Brooks, Aidan Francis

18 Pages·2013·1.24 MB·English

PUBLISHED VERSION
Abbott, B.;...; Brooks, Aidan Francis; Brown, D. A.;...; Hosken, David John; Hough, J.;...;
Mudge, Damien Troy; Mueller, G.;...; Munch, Jesper; Murray, P. G.;...; Veitch, Peter John;
... et al.; LIGO Scientific Collaboration; ALLEGRO Collaboration
First cross-correlation analysis of interferometric and resonant-bar gravitational-wave data
for stochastic backgrounds Physical Review D, 2007; 76(2):022001
©2007 American Physical Society
http://link.aps.org/doi/10.1103/PhysRevD.76.022001

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23rd April 2013

http://hdl.handle.net/2440/47208

See Also: Publisher's Note: B Abbott et al., Publisher’s Note: First cross-correlation analysis
of interferometric and resonant-bar gravitational-wave data for stochastic backgrounds [Phys. Rev.
D 76, 022001 (2007)], Phys. Rev. D 76, 029905 (2007).
Publisher's Note: B. Abbott et al. LIGO Scientific Collaboration, ALLEGRO Collaboration,
Publisher’s Note: First cross-correlation analysis of interferometric and resonant-bar gravitational-
wave data for stochastic backgrounds [Phys. Rev. D 76, 022001 (2007)], Phys. Rev. D 77, 069904
(2008).
PHYSICALREVIEW D 76, 022001 (2007)
First cross-correlation analysisofinterferometric andresonant-bar gravitational-wave data for
stochastic backgrounds
B.Abbott,14 R.Abbott,14 R.Adhikari,14 J. Agresti,14 P.Ajith,2 B. Allen,2,51R. Amin,18 S.B.Anderson,14
W.G.Anderson,51M.Arain,39M.Araya,14H.Armandula,14M.Ashley,4S.Aston,38P.Aufmuth,36C.Aulbert,1S.Babak,1
S.Ballmer,14H.Bantilan,8B.C.Barish,14C.Barker,15D.Barker,15B.Barr,40P.Barriga,50M.A.Barton,40K.Bayer,17
K.Belczynski,24 J.Betzwieser,17 P.T.Beyersdorf,27 B. Bhawal,14 I.A.Bilenko,21 G.Billingsley,14 R.Biswas,51
E.Black,14K.Blackburn,14L.Blackburn,17D.Blair,50B.Bland,15J.Bogenstahl,40L.Bogue,16R.Bork,14V.Boschi,14
S.Bose,52P.R.Brady,51V.B.Braginsky,21J.E.Brau,43M.Brinkmann,2A.Brooks,37D.A.Brown,14,6A.Bullington,30
A.Bunkowski,2A.Buonanno,41M.Burgamy,18,*O.Burmeister,2D.Busby,14,†R.L.Byer,30L.Cadonati,17G.Cagnoli,40
J.B. Camp,22 J.Cannizzo,22 K.Cannon,51 C.A.Cantley,40 J.Cao,17 L. Cardenas,14 M.M.Casey,40 G.Castaldi,46
C.Cepeda,14 E. Chalkey,40 P.Charlton,9 S.Chatterji,14S. Chelkowski,2 Y. Chen,1 F. Chiadini,45 D.Chin,42 E. Chin,50
J.Chow,4N.Christensen,8J.Clark,40P.Cochrane,2T.Cokelaer,7C.N.Colacino,38R.Coldwell,39R.Conte,45D.Cook,15
T. Corbitt,17 D.Coward,50 D.Coyne,14 J.D.E. Creighton,51T.D.Creighton,14 R.P.Croce,46 D.R.M.Crooks,40
A.M.Cruise,38 A.Cumming,40 J. Dalrymple,31 E. D’Ambrosio,14 K.Danzmann,2,36G. Davies,7 D.DeBra,30
J.Degallaix,50 M.Degree,30 T.Demma,46 V. Dergachev,42S. Desai,32 R. DeSalvo,14 S. Dhurandhar,13 M.D´ıaz,33
J.Dickson,4A.DiCredico,31G.Diederichs,36A.Dietz,7E.E.Doomes,29R.W.P.Drever,5J.-C.Dumas,50R.J.Dupuis,14
J.G.Dwyer,10 P.Ehrens,14 E. Espinoza,14 T.Etzel,14 M.Evans,14T. Evans,16 S. Fairhurst,7,14 Y. Fan,50 D.Fazi,14
M.M.Fejer,30L.S. Finn,32 V. Fiumara,45 N.Fotopoulos,51 A.Franzen,36 K.Y.Franzen,39 A.Freise,38 R.Frey,43
T. Fricke,44 P.Fritschel,17V.V. Frolov,16 M.Fyffe,16 V.Galdi,46 J. Garofoli,15 I. Gholami,1 J.A. Giaime,16,18
S.Giampanis,44K.D.Giardina,16K.Goda,17E.Goetz,42L.Goggin,14G.Gonza´lez,18S.Gossler,4A.Grant,40S.Gras,50
C. Gray,15 M.Gray,4J. Greenhalgh,26 A.M.Gretarsson,11 R. Grosso,33 H.Grote,2S. Grunewald,1M.Guenther,15
R.Gustafson,42 B. Hage,36W.O.Hamilton,18,* D.Hammer,51 C.Hanna,18 J. Hanson,16,† J. Harms,2 G.Harry,17,†
E.Harstad,43T.Hayler,26J.Heefner,14I.S.Heng,40,†A.Heptonstall,40M.Heurs,2M.Hewitson,2S.Hild,36E.Hirose,31
D.Hoak,16D.Hosken,37J.Hough,40E.Howell,50D.Hoyland,38S.H.Huttner,40D.Ingram,15E.Innerhofer,17M.Ito,43
Y. Itoh,51 A.Ivanov,14 D.Jackrel,30 B.Johnson,15 W.W.Johnson,18,† D.I. Jones,47 G.Jones,7 R. Jones,40L. Ju,50
P.Kalmus,10V.Kalogera,24D.Kasprzyk,38E.Katsavounidis,17K.Kawabe,15S.Kawamura,23F.Kawazoe,23W.Kells,14
D.G.Keppel,14 F.Ya. Khalili,21 C.Kim,24 P. King,14J.S. Kissel,18S. Klimenko,39 K.Kokeyama,23 V.Kondrashov,14
R.K.Kopparapu,18D.Kozak,14B.Krishnan,1P.Kwee,36P.K.Lam,4M.Landry,15B.Lantz,30A.Lazzarini,14B.Lee,50
M.Lei,14J. Leiner,52 V.Leonhardt,23I. Leonor,43 K.Libbrecht,14 P. Lindquist,14 N.A.Lockerbie,48 M.Longo,45
M.Lormand,16 M.Lubinski,15 H.Lu¨ck,2,36B. Machenschalk,1 M.MacInnis,17 M.Mageswaran,14 K.Mailand,14
M.Malec,36 V. Mandic,14 S. Marano,45 S.Ma´rka,10 J. Markowitz,17 E. Maros,14 I. Martin,40 J.N.Marx,14 K.Mason,17
L. Matone,10 V. Matta,45 N.Mavalvala,17R. McCarthy,15 B.J. McCaulley,20 D.E. McClelland,4 S.C. McGuire,29
M.McHugh,20,† K.McKenzie,4 J.W.C. McNabb,32 S.McWilliams,22 T.Meier,36 A.Melissinos,44 G.Mendell,15
R.A.Mercer,39S.Meshkov,14E.Messaritaki,14C.J.Messenger,40D.Meyers,14E.Mikhailov,17P.Miller,18,*S.Mitra,13
V.P.Mitrofanov,21 G.Mitselmakher,39 R. Mittleman,17 O.Miyakawa,14 S. Mohanty,33 V.Moody,41,*G. Moreno,15
K.Mossavi,2C.MowLowry,4A.Moylan,4D.Mudge,37G.Mueller,39S.Mukherjee,33H.Mu¨ller-Ebhardt,2J.Munch,37
P.Murray,40E.Myers,15J.Myers,15T.Nash,14D.Nettles,18,*G.Newton,40A.Nishizawa,23K.Numata,22B.O’Reilly,16
