Toward faithful templates for non-spinning binary black holes using the effective-one-body approach

0.94 MB·English

by  Alessandra Buonanno

Toward faithful templates for non-spinning binary black holes using the
effective-one-body approach
Alessandra Buonanno,1 Yi Pan,1 John G. Baker,2 Joan Centrella,2
Bernard J. Kelly,2 Sean T. McWilliams,3 and James R. van Meter2,4
1Maryland Center for Fundamental Physics, Department of Physics,
University of Maryland, College Park, MD 20742
2Gravitational Astrophysics Laboratory, NASA Goddard Space Flight Center, 8800 Greenbelt Rd., Greenbelt, MD 20771
3Department of Physics, University of Maryland, College Park, MD 20742
4Center for Space Science & Technology, University of Maryland Baltimore County,
Physics Department, 1000 Hilltop Circle, Baltimore, MD 21250
(Dated: February 1, 2008)
Wepresent an accurate approximation of thefull gravitational radiation waveforms generated in
8 the merger of non-eccentric systems of two non-spinning black holes. Utilizing information from
0 recent numerical relativity simulations and the natural flexibility of the effective-one-body (EOB)
0 model, we extend the latter so that it can successfully match the numerical relativity waveforms
2 duringthelaststagesofinspiral,mergerandringdown. By“successfully”here,wemeanwithphase
differences<8%ofagravitational-wavecycleaccumulated bytheendoftheringdownphase,max-
n imizingonly∼overtimeofarrivalandinitialphase. Weobtainthisresultbysimplyaddinga4-post-
a
NewtonianordercorrectionintheEOBradialpotentialanddeterminingthe(constant)coefficientby
J
imposing high-matching performances with numerical waveforms of mass ratios m1/m2 =1,3/2,2
4
and 4, m1 and m2 being the individual black-hole masses. The final black-hole mass and spin

predicted by the numerical simulations are used to determine the ringdown frequency and decay
]
c timeof threequasi-normal-mode damped sinusoids that are attached to theEOB inspiral-(plunge)
q waveform at the EOB light-ring. The EOB waveforms might be tested and further improved in
- thefuturebycomparisonwithextremelylongandaccurateinspiralnumerical-relativitywaveforms.
r
g They may already be employed for coherent searches and parameter estimation of gravitational
[ wavesemittedbynon-spinningcoalescingbinaryblackholeswithground-basedlaser-interferometer
detectors.

3
v PACSnumbers: 04.25.Dm,04.30.Db,04.70.Bw,x04.25.Nx,04.30.-w,04.80.Nn
2
3
7 I. INTRODUCTION tually thousands of waveform templates may be needed
3 to extract the GW signal from the noise, an impossible
.
6 demand for numerical relativity (NR) alone.
The network of ground-based laser-interferometer
0 The best-developed analytic method for describing
7 gravitational-wave (GW) detectors, such as LIGO [1],
the two-body dynamics of comparable-mass BHs and
0 VIRGO [2], GEO [3] and TAMA [4], are currently op-
predicting the GW signal is undoubtedly the PN
: erating at design sensitivity (except for VIRGO which is
v method [14], which for compact bodies is essentially an
expectedtoreachdesignsensitivitywithinoneyear)and
i expansioninthecharacteristicorbitalvelocityv/c. Tem-
X are searching for GWs in the frequency range of 10–103
plate predictions are currently available through 3.5PN
r Hz. Within the next decade these detectors will likely
a order (v7/c7) [15, 16, 17, 18], if the compact objects do
be complemented by the laser-interferometer space an-
notcarryspin,and2.5PNorder(v5/c5)[19]iftheycarry
tenna (LISA) [5], a joint venture between NASA and
spin. Resummation of the PN expansionaimed at push-
ESA, which will search for GWs in the frequency range
3 10−5–10−1 Hz. ing analytic calculations until the final stage of evolu-
× tion, including the transition inspiral–merger–ringdown,
Binarysystemscomposedofblackholes (BHs)and/or havebeenproposed. In1999,Buonanno&Damourintro-
neutron stars (NSs) are among the most promising GW duced a non-perturbative resummation of the two-body
sources. ThesearchforGWs fromcoalescingbinarysys- conservative dynamics, the so-called effective-one-body
tems and the extraction of parameters are based on the (EOB)approach[20]. TheoriginalEOBmodelwascom-
matched-filtering technique [6], which requires accurate putedusing the 2PNconservativedynamics. Itwasthen
knowledge of the waveform of the incoming signal. Re- extended to 3PN order [21] when the 3PN calculation
centcomparisons[7, 8, 9, 10, 11, 12, 13] betweennumer- was completed [18] and then to spinning BHs [22]. The
ical and post-Newtonian (PN) analytic waveforms emit- EOB approachhas been the only analytic approachable
ted by non-spinning binary BH systems suggest that it topredict,within 10%ofaccuracy,thespinofthefinal
∼
should be possible to design purely analytic templates BH[23]. Recently,bycombiningtheEOBapproachwith
withthefullnumericsusedtoguidethepatchingtogether test-masslimitpredictionsfortheenergyreleasedduring
of the inspiral and ringdown (RD) waveforms. This is the merger-ringdown phases, Ref. [25] has refined this
an important avenue to template construction as even- prediction obtaining 2% of accuracy. The EOB ap-
∼
2
proachalsoprovideda complete waveform,frominspiral II. THE EFFECTIVE-ONE-BODY MODEL FOR
to ringdown,for non-spinning[23, 24]andspinning, pre- NON-SPINNING BLACK HOLE BINARIES
cessing binary systems [26]. To include accurately the
radiation-reaction contribution, the EOB approach uses
the Pad´e resummation of the GW flux, as proposed in
Attheendofthe90s,intheabsenceofNRresultsand
Ref. [27].
