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Preprint typeset using LATEX style emulateapj v. 5/2/11

ON THE MASS RADIATED BY COALESCING BLACK-HOLE BINARIES

E. BARAUSSE1

Department of Physics, University of Guelph, Guelph, Ontario, N1G 2W1, Canada

V. MOROZOVA

Max-Planck-Institut f¨ ur Gravitationsphysik, Albert Einstein Institut, Potsdam, Germany

AND

L. REZZOLLA

Max-Planck-Institut f¨ ur Gravitationsphysik, Albert Einstein Institut, Potsdam, Germany and

Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA, USA

ABSTRACT

We derive an analytic phenomenological expression that predicts the final mass of the black-hole remnant

resulting from the merger of a generic binary system of black holes on quasi-circular orbits. Besides recovering

the correct test-particle limit for extreme mass-ratio binaries, our formula reproduces well the results of all the

numerical-relativity simulations published so far, both when applied at separations of a few gravitational radii,

and when applied at separations of tens of thousands of gravitational radii. These validations make our formula

a useful tool in a variety of contexts ranging from gravitational-wave physics to cosmology. As representative

examples, we first illustrate how it can be used to decrease the phase error of the effective-one-body waveforms

during the ringdown phase. Second, we show that, when combined with the recently computed self-force

correction to the binding energy of nonspinning black-hole binaries, it provides an estimate of the energy

emitted during the merger and ringdown. Finally, we use it to calculate the energy radiated in gravitational

waves by massive black-hole binaries as a function of redshift, using different models for the seeds of the

black-hole population.

Subject headings: black-hole physics — relativity — gravitational waves — galaxies

1. INTRODUCTION

mergers Black-hole

gravitational-wave (GW) astrophysics, because they are

expected to be among the main sources for existing and

future detectors. More specifically, the LIGO/Virgo detec-

tors (Abbott et al. 2009; Acernese et al. 2008) are expected

to detect mergers of stellar-mass BHs happening within

several hundred Mpc, when operating in their advanced

configurations. Similarly, future space-based detectors such

as LISA (Amaro-Seoane et al. 2012) or DECIGO (Kawamura

et al. 2011) will detect mergers of massive BHs (MBHs) up

to redshifts as high as z ∼ 10 or beyond. Even intermediate-

mass BHs (IMBHs), provided they exist, will be within reach

of GW detectors, e.g. IMBH-MBH binaries will be detectable

by LISA or DECIGO, while IMBH-IMBH binaries will be

detectable with DECIGO or with the planned ground-based

Einstein Telescope (Punturo et al. 2012; Sathyaprakash et al.

2012).

Given their relevance for GW astrophysics, it is not sur-

prising that BH binaries have received widespread attention

over the past few years. Because a detailed understanding

of the dynamics of these systems is crucial in order to pre-

dict accurately the gravitational waveforms, which, in turn,

is necessary to detect the signal and extract information on

the physical parameters of the binaries, numerical simulations

have been performed by a number of groups for a variety of

massratios, BHspinmagnitudesandorientations[seePfeiffer

(2012) for a recent review].

However, even today, numerical-relativity simulations are

computationally very expensive and not able to cover the full

(BH) playa centralrole in

1CITA National Fellow

seven-dimensional space of parameters of quasi-circular BH

binaries. Fortunately, phenomenological models have been

very successful at reproducing many aspects of the dynamics

of BH binaries as revealed by the numerical simulations. For

instance, hybrid “phenomenological waveforms” (Ajith et al.

2008; Santamar´ ıa et al. 2010), i.e., templates that represent

phenomenological combinations of Post-Newtonian (PN) and

numerical-relativity (NR) waveforms, can reproduce with

high precision the NR waveforms for a wide range of binary

parameters. Similar results are achieved by the effective-one-

body (EOB) model, which attempts to reproduce not only the

gravitational waveforms, but also the full dynamics of BH bi-

naries during the inspiral, merger and ringdown phases, by

resumming the PN dynamics (Buonanno & Damour 1999;

Damour et al. 2009), and more recently the self-force dynam-

ics (Barausse et al. 2012).

