of 17
First cross-correlation analysis of interferometric and resonant-bar gravitational-wave data for
stochastic backgrounds
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*
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*
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̈
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2
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PHYSICAL REVIEW D
76,
022001 (2007)
1550-7998
=
2007
=
76(2)
=
022001(17)
022001-1
©
2007 The American Physical Society
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33
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35
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H. zur Mu
̈
hlen,
36
and J. Zweizig
14
(LIGO Scientific Collaboration and ALLEGRO Collaboration)
1
Albert-Einstein-Institut, Max-Planck-Institut fu
̈
r Gravitationsphysik, D-14476 Golm, Germany
2
Albert-Einstein-Institut, Max-Planck-Institut fu
̈
r Gravitationsphysik, D-30167 Hannover, Germany
3
Andrews University, Berrien Springs, Michigan 49104, USA
4
Australian National University, Canberra, 0200, Australia
5
California Institute of Technology, Pasadena, California 91125, USA
6
Caltech-CaRT, Pasadena, California 91125, USA
7
Cardiff University, Cardiff, CF24 3AA, United Kingdom
8
Carleton College, Northfield, Minnesota 55057, USA
9
Charles Sturt University, Wagga Wagga, NSW 2678, Australia
10
Columbia University, New York, New York 10027, USA
11
Embry-Riddle Aeronautical University, Prescott, Arizona 86301, USA
12
Hobart and William Smith Colleges, Geneva, New York 14456, USA
13
Inter-University Centre for Astronomy and Astrophysics, Pune-411007, India
14
LIGO-California Institute of Technology, Pasadena, California 91125, USA
15
LIGO Hanford Observatory, Richland, Washington 99352, USA
16
LIGO Livingston Observatory, Livingston, Louisiana 70754, USA
17
LIGO-Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
18
Louisiana State University, Baton Rouge, Louisiana 70803, USA
19
Louisiana Tech University, Ruston, Louisiana 71272, USA
20
Loyola University, New Orleans, Louisiana 70118, USA
21
Moscow State University, Moscow, 119992, Russia
22
NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
23
National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
24
Northwestern University, Evanston, Illinois 60208, USA
25
Rochester Institute of Technology, Rochester, New York 14623, USA
26
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX United Kingdom
27
San Jose State University, San Jose, California 95192, USA
28
Southeastern Louisiana University, Hammond, Louisiana 70402, USA
29
Southern University and A&M College, Baton Rouge, Louisiana 70813, USA
30
Stanford University, Stanford, California 94305, USA
31
Syracuse University, Syracuse, New York 13244, USA
32
The Pennsylvania State University, University Park, Pennsylvania 16802, USA
33
The University of Texas at Brownsville and Texas Southmost College, Brownsville, Texas 78520, USA
34
Trinity University, San Antonio, Texas 78212, USA
35
Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
36
Universita
̈
t Hannover, D-30167 Hannover, Germany
37
University of Adelaide, Adelaide, SA 5005, Australia
38
University of Birmingham, Birmingham, B15 2TT, United Kingdom
39
University of Florida, Gainesville, Florida 32611, USA
40
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
41
University of Maryland, College Park, Maryland 20742 USA
42
University of Michigan, Ann Arbor, Michigan 48109, USA
43
University of Oregon, Eugene, Oregon 97403, USA
44
University of Rochester, Rochester, New York 14627, USA
45
University of Salerno, 84084 Fisciano (Salerno), Italy
46
University of Sannio at Benevento, I-82100 Benevento, Italy
47
University of Southampton, Southampton, SO17 1BJ, United Kingdom
48
University of Strathclyde, Glasgow, G1 1XQ, United Kingdom
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
022001 (2007)
022001-2
49
University of Washington, Seattle, Washington 98195, USA
50
University of Western Australia, Crawley, WA 6009, Australia
51
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA
52
Washington State University, Pullman, Washington 99164, USA
(Received 28 March 2007; published 9 July 2007; publisher error corrected 23 July 2007; publisher error corrected 4 March 2008)
Data from the LIGO Livingston interferometer and the ALLEGRO resonant-bar detector, taken during
LIGO’s fourth science run, were examined for cross correlations indicative of a stochastic gravitational-
wave background in the frequency range 850 – 950 Hz, with most of the sensitivity arising between 905
and 925 Hz. ALLEGRO was operated in three different orientations during the experiment to modulate the
relative sign of gravitational-wave and environmental correlations. No statistically significant correlations
were seen in any of the orientations, and the results were used to set a Bayesian 90% confidence level
upper limit of

