Page 1
arXiv:gr-qc/0509129v1 30 Sep 2005
Search for gravitational waves from binary black hole inspirals in LIGO data
B. Abbott,13R. Abbott,13R. Adhikari,13A. Ageev,21,28J. Agresti,13P. Ajith,2B. Allen,41J. Allen,14R. Amin,17
S. B. Anderson,13W. G. Anderson,30M. Araya,13H. Armandula,13M. Ashley,29F. Asiri,13, *P. Aufmuth,32C. Aulbert,1
S. Babak,7R. Balasubramanian,7S. Ballmer,14B. C. Barish,13C. Barker,15D. Barker,15M. Barnes,13, +B. Barr,36
M. A. Barton,13K. Bayer,14R. Beausoleil,27, $K. Belczynski,24R. Bennett,36, #S. J. Berukoff,1, †J. Betzwieser,14B. Bhawal,13
I. A. Bilenko,21G. Billingsley,13E. Black,13K. Blackburn,13L. Blackburn,14B. Bland,15B. Bochner,14, ‡L. Bogue,16
R. Bork,13S. Bose,43P. R. Brady,41V. B. Braginsky,21J. E. Brau,39D. A. Brown,13A. Bullington,27A. Bunkowski,2,32
A. Buonanno,37R. Burgess,14D. Busby,13W. E. Butler,40R. L. Byer,27L. Cadonati,14G. Cagnoli,36J. B. Camp,22
J. Cannizzo,22K. Cannon,41C. A. Cantley,36J. Cao,14L. Cardenas,13K. Carter,16M. M. Casey,36J. Castiglione,35
A. Chandler,13J. Chapsky,13, +P. Charlton,13, §S. Chatterji,13S. Chelkowski,2,32Y. Chen,1V. Chickarmane,17, ¶D. Chin,38
N. Christensen,8D. Churches,7T. Cokelaer,7C. Colacino,34R. Coldwell,35M. Coles,16, ?D. Cook,15T. Corbitt,14D. Coyne,13
J. D. E. Creighton,41T. D. Creighton,13D. R. M. Crooks,36P. Csatorday,14B. J. Cusack,3C. Cutler,1J. Dalrymple,28
E. D’Ambrosio,13K. Danzmann,32,2G. Davies,7E. Daw,17,aD. DeBra,27T. Delker,35, •V. Dergachev,38S. Desai,29
R. DeSalvo,13S. Dhurandhar,12A. Di Credico,28M. D´iaz,30H. Ding,13R. W. P. Drever,4R. J. Dupuis,13J. A. Edlund,13, +
P. Ehrens,13E. J. Elliffe,36T. Etzel,13M. Evans,13T. Evans,16S. Fairhurst,41C. Fallnich,32D. Farnham,13M. M. Fejer,27
T. Findley,26M. Fine,13L. S. Finn,29K. Y. Franzen,35A. Freise,2, ⋆R. Frey,39P. Fritschel,14V. V. Frolov,16M. Fyffe,16
K. S. Ganezer,5J. Garofoli,15J. A. Giaime,17A. Gillespie,13, ♣K. Goda,14L. Goggin,13G. Gonz´ alez,17S. Goßler,32
P. Grandcl´ ement,24, ♠A. Grant,36C. Gray,15A. M. Gretarsson,10D. Grimmett,13H. Grote,2S. Grunewald,1M. Guenther,15
E. Gustafson,27, &&R. Gustafson,38W. O. Hamilton,17M. Hammond,16C. Hanna,17J. Hanson,16C. Hardham,27J. Harms,20
G. Harry,14A. Hartunian,13J. Heefner,13Y. Hefetz,14G. Heinzel,2I. S. Heng,32M. Hennessy,27N. Hepler,29A. Heptonstall,36
M. Heurs,32M. Hewitson,2S. Hild,2N. Hindman,15P. Hoang,13J. Hough,36M. Hrynevych,13,W. Hua,27M. Ito,39
Y. Itoh,1A. Ivanov,13O. Jennrich,36, **B. Johnson,15W. W. Johnson,17W. R. Johnston,30D. I. Jones,29G. Jones,7
L. Jones,13D. Jungwirth,13, ++V. Kalogera,24E. Katsavounidis,14K. Kawabe,15S. Kawamura,23W. Kells,13J. Kern,16, $$
A. Khan,16S. Killbourn,36C. J. Killow,36C. Kim,24C. King,13P. King,13S. Klimenko,35S. Koranda,41K. K¨ otter,32
J. Kovalik,16, +D. Kozak,13B. Krishnan,1M. Landry,15J. Langdale,16B. Lantz,27R. Lawrence,14A. Lazzarini,13M. Lei,13
I. Leonor,39K. Libbrecht,13A. Libson,8P. Lindquist,13S. Liu,13J. Logan,13, ##M. Lormand,16M. Lubinski,15H. L¨ uck,32,2
M. Luna,33T. T. Lyons,13, ##B. Machenschalk,1M. MacInnis,14M. Mageswaran,13K. Mailand,13W. Majid,13, +
M. Malec,2,32V. Mandic,13F. Mann,13A. Marin,14, ††S. M´ arka,9E. Maros,13J. Mason,13, ‡‡K. Mason,14O. Matherny,15
L. Matone,9N. Mavalvala,14R. McCarthy,15D. E. McClelland,3M. McHugh,19J. W. C. McNabb,29A. Melissinos,40
G. Mendell,15R. A. Mercer,34S. Meshkov,13E. Messaritaki,41C. Messenger,34E. Mikhailov,14S. Mitra,12V. P. Mitrofanov,21
G. Mitselmakher,35R. Mittleman,14O. Miyakawa,13S. Miyoki,13, §§S. Mohanty,30G. Moreno,15K. Mossavi,2G. Mueller,35
S. Mukherjee,30P. Murray,36E. Myers,42J. Myers,15S. Nagano,2T. Nash,13R. Nayak,12G. Newton,36F. Nocera,13
J. S. Noel,43P. Nutzman,24T. Olson,25B. O’Reilly,16D. J. Ottaway,14A. Ottewill,41, ¶¶D. Ouimette,13, ++H. Overmier,16
B. J. Owen,29Y. Pan,6M. A. Papa,1V. Parameshwaraiah,15C. Parameswariah,16M. Pedraza,13S. Penn,11M. Pitkin,36
M. Plissi,36R. Prix,1V. Quetschke,35F. Raab,15H. Radkins,15R. Rahkola,39M. Rakhmanov,35S. R. Rao,13K. Rawlins,14, ??
S. Ray-Majumder,41V. Re,34D. Redding,13, +M. W. Regehr,13, +T. Regimbau,7S. Reid,36K. T. Reilly,13K. Reithmaier,13
D. H. Reitze,35S. Richman,14,aaR. Riesen,16K. Riles,38B. Rivera,15A. Rizzi,16, ••D. I. Robertson,36N. A. Robertson,27,36
C. Robinson,7L. Robison,13S. Roddy,16A. Rodriguez,17J. Rollins,9J. D. Romano,7J. Romie,13H. Rong,35, ♣D. Rose,13
E. Rotthoff,29S. Rowan,36A. R¨ udiger,2L. Ruet,14P. Russell,13K. Ryan,15I. Salzman,13V. Sandberg,15G. H. Sanders,13, ***
V. Sannibale,13P. Sarin,14B. Sathyaprakash,7P. R. Saulson,28R. Savage,15A. Sazonov,35R. Schilling,2K. Schlaufman,29
V. Schmidt,13, + + +R. Schnabel,20R. Schofield,39B. F. Schutz,1,7P. Schwinberg,15S. M. Scott,3S. E. Seader,43
A. C. Searle,3B. Sears,13S. Seel,13F. Seifert,20D. Sellers,16A. S. Sengupta,12C. A. Shapiro,29, $$$P. Shawhan,13
D. H. Shoemaker,14Q. Z. Shu,35, ##A. Sibley,16X. Siemens,41L. Sievers,13, +D. Sigg,15A. M. Sintes,1,33J. R. Smith,2
M. Smith,14M. R. Smith,13P. H. Sneddon,36R. Spero,13, +O. Spjeld,16G. Stapfer,16D. Steussy,8K. A. Strain,36
D. Strom,39A. Stuver,29T. Summerscales,29M. C. Sumner,13M. Sung,17P. J. Sutton,13J. Sylvestre,13, ††A. Takamori,13, ‡‡
D. B. Tanner,35H. Tariq,13M. Tarallo,13I. Taylor,7R. Taylor,36R. Taylor,13K. A. Thorne,29K. S. Thorne,6M. Tibbits,29
S. Tilav,13, §§§M. Tinto,4, +K. V. Tokmakov,21C. Torres,30C. Torrie,13G. Traylor,16W. Tyler,13D. Ugolini,31C. Ungarelli,34
M. Vallisneri,6, ¶¶¶M. van Putten,14S. Vass,13A. Vecchio,34J. Veitch,36C. Vorvick,15S. P. Vyachanin,21L. Wallace,13
H. Walther,20H. Ward,36R. Ward,13B. Ware,13, +K. Watts,16D. Webber,13A. Weidner,20,2U. Weiland,32A. Weinstein,13
R. Weiss,14H. Welling,32L. Wen,1S. Wen,17K. Wette,3J. T. Whelan,19S. E. Whitcomb,13B. F. Whiting,35S. Wiley,5
C. Wilkinson,15P. A. Willems,13P. R. Williams,1, ???R. Williams,4B. Willke,32,2A. Wilson,13B. J. Winjum,29, †W. Winkler,2
S. Wise,35A. G. Wiseman,41G. Woan,36D. Woods,41R. Wooley,16J. Worden,15W. Wu,35I. Yakushin,16H. Yamamoto,13
S. Yoshida,26K. D. Zaleski,29M. Zanolin,14I. Zawischa,32,aaaL. Zhang,13R. Zhu,1N. Zotov,18M. Zucker,16and J. Zweizig13
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(The LIGO Scientific Collaboration, http://www.ligo.org)
1Albert-Einstein-Institut, Max-Planck-Institut f¨ ur Gravitationsphysik, D-14476 Golm, Germany
2Albert-Einstein-Institut, Max-Planck-Institut f¨ ur Gravitationsphysik, D-30167 Hannover, Germany
3Australian National University, Canberra, 0200, Australia
4California Institute of Technology, Pasadena, CA 91125, USA
5California State University Dominguez Hills, Carson, CA 90747, USA
6Caltech-CaRT, Pasadena, CA 91125, USA
7Cardiff University, Cardiff, CF2 3YB, United Kingdom
8Carleton College, Northfield, MN 55057, USA
9Columbia University, New York, NY 10027, USA
10Embry-Riddle Aeronautical University, Prescott, AZ 86301 USA
11Hobart and William Smith Colleges, Geneva, NY 14456, USA
12Inter-University Centre for Astronomy and Astrophysics, Pune - 411007, India
13LIGO - California Institute of Technology, Pasadena, CA 91125, USA
14LIGO - Massachusetts Institute of Technology, Cambridge, MA 02139, USA
15LIGO Hanford Observatory, Richland, WA 99352, USA
16LIGO Livingston Observatory, Livingston, LA 70754, USA
17Louisiana State University, Baton Rouge, LA 70803, USA
18Louisiana Tech University, Ruston, LA 71272, USA
19Loyola University, New Orleans, LA 70118, USA
20Max Planck Institut f¨ ur Quantenoptik, D-85748, Garching, Germany
21Moscow State University, Moscow, 119992, Russia
22NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
23National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
24Northwestern University, Evanston, IL 60208, USA
25Salish Kootenai College, Pablo, MT 59855, USA
26Southeastern Louisiana University, Hammond, LA 70402, USA
27Stanford University, Stanford, CA 94305, USA
28Syracuse University, Syracuse, NY 13244, USA
29The Pennsylvania State University, University Park, PA 16802, USA
30The University of Texas at Brownsville and Texas Southmost College, Brownsville, TX 78520, USA
31Trinity University, San Antonio, TX 78212, USA
32Universit¨ at Hannover, D-30167 Hannover, Germany
33Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
34University of Birmingham, Birmingham, B15 2TT, United Kingdom
35University of Florida, Gainesville, FL 32611, USA
36University of Glasgow, Glasgow, G12 8QQ, United Kingdom
37University of Maryland, College Park, MD 20742 USA
38University of Michigan, Ann Arbor, MI 48109, USA
39University of Oregon, Eugene, OR 97403, USA
40University of Rochester, Rochester, NY 14627, USA
41University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
42Vassar College, Poughkeepsie, NY 12604
43Washington State University, Pullman, WA 99164, USA
( RCS ; compiled 27 April 2009)
We report on a search for gravitational waves from binary black hole inspirals in the data from the second
sciencerunof theLIGOinterferometers. Thesearch focused onbinary systemswithcomponent masses between
3 and 20 M⊙. Optimally oriented binaries with distances up to 1 Mpc could be detected with efficiency of at
least 90%. We found no events that could be identified as gravitational waves in the 385.6 hours of data that we
searched.
