Upper limit map of a background of gravitational waves
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PHYSICAL REVIEW D
76,
082003 (2007)
1550-7998
=
2007
=
76(8)
=
082003(11)
082003-1
©
2007 The American Physical Society
S. Vass,
14
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38
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40
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37
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14
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15
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21
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42
L. Zhang,
14
C. Zhao,
50
N. Zotov,
19
M. Zucker,
17
H. zur Mu
̈
hlen,
36
and J. Zweizig
14
(LIGO Scientific 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, CF2 3YB, 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
49
University of Washington, Seattle, Washington, 98195, USA
50
University of Western Australia, Crawley, WA 6009, Australia
B. ABBOTT
PHYSICAL REVIEW D
76,
082003 (2007)
082003-2
51
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA
52
Washington State University, Pullman, Washington 99164, USA
(Received 31 January 2007; published 29 October 2007; publisher error corrected 4 March 2008)
We searched for an anisotropic background of gravitational waves using data from the LIGO S4 science
run and a method that is optimized for point sources. This is appropriate if, for example, the gravitational
wave background is dominated by a small number of distinct astrophysical sources. No signal was seen.
Upper limit maps were produced assuming two different power laws for the source strain power spectrum.
For an
f
3
power law and using the 50 Hz to 1.8 kHz band the upper limits on the source strain power
spectrum vary between
1
:
2
10
48
Hz
1
100 Hz
=f
3
and
1
:
2
10
47
Hz
1
100 Hz
=f
3
, depending
on the position in the sky. Similarly, in the case of constant strain power spectrum, the upper limits vary
between
8
:
5
10
49
Hz
1
and
6
:
1
10
48
Hz
1
. As a side product a limit on an isotropic background
of gravitational waves was also obtained. All limits are at the 90% confidence level. Finally, as an
application, we focused on the direction of Sco-X1, the brightest low-mass x-ray binary. We compare
the upper limit on strain amplitude obtained by this method to expectations based on the x-ray flux from
Sco-X1.
DOI:
10.1103/PhysRevD.76.082003
PACS numbers: 04.80.Nn, 02.50.Ey, 04.30.Db, 07.05.Kf
I. INTRODUCTION
A stochastic background of gravitational waves can be
anisotropic if, for example, the dominant source of sto-
chastic gravitational waves comes from an ensemble of
astrophysical sources (e.g., [
1
,
2
]), and if this ensemble is
dominated by its strongest members. So far the LIGO
Scientific Collaboration has analyzed the data from the
first science runs for a stochastic background of gravita-
tional waves [
3
–
5
], assuming that this background is
iso-
tropic
. If astrophysical sources indeed dominate this
background, one should look for anisotropies.
A method that is optimized for extreme anisotropies,
namely, point sources of stochastic gravitational radiation,
was presented in [
6
]. It is based on the cross correlation of
the data streams from two spatially separated gravitational
wave interferometers, and is referred to as the radiometer
analysis. We have analyzed the data of the 4th LIGO
science run using this method.
Section II is a short description of the radiometer analy-
sis method. The peculiarities of the S4 science run are
summarized in Sec. III, and we discuss the results in
Sec. IV.
II. METHOD DESCRIPTION
A stochastic background of gravitational waves can be
distinguished from other sources of detector noise by cross
correlating two independent detectors. Thus we cross cor-
relate the data streams from a pair of detectors with a cross
correlation kernel
Q
, chosen to be optimal for a source
which is specified by an assumed strain power spectrum
H
f
and angular power distribution
P
^
. Specifically,
with the data stream divided into intervals labeled by
t
, and
with
~
s
1
;t
f
and
~
s
2
;t
f
representing the Fourier transforms
of the strain outputs of two detectors, this cross correlation
is computed in the frequency domain interval by interval as
Y
t
Z
1
1
df
~
s
1
;t
f
Q
t
f
~
s
2
;t
f
:
(1)
In contrast to the isotropic analysis the optimal filter
Q
t
is
now sidereal time dependent. It has the general form
Q
t
f
t
R
S
2
d
^
^
;t
f
P
^
H
j
f
j
P
1
f
P
2
f
(2)
where
t
is a normalization factor,
P
1
and
P
2
are the strain
noise power spectra of the two detectors.
H
j
f
j
and
P
^
are defined by
h
h
Af
^
h
A
0
f
0
^
0
i
AA
0
f
f
0
2
^
;
^
0
P
^
H
j
f
j
4
(3)
where
H
j
f
j
is the one-sided (positive frequencies only)
spectrum of strain power, summed over both polarizations.
