of 13
SEARCHING FOR A STOCHASTIC BACKGROUND OF GRAVITATIONAL WAVES WITH THE LASER
INTERFEROMETER GRAVITATIONAL-WAVE OBSERVATORY
B. Abbott,
1
R. Abbott,
1
R. Adhikari,
1
J. Agresti,
1
P. Ajith,
2
B. Allen,
3
R. Amin,
4
S. B. Anderson,
1
W. G. Anderson,
3
M. Araya,
1
H. Armandula,
1
M. Ashley,
5
S Aston,
6
C. Aulbert,
7
S. Babak,
7
S. Ballmer,
8
B. C. Barish,
1
C. Barker,
9
D. Barker,
9
B. Barr,
10
P. Barriga,
11
M. A. Barton,
1
K. Bayer,
8
K. Belczynski,
12
J. Betzwieser,
8
P. Beyersdorf,
13
B. Bhawal,
1
I. A. Bilenko,
14
G. Billingsley,
1
E. Black,
1
K. Blackburn,
1
L. Blackburn,
8
D. Blair,
11
B. Bland,
9
L. Bogue,
15
R. Bork,
1
S. Bose,
16
P. R. Brady,
3
V. B. Braginsky,
14
J. E. Brau,
17
A. Brooks,
18
D. A. Brown,
1
A. Bullington,
13
A. Bunkowski,
2
A. Buonanno,
19
R. Burman,
11
D. Busby,
1
R. L. Byer,
13
L. Cadonati,
8
G. Cagnoli,
10
J. B. Camp,
20
J. Cannizzo,
20
K. Cannon,
3
C. A. Cantley,
10
J. Cao,
8
L. Cardenas,
1
M. M. Casey,
10
C. Cepeda,
1
P. Charlton,
1
S. Chatterji,
1
S. Chelkowski,
2
Y. Chen,
7
D. Chin,
21
E. Chin,
11
J. Chow,
5
N. Christensen,
22
T. Cokelaer,
23
C. N. Colacino,
6
R. Coldwell,
24
D. Cook,
9
T. Corbitt,
8
D. Coward,
11
D. Coyne,
1
J. D. E. Creighton,
3
T. D. Creighton,
1
D. R. M. Crooks,
10
A. M. Cruise,
6
A. Cumming,
10
C. Cutler,
25
J. Dalrymple,
26
E. D’Ambrosio,
1
K. Danzmann,
27,2
G. Davies,
23
G. de Vine,
5
D. DeBra,
13
J. Degallaix,
11
V. Dergachev,
21
S. Desai,
28
R. DeSalvo,
1
S. Dhurandar,
29
A. Di Credico,
26
M. D
I
́
ı
́
az,
30
J. Dickson,
5
G. Diederichs,
27
A. Dietz,
4
E. E. Doomes,
31
R. W. P. Drever,
32
J.-C. Dumas,
11
R. J. Dupuis,
1
P. Ehrens,
1
E. Elliffe,
10
T. Etzel,
1
M. Evans,
1
T. Evans,
15
S. Fairhurst,
3
Y. Fan,
11
M. M. Fejer,
13
L. S. Finn,
28
N. Fotopoulos,
8
A. Franzen,
27
K. Y. Franzen,
24
R. E. Frey,
17
T. Fricke,
33
P. Fritschel,
8
V. V. Frolov,
15
M. Fyffe,
15
J. Garofoli,
9
I. Gholami,
7
J. A. Giaime,
4
S. Giampanis,
33
K. Goda,
8
E. Goetz,
21
L. Goggin,
1
G. Gonza
́
lez,
4
S. Gossler,
5
A. Grant,
10
S. Gras,
11
C. Gray,
9
M. Gray,
5
J. Greenhalgh,
34
A. M. Gretarsson,
35
D. Grimmett,
1
R. Grosso,
30
H. Grote,
2
S. Grunewald,
7
M. Guenther,
9
R. Gustafson,
21
B. Hage,
27
C. Hanna,
4
J. Hanson,
15
C. Hardham,
13
J. Harms,
2
G. Harry,
8
E. Harstad,
17
T. Hayler,
34
J. Heefner,
1
I. S. Heng,
10
A. Heptonstall,
10
M. Heurs,
27
M. Hewitson,
2
S. Hild,
27
N. Hindman,
9
E. Hirose,
26
D. Hoak,
15
P. Hoang,
1
D. Hosken,
18
J. Hough,
10
E. Howell,
11
D. Hoyland,
6
W. Hua,
13
S. Huttner,
10
D. Ingram,
9
M. Ito,
17
Y. Itoh,
3
A. Ivanov,
1
D. Jackrel,
13
B. Johnson,
9
W. W. Johnson,
4
D. I. Jones,
10
G. Jones,
23
R. Jones,
10
L. Ju,
11
P. Kalmus,
36
V. Kalogera,
12
D. Kasprzyk,
6
E. Katsavounidis,
8
K. Kawabe,
9
S. Kawamura,
37
F. Kawazoe,
37
W. Kells,
1
F. Ya. Khalili,
14
A. Khan,
15
C. Kim,
12
P. King,
1
S. Klimenko,
24
K. Kokeyama,
37
V. Kondrashov,
1
S. Koranda,
3
D. Kozak,
1
B. Krishnan,
7
P. Kwee,
27
P. K. Lam,
5
M. Landry,
9
B. Lantz,
13
A. Lazzarini,
1
B. Lee,
11
M. Lei,
1
V. Leonhardt,
37
I. Leonor,
17
K. Libbrecht,
1