PACS numbers: 95.85.Sz, 04.80.Nn, 07.05.Kf, 97.80.–d
*Currently at Stanford Linear Accelerator Center
+Currently at Jet Propulsion Laboratory
$Permanent Address: HP Laboratories
#Currently at Rutherford Appleton Laboratory
†Currently at University of California, Los Angeles
‡Currently at Hofstra University
§Currently at Charles Sturt University, Australia
¶Currently at Keck Graduate Institute
?Currently at National Science Foundation
aCurrently at University of Sheffield
•Currently at Ball Aerospace Corporation
⋆Currently at European Gravitational Observatory
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3
I.INTRODUCTION
The Laser Interferometric Gravitational Wave Observatory
(LIGO) [1] consists of three Fabry-Perot-Michelson interfer-
ometers, which are sensitive to the minute changes that would
beinducedin therelativelengthsoftheirorthogonalarmsbya
passing gravitational wave. These interferometers are nearing
theendof theircommissioningphaseandwereclose to design
sensitivity as of March 2005. During the four science runs
that have been completed until now (first (S1) during 2002,
second (S2) and third (S3) during 2003 and fourth (S4) dur-
ing 2005)all three LIGO interferometerswere operatedstably
and in coincidence. Although these science runs were per-
formed during the commissioning phase they each represent
the best broad-bandsensitivity to gravitational waves that had
been achieved up to that date.
In this paper we report the results of a search for gravi-
tational waves from the inspiral phase of stellar mass binary
black hole (BBH) systems, using the data from the second
science run of the LIGO interferometers. These BBH sys-
tems are expected to emit gravitational waves at frequencies
detectable by LIGO during the final stages of inspiral (decay
of the orbit due to energy radiated as gravitational waves),
the merger (rapid infall) and the subsequent ringdown of the
quasi-normal modes of the resulting single black hole.
The rate of BBH coalescences in the Universe is highly un-
certain. In contrast to searches for gravitational waves from
the inspiral phase of binary neutron star (BNS) systems [2],
it is not possible to set a reliable upper limit on astrophys-
ical BBH coalescences. That is because the distribution of
the sources in space, in the component mass space and in the
♣Currently at Intel Corp.
♠Currently at University of Tours, France
&&Currently at Lightconnect Inc.
Currently at W.M. Keck Observatory
**Currently at ESA Science and Technology Center
++Currently at Raytheon Corporation
$$Currently at New Mexico Institute of Mining and Technology / Magdalena
Ridge Observatory Interferometer
##Currently at Mission Research Corporation
††Currently at Harvard University
‡‡Currently at Lockheed-Martin Corporation
§§Permanent Address: University of Tokyo, Institute for Cosmic Ray Re-
search
¶¶Permanent Address: University College Dublin
??Currently at University of Alaska Anchorage
aaCurrently at Research Electro-Optics Inc.
••Currently at Institute of Advanced Physics, Baton Rouge, LA
***Currently at Thirty Meter Telescope Project at Caltech
+ + +Currently at European Commission, DG Research, Brussels, Belgium
$$$Currently at University of Chicago
##Currently at LightBit Corporation
††Permanent Address: IBM Canada Ltd.
‡‡Currently at The University of Tokyo
§§§Currently at University of Delaware
¶¶¶Permanent Address: Jet Propulsion Laboratory
???Currently at Shanghai Astronomical Observatory
aaaCurrently at Laser Zentrum Hannover
spin angular momentum space is not reliably known. Addi-
tionally, the gravitational waveforms for the inspiral phase of
stellar-mass BBH systems which mergein the frequencyband
oftheLIGOinterferometersarenotknownwithprecision. We
perform a search that aims at detection of BBH inspirals. In
the absence of a detection, we use a specific nominal model
for the BBH population in the Universe and the gravitational
waveforms given in the literature to calculate an upper limit
for the rate of BBH coalescences.
The rest of the paper is organized as follows. Sec. II pro-
vides a short description of the data that was used for the
search. In Sec. III we discuss the target sources of the search
and we explain the motivation for using a family of phe-
nomenological templates to search the data. In Sec. IV we
give a detailed discussion of the templates and the filtering
methods. In Sec. VA we provide information on various data
quality checks that we performed, in Sec. VB we describe in
detail the analysis method that we used and in Sec. VC we
provide details on the parameter tuning. In Sec. VI we de-
scribe the estimation of the background and in Sec. VII we
present the results of the search. We finally show the calcula-
tion of the rate upper limit on BBH coalescences in Sec. VIII
and we provide a brief summary of the results in Sec. IX.
II.DATA SAMPLE
Duringthesecondsciencerun,thethreeLIGOinterferome-
ters were operating in science mode (see Sec. VA). The three
interferometersare based at two observatories. We referto the
observatory at Livingston, LA, as LLO and the observatory at
Hanford, WA as LHO. A total of 536 hours of data from the
LLO 4 km interferometer (hereafter L1), 1044 hours of data
from the LHO 4 km (hereafter H1) interferometer, and 822
hours of data from the LHO 2 km (hereafter H2) interferom-
eter was obtained. The data was subjected to several quality
checks. In this search, we used only data from times when
the L1 interferometerwas runningin coincidencewith at least
one of H1 and H2, and we only used continuous data of dura-
tion longer than 2048 s (see Sec. VB). After the data quality
cuts, there was a total of 101.7 hours of L1-H1 double coinci-
dent data (when both L1 and H1 but not H2 were operating),
33.3 hours of L1-H2 double coincident data (when both L1
and H2 but not H1 were operating) and 250.6 hours of L1-
H1-H2 triple coincident data (when all three interferometers
were operating)from the S2 data set, fora total of 385.6hours
of data.
A fraction (approximately 9%) of this data (chosen to be
representativeof the wholerun)was set aside as “playground”
data where the various parameters of the analysis could be
tunedandwherevetoes effectivein eliminatingspuriousnoise
events could be identified. The fact that the tuning was per-
formedusing this subset of data does not excludethe possibil-
ity that a detection could be made in this subset. However, to
avoid biasing the upper limit, those times were excluded from
the upper limit calculation.
As with earlier analyses of LIGO data, the output of the an-
tisymmetricportofthe interferometerwas calibratedto obtain
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4
a measure of the relative strain ∆L/L of the interferometer
arms, where ∆L = Lx− Lyis the difference in length be-
tweenthexarmandthey armandLis theaveragearmlength.
Thecalibrationwas measuredbyapplyingknownforcestothe
end mirrors of the interferometers before, after and occasion-
ally during the science run. In the frequency band between
100Hz and1500Hz, the calibrationaccuracywas within10%
in amplitude and 10◦of phase.
III.TARGET SOURCES
The target sources for the search described in this paper
are binary systems that consist of two black holes with com-
ponent masses between 3 and 20 M⊙, in the last seconds
before coalescence. Coalescences of binary systems consist
of three phases: the inspiral, the merger and the ringdown.
We performed the search by matched filtering the data us-
ing templates for the inspiral phase of the evolution of the
binaries. The exact duration of the inspiral signal depends on
the masses of the binary. Given the low-frequency cutoff of
100 Hz that needed to be imposed on the data (see Sec. VB)
the expected duration of the inspiral signals in the S2 LIGO
band as predicted by post-Newtonian calculations varies from
0.607 s for a 3 − 3 M⊙binary to 0.013 s for a 20 − 20 M⊙
binary.
The gravitational wave signal is dominated by the merger
phase which potentially may be computed using numerical
solutions to Einstein’s equations. Searching exclusively for
the merger using matched-filter techniques is not appropri-
ate until the merger waveforms are known. BBH mergers are
usually searched for by using techniques developed for de-
tection of unmodeled gravitational wave bursts [3]. However,
for reasons that will be explained below, it is possible that the
search described in this paper was also sensitive to at least
part of the merger of the BBH systems of interest. Certain
re-summation techniques have been applied to model the late
time evolution of BBH systems which makes it possible to
evolve those systems beyond the inspiral and into the merger
phase [4, 5, 6, 7, 8, 9, 10] and the templates that we used
for matched filtering incorporate the early merger features (in
addition to the inspiral phase) of those waveforms.
