADM
= 0.0225
?L
M2
?
+ 0.0381
?L
M2
?2
+ 0.5589
?L
M2
?7
.(4.2)
The large exponent in the final term is an artifact of the phenomenological nature of the fit and yields optimal
agreement with the data. We found that the following alternative fitting functions, with two unknowns, also perform
very well, yielding Erad/MADM= 3.18 × 10−4× e5.54L/Mand Jrad/M2
An understanding of the apparent threshold in the orbital angular momentum L/M2separating inspiraling and
plunging orbits can be obtained by considering geodesic orbits in a Schwarzschild background. The key observation
here is that particles of mass mpin closed orbits must satisfy the condition L/mp> 2√3M. A rough extrapolation of
this result to comparablemass binaries would yield L/(ηM) > 2√3M, or L/M2> 0.866. There is no firm theoretical
ADM= 7.34 × 10−4e6.74L/M.
Page 15
15
00.020.04
0.06
0.08 0.10.12 0.14
P/M
107
106
105
104
103
102
101
energy distribution
0.810.650.480.310.180.100.10
et
Etot
E2
E3
E4
E5
E6
E22
E20
no initial data burst
Eb/M=0.014465
00.020.04
0.06
P/M
0.080.10.12
107
106
105
104
103
102
101
energy distribution
0.810.640.450.270.150.10
et
Etot
E2
E3
E4
E5
E6
E22
E20
no initial data burst
Eb/M=0.013229
00.020.04
P/M
0.06
0.08
107
106
105
104
103
102
101
energy distribution
0.800.680.54
0.39
0.26 0.150.08
et
Etot
E2
E3
E4
E5
E22
E20
no initial data burst
Eb/M=0.008861
FIG. 7: Multipolar energy distribution as a function of the initial momentum P for sequence 1 (upper left), sequence 2 (upper
right) and sequence 3 (lower). Eℓ denotes the energy radiated in all multipoles with indices ℓ and m = −ℓ,...,ℓ. We remove
the initial data burst by ignoring all data with t < rext+ 50M.
justification for this extrapolation, and yet the pointparticle threshold L/M2= 0.866 is remarkably close to the
observed transition range of L/M2≈ 0.8.
The agreement gets even better if, in the spirit of Ref. [2], we use a slightly improved perturbative model, considering
particle orbits around a Kerr black hole with spin given by the final Kerr parameter of our simulations (see Sec. V and
Appendix A for details). For eccentric inspirals, the minimum allowed angular momentum should be attained at the
socalled separatrix corresponding to the maximal allowed eccentricity, ep= 1, where epis the eccentricity defined in
0 0.20.4
0.6
0.81
L / M2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
jfin
Eb/M=0.014465
Eb/M=0.013229
Eb/M=0.008861
FIG. 8: Final angular momentum as a function of the initial momentum P for sequence 1 (black solid, cross), 2 (red dashed,
plus) and 3 (blue dotted curve, diamond).
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