withtheurgentneedofprovidingtemplatestosearchfor
By construction the EOB approach recovers ex-
comparable-mass BHs, some resummation techniques of
actly geodesic motion in the test-mass limit. In the
the post-Newtonian series were proposed. The general
comparable-mass limit the EOB approach provides a
philosophy underlying these techniques [20, 27] was to
non-perturbative resummation of the dynamics, which
firstresuminthe test-mass–limitcasethetwocrucialin-
today can be tested and improved by comparing it
gredients determining the gravitational-wave signal: the
to NR results. NR simulations are in fact the best
two-body energyand the gravitational-waveenergy flux.
tool to describe the non-linear, strong-gravity regime
In fact, in the test-mass–limit case these ingredients are
of comparable-mass binary coalescences. As we shall
known exactly. Secondly, it was assumed that the re-
see below, because of the reduction of the dynamics to
summed quantities will also be a good representation of
a few crucial functions determining the inspiral evolu-
the comparable-mass case, viewed as a smooth deforma-
tion [20, 22, 23], and because of the rather simple proce-
tion of the test-mass–limit case.
dure for matching the inspiral(-plunge) waveform to the
ringdownwaveform,the EOB model is particularly suit-
The resummation technique discussed in this section,
able for fitting to the numerical results [28, 29]. In this
the EOBapproach[20], wasoriginallyinspiredby asim-
paper we shall employ its flexibility [22, 30] to obtain
ilar approach introduced by Br´ezin, Itzykson and Zinn-
accurate waveformsfor potentially the full range of non-
Justin [31] to study two electromagnetically interacting
spinning binary BHs. We shall test the analytic wave-
particles with comparable masses. The basic idea of the
forms against the numerical ones for mass ratios rang-
EOB approach is to map the real conservative two-body
ing between m /m = 1 and m /m = 4, with m and
1 2 1 2 1
dynamics up to the highest PN order available, onto
m being the BH masses. The method also allows us
2
an effective one-body problem, where a test particle of
to predict the waveforms for mass ratios m /m > 4.
1 2
mass µ = m m /M, with m , m the BH masses and
Thesewaveformswillbe testedagainstnumericalresults 1 2 1 2
M =m +m , moves in some effective backgroundmet-
when accurate long numerical simulations for mass ra- 1 2
ric geff. This mapping has been worked out within the
tios m1/m2 > 4 become available. In this paper the µν
Hamilton-Jacobi formalism, by requiring that while the
comparisons are carried out using simulations from the
actionvariablesoftherealandeffectivedescriptioncoin-
NASA-Goddard group.
cide(i.e.L =L , = ,whereLdenotestheto-
The paper is organizedasfollows. In Sec.II we briefly real eff real eff
I I
review the EOB model. In Sec. III we improve the EOB talangularmomentum,and theradialactionvariable),
I
the energy axis is allowed to change: E = f(E ),
modelbyadding a4PNorderunknowncoefficientto the real eff
where f is a generic function determined by the map-
two-body conservative dynamics. In Sec. IV we com-
ping. By applying the above rules defining the mapping,
plete the EOB model using inputs from NR simulations.
itwasfound[20]inthe non-spinningcasethataslongas
In Sec. V we compare the improved EOB model to two
radiation-reactioneffects are not taken into account, the
accurate,longnumericalsimulationswithmassratios1:1
effectivemetricisjustadeformationoftheSchwarzschild
and4:1,determine the best-fit 4PNordercoefficient and
metric, with deformation parameter η =µ/M.
discuss the matching performances for several dominant
modes. Section VI summarizes our main conclusions.