The frequencies of the ringdown radiation from BBH sys-
tems with component masses between 3 and 20 M⊙range
from 295 Hz to 1966 Hz [11, 12, 13] and the gravitational
wave forms are known. Based on the frequencies of these sig-
nals, some of the signals are in the S2 LIGO frequency band
of goodsensitivity and some are not. At the time of the search
presented in this paper, the matched-filtering tools necessary
to search for the ringdown phase of BBH were being devel-
oped. In future searches we will look for ringdown signals
associated with inspiral candidates.
Finally, we have verified through simulations that the pres-
ence of the merger and the ringdown phases of the gravita-
tional wave signal in the data does not degrade our ability to
detect the inspiral phase, when we use matched filter tech-
niques.
A. Characteristics of BNS and BBH inspirals
Weusethestandardconventionc = G = 1intheremainder
of this paper.
The standard approach to solving the BBH evolution prob-
lem uses the post-Newtonian (PN) expansion [10] of the Ein-
stein equations to compute the binding energyE of the binary
and the flux F of the radiation at infinity, both as series ex-
pansions in the invariant velocity v (or the orbital frequency)
of the system. This is supplemented with the energy balance
equation (dE/dt = −F) which in turn gives the evolution
of the orbital phase and hence the gravitational wave phase
which, to the dominant order, is twice the orbital phase. This
methodworkswell whenthevelocitiesinthesystem aremuch
smaller compared to the speed of light, v ≪ 1. Moreover, the
post-Newtonian expansion is now complete to order v7giv-
ing us the dynamics and orbital phasing to a high accuracy
[14, 15]. Whether the waveform predicted by the model to
such high orders in the post-Newtonian expansion is reliable
for use as a matched filter depends on how relativistic the sys-
tem is in the LIGO band. For the second science run of LIGO,
the interferometershadverygoodsensitivity between100and
800 Hz so we calculate how relativistic BNS and BBH sys-
tems are at those two frequencies.
The velocity in a binary system of total mass M is related
to the frequency f of the gravitational waves by
v = (πMf)1/3.
(1)
When a BNS system that consists of two 1.4M⊙ compo-
nents enters the S2 LIGO band, the velocity in the system is
v ≃ 0.16; when it leaves the S2 LIGO band at 800 Hz, it is
v ≃ 0.33 and the system is mildly relativistic. Thus, relativis-
tic corrections are not too important for the inspiral phase of
BNS.
BBH systems of high mass, however, would be quite rela-
tivisticintheS2 LIGOband. Forinstance, whena10−10M⊙
BBH enters the S2 LIGO band the velocity would be v ≃
0.31. At a frequency of 200 Hz (smaller than the frequency
of the innermost stable circular orbit, explained below, which
is 220 Hz according to the test-mass approximation) the ve-
locity would be v ≃ 0.40. Such a binary is expected to
merge producing gravitational waves within the LIGO fre-
quency band. Therefore LIGO would observe BBH systems
in the most non-linear regime of their evolution and thereby
witness highly relativistic phenomena for which the perturba-
tive expansion is unreliable.
Numerical relativity is not yet in a position to fully solve
the late time phasing of BBH systems. For this reason, in
recent years, (non-perturbative) analytical resummation tech-
niques of the post-Newtonian series have been developed to
speed up its convergence and obtain information on the late
stages of the inspiral and the merger [16]. These resummation
techniqueshavebeenappliedto the post-Newtonianexpanded
conservative and non-conservative part of the dynamics and
are called effective-one-body (EOB) and P-approximants
(also referredto as Pad´ e approximants)[4, 5, 6, 7, 8, 9]. Some
insights into the merger problem have been also provided in
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5
[17, 18] by combiningnumericaland perturbativeapproxima-
tion schemes.
The amplitude and the phase of the standard post-
Newtonian(TaylorT3,[16]), EOB, andPad´ e waveforms,eval-
uated at different post-Newtonian orders, differ from each
other in the last stages of inspiral, close to the innermost sta-
ble circular orbit (ISCO, [16]). The TaylorT3 and Pad´ e wave-
forms are derived assuming that the two black holes move
along a quasi-stationary sequence of circular orbits. The EOB
waveforms, extending beyond the ISCO, contain features of
the merger dynamics. All those model-based waveforms are
characterized by different ending frequencies. For the quasi-
stationary two-body models the ending frequency is deter-
mined by the minimum of the energy. For the models that
extend beyond the ISCO, the ending frequency is fixed by the
light-ring [6, 19] of the two-body dynamics.
We could construct matched filters using waveforms from
each of these families to search for BBH inspirals but yet the
true gravitationalwave signal mightbe “in between” the mod-
els we search for. In order not to miss the true gravitational
wave signal it is desirable to search a space that encompasses
all the different families and to also search the space “in be-
tween” them.
B. Scope of the search
Recent work by Buonanno, Chen and Vallisneri [19] (here-
after BCV) has unified the different approximation schemes
into one family of phenomenologicalwaveforms by introduc-
ing two new parameters, one of which is an amplitude cor-
rection factor and the other a variable frequency cutoff, in
order to model the different post-Newtonian approximations
and their variations. Additionally, in order to achieve high
signal-matching performance, they introduced unphysical pa-
rameters in the phase evolution of the waveform.
In this work we used a specific implementation of the phe-
nomenological templates. As these phenomenological wave-
forms are not guaranteed to have a good overlap with the true
gravitational wave signal it is less meaningful to set upper
limits on either the strength of gravitational waves observed
during our search or on the coalescence rate of BBH in the
Universe than it was for the BNS search in the S2 data [2].
However, in order to give an interpretation of the result of
our search, we did calculate an upper limit on the coalescence
rate of BBH systems, based on two assumptions: (1) that the
model-based waveforms that exist in the literature have good
overlap with a true gravitational wave signal and (2) that the
phenomenological templates used have a good overlap with
the majorityof the model-basedBBH inspiral waveformspro-
posed in the literature [19].
To set the stage for later discussion we plot in Fig. 1 the
distance at which a binary of two components of equal mass
that is optimally oriented (positioned directly above the in-
terferometer and with its orbital plane perpendicular to the
line of sight from the interferometer to the binary) would pro-
duce a signal-to-noise ratio (SNR, see Sec. IV) of 8 in the
LIGO interferometers during the second science run. We re-
fertothis distanceas “range”oftheinterferometers. Thesolid
line shows the range of the LIGO interferometersfor matched
filtering performed with the standard post-Newtonian (Tay-
lorT3) waveforms, which predict the evolution of the system
up to the ISCO [16], at a gravitational wave frequency of
fGW ∼ 110(M/40M⊙)−1Hz. The dashed line shows the
range of the interferometers for matched filtering performed
with the EOB waveforms, which predict the evolution of the
system up to the light ring orbit [6, 19], at a gravitational
wave frequencyof fGW∼ 218(M/40M⊙)−1Hz (notice that
both these equations for fGWare for binaries of equal com-
ponent masses). Since the EOB waveforms extend beyond
the ISCO, they have longer duration and greater energy in the
LIGO band which explains why the range for the EOB wave-
forms is greater than the rangefor the TaylorT3or Pad´ e wave-
forms(calculationsperformedwith thePad´ ewaveformsresult
in ranges similar to those given by the TaylorT3 waveforms).
10 2050
10
0
10
1
Range (Mpc)
L1−EOB
L1−TaylorT3
1020 50
10
0
10
1
Range (Mpc)
H1−EOB
H1−TaylorT3
510 20 50100
10
0
10
1
Total Mass (Mo)
Range (Mpc)
H2−EOB
H2−TaylorT3
FIG. 1: Range (distance at which an optimally oriented inspiraling
binary of given total mass would produce a signal-to-noise ratio of 8)
of the LIGO interferometers during S2. The error bars are calculated
from the fluctuations of the noise in the LIGO interferometers during
S2.
During S2 the L1 interferometer was the most sensitive
with a range of 7 Mpc for a 10 − 10M⊙binary (calculated
using the TaylorT3 waveform). However, since for the search
describedin this paperwe demandedthat ourcandidateevents
are seen in coincidence between the two LIGO observatories
(as described in Sec. VB) the overall range of the search was
determinedbythe less sensitiveLHO interferometersandthus
was smaller than this maximum.
IV.FILTERING
A.Detection template family
As was mentioned in Sec. III, the gravitational wave sig-
nal from inspiraling black hole binaries of high masses enters
the LIGO frequency sensitivity band in the later stages, when
the post-Newtonian approximation is beginning to lose valid-
Page 6
6
ity and different versions of the approximation are beginning
to substantially differ from each other. In order to detect these
inspiral signals we need to use filters based on phenomeno-
logical waveforms (instead of model-based waveforms) that
cover the function space spanned by different versions of the
late-inspiral post-Newtonian approximation.
It must be emphasized at this point that black hole binaries
with small component masses (corresponding to total mass
up to 10 M⊙) enter the S2 LIGO sensitivity band at an early
enough stage of the inspiral that the signal can be adequately
approximated by the stationary phase approximation to the
standard post-Newtonian approximation. For those binaries
it is not necessary to use phenomenological templates for the
matched filtering; the standard post-Newtonian waveforms
can be used as in the search for BNS inspirals. However, us-
ing the phenomenological waveforms for those binaries does
not limit the efficiency of the search [19]. In this search, in or-
der to treat all black hole binaries uniformly, we chose to use
the BCV templates with parameters that span the component
mass range from 3 to 20 M⊙.
The phenomenological templates introduced in [19] match
very well most physicalwaveformmodels that have been sug-
gested in the literature for BBH coalescences. Even though
they are not derived by calculations based on a specific phys-
ical model they are inspired by the standard post-Newtonian
inspiral waveforms. In the frequency domain, they are
˜h(f) ≡ A(f)eiψ(f), f > 0,
(2)
where the amplitude A(f) is
A(f) ≡ f−7/6?
1 − αf2/3?
θ(fcut− f)
(3)
and the phase ψ(f) is
ψ(f) ≡ φ0+ 2πft0+ f−5/3
∞
?
n=0
fn/3ψn.