gw

f

1
:
02
, which corresponds to a gravitational-wave strain at 915 Hz of
1
:
5

10

23
Hz

1
=
2
. In the traditional units of
h
2
100

gw

f

, this is a limit of 0.53, 2 orders of magnitude better
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
One of the signals targeted by the current generation of
ground-based gravitational-wave (GW) detectors is a sto-
chastic gravitational-wave background (SGWB) [
1
3
].
Such a background is analogous to the cosmic microwave
background, although the dominant contribution is un-
likely to have a blackbody spectrum. A SGWB can be
characterized as cosmological or astrophysical in origin.
Cosmological backgrounds can arise from, for example,
pre-big-bang models [
4
6
], amplification of quantum vac-
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,
which can include rotating neutron stars [
15
,
16
], super-
novae [
17
], and low-mass x-ray binaries [
18
].
The standard cross-correlation search [
19
] for a SGWB
necessarily requires two or more GW detectors. Such
searches have been performed using two resonant-bar de-
tectors [
20
] and also using two or more kilometer-scale
GW interferometers (IFOs) [
21
23
]. The present work
describes the results of the first cross-correlation analysis
carried out between an IFO [the 4 km IFO at the LIGO
Livingston Observatory (LLO), known as L1] and a bar
(the cryogenic ALLEGRO detector, referred to as A1).
This pair of detectors is separated by only 40 km, the
closest pair among modern ground-based GW detector
sites, which allows it to probe the stochastic GW spectrum
around 900 Hz. In addition, the ALLEGRO bar can be
rotated, changing the response of the correlated data
streams to stochastic GWs and thus providing a means to
distinguish correlations due to a SGWB from those due to
correlated environmental noise [
24
]. This paper describes
cross-correlation analysis of L1 and A1 data taken between
February 22 and March 23, 2005, during LIGO’s fourth
science run (S4). Average sensitivities of L1 and A1 during
S4 are shown in Fig.
1
. ALLEGRO was operated in three
orientations, which modulated the GW response of the
LLO-ALLEGRO pair through 180

of phase.
The LLO-ALLEGRO correlation experiment is comple-
mentary to experiments using data from the two LIGO
sites, in that it is sensitive to a SGWB at frequencies of
around 900 Hz rather than 100 Hz. Targeted sources are
thus those with a relatively narrow-band spectrum peaked
near 900 Hz. Spectra with such shapes can arise from
850
860
870
880
890
900
910
920
930
940
950
10
−23
10
−22
10
−21
10
−20
10
−19
10
−18
Avg Calibrated ASD from S4 non−NULL non−PG
Frequency (Hz)
Amplitude Spectrum (Strain /
Hz)
L1 ASD
A1 ASD
Spectrum for
gw
(f)=1.02
FIG. 1. Sensitivity of the LLO IFO (L1) and the ALLEGRO
bar (A1) during S4, along with strain associated with

gw

f

1
:
02
(assuming a Hubble constant of
H
0

72 km
=
s
=
Mpc
).
[There are two

gw

f

1
:
02
curves, corresponding to the
different strain levels such a background would generate in an
IFO and a bar, as explained in Sec. II and [
26
].] The quantity
plotted is amplitude spectral density (ASD), the square root of
the one-sided power spectral density defined in (
4.2
), at a
resolution of 0.25 Hz.
*
Member of ALLEGRO Collaboration.
Member of LIGO Scientific Collaboration and ALLEGRO
Collaboration.
http://www.ligo.org/
FIRST CROSS-CORRELATION ANALYSIS OF
...
PHYSICAL REVIEW D
76,
022001 (2007)
022001-3
exotic cosmological models, as described in Sec. II, or
from astrophysical populations [
16
].
The organization of this paper is as follows. Section II
reviews the properties and characterization of a SGWB.
Section III describes the LLO and ALLEGRO experimen-
tal arrangements, including the data acquisition and strain
calibration for each instrument. Section IV describes the
cross-correlation method and its application to the present
situation. Section V describes the details of the postpro-
cessing methods and statistical interpretation of the cross-
correlation results. Section VI describes the results of the
cross-correlation measurement and the corresponding
upper limit on the SGWB strength in the range 850 –
950 Hz. Section VII describes the results of our analysis
pipeline when applied to simulated signals injected both
within the analysis software and in the hardware of the
instruments themselves. Section VIII compares our results
to those of previous experiments and to the sensitivities of
other operating detector pairs. Section IX considers the
prospects for future work.
II. STOCHASTIC GRAVITATIONAL-WAVE
BACKGROUNDS
A gravitational wave (GW) is described by the metric
tensor perturbation
h
ab

