of 37
arXiv:astro-ph/0608606v2 21 Sep 2006
Searching for a Stochastic Background of Gravitational Wav
es
with LIGO
B. Abbott
12
, R. Abbott
12
, R. Adhikari
12
, J. Agresti
12
, P. Ajith
2
, B. Allen
42
, R. Amin
16
,
S. B. Anderson
12
, W. G. Anderson
42
, M. Araya
12
, H. Armandula
12
, M. Ashley
3
, S Aston
34
,
C. Aulbert
1
, S. Babak
1
, S. Ballmer
13
, B. C. Barish
12
, C. Barker
14
, D. Barker
14
, B. Barr
36
,
P. Barriga
41
, M. A. Barton
12
, K. Bayer
13
, K. Belczynski
22
, J. Betzwieser
13
, P. Beyersdorf
26
,
B. Bhawal
12
, I. A. Bilenko
19
, G. Billingsley
12
, E. Black
12
, K. Blackburn
12
, L. Blackburn
13
,
D. Blair
41
, B. Bland
14
, L. Bogue
15
, R. Bork
12
, S. Bose
43
, P. R. Brady
42
, V. B. Braginsky
19
,
J. E. Brau
39
, A. Brooks
33
, D. A. Brown
12
, A. Bullington
26
, A. Bunkowski
2
, A. Buonanno
37
,
R. Burman
41
, D. Busby
12
, R. L. Byer
26
, L. Cadonati
13
, G. Cagnoli
36
, J. B. Camp
20
,
J. Cannizzo
20
, K. Cannon
42
, C. A. Cantley
36
, J. Cao
13
, L. Cardenas
12
, M. M. Casey
36
,
C. Cepeda
12
, P. Charlton
12
, S. Chatterji
12
, S. Chelkowski
2
, Y. Chen
1
, D. Chin
38
, E. Chin
41
,
J. Chow
3
, N. Christensen
7
, T. Cokelaer
6
, C. N. Colacino
34
, R. Coldwell
35
, D. Cook
14
,
T. Corbitt
13
, D. Coward
41
, D. Coyne
12
, J. D. E. Creighton
42
, T. D. Creighton
12
,
D. R. M. Crooks
36
, A. M. Cruise
34
, A. Cumming
36
, C. Cutler
5
, J. Dalrymple
27
,
E. D’Ambrosio
12
, K. Danzmann
31
,
2
, G. Davies
6
, G. de Vine
3
, D. DeBra
26
, J. Degallaix
41
,
V. Dergachev
38
, S. Desai
28
, R. DeSalvo
12
, S. Dhurandar
11
, A. Di Credico
27
, M. D
́
iaz
29
,
J. Dickson
3
, G. Diederichs
31
, A. Dietz
16
, E. E. Doomes
25
, R. W. P. Drever
4
, J.-C. Dumas
41
,
R. J. Dupuis
12
, P. Ehrens
12
, E. Elliffe
36
, T. Etzel
12
, M. Evans
12
, T. Evans
15
, S. Fairhurst
42
,
Y. Fan
41
, M. M. Fejer
26
, L. S. Finn
28
, N. Fotopoulos
13
, A. Franzen
31
, K. Y. Franzen
35
,
R. E. Frey
39
, T. Fricke
40
, P. Fritschel
13
, V. V. Frolov
15
, M. Fyffe
15
, J. Garofoli
14
,
I. Gholami
1
, J. A. Giaime
16
, S. Giampanis
40
, K. Goda
13
, E. Goetz
38
, L. Goggin
12
,
G. Gonz ́alez
16
, S. Gossler
3
, A. Grant
36
, S. Gras
41
, C. Gray
14
, M. Gray
3
, J. Greenhalgh
23
,
A. M. Gretarsson
9
, D. Grimmett
12
, R. Grosso
29
, H. Grote
2
, S. Grunewald
1
, M. Guenther
14
,
R. Gustafson
38
, B. Hage
31
, C. Hanna
16
, J. Hanson
15
, C. Hardham
26
, J. Harms
2
, G. Harry
13
,
E. Harstad
39
, T. Hayler
23
, J. Heefner
12
, I. S. Heng
36
, A. Heptonstall
36
, M. Heurs
31
,
M. Hewitson
2
, S. Hild
31
, N. Hindman
14
, E. Hirose
27
, D. Hoak
15
, P. Hoang
12
, D. Hosken
33
,
J. Hough
36
, E. Howell
41
, D. Hoyland
34
, W. Hua
26
, S. Huttner
36
, D. Ingram
14
, M. Ito
39
,
Y. Itoh
42
, A. Ivanov
12
, D. Jackrel
26
, B. Johnson
14