Other aspects of the dynamics of BH binaries have been

phenomenologically understood by using combinations of

PN theory, symmetry arguments, as well as hints from the

test-particle limit and fits to numerical simulations. For in-

stance, the final spin magnitude of the BH remnant can be

predicted by a number of phenomenological formulas (Rez-

zolla et al. 2008a,b; Tichy & Marronetti 2008; Buonanno et

al. 2008; Kesden 2008; Rezzolla et al. 2008c; Barausse &

Rezzolla 2009), starting from the configuration of the binary

either at small separations r ? 10M, or at large separa-

tions2r ∼ 104M. These formulas also predict the orientation

2For MBHs, the latter are roughly the separations at which the dynamics

starts being dominated by GW emission, and represent therefore the separa-

tions at which these phenomenological formulas should work in order to be

useful in cosmological contexts.

arXiv:1206.3803v2 [gr-qc] 5 Sep 2012

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2 Barausse, Morozova and Rezzolla

of the final spin with good accuracy when applied to small-

separation binaries, while the formula of Barausse & Rezzolla

(2009) is also accurate when the binary has a large separation,

e.g. r ∼ 104M, in a large portion of the parameter space (Ba-

rausse & Rezzolla 2009; Kesden et al. 2010). Similar phe-

nomenological formulas have also been proposed for the re-

coil imparted to the final BH remnant from the anisotropic

emission of GWs (Herrmann et al. 2007b; Koppitz et al. 2007;

Rezzolla et al. 2008b; Campanelli et al. 2007a; Gonzalez et

al. 2007; Campanelli et al. 2007b; Lousto & Zlochower 2009,

2011; Baker et al. 2007, 2008a; van Meter et al. 2010). Be-

cause most of the anisotropic GW emission takes place as a

result of the strongly nonlinear merger dynamics, these recoil

formulas are not predictive, as they depend on quantities that

can only be derived with full NR simulations, but they are still

useful in the statistical studies usually performed in a cosmo-

logical context (Barausse 2012; Lousto et al. 2010b, 2012).

The dependence of the final mass of the BH remnant on

the binary’s initial parameters has also been investigated sys-

tematically in the literature (Tichy & Marronetti 2008; Boyle

& Kesden 2008; Reisswig et al. 2009; Kesden 2008; Lousto

et al. 2010a)3, but the knowledge of this dependence is far

less detailed. For instance, the formula of Tichy & Marronetti

(2008) [who built upon previous work by Boyle & Kesden

(2008)] is calibrated to reproduce NR results for comparable-

mass binaries, but does not have the correct test-particle limit

and is therefore inaccurate for binaries with small mass ratios.

Theformulaof Kesden(2008), on thecontrary, has thecorrect

test-particle limit, but does not reproduce accurately the NR

results for comparable-mass binaries. Finally, the formula of

Lousto et al. (2010a) depends, for generic binary configura-

tions, on quantities that can only be calculated using full NR

simulations, and is therefore only useful in statistical studies.

We here introduce a new phenomenological formula for the

final mass of the BH remnant (Section 2), which, by construc-

tion, reproduces both the test-particle limit and the regime of

binaries with comparable masses and aligned or antialigned

spins, which has been extensively investigated by NR calcu-

lations. In Section 3 we show that this novel formula repro-

duces accurately all of the available NR data (even for generic

spinorientationsandmassratios), bothwhenappliedtosmall-

and large-separation binary configurations. Furthermore, in

Section 4, we consider three different areas where our for-

mula can be useful: (i) we show that it can help reduce the

phase error of the EOB waveforms during the ringdown; (ii)

we combine it with the results of Le Tiec et al. (2012) for the

self-force correction to the binding energy of nonspinning BH

binaries and derive an estimate for the energy emitted during

the merger and ringdown by nonspinning binaries; (iii) us-

ing a semi-analytical galaxy-formation model to follow the

coevolution of MBHs and their host galaxies, we use our for-

mula to predict the energy emitted in GWs by MBH binaries

as a function of redshift, and show that these predictions are

strongly dependent on the model for the seeds of the MBH

population at high redshifts. Our final conclusions are drawn

in Section 5.