Appendix A refersto shorternumericalsimulations with Theexplicitexpressionofthenon-spinningEOBeffec-
mass ratios 2:1 and 3:2. tive Hamiltonian through 3PN order is [20, 21]:
A(r) 1
H (r,p)=µH (r,p)=µ A(r) 1+p2+ 1 (n p)2+ (z (p2)2+z p2(n p)2+z (n p)4) , (1)
eff eff s D(r) − · r2 1 2 · 3 ·
(cid:20) (cid:18) (cid:19) (cid:21)
b
with r and p being the reduced dimensionless variables; the motion is constrained to a plane. Introducing polar
n = r/r where we set r = r. In the absence of spins coordinates (r,ϕ,p ,p ), the EOB effective metric reads
r ϕ
| |
3
D(r)
ds2 geffdxµdxν = A(r)c2dt2+ dr2+r2(dθ2+sin2θdϕ2). (2)
eff ≡ µν − A(r)
6η 1
The EOB real Hamiltonian is D3PN(r)=1 +[7z +z +2η(3η 26)] . (6)
T − r2 1 2 − r3
H µ
eff
H =M 1+2η − M, (3)
real
s (cid:18) µ (cid:19)− Inprinciple we couldexplore the possibility ofdetermin-
ing some of the z coefficients through a fit with the nu-
i
and we define Hˆreal = Hreal/µ. Remarkably, as origi- mericalresults. However,here we do notfollowthis pos-
nallyobservedinRef.[20],themappingbetweenthereal sibility and, as in previous works, except for Ref. [32],
andtheeffectiveHamiltoniansgivenbyEq.(3)coincides we set z = z = 0, z = 2(4 3η)η. The EOB ef-
1 2 3
with the mapping obtained in the context of quantum fective potential A3PN(r) does no−t lead to a last-stable
T
electrodynamics in Ref. [31], where the authors mapped circular orbit (LSO), contrary to what happens in the
the one-body relativistic Balmer formula onto the two- 2PN-accurate case [20]. This is due to the rather large
body energy formula. Moreover, Eq. (3) holds at 2PN value of the 3PN coefficient 94/3 41/32π2 18.688
and 3PN order [21]. The coefficients z1,z2 and z3 in entering the PN expansion of A(r)−. Replacing≃the PN-
Eq. (1) are arbitrary, subject to the constraint expandedformofA(r)byaPad´eapproximantcuresthis
problem [21]. The Pad´e approximant is
8z +4z +3z =6(4 3η)η. (4)
1 2 3
−
The coefficients A(r) and D(r) in Eq. (1) have been cal-
r( 4+2r+η)
culated through 3PN order [20, 21]. In Taylor-expanded A2PP21N(r)= 2r−2+2η+rη , (7)
form they read:
2 2η 94 41 1
A3PN(r)=1 + + π2 η z , (5) at 2PN order and
T −r r3 3 − 32 − 1 r4
(cid:20)(cid:18) (cid:19) (cid:21)
r2[(a (η,0)+8η 16)+r(8 2η)]
A3PP31N(r)= r3(8 2η)+r2[a4(η4,0)+4η]+r−[2a4(η,0)+−8η]+4[η2+a4(η,0)], (8)
−
at 3PN order where dr ∂H
= (r,p ,p ), (11)
r ϕ
dt ∂pr
94 41
a4(η,z1)= π2 η z1 . (9) dϕ b ∂H
3 − 32 − ω = (r,p ,p ), (12)
(cid:20)(cid:18) (cid:19) (cid:21) dtb ≡ ∂pϕ r ϕ
For the coefficient D(r) at 3PN order we use the Pad´e b
dp ∂H
r b
approximant b = (r,pr,pϕ), (13)
dt − ∂r
dp b
r3 ϕ = Fϕ[ω(r,p ,p )], (14)
DP3P30N(r)= r3+6ηr+2η(26 3η). (10) dtb r ϕ
−
b b
To include radiation-reaction effects we write the EOB b
Hamilton equations in terms of the reduced quantities
wherefortheϕcomponentoftheradiation-reactionforce
H, t=t/M, ω =ωM [23], as 1
b b b
arevalidinanycanonicalcoordinatesystem,whenweevolvethe
1 When using the EOB real Hamiltonian we should in principle EOBdynamicswewritetheHamiltonequationsintermsofthe
considerthe(generalized)canonicaltransformationbetweenthe effective variables. When comparingto NRresults,there might
realandeffective variableswhichisexplicitlygivenas aPNex- be some differences in the time variable, though. In any case
pansioninRefs.[20,21]. However,sincetheHamiltonequations Eqs.(11)–(14)defineourEOBmodel.
4
5.5
5
4.5
4
3.5
M LSO
r/ 3 Light-ring
2.5
2
1.5
1
0 30 60 90 120 150
λ
FIG. 1: In the left panel we show the position of the LSO and light-ring as function of the parameter λ, for different mass
ratios: 4:1 (dotted line), 2:1 (dot-dashed line), 3:2 (dashed line) and 1:1 (continuous line). In the right panel we show: (top
part) the energy for circular orbits as a function of the frequency evaluated from the EOB Hamiltonian, (bottom part) the
radial potential as function of the radial coordinate for a massless particle in the EOB model. The various curves refer to
different PN orders.
we shall use the P-approximant[23, 27] waveformtoadampedsinusoidatthelight-ringposition.
The frequency and decay time were computed estimat-
1 32 f (v ;η)
Fϕ [v ]= ηv7 PN ω , ingthefinalBHmassandspinfromtheEOBenergyand
PN ≡−ηv3 FPN ω − 5 ω 1 v /v (η)
ω − ω pole angular momentum at the matching point. The match-
(15)
b ing procedure has then been improved, by adding more
where v ω1/3 (dϕ/dt)1/3. The coefficients f can
ω ≡ ≡ PN QNMs,extendingittoseveralmultipolemoments[7,38],
bereadfromEqs.(50)–(54)inRef.[32],whileforv we
pole and applying it over a time-interval instead of one point
use the exprbession given bby Eq. (55) in Ref. [32]. Initial in time [13].
conditions for Eqs. (11)–(14) are discussed in Ref. [23].
InRef.[12,38],the authorspointedoutthatinprinciple
amoreaccurateexpressionoftheradiation-reactionforce
III. THE PSEUDO 4PN
should not use the Keplerian relation between r and ω
EFFECTIVE-ONE-BODY MODEL
whenthebinaryevolvesinsidetheLSO.However,asalso
noticedanddiscussedinRef.[13],thismodificationofthe
Previous investigations [7, 9] focusing on comparable-
radiation-reactioneffectshaslittleeffectonthewaveform
mass binaries, have shown that a non-negligible dephas-
amplitude. Sinceitisnotoneofthegoalsofthispaperto
ing can accumulate at the transition inspiral(-plunge)to
improvetheamplitudeagreementbetweenthenumerical
ringdownbetweenthe3.5PN-EOBwaveformandtheNR
and EOB waveforms,we do not include it.