, W. W. Johnson
16
, D. I. Jones
36
,
G. Jones
6
, R. Jones
36
, L. Ju
41
, P. Kalmus
8
, V. Kalogera
22
, D. Kasprzyk
34
,
E. Katsavounidis
13
, K. Kawabe
14
, S. Kawamura
21
, F. Kawazoe
21
, W. Kells
12
,
F. Ya. Khalili
19
, A. Khan
15
, C. Kim
22
, P. King
12
, S. Klimenko
35
, K. Kokeyama
21
,
V. Kondrashov
12
, S. Koranda
42
, D. Kozak
12
, B. Krishnan
1
, P. Kwee
31
, P. K. Lam
3
,
M. Landry
14
, B. Lantz
26
, A. Lazzarini
12
, B. Lee
41
, M. Lei
12
, V. Leonhardt
21
, I. Leonor
39
,
K. Libbrecht
12
, P. Lindquist
12
, N. A. Lockerbie
34
, M. Lormand
15
, M. Lubinski
14
,
H. L ̈uck
31
,
2
, B. Machenschalk
1
, M. MacInnis
13
, M. Mageswaran
12
, K. Mailand
12
,
M. Malec
31
, V. Mandic
12
, S. M ́arka
8
, J. Markowitz
13
, E. Maros
12
, I. Martin
36
, J. N. Marx
12
,
– 2 –
K. Mason
13
, L. Matone
8
, N. Mavalvala
13
, R. McCarthy
14
, D. E. McClelland
3
,
S. C. McGuire
25
, M. McHugh
18
, K. McKenzie
3
, J. W. C. McNabb
28
, T. Meier
31
,
A. Melissinos
40
, G. Mendell
14
, R. A. Mercer
35
, S. Meshkov
12
, E. Messaritaki
42
,
C. J. Messenger
36
, D. Meyers
12
, E. Mikhailov
13
, S. Mitra
11
, V. P. Mitrofanov
19
,
G. Mitselmakher
35
, R. Mittleman
13
, O. Miyakawa
12
, S. Mohanty
29
, G. Moreno
14
,
K. Mossavi
2
, C. MowLowry
3
, A. Moylan
3
, D. Mudge
33
, G. Mueller
35
, H. M ̈uller-Ebhardt
2
,
S. Mukherjee
29
, J. Munch
33
, P. Murray
36
, E. Myers
14
, J. Myers
14
, G. Newton
36
,
K. Numata
20
, B. O’Reilly
15
, R. O’Shaughnessy
22
, D. J. Ottaway
13
, H. Overmier
15
,
B. J. Owen
28
, Y. Pan
5
, M. A. Papa
1
,
42
, V. Parameshwaraiah
14
, M. Pedraza
12
, S. Penn
10
,
M. Pitkin
36
, M. V. Plissi
36
, R. Prix
1
, V. Quetschke
35
, F. Raab
14
, D. Rabeling
3
,
H. Radkins
14
, R. Rahkola
39
, M. Rakhmanov
28
, K. Rawlins
13
, S. Ray-Majumder
42
, V. Re
34
,
H. Rehbein
2
, S. Reid
36
, D. H. Reitze
35
, L. Ribichini
2
, R. Riesen
15
, K. Riles
38
, B. Rivera
14
,
D. I. Robertson
36
, N. A. Robertson
26
,
36
, C. Robinson
6
, S. Roddy
15
, A. Rodriguez
16
,
A. M. Rogan
43
, J. Rollins
8
, J. D. Romano
6
, J. Romie
15
, R. Route
26
, S. Rowan
36
,
A. R ̈udiger
2
, L. Ruet
13
, P. Russell
12
, K. Ryan
14
, S. Sakata
21
, M. Samidi
12
,
L. Sancho de la Jordana
32
, V. Sandberg
14
, V. Sannibale
12
, S.Saraf
26
, P. Sarin
13
,
B. S. Sathyaprakash
6
, S. Sato
21
, P. R. Saulson
27
, R. Savage
14
, S. Schediwy
41
, R. Schilling
2
,
R. Schnabel
2
, R. Schofield
39
, B. F. Schutz
1
,
6
, P. Schwinberg
14
, S. M. Scott
3
, S. E. Seader
43
,
A. C. Searle
3
, B. Sears
12
, F. Seifert
2
, D. Sellers
15
, A. S. Sengupta
6
, P. Shawhan
12
,
B. Sheard
3
, D. H. Shoemaker
13
, A. Sibley
15
, X. Siemens
42
, D. Sigg
14
, A. M. Sintes
32
,
1
,
B. Slagmolen
3
, J. Slutsky
16
, J. Smith
2
, M. R. Smith
12
, P. Sneddon
36
, K. Somiya
2
,
1
,
C. Speake
34