~
r; t

. A given GW detector, located
at position
~
r
det
on the Earth, will measure a GW strain
which, in the long-wavelength limit, is some projection of
this tensor:
h

t

h
ab

~
r
det
;t

d
ab
;
(2.1)
where
d
ab
is the detector response tensor, which is
d
ab

ifo


1
2

^
x
a
^
x
b

^
y
a
^
y
b

(2.2)
for an interferometer with arms parallel to the unit vectors
^
x
and
^
y
and
d
ab

bar


^
u
a
^
u
b
(2.3)
for a resonant bar with long axis parallel to the unit vector
^
u
.
A stochastic GW background (SGWB) can arise from a
superposition of uncorrelated cosmological or astrophysi-
cal sources. Such a background, which we assume to be
isotropic, unpolarized, stationary, and Gaussian, will gen-
erate a cross correlation between the strains measured by
two detectors. In terms of the continuous Fourier transform
defined by
~
a

f

R
1
1
dt a

t

exp

i
2
ft

, the expected
cross correlation is
h
~
h

1

f

~
h
2

f
0
i 
1
2


f

f
0

S
gw

f


12

f

;
(2.4)
where

12

f

d
1
ab
d
cd
2
5
4

Z
Z
d
2

^
n
P
TT
^
nab
cd
e
i
2
f
^
n

~
r
2

~
r
1

=c
(2.5)
is the overlap reduction function (ORF) [
25
] between the
two detectors, defined in terms of the projector
P
TT
^
nab
cd
onto traceless symmetric tensors transverse to the unit
vector
^
n
. The ORF for several detector pairs of interest is
shown in Fig.
2
.
S
gw

f

is the one-sided spectrum of the SGWB. This is
the one-sided power spectral density (PSD) the background
would generate in an interferometer with perpendicular
arms, which can be seen from (
2.4
) and the fact that the
ORF of such an interferometer with itself is unity. Since the
ORF of a resonant bar with itself is
4
=
3
(see [
26
] and
Sec. VII A for more details), the PSD of the strain mea-
sured by a bar detector due to the SGWB would be

4
=
3

S
gw

f

.
A related measure of the spectrum is the dimensionless
quantity

gw

f

, the GW energy density per unit logarith-
mic frequency divided by the critical energy density

c
needed to close the universe:

gw

f

f

c
d
gw
df

10

2
3
H
2
0
f
3
S
gw

f

:
(2.6)
Note that the definition

gw

f

thus depends on the value
0
100
200
300
400
500
600
700
800
900
1000
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Frequency (Hz)
Overlap Reduction Function
LLO−LHO
LLO−ALLEGRO (N108
°
W) "XARM"
LLO−ALLEGRO (N18
°
W) "YARM"
LLO−ALLEGRO (N63
°
W) "NULL"
FIG. 2.
The overlap reduction function for LIGO Livingston
Observatory (LLO) with ALLEGRO and with LIGO Hanford
Observatory (LHO). The three LLO-ALLEGRO curves corre-
spond to the three orientations in which ALLEGRO was oper-
ated during LIGO’s S4 run: ‘‘XARM’’ (N108

W) is nearly
parallel to the
x
-arm of LLO (‘‘aligned’’); ‘‘YARM’’ (N18

W)
is nearly parallel to the
y
-arm of LLO (‘‘antialigned’’); NULL
(N63

W) is halfway in between these two orientations (a ‘‘null
alignment’’ midway between the two LLO arms). Note that for
nonzero frequencies, the separation vector between the two sites
breaks the symmetry between the XARM and YARM align-
ments, and leads to an offset of the NULL curve, as described in
[
26
]. The LLO-LHO overlap reduction function is shown for
reference. The frequency band of the present analysis,
850 Hz

f

950 Hz
, is indicated with dashed vertical lines.
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
022001 (2007)
022001-4
of the Hubble constant
H
0
. Most SGWB literature avoids
this artificial uncertainty by working in terms of
h
2
100

gw

f


H
0
100 km
=
s
=
Mpc

2

gw

f

(2.7)
rather than

gw

f

itself. We will instead follow the pre-
cedent set by [
22
] and quote numerical values for

gw

f

assuming a Hubble constant of
72 km
=
s
=
Mpc
.
A variety of spectral shapes have been proposed for

gw
,
for both astrophysical and cosmological stochastic back-
grounds [
3
,
27
,
28
]. For example, whereas the slow-roll
inflationary model predicts a constant