R. O’Shaughnessy,24 D.J. Ottaway,17 H.Overmier,16B.J. Owen,32H.-J.Paik,41,* Y.Pan,41 M.A.Papa,1,51
V. Parameshwaraiah,15 P. Patel,14M.Pedraza,14 S.Penn,12 V. Pierro,46 I.M.Pinto,46 M.Pitkin,40 H.Pletsch,51
M.V.Plissi,40F.Postiglione,45R.Prix,1V.Quetschke,39F.Raab,15D.Rabeling,4H.Radkins,15R.Rahkola,43N.Rainer,2
M.Rakhmanov,32M.Ramsunder,32 K.Rawlins,17 S.Ray-Majumder,51V.Re,38 H.Rehbein,2 S.Reid,40 D.H. Reitze,39
L. Ribichini,2R. Riesen,16 K.Riles,42 B. Rivera,15 N.A.Robertson,14,40 C.Robinson,7 E.L. Robinson,38 S. Roddy,16
A.Rodriguez,18A.M.Rogan,52J.Rollins,10J.D.Romano,7J.Romie,16R.Route,30S.Rowan,40A.Ru¨diger,2L.Ruet,17
P.Russell,14K.Ryan,15 S.Sakata,23 M.Samidi,14L. Sancho de la Jordana,35V.Sandberg,15 V. Sannibale,14 S.Saraf,25
P. Sarin,17 B.S.Sathyaprakash,7S. Sato,23 P.R. Saulson,31 R.Savage,15 P.Savov,6 S.Schediwy,50 R. Schilling,2
R.Schnabel,2 R. Schofield,43B.F. Schutz,1,7 P. Schwinberg,15 S.M.Scott,4 A.C.Searle,4 B. Sears,14 F. Seifert,2
D.Sellers,16 A.S. Sengupta,7 P.Shawhan,41 D.H. Shoemaker,17 A.Sibley,16 J.A.Sidles,49 X.Siemens,6,14 D.Sigg,15
S. Sinha,30 A.M.Sintes,1,35B.J.J. Slagmolen,4J. Slutsky,18 J.R. Smith,2M.R. Smith,14K. Somiya,1,2 K.A.Strain,40
D.M.Strom,43A.Stuver,32T.Z. Summerscales,3K.-X.Sun,30M.Sung,18P.J.Sutton,14H.Takahashi,1D.B.Tanner,39
M.Tarallo,14 R.Taylor,14 R. Taylor,40 J. Thacker,16 K.A.Thorne,32 K.S.Thorne,6 A.Thu¨ring,36 K.V. Tokmakov,40
1550-7998=2007=76(2)=022001(17) 022001-1 © 2007 The American Physical Society
B. ABBOTTet al. PHYSICALREVIEW D76, 022001 (2007)
C.Torres,33C.Torrie,40G.Traylor,16M.Trias,35W.Tyler,14D.Ugolini,34C.Ungarelli,38K.Urbanek,30H.Vahlbruch,36
M.Vallisneri,6 C. Van Den Broeck,7M.Varvella,14 S.Vass,14 A.Vecchio,38 J. Veitch,40 P.Veitch,37A. Villar,14
C.Vorvick,15 S.P. Vyachanin,21 S.J.Waldman,14 L. Wallace,14 H.Ward,40 R.Ward,14K. Watts,16 J. Weaver,18,*
D. Webber,14 A.Weber,18,* A.Weidner,2 M.Weinert,2A. Weinstein,14 R.Weiss,17 S.Wen,18K. Wette,4J.T. Whelan,1
D.M.Whitbeck,32 S.E. Whitcomb,14 B.F.Whiting,39 C. Wilkinson,15 P.A.Willems,14 L. Williams,39 B.Willke,2,36
I.Wilmut,26W.Winkler,2C.C.Wipf,17S.Wise,39A.G.Wiseman,51G.Woan,40D.Woods,51R.Wooley,16J.Worden,15
W.Wu,39 I. Yakushin,16 H.Yamamoto,14 Z. Yan,50 S.Yoshida,28 N.Yunes,32 M.Zanolin,17 J.Zhang,42 L. Zhang,14
P. Zhang,18,* C. Zhao,50 N.Zotov,19 M.Zucker,17 H.zur Mu¨hlen,36 and J. Zweizig14
(LIGO Scientific Collaboration and ALLEGROCollaboration)‡
1Albert-Einstein-Institut, Max-Planck-Institut fu¨r Gravitationsphysik, D-14476Golm,Germany
2Albert-Einstein-Institut, Max-Planck-Institut fu¨r Gravitationsphysik, D-30167Hannover, Germany
3Andrews University, Berrien Springs, Michigan 49104,USA
4AustralianNational University, Canberra, 0200, Australia
5California Institute of Technology, Pasadena, California 91125, USA
6Caltech-CaRT, Pasadena,California 91125,USA
7Cardiff University, Cardiff, CF24 3AA, United Kingdom
8Carleton College, Northfield, Minnesota 55057,USA
9Charles Sturt University, Wagga Wagga, NSW 2678, Australia
10Columbia University, New York, New York 10027, USA
11Embry-Riddle Aeronautical University, Prescott, Arizona 86301, USA
12Hobartand William Smith Colleges, Geneva, New York 14456,USA
13Inter-University Centre for Astronomy andAstrophysics, Pune-411007, India
14LIGO-CaliforniaInstitute of Technology, Pasadena, California 91125, USA
15LIGOHanford Observatory, Richland, Washington 99352,USA
16LIGOLivingston Observatory, Livingston, Louisiana 70754,USA
17LIGO-Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,USA
18Louisiana State University, Baton Rouge, Louisiana 70803, USA
19Louisiana Tech University, Ruston, Louisiana 71272,USA
20Loyola University, NewOrleans, Louisiana 70118,USA
21Moscow State University, Moscow, 119992,Russia
22NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
23National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
24Northwestern University, Evanston, Illinois 60208,USA
25Rochester Institute of Technology, Rochester, New York 14623,USA
26Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX United Kingdom
27San Jose State University, San Jose, California 95192,USA
28Southeastern Louisiana University, Hammond,Louisiana 70402, USA
29Southern University and A&M College, Baton Rouge, Louisiana 70813,USA
30Stanford University, Stanford, California 94305,USA
31Syracuse University, Syracuse, New York 13244, USA
32The Pennsylvania State University, University Park, Pennsylvania 16802,USA
33TheUniversity of Texas at Brownsville and Texas SouthmostCollege, Brownsville, Texas 78520, USA
34Trinity University, SanAntonio, Texas 78212, USA
35Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
36Universita¨t Hannover, D-30167Hannover, Germany
37University of Adelaide, Adelaide, SA 5005, Australia
38University of Birmingham, Birmingham, B15 2TT, United Kingdom
39University of Florida, Gainesville, Florida 32611,USA
40University of Glasgow, Glasgow, G12 8QQ,United Kingdom
41University of Maryland, College Park, Maryland 20742 USA
42University of Michigan, Ann Arbor, Michigan 48109,USA
43University of Oregon,Eugene, Oregon 97403, USA