, O. Spjeld
15
, K. A. Strain
36
, D. M. Strom
39
, A. Stuver
28
, T. Summerscales
28
,
K. Sun
26
, M. Sung
16
, P. J. Sutton
12
, D. B. Tanner
35
, M. Tarallo
12
, R. Taylor
12
, R. Taylor
36
,
J. Thacker
15
, K. A. Thorne
28
, K. S. Thorne
5
, A. Th ̈uring
31
, K. V. Tokmakov
19
, C. Torres
29
,
C. Torrie
12
, G. Traylor
15
, M. Trias
32
, W. Tyler
12
, D. Ugolini
30
, C. Ungarelli
34
,
H. Vahlbruch
31
, M. Vallisneri
5
, M. Varvella
12
, S. Vass
12
, A. Vecchio
34
, J. Veitch
36
,
P. Veitch
33
, S. Vigeland
7
, A. Villar
12
, C. Vorvick
14
, S. P. Vyachanin
19
, S. J. Waldman
12
,
L. Wallace
12
, H. Ward
36
, R. Ward
12
, K. Watts
15
, D. Webber
12
, A. Weidner
2
,
A. Weinstein
12
, R. Weiss
13
, S. Wen
16
, K. Wette
3
, J. T. Whelan
18
,
1
, D. M. Whitbeck
28
,
S. E. Whitcomb
12
, B. F. Whiting
35
, C. Wilkinson
14
, P. A. Willems
12
, B. Willke
31
,
2
,
I. Wilmut
23
, W. Winkler
2
, C. C. Wipf
13
, S. Wise
35
, A. G. Wiseman
42
, G. Woan
36
,
D. Woods
42
, R. Wooley
15
, J. Worden
14
, W. Wu
35
, I. Yakushin
15
, H. Yamamoto
12
, Z. Yan
41
,
S. Yoshida
24
, N. Yunes
28
, M. Zanolin
13
, L. Zhang
12
, C. Zhao
41
, N. Zotov
17
, M. Zucker
15
,
H. zur M ̈uhlen
31
, J. Zweizig
12
,
The LIGO Scientific Collaboration, http://www.ligo.org
– 3 –
1
Albert-Einstein-Institut, Max-Planck-Institut f ̈ur Gra
vitationsphysik, D-14476 Golm, Germany
2
Albert-Einstein-Institut, Max-Planck-Institut f ̈ur Gra
vitationsphysik, D-30167 Hannover, Germany
3
Australian National University, Canberra, 0200, Australi
a
4
California Institute of Technology, Pasadena, CA 91125, US
A
5
Caltech-CaRT, Pasadena, CA 91125, USA
6
Cardiff University, Cardiff, CF2 3YB, United Kingdom
7
Carleton College, Northfield, MN 55057, USA
8
Columbia University, New York, NY 10027, USA
9
Embry-Riddle Aeronautical University, Prescott, AZ 86301
USA
10
Hobart and William Smith Colleges, Geneva, NY 14456, USA
11
Inter-University Centre for Astronomy and Astrophysics, P
une - 411007, India
12
LIGO - California Institute of Technology, Pasadena, CA 911
25, USA
13
LIGO - Massachusetts Institute of Technology, Cambridge, M
A 02139, USA
14
LIGO Hanford Observatory, Richland, WA 99352, USA
15
LIGO Livingston Observatory, Livingston, LA 70754, USA
16
Louisiana State University, Baton Rouge, LA 70803, USA
17
Louisiana Tech University, Ruston, LA 71272, USA
18
Loyola University, New Orleans, LA 70118, USA
19
Moscow State University, Moscow, 119992, Russia
20
NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
21
National Astronomical Observatory of Japan, Tokyo 181-858
8, Japan
22
Northwestern University, Evanston, IL 60208, USA
23
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX1
1 0QX United Kingdom
24
Southeastern Louisiana University, Hammond, LA 70402, USA
25
