Eccentric binary black-hole mergers: The transition from inspiral to plunge in general relativity

Abstract

We study the transition from inspiral to plunge in general relativity by computing gravitational waveforms of non-spinning, equal-mass black-hole binaries. We consider three sequences of simulations, starting with a quasi-circular 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 head-on collision. We analyze in detail the radiation of energy and angular momentum in gravitational waves, the contribution of different multipolar components and the final spin of the remnant. We find that the motion transitions from inspiral to plunge when the orbital angular momentum L=L_crit is about 0.8M^2. For L<L_crit the radiated energy drops very rapidly. Orbits with L of about L_crit produce our largest dimensionless Kerr parameter for the remnant, j=J/M^2=0.724. Generalizing a model recently proposed by Buonanno, Kidder and Lehner to eccentric binaries, we conjecture that (1) j=0.724 is the maximal Kerr parameter that can be obtained by any merger of non-spinning 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.

Full text

Page 1

arXiv:0710.3823v2 [gr-qc] 19 Sep 2008
Eccentric binary black-hole 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 38677-1848, USA
5Centro Multidisciplinar de Astrof´ ısica - CENTRA, Departamento de F´ ısica,
Instituto Superior T´ ecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
6Instituto de F´ ısica y Matem´ aticas, Universidad Michoacana de San Nicol´ as de Hidalgo,
Edificio C-3, Cd. Universitaria. C. P. 58040 Morelia, Michoac´ an, M´ exico and
7Max-Planck-Institut f¨ ur Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany
(Dated: September 19, 2008)
We study the transition from inspiral to plunge in general relativity by computing gravitational
waveforms of non-spinning, equal-mass black-hole binaries. We consider three sequences of simula-
tions, starting with a quasi-circular 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 head-on 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 post-Newtonian approximation and
with extrapolations of point-particle 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 quasi-circular 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 non-spinning 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. Earth-based laser-interferometric 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 space-based 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 black-hole 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 post-Newtonian (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@uni-jena.de

Page 2

2
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 non-spinning, equal-mass black-hole binaries, and
we focus on the effect of the orbital eccentricity on the merger waveforms. We consider three sequences, starting with
quasi-circular 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 head-on 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 point-particle results.
Non-eccentric 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 solar-mass black hole
binaries, the orbit will usually be circular by the time the GW signal enters the best-sensitivity 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 Earth-based
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 gas-dynamical processes. Three-body 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 large-scale action of the gaseous and/or
stellar disc, possibly leaving the binary with non-zero 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 disc-driven and GW-driven
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 non-spinning black-hole 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 head-on
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 two-body
dynamics, i.e. the so-called “Effective One Body” (EOB) model. Ori and Thorne [46] provided a semi-analytical
treatment of the transition in the extreme mass ratio limit. Waveforms comprising inspiral, merger and ringdown
for comparable-mass 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, equal-mass, non-spinning 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.

Page 3

3
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 post-Newtonian predictions was presented in [52].
Our focus in this work is on the high-eccentricity region of the parameter space, which always leads to merger. In
particular, the near-head-on limit of our study is of interest as a first step to compute the energy loss and production
cross-section of mini-black holes in TeV-scale gravity scenarios (possibly at the upcoming LHC [53]), and trans-
Planckian scattering in general [54, 55]. Present semi-analytical techniques (including a trapped surface search in
the union of Aichelburg-Sexl shock waves, close-limit approximation calculations and perturbation theory) only give
rough estimates of the emitted energy and production cross-section [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);
• PN-based 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 point-particle results (see also [2]), we conjecture that (1) the maximal
Kerr parameter that can be obtained by any merger of non-spinning 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 point-particle
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. POST-NEWTONIAN 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 non-linear
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 quasi-Keplerian parametrization originally proposed by Damour and Deruelle
[59], and a definition in terms of observable quantities recently introduced by Mora and Will [60].

Page 4

4
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 ADM-type 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.Quasi-Keplerian parametrization
A quasi-Keplerian 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 ADM-type 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 quasi-circularity. 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 head-on 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

Page 5

5
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 head-on 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 head-on 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 quasi-circular 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 Arnowitt-Deser-Misner-Transverse-Traceless (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 quasi-circular configurations of Table I.
A third noteworthy feature of the “quasi-Keplerian” 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 quasi-circular
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 quasi-Keplerian 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 head-on 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. Mora-Will 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 quasi-Keplerian parametrization with
different eccentricities for t, r and φ, MW define a single eccentricity parameter eMWand a PN expansion parameter