waveform. The dephasing is caused by the much faster
ThelastcrucialingredientoftheEOBmodelisthein-
increase of the GW frequency in the 3.5PN-EOB model
clusionoftheringdownphase. AfterthetwoBHsmerge,
than in the NR simulation when approaching the light-
the system settles down to a Kerr BH and emits quasi-
ring position. Although the dephasing would prevent an
normal modes (QNMs) [34, 35]. In the test-mass limit,
accuratedeterminationofthebinaryparametersinappli-
η 1, Refs. [35, 36] realized that when a test parti-
≪ cations to gravitational wave observations, it would not
cle falls radially below 3M (the unstable light-ring of
prevent detection of the signals. In Ref. [9] the authors
Schwarzschild),itimmediatelytriggerstheproductionof
built effectual 2 EOB templates which match the EOB
QNMs,thusproducingauniversalmergersignal(bycon-
inspiral(-plunge) to ringdown through three parameters.
trastthe directgravitationalradiationfromthe sourceis
The latter describe the time of matching, and the dif-
strongly filtered by the curvature potential barrier cen-
ference between the final BH mass (spin) and the EOB
teredaroundit,seeFig.7). Inthecomparable-masscase
η<1/4,toapproximatethelatepartofthemergerwave-
fo∼rms,Ref.[37]proposedtheso-calledclose-limitapprox-
imation, which consists of switching from the two-body
description to the one-body description (perturbed-BH) 2 Following Ref. [27], by effectual templates we mean templates
that have large overlaps, say >∼96.5%, withthe expected signal
close to the light-ring location. Based on these obser-
after maximizing over the initial phase, time of arrival and BH
vations, Ref. [20] modeled the merger as a very short
masses. Effectual templates can be used for detection but may
(instantaneous)phaseandmatchedthe inspiral(-plunge) leadtolargebiasedinestimatingthebinaryparameters.
5
energy(angularmomentum) atthe matchingpoint3. In with fixed angular-momentum p and energy Hˆ and
ϕ real
thispaperweimproveonRef.[9]implementingamatch- Hˆ . Thus, we can write
eff
ing procedure that does not require the introduction of
anynewparameterandthatagreeswithnumericalsimu-
A(r)
lation waveforms within a rather small phase difference, ωplunge(t) = r2 const, (18)
<0.08 of a GW cycles, thus providing accurate or faith-
p
∼ful 4 templates. const = ϕ . (19)
To decrease the differences between the EOB and NR (cid:20)ηHˆrealHˆeff(cid:21)LSO
waveforms during the last stages of inspiral and plunge,
we introduce a 4PN order term in the effective potential TheaboveEq.(19)clearlyshowshowthecoefficientA(r)
A(r), given by Eq. (5), that is determinesthefrequencyduringtheplunge,i.e.,fromthe
LSO until the light-ring position. The latter happens at
Ap4PN(r)=A3PN(r)+ a5(η), (16) the maximum of A(r)/r2. A direct test has shown that
T T r5 the relative difference between ω and the exact ω
plunge
fromthe EOB-LSOtothelight-ringisatmost5%inthe
and Pad´e-approximateit using the approximantAP41. A case 4:1.
similar modification was employed in Ref. [30] to obtain
Sincethe4PNtermhasnotbeencalculatedinPNthe-
better matches of the EOB model to quasi-equilibrium
ory, we shall denote it as “p4PN”, where “p” stands for
initial-data configurations [39] and was also pointed out
pseudo. We do not claim that when the 4PN order term
in Ref. [22]. An interesting motivation for this change is
is calculated it will agree with the p4PN order term. In
thefollowing[20,23]: FromEq.(12)itisstraightforward
fact, the latter should be consideredas a phenomenolog-
to write the EOB instantaneous frequency as
ical term. We have
A(r) p
ϕ
ω(t)= r2 ηHˆrealHˆeff . (17) Ap4PN(r)= Num(ApP441PN), (20)
P41 Den(Ap4PN)
It is reasonable to assume that during the plunge, the P41
two-body dynamics is no longer driven by radiation-
reaction effects [24], but occurs mostly along a geodesic, with
Num(Ap4PN)=r3[32 24η 4a (η,0) a (η,λ)]+r4[a (η,0) 16+8η], (21)
P41 − − 4 − 5 4 −
Den(Ap4PN) = a2(η,0) 8a (η,λ) 8a (η,0)η+2a (η,λ)η 16η2+r[ 8a (η,0) 4a (η,λ) 2a (η,0)η
P41 − 4 − 5 − 4 5 − − 4 − 5 − 4 −
16η2]+r2[ 4a (η,0) 2a (η,λ) 16η]+r3[ 2a (η,0) a (η,λ) 8η]+r4( 16+a (η,0)+8η),
4 5 4 5 4
− − − − − − −
(22)
where light-ring(the lastunstable orbitfor amasslessparticle)
forseveralbinarymassratios. Lateronweshallseethat
a (η,λ)=λη, (23)
5 the value of λ that best fits the NR results (see Sec. IV)
is λ = 60. It always guarantees the presence of a LSO
and λ will be determined by comparison with numerical