(4)
In Eq. (3) θ is the Heaviside step function and in Eq. (4) t0
and φ0are offsets on the time of arrival and on the phase of
the signal respectively. Also, α, fcutand ψnare parameters
of the phenomenologicalwaveforms.
Two components can be identified in the amplitude part
of the BCV templates. The f−7/6term comes from the
restricted-Newtonian amplitude in the Stationary Phase Ap-
proximation (SPA) [20, 21, 22]. The term αf2/3× f−7/6=
αf−1/2is introduced to capture any post-Newtonian ampli-
tude corrections and to give high overlaps between the BCV
templates and the various models that evolve the binary past
the ISCO frequency. Additionally, in order to obtain high
matches with the various post-Newtonian models that predict
different terminating frequencies, a cutoff frequency fcutis
imposed to terminate the waveform.
It has beenshown[19] that in orderto achievehighmatches
with the various model-derived BBH inspiral waveforms it is
in fact sufficient to use only the parameters ψ0and ψ3in the
phase expression in Eq. (4), if those two parameters are al-
lowed to take unphysical values. Thus, we set all other ψn
coefficients equal to 0 and simplify the phase to
ψ(f) = φ0+ 2πft0+ f−5/3(ψ0+ ψ3f)
≡ φ0+ ψs(f)
(5)
(6)
where the subscript s stands for “simplified”.
For the filtering of the data, a bank of BCV templates was
constructedoverthe parametersfcut, ψ0andψ3(intrinsictem-
plate parameters). For details on how the templates in the
bank were chosen see Sec. VB. For each template, the signal-
to-noise ratio (defined in Sec. IVB) is maximized over the
parameters t0, φ0and α (extrinsic template parameters).
B.Filtering and signal-to-noise ratio maximization
For a signal s, the signal-to-noise ratio (SNR) resulting
from matched-filtering with a template h is
ρ(h) =
?s,h?
??h,h?,
(7)
with the inner product ?s,h? being
?s,h? = 2
?∞
−∞
˜ s(f)˜h∗(f)
Sh(|f|)
df = 4ℜ
?∞
0
˜ s(f)˜h∗(f)
Sh(f)
df (8)
and Sh(f) being the one-sided noise power spectral density.
Various manipulations (given in detail in App. A) give the
expressionfortheSNR (maximizedovertheextrinsicparame-
ters φ0, α andt0) that was usedin this search. Thatexpression
is
ρmaximized =1
2
?|F1|2+ |F2|2+ 2ℑ(F1F∗
?|F1|2+ |F2|2− 2ℑ(F1F∗
2)
(9)
+1
2
2).
where
F1=
?fcut
0
4˜ s(f)a1f−7
Sh(f)
6
e−iψs(f)df
(10)
F2=
?fcut
0
4˜ s(f)(b1f−7
6 + b2f−1
Sh(f)
2)
e−iψs(f)df.
(11)
Thequantitiesa1, b1andb2aredependentonthenoiseandthe
cutoff frequency fcutand are defined in App. A. The original
suggestion of Buonanno, Chen and Vallisneri was that for the
SNR maximization over the parameter α the values of (α ×
f2/3
that will be explained in Sec. VC1. However, in order to be
able to perform various investigations on the values of α we
leave its value unconstrained in this maximization procedure.
More details on this can be found in Sec. VC1.
cut) should be restricted within the range [0, 1], for reasons
Page 7
7
V.SEARCH FOR EVENTS
A. Data quality and veto study
The matched filtering algorithm is optimal for data with
a known calibrated noise spectrum that is Gaussian and
stationary over the time scale of the data blocks analyzed
(2048 s, described in Section VB), which requires stable,
well-characterized interferometer performance. In practice,
the performanceis influenced by non-stationaryoptical align-
ment, servo control settings, and environmental conditions.
We used two strategies to avoid problematic data. The first
strategy was to evaluate data quality over relatively long time
intervals using several different tests. As in the BNS search,
time intervals identified as being unsuitable for analysis were
skipped when filtering the data. The second strategy was to
look for signatures in environmental monitoring channels and
auxiliary interferometer channels that would indicate an envi-
ronmental disturbance or instrumental transient, allowing us
to veto any candidate events recorded at that time.
The most promising candidate for a veto channel was
L1:LSC-POB I (hereafterreferredto as “POBI”), an auxiliary
channel measuring signals proportional to the length fluctua-
tions of the power recycling cavity. This channel was found
to have highly variable noise at 70 Hz which coupled into the
gravitational wave channel. Transients found in this channel
were used as vetoes for the BNS search in the S2 data [2].
Hardware injections of simulated inspiral signals [23] were
used to prove that signals in POBI would not veto true inspi-
ral gravitational waves present in the data.
Investigations showed that using the correlations between
POBI and the gravitational wave channel to veto candidate
eventswouldbeless efficaciousthanit was inthe BNS search.
ThereforePOBI was not used a an a-priori veto. However, the
fact that correlations were proven to exist between the POBI
signals and the BBH inspiral signals made it worthwhile to
follow-up the BBH inspiral events that resulted at the end of
our analysis and check if they were correlated with POBI sig-
nals (see Sec. VIIB).
As in the BNS search in the S2 data, no instrumentalvetoes
were found for H1 and H2. A more extensive discussion of
the LIGO S2 binary inspiral veto studies can be found in [24].
B. Analysis Pipeline
In order to increase the confidence that a candidate event
coming out of our analysis is a true gravitationalwave and not
due to environmental or instrumental noise we demanded the
candidate event to be present in the L1 interferometer and at
least one of the LHO interferometers. Such an event would
then be characterized as a potential inspiral event and be sub-
ject to thorough examination.
The analysis pipeline that was used to perform the BNS
search (and was described in detail in [2]) was the starting
structure for constructing the pipeline used in the BBH inspi-
ral search described in this paper. However, due to the dif-
ferent nature of the search, the details of some components of
the pipeline needed to be modified. In order to highlight the
differences of the two pipelines and to explain the reasons for
those, we describe our pipeline below.
First, various data quality cuts were applied on the data and
the segments of gooddata for each interferometerwere inden-
tified. The times when each interferometer was in stable op-
eration (called science segments) were used to construct three
data sets corresponding to: (1) times when all three interfer-
ometerswere operating(L1-H1-H2triplecoincidentdata),(2)
times when only the L1 and H1 (and not the H2) interfer-
ometers were operating (L1-H1 double coincident data) and
(3) times when only the L1 and H2 (and not the H1) interfer-
ometers were operating (L1-H2 double coincident data). The
analysis pipeline produced a list of coincident triggers (times
and template parameters for which the SNR thresholdwas ex-
ceeded and all cuts mentioned below were passed) for each of
the three data sets.
Thescience segmentswere analyzedin blocksof2048s us-
ing the FINDCHIRP implementation [25] of matched filtering
for inspiral signals in the LIGO Algorithm library [26]. The
original version of FINDCHIRP had been coded for the BNS
search and thus had to be modified to allow filtering of the
data with the BCV templates described in Sec. IV.
The data for each 2048 s block was first down-sampled
from 16384 Hz to 4096 Hz. It was subsequently high-pass
filtered at 90 Hz in the time domain and a low frequency cut-
off of 100 Hz was imposed in the frequency domain. The
instrumental response for the block was calculated using the
average value of the calibration (measured every minute) over
the duration of the block.
The breaking up of each segment for power spectrum es-
timation and for matched-filtering was identical to the BNS
search [2] and is briefly mentioned here so that the terminol-
ogy is established for the pipeline description that follows.
Triggers were not searched for within the first and last 64 s
of a given block, so subsequent blocks were overlapped by
128 s to ensure that all of the data in a continuousscience seg-
ment (except for the first and last 64 s) was searched. Any
science segments shorter than 2048 s were ignored. If a sci-
ence segment could not be exactly divided into overlapping
blocks (as was usually the case) the remainder of the segment
was covered by a special 2048 s block which overlapped with
the previous block as much as necessary to allow it to reach
the end of the segment. For this final block, a parameter was
set to restrict the inspiral search to the time interval not cov-
ered by any previous block, as shown in Fig. 2.
Each block was further split into 15 analysis segments of
length 256 s overlapped at the beginning and at the end by 64
s. The average power spectrum Sh(f) for the 2048 s of data
was estimated by taking the median of the power spectra of
the 15 segments. We used the median instead of the mean to
avoid biased estimates due to large outliers, producedby non-
stationarydata. Thecalibrationwas appliedto thedatain each
analysis segment.
In order to avoid end-effects due to wraparound of the dis-
crete Fourier transform when performing the matched filter,
the frequency-weighting factor 1/Sh(f) was truncated in the
time domain so that its inverse Fourier transform had a max-
Page 8
8
???????????????
?????????????
????
???????????????????
?????????????
????????????
????????????
FIG.2: Thealgorithmused todivide science segments intodata anal-
ysis segments. Science segments are divided into 2048 sblocks over-
lapped by 128 s. (Science segments shorter than 2048 s are ignored.)
An additional block with a larger overlap is added to cover any re-
maining data at the end of a science segment. Each block is divided
into 15 analysis sesgments of length 256 s for filtering. The first
and last 64 s of each analysis segment are ignored, so the segments
overlap by 128 s. Areas shaded black are searched for triggers by
the search pipeline. The gray area in the last block of the science
segment is not searched for triggers as this time is covered by the
preceding block, although these data points are used in estimating
the noise power spectral density for the final block.
imum duration of ±16 s. The output of the matched filter
near the beginning and end of each segment was corrupted
by end-effects due to the finite duration of the power spec-
trum weighting and the template. By ignoringthe filter output
within 64 s of the beginning and end of each segment, we
ensured that only uncorrupted filter output is searched for in-
spiral triggers. This necessitated the overlapping of segments
and blocks described above.
The single-sided power spectral density (PSD) of the noise
Sh(f) in the L1 interferometer was estimated independently
for each L1 block that was coincident with operation of at
least one LHO interferometer. The PSD was used to construct
a template bank for filtering that block. The bank was con-
structed over the parameters ψ0and ψ3so that there was no
less than 95% overlap (defined in the sense of Eq. (A16) be-
tween two neighboring (in {ψ0, ψ3, fcut} parameter space)
templates, if the value of α of those templates was equal to
0. The ψ0− ψ3space was tiled using a square grid based on
the metric in Eq. (117) of [19]. For each pair of ψ0and ψ3,
three values of the cutoff frequencyfcutwere generated. This
process gave the three intrinsic parameters for each template.