44University of Rochester, Rochester, New York 14627, USA
45University of Salerno, 84084 Fisciano (Salerno), Italy
46University of Sannio at Benevento, I-82100 Benevento, Italy
47University of Southampton, Southampton, SO17 1BJ, United Kingdom
48University of Strathclyde, Glasgow, G1 1XQ, United Kingdom
022001-2
FIRST CROSS-CORRELATIONANALYSISOF... PHYSICALREVIEW D76, 022001 (2007)
49University of Washington, Seattle, Washington 98195,USA
50University of Western Australia, Crawley, WA 6009, Australia
51University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA
52Washington State University, Pullman, Washington 99164,USA
(Received 28 March 2007; published 9 July2007; publisher error corrected 23 July 2007; publisher error corrected 4 March 2008)
DatafromtheLIGOLivingstoninterferometerandtheALLEGROresonant-bardetector,takenduring
LIGO’sfourthsciencerun,wereexaminedforcrosscorrelationsindicativeofastochasticgravitational-
wavebackgroundin the frequencyrange 850–950Hz,withmostofthe sensitivity arisingbetween 905
and925Hz.ALLEGROwasoperatedinthreedifferentorientationsduringtheexperimenttomodulatethe
relativesignofgravitational-waveandenvironmentalcorrelations.Nostatisticallysignificantcorrelations
were seen in any of the orientations, and the results were used to set a Bayesian 90% confidence level
upper limit of (cid:1) (cid:1)f(cid:2)(cid:3)1:02, which corresponds to a gravitational-wave strain at 915 Hz of 1:5(cid:4)
gw
10(cid:5)23 Hz(cid:5)1=2.Inthetraditional unitsofh2 (cid:1) (cid:1)f(cid:2),thisisalimitof0.53,2ordersofmagnitudebetter
100 gw
than the previous direct limit at these frequencies. The method was also validated with successful
extraction of simulated signals injected in hardware and software.
DOI: 10.1103/PhysRevD.76.022001 PACS numbers: 04.80.Nn, 04.30.Db, 07.05.Kf, 95.55.Ym
I. INTRODUCTION February 22 and March 23, 2005, during LIGO’s fourth
sciencerun(S4).AveragesensitivitiesofL1andA1during
Oneofthe signalstargeted bythe current generation of
S4 are shown in Fig. 1. ALLEGRO was operated in three
ground-based gravitational-wave (GW) detectors is a sto-
orientations, which modulated the GW response of the
chastic gravitational-wave background (SGWB) [1–3].
LLO-ALLEGROpair through 180(cid:6) of phase.
Such a backgroundisanalogousto the cosmic microwave
TheLLO-ALLEGROcorrelationexperimentiscomple-
background, although the dominant contribution is un-
mentary to experiments using data from the two LIGO
likely to have a blackbody spectrum. A SGWB can be
sites, in that it is sensitive to a SGWB at frequencies of
characterized as cosmological or astrophysical in origin.
around 900 Hz rather than 100 Hz. Targeted sources are
Cosmological backgrounds can arise from, for example,
thus thosewith a relatively narrow-band spectrum peaked
pre-big-bangmodels[4–6],amplificationofquantumvac-
near 900 Hz. Spectra with such shapes can arise from
uum fluctuations during inflation [7–9], phase transitions
[10,11], and cosmic strings [12–14]. Astrophysical back-
grounds consist of a superposition of unresolved sources,
Avg Calibrated ASD from S4 non−NULL non−PG
which can include rotating neutron stars [15,16], super- 10−18
novae [17], and low-mass x-ray binaries [18]. L1 ASD
The standard cross-correlation search [19] for a SGWB AS1p eAcStrDum for Ω (f)=1.02
gw
necessarily requires two or more GW detectors. Such z)10−19
searches have been performed using two resonant-bar de- √H
tectors [20] and also using two or more kilometer-scale ain /
GW interferometers (IFOs) [21–23]. The present work Str10−20
m (
describes the results of the first cross-correlation analysis u
ctr
carried out between an IFO [the 4 km IFO at the LIGO e
Sp10−21
Livingston Observatory (LLO), known as L1] and a bar e
d
u
(the cryogenic ALLEGRO detector, referred to as A1). plit
m
This pair of detectors is separated by only 40 km, the A10−22
closest pair among modern ground-based GW detector
sites,whichallowsittoprobethestochasticGWspectrum
around 900 Hz. In addition, the ALLEGRO bar can be 10−23
850 860 870 880 890 900 910 920 930 940 950
rotated, changing the response of the correlated data Frequency (Hz)
streams to stochastic GWs and thus providing a means to
FIG. 1. Sensitivity of the LLO IFO (L1) and the ALLEGRO
distinguishcorrelationsduetoaSGWBfromthosedueto
bar(A1)duringS4,alongwithstrainassociatedwith(cid:1) (cid:1)f(cid:2)(cid:7)
gw
correlated environmental noise [24]. This paper describes
1:02 (assuming a Hubble constant of H (cid:7)72 km=s=Mpc).
0
cross-correlationanalysisofL1andA1datatakenbetween [There are two (cid:1) (cid:1)f(cid:2)(cid:7)1:02 curves, corresponding to the
gw
different strain levels such a background would generate in an
*Member of ALLEGROCollaboration. IFO and a bar, as explained in Sec. II and [26].] The quantity
†Member of LIGO Scientific Collaboration and ALLEGRO plotted is amplitude spectral density (ASD), the square root of
Collaboration. the one-sided power spectral density defined in (4.2), at a
‡http://www.ligo.org/ resolution of 0.25 Hz.