Southern University and A&M College, Baton Rouge, LA 70813,
USA
26
Stanford University, Stanford, CA 94305, USA
27
Syracuse University, Syracuse, NY 13244, USA
28
The Pennsylvania State University, University Park, PA 168
02, USA
29
The University of Texas at Brownsville and Texas Southmost C
ollege, Brownsville, TX 78520, USA
30
Trinity University, San Antonio, TX 78212, USA
– 4 –
ABSTRACT
The Laser Interferometer Gravitational-wave Observatory
(LIGO) has per-
formed the fourth science run, S4, with significantly improv
ed interferometer sen-
sitivities with respect to previous runs. Using data acquir
ed during this science
run, we place a limit on the amplitude of a stochastic backgro
und of gravitational
waves. For a frequency independent spectrum, the new limit i
s Ω
GW
<
6
.
5
×
10
5
.
This is currently the most sensitive result in the frequency
range 51-150 Hz, with
a factor of 13 improvement over the previous LIGO result. We d
iscuss comple-
mentarity of the new result with other constraints on a stoch
astic background of
gravitational waves, and we investigate implications of th
e new result for different
models of this background.
Subject headings:
gravitational waves
1. Introduction
A stochastic background of gravitational waves (GWs) is exp
ected to arise as a super-
position of a large number of unresolved sources, from differ
ent directions in the sky, and
31
Universit ̈at Hannover, D-30167 Hannover, Germany
32
Universitat de les Illes Balears, E-07122 Palma de Mallorca
, Spain
33
University of Adelaide, Adelaide, SA 5005, Australia
34
University of Birmingham, Birmingham, B15 2TT, United King
dom
35
University of Florida, Gainesville, FL 32611, USA
36
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
37
University of Maryland, College Park, MD 20742 USA
38
University of Michigan, Ann Arbor, MI 48109, USA
39
University of Oregon, Eugene, OR 97403, USA
40
University of Rochester, Rochester, NY 14627, USA
41
University of Western Australia, Crawley, WA 6009, Austral
ia
42
University of Wisconsin-Milwaukee, Milwaukee, WI 53201, U
SA
43
Washington State University, Pullman, WA 99164, USA
– 5 –
with different polarizations. It is usually described in ter
ms of the GW spectrum:
GW
(
f
) =
f
ρ
c
GW
df
,
(1)
where
GW
is the energy density of gravitational radiation contained
in the frequency range
f
to
f
+
df
(Allen & Romano 1999),
ρ
c
is the critical energy density of the Universe, and
f
is frequency (for an alternative and equivalent definition o
f Ω
GW
(
f
) see, for example,
(Baskaran et al. 2006)). In this paper, we will focus on power
-law GW spectra:
GW
(
f
) = Ω
α
(
f
100 Hz
)
α
.
(2)
Here, Ω
α
is the amplitude corresponding to the spectral index
α
. In particular, Ω
0
denotes
the amplitude of the frequency-independent GW spectrum.