Throughout this paper, geometrized units G = c = 1 are

used.

2. THE DEPENDENCE OF THE FINAL MASS ON THE SPINS AND

THE MASS RATIO

3An initial expression for the radiated energy was also suggested by Buo-

nanno et al. (2007a), but was restricted to nonspinning binaries and based on

early NR calculations.

When deriving a simple algebraic formula that expresses,

with a given precision, the mass/energy radiated by a binary

system of BHs, two regimes are particularly well-understood.

On the analytic side, in fact, the test-particle limit yields pre-

dictions that are well-known and simple to derive. On the

numerical side, the simulations of binaries with equal-masses

and spins aligned or antialigned with the orbital angular mo-

mentum are comparatively simpler to study, and have been

explored extensively over the last few years. Hence, it is nat-

ural that any attempt to derive an improved expression for the

radiated energy should try and match both of these regimes.

This is indeed what our formula will be built to do.

Let us therefore start by considering the test-particle limit

and, in particular, a Kerr spacetime with mass m1and spin pa-

rameter a ≡ S1/m2

m2on a equatorial circular orbit with radius r ? m14. To

first approximation (i.e., for mass ratios q ≡ m2/m1? 1),

the particle will inspiral towards the BH under the effect of

GW emission, moving slowly (“adiabatically”) through a se-

quence of equatorial circular orbits (Kennefick & Ori 1996)

until it reaches the innermost stable circular orbit (ISCO),

where it starts plunging, eventually crossing the horizon. The

energy Erademitted by the particle during the inspiral from

r ? m1to the moment it merges with the central BH can be

written as

Erad

M

?

3˜ req

˜ req

1, and a particle (or small BH) with mass

= [1 −˜Eeq

ISCO(a)]ν + o(ν),

(1)

˜Eeq

ISCO(a) =

1 −

2

ISCO(a),

(2)

ISCO(a) = 3 + Z2− sign(a)

Z1= 1 + (1 − a2)1/3?

?

Here, M ≡ m1+ m2is the total mass, ν ≡ m1m2/M2is

the symmetric mass ratio,˜EISCOand ˜ rISCOare respectively

the energy per unit mass at the ISCO and the ISCO radius

in units of m1(Bardeen et al. 1972), while the remainder,

o(ν), contains the higher-order corrections to the radiated en-

ergy5. These corrections account, for instance, for the conser-

vative self-force effects, which affect the ISCO position and

energy (Barack & Sago 2009; Le Tiec et al. 2012), but also

for the deviations from adiabaticity, which arise because of

the finiteness of the mass m2and which blur the sharp transi-

tion between inspiral and plunge (Buonanno & Damour 2000;

Ori & Thorne 2000; Kesden 2011), and, more in general, for

the energy emitted during the plunge and merger (Berti et al.

2007; Buonanno et al. 2007a,b).

If the particle is initially on an inclined (i.e., non-equatorial)

circular orbit, GW emission will still cause it to adiabatically

inspiral through a sequence of circular orbits (Kennefick &

Ori 1996). Also, the inclination of these orbits relative to the

?

(1 + a)1/3+ (1 − a)1/3?

(3 − Z1)(3 + Z1+ 2Z2),

(3)

,

(4)

Z2=

3a2+ Z2

1.

(5)

4Without loss of generality, we can assume that the particle moves on a

prograde orbit (i.e. in the positive-φ direction), and let the spin of the Kerr

BH point up (a > 0) or down (a < 0).

5We here use the Landau symbol o, so that f = o(g) indicates that

f/g → 0 when g → 0. Similarly, we will also use the Landau symbol

O, where instead f = O(g) indicates that f/g → const when g → 0.