and a light ring. In the right panel of Fig 1, we show
results. Wecouldalsointroducea4PNorderterminthe
thecircular-orbitenergycomputedwiththeEOBHamil-
coefficient D(r). However, we find that the effect on the
tonian, and the radial potential for a massless particle,
dynamics is relatively small and decide to use the Pad´e
at different PN orders with fixed λ = 60. We notice
approximant
that the LSO energy (EEOB/M = 0.0185) and fre-
DPp440PN(r)= r4+6ηr2+2ηr(246 3η)r+36η . (24) qthueenccoyrre(MspoωnpEd4OPiBnNg=va0lu.1e0s4op7b4)PtaaNitnped4PuNsino−grdthere3aPreNc-lToaseyrlotro-
−
expanded model for quasi-circular adiabatic orbits [40],
ThedifferencebetweenDp4PN(r)andD3PN(r),Eq.(10),
P40 P30 and to the quasi-equilibrium initial-data approach [41]
causes a negligible change in all our results. (see Fig. 16 and Table II in Ref. [41]). This could
In the left panel of Fig. 1 we show how the p4PN or- be a pure accident. In fact, it should be kept in mind
der term λ modifies the position of the EOB LSO and
6
η [Mf/M]Goddard [af/Mf]Goddard [Mf/M]fit [af/Mf]fit [Mf/Mo]Jena [af/Mf]Jena
0.25 0.9526 0.687 0.9546 0.685 0.9628 0.684
0.24 0.9561 0.670 0.9576 0.664 0.9660 0.664
0.2222 0.9668 0.621 0.9633 0.623 0.9714 0.626
0.2041 0.9676 0.586 0.9762 0.581
··· ···
0.1875 0.9618 0.548 0.9800 0.544
··· ···
0.1728 0.9752 0.512 0.9831 0.509
··· ···
0.16 0.9783 0.472 0.9781 0.480 0.9855 0.478
0.12 0.9860 0.374
··· ··· ··· ···
0.08 0.9922 0.259
··· ··· ··· ···
0.04 0.9969 0.134
··· ··· ··· ···
0.01 0.9994 0.034
··· ··· ··· ···
TABLE I: For several values of η, we list in the second and third columns the values of M /M and a /M computed from
f f f
the energy released and by extracting the fundamental QNM from −2C22, respectively. In the fourth and fifth columns we
list the values obtained using the one-parameter fits Mf/M = 1+(p8/9 1)η 0.498( 0.027)η2, and af/Mf = √12η
2.900( 0.065)η2, where the terms linear in η have been fixed to the test-m−ass lim−it value±s. In the last two columns we lis−t
±
thevaluesfrom theJena group (see Table V of Ref.[46]). Note that theJena mass valuesare scaled differently,against ADM
mass, Mo of thesystem at thebeginning of the numerical simulations, which should result slightly larger values.
1.00
0.98
0.96
h
F0.94 h 22
F 33
h
44
0.92
mass ratio 1:1
mass ratio 3:2
mass ratio 2:1
0.90 mass ratio 4:1
0.88
0 25 50 75 100 125 150
λ
FIG.2: Intheleftpanelweshowthe(minmax)FFbetweenthehighandmediumresolutionruns, hNR,h,hNR,m ,asafunction
h i
ofthebinarytotal-massM. TheFFsareevaluatedusingLIGO’sPSD.Ifweusewhitenoisewefind0.9922and0.9920formass
ratios 1 : 1 and 4 : 1, respectively. In the right panel, for different mass ratios, we show how the (minmax) FF hNR,hEOB
(computed using white noise) depends on the parameter λ. For mass ratios 1 : 1, 2 : 1, and 3 : 2 we compute hhNR,hEOBi,
while for 4 : 1 we show also results for hNR,hEOB and hNR,hEOB . The vertical line refers to the value λ =h6022whic2h2 wie
h 33 33 i h 44 44 i
employ in all subsequentanalyses.
thatthe LSO frequencycomputedfromthe 3PN-Taylor- IV. EFFECTIVE-ONE-BODY WAVEFORMS
expanded conservative dynamics is [40] MωT 0.129 FOR INSPIRAL, MERGER AND RINGDOWN
(ET /M = 0.0193), quite close to the f3oPrNma∼tion of
3PN −
thecommonapparenthorizonintheNRsimulation,and
Integrating the p4PN-EOB Hamiltonian equations
quite far from the frequency 0.08 at which the indis-
∼ provides a description of the binary’s dynamical evo-
tinctplungeoccurs[7]. Whatwecancertainlysayisthat
lution. As described below, we derive our waveforms
the p4PN-EOB conservative dynamics is closer than at
directly from the EOB dynamics until the system ap-
3PNorderto the 3PNTaylor-expandedconservativedy-
proaches the light-ring. Thereafter we complete each
namics of quasi-circular adiabatic orbits [40].
spherical harmonic waveform component by matching it
to a set of quasinormal ringdown modes. Because the
ringdown mode frequencies depend on the mass M and
f
spinparametera ofthe finalBHformedby the merger,
f
thispartofthemodelwillrequireanadditionalprescrip-
tion for accurately determining these values.
Following the tradition in NR we describe the wave-
7
FIG.3: Weshowthedifferencesintheorbitalfrequencybetweenthe3.5PN-Tt3modeland3.5PN-T,3.5PN-Tt1,3.5PN-EOB,
p4PN-EOBmodels formassratios 1:1(left panel) and4:1(right panel). ThePN frequenciescoincideatωM =0.017 att=0,
and end at ωM =0.035.