022001-3
B. ABBOTTet al. PHYSICALREVIEW D76, 022001 (2007)
exotic cosmological models, as described in Sec. II, or is the overlap reduction function (ORF) [25] between the
from astrophysical populations [16]. two detectors, defined in terms of the projector PTTn^ab
cd
The organization of this paper is as follows. Section II onto traceless symmetric tensors transverse to the unit
reviews the properties and characterization of a SGWB. vector n^. The ORF for several detector pairs of interest is
SectionIIIdescribestheLLOandALLEGROexperimen- shownin Fig.2.
tal arrangements, including the data acquisition andstrain S (cid:1)f(cid:2) is the one-sided spectrum of the SGWB. This is
gw
calibration for each instrument. Section IV describes the theone-sidedpowerspectraldensity(PSD)thebackground
cross-correlation methodanditsapplication tothepresent would generate in an interferometer with perpendicular
situation. Section V describes the details of the postpro- arms, which can be seen from (2.4) and the fact that the
cessingmethods andstatistical interpretation of the cross- ORFofsuchaninterferometerwithitselfisunity.Sincethe
correlation results. Section VI describes the results of the ORF of a resonant bar with itself is 4=3 (see [26] and
cross-correlation measurement and the corresponding Sec. VIIA for more details), the PSD of the strain mea-
upper limit on the SGWB strength in the range 850– sured by a bar detector due to the SGWB would be
950 Hz. Section VII describes the results of our analysis (cid:1)4=3(cid:2)S (cid:1)f(cid:2).
gw
pipeline when applied to simulated signals injected both A related measure of the spectrum is the dimensionless
within the analysis software and in the hardware of the quantity(cid:1) (cid:1)f(cid:2),theGWenergydensityperunitlogarith-
gw
instrumentsthemselves.SectionVIIIcomparesourresults
mic frequency divided by the critical energy density (cid:4)
c
tothoseofpreviousexperimentsandtothesensitivitiesof
needed to close the universe:
other operating detector pairs. Section IX considers the
prospects for futurework. f d(cid:4) 10(cid:1)2
(cid:1) (cid:1)f(cid:2)(cid:7) gw (cid:7) f3S (cid:1)f(cid:2): (2.6)
gw (cid:4) df 3H2 gw
c 0
II.STOCHASTICGRAVITATIONAL-WAVE
BACKGROUNDS Note that the definition (cid:1) (cid:1)f(cid:2) thus depends on thevalue
gw
A gravitational wave (GW) is described by the metric
Overlap Reduction Function
tensorperturbationh (cid:1)r~;t(cid:2).AgivenGWdetector,located
ab
at position r~det on the Earth, will measure a GW strain 1
which,inthelong-wavelengthlimit, issomeprojectionof LLO−LHO
0.8
this tensor: °
LLO−ALLEGRO (N108W) "XARM"
0.6
h(cid:1)t(cid:2)(cid:7)h (cid:1)r~ ;t(cid:2)dab; (2.1) LLO−ALLEGRO (N18°W) "YARM"
ab det 0.4 °
LLO−ALLEGRO (N63W) "NULL"
where dab is the detector response tensor, which is 0.2
dab (cid:7)1(cid:1)x^ax^b(cid:5)y^ay^b(cid:2) (2.2) 0
(cid:1)ifo(cid:2) 2
−0.2
foraninterferometerwitharmsparalleltotheunitvectors
−0.4
x^ and y^ and
−0.6
dab (cid:7)u^au^b (2.3)
(cid:1)bar(cid:2) −0.8
foraresonantbarwithlongaxisparalleltotheunitvector −1
u^.
0 100 200 300 400 500 600 700 800 900 1000
AstochasticGWbackground(SGWB)canarisefroma Frequency (Hz)
superposition of uncorrelated cosmological or astrophysi-
FIG. 2. The overlap reduction function for LIGO Livingston
cal sources. Such a background, which we assume to be
Observatory (LLO) with ALLEGRO and with LIGO Hanford
isotropic, unpolarized, stationary, and Gaussian, will gen-
Observatory (LHO). The three LLO-ALLEGRO curves corre-
erate a cross correlation between the strains measured by spond to the three orientations in which ALLEGRO was oper-
twodetectors.IntermsofthecontinuousFouriertransform ated during LIGO’s S4 run: ‘‘XARM’’ (N108(cid:6)W) is nearly
defined by a~(cid:1)f(cid:2)(cid:7)R1 dta(cid:1)t(cid:2)exp(cid:1)(cid:5)i2(cid:1)ft(cid:2),the expected parallel to the x-arm of LLO (‘‘aligned’’); ‘‘YARM’’ (N18(cid:6)W)
(cid:5)1
crosscorrelation is is nearly parallel to the y-arm of LLO (‘‘antialigned’’); NULL
(N63(cid:6)W)ishalfway inbetween these twoorientations (a ‘‘null
hh~(cid:8)1(cid:1)f(cid:2)h~2(cid:1)f0(cid:2)i(cid:7)12(cid:2)(cid:1)f(cid:5)f0(cid:2)Sgw(cid:1)f(cid:2)(cid:3)12(cid:1)f(cid:2); (2.4) alignment’’ midway between the twoLLO arms). Note that for
nonzerofrequencies,theseparationvectorbetweenthetwosites
where
breaks the symmetry between the XARM and YARM align-
5 ZZ ments,andleadstoanoffsetoftheNULLcurve,asdescribedin
(cid:3)12(cid:1)f(cid:2)(cid:7)d1abdc2d4(cid:1) d2(cid:1)n^PTTn^abcdei2(cid:1)fn^(cid:9)(cid:1)r~2(cid:5)r~1(cid:2)=c [26]. The LLO-LHO overlap reduction function is shown for
reference.Thefrequencybandofthepresentanalysis,850 Hz(cid:3)
(2.5) f(cid:3)950 Hz, is indicated with dashed vertical lines.
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FIRST CROSS-CORRELATIONANALYSISOF... PHYSICALREVIEW D76, 022001 (2007)
of the Hubble constant H . Most SGWB literature avoids length nearly zero. It is this error signal q(cid:1)t(cid:2) which is
0
this artificial uncertainty by working in terms of recorded, and its relationship to the strain estimate s(cid:1)t(cid:2) is
most easily described in the Fourier domain:
(cid:1) H (cid:2)2
h2 (cid:1) (cid:1)f(cid:2)(cid:7) 0 (cid:1) (cid:1)f(cid:2) (2.7)
100 gw 100 km=s=Mpc gw s~(cid:1)f(cid:2)(cid:7)R~(cid:1)f(cid:2)q~(cid:1)f(cid:2): (3.2)

rather than (cid:1) (cid:1)f(cid:2) itself. We will instead follow the pre- TheresponsefunctionR~(cid:1)f(cid:2)isestimated byacombination
gw
cedent set by [22] and quote numerical values for (cid:1) (cid:1)f(cid:2) ofmodelingandmeasurement[32]andvariesslowlyover
gw
assuming a Hubble constant of 72 km=s=Mpc. the course of the experiment.