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On the mass radiated by coalescing black-hole binaries3

equatorial plane, which can be defined as (Hughes 2001)6

cos(ι) ≡

Lz

?Q + L2

z

,

(6)

with Q and Lz being respectively the Carter constant and

the azimuthal angular momentum, will remain approximately

constant during the inspiral (Hughes 2001; Barausse et al.

2007). As in the equatorial case, the particle plunges when

it reaches the ISCO corresponding to its inclination ι. Un-

like in the equatorial case, though, the radius of the ISCO as a

function of a and ι can only be found numerically. An analyt-

ical expression, however, can be derived if one considers only

the spin-orbit coupling of the particle to the Kerr BH, i.e., if

one considers small spins a ? 1. In that case, in fact, one can

explicitly check [using, for instance, equations (4)–(5) of Ba-

rausse et al. (2007)] that the ISCO location and energy depend

only on the combination acos(ι), so that at O(a)2, the gen-

eralization of expressions (1)–(5) to inclined orbits is given

by

Erad

M

= [1 −˜EISCO(a,ι)]ν + o(ν),

?

˜ rISCO(a,ι) ≈ ˜ req

where ˜ req

equations (1)–(5) in the case of equatorial orbits (ι = 0) and

are therefore exact in that limit, with the exception of the

higher-order terms in ν.

As mentioned above, another case in which we know ac-

curately the total energy emitted in GWs is given by binaries

of BHs with equal masses and spins aligned or antialigned

with the orbital angular momentum. Reisswig et al. (2009),

for instance, showed that the energy emitted by these bina-

riesduringtheirinspiral(frominfiniteseparation), mergerand

ringdown can be well described by a polynomial fit

(7)

˜EISCO(a,ι) ≈

1 −

2

3˜ rISCO(a,ι),

(8)

ISCO(acos(ι)),

(9)

ISCOis given by (3). Expressions (7)–(9) reduce to

Erad

M

= p0+ p1(a1+ a2) + p2(a1+ a2)2,

(10)

where a1and a2are the projections of the spin parameters

along the directionˆL of the orbital angular momentum (ai

is therefore respectively positive/negative when the spin is

aligned/antialigned withˆL), and where the fitting coefficients

were found to be (Reisswig et al. 2009) p0= 0.04826, p1=

0.01559 and p2= 0.00485, with uncertainties on the order of

∼ 5% (Reisswig et al. 2009). We recall that the coefficient

p0can be interpreted as the nonspinning orbital contribution

to the energy loss (which is the largest one and ∼ 50% of the

largest possible mass loss, which happens for a1= a2= 1),

p1 can instead be interpreted as the spin-orbit contribution

(which is ? 30% of the largest possible loss), while p2can be

associated to the spin-spin contribution (which is ? 20% of

the largest possible loss). Although the fit proposed by Reis-

swig et al. (2009) predicts a (shallow) minimum for the ra-

diated energy Eradat (a1+ a2)/2 ∼ −0.8, this minimum

is (very likely) just an artifact of the fit due to the scarce data

availableatthattime(Reisswigetal.2009). Havingnowmore

data to analyze, we can enforce the monotonicity of Eradas a

6As in the equatorial case, we can consider only prograde orbits (0 ≤ ι ≤

π/2) and allow a to be either positive or negative.

FIG. 1.— Top panel: Radiated energy, Erad/M, as a function of the to-

tal spin of the system along the orbital angular momentum, |a1|cosβ +

|a2|cosγ, for all published NR simulations with q

aligned/antialigned spins (in red) and for misaligned spins (in blue). Shown

instead with a black solid line is the prediction of expression (14) with the

coefficients fitted from aligned/antialigned binaries. Bottom panels: resid-

uals of the NR data from the fitting expression and the corresponding error

relative to Erad/M.