gw

f

in the bands
of LIGO or ALLEGRO, certain alternative cosmological
models predict broken-power-law spectra, where the rising
and falling slopes and the peak frequency are determined
by model parameters [
3
]. String-inspired pre-big-bang
cosmological models belong to this category [
5
,
29
]. For
certain ranges of these three parameters, the LLO-
ALLEGRO correlation measurement offers the best con-
straints on theory that can be inferred from any contempo-
rary observation. This can happen, e.g., if the power-law
exponent on the rising spectral slope is greater than 3 and
the peak frequency is sufficiently close to 900 Hz [
30
].
III. EXPERIMENTAL SETUP
A. The LIGO Livingston interferometer
The experimental setup of the LIGO observatories has
been described at length elsewhere [
31
]. Here we provide a
brief review, with particular attention paid to details sig-
nificant
for
the
LLO-ALLEGRO
cross-correlation
measurement.
The LIGO Livingston Observatory (LLO) is an inter-
ferometric GW detector with perpendicular 4-km arms.
The laser interferometer senses directly any changes in
the differential arm length. It does this by splitting a light
beam at the vertex, sending the separate beams into 4-km
long optical cavities of their respective arms, and then
recombining the beams to detect any change in the optical
phase difference between the arms, which is equivalent to a
difference in light travel time. This provides a measure-
ment of
h

t

as defined in (
2.1
) and (
2.2
). However, the
measured quantity is not exactly
h

t

for two reasons.
First, there are local forces which perturb the test
masses, and so produce changes in arm length. There are
also optical and electronic fluctuations that mimic real
strains. The combination of these effects causes a strain
noise
n

t

to always be present in the output, producing a
measurement of
s

t

h

t

n

t

:
(3.1)
Second, the test masses are not really free. There is a
servo system, which uses changes in the differential arm
length as its error signal
q

t

, and then applies extra (‘‘con-
trol’’) forces to the test masses to keep the differential arm
length nearly zero. It is this error signal
q

t

which is
recorded, and its relationship to the strain estimate
s

t

is
most easily described in the Fourier domain:
~
s

f

~
R

f

~
q

f

:
(3.2)
The response function
~
R

f

is estimated by a combination
of modeling and measurement [
32
] and varies slowly over
the course of the experiment.
Because the error signal
q

t

has a smaller dynamic
range than the reconstructed strain
s

t

, our analysis starts
from the digitized time series
q
k

q

t
k

(sampled
2
14

16 384
times per second, and digitally downsampled to
4096 Hz in the analysis) and reconstructs the LLO strain
only in the frequency domain.
B. The ALLEGRO resonant-bar detector
The ALLEGRO resonant detector, operated by a group
from Louisiana State University [
33
], is a two-ton alumi-
num cylinder coupled to a niobium secondary resonator.
The secondary resonator is part of an inductive transducer
[
34
] which is coupled to a DC SQUID. Strain along the
cylindrical axis excites the first longitudinal vibrational
mode of the bar. The transducer is tuned for sensitivity to
this mechanical mode. Raw data acquired from the detector
thus reflect the high-
Q
resonant mechanical response of the
system. A major technical challenge of this analysis is due
to the extent to which the bar data differ from those of the
interferometer.
1. Data acquisition, heterodyning, and sampling
The ALLEGRO detector has a relatively narrow sensi-
tive band of
100 Hz
centered around
900 Hz
near the
two normal modes of the mechanical bar-resonator system.
For this reason, the output of the detector can first be
heterodyned with a commercial lock-in amplifier to greatly
reduce the sampling rate, which is set at
250 samples
=
s
.
Both the in-phase and quadrature outputs of the lock-in are
recorded and the detector output can thus be represented as
a complex time series which covers a 250 Hz band centered
on the lock-in reference oscillator frequency. This refer-
ence frequency is chosen to be near the center of the
sensitive band, and during the S4 run it was set to
904 Hz. The overall timing of data heterodyned in this
fashion is provided by both the sampling clock and the
reference oscillator. Both time bases were locked to the
Global Positioning System (GPS) time reference.
The nature of the resonant detector and its data acquis-
ition system gives rise to a number of timing issues:
heterodyning, filter delays of the electronics, and the tim-
ing of the data acquisition system itself [
35
]. It is of critical
importance that the timing be fully understood so that the
phase of any potential signal may be recovered.
Convincing evidence that all of the issues are accounted
for is demonstrated by the recovery and cross correlation of
test signals simultaneously injected into both detectors.
FIRST CROSS-CORRELATION ANALYSIS OF
...
PHYSICAL REVIEW D
76,
022001 (2007)
022001-5
The signals were recovered at the expected phase as pre-
sented in Sec. VII.
2. Strain calibration
The raw detector output is proportional to the displace-
ment of the secondary resonator, and thus has a spectrum
with sharp line features due to the high-
Q
resonances of the
bar-resonator system as can be seen in Fig.
3
. The desired
GW signal is the effective strain on the bar, and recovering
this means undoing the resonant response of the detector.
This response has a long coherence time — thus long
stretches of data are needed to resolve the narrow lines in
the raw data. The strain data have a much flatter spectrum,
as shown in Fig.
1
. Therefore it is practical to generate the
calibrated strain time series,
s