Avarietyofspectralshapeshavebeenproposedfor(cid:1) , Because the error signal q(cid:1)t(cid:2) has a smaller dynamic
gw
range than the reconstructed strain s(cid:1)t(cid:2),ouranalysis starts
for both astrophysical and cosmological stochastic back-
fromthedigitizedtimeseriesq(cid:11)k(cid:12)(cid:7)q(cid:1)t (cid:2)(sampled214 (cid:7)
grounds [3,27,28]. For example, whereas the slow-roll k
16384 times per second, and digitally downsampled to
inflationarymodelpredictsaconstant(cid:1) (cid:1)f(cid:2)inthebands
gw
4096 Hz in the analysis) and reconstructs the LLO strain
of LIGO or ALLEGRO, certain alternative cosmological
only in the frequency domain.
modelspredictbroken-power-lawspectra,wheretherising
and falling slopes and the peak frequency are determined
B.The ALLEGRO resonant-bardetector
by model parameters [3]. String-inspired pre-big-bang
cosmological models belong to this category [5,29]. For The ALLEGRO resonant detector, operated by a group
certain ranges of these three parameters, the LLO- from Louisiana State University [33], is a two-ton alumi-
ALLEGRO correlation measurement offers the best con- num cylinder coupled to a niobium secondary resonator.
straintsontheorythatcanbeinferredfromanycontempo- Thesecondaryresonatorispartofaninductivetransducer
rary observation. This can happen, e.g., if the power-law [34] which is coupled to a DC SQUID. Strain along the
exponent on the rising spectral slope is greater than 3and cylindrical axis excites the first longitudinal vibrational
the peak frequency is sufficiently close to 900 Hz [30]. mode of the bar. The transducer is tuned for sensitivity to
thismechanicalmode.Rawdataacquiredfromthedetector
thusreflectthehigh-Qresonantmechanicalresponseofthe
III.EXPERIMENTALSETUP
system.Amajortechnicalchallengeofthisanalysisisdue
A. TheLIGO Livingstoninterferometer totheextenttowhichthebardatadifferfromthoseofthe
interferometer.
The experimental setup of the LIGO observatories has
beendescribedatlengthelsewhere[31].Hereweprovidea
brief review, with particular attention paid to details sig- 1.Data acquisition, heterodyning, andsampling
nificant for the LLO-ALLEGRO cross-correlation The ALLEGRO detector has a relatively narrow sensi-
measurement. tive band of (cid:13)100 Hz centered around (cid:13)900 Hz near the
The LIGO Livingston Observatory (LLO) is an inter- twonormalmodesofthemechanicalbar-resonatorsystem.
ferometric GW detector with perpendicular 4-km arms. For this reason, the output of the detector can first be
The laser interferometer senses directly any changes in heterodynedwithacommerciallock-inamplifiertogreatly
the differential arm length. It does this by splitting a light reduce the sampling rate, which is set at 250 samples=s.
beam at thevertex, sending the separate beams into 4-km Boththein-phaseandquadratureoutputsofthelock-inare
long optical cavities of their respective arms, and then recordedandthedetectoroutputcanthusberepresentedas
recombiningthebeamstodetectanychangeintheoptical acomplextimeserieswhichcoversa250Hzbandcentered
phasedifferencebetweenthearms,whichisequivalenttoa on the lock-in reference oscillator frequency. This refer-
difference in light travel time. This provides a measure- ence frequency is chosen to be near the center of the
ment of h(cid:1)t(cid:2) as defined in (2.1) and (2.2). However, the sensitive band, and during the S4 run it was set to
measured quantity is not exactly h(cid:1)t(cid:2) for two reasons. 904 Hz. The overall timing of data heterodyned in this
First, there are local forces which perturb the test fashion is provided by both the sampling clock and the
masses, and so produce changes in arm length. There are reference oscillator. Both time bases were locked to the
also optical and electronic fluctuations that mimic real Global Positioning System (GPS)time reference.
strains. The combination of these effects causes a strain The nature of the resonant detector and its data acquis-
noise n(cid:1)t(cid:2) to always be present in the output, producing a ition system gives rise to a number of timing issues:
measurement of heterodyning, filter delays of the electronics, and the tim-
ingofthedataacquisitionsystemitself[35].Itisofcritical
s(cid:1)t(cid:2)(cid:7)h(cid:1)t(cid:2)(cid:10)n(cid:1)t(cid:2): (3.1)
importance that the timing be fully understood so that the
Second, the test masses are not really free. There is a phase of any potential signal may be recovered.
servo system, which uses changes in the differential arm Convincing evidence that all of the issues are accounted
lengthasitserrorsignalq(cid:1)t(cid:2),andthenappliesextra(‘‘con- forisdemonstratedbytherecoveryandcrosscorrelationof
trol’’)forcestothetestmassestokeepthedifferentialarm test signals simultaneously injected into both detectors.
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B. ABBOTTet al. PHYSICALREVIEW D76, 022001 (2007)
Raw spectrum, S4 run − averaged over orientation calibrated strain time series,s(cid:1)t(cid:2),andusethat astheinput
104
to the cross-correlation analysis.
XARM
YARM Thecalibrationprocedure,describedindetailin[35],is
NULL carried out in the frequency domain and consists of the
Hz)103 following: A 30 min stretch of clean ALLEGRO data is
√
s / windowedandFouriertransformed.Themechanicalmode
unt frequencies drift slightly due to small temperature varia-
o
c
m ( tions,sothesefrequencies aredetermined foreach stretch
ctru102 andthoseareincorporatedintothemodelofthemechani-
e
p calresponseofthesystemtoastrain.Themodelconsistsof
S
de two double poles at these normal mode frequencies. In
u
mplit101 addition to this response, we must then account for the
A phase shifts due to the time delays in the lock-in and
antialiasing filters.
After applying the full response function, the data are
100 theninverseFouriertransformedbacktothetimedomain.
840 860 880 900 920 940 960
frequency (Hz) The next 50% overlapping 30 min segment is then taken.
The windowed segments are stitched together until the
FIG. 3. The graph displays the amplitude spectral density of
entire continuous stretch of good data is completed. The
raw ALLEGRO detector output during S4, at a frequency
firstandlast15minutesaredropped.Theresultrepresents
resolution of 0.1 Hz. For this graph these data have not been
transformed to strain via the calibrated rpesponse function. The aheterodynedcomplextimeseriesofstrain,whoseampli-
vertical scale represents digital counts= (cid:3)H(cid:3)(cid:3)(cid:3)z(cid:3)(cid:3). The normal me- tude spectral density is shownin Fig.1.
chanical modes where the detector is most sensitive are at Theoverallscaleofthedetectoroutputintermsofstrain
880.78 Hz and 917.81 Hz. There is an injected calibration line is determined by applying a known signal to the bar. A
at837Hz.Alsoprominentareanextramechanicalresonanceat forceappliedtooneendofthebarhasasimpletheoretical
885.8 Hz and a peak at 904 Hz (DC in the heterodyned data relationship to an equivalent gravitational strain [35–37].
stream).