P. Lindquist,
1
N. A. Lockerbie,
6
M. Lormand,
15
M. Lubinski,
9
H. Lu
̈
ck,
27,2
B. Machenschalk,
7
M. MacInnis,
8
M. Mageswaran,
1
K. Mailand,
1
M. Malec,
27
V. Mandic,
1
S. Ma
́
rka,
36
J. Markowitz,
8
E. Maros,
1
I. Martin,
10
J. N. Marx,
1
K. Mason,
8
L. Matone,
36
N. Mavalvala,
8
R. McCarthy,
9
D. E. McClelland,
5
S. C. McGuire,
31
M. McHugh,
38
K. McKenzie,
5
J. W. C. McNabb,
28
T. Meier,
27
A. Melissinos,
33
G. Mendell,
9
R. A. Mercer,
24
S. Meshkov,
1
E. Messaritaki,
3
C. J. Messenger,
10
D. Meyers,
1
E. Mikhailov,
8
S. Mitra,
29
V. P. Mitrofanov,
14
G. Mitselmakher,
24
1
Laser Interferometer Gravitational-Wave Observatory, California Institute
of Technology, Pasadena, CA.
2
Albert-Einstein-Institut, Max-Planck-Institut fu
̈
r Gravitationsphysik,
Hannover, Germany.
3
University of Wisconsin-Milwaukee, Milwaukee, WI.
4
Louisiana State University, Baton Rouge, LA.
5
Australian National University, Canberra, Australia.
6
University of Birmingham, Birmingham, UK.
7
Albert-Einstein-Institut, Max-Planck-Institut fu
̈
r Gravitationsphysik,
Golm, Germany.
8
Laser Interferometer Gravitational-Wave Observatory, Massachusetts In-
stitute of Technology, Cambridge, MA.
9
Laser Interferometer Gravitational-Wave Observatory, Hanford Observa-
tory, Richland, WA.
10
University of Glasgow, Glasgow, UK.
11
University of Western Australia, Crawley, Australia.
12
Northwestern University, Evanston, IL.
13
Stanford University, Stanford, CA.
14
Moscow State University, Moscow, Russia.
15
Laser Interferometer Gravitational-Wave Observatory, Livingston Obser-
vatory, Livingston, LA.
16
Washington State University, Pullman, WA.
17
University of Oregon, Eugene, OR.
18
University of Adelaide, Adelaide, Australia.
19
University of Maryland, College Park, MD.
20
NASA/Goddard Space Flight Center, Greenbelt, MD.
21
University of Michigan, Ann Arbor, MI.
22
Carleton College, Northfield, MN.
23
Cardiff University, Cardiff, UK.
24
University of Florida, Gainesville, FL.
25
California Institute of Technology-CaRT, Pasadena, CA.
26
Syracuse University, Syracuse, NY.
27
Universita
̈t Hannover, Hannover, Germany.
28
Pennsylvania State University, University Park, PA.
29
Inter-University Centre for Astronomy and Astrophysics, Pune, India.
30
University of Texas at Brownsville and Texas Southmost College,
Brownsville, TX.
31
Southern University and A&M College, Baton Rouge, LA.
32
California Institute of Technology, Pasadena, CA.
33
University of Rochester, Rochester, NY.
34
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, UK.
35
Embry-Riddle Aeronautical University, Prescott, AZ.
36
Columbia University, New York, NY.
37
National Astronomical Observatory of Japan, Tokyo, Japan.
38
Loyola University, New Orleans, LA.
918
The Astrophysical Journal
, 659:918
Y
930, 2007 April 20
#
2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.
R. Mittleman,
8
O. Miyakawa,
1
S. Mohanty,
30
G. Moreno,
9
K. Mossavi,
2
C. MowLowry,
5
A. Moylan,
5
D. Mudge,
18
G. Mueller,
24
H. Mu
̈
ller-Ebhardt,
2
S. Mukherjee,
30
J. Munch,
18
P. Murray,