t

, and use that as the input
to the cross-correlation analysis.
The calibration procedure, described in detail in [
35
], is
carried out in the frequency domain and consists of the
following: A 30 min stretch of clean ALLEGRO data is
windowed and Fourier transformed. The mechanical mode
frequencies drift slightly due to small temperature varia-
tions, so these frequencies are determined for each stretch
and those are incorporated into the model of the mechani-
cal response of the system to a strain. The model consists of
two double poles at these normal mode frequencies. In
addition to this response, we must then account for the
phase shifts due to the time delays in the lock-in and
antialiasing filters.
After applying the full response function, the data are
then inverse Fourier transformed back to the time domain.
The next 50% overlapping 30 min segment is then taken.
The windowed segments are stitched together until the
entire continuous stretch of good data is completed. The
first and last 15 minutes are dropped. The result represents
a heterodyned complex time series of strain, whose ampli-
tude spectral density is shown in Fig.
1
.
The overall scale of the detector output in terms of strain
is determined by applying a known signal to the bar. A
force applied to one end of the bar has a simple theoretical
relationship to an equivalent gravitational strain [
35
37
].
A calibrated force can be applied via a capacitive ‘‘force
generator’’ which also provides the mechanism used for
hardware signal injections. A reciprocal measurement —
excitation followed by measurement with the same trans-
ducer — along with known properties of the mechanical
system, allows the determination of the force generator
constant. With that constant determined (with units of
newtons per volt), a calibrated force is applied to the bar
and the overall scale of the response determined.
3. Orientation
A unique feature of this experiment is the ability to
rotate the ALLEGRO detector and modulate the response
of the ALLEGRO-LLO pair to a GW background [
24
].
Data were taken in three different orientations of
ALLEGRO, known as XARM, YARM, and NULL, de-
tailed in Table
I
. As shown in Fig.
2
and (
2.4
), these
840
860
880
900
920
940
960
10
0
10
1
10
2
10
3
10
4
Raw spectrum, S4 run − averaged over orientation
frequency (Hz)
Amplitude Spectrum (counts /
Hz)
XARM
YARM
NULL
FIG. 3.
The graph displays the amplitude spectral density of
raw ALLEGRO detector output during S4, at a frequency
resolution of 0.1 Hz. For this graph these data have not been
transformed to strain via the calibrated response function. The
vertical scale represents digital counts
=

Hz
p
. The normal me-
chanical modes where the detector is most sensitive are at
880.78 Hz and 917.81 Hz. There is an injected calibration line
at 837 Hz. Also prominent are an extra mechanical resonance at
885.8 Hz and a peak at 904 Hz (DC in the heterodyned data
stream).
TABLE I. Orientations of ALLEGRO during the LIGO S4 Run, including overlap reduction function evaluated at the extremes of
the analyzed frequency range, and at the frequency of peak sensitivity. Note that, while the NULL orientation represents perfect
misalignment (


0
) at 0 Hz, it is not quite perfect at the frequencies of interest. This is primarily because of an azimuth-independent
offset term in


f

which contributes at nonzero frequencies [
24
,
26
]. Because of this term, it is impossible to orient ALLEGRO so that


f

0
at all frequencies, and to set it to zero around 915 Hz one would have to use an azimuth of N62

W rather than N63

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


850 Hz



915 Hz



950 Hz

2005 Feb 22 – 2005 Mar 4
YARM
N108

W

0
:
9087

0
:
8947

0
:
8867
2005 Mar 4 – 2005 Mar 18
XARM
N18

W
0.9596
0.9533
0.9498
2005 Mar 18– 2005 Mar 23
NULL
N63

W
0.0280
0.0318
0.0340
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
022001 (2007)
022001-6
orientations correspond to different pair responses due to
different overlap reduction functions. In the XARM ori-
entation — the bar axis parallel to the x-arm of the inter-
ferometer — a GW signal produces positive correlation
between the data in the two detectors. In the YARM
orientation a GW signal produces an anticorrelation. In
the NULL orientation — the bar aligned halfway between
the two arms of the interferometer — the pair has very
nearly zero sensitivity as a GW signal produces almost
zero correlation between the detectors’ data. A real signal
is thus modulated whereas many types of instrumental
correlation would not have the same dependence on
orientation.
IV. CROSS-CORRELATION METHOD
This section describes the method to used to search for a
SGWB by cross-correlating detector outputs. In the case of
L1-A1 correlation measurements, it is complicated by the
different sampling rates for the two instruments and the
fact that the A1 data are heterodyned at 904 Hz prior to
digitization.
A. Continuous-time idealization
Both ground-based interferometric and resonant-mass
detectors produce a time-series output which can be related
to a discrete sampling of the signal
s
i