formsinterms ofaspin-weightedsphericalharmonicde- They read:
composition. Fromournumericalsimulationswedirectly
compute the Weyl tensor Ψ4, which in terms of spin- hEOB = 8 π δmη(Mω) e−iϕ, (30)
weight−2sphericalharmonics−2Ylm(θ,φ)[43]reads[see 21 −3r5 M
Ref. [42] for details] π
hEOB = 8 η(Mω)2/3 e−2iϕ, (31)
22 − 5
r
MRΨ4 = −2Clm(t)−2Ylm(θ,φ), (25) 2 π δm
Xlm hE31OB = −3 70η M (Mω)e−iϕ, (32)
r
R being the extraction radius. In terms of the + and 8 π
GW polarizations we have 5. × hE32OB = −3 7 η(1−3η)(Mω)4/3e−2iϕ, (33)
r
Ψ4 =−(h¨+−ih¨×). (26) hE33OB = −3 67π δMmη(Mω)e−3iϕ, (34)
r
8
Thus, we can write hEOB = √πη(1 3η)(Mω)4/3e−2iϕ, (35)
42 −63 −
−2Clm =−MR dΩ−2Yl∗m(θ,φ)(h¨+−ih¨×). (27) hE44OB = −694 π7 η(1−3η)(Mω)4/3e−4iϕ, (36)
Z r
In the adiabatic approximation(ω˙/ω2 1), we obtain where δm = m1 m2 > 0, and ϕ is the binary orbital
≪ phase. Note that−hl−m =(−1)lh∗lm. The RD modes are
−2Clm = m2ω2hlm, (28) attached at the time when the orbital frequency reaches
− itsmaximumandthisoccursslightlybeforethelight-ring
position, r = 1.651, Mω = 0.1883 (η = 0.25) and
where match
r = 2.089, Mω = 0.1665 (η = 0.16). It is useful
match
tohaveananalyticformularelatingthe maximumofthe
∗
hlm ≡−(h+−ih×)lm =− dΩ−2Ylm(θ,φ)(h+−ih×). orbitalfrequency,i.e.,thematchingpoint,toη. Asimple
Z (29) fit gives
We compute the EOB h in the so-calledrestricted ap-
lm Mω =0.133+0.183η+0.161η2. (37)
proximation, i.e., at leading order in the PN expansion. match
Wefindthattheabovefittinghas<0.35%errorcompar-
ing to numerical values in the range η =0.05–0.25. The
matching to QNMs is obtainedby imposing the continu-
5 Note that this definition of Ψ4 is tetrad-dependent. Here we ityofh (t)andallthehighertimederivativesneededto
lm
assumethetetradgiveninRef.[44],Eqs.(5.6). Itisalsocommon
fix the six unknown amplitudes and phases of the three
for Ψ4 to be scaled according to an asymptotically Kinnersley
RD modes [7, 23]. Following Ref. [7], we attach the fun-
tetrad (Ref.[44], Eqs.(5.9)) whichintroduces afactor of 2 as in
Ref.[42] damental mode, and two overtones. We find that the
8
0.30 3
NR waveform
0.25 NR frequency 2 EOB waveform
2π 0.20 ENORB a mfrpeqliutuednecy 1
/W EOB amplitude
G Phase difference
∆ϕ 0.15 h22 0
r
o
ω 0.10 -1
M
0.05
-2
0.00
-3
-600 -500 -400 -300 -200 -100 0 -200 -150 -100 -50 0 50
t/M t/M
FIG. 4: Equal-mass binary. In the left panel we plot the NR and p4PN-EOB frequencies and amplitudes, and the phase
differencebetweentheEOBandNRh22. IntherightpanelwecomparetheEOBandNRh22. Wemaximizeonlyontheinitial
phase and time of arrival. Notethat we show only thelast few cycles. The complete inspiral waveform has 14 GW cycles.
matching-performance does not improve significantly if the one-parameter fit, we stick with the latter. The ex-
we add more overtones. The frequency and decay time trapolation to smaller values of η is consistent with the
of the RD modes are computed using the mass M and values obtained in Ref. [46] using NR simulations, also
f
spin a of the final BH, using Refs. [45]. listed in Table I, and in Ref. [25] using a combination
f
of test-mass limit predictions and the EOB approach.
For non-spinning binary systems, it is now possible to
Henceforth, when computing the frequency and decay
determine how the final-BH mass and spin depend on
time of the QNMs we use the results obtained from the
the mass ratio. Here we apply an empirical estimate
one-parameter fit in Table I.
for the functions M (η) as and a (η) based on a combi-
f f This completes the specification of our waveform
nation of numerical simulations in the range η > 0.16
model, which can be applied to provide full waveform
and expectations from the test particle limit η 0.
→ predictionsfornon-eccentricandnon-spinningbinaryBH
In Table I we list the final BH masses and spins for
mergers of arbitrary mass-ratio.
η = 0.25,0.24,0.22,0.16, extracted from the NR simu-
lations. The values are compatible with Ref. [46, 49].
The final mass was computed from the difference of the
V. COMPARISON WITH NUMERICAL
total radiated energy and the initial ADM mass. To ob-
RELATIVITY
tain the final spin of the merged BH we first calculate
the complex QNM frequency of the l = 2, m = 2 mode
Inthissectionweexaminehowcloselyourmodelwave-
by a linear fit to the phase and log amplitude; the two
forms fit with the results of our numerical simulations.
slopesgivethe realandimaginarypartsofthe frequency
While we have already applied some limited information
which are uniquely related to the final spin [45].
from these numerical simulations in defining our model,
In the absence of NR results, to determine the final suchasinderivingourfunctionalfitsforM (η)anda (η)
f f
BH masses and spins to lower values of η we apply a fit andinselectingtheoptimalvalueforλ,wecannowcom-
to the data η = 0.25,0.24,0.22,0.16. For the BH mass parethefullwaveforms. Inparticular,thoughourmodel
we consider the one-parameterfit function Mf/M =1+ wasdevelopedprimarilyinconsiderationoftheh22wave-
( 8/9 1)η 0.498( 0.027)η2, where the coefficient of forms,wefindherethatothermultipolarwaveformcom-
− − ±
the linearterm inη has been fixedto the test-mass limit ponents are also well described.