10
E. Myers,
9
J. Myers,
9
G. Newton,
10
K. Numata,
20
B. O’Reilly,
15
R. O’Shaughnessy,
12
D. J. Ottaway,
8
H. Overmier,
15
B. J. Owen,
28
Y. Pan,
25
M. A. Papa,
7,3
V. Parameshwaraiah,
9
M. Pedraza,
1
S. Penn,
39
M. Pitkin,
10
M. V. Plissi,
10
R. Prix,
7
V. Quetschke,
24
F. Raab,
9
D. Rabeling,
5
H. Radkins,
9
R. Rahkola,
17
M. Rakhmanov,
28
K. Rawlins,
8
S. Ray-Majumder,
3
V. Re,
6
H. Rehbein,
2
S. Reid,
10
D. H. Reitze,
24
L. Ribichini,
2
R. Riesen,
15
K. Riles,
21
B. Rivera,
9
D. I. Robertson,
10
N. A. Robertson,
13,10
C. Robinson,
23
S. Roddy,
15
A. Rodriguez,
4
A. M. Rogan,
16
J. Rollins,
36
J. D. Romano,
23
J. Romie,
15
R. Route,
13
S. Rowan,
10
A. Ru
̈
diger,
2
L. Ruet,
8
P. Russell,
1
K. Ryan,
9
S. Sakata,
37
M. Samidi,
1
L. Sancho de la Jordana,
40
V. Sandberg,
9
V. Sannibale,
1
S. Saraf,
13
P. Sarin,
8
B. S. Sathyaprakash,
23
S. Sato,
37
P. R. Saulson,
26
R. Savage,
9
S. Schediwy,
11
R. Schilling,
2
R. Schnabel,
2
R. Schofield,
17
B. F. Schutz,
7,23
P. Schwinberg,
9
S. M. Scott,
5
S. E. Seader,
16
A. C. Searle,
5
B. Sears,
1
F. Seifert,
2
D. Sellers,
15
A. S. Sengupta,
23
P. Shawhan,
1
B. Sheard,
5
D. H. Shoemaker,
8
A. Sibley,
15
X. Siemens,
3
D. Sigg,
9
A. M. Sintes,
40,7
B. Slagmolen,
5
J. Slutsky,
4
J. Smith,
2
M. R. Smith,
1
P. Sneddon,
10
K. Somiya,
2,7
C. Speake,
6
O. Spjeld,
15
K. A. Strain,
10
D. M. Strom,
17
A. Stuver,
28
T. Summerscales,
28
K. Sun,
13
M. Sung,
4
P. J. Sutton,
1
D. B. Tanner,
24
M. Tarallo,
1
R. Taylor,
1
R. Taylor,
10
J. Thacker,
15
K. A. Thorne,
28
K. S. Thorne,
25
A. Thu
̈
ring,
27
K. V. Tokmakov,
14
C. Torres,
30
C. Torrie,
1
G. Traylor,
15
M. Trias,
40
W. Tyler,
1
D. Ugolini,
41
C. Ungarelli,
6
H. Vahlbruch,
27
M. Vallisneri,
25
M. Varvella,
1
S. Vass,
1
A. Vecchio,
6
J. Veitch,
10
P. Veitch,
18
S. Vigeland,
22
A. Villar,
1
C. Vorvick,
9
S. P. Vyachanin,
14
S. J. Waldman,
1
L. Wallace,
1
H. Ward,
10
R. Ward,
1
K. Watts,
15
D. Webber,
1
A. Weidner,
2
A. Weinstein,
1
R. Weiss,
8
S. Wen,
4
K. Wette,
5
J. T. Whelan,
38,7
D. M. Whitbeck,
28
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1
B. F. Whiting,
24
C. Wilkinson,
9
P. A. Willems,
1
B. Willke,
27,2
I. Wilmut,
34
W. Winkler,
2
C. C. Wipf,
8
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24
A. G. Wiseman,
3
G. Woan,
10
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3
R. Wooley,
15
J. Worden,
9
W. Wu,
24
I. Yakushin,
15
H. Yamamoto,
1
Z. Yan,
11
S. Yoshida,
42
N. Yunes,
28
M. Zanolin,
8
L. Zhang,
1
C. Zhao,
11
N. Zotov,
43
M. Zucker,
15
H. zur Mu
̈
hlen,
27
and J. Zweizig
1
( The LIGO Scientific Collaboration)
44
Recei
v
ed 2006 September 21; accepted 2006 No
v
ember 30
ABSTRACT
The Laser Interferometer Gravitational-Wave Observatory (LIGO) has performed the fourth science run, S4, with
significantly improved interferometer sensitivities with respect to previous runs. Using data acquired during this
science run, we place a limit on the amplitude of a stochastic background of gravitational waves. For a frequency in-
dependent spectrum, the new Bayesian 90% upper limit is