t

h
i

t

n
i

t

;
(4.1)
where
i
labels the detector (1 or 2 in this case),
h
i

t

is the
gravitational-wave strain defined in (
2.1
), and
n
i

t

is the
instrumental noise associated with each detector, converted
into an equivalent strain. The detector output is character-
ized by its power spectral density
P
i

f

,
h
~
s

i

f

~
s
i

f
0
i 
1
2


f

f
0

P
i

f

;
(4.2)
which should be dominated by the autocorrelation of the
noise [
h
~
s

i

f

~
s
i

f
0
i  h
~
n

i

f

~
n
i

f
0
i
]. If the instrument
noise is approximately uncorrelated, the expected cross
correlation of the detector outputs is [cf. (
2.4
)]
h
~
s

1

f

~
s
2

f
0
i  h
~
h

1

f

~
h
2

f
0
i 
1
2


f

f
0

S
gw

f


12

f

(4.3)
which can be used along with the autocorrelation (
4.2
)to
determine the statistical properties of the cross-correlation
statistic defined below.
We use the optimally filtered cross-correlation method
described in [
19
,
21
] to calculate a cross-correlation statis-
tic which is an approximation to the continuous-time cross-
correlation statistic
Y
c

Z
dt
1
dt
2
s
1

t
1

Q

t
1

t
2

s
2

t
2


Z
df
~
s

1

f

~
Q

f

~
s
2

f

:
(4.4)
In the continuous-time idealization, such a cross-
correlation statistic, calculated over a time
T
, has an ex-
pected mean

Y
c
h
Y
c
i
T
2
Z
1
1
df
j
f
j
S
gw

f

~
Q

f

(4.5)
and variance

2
Y
c
h
Y
c


Y
c

2
i
T
4
Z
1
1
dfP
1

f

P
2

f
j
~
Q

f
j
2
:
(4.6)
Using (
4.5
) and (
4.6
), the optimal choice for the filter
~
Q

f

,
given a predicted shape for the spectrum
S
gw

f

can be
shown [
19
]tobe
~
Q

f
/

j
f
j
S
gw

f

P
1

f

P
2

f

:
(4.7)
If the spectrum of gravitational waves is assumed to be a
power law over the frequency band of interest, a conve-
nient parametrization of the spectrum, in terms of

gw

f

defined in (
2.6
), is

gw

f


R

f
f
R


;
(4.8)
where
f
R
us a conveniently chosen reference frequency
and

R


gw

f
R

. The cross-correlation measurement is
then a measurement of

R
, and if the optimal filter is
normalized according to
~
Q

f

N


f

f=f
R


j
f
j
3
P
1

f

P
2

f

;
(4.9a)
where
N

20

2
3
H
2
0

Z
1
1
df
f
6


f

f=f
R


2
P
1

f

P
2

f



1
;
(4.9b)
then the expected statistics of
Y
c
in the presence of a
background of actual strength

R
are

Y
c


R
T
(4.10)
and

2
Y
c

T

10

2
3
H
2
0

2

Z
1
1
df
f
6


f

f=f
R


2
P
1

f

P
2

f



1
(4.11)
and a measurement of
Y
c
=T
provides a
point estimate
of
the background strength

R
with associated estimated
error bar

Y
c
=T
.
B. Discrete-time method
1. Handling of different sampling rates and heterodyning
Stochastic-background measurements using pairs of
LIGO interferometers [
21
] have implemented (
4.4
) from
discrete samplings
s
i
k

s

t
0
kt

as follows: First the
continuous Fourier transforms
~
s

f

were approximated
FIRST CROSS-CORRELATION ANALYSIS OF
...
PHYSICAL REVIEW D
76,
022001 (2007)
022001-7
using the discrete Fourier transforms of windowed and
zero-padded versions of the discrete time series; then an
optimal filter was constructed using an approximation to
(
4.7
), and finally the product of the three was summed bin-
by-bin to approximate the integral over frequencies. The
discrete version of
~
Q

f

was simplified in two ways: first,
because of the averaging used in calculating the power
spectrum, the frequency resolution on the optimal filter
was generally coarser than that associated with the discrete
Fourier transforms of the data streams, and second, the
value of the optimal filter was arbitrarily set to zero outside
some desired range of frequencies
f
min

f

f
max
. This
was justified because the optimal filter tended to have little
support for frequencies outside that range.
The present experiment has two additional complica-
tions associated with the discretization of the time-series
data. First, the A1 data are not a simple time sampling of
the gravitational-wave strain, but are base banded with a
heterodyning frequency
f
h
2