A calibrated force can be applied via a capacitive ‘‘force
generator’’ which also provides the mechanism used for
The signals were recovered at the expected phase as pre- hardware signal injections. A reciprocal measurement—
sented in Sec. VII. excitation followed by measurement with the same trans-
ducer—along with known properties of the mechanical
system, allows the determination of the force generator
2.Straincalibration constant. With that constant determined (with units of
newtons per volt), a calibrated force is applied to the bar
The rawdetector output is proportional to the displace-
and the overall scale of the response determined.
ment of the secondary resonator, and thus has a spectrum
withsharplinefeaturesduetothehigh-Qresonancesofthe
bar-resonatorsystem ascan beseen in Fig.3.The desired 3. Orientation
GWsignalistheeffectivestrainonthebar,andrecovering A unique feature of this experiment is the ability to
this means undoing the resonant response of the detector. rotate the ALLEGRO detector and modulate the response
This response has a long coherence time—thus long of the ALLEGRO-LLO pair to a GW background [24].
stretches of data are needed to resolve the narrow lines in Data were taken in three different orientations of
therawdata.Thestraindatahaveamuchflatterspectrum, ALLEGRO, known as XARM, YARM, and NULL, de-
asshowninFig.1.Thereforeitispracticaltogeneratethe tailed in Table I. As shown in Fig. 2 and (2.4), these
TABLE I. OrientationsofALLEGROduringtheLIGOS4Run,includingoverlapreduction functionevaluated attheextremesof
the analyzed frequency range, and at the frequency of peak sensitivity. Note that, while the NULL orientation represents perfect
misalignment((cid:3)(cid:7)0)at0Hz,itisnotquiteperfectatthefrequenciesofinterest.Thisisprimarilybecauseofanazimuth-independent
offsettermin(cid:3)(cid:1)f(cid:2)whichcontributesatnonzerofrequencies[24,26].Becauseofthisterm,itisimpossibletoorientALLEGROsothat
(cid:3)(cid:1)f(cid:2)(cid:7)0atallfrequencies,andtosetittozeroaround915HzonewouldhavetouseanazimuthofN62(cid:6)WratherthanN63(cid:6)W.This
subtlety was not incorporated into the choice of orientations in S4, but the approximate cancellation is adequate for our purposes.
Dates Orientation Azimuth (cid:3)(cid:1)850 Hz(cid:2) (cid:3)(cid:1)915 Hz(cid:2) (cid:3)(cid:1)950 Hz(cid:2)
2005 Feb 22–2005 Mar 4 YARM N108(cid:6)W (cid:5)0:9087 (cid:5)0:8947 (cid:5)0:8867
2005 Mar4–2005 Mar18 XARM N18(cid:6)W 0.9596 0.9533 0.9498
2005 Mar18–2005 Mar 23 NULL N63(cid:6)W 0.0280 0.0318 0.0340
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FIRST CROSS-CORRELATIONANALYSISOF... PHYSICALREVIEW D76, 022001 (2007)
orientations correspond to different pair responses due to In the continuous-time idealization, such a cross-
different overlap reduction functions. In the XARM ori- correlation statistic, calculated over a time T, has an ex-
entation—the bar axis parallel to the x-arm of the inter- pected mean
ferometer—a GW signal produces positive correlation
T Z1
between the data in the two detectors. In the YARM (cid:5) (cid:7)hYci(cid:14) df(cid:3)(cid:1)jfj(cid:2)S (cid:1)f(cid:2)Q~(cid:1)f(cid:2) (4.5)
orientation a GW signal produces an anticorrelation. In Yc 2 (cid:5)1 gw
the NULL orientation—the bar aligned halfway between and variance
the two arms of the interferometer—the pair has very
T Z1
nearly zero sensitivity as a GW signal produces almost (cid:6)2 (cid:7)h(cid:1)Yc(cid:5)(cid:5) (cid:2)2i(cid:14) dfP (cid:1)f(cid:2)P (cid:1)f(cid:2)jQ~(cid:1)f(cid:2)j2:
zero correlation between the detectors’data. Areal signal Yc Yc 4 (cid:5)1 1 2
is thus modulated whereas many types of instrumental (4.6)
correlation would not have the same dependence on Using(4.5)and(4.6),theoptimalchoiceforthefilterQ~(cid:1)f(cid:2),
orientation.
given a predicted shape for the spectrum S (cid:1)f(cid:2) can be
gw
shown[19] to be
IV.CROSS-CORRELATIONMETHOD
(cid:3)(cid:1)jfj(cid:2)S (cid:1)f(cid:2)
Thissectiondescribesthemethodtousedtosearchfora Q~(cid:1)f(cid:2)/ gw : (4.7)
P (cid:1)f(cid:2)P (cid:1)f(cid:2)
1 2
SGWBbycross-correlatingdetectoroutputs.Inthecaseof
L1-A1 correlation measurements, it is complicated by the If the spectrum of gravitational waves is assumed to be a
different sampling rates for the two instruments and the power law over the frequency band of interest, a conve-
fact that the A1 data are heterodyned at 904 Hz prior to nient parametrization of the spectrum, in terms of (cid:1) (cid:1)f(cid:2)
gw
digitization. defined in (2.6), is
(cid:1)f(cid:2)(cid:7)
A.Continuous-timeidealization (cid:1)gw(cid:1)f(cid:2)(cid:7)(cid:1)R f ; (4.8)
R
Both ground-based interferometric and resonant-mass
where f us a conveniently chosen reference frequency
detectorsproduceatime-seriesoutputwhichcanberelated R
and(cid:1) (cid:7)(cid:1) (cid:1)f (cid:2).Thecross-correlationmeasurementis
to a discrete sampling of the signal R gw R
then a measurement of (cid:1) , and if the optimal filter is
R
si(cid:1)t(cid:2)(cid:7)hi(cid:1)t(cid:2)(cid:10)ni(cid:1)t(cid:2); (4.1) normalized according to

wgrhaevrietaitiloanbaell-swtahveedsettreacitnord(e1finoerd2iinn(t2h.i1s)c,aasned),nhi(cid:1)(cid:1)tt(cid:2)(cid:2)iisstthhee Q~(cid:1)f(cid:2)(cid:7)N (cid:3)(cid:1)f(cid:2)(cid:1)f=fR(cid:2)(cid:7) ; (4.9a)
i jfj3P (cid:1)f(cid:2)P (cid:1)f(cid:2)
1 2
instrumentalnoiseassociatedwitheachdetector,converted
into anequivalent strain.The detector output ischaracter- where
ized by its power spectral density P(cid:1)f(cid:2),
i 20(cid:1)2(cid:1)Z1 df (cid:11)(cid:3)(cid:1)f(cid:2)(cid:1)f=f (cid:2)(cid:7)(cid:12)2(cid:2)(cid:5)1
N (cid:7) R ; (4.9b)
hs~(cid:8)i(cid:1)f(cid:2)s~i(cid:1)f0(cid:2)i(cid:7)12(cid:2)(cid:1)f(cid:5)f0(cid:2)Pi(cid:1)f(cid:2); (4.2) 3H02 (cid:5)1 f6 P1(cid:1)f(cid:2)P2(cid:1)f(cid:2)
which should be dominated by the autocorrelation of the then the expected statistics of Yc in the presence of a
noise [hs~(cid:8)(cid:1)f(cid:2)s~(cid:1)f0(cid:2)i(cid:14)hn~(cid:8)(cid:1)f(cid:2)n~(cid:1)f0(cid:2)i]. If the instrument background of actual strength (cid:1) are
i i i i R
noise is approximately uncorrelated, the expected cross
(cid:5) (cid:7)(cid:1) T (4.10)
correlation of the detector outputs is [cf. (2.4)] Yc R
and
hs~(cid:8)(cid:1)f(cid:2)s~ (cid:1)f0(cid:2)i(cid:14)hh~(cid:8)(cid:1)f(cid:2)h~ (cid:1)f0(cid:2)i(cid:7)1(cid:2)(cid:1)f(cid:5)f0(cid:2)S (cid:1)f(cid:2)(cid:3) (cid:1)f(cid:2)
1 2 1 2 2 gw 12(4.3) (cid:6)2 (cid:7)T(cid:1)10(cid:1)2(cid:2)2(cid:1)Z1 df (cid:11)(cid:3)(cid:1)f(cid:2)(cid:1)f=fR(cid:2)(cid:7)(cid:12)2(cid:2)(cid:5)1 (4.11)
which can be used along with the autocorrelation (4.2) to Yc 3H02 (cid:5)1 f6 P1(cid:1)f(cid:2)P2(cid:1)f(cid:2)
determinethestatisticalpropertiesofthecross-correlation and a measurement of Yc=T provides a point estimate of
statistic defined below. the background strength (cid:1) with associated estimated
R
We use the optimally filtered cross-correlation method error bar (cid:6) =T.