Details on the exact values of the parameters used are given in
Sec. VC1. The number of templates in the bank varied with
the PSD. For this search the number of templates ranged be-
tween 741 and 1296 templates per 2048 s L1 analysis block,
with the average number being 958.
The data from the L1 interferometer for the block was then
matched-filtered, as described previously, against the bank of
templates, with a SNR threshold ρthresh
triggers. As will be explained below, the χ2-veto [27] that
was used for the BNS search was not used in this search. For
each block in the LHO interferometers, a triggered bank was
L
to produce a list of
created consisting of everytemplate that producedat least one
trigger in the L1 data during the time of the LHO block. This
triggered bank was used to matched-filter the data from the
LHOinterferometers. Fortimes whenonlytheH2interferom-
eter was operating in coincidence with L1, the triggered bank
was used to matched-filter the H2 blocks that overlappedwith
L1 data. For all other times, all H1 data that overlapped with
L1 data was matched-filtered using the triggered bank for that
block. For H1 triggers produced during times when all three
interferometers were operating, a second triggered bank was
produced for each H2 block. This triggered bank consisted of
every template which produced at least one trigger found in
coincidence in L1 and H1 during the time of the H2 block.
The H2 block was matched-filtered with this bank.
Before any triggers were tested for coincidence, all triggers
with (α ×f2/3
The reason for this veto will be explained in Sec. VC.
cut) greater than a threshold αthresh
F
were rejected.
For triggers to be considered coincident between two in-
terferometers they had to be observed in both interferometers
within a time window that allowed for the error in measure-
ment of the time of the trigger. If the interferometerswere not
co-located, this parameter was increased by the light travel
time between the two LIGO observatories (10 ms). We then
ensured that the triggers had consistent waveform parameters
by demanding that the two parameters ψ0and ψ3for the tem-
plate were exactly equal to each other.
Triggers that were generated from the triple coincident data
were required to be found in coincidence in the L1 and H1
interferometers. We searched the H2 data from these triple
coincident times but did not reject L1-H1 coincident triggers
that were not found in the H2 data, even if, based on the SNR
observed in H1 they would be expected to be found in H2.
This was a looser rejectionalgorithmthan the one used for the
BNS search [2] and could potentially increase the number of
false alarms in the triple coincident data. The reason for using
this algorithm in the BBH search is explained in Sec. VC2,
where the coincidence parameter tuning is discussed.
The last step of the pipeline was the clustering of the trig-
gers. The clustering is necessary because both large astro-
physical signals and instrumental noise bursts can produce
many triggers with coalescence times within a few ms of each
other and with different template parameters. Triggers sepa-
rated by more than 0.25 s were considered distinct. This time
was approximately half of the duration of the longest signal
that we could detect in this search. We chose the trigger with
the largest combined SNR from each cluster, where the com-
bined SNR is defined in Sec. VI.
To perform the search on the full data set, a Directed
Acyclic Graph (DAG) was constructed to describe the work
flow,andexecutionofthepipelinetaskswas managedbyCon-
dor [28] on the various Beowulf clusters of the LIGO Scien-
tific Collaboration. The software to perform all steps of the
analysis pipeline and construct the DAG is available in the
package LALAPPS [26].
Page 9
9
C.Parameter Tuning
An important part of the analysis was to decide on the val-
ues of the various parameters of the search, such as the SNR
thresholds and the coincidence parameters. The parameters
were chosen so as to compromise between increasing the de-
tection efficiency and lowering the number of false alarms.
The tuning of all the parameters was done by studying the
playground data only. In order to tune the parameters we per-
formed a number of Monte-Carlo simulations, in which we
addedsimulatedBBH inspiralsignals in the data andsearched
forthemwith ourpipeline. Whilewe usedthephenomenolog-
ical detection templates to perform the matched filtering, we
used various model-based waveforms for the simulated sig-
nals that we added in the data. Specifically, we chose to inject
effective-one-body(EOB, [4, 5, 6, 8]), Pad´ e (Pad´ eT1,[4]) and
standard post-Newtonian waveforms (TaylorT3, [16]), all of
second post-Newtonian order. Injecting waveforms from dif-
ferent families allowed us to additionally test the efficiency of
the BCV templates for recovering signals predicted by differ-
ent models.
In contrast to neutron stars, there are no observation-based
predictions about the population of BBH systems in the Uni-
verse. Forthepurposeoftuningthe parametersofourpipeline
we decided to draw the signals to be added in the data from a
population with distances between 10 kpc and 20 Mpc from
the Earth. The random sky positions and orientations of the
binaries resulted in some signals having much larger effec-
tive distances (distance from which the binary would give the
same signal in the data if it were optimally oriented). It was
determined that using a uniform-distance or uniform-volume
distributionfor the binarieswould overpopulatethe largerdis-
tances(forwhichtheLIGOinterferometerswerenotverysen-
sitive during S2) and only give a small number of signals in
the small-distance region, which would be insufficient for the
parameter tuning. For that reason we decided to draw the sig-
nals from a population that was uniform in log(distance). For
the mass distribution, we limited each component mass be-
tween 3 and 20 M⊙. Populations with uniform distribution of
total mass were injected for the tuning part of the analysis.
There were two sets of parameters that we could tune in
thepipeline: thesingle interferometerparameters,whichwere
used in the matched filtering to generate inspiral triggers in
each interferometer, and the parameters which were used to
determine if triggers from different interferometers were co-
incident. The single interferometer parameters that needed to
be tuned were the ranges of values for ψ0and ψ3in the tem-
plate bank, the number of fcutfrequencies for each pair of
{ψ0,ψ3} in the template bank and the SNR threshold ρthresh.
The coincidence parameters were the time coincidence win-
dow for triggers from different interferometers, δt, and the
coincidence window for the template parameters ψ0and ψ3.
Due to the nature of the triggered search pipeline, parameter
tuning was carried out in two stages. We first tuned the sin-
gle interferometer parameters for the primary interferometer
(L1). We then used the triggered template banks (generated
from the L1 triggers) to explore the single interferometer pa-
rameters for the less sensitive LHO interferometers. Finally
the parameters of the coincidence test were tuned.
1. Single interferometer tuning
Based on the playground injection analysis it was de-
termined that the range of values for ψ0 had to be
[10, 550000] Hz5/3and the range of values for ψ3had to be
[−4000, −10]Hz2/3inordertohavehighdetectionefficiency
for binaries of total mass between 6 and 40 M⊙.
Our numerical studies showed that using between 3 and 5
cutoff frequencies per {ψ0, ψ3} pair would yield very high
detection efficiency. Consideration of the computational cost
of the search led us to use 3 cutoff frequencies per pair,
thus reducing the number of templates in each bank by 40%
compared to a template bank with 5 cutoff frequencies per
{ψ0, ψ3} pair.
Our Monte-Carlo simulations showed that, in order to be
able to distinguish an inspiral signal from an instrumental or
environmental noise event in the data, the minimum require-
ment should be that the trigger has SNR of at least 7 in each
interferometer. A threshold of 6 (that was used in the BNS
search) resulted in a very large number of noise triggers that
needlessly complicated the data handling and post-pipeline
processing.
A standard part of the matched-filtering process is the χ2-
veto [27]. The χ2-veto compares the SNR accumulated in
each of a number of frequency bands of equal inspiral tem-
plate power to the expected amount in each band. Gravita-
tional waves from inspiraling binaries give small χ2values
while instrumental artifacts give high χ2values. Thus, the
triggers resulting from instrumental artifacts can be vetoed by
requiring the value of χ2for a trigger to be below a threshold.
The test is very efficient at distinguishing BNS inspiral sig-
nals from loud non-Gaussian noise events in the data and was
used in the BNS inspiral search [2] in the S2 data. However,
we found that the χ2-veto was not suitable for the search for
gravitational waves from BBH inspirals in the S2 data. The
expected short duration, low bandwidth and small number of
cycles in the S2 LIGO frequency band for many of the possi-
ble BBH inspiral signals made such a test unreliable unless a
very high threshold on the values of χ2were to be set. A high
threshold, on the other hand, resulted in only a minimal re-
duction in the numberof noise events picked up. Additionally
the χ2-veto is computationally very costly. We thus decided
to not use it in this search.
As mentioned earlier, the SNR calculated using the BCV
templates was maximized over the template parameter α. For
every value of α, there is a frequency f0for which the ampli-
tude factor (1 − αf2/3) becomes zero:
f0= α−3/2.
(12)
If the value of α associated with a trigger is such that the fre-
quencyf0is greater than the cutoff frequencyfcutof the tem-
plate (and consequentlyα f2/3
behavior of the phenomenological template is as expected for
an inspiral gravitational waveform. If the value of α is such
cut≤ 1), then the high-frequency
Page 10
10
0.4
αF = 1
0.5 0.60.7 0.80.91
0
αF = 0
0.4
αF = 20.50.60.70.80.91
0
0.3 0.40.50.60.70.8 0.91
0
time (s)
FIG. 3: Time domain plots of BCV waveforms for different values
of αF. The top plot is for αF = 0, the middle is for αF = 1 and the
bottom is for αF = 2. For all three waveforms ψ0 = 150000 Hz5/3,
ψ3 = −1500 Hz2/3and fcut = 500 Hz. It can be seen that the
behavior is not that of a typical inspiral waveform for αF = 2.
that f0is smaller than the cutoff frequency fcut(and conse-
quently α f2/3
ical waveform becomes zero before the cutoff frequency is
reached. For such a waveform, the high-frequency behavior
does not resemble that of a typical inspiral gravitationalwave-
form. For simplicity we define
cut > 1), the amplitude of the phenomenolog-
αF ≡ αf2/3
cut.
(13)
The behaviorof the BCV waveformsfor three differentvalues
of αFis shown in Fig. 3, where it can be seen that for the case
of αF = 2 the amplitude becomes zero and then increases
again.
Despite that fact, many of the simulated signals that we
added in the data were in fact recovered with values of αF >
1, with a higher SNR than the SNR they would have been re-
coveredwith, had we imposeda restrictionon α. Additionally
some signals gave SNR smaller than the threshold for all val-
ues of α that gave αF ≤ 1. Multiple studies showed that this
was dueto thefact that we onlyhada limited numberofcutoff
frequencies in our template bank and in many cases the lack
of the appropriate ending frequency was compensated for by
a value of α that corresponded to an untypical inspiral gravi-
tational waveform.