904 Hz
as described in
Sec. III B 1 and III B 2. Second, the A1 data are sampled
at

t
2


1

250 Hz
, while the L1 data are sampled at
16384 Hz, and subsequently downsampled to

t
1


1

4096 Hz
. This would make a time domain cross correlation
extremely problematic, as it would necessitate a large
variety of time offsets
t
1

t
2
. In the frequency domain,
it means that the downsampled L1 data, once calibrated,
represent frequencies ranging from

2048 Hz
to 2048 Hz,
while the calibrated A1 data represent frequencies ranging
from

904

125

Hz

779 Hz
to

904
125

Hz

1029 Hz
. These different frequency ranges do not pose a
problem, as long as the range of frequencies chosen for the
integral satisfies
f
min
>
779 Hz
and
f
max
<
1029 Hz
.
Another requirement is that, for the chosen frequency
resolution, the A1 data heterodyne reference frequency
must align with a frequency bin in the L1 data. This is
satisfied for integer-second data spans and integer-hertz
reference frequencies. With these conditions, the Fourier
transforms of the A1 and L1 data are both defined over a
common set of frequencies, as detailed in [
38
]. Looking at
the A1 spectrum in Fig.
1
, a reasonable range of frequen-
cies should be a subset of the range
850 Hz
&
f
&
950 Hz
.
2. Discrete-time cross correlation
Explicitly, the time series inputs to the analysis pipeline,
from each
T

60 sec
of analyzed data, are:
(i) For L1, a real time series
f
q
1
j
j
j

0
;
...
N
1

1
g
,
sampled at

t
1


1

4096 Hz
, consisting of
N
1

T=t
1

245 760
points. This is obtained by down-
sampling the raw data stream by a factor of 4. The
data are downsampled to 4096 Hz rather than
2048 Hz to ensure that the roll-off of the associated
antialiasing filter is outside the frequency range
being analyzed. The raw L1 data are related to
gravitational-wave strain by the calibration response
function
~
R
1

f

constructed as described in Sec. III A.
(ii) For A1, a complex time series
f
s
h
2
k
j
k

0
;
...
N
2

1
g
, sampled at

t
2


1

250 Hz
, consisting of
N
2

T=t
2

150 00
points. This is calibrated to
represent strain data as described in Sec. III B 2, but
still heterodyned.
To produce an approximation of the Fourier transform of
the data from detector
i
, the data are multiplied by an
appropriate windowing function, zero padded to twice their
original length, discrete-Fourier-transformed, and multi-
plied by
t
i
. In addition, the L1 data are multiplied by a
calibration response function, while the A1 data are inter-
preted as representing frequencies appropriate in light of
the heterodyne. For L1,
~
s
1

f

~
s
1
:

~
R
1

f

X
N
1

1
j

0
w
1
j
q
1
j

exp


i
2
‘j
2
N
1

t
1
;

N
1
;
...
;N
1

1
;
(4.12)
where
f

2
T
is the frequency associated with the
th
frequency bin. In the case of A1, the identification is offset
by
h
2

f
h

2
T

107 880
:
~
s
2

f

~
s
2
:

X
N
2

1
k

0
w
2
k
q
2
k

exp


i
2



h
2

k
2
N
2

t
2
;

h
2

N
2
;
...
;‘
h
2
N
2

1
:
(4.13)
As is shown in [
38
], if we construct a cross-correlation
statistic
Y
:

X
max

min
1
2
T
~
s
1

f


~
Q

f

~
s
2

f

;
(4.14)
the expected mean value in the presence of a stochastic
background with spectrum
S
gw

f

is

:
h
Y
i
w
1
w
2
T
2
X
max

min
1
2
T
~
Q

f



f

S
gw

f

;
(4.15)
where
w
1
w
2
is an average of the product of the two
windows, calculated using the points which exist at both
sampling rates; specifically, if
r
1
and
r
2
are the smallest
integers such that
t
1
=t
2

r
1
=r
2

125
=
2048
,
w
1
w
2

r
1
r
2
N
X
N=

r
1
r
2

1
n

0
w
1
nr
2
w
2
nr
1
:
(4.16)
Note that, while the average value given by (
4.15
)is
manifestly real, any particular measurement of
Y
will be
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
022001 (2007)
022001-8
complex, because of the bandpass associated with the
heterodyning of the A1 data. As shown in [
38
], the real
and imaginary parts of the cross-correlation statistic each
have expected variance