= 1, both with

function of a1+a2by assuming p2= p1/4, which constrains

the minimum of Eradto be at (a1+ a2)/2 = −1. Interest-

ingly, a fitting expression of the type

Erad

M

= p0+ p1(a1+ a2) +p1

4(a1+ a2)2,

(11)

provides an estimate of the radiated energy which is as accu-

rate as the one obtained with (10). Indeed, with fitting param-

eters

p1= 0.01707±0.00032, (12)

this expression reproduces all of the available NR data7for

the energy emitted by equal-mass binaries with aligned or an-

tialigned spins, to within ∼ 0.005M (except for almost maxi-

mal spins, see below). Such an accuracy is comparable to the

typical accuracy of the data themselves, so we can conclude

that expressions (11)–(12) summarize our complete knowl-

edge of the GW emission from this class of binaries to date.

We note, however, that higher-order terms in the spins may

be needed in equation (11) to reproduce the data for nearly

extremal spins. In fact, the maximum value for the radiated

energy predicted by our fit, i.e., 9.95% of the total mass of

the binary at infinite separation when a1= a2= 1, is signif-

icantly less than the 10.95% found by Lovelace et al. (2012)

for a1 = a2 ≈ 0.97. Such a large value for Eradis some-

what off the general trend shown by the other NR data for

large aligned spins. However, it is clear that higher-order spin

p0= 0.04827±0.00039,

7The NR data considered is relative to the following references listed in

alphabetical order: Baker et al. (2008b); Berti et al. (2007, 2008); Campanelli

et al. (2006); Chu et al. (2009); Chu (2012); Hannam et al. (2008, 2010);

Kelly et al. (2011); Lovelace et al. (2011, 2012); Marronetti et al. (2008);

Pollney et al. (2007); Pollney & Reisswig (2011); Reisswig et al. (2009).

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4 Barausse, Morozova and Rezzolla

terms may have to be added if more numerical data for high-

spin configurations becomes available and confirm this result.

Using therefore the knowledge of the radiated en-

ergy from the test-particle limit and from the equal-mass

aligned/antialigned configurations, we derive an expression

valid for generic binaries. As a first step, let us note that the

PN binding energy of an equal-mass binary of spinning BHs

depends on the spins, at 1.5 PN order, i.e., at leading order in

the spins (Barker & O’Connell 1975), only through the com-

bination

ˆL · (S1+ S2)

M2

where |a1| and |a2| are the spin magnitudes, and β, γ are the

angles between the orbital angular momentum unit vectorˆL

and the spins of the first and second BH, respectively. We can

therefore attempt to extend expression (11) to generic equal-

mass binaries simply by replacing a1+ a2with |a1|cosβ +

|a2|cosγ, i.e.,

Erad

M

+p1

4(|a1|cosβ + |a2|cosγ)2.

As a check of this ansatz, in the top panel of Fig. 1 we have

plotted the radiated energy, Erad/M, as a function of the to-

tal spin along the orbital angular momentum, |a1|cosβ +

|a2|cosγ, for all published NR simulations with q = 1,

both with aligned/antialigned spins (in red; see footnote 7)

and with misaligned spins (in blue8). Also, we show with

a black solid line the prediction of expression (14) with the

coefficients fitted from aligned/antialigned binaries [equation

(12)]. In the bottom panels we show instead the residuals of

the NR data from the same curve and the corresponding errors

relative to Erad/M. Clearly, while future simulations that

are more accurate or describe more involved configurations

may present deviations from our simple ansatz, all published

simulations for equal-mass binaries are in reasonable agree-

ment with equation (14), with residuals of ? 1% and errors

of ? 10% relative to the radiated mass. Note that these errors

are comparable with the intrinsic scatter of the different NR

data.

Because in the test-particle limit the angle β becomes the

angle between the spin S1of the Kerr BH and the orbital

angular momentum of the particle, thus coinciding with the

angle ι defined in (6), it is natural to rewrite equations (7)–(9)

as

Erad

M

?

where we have defined

ˆL · (S1+ S2)

M2

=|a1|cosβ + |a2|cosγ

4

,

(13)

= p0+ p1(|a1|cosβ + |a2|cosγ)

(14)

= [1 −˜EISCO(˜ a)]ν + o(ν),

2

3˜ req

(15)

˜EISCO(˜ a) =1 −

ISCO(˜ a),

(16)

˜ a ≡

=|a1|cosβ + q2|a2|cosγ

(1 + q)2

.