Many possible sources of stochastic GW background have been
proposed and several
experiments have searched for it (see (Maggiore 2000; Allen
1996) for reviews). Some of the
proposed theoretical models are cosmological in nature, su
ch as the amplification of quantum
vacuum fluctuations during inflation (Grishchuk 1975), (Gri
shchuk 1997), (Starobinsky 1979),
pre-big-bang models (Gasperini & Veneziano 1993), (Gasper
ini & Veneziano 2003), (Buonanno et al. 1997),
phase transitions (Kosowsky et al. 1992), (Apreda et al. 200
2), and cosmic strings (Caldwell & Allen 1992),
(Damour & Vilenkin 2000), (Damour & Vilenkin 2005). Others a
re astrophysical in nature,
such as rotating neutron stars (Regimbau & de Freitas Pachec
o 2001), supernovae (Coward et al. 2002)
or low-mass X-ray binaries (Cooray 2004).
A number of experiments have been used to constrain the spect
rum of GW background
at different frequencies. Currently, the most stringent con
straints arise from large-angle cor-
relations 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 r
esonant bar GW detectors, such
as Explorer and Nautilus (Astone et al. 1999). An indirect bo
und can be placed on the total
energy carried by gravitational waves at the time of the Big-
Bang Nucleosynthesis (BBN)
using the BBN model and observations (Kolb & Turner 1990; Mag
giore 2000; Allen 1996).
Similarly, (Smith et al. 2006a) used the CMB and matter spect
ra to constrain the total
energy density of gravitational waves at the time of photon d
ecoupling.
Ground-based interferometer networks can directly measur
e the GW strain spectrum in
the frequency band 10 Hz - few kHz, by searching for correlate
d signal beneath uncorrelated
detector noise. LIGO has built three power-recycled Michel
son interferometers, with a Fabry-
Perot cavity in each orthogonal arm. They are located at two s
ites, Hanford, WA, and
Livingston Parish, LA. There are two collocated interferom
eters at the WA site: H1, with
– 6 –
4km long arms, and H2, with 2km arms. The LA site contains L1, a
4km interferometer,
similar in design to H1. The detector configuration and perfo
rmance during LIGO’s first
science run (S1) was described in (Abbott et al. 2004a). The d
ata acquired during that run
was used to place an upper limit of Ω
0
<
44
.
4 on the amplitude of a frequency independent
GW spectrum, in the frequency band 40-314 Hz (Abbott et al. 20
04b). This limit, as well as
the rest of this paper, assumes the present value of the Hubbl
e parameter
H
0
= 72 km/s/Mpc
or, equivalently,
h
100
=
H
0
/
(100 km
/
s
/
Mpc) = 0
.
72 (Bennet et al. 2003). The most recent
bound on the amplitude of the frequency independent GW spect
rum from LIGO is based on
the science run S3: Ω
0
<
8
.
4
×
10
4
for a frequency-independent spectrum in the 69-156 Hz
band (Abbott et al. 2005).
In this paper, we report much improved limits on the stochast
ic GW background around
100 Hz, using the data acquired during the LIGO science run S4
, which took place between
February 22, 2005 and March 23, 2005. The sensitivity of the i
nterferometers during S4,
shown in Figure 1, was significantly better as compared to S3 (
by a factor 10
×
at certain
frequencies), which leads to an order of magnitude improvem
ent in the upper limit on the
amplitude of the stochastic GW background: Ω
0
<
6
.
5
×
10
5
for a frequency-independent
spectrum over the 51-150 Hz band.
This limit is beginning to probe some models of the stochasti
c GW background. As
examples, we investigate the implications of this limit for
cosmic strings models and for
pre-big-bang models of the stochastic gravitational radia
tion. 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 Section 2 we r
eview the analysis procedure
and present the results in Section 3. In Section 4, we discuss
some of the implications of our
results for models of a stochastic GW background, as well as t
he complementarity between
LIGO and other experimental constraints on a stochastic GW b
ackground. We conclude
with future prospects in Section 5.