p
value. For the BH spin we employ the one-parameter fit To measure the differences between the NR and EOB
functiona /M =√12η 2.900( 0.065)η2,whereagain waveforms we compute the fitting factor (FF), or am-
f f
− ±
the linearterm inη has been fixedto the test-mass limit biguity function [9, 27, 32]. We recall that the overlap
prediction. Ifwewereusingatwo-parameterfitwewould h1(t),h2(t) between the waveforms h1(t) and h2(t) is
h i
obtain: M /M =1 0.024( 0.057)η 0.641( 0.249)η2, defined by:
f
and a /M =3.29(−0.08)η± 2.13( −0.33)η2.±We notice
that tfhe vaflue of 3.±29 is quit−e close±to the LSO angular- h (t),h (t) 4Re ∞ h˜1(f)h˜∗2(f)df, (38)
momentum for a test particle in Schwarzschild, i.e., h 1 2 i≡ S (f)
Z0 h
√12 3.4641 [46]. However, since the two-parameter
fit giv≃es larger errors for the BH mass with respect to where h˜ (f) is the Fourier transform of h (t), and S (f)
i i h
9
0.8
0.25
NR frequency 0.6 NR waveform
EOB waveform
EOB frequency
0.20
π NEORB a mamplpitluitduede 0.4
2
/W0.15 Phase difference 0.2
G
∆ϕ0.10 h220.0
r
ω o -0.2
M0.05
-0.4
0.00 -0.6
-0.8
-0.05
-300 -200 -100 0 -300 -250 -200 -150 -100 -50 0 50
t/M t/M
FIG. 5: Binary with mass ratio 4 : 1. In the left panel we plot the NR and p4PN-EOB frequencies and amplitudes, and the
phase difference between the EOB and NR h22. In theright panel we compare the EOB and NR h22. Wemaximize only over
the initial phase and time of arrival. Note that we show only the last few cycles. The complete inspiral waveform has 9 GW
cycles.
isthedetector’spowerspectraldensity(PSD).TheFFis maximized only over the initial time t and initial phase
0
thenormalizedoverlapbetweentheNRwaveformhNR(t) ϕ , and minimized over the initial phase ϕ of the target
0
(target)andthe EOBwaveformhEOB(t ,ϕ ) (template) (the so-called minmax [27]), that is
0 0
hNR(ϕ;λi),hEOB(t ,ϕ ;λi)
0 0
FF minmax h i , (39)
≡ ϕ t0,ϕ0 hNR(ϕ;λi),hNR(ϕ;λi) hEOB(t0,ϕ0;λi),hEOB(t0,ϕ0;λi)
h ih i
p
where λi are the binary parameters. For the detector waveforms h are given by
i
PSD we shall consider either white noise or the LIGO
noise. hi =h0+xnihd, (40)
where n is the convergence factor of the waveform,h is
0
The equal-mass run lasts for 14 GW cycles before
∼ theexactwaveformgeneratedfromtheinfiniteresolution
merger. ThisrunwaspublishedoriginallyinRefs.[8,42],
run(x 0),andh istheleadingordertruncationerror
andfurtherstudiedfordata-analysispurposesinRef.[9]. 0 → d
contribution to the waveform and is independent of the
The unequal-mass runs, m /m =3/2,2,4,last for 5, 5,
1 2 mesh spacing x . The mismatch between the waveforms
and 9 GW cycles before merger. The m /m = 3/2,2 i
1 2 h and h , 1 FF , then scales as
cases were published in Refs. [9, 10] and the m1/m2 = i j − ij
4 has recently been computed by the NASA-Goddard 1 FF (xn xn)2. (41)
group. Adaptive mesh refinement was employed for this − ij ∝ i − j
caseasintheprevioussimulations,withafinestmeshres- In the Goddard simulations, the high and medium reso-
olutionofhm =3M/160usedinonerunandhh =M/64 lution runs have mesh-spacing ratio xh/xm = 5/6, and
used in a second run. Adequate convergence of the the waveform convergence rate was shown to be n = 4
Hamiltonianandmomentumconstraintswerefound; the in the 1:1 case [42]. The FF between the high resolution
numericaldetailswillbereportedinafuturepublication. and exact waveforms h and h is given by
h 0
Based on the comparisons between high- and medium-
resolution waveforms, we estimated in Ref. [9] the FFs FF =1 0.87(1 FF ), (42)
0h hm
between high resolution and exact waveforms for the − −
m /m = 1 case. Here we apply the same procedure where FF is the FF between the high and medium
1 2 hm
for the m /m = 4 case. If we have several simulations resolution waveforms h and h . Thus, the mismatch
1 2 h m
withdifferentresolutions,specifiedbythemesh-spacings between h and h is slightly smaller than that between
h 0
x , and x are sufficiently small, we can assume that the h andh ,wherethelattercanbederivedfromtheFFs
i i h m
10
shownin the left panel of Fig. 2 computed using LIGO’s Appendix Awepresentsimilarplotsforthel=2,m=2
PSD. If we use white noise we find FF = 0.9922 and mode of the shorter runs with mass ratios 2:1 and 3:2.