(17)

If we now assume that the higher-order term o(ν) in equation

(15) is quadratic in ν, we can determine it by imposing that

8The NR data for equal-mass misaligned binaries are relative to the

following references listed in alphabetical order: Herrmann et al. (2007a);

Lousto et al. (2012); Tichy & Marronetti (2007, 2008).

we recover the equal-mass expression (14) for q = 1, thus

obtaining the final expression

Erad

M

=[1 −˜EISCO(˜ a)]ν

+ 4ν2[4p0+ 16p1˜ a(˜ a + 1) +˜EISCO(˜ a) − 1],

(18)

where˜EISCO(˜ a) is given by (16). By construction, there-

fore, expression (18) has the correct behavior both in the test-

particle limit and for equal-mass binaries. Also, we stress that

the fitting coefficients [given by (12)] are obtained using only

a subset of the NR data (i.e., those for equal-mass binaries

with aligned/antialigned spins).

3. COMPARISON TO DATA: BINARIES AT SMALL AND LARGE

SEPARATIONS

In order to test the accuracy of expression (18), we used the

data of 186 numerical simulations of inspiralling and merg-

ing BH binaries9, which have reported the ratio Mf/M ≡

1 − Erad/M between the final mass of the BH remnant, Mf,

and the mass M = m1+ m2of the binary at infinite separa-

tion. In cases where this ratio was not reported explicitly, we

have reconstructed it from the energy radiated during the nu-

merical simulation using PN expressions. More specifically,

we calculate the radiated energy as

Erad= E

NR

rad+ |E

3PN

bind(Ω0)|,

(19)

where E

the orbital frequency Ω (Buonanno et al. 2003b), and Ω0is

the initial orbital frequency of the simulation (either reported

explicitly or, when unavailable, reconstructed from the ini-

tial puncture data). In addition, in those cases where the

simulation results are normalized in terms of the Arnowitt-

Deser-Misner mass MADM, we approximate it as MADM=

M − |E

For each binary, we apply our expression (18) to the ini-

tial configuration of the numerical simulation (where the bi-

nary typically has a “ small separation” r ? 10M). How-

ever, in order to check whether our expression predicts the

final mass correctly also for widely separated binaries, we

have also integrated the initial binary back to a “large sepa-

ration” r = 2 × 104M using the quasi-circular PN evolu-

tion equations of Buonanno et al. (2003b) (which are accu-

rate through 3.5PN order for the adiabatic evolution of the

orbital frequency, and through 2PN order for the dynamics of

the spins). For massive BHs, this is roughly the separation at

which the dynamics starts being dominated by GW emission,

and is therefore the separation at which our expression (18)

ought to work if we want it to be useful in cosmological con-

texts [cf. Barausse & Rezzolla (2009) and the discussion in

Barausse (2012)].

The results of these comparisons are summarized in Fig. 2,

which shows the difference between the data and our expres-

sion, both for small (left panel) and large separations (right

3PN

bind(Ω) is the 3PN binding energy as function of

3PN

bind(Ω0)|.

9The data is relative to the following references listed in alphabetical or-

der Baker et al. (2008b); Berti et al. (2008, 2007); Buchman et al. (2012);

Campanelli et al. (2006, 2009); Chu et al. (2009); Chu (2012); Gonzalez et

al. (2009); Hannam et al. (2008, 2010); Herrmann et al. (2007a); Kelly et

al. (2011); Lousto & Zlochower (2009); Lousto et al. (2012); Lovelace et al.

(2011, 2012); Marronetti et al. (2008); Nakano et al. (2011); Pollney et al.

(2007); Pollney & Reisswig (2011); Reisswig et al. (2009); Tichy & Mar-

ronetti (2007, 2008).