2
:

1
2
h
Y

Y
i

T
8
w
2
1
w
2
2
X
max

min
1
2
T
j
~
Q

f
j
2
P
1
j
f
j
P
2
j
f
j
;
(4.17)
where once again
w
2
1
w
2
2
is an average over the time samples
the two windows have ‘‘in common’’:
w
2
1
w
2
2

r
1
r
2
N
X
N=

r
1
r
2

1
n

0

w
1
nr
2

2

w
2
nr
1

2
:
(4.18)
3. Construction of the optimal filter
To perform the cross correlation in (
4.14
), we need to
construct an optimal filter by a discrete approximation to
(
4.9
). We approximate the power spectra
P
1

f

using
Welch’s method [
39
]; as a consequence of the averaging
of periodograms constructed from shorter stretches of data,
the power spectra are estimated with a frequency resolution
f
which is coarser than the
1
=
2
T
associated with the
construction in (
4.14
). As detailed in [
21
], we handle this
by first multiplying together
~
s
1

f


and
~
s
2

f

at the
finer frequency resolution, then summing together sets of
2
Tf
bins and multiplying them by the coarser-grained
optimal filter. For our search,
f

0
:
25 Hz
and
T

60 sec
,so
2
Tf

30
.
4. Power spectrum estimation
Because the noise power spectrum of the LLO can vary
with time, we continuously update the optimal filter used in
the cross-correlation measurement. However, using an op-
timal filter constructed from power spectra calculated from
the same data to be analyzed leads to a bias in the cross-
correlation statistic
Y
, as detailed in [
40
]. To avoid this, we
analyze each
T

60 sec
segment of data using an optimal
filter constructed from the average of the power spectra
from the segments before and after the segment to be
analyzed. This method is known as ‘‘sliding power spec-
trum estimation’’ because, as we analyze successive seg-
ments of data, the segments used to calculated the PSDs for
the optimal filter slide through the data to remain adjacent.
The
f

0
:
25 Hz
resolution is obtained by calculating
the power spectra using Welch’s method with 29 over-
lapped 4-second subsegments in each 60-second segment
of data, for a total of 58 subsegments.
V. POSTPROCESSING TECHNIQUES
A. Stationarity cut
The sliding power spectrum estimation method de-
scribed in Sec. IV B 4 can lead to inaccurate results if the
noise level of one or both instruments varies widely over
successive intervals. Most problematically, if the data are
noisy only within a single analysis segment, consideration
of the power spectrum constructed from the segments
before and after, which are not noisy, will cause the seg-
ment to be overweighted when combining cross-
correlation data from different segments. To avoid this,
we calculate for each segment both the usual estimated
standard deviation

I
using the ‘‘sliding’’ PSD estimator
and the ‘‘naive’’ estimated standard deviation

0
I
using the
data from the segment itself. If the ratio of these two is too
far from unity, the segment is omitted from the cross-
correlation analysis. The threshold used for this analysis
was 20%. The amount of data excluded based on this cut
was between 1% and 2% in each of the three orientations,
and subsequent investigations show the final results would
not change significantly for any reasonable choice of
threshold.
B. Bias correction of estimated error bars
Although use of the sliding power spectrum estimator
removes any bias from the optimally filtered cross-
correlation measurement, the methods of [
40
] show that
there is still a slight underestimation of the estimated
standard deviation associated with the finite number of
periodograms averaged together in calculating the power
spectrum. To correct for this, we have to scale up the error
bars by a factor of
1
1
=

N
avg

9
=
11

, where
N
avg
is the
number subsegments whose periodograms are averaged
together in estimating the power spectrum for the optimal
filter. For the data analyzed with the sliding power spec-
trum estimator, 29 overlapping four-second subsegments
are averaged from each of two 60-second segments, for a
total
N
avg

58
. This gives a correction factor of
1
1
=

58

9
=
11

1
:
021
. The naive estimated sigmas, de-
rived from power spectra calculated using 29 averages in a
single 60-second segment, are scaled up by a factor of
1
1
=

29

9
=
11

1
:
042
.
C. Combination of analysis segment results
As shown in [
19
], the optimal way to combine a series of
independent cross-correlation measurements
f
Y
I
g
with as-
sociated one-sigma error bars is
Y
opt

P
I


2
I
Y
I
P
I


2
I
(5.1a)

Y
opt


X
I


2
I


1
=
2
:
(5.1b)
FIRST CROSS-CORRELATION ANALYSIS OF
...
PHYSICAL REVIEW D
76,
022001 (2007)
022001-9