=0.9920formassratios1:1and4:1,respectively. Hence- η lm hhNlmR,hlpm4PN−EOBi ∆ϕGW/(2π)
forth, we shall always use high-resolution waveforms for 0.25 22 0.9907 0.030
±
the m1/m2 =4 and 1 cases. 0.24 22 0.9881 0.058
±
For different mass ratios, we show in the right panel 0.22 22 0.9878 0.078
±
of Fig.2 how the FF computed for the dominant modes, 0.16 22 0.9925 0.035
±
usingwhitenoise,dependsontheparameterλ. Basedon 0.16 33 0.9860 0.055
±
this plot we identified λ = 60 as the best model, which 0.16 44 0.9436 0.065
±
0.16 21 0.9092 0.050
we use in the rest of the paper. ±
The p4PN-EOB model has better matching per-
formances at the transition inspiral/merger/ringdown.
TABLEII: Forseveral mass configurations, we list the(min-
However, the introduction of the 4PN order term in-
max)FFobtainedusingwhitenoiseandmaximizing onlyon
evitably affects the inspiral waveform. To understand thetimeofarrivalandinitial phase,andthephasedifference
the differences between the p4PN-EOB model and NR in one GW cycles. For comparison, using the 3.5PN-EOB
and other PN models during the long inspiral phase we model we find hNR,h3.5PN−EOB =0.8718 and = 0.9569 for
h 22 22 i
plot in Fig. 3 the frequency difference between several mass ratios 1 and 4, respectively.
PN-approximants. At the time this paper is written,
preliminary results from Caltech/Cornell group suggest
that the 3PN-Tt3 approximant model fits well with an In Tables II, III and IV we list the FFs and the phase
accurate 3000M-longequal-mass simulation [48], we use difference(inoneGWcycle)betweenseveralp4PN-EOB
this as the fiducial PN-model. The 3.5PN-T, 3PN-Tt3 modes and the NR modes, for white noise and LIGO’s
and3.5Tt1modelsaretheso-calledTaylor-expandedPN PSD, respectively. The dominant frequencies associated
models, widely used in the data-analysis literature (see with each l, m mode are rather similar all along the in-
e.g., Ref. [9, 32, 47]). The plots are obtained by impos- spiral, as seen in the left panel of Fig. 8. However, due
ing that the PN frequencies agree at ωM = 0.017. The to the different frequency of the fundamental QNM [10],
3PN-Tt3 model is an analytic model that uses the PN- the frequencies associated with the l = 2, m = 1 and
expanded phase as function of time. It was also used in l =3, m=2 modes decouple from the other frequencies
Ref. [7] [see Eq. (31) there], where it was found to best- during the transition inpiral(-plunge) to ringdown.
match(togetherwiththe3.5PN-Tmodel)theequal-mass TablesII,IIIshowthattheFFsareratherhighexcept
waveform computed with generalized harmonic coordi- for a few modes, like the l =4, m = 4 mode with mass-
nates by Pretorius. The 3.5PN-T model uses the energy ratio 4:1 [see the left panel in Fig. 6]. In this case we
balance equation re-expanded in powers of the orbital findthatourmatchingprocedureisnotsoefficientinre-
frequency. It was found to best-match the equal-mass producing the amplitude of the NR ringdown waveform.
waveforms in Refs. [7, 8, 9]. The 3.5PN-Tt1 model is a More studies extended to different mass ratios may shed
numericalPNmodelthatsolvestheenergybalanceequa- light on this anomaly. The l = 3, m = 2 mode contains
tion without re-expanding the flux and energy function, a mode mixing between the l = 2, m = 2 and l = 3,
as done in the 3.5PN-T model. From the plots we con- m = 2 modes [7], and the matching procedure that we
clude that while for the equal-mass case the p4PN-EOB adopt does not accurately reproduce it. Better knowl-
model is rather close to our fiducial model (and thus to edge of how these modes are excited during the inspiral
preliminary results fromlong numericalsimulations), we to ringdown transition is required to solve this problem.
cannot draw definite conclusions for generic mass ratios. Finally, because the dominant frequency associatedwith
Indeed, the right panel in Fig. 3 shows that for unequal the l = 2, m = 1 mode departs from the dominant fre-
masses,theclosenessofthePN-approximantsisdifferent quency associated with the l = 2, m = 2 mode quite
than in the equal-mass case. before the merger (see the left panel in Fig. 8), and it
In Figs. 4, 5, and 6 we show the comparison between rises to a much higher QNM frequency, we find that the
the p4PN-EOB and NR waveforms, orbital frequencies, FF associated with the l = 2, m = 1 mode is not very
andphase differencesfor the mostaccurate,longnumer- high. Infact,ourmatchingprocedureisoptimizedtore-
ical simulations, notably the 1:1 and 4:1 cases. We show produce the increase of frequency of the dominantl =2,
results for the l = 2,m = 2 mode, and also for the m=2 and l=3, m=3 modes.
l = 4,m = 4 and l = 2,m = 1 modes. The p4PN- WhiletheFFslistedinTablesII,IIIareobtainedmax-
EOB l = 3,m = 3 mode matches rather well the NR imizing independently over the phase and time of arrival
mode, similarly to the l = 2,m = 2 case, thus we do ofeachl,m mode, the FFs inTable IVarecomputedby
not show it. When the mass-ratio increases the other matching the full waveform containing all-together the
modes are no-longer so subdominant with respect to the leading modes (see Fig. 8). To achieve this we build the
l=2,m=2mode,asseenintherightpanelofFig.8. In numerical and EOB expressions for