Yc
described in[19,21]tocalculate across-correlation statis-
ticwhichisanapproximationtothecontinuous-timecross-
B. Discrete-time method
correlation statistic
1.Handlingofdifferentsamplingratesandheterodyning
Z
Yc (cid:7) dt dt s (cid:1)t (cid:2)Q(cid:1)t (cid:5)t (cid:2)s (cid:1)t (cid:2) Stochastic-background measurements using pairs of
1 2 1 1 1 2 2 2

LIGO interferometers [21] have implemented (4.4) from
(cid:7)Z dfs~(cid:8)(cid:1)f(cid:2)Q~(cid:1)f(cid:2)s~ (cid:1)f(cid:2): (4.4) discretesamplingssi(cid:11)k(cid:12)(cid:7)s(cid:1)t0(cid:10)k(cid:2)t(cid:2)asfollows:Firstthe
1 2 continuous Fourier transforms s~(cid:1)f(cid:2) were approximated
022001-7
B. ABBOTTet al. PHYSICALREVIEW D76, 022001 (2007)
using the discrete Fourier transforms of windowed and functionR~ (cid:1)f(cid:2)constructedasdescribedinSec.IIIA.
1
zero-padded versions of the discrete time series; then an (ii) ForA1,acomplextimeseriesfsh(cid:11)k(cid:12)jk(cid:7)0;...N (cid:5)
2 2
optimal filter was constructed using an approximation to 1g, sampled at (cid:1)(cid:2)t (cid:2)(cid:5)1 (cid:7)250 Hz, consisting of
2
(4.7),andfinallytheproductofthethreewassummedbin- N (cid:7)T=(cid:2)t (cid:7)15000 points. This is calibrated to
2 2
by-bin to approximate the integral over frequencies. The represent strain data asdescribed inSec.IIIB2,but
discreteversionof Q~(cid:1)f(cid:2) was simplified in twoways: first, still heterodyned.
because of the averaging used in calculating the power To produce an approximation of the Fourier transform of
spectrum, the frequency resolution on the optimal filter the data from detector i, the data are multiplied by an
wasgenerallycoarserthanthatassociatedwiththediscrete appropriatewindowingfunction,zeropaddedtotwicetheir
Fourier transforms of the data streams, and second, the original length, discrete-Fourier-transformed, and multi-
valueoftheoptimalfilterwasarbitrarilysettozerooutside plied by (cid:2)ti. In addition, the L1 data are multiplied by a
some desired range of frequencies f (cid:3)f (cid:3)f . This calibration responsefunction, while the A1data are inter-
min max
wasjustifiedbecausetheoptimalfiltertendedtohavelittle preted as representing frequencies appropriate in light of
support for frequencies outside that range. the heterodyne. For L1,
The present experiment has two additional complica-
N (cid:5)1
tions associated with the discretization of the time-series s~ (cid:1)f (cid:2)(cid:13)s~ (cid:11)‘(cid:12):(cid:7)R~ (cid:1)f (cid:2) X1 w (cid:11)j(cid:12)q (cid:11)j(cid:12)
1 ‘ 1 1 ‘ 1 1
data. First, the A1 data are not a simple time sampling of
j(cid:7)0
the gravitational-wave strain, but are base banded with a (cid:1)(cid:5)i2(cid:1)‘j(cid:2)
heterodyning frequency fh (cid:7)904 Hz as described in (cid:4)exp (cid:2)t1;
2 2N
Sec. IIIB1 and IIIB2. Second, the A1 data are sampled 1
at (cid:1)(cid:2)t2(cid:2)(cid:5)1 (cid:7)250 Hz, while the L1 data are sampled at ‘(cid:7)(cid:5)N1;...;N1(cid:5)1; (4.12)
16384 Hz, and subsequently downsampled to (cid:1)(cid:2)t (cid:2)(cid:5)1 (cid:7)
1 where f (cid:7) ‘ is the frequency associated with the ‘th
4096 Hz.Thiswouldmakeatimedomaincrosscorrelation ‘ 2T
frequencybin.InthecaseofA1,theidentificationisoffset
extremely problematic, as it would necessitate a large
by ‘h (cid:7)f (cid:9)(cid:1)2T(cid:2)(cid:7)107880:
variety of time offsets t1(cid:5)t2. In the frequency domain, 2 h
it means that the downsampled L1 data, once calibrated,
N (cid:5)1
representfrequenciesrangingfrom(cid:5)2048 Hzto2048Hz, s~ (cid:1)f (cid:2)(cid:13)s~ (cid:11)‘(cid:12):(cid:7) X2 w (cid:11)k(cid:12)q (cid:11)k(cid:12)
2 ‘ 2 2 2
whilethecalibratedA1datarepresentfrequenciesranging k(cid:7)0
from (cid:1)904(cid:5)125(cid:2) Hz(cid:7)779 Hz to (cid:1)904(cid:10)125(cid:2) Hz(cid:7) (cid:1)(cid:5)i2(cid:1)(cid:1)‘(cid:5)‘h(cid:2)k(cid:2)
1029 Hz. These different frequency ranges do not pose a (cid:4)exp 2N 2 (cid:2)t2;
problem,aslongastherangeoffrequencieschosenforthe 2
integral satisfies fmin>779 Hz and fmax<1029 Hz. ‘(cid:7)‘h2 (cid:5)N2;...;‘h2 (cid:10)N2(cid:5)1: (4.13)
Another requirement is that, for the chosen frequency
As is shown in [38], if we construct a cross-correlation
resolution, the A1 data heterodyne reference frequency
statistic
must align with a frequency bin in the L1 data. This is
sreafteisrfienedcefofrreqinuteengceire-ss.ecWonitdh dthaetasescpoanndsitainodns,inttheegeFro-huerrietzr Y :(cid:7) ‘Xmax 1 (cid:11)s~ (cid:1)f (cid:2)(cid:12)(cid:8)Q~(cid:1)f (cid:2)s~ (cid:1)f (cid:2); (4.14)
2T 1 ‘ ‘ 2 ‘
transforms of the A1 and L1 data are both defined over a ‘(cid:7)‘
min
commonsetoffrequencies,asdetailedin[38].Lookingat
the expected mean value in the presence of a stochastic
the A1 spectrum in Fig. 1, a reasonable range offrequen-
background with spectrum S (cid:1)f(cid:2) is
cies should be a subset of the range 850 Hz&f & gw
950 Hz. (cid:5):(cid:7)hYi(cid:14)w w T ‘Xmax 1 (cid:11)Q~(cid:1)f (cid:2)(cid:12)(cid:3)(cid:1)f (cid:2)S (cid:1)f (cid:2);
1 22 2T ‘ ‘ gw ‘
2.Discrete-time crosscorrelation ‘(cid:7)‘min
(4.15)
Explicitly,thetimeseriesinputstotheanalysispipeline,
from each T (cid:7)60 sec of analyzed data, are: where w w is an average of the product of the two
1 2
(i) For L1, a real time series fq1(cid:11)j(cid:12)jj(cid:7)0;...N1(cid:5)1g, windows, calculated using the points which exist at both
sampled at (cid:1)(cid:2)t1(cid:2)(cid:5)1 (cid:7)4096 Hz, consisting of N1 (cid:7) sampling rates; specifically, if r1 and r2 are the smallest
T=(cid:2)t1 (cid:7)245760 points. This is obtained by down- integers suchthat (cid:2)t1=(cid:2)t2 (cid:7)r1=r2 (cid:7)125=2048,