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On the mass radiated by coalescing black-hole binaries5

FIG. 2.— Residuals for the fitting formula at small and large separations as a function of a dummy index representing the binaries in our dataset. Binaries

with spins aligned/antialigned with the orbital angular momentum are plotted in red, while binaries with misaligned spins are shown in blue. Cyan and violet

lines represent the 1σ and 2σ errors (estimated a posteriori) of the data with spins aligned/antialigned with the orbital angular momentum. The first 50 points

correspond to the simulations performed after 2010.

panel) as a function of a dummy index representing the or-

dering of the binaries in our dataset. As can be seen the re-

sults at small and large separations are almost indistinguish-

able. This does not come as a surprise, because the projection

of the total spin on the direction of the angular momentum

[equation (13)] is approximately conserved during the inspi-

ral, in most of the parameter space of quasi-circular binaries

(see discussion in Barausse & Rezzolla (2009) for more de-

tails). Binaries with spins aligned/antialigned with the orbital

angularmomentumareplottedinred, whilebinarieswithmis-

aligned spins are shown in blue. Also shown are cyan and vi-

olet lines representing the 1σ and 2σ errors of the data with

spins aligned/antialigned with the orbital angular momentum,

as obtained a posteriori by comparing them to the fit (11) (cf.

Fig. 1). Also, the first 50 points correspond to the simula-

tions performed after 2010, while others correspond to the

simulations performed in 2006-2009. Although the quality

of numerical simulations improved substantially in last few

years, the “old” data give residuals comparable to the “new”

ones. Furthermore, the residuals for the binaries with spins

aligned/antialigned with the orbital angular momentum ap-

pear to be comparable with those for the binaries with mis-

aligned spins.

We stress that while our expression is in reasonable agree-

ment with all the published data, both at large and small sep-

arations, there are still large gaps in the parameter space of

BH binaries that prevent us from testing our approach more

thoroughly. This is best seen in Fig. 3, where we plot the fi-

nal mass of the remnant for all the published data for binaries

with a1cosβ = a2cosγ (blue circles), as well as the predic-

tionsofourexpressionwhenappliedtothe“small-separation”

initial data of the simulations (meshed surface). Clearly, spin-

ning binaries with unequal mass ratios are essentially absent,

and simulations for such binaries will provide a very signifi-

cant check of our expression (18). Nevertheless, the simple

functional dependence shown by the available data, whose

behaviour can be well captured with low-order polynomials

(with the possible exception, as we already stressed, of almost

maximally spinning configurations), is quite remarkable.

The graphical representation of the data in Fig. 3 also al-

lows to reinforce a remark already made by Reisswig et al.

(2009), namely, that the largest radiated energy, Erad(a =

1)/M = 9.95%, is lost by binaries with equal-mass and max-

imally spinning BHs with spins aligned with the orbital an-

gular momentum. Hence, BH binaries on quasi-circular or-

bits are among the most efficient sources of energy in the

Universe. Note, however, that equal-mass binaries are not

always the systems that lose the largest amount of energy.

Indeed, unequal-mass systems with sufficiently large spins

aligned with the angular momentum can lead to emissions

larger than those from equal-mass binaries but with large an-

tialigned spins. For instance, a binary with ν = 0.15 and

a1= a2= 1 will radiate more than a binary with ν = 0.25

and a1= −a2.

Notwithstanding the limited coverage of the parameter

space, we can note that our approach substantially improves

upon earlier formulas for the final mass. For instance, Tichy

& Marronetti (2008) [building on the work of Boyle & Kes-

den (2008)] suggested a formula linear in the symmetric mass

ratio ν, but the coefficients needed to fit NR results are such

that the test-particle limit (1)–(9) is not recovered. As men-

tioned earlier, because we recover the test-particle limit ex-

actly, our expression reproduces the published data more ac-

curately, especially for small mass-ratio configurations (cf.

the discussion on the effective-one-body model in the next

section). Another example is given by the formula of Lousto

et al. (2010a), which has the correct test-particle limit but de-

pends, for generic configurations, on parameters that describe

the binary’s plunge and merger and which can only be cal-

culated with numerical simulations. Our algebraic formula,

instead, allows one to calculate the final mass with reasonable

accuracy, using only information on the initial binary config-