We performedvarious investigationswhich showed that re-
jecting triggers with αF > 1 allowed us to still have a very
high efficiency in detecting BBH inspiral signals (although
not as high as if we did not impose that cut) and the cut pri-
marily affected signals that were recoveredwith SNR close to
the threshold. It was also proven that such a cut reduced the
number of noise triggers significantly, so that the false-alarm
probability was significantly reduced as well. The result was
thatsuchacut provideda clearerdistinctionbetweenthenoise
triggers that resulted from our pipeline and the triggers that
came from simulated signals injected in the data. In order to
increase our confidence in the triggers that came out of the
pipeline being BBH inspiral signals, we rejected all triggers
with αF > 1 in this search.
As mentioned in Sec. IV, the initial suggestion of Buo-
nanno,Chen and Vallisneri was that the parameterαFbe con-
strained from below to not take values less than zero. This
suggestion was based on the fact that for values of α < 0,
the amplitude factor (1 − αf2/3
from the predictions of the post-Newtoniantheory at high fre-
quencies. Investigations similar to those described for the cut
αF ≤ 1 did not justify rejecting the triggers with αF < 0, so
we set no low threshold for αF.
cut) can substantially deviate
2. Coincidence parameter tuning
After the single interferometer parameters had been se-
lected, the coincidence parameters were tuned using the trig-
gers from the single interferometers.
The time of arrival of a simulated signal at an interferome-
ter could be measured within ±10 ms. Since the H1 and H2
interferometers are co-located, δt was chosen to be 10 ms for
H1-H2 coincidence. Since the light-travel time between the
two LIGO observatories is 10 ms, δt was chosen to be 20 ms
for LHO-LLO coincidence.
Because we performed a triggered search, the data from all
three LIGO interferometers was filtered with the same tem-
plates for each 2048 s segment. That led us to set the values
for the template coincidence parameters ∆ψ0and ∆ψ3equal
to 0. We found that that was sufficient for the simulated BBH
inspiral signals to be recovered in coincidence.
The slight misalignment of the L1 interferometer with re-
spect to the LHO interferometers led us to choose to not im-
pose an amplitude cut in triggers that came from the two dif-
ferent observatories. This choice was identical to the choice
made for the BNS search [2].
We considered imposing an amplitude cut on the triggers
that came from triple coincident data and were otherwise co-
incident between H1 and H2. A similar cut was imposed on
the equivalent triggers in the BNS search. The cut relied on
the calculation of the “BNS range” for H1 and H2. The BNS
range is defined as the distance at which an optimally oriented
neutron star binary, consisting of two components each of 1.4
M⊙, wouldbe detectedwith a SNR of8 in the data. Thevalue
of the range depends on the PSD. For binary neutron stars the
valueof the rangecan be calculatedfor the 1.4-1.4M⊙binary
and then be rescaled for all masses. That is because the BNS
inspiral signals always terminate at frequencies above 733 Hz
(ISCO frequency for a 3-3 M⊙binary, according to the test-
mass approximation) and thus for BNS the larger part of the
SNR comesfromthehigh-sensitivitybandofLIGO.Forblack
hole binaries, on the other hand, the ending frequency of the
inspiral varies from 110 Hz for a 20-20 M⊙binary up to 733
Hz for a 3-3 M⊙binary (according to the test-mass approxi-
mation). That means that the range depends not only on the
Page 11
11
PSD but also on the binary that is used to calculate it. That
can make a cut based on the range very unreliable and force
rejection of triggers that should not be rejected. In order to
be sure that we would not miss any BBH inspiral signals, we
decided to not impose the amplitude consistency cut between
H1 and H2.
VI.BACKGROUND ESTIMATION
We estimated the rate of accidental coincidences (also
known as background rate) for this search by introducing
an artificial time shift ∆t to the triggers coming from the
L1 interferometer relative to the LHO interferometers. The
time-shift triggers were fed into the coincidence steps of the
pipeline and, for the triple coincident data, to the step of the
filteringof the H2 data andthe H1-H2coincidence. By choos-
ing a shift larger than 20 ms (the time coincidence window
between the two observatories), we ensured that a true gravi-
tational wave could never produce coincident triggers in the
time-shifted data streams. To avoid correlations, we used
shifts longer than the duration of the longest waveform that
we could detect (0.607 s given the low-frequency cutoff im-
posed, as explainedin Sec. III). We chose to not time-shift the
data fromthe two LHO interferometersrelativeto oneanother
since there could be true correlationsproducingaccidentalco-
incidencetriggersduetoenvironmentaldisturbancesaffecting
both of them. The resulting time-shift triggers corresponded
only to accidental coincidences of noise triggers. For a given
time shift, the triggers that emerged from the pipeline were
considered as one single trial representation of an output from
a search if no signals were present in the data.
A total of 80 time-shifts were performed and analyzed in
orderto estimate the background. The time shifts rangedfrom
∆t = −407 s up to ∆t = +407 s in increments of 10 s. The
time shifts of ±7s were not performed.
A.Distribution of background events
Fig. 4 shows the time shift triggers that resulted from our
pipeline (crosses) in the LHO SNR (ρH) versus the LLO SNR
(ρL) plane. There were only double coincident triggers in
both the double and triple coincident data. Specifically, all
the triggers present in the triple coincident data were L1-H1
coincident and were not seen in H2. Thus, ρHis defined as
the SNR of either H1 or H2, depending on which of the three
S2 data sets described in Sec. VB the trigger came from.
There is a group of triggers at the lower left corner of the
plot, which correspond to coincidences with SNR of no more
than 15 in each observatory. There are also long “tails” of
triggers which have high SNR (above 15) in one of the inter-
ferometersandlow SNR (below10)in the other. The distribu-
tion is quite different from the equivalent distribution that was
observedin the BNS search [2]. The presence of the tails (and
their absence from the corresponding distribution in the BNS
search) can be attributed to the fact that the χ2-veto was not
applied in this search and thus some of the loud noise events
7101520253035 40 4550
7
10
15
20
25
30
35
40
45
50
LLO signal−to−noise ratio
LHO signal−to−noise ratio
accidental coincidences
simulated signals
FIG. 4: The LLO and LHO SNRs of the accidental coincidences
fromthe time-shifted triggers(crosses) and thetriggers from thesim-
ulated signal injections (circles) are shown. The dashed lines show
the equal false-alarm contours.
that would have been eliminated by that test have instead sur-
vived.
For comparison, the triggers from some recovered injected
signals are also plotted in Fig. 4 (circles). The distribution
of those triggers is quite different from the distribution of
the triggers resulting from accidental coincidences of noise
events. The noise triggers are concentratedalong the two axes
of the (ρL, ρH) plane and the injection triggers are spread
in the region below the equal-SNR diagonal line of the same
plane. This distribution of the injection triggers is due to the
fact that during the second science run the L1 interferome-
ter was more sensitive than the LHO interferometers. Con-
sequently, a true gravitational wave signal that had compara-
ble LLO and LHO effective distances would be observed with
higher SNR in LLO than in LHO. The few injections that pro-
duced triggers above the diagonal of the graph correspond to
BBH systems that are better oriented for the LHO interferom-
eters thanforthe L1 interferometer,andthus havehigherSNR
at LHO than at LLO.
B.Combined SNR
We define a “combined SNR” for the coincident triggers
that come out of the time-shift analysis. The combined SNR
is a statistic based on the accidental coincidences and is de-
fined so that the higher it is for a trigger, the less likely it is
that the trigger is due to an accidental coincidence of single-
interferometeruncorrelatednoise triggers. Lookingat the plot
of the ρHversus the ρLof the background triggers we notice
that the appropriate contours for the triggers at the lower left
Page 12
12
corner of the plot are concentric circles with the center at the
origin. However, for the tails along the axes the appropriate
contours are “L” shaped. The combination of those two kinds
of contours gives the contours plotted with dashed lines in
Fig. 4. Based on these contours, we define the combinedSNR
of a trigger to be
ρC= min{
?
ρ2
L+ ρ2
H, 2ρH− 3, 2ρL− 3}.
(14)
After the combined SNR is assigned to each pair of triggers,
the triggers are clustered by keeping the one with the high-
est combined SNR within 0.25 s, thus keeping the “highest
confidence” trigger for each event.
VII.RESULTS
In this section we present the results of the search in the
S2 data with the pipeline described in Sec. VB. The com-
bined SNR was assigned to the candidate events according to
Eq. (14).
A. Comparison of the unshifted triggers to the background
Therewere 25 distinct candidateeventsthat survivedall the
analysiscuts. Ofthose, 7wereintheL1-H1doublecoincident
data, 10 were in the L1-H2 double coincident data and 8 were
in the L1-H1-H2 triple coincident data. Those 8 events ap-
peared only in the L1 and H1 data streams and even though
they were not seen in H2 they were still kept, according to the
procedure described in Sec. VB.
90130170210
ρ2
250290330
−3
−2
−1
0
1
2
log10(# of events) with ρC
2 > ρ2, per s2
accidental coincidences
s2 result
FIG. 5:
one standard deviation bars. The number of events in the S2 (circles)
is overlayed.
Expected accidental coincidences per S2 (triangles) with
In order to determine if there was an excess of candidate
events above the backgroundin the S2 data, we compared the
number of zero-shift events to the expected number of acci-
dental coincidences in S2, as predicted by the time-shift anal-
ysis described in Sec. VI. Fig. 5 shows the mean cumula-
tive number of accidental coincidences (triangles) versus the
combinedSNR squaredof those accidental coincidences. The
bars indicate one standard deviation. The cumulative number
of candidate events in the zero-shift S2 data is overlayed (cir-
cles). It is clear that the candidate events are consistent with
the background.
B.Investigations of the zero-shift candidate events
Even though the zero-shift candidate events are consistent
with the background, we investigated them carefully. We first
looked at the possibility of those candidate events being cor-
related with events in the POBI channel, for the reasons de-
scribed in Sec. VA.
It was determinedthat the loudest candidateeventand three
of the remaining candidate events that resulted from our anal-
ysis were coincident with noise transients in POBI. That led
us to believe that the source of these candidate events was in-
strumental and that they were not due to gravitational waves.
The rest of the candidate events were indistinguishable from
the background events.