sampling the raw data stream by a factor of 4. The
data are downsampled to 4096 Hz rather than w w (cid:7)r1r2 N=(cid:1)rX1r2(cid:2)(cid:5)1w (cid:11)nr (cid:12)w (cid:11)nr (cid:12): (4.16)
2048 Hz to ensure that the roll-off of the associated 1 2 N 1 2 2 1
n(cid:7)0
antialiasing filter is outside the frequency range
being analyzed. The raw L1 data are related to Note that, while the average value given by (4.15) is
gravitational-wavestrainbythecalibrationresponse manifestly real, any particular measurement of Y will be
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FIRST CROSS-CORRELATIONANALYSISOF... PHYSICALREVIEW D76, 022001 (2007)
complex, because of the bandpass associated with the V.POSTPROCESSINGTECHNIQUES
heterodyning of the A1 data. As shown in [38], the real
A.Stationarity cut
and imaginary parts of the cross-correlation statistic each
have expected variance The sliding power spectrum estimation method de-
scribed in Sec. IVB4 can lead to inaccurate results if the
noise level of one or both instruments varies widely over
1 successive intervals. Most problematically, if the data are
(cid:6)2 :(cid:7) hY(cid:8)Yi
2 noisyonlywithinasingleanalysissegment,consideration
(cid:14)Tw2w2 ‘Xmax 1 jQ~(cid:1)f (cid:2)j2P (cid:1)jf j(cid:2)P (cid:1)jf j(cid:2); (4.17) obfefothree apnodwaefrtesr,pewchtriucmh arceonnsottruncoteisdy,fwroimll ctahuesesethgemseengts-
8 1 2 2T ‘ 1 ‘ 2 ‘
‘(cid:7)‘ ment to be overweighted when combining cross-
min
correlation data from different segments. To avoid this,
we calculate for each segment both the usual estimated
whereonceagainw2w2isanaverageoverthetimesamples standard deviation (cid:6) using the ‘‘sliding’’ PSD estimator
1 2 I
the twowindowshave ‘‘in common’’: andthe‘‘naive’’estimatedstandarddeviation(cid:6)0 usingthe
I
datafromthesegmentitself.Iftheratioofthesetwoistoo
far from unity, the segment is omitted from the cross-
w2w2 (cid:7)r1r2 N=(cid:1)rX1r2(cid:2)(cid:5)1(cid:1)w (cid:11)nr (cid:12)(cid:2)2(cid:1)w (cid:11)nr (cid:12)(cid:2)2: (4.18) correlation analysis. The threshold used for this analysis
1 2 N 1 2 2 1 was 20%. The amount of data excluded based on this cut
n(cid:7)0
wasbetween1%and2%ineach ofthethreeorientations,
andsubsequentinvestigationsshowthefinalresultswould
not change significantly for any reasonable choice of
3. Constructionof the optimal filter threshold.
To perform the cross correlation in (4.14), we need to
B.Bias correction of estimated error bars
construct an optimal filter by a discrete approximation to
(4.9). We approximate the power spectra P (cid:1)f(cid:2) using Although use of the sliding power spectrum estimator
1
Welch’s method [39]; as a consequence of the averaging removes any bias from the optimally filtered cross-
ofperiodogramsconstructedfromshorterstretchesofdata, correlation measurement, the methods of [40] show that
thepowerspectraareestimatedwithafrequencyresolution there is still a slight underestimation of the estimated
(cid:2)f which is coarser than the 1=2T associated with the standard deviation associated with the finite number of
construction in (4.14). As detailed in [21], we handle this periodograms averaged together in calculating the power
by first multiplying together (cid:11)s~ (cid:1)f (cid:2)(cid:12)(cid:8) and s~ (cid:1)f (cid:2) at the spectrum.Tocorrectforthis,wehavetoscaleuptheerror
1 ‘ 2 ‘
finer frequency resolution, then summing together sets of barsbyafactorof1(cid:10)1=(cid:1)N (cid:4)9=11(cid:2),whereN isthe
avg avg
2T (cid:2)f bins and multiplying them by the coarser-grained number subsegments whose periodograms are averaged
optimal filter. For our search, (cid:2)f (cid:7)0:25 Hz and T (cid:7) together in estimating the power spectrum for the optimal
60 sec, so2T(cid:2)f (cid:7)30. filter. For the data analyzed with the sliding power spec-
trum estimator, 29 overlapping four-second subsegments
4. Power spectrum estimation are averaged from each of two 60-second segments, for a
total N (cid:7)58. This gives a correction factor of 1(cid:10)
BecausethenoisepowerspectrumoftheLLOcanvary avg
1=(cid:1)58(cid:4)9=11(cid:2)(cid:7)1:021. The naive estimated sigmas, de-
withtime,wecontinuouslyupdatetheoptimalfilterusedin
rivedfrompowerspectracalculatedusing29averagesina
thecross-correlationmeasurement. However,usinganop-
single60-secondsegment,arescaledupbyafactorof1(cid:10)
timalfilterconstructedfrompowerspectracalculatedfrom
1=(cid:1)29(cid:4)9=11(cid:2)(cid:7)1:042.
the same data to be analyzed leads to a bias in the cross-
correlationstatisticY,asdetailedin[40].Toavoidthis,we
analyzeeachT (cid:7)60 secsegmentofdatausinganoptimal C.Combinationof analysissegment results
filter constructed from the average of the power spectra Asshownin[19],theoptimalwaytocombineaseriesof
from the segments before and after the segment to be independent cross-correlation measurements fY g with as-
I
analyzed. This method is known as ‘‘sliding power spec- sociated one-sigma error bars is
trum estimation’’ because, as we analyze successive seg-
P(cid:6)(cid:5)2Y
mentsofdata,thesegmentsusedtocalculatedthePSDsfor I I
theoptimalfilterslidethroughthedatatoremainadjacent. Yopt (cid:7) I (5.1a)
P(cid:6)(cid:5)2
The (cid:2)f (cid:7)0:25 Hz resolution is obtained by calculating I
I
the power spectra using Welch’s method with 29 over- (cid:1)X (cid:2)(cid:5)1=2
lapped 4-second subsegments in each 60-second segment (cid:6)Yopt (cid:7) (cid:6)(cid:5)I 2 : (5.1b)
of data, for a total of 58 subsegments. I
022001-9