GW
;
H
0
/72kms

1
Mpc

1


2
<
6
:
5
;
10

5
.This
is currently the most sensitive r
esult in the frequency range 51
Y
150 Hz, with a factor of 13 improvement over the
previous LIGO result. We discuss the
complementarity of the new result wi
th other constraints on a stochastic
background of gravitational waves, and we investigate implications of the new result for different models of this
background.
Subject headin
g
g
:
gravitational waves
1. INTRODUCTION
A stochastic background of gravitational waves (GWs) is ex-
pected to arise as a superposition of a large number of unresolved
sources, from different directions in the sky and with different
polarizations. It is usually described in terms of the GW spectrum,

GW
(
f
)
¼
f

c
d

GW
df
;
ð
1
Þ
where
d

GW
is the energy density of gravitational radiation
contained in the frequency range
f
to
f
þ
df
(Allen & Romano
1999),

c
is the critical energy density of the universe, and
f
is frequency (for an a
lternative and equi
valent definition of

GW
(
f
) see, e.g., Baskaran et al. [2006]; Grishchuk et al.
[2006]).
Many possible sources of stochastic GW background have
been proposed, and several experiments have searched for it (see
Maggiore 2000; Allen 1997 for reviews). Some of the proposed
theoretical models are cosmological in nature, such as the amplifi-
cation of quantum vacuum fluctuations during inflation (Grishchuk
1975, 1997; Starobinsky 1979), pre
Y
big bang models (Gasperini
& Veneziano 1993, 2003; Buonanno et al. 1997), phase transitions
(Kosowsky et al. 1992; Apreda et al. 2002), and cosmic strings
(Caldwell & Allen 1992; Damour & Vilenkin 2000, Damour
& Vilenkin 2005). Others are astrophysical in nature, such as
rotating neutron stars (Regimb
au & de Freitas Pacheco 2001),
supernovae (Coward et al. 2002
), or low-mass X-ray binaries
(Cooray 2004).
While some of these models pre
dict complex GW spectra,
most of them can be well approximated with power laws in the
39
Hobart and William Smith Colleges, Geneva, NY.
40
Universitat de les Illes Balears, Palma de Mallorca, Spain.
41
Trinity University, San Antonio, TX.
42
Southeastern Louisiana University, Hammond, LA.
43
Louisiana Tech University, Ruston, LA.
44
See http://www.ligo.org.
STOCHASTIC BACKGROUND OF GRAVITATIONAL WAVES
919
Laser Interferometer Gravita
tional-Wave Obse
rvatory (LIGO)
frequency band. Hence, we focus on power-law GW spectra:

GW
(
f
)
¼


f
100 Hz
!

;
ð
2
Þ
where


is the amplitude corresponding to the spectral in-
dex

. In particular,

0
denotes the amplitude of the frequency-
independent GW spectrum. We consider the range

3
<<
3.
A number of experiments have been used to constrain the
spectrum of GW background at different frequencies. Currently,
the most stringent constraints arise from large-angle correlations
in the cosmic microwave background (CMB; Allen & Koranda
1994; Turner 1997), from the arrival times of millisecond pulsar
signals (Jenet et al. 2006), from Doppler tracking of the
Cassini
spacecraft (Armstrong et al. 2003), and from resonant bar GW
detectors, such as Explorer and Nautilus (Astone et al. 1999). An
indirect bound can be placed on the total energy carried by gravi-
tational waves at the time of the big bang nucleosynthesis (BBN)
using the BBN model and observations (Kolb & Turner 1990;
Maggiore 2000; Allen 1997). Similarly, Smith et al. (2006b) used
the CMB and matter spectra to constrain the total energy density
of gravitational waves at the time of photon decoupling.
Ground-based interferometer networks can directly measure
the GW strain spectrum in the frequency band 10 Hz to a few kHz
by searching for correlated signal beneath uncorrelated detector
noise. LIGO has built three power-recycled Michelson interfer-
ometers, with a Fabry-Perot cavity in each orthogonal arm. They
are located at two sites: Hanford, Washington, and Livingston
Parish, Louisiana. There are two collocated interferometers at
the Washington site: H1, with 4 km long arms, and H2, with 2 km
arms. The Louisiana site contains L1, a 4 km interferometer, sim-
ilar in design to H1. The detector configuration and performance
during LIGO’s first science run (S1) was described in Abbott
et al. (2004a). The data acquired during that run were used to place
an upper limit of

0
<
44
:
4 on the amplitude of a frequency
independent GW spectrum, in the frequency band 40
Y
314 Hz
(Abbott et al. 2004b). For this limit, as well as in the rest of this
paper, we assume the present value of the Hubble parameter
H
0
¼
72 km s

1
Mpc

1
(Bennet et al. 2003), i.e., when writing

0
,
we implicitly mean

0
;
H
0
/72kms

1
Mpc

1


2
. The most
recent bound on the amplitude of the frequency-independent
GW spectrum from LIGO is based on the science run S3;

0
<
8
:
4
;
10

4
for a frequency-independent spectrum in the 69
Y
156 Hz band (Abbott et al. 2005).
In this paper, we report much-improved limits on the stochas-
tic GW background around 100 Hz, using the data acquired
during the LIGO science run S4, which took place between 2005
February 22 and March 23. The sensitivity of the interferometers
during S4, shown in Figure 1, was significantly better than S3
(by a factor 10 at certain frequencies), which leads to an order-of-
magnitude improvement in the upper limit on the amplitude of
the stochastic GW background:

0
<
6
:
5
;
10

5
for a frequency-
independent spectrum over the 51
Y
150 Hz band.
This limit is beginning to probe some models of the stochastic
GW background. As examples, we investigate the implications of
this limit for cosmic strings models and for pre
Y
big bang models
of the stochastic gravitational r
adiation. In both cases, the new
LIGO result excludes parts of the parameter space of these models.
The organization of this paper is as follows. In
x
2 we review
the analysis procedure and present the results in
x
3. In
x
4, we
discuss some of the implications of our results for models of a
stochastic GW background, as well as the complementarity be-
tween LIGO and other experimental constraints on a stochastic
GW background. We conclude with future prospects in
x
5.
2. ANALYSIS
2.1.
Cross-Correlation Method
The cross-correlation method for searching for a stochastic
GW background with pairs of ground-based interferometers is
described in Allen & Romano (1999). We define the following
cross-correlation estimator:
Y
¼
Z
þ1
0
df Y
(
f
)
¼
Z
þ1
1
df
Z
þ1
1
df
0

T
(
f

f
0
)
̃
s
1
(
f
)

̃
s
2
(
f
0
)
̃
Q
(
f
0
)
;
ð
3
Þ
where

T
is a finite-time approximation to the Dirac delta func-
tion,
̃
s
1
and
̃
s
2
are the Fourier transforms of the strain time series
of two interferometers, and
̃
Q
is a filter function. Assuming that
the detector noise is Gaussian, stationary, uncorrelated between
the two interferometers, and much larger than the GW signal, the
variance of the estimator
Y
is given by

2
Y
¼
Z
þ1
0
df

2
Y
(
f
)

T
2
Z
þ1
0
dfP
1
(
f
)
P
2
(
f
)
̃
Qf
ðÞ




2
;
ð
4
Þ
where
P
i
(
f
) are the one-sided power spectral densities (PSDs)
of the two interferometers, and
T
is the measurement time. Op-
timization of the signal-to-noise ratio leads to the following form
of the optimal filter (Allen & Romano 1999):
̃
Q
(
f
)
¼
N

(
f
)
S
GW
(
f
)
P
1
(
f
)
P
2
(
f
)
;
ð
5
Þ
Fig.
1.—Typical strain amplitude spectra of LIGO interferometers during
the science run S4 (solid curves top to bottom at 70 Hz: H2, L1, H1). The black
dashed curve is the LIGO sensitivity goal for the 4 km interferometers H1 and L1.
The black dotted curve is the expected design LIGO sensitivity of the 2 km inter-
ferometer H2. The gray dashed curve is the strain amplitude spectrum correspond-
ing to the limit presented in this paper for the frequency-independent GW spectrum

0
<
6
:
5
;
10

5
.
ABBOTT ET AL.
920
Vol. 659
where
S
GW
(
f
)
¼
3
H
2
0
10

2

GW
(
f
)
f
3
;
ð
6
Þ
and

(
f
) is the overlap reduction fun
ction arising from the dif-
ferent locations and orientatio
ns of the two interferometers.
As shown in Figure 2, the identical antenna patterns of the collo-
cated Hanford interferometers imply

(
f
)
¼
1. For the Hanford-
Livingston pair the overlap reduction is significant above 50 Hz.
In equations (5) and (6),
S
GW
(
f
) is the strain power spectrum of
the stochastic GW background to be searched. Assuming a power-
law template GW spectrum with index

(see eq. [2]), the normal-
ization constant
N
in equation (5) is chosen such that
Y
hi
¼


T
.
In order to deal with data nonstationarity, and for purposes
of computational feasibility, the data for an interferometer pair
are divided into many intervals of equal duration, and
Y
I
and

Y
I
are calculated for each interval
I
. The data in each interval are
decimated from 16,384 to 1024 Hz and high-pass filtered with a
40 Hz cutoff. As we discuss below, most of the sensitivity of this
search lies below 300 Hz, safely below the Nyquist frequency of
512 Hz. The intervals are also Hann windowed to avoid spectral
leakage from strong lines present in the data. Since Hann win-
dowing effectively reduces the interval length by 50%, the data
intervals are overlapped by 50% to recover the original signal-to-
noise ratio. The effects of windowing are taken into account as
discussed in Abbott et al. (2004b).
The PSDs for each interval [needed for the calculation of
Q
I
(
f
) and of