2. Analysis
2.1. Cross-Correlation Method
The cross-correlation method for searching for a stochasti
c GW background with pairs
of ground-based interferometers is described in (Allen & Ro
mano 1999). We define the fol-
lowing cross-correlation estimator:
Y
=
+
0
df Y
(
f
)
– 7 –
50
70
100
200
300
500
10
−24
10
−23
10
−22
10
−21
0
= 6.5
×
10
−5
Frequency (Hz)
Strain (1/
Hz)
H1
H2
L1
Goal
Fig. 1.— Typical strain amplitude spectra of LIGO interfero
meters during the science run
S4 (solid curves top-to-bottom at 70 Hz: H2, L1, H1). The blac
k dashed curve is the LIGO
sensitivity goal. The gray dashed curve is the strain amplit
ude spectrum corresponding to the
limit presented in this paper for the frequency-independen
t GW spectrum Ω
0
<
6
.
5
×
10
5
.
– 8 –
=
+
−∞
df
+
−∞
df
δ
T
(
f
f
) ̃
s
1
(
f
)
̃
s
2
(
f
)
̃
Q
(
f
)
,
(3)
where
δ
T
is a finite-time approximation to the Dirac delta function, ̃
s
1
and ̃
s
2
are the Fourier
transforms of the strain time-series of two interferometer
s, and
̃
Q
is a filter function. As-
suming that the detector noise is Gaussian, stationary, unc
orrelated between the two inter-
ferometers, and much larger than the GW signal, the variance
of the estimator
Y
is given
by:
σ
2
Y
=
+
0
df σ
2
Y
(
f
)
T
2
+
0
dfP
1
(
f
)
P
2
(
f
)
|
̃
Q
(
f
)
|
2
,
(4)
where
P
i
(
f
) are the one-sided power spectral densities (PSDs) of the tw
o interferometers and
T
is the measurement time. Optimization of the signal-to-noi
se ratio leads to the following
form of the optimal filter (Allen & Romano 1999):
̃
Q
(
f
) =
N
γ
(
f
)
S
GW
(
f
)
P
1
(
f
)
P
2
(
f
)
,
(5)
where
S
GW
(
f
) =
3
H
2
0
10
π
2
GW
(
f
)
f
3
,
(6)
and
γ
(
f
) is the overlap reduction function, arising from the differe
nt locations and orien-
tations of the two interferometers. As shown in Figure 2, the
identical antenna patterns of
the collocated Hanford interferometers imply
γ
(
f
) = 1. For the Hanford-Livingston pair the
overlap reduction is significant above 50 Hz. In Equations 5 a
nd 6,
S
GW
(
f
) is the strain
power spectrum of the stochastic GW background to be searche
d. Assuming a power-law
template GW spectrum with index
α
(see Equation 2), the normalization constant
N
in
Equation 5 is chosen such that
< Y >
= Ω
α
T
.
In order to deal with data non-stationarity, and for purpose
s of computational feasibility,
the data for an interferometer pair are divided into many int
ervals of equal duration, and
Y
I
and
σ
Y
I
are calculated for each interval
I
. The data in each interval are decimated from 16384
Hz to 1024 Hz and high-pass filtered with a 40 Hz cut-off. They ar
e also Hann-windowed
to avoid spectral leakage from strong lines present in the da
ta. Since Hann-windowing
effectively reduces the interval length by 50%, the data inte
rvals are overlapped by 50% to
recover the original signal-to-noise ratio. The effects of w
indowing 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
σ
Y
I
) are calculated
using the two neighboring intervals. This approach avoids a
bias that would otherwise exist
– 9 –
0
50
100
150
200
250
300
350
400
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Frequency (Hz)
γ
(f)
H1−H2
H1−L1
Fig. 2.— Overlap reduction function for the Hanford-Hanfor
d pair (black solid) and for the
Hanford-Livingston pair (gray dashed).
– 10 –
due to a non-zero 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
will describe in more detail below.
We consider two interval durations and frequency resolutio
ns:
60-sec duration with 1
/
4 Hz resolution: the PSDs are calculated by averaging 58 50%
overlapping periodograms (based on the two neighboring 60-
sec intervals) in Welch’s
modified periodogram method.