C.Results of the Monte-Carlo simulations
As mentioned in Sec. VB, the Monte-Carlo simulations al-
low us not only to tune the parameters of the pipeline, but also
to measure the efficiency of our search method. In this section
we look in detail at the results of Monte-Carlo simulations in
the full data set of the second science run.
Due to the lack of observation-based predictions for the
population of BBH systems in the Universe, the inspi-
ral signals that we injected were uniformly distributed in
log(distance), with distance varying from 10 kpc to 20 Mpc
and uniformly distributed in component mass (this mass dis-
tribution was proposed by [29]), with each component mass
varying between 3 and 20 M⊙.
Fig. 6 shows the efficiency of recovering the injected sig-
nals (number of found injections of a given effective distance
divided by the total number of injections of that effective dis-
tance) versus the injected LHO-effective distance. We chose
to plot the efficiency versus the LHO-effective distance rather
than versus the LLO-effective distance since H1 and H2 were
less sensitive than L1 during S2.
Our analysis method had efficiency of at least 90% for re-
covering BBH inspiral signals with LHO-effective distance
less than 1 Mpc for the mass range we were exploring. It
should be noted how the efficiency of our pipeline varied for
different injected waveforms. It is clear from Fig. 6 that the
efficiencyforrecoveringEOBwaveformswashigherthanthat
for TaylorT3 or Pad´ eT1 waveforms for all distances. This
is expected because the EOB waveforms have more power
(longer duration and larger number of cycles) in the LIGO
Page 13
13
10
−2
10
−1
10
0
10
1
0
0.2
0.4
0.6
0.8
0.9
1
Injected LHO−effective distance (Mpc)
Efficiency
EOB
TaylorT3
PadeT1
FIG. 6: The efficiency versus the LHO-effective distance for the dif-
ferent families of injected waveforms is shown. The dashed line rep-
resentsthe efficiencyfor Pad´ eT1injections, the dottedlinerepresents
the efficiency for standard post-Newtonian time-domain waveforms
and the solid line represents the efficiency for effective-one-body
waveforms. All injected waveforms were of second post-Newtonian
order. Binomial error bars are shown.
frequency band compared to the Pad´ eT1 and TaylorT3 wave-
forms.
Even though the main determining factor of whether a sim-
ulated signal is recovered or not is its effective distance in the
least sensitive interferometer, it is worth investigating the ef-
ficiency of recovering injections as a function of the injected
total mass. In order to limit the effect of the distance in the
efficiency, we chose all the injections with LHO-effective dis-
tance between 1 and 3 Mpc and plotted the efficiency versus
total mass in Fig.7. The plotshows a decreaseinthe detection
efficiencyas the total mass increases, for the TaylorT3and the
Pad´ eT1 waveforms, but not for the EOB waveforms. The rea-
son for this is the same as the one mentioned previously. For
the TaylorT3 and the Pad´ eT1 waveforms, the higher the total
mass of the binary system, the fewer the cycles in the LIGO
band. That leads to the reduction in efficiency for those wave-
forms. The EOB waveforms, on the other hand, extend be-
yond the ISCO and thus have more cycles in the LIGO band
than the TaylorT3 or the Padi´ eT1 waveforms. The power of
the signal in band is sufficient for those waveforms to be de-
tected with an approximately equal probability for the higher
masses.
VIII.UPPER LIMIT ON THE RATE OF BBH INSPIRALS
As was mentioned previously, a reliable upper limit cannot
be calculated for the rate of BBH inspirals as was calculated
for the rate of BNS inspirals. The reason for this is two-fold.
Firstly, thecharacteristicsoftheBBH population(suchas spa-
tial, mass andspin distributions)are notknown,sinceno BBH
610 15 2025303540
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Injected total mass (Mo)
Efficiency
EOB
TaylorT3
PadeT1
FIG. 7: The efficiency versus the injected total mass of all injected
signals withLHO-effectivedistance between 1 and 3Mpc for thedif-
ferent families of injected waveforms is shown. The dashed line rep-
resentstheefficiency for Pad´ eT1injections, thedotted linerepresents
the efficiency for standard post-Newtonian time-domain waveforms
and the solid line represents the efficiency for effective-one-body
waveforms. All injected waveforms were of second post-Newtonian
order. Binomial error bars are shown.
systems have ever been observed. In addition, the BCV tem-
plates are not guaranteed to have a good overlap with the true
BBH inspiral gravitational wave signals. However, work by
various groups has given insights on the possible spatial dis-
tribution and mass distribution of BBH systems [29]. Assum-
ing that the model-based inspiral waveforms proposed in the
literature have good overlap with a true inspiral gravitational
wavesignal andbecausethe BCV templates havea goodover-
lap withthose model-basedBBH inspiralwaveforms,we used
someofthosepredictionstogiveaninterpretationofthe result
of our search.
A. Upper limit calculation
Following the notation used in [30], let R indicate the
rate of BBH coalescences per year per Milky Way Equiv-
alent Galaxy (MWEG) and NG(ρ∗) indicate the number of
MWEGs which our search probes at ρ ≥ ρ∗. The probability
of observing an inspiral signal with ρ > ρ∗in an observation
time T is
P(ρ > ρ∗;R) = 1 − e−RTNG(ρ∗).
(15)
A trigger can arise from either an inspiral signal in the data or
from background. If Pbdenotes the probability that all back-
groundtriggers have SNR less than ρ∗, then the probabilityof
observing one or more triggers with ρ > ρ∗is
P(ρ > ρ∗;R,b) = 1 − Pbe−RTNG(ρ∗).
(16)
Page 14
14
Given the probability Pb, the total observation time T, the
SNR of the loudest event ρmax, and the number of MWEGs
NG(ρmax) to which the search is sensitive, we find that a fre-
quentist upper limit, at 90% confidence level, is
R < R90%=2.303 + lnPb
TNG(ρmax).
(17)
For R > R90%, there is more than 90% probability that at
least one event would be observed with SNR greater than
ρmax. Details of this method of determining an upper limit
can be found in [31]. In particular, one obtains a conservative
upper limit by setting Pb= 1. We adopt this approach below
because of uncertainties in our background estimate.
During the 350.4 hours of non-playground data used in
this search, the highest combined SNR that was observed
was 16.056. The number NG can be calculated, as in the
BNS search [2], using the Monte-Carlo simulations that were
performed. The difference from the BNS search is that
the injected signals were not drawn from an astrophysical
population, but from a population that assumes a uniform
log(distance) distribution. The way this difference is handled
is explained below.
Our model for the BBH population carried the following
assumptions:
(1) Black holes of mass between 3 and 20 M⊙result exclu-
sively from the evolutionof stars, so that our BBH sources are
present only in galaxies.
(2) The field population of BBHs is distributed amongst
galaxies in proportion to the galaxies’ blue light (as was as-
sumed for BNS systems [2]).
(3) The component mass distribution is uniform, with val-
ues ranging from 3 to 20 M⊙[29, 32].
(4) The component spins are negligibly small.
(5) The waveforms are an equal mixture of EOB, Pad´ eT1
and TaylorT3 waveforms.
(6) The sidereal times of the coalescences are distributed
uniformly throughout the S2 run.
Assumption (4) was made because the BCV templates used
in this search were not intended to capture the amplitude
modulations of the gravitational waveforms expected to re-
sult from BBH systems with spinning components. However,
studies performed by BCV [33] showed that the BCV tem-
plates do have high overlaps (90% on average, the average
taken over one thousand initial spin orientations) with wave-
forms of spinning BBH systems of comparable component
masses. Templatesthataremoresuitablefordetectionofspin-
ning BBH have been developed [33] (BCV2) and are being
used in a search for the inspiral of such binaries in the S3
LIGO data.
Assumption (5) was probably the most ad-hoc assumption
in our upper limit calculation. Since this calculation was pri-
marily intended to be illustrative of how our results can be
used to set an upper limit, the mix of the waveforms was cho-
sen for simplicity. It should be apparent how to modify the
calculation to fit a different population model.
With assumptions (1) and (2) we determined the number
of MWEGs in each logarithmicbin of LHO-effectivedistance
10
−2
10
−1
10
0
10
1
0
0.25
0.5
0.75
1
1.25
1.5
1.75
LHO−effective distance (Mpc)
f
NG
NG
nc
FIG. 8: The number of Milky Way equivalent galaxies (crosses) and
the efficiency of the search (circles) as calculated by Eq. (19) ver-
sus the LHO-effective distance (in Mpc) are shown. The cumulative
number of MWEGs NG (diamonds) versus LHO effective distance
is overlayed. The horizontal axis has a logarithmic scale, in accor-
dance with the uniform-log distance distribution of the injected BBH
inspiral signals. Nounitsaregiven for theverticalaxisbecause itcor-
responds to three different quantities plotted against LHO-effective
distance.
(Nnc
tions to determine the efficiency for detection of a source at
LHO-effective distance d with combined SNR ρC > ρmax,
ρmaxbeing the combined SNR of the loudest event observed
in our search. Specifically, we define
G, crosses in Fig. 8). We used our Monte-Carlo simula-
f(d;ρmax) =1
3
?Nf(d;ρmax)
?Nf(d;ρmax)
Ninj(d)
?
EOB+1
TaylorT3
(18)
+1
3Ninj(d)
?
3
?Nf(d;ρmax)
Ninj(d)
?
PT1
and that is also plotted in Fig. 8 (circles). The efficiencies
for each waveform family individually are given in Table I.
Finally, NG(ρmax) was calculated as
NG(ρmax) =
∞
?
d=0
f(d;ρmax) × Nnc
G(d).
(19)
Evaluating that sum we obtained NG = 1.6603.
Eq. (17) we obtained R90%= 35 year−1MWEG−1.
Using
B.Error analysis
A detailed error analysis is necessary and was carefully
done for the rate upper limit calculated for BNS inspirals in
[2]. For the upper limit calculated here, such a detailed analy-
sis was not possible due to the lack of a reliable BBH popula-
tion. However, we estimated the errors coming from calibra-
Page 15
15
TABLE I: Efficiencies of recovering simulated BBH inspiral signals from three different waveform families for each LHO-effective distance
bin.