Y
I
] are calculated using the two neighboring inter-
vals. This approach avoids a bias that would otherwise exist due
to a nonzero covariance between the cross-power
Y
(
f
) and the
power spectra estimated from the same data (Bendat & Piersol
2000). It also allows for a stationarity cut, which we describe in
more detail below.
We consider two interval durations and frequency resolutions:
1. 60 s
duration with
1/4 Hz
resolution
.—The PSDs are
calculated by averaging 58 50% overlapping periodograms
(based on the two neighboring 60 s intervals) in Welch’s mod-
ified periodogram method.
2. 192 s
duration with
1/32 Hz
resolution
.—The PSDs are
calculated by averaging 22 50% overlapping periodograms (based
on the two neighboring 192 s intervals) in Welch’s modified
periodogram method.
As we discuss below, the 60 s intervals allow better sensitivity
to noise transients and are better suited for data-stationarity cuts.
We used this interval duration in the blind analysis. However,
after unblinding we discovered a comb of correlated sharp 1 Hz
harmonic lines between interferometers. To remove these sharp
lines from our analysis without significantly affecting the sen-
sitivity, we performed the second analysis with 192 s intervals.
The 192 s intervals allow higher frequency resolution of the power
and cross-power spectra and are better suited for removing sharp
lines from the analysis.
Thedataforagiveninterval
I
are Fourier transformed (with
frequency resolution 1/60 or 1/1
92 Hz) and rebinned to the fre-
quency resolution of the PSDs and of the optimal filter (1/4 or
1/32 Hz, respectively) to complete the calculation of
Y
I
(eq. [3]).
Both the PSDs and the Fourier transforms of the data are cal-
ibrated using interferometer response functions, determined for
every minute of data using a measurement of the interferom-
eter response to a sinusoidal calibration force. To maximize the
signal-to-noise ratio, the inte
rvals are combined by performing
a weighted average (with weights 1/

2
Y
I
), properly accounting
for the 50% overlapping as discussed in Lazzarini & Romano
(2004).
2.2.
Identification of Correlated Instrumental Lines
The results of this paper are based on the Hanford-Livingston
interferometer pairs, for which the broadband instrumental cor-
relations are minimized. Nevertheless, it is still necessary to in-
vestigate whether there are any remaining periodic instrumental
correlations. We do this by calculating the coherence over the
whole S4 run. The coherence is defined as

(
f
)
¼
P
12
(
f
)
jj
2
P
1
(
f
)
P
2
(
f
)
:
ð
7
Þ
The numerator is the square of the cross-spectral density (CSD)
between the two interferometers, and the denominator contains
the two power spectral densities (PSDs). We average the CSD
and the PSDs over the whole run at two different resolutions: 1
and 100 mHz. Figure 3 shows the results of this calculation for
the H1-L1 pair and for the H2-L1 pair.
At 1 mHz resolution, a forest of sharp 1 Hz harmonic lines
can be observed. These lines were likely caused by the sharp
ramp of a one-pulse-per-second signal, injected into the data ac-
quisition system to synchronize it with the Global Positioning
System (GPS) time reference. Since the coupling of this signal to
the gravitational-wave channel of H1 was much stronger than
that of H2, the forest of 1 Hz harmonics is much clearer in the
H1-L1 coherence. After the S4 run ended, the sharp ramp signal
was replaced by smooth sinusoidal signals, with the goal of sig-
nificantly reducing the 1 Hz harmonic lines in future LIGO data
runs. In addition to the 1 Hz lines, the 1 mHz coherence plots in
Figure 3 also include some of the simulated pulsar lines, which
were injected into the differential-arm servo of the interferom-
eters by physically moving the mirrors. Both the 1 Hz harmonics
and the simulated pulsar lines can be removed in the final anal-
ysis, and we discuss this further in
x
3. Figure 4 shows that the
histogram of the coherence at 1 mHz resolution follows the ex-
pected exponential distribution, if one ignores the 1 Hz harmonics
and the simulated pulsar lines.
Fig.
2.—Overlap reduction function for the Hanford-Hanford pair (
black
solid line
) and for the Hanford-Livingston pair (
g
ray dashed line
).
STOCHASTIC BACKGROUND OF GRAVITATIONAL WAVES
921
No. 2, 2007