192-sec duration with 1
/
32 Hz resolution: the PSDs are calculated by averaging 22 50%
overlapping periodograms (based on the two neighboring 192
-sec intervals) in Welch’s
modified periodogram method.
As we will discuss below, the 60-sec intervals allow better s
ensitivity to noise transients
and are better suited for data-stationarity cuts, while the
192-sec intervals allow higher
frequency resolution of the power and cross-power spectra a
nd are better suited for removing
sharp lines from the analysis.
The data for a given interval
I
are Fourier transformed and rebinned to the frequency
resolution of the optimal filter to complete the calculation
of
Y
I
(Eq. 3). Both the PSDs and
the Fourier transforms of the data are calibrated using inte
rferometer response functions,
determined for every minute of data using a measurement of th
e interferometer response
to a sinusoidal calibration force. To maximize the signal-t
o-noise ratio, the intervals 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-Livingsto
n interferometer pairs,
for which the broadband instrumental correlations are mini
mized. Nevertheless, it is still
necessary to investigate if there are any remaining periodi
c instrumental correlations. We
do this by calculating the coherence over the whole S4 run. Th
e coherence is defined as:
Γ(
f
) =
|
s
1
(
f
)
s
2
(
f
)
|
2
P
1
(
f
)
P
2
(
f
)
.
(7)
The numerator is the square of the cross-spectral density (C
SD) between the two interfer-
ometers, and the denominator contains the two power spectra
l densities (PSDs). We average
– 11 –
10
−3
10
−2
10
−1
10
0
H1L1
Γ
at 1 mHz
Notched Frequencies
Accepted Frequencies
50
100
150
200
250
300
350
400
450
500
10
−6
10
−5
10
−4
Freq. (Hz)
Γ
at 0.1 Hz
10
−3
10
−2
10
−1
10
0
H2L1
Γ
at 1 mHz
Notched Frequencies
Accepted Frequencies
50
100
150
200
250
300
350
400
450
500
10
−6
10
−5
10
−4
Freq. (Hz)
Γ
at 0.1 Hz
Fig. 3.— Coherence calculated for the H1L1 pair (top) and for
the H2L1 pair (bottom) over
all of S4 data for 1 mHz resolution and 100 mHz resolution. The
horizontal dashed lines
indicate 1
/N
avg
- the expected level of coherence after averaging over
N
avg
time-periods with
uncorrelated spectra. The line at 376 Hz is one of the simulat
ed pulsar lines.
– 12 –
the CSD and the PSDs over the whole run at two different resolut
ions: 1 mHz and 100 mHz.
Figure 3 shows the results of this calculation for the H1L1 pa
ir and for the H2L1 pair.
At 1 mHz resolution, a forest of sharp 1 Hz harmonic lines can b
e observed. These
lines were likely caused by the sharp ramp of a one-pulse-per
-second signal, injected into the
data acquisition system to synchronize it with the Global Po
sitioning System (GPS) time
reference. After the S4 run ended, the sharp ramp signal was r
eplaced by smooth sinusoidal
signals, with the goal of significantly reducing the 1 Hz harm
onic lines in future LIGO data
runs. In addition to the 1 Hz lines, the 1 mHz coherence plots i
n Figure 3 also include
some of the simulated pulsar lines, which were injected into
the differential-arm servo of
the interferometers by physically moving the mirrors. Both
the 1 Hz harmonics and the
simulated pulsar lines can be removed in the final analysis, a
nd we will discuss this further
in Section 3. Figure 4 shows that the histogram of the coheren
ce at 1 mHz resolution follows
the expected exponential distribution, if one ignores the 1
Hz harmonics and the simulated
pulsar lines.
2.3. Data Quality Cuts
In our analysis, we include time periods during which both in
terferometers are in low-
noise, science mode. We exclude:
Time-periods when digitizer signals saturate.
30-sec intervals prior to each lock loss. These intervals ar
e known to be particularly
noisy.