LHO-Effective Distance Range (Mpc)MWEGs Detected with ρ > 16.056 versus Injected
EOBTaylorT3
1 / 12 / 2
5 / 58 / 9
30 / 30 21 / 21
44 / 4437 / 39
58 / 5858 / 60
81 / 8177 / 77
61 / 6178 / 81
64 / 64 81 / 82
68 / 68 76 / 77
83 / 83 64 / 67
82 / 82 66 / 67
77 / 77 90 / 92
74 / 7461 / 64
53 / 55 81 / 101
68 / 8538 / 75
28 / 6714 / 74
10 / 831 / 69
0 / 88 0 / 67
0 / 83 0 / 65
0 / 71 0 / 65
NG
Pad´ eT1
2 / 2
25 / 25
57 / 57
85 / 86
100 / 103
117 / 118
146 / 146
155 / 155
144 / 144
152 / 155
150 / 152
126 / 133
148 / 165
115 / 154
61 / 156
11 / 147
0 / 133
0 /156
0 /170
0 /150
0.0100 - 0.0141
0.0141 - 0.0200
0.0200 - 0.0282
0.0282 - 0.0398
0.0398 - 0.0562
0.0562 - 0.0794
0.0794 - 0.1122
0.1122 - 0.1585
0.1585 - 0.2239
0.2239 - 0.3162
0.3162 - 0.4467
0.4467 - 0.6310
0.6310 - 0.8913
0.8913 - 1.2589
1.2589 - 1.7783
1.7783 - 2.5119
2.5119 - 3.5481
3.5481 - 5.0119
5.0119 - 7.0795
7.0795 - 10.000
0.0814
0.1415
0.1850
0.1844
0.1404
0.0999
0.0722
0.0515
0.0338
0.0172
0.0075
0.0038
0.0516
0.2490
0.4505
0.5267
0.5222
0.6005
0.8510
0.9507
0.0814
0.2177
0.4027
0.5832
0.7207
0.8203
0.8916
0.9429
0.9766
0.9934
1.0009
1.0046
1.0536
1.2622
1.5171
1.6368
1.6603
1.6603
1.6603
1.6603
tion uncertainties and the errors due to the limited number of
injections performed.
The effect of the calibration uncertainties was calculated as
was done in Sec. IX A 2 of [2]. In principle, those uncertain-
ties affect the combined SNR ρCas
δρC≤ max
??ρ2
L
ρ2
C
(δρL)2+ρ2
H
ρ2
C
(δρH)2?1/2
, 2(δρH), 2(δρL)
?
(20)
where we modified Eq. (23) of [2] based on our Eq. (14) for
the combined SNR. However, for the calculation of the effect
of this error on the rate upper limit, we were interested in how
the calibration uncertainties would affect the combined SNR
of the loudest event. Careful examination of the combined
SNR of the loudest event showed that for that event the min-
imum of the three possible values in Eq. (14) was the value
(ρ2
uncertainties by
L+ ρ2
H)1/2, so we calculated the error due to calibration
δρC≤
?ρ2
L
ρ2
C
(δρL)2+ρ2
H
ρ2
C
(δρH)2?1/2
.
(21)
We simplified the calculation by being more conservative and
using
δρC≤
?
(δρL)2+ (δρH)2?1/2
.
(22)
We also used the fact that the maximum calibration errors at
each site were 8.5% for L1 and 4.5% for LHO (as explained
in [2]) to obtain
δρC≤
?
(0.085 ρmax
L
)2+ (0.045ρmax
H
)2?1/2
.
(23)
The resulting error in NGwas
δNG|cal= ±0.0859 MWEG.
(24)
The errors due to the limited number of injections in our
Monte-Carlo simulations had to be calculated for each loga-
rithmic distance bin and the resulting errors to be combinedin
quadrature. Specifically,
δNG|MC=
? ∞
?
d=0
?
δf(d;ρmax) × Nnc
G(d)
?2?1/2
.
(25)
Because f was calculatedusingthe 3 differentwaveformfam-
ilies, the error δf(d;ρmax) is
δf(d;ρmax) =
??
appr
Nf(d;ρmax)(Ninj(d) − Nf(d;ρmax))
32[Ninj(d)]3
?1/2
(26)
where the sum was calculated over the three waveform fami-
lies: EOB, Pad´ eT1 and TaylorT3. This gave
δNG|MC= ±0.0211 MWEG.
(27)
Finally we added the errors in NGin quadrature and ob-
tained
δNG= ±0.0885 MWEG.
(28)
Page 16
16
Both contributions to this error can be thought of a 1-σ vari-
ations. In order to calculate the 90% level of the systematic
errors we multiplied δNGby 1.6, so
δNG|90%= ±0.1415 MWEG.
(29)
To be conservative,we assumed a downwardexcursionNG=
(1.6603−0.1415)MWEG = 1.5188 MWEG. When substi-
tuted into the rate upper limit equation this gave
R90%= 38 year−1MWEG−1.
(30)
IX.CONCLUSIONS AND FUTURE PROSPECTS
We performed the first search for binary black hole inspiral
signals in data from the LIGO interferometers. This search,
even though similar in some ways to the binary neutron star
inspiral search, has some significant differences and presents
unique challenges. There were no events that could be inden-
tified as gravitational waves.
The fact that the performance and sensitivity of the LIGO
interferometers is improving and the frequency sensitivity
band is being extendedto lower frequencies makes us hopeful
that the first detection of gravitational waves from the inspi-
ral phase of binary black hole coalescences may happen in
the near future. In the absense of a detection, astrophysically
interesting results can be expected by LIGO very soon. The
current most optimistic rates for BBH coalescences are of the
order of 10−4year−1MWEG−1[34]. It is estimated that at
design sensitivity the LIGO interferometerswill be able to de-
tect binary black hole inspirals in at least 5600 MWEGs with
the most optimistic calculations giving up to 13600 MWEGs
[35]. A science run of two years at design sensitivity is ex-
pected to give BBH coalescence rate upper limits of less than
10−4year−1MWEG−1.
APPENDIX A: FILTERING DETAILS
The amplitude part of the BCV templates˜h(f) can be de-
composed into two pieces, which are linear combinations of
f−7/6and f−1/2. Those expressions can be used to construct
an orthogonal basisˆhjfor the 4-dimensional linear subspace
of templates with φ0∈ [0,2π) and α ∈ (−∞,+∞). Specifi-
cally, we want the basis vectors to satisfy
?ˆhk,ˆhj
?
= δkj.
(A1)
To do this we construct two real functions A1(f) and A2(f),
linear combinations of f−7/6and f−1/2, which are related to
the four basis vectors via:
ˆh1,2(f) = A1,2(f) eiψs(f)
ˆh3,4(f) = A1,2(f) i eiψs(f).
(A2)
(A3)
Then, Eq. (A1) becomes:
4ℜ
?∞
0
Ak(f)Aj(f)
Sh(f)
df = δkj.
(A4)
Since the templates have to be normalized, A1(f) and A2(f)
must satisfy
?
A1(f)
A2(f)
?
=
?
a1 0
b1 b2
??
f−7/6
f−1/2
?
.
Imposing condition (A4) gives the numerical values of the
normalization factors. Those are
a1 = I−1/2
7/3,
(A5)
b1 = −I5/3
I7/3
?I1−
I2
5/3
I7/3
I2
5/3
I7/3
?−1/2,
(A6)
b2 =
?I1−
?−1/2,
(A7)
(A8)
where the integrals Ikare
Ik= 4
?fcut
0
df
fkSh(f).
(A9)
The next step is to write the normalized template in terms
of the 4 basis vectors
ˆh(f) = c1ˆh1(f) + c2ˆh2(f) + c3ˆh3(f) + c4ˆh4(f)
(A10)
with
c1 = cosφ0cosω,
c2 = cosφ0sinω,
c3 = sinφ0cosω,
c4 = sinφ0sinω,
(A11)
(A12)
(A13)
(A14)
where ω is related to α by
tanω = −
a1α
b2+ b1α.
(A15)
Once the filters are designed, the overlap is calculated and
is equal to
ρ =
?
s,ˆh
?
= K1cosω cosφ0+ K2sinω cosφ0(A16)
+K3cosωsinφ0+ K4sinω sinφ0
where Kj=
are necessary, namely
?
s,ˆhj
?
,k = 1,2,3,4 are the four integrals that
K1= ℜ
?fcut
0
4˜ s(f)a1f−7/6
Sh(f)
e−iψs(f)df,
(A17)
K2= ℜ
?fcut
0
4˜ s(f)(b1f−7/6+ b2f−1/2)
Sh(f)
e−iψs(f)df,
(A18)
K3= −ℑ
?fcut
0
4˜ s(f)a1f−7/6
Sh(f)
e−iψs(f)df,
(A19)
Page 17
17
K4= −ℑ
?fcut
0
4˜ s(f)(b1f−7/6+ b2f−1/2)
Sh(f)
e−iψs(f)df.
(A20)
Maximizing the SNR over φ0and ω we get
ρmaximized =1
2
?(K1− K4)2+ (K2+ K3)2.
?(K1+ K4)2+ (K2− K3)2(A21)
+
1
2
The values of φ0and α that give the maximized SNR are
φmax
0
=1
2arctanK2+ K3
K1− K4
−1
2arctanK2− K3
K1+ K4, (A22)
αmax= −
b2tanωmax
a1+ b1tanωmax,
(A23)
where
ωmax=1
2arctanK2− K3
K1+ K4
+1
2arctanK2+ K3
K1− K4. (A24)
An extensive discussion on the values of α is provided in
Sec. VC1.
An equivalent expression for the SNR, which is computa-
tionally less costly, can be produced if we define
F1= K1− iK3,
F2= K2− iK4.
(A25)
Eq. A22 can then be written as
ρmaximized =1
2
?|F1|2+ |F2|2+ 2ℑ(F1F∗
?|F1|2+ |F2|2− 2ℑ(F1F∗
2)
(A26)
+1
2
2).
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support of the
United States National Science Foundation for the construc-
tion and operation of the LIGO Laboratory and the Particle
PhysicsandAstronomyResearch Councilofthe UnitedKing-
dom, the Max-Planck-Society and the State of Niedersach-
sen/Germany for support of the construction and operation of
the GEO600 detector. The authors also gratefully acknowl-
edge the support of the research by these agencies and by the
Australian Research Council, the Natural Sciences and Engi-
neeringResearch Councilof Canada, the CouncilofScientific
and Industrial Research of India, the Department of Science
andTechnologyofIndia,theSpanishMinisteriodeEducacion
y Ciencia, the John Simon Guggenheim Foundation, the Lev-
erhulme Trust, the David and Lucile Packard Foundation, the
Research Corporation, and the Alfred P. Sloan Foundation.
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