We then proceed to calculate
Y
I
and
σ
Y
I
for each interval
I
, and define three data-quality
cuts. First, we reject intervals known to contain large glit
ches in one interferometer. These
intervals were identified by searching for discontinuities
in the PSD trends over the whole
S4 run. Second, we reject intervals for which
σ
Y
I
is anomalously large. In particular, for the
192-sec analysis, we require
σ
Y
I
<
1 sec for the H1L1 pair, and
σ
Y
I
<
2 sec for the H2L1 pair
(recall that
Y
is normalized such that
< Y >
= Ω
α
T
, with
T
= 192 sec in this case). The
glitch cut and the large-sigma cut largely overlap, and are d
esigned to remove particularly
noisy time-periods from the analysis. Note, also, that due t
o the weighting with 1
2
Y
I
, the
contribution of these intervals to the final result would be s
uppressed, but we reject them from
the analysis nevertheless. Third, we reject the intervals f
or which ∆
σ
=
|
σ
Y
I
σ
Y
I
|
Y
I
> ζ
.
Here,
σ
Y
I
is calculated using the two intervals neighboring interval
I
, and
σ
Y
I
is calculated
using the interval
I
itself. The optimization of threshold
ζ
is discussed below. The goal of
– 13 –
0
0.01
0.02
0.03
0.04
0.05
10
−1
10
0
10
1
10
2
10
3
10
4
Γ
Number of Frequency Bins
H1L1
1/N
avg
= 0.0013755
Before Notching
After Notching
exp(−
Γ
N
avg
)
0
0.01
0.02
0.03
0.04
0.05
10
−1
10
0
10
1
10
2
10
3
10
4
Γ
Number of Frequency Bins
H2L1
1/N
avg
= 0.0010764
Before Notching
After Notching
exp(−
Γ
N
avg
)
Fig. 4.— Histogram of the coherence for H1L1 (top) and H2L1 (b
ottom) at 1 mHz resolution
follows the expected exponential distribution, with expon
ent coefficient
N
avg
(the number of
time-periods over which the average is made).
– 14 –
this cut is to capture noise-transients in the data, and reje
ct them from the analysis. Figure
5 shows the impact of these cuts for the H1L1 pair, analyzed wi
th 192-sec segments, 1/32
Hz resolution, and with
ζ
= 0
.
3. This Figure also shows daily variation in the sensitivity
to stochastic GW background, arising from the variation in t
he strain sensitivity of the
interferometers, which is typically worse during the week-
days than during the weekends or
nights.
Figure 6 shows the distribution of the residuals for the same
analysis. For a given
interval
I
, the residual is defined as
Y
I
< Y >
σ
Y
I
.
(8)
Note that the data quality cuts remove outliers from the resi
dual distribution, hence making
the data more stationary. After the cuts, the Kolmogorov-Sm
irnov test indicates that the
residual distribution is consistent with a Gaussian, for bo
th H1L1 and H2L1 analyses with
192-sec intervals, 1/32 Hz resolution, and
ζ
= 0
.
3.
3. Results
3.1. New Upper Limit
We performed a “blind” analysis for the H1L1 and the H2L1 pair
s with 60-sec intervals,
1
/
4 Hz resolution, and
ζ
= 0
.
2. To avoid biasing the results, all data-quality cuts were
defined based on studies done with a 0.1 sec time-shift betwee
n the two interferometers in
a pair. Such a time-shift removes any GW correlations, witho
ut significantly affecting the
instrumental noise performance. After the data quality cut
s were finalized, we made one
last pass through the data, with zero time-shift, and obtain
ed the final results of the blind
analysis.
The results from the blind analysis for the frequency-indep
endent template spectrum
(
α
= 0) are listed in the first row of Table 1 for H1L1 and in the first
row of Table 2 for
H2L1. These results show no evidence of a stochastic GW backg
round. After completing
the blind analysis, we discovered that the instrumental 1 Hz
harmonic lines, discussed in
Section 2.2, are correlated between the two sites. We felt co
mpelled on scientific grounds not
to ignore these correlations, even though they had been disc
overed after our initial, blind,
analysis was complete.
In order to remove from our results any possible influence of t
he correlated lines, we
repeated our analysis with refined frequency resolution of 1
/
32 Hz. We increased the interval