This explains the factor of
1
4
and is appropriate for the
unpolarized stochastic background we search for.
h
Af
^
is
the gravitational wave strain in polarization
A
at frequency
f
arriving from the direction
^
. Finally, the factor
^
;t
in
Eq. (
2
) takes into account the sidereal time dependent time
delay due to the detector separation and the directionality
of the acceptance of the detector pair. Assuming that the
source is unpolarized,
^
;t
is given by
^
;t
f
1
2
X
A
e
i
2
f
^
x
*
t
=c
F
1
;t
A
^
F
2
;t
A
^
(4)
where
x
*
t
x
*
2
;t
x
*
1
;t
is the detector separation vector,
^
is the unit vector specifying the sky position, and
F
i;t
A
^
e
A
ab
^
1
2
^
X
i;t
a
^
X
i;t
b
^
Y
i;t
a
^
Y
i;t
b
(5)
is the response of detector
i
to a zero frequency, unit
amplitude,
A
or
polarized gravitational wave.
e
A
ab
^
is the spin-two polarization tensor for polarization
UPPER LIMIT MAP OF A BACKGROUND OF
...
PHYSICAL REVIEW D
76,
082003 (2007)
082003-3
A
and
^
X
i;t
a
and
^
Y
i;t
a
are unit vectors pointing in the
directions of the detector arms (see [
7
] for details). The
sidereal time dependence enters through the rotation of the
earth, affecting
^
X
i;t
a
,
^
Y
i;t
a
, and
x
*
t
.
The optimal filter
Q
t
is derived assuming that the intrin-
sic detector noise is Gaussian and stationary over the
measurement time, uncorrelated between detectors, and
uncorrelated with and much greater in power than the
stochastic gravitational wave signal. Under these assump-
tions the expected variance,
2
Y
t
, of the cross correlation is
dominated by the noise in the individual detectors, whereas
the expected value of the cross correlation
Y
t
depends on
the stochastic background power spectrum:
2
Y
t
h
Y
2
t
ih
Y
t
i
2
T
4
Q
t
;Q
t
(6)
h
Y
t
i
T
Q
t
;
R
S
2
d
^
^
;t
P
^
H
P
1
P
2
:
(7)
Here the scalar product
;
is defined as
A; B
R
1
1
A
f
B
f
P
1
f
P
2
f
df
and
T
is the duration of the
measurement.
Equation (
2
) defines the optimal filter
Q
t
for any arbi-
trary choice of
P
^
. To optimize the method for finite, but
unresolved astrophysical sources one should use a
P
^
that covers only a localized patch in the sky. But the
diffraction limit
of two detectors separated by
d
3000 km
is given by
c
2
fd
50 Hz
f
:
(8)
The relevant frequency depends on the assumed source
power spectrum
H
f
as well as on the noise power spectra
P
1
and
P
2
, but for a typical frequency
f
of 300 Hz
is
about 10
. Thus astrophysical sources will not be spatially
resolved and we can choose to optimize the method for true
point sources, i.e.,
P
^
2
^
;
^
0
, which also allows
for a more efficient implementation (see [
6
]).
We defined the strain power spectrum
H
f
of a point
source as one-sided (positive frequencies only) and in-
cluded the power in both polarizations. Thus
H
f
is
related to the gravitational wave energy flux
F
GW
through
F
GW
Z
f
max
f
min
F
f
df
c
3
4
G
Z
f
max
f
min
H
f
f
2
df;
(9)
with
F
f
the gravitational wave energy flux per unit
frequency,
c
the light speed, and
G
Newton’s constant.
We look for strain power spectra
H
f
in the form of a
power law with exponent
. The amplitude at the pivot
point of 100 Hz is described by
H
, i.e.,
H
f
H
f
100 Hz
:
(10)
With this definition we can choose the normalization of the
optimal filter
Q
t
such that Eq. (
7
) reduces to
h
Y
t
i
H
:
(11)
The data set from a given interferometer pair is divided
into equal-length intervals, and the cross correlation
Y
t
and
theoretical
Y
t
are calculated for each interval, yielding a
set
f
Y
t
;
Y
t
g
of such values for each sky direction
^
, with
t
labeling the intervals. The optimal filter
Q
t
is kept constant
and equal to its midinterval value for the whole interval.
The remaining error due to this discretization is of second
order in (
T
seg
=
1 day
) and is given by
Y
err
T
seg
=Y
T
2
seg
24
R
1
1
@
2
^
0
=@t
2
^
0
H
2
P
1
1
P
1
2
df
R
1
1
j
^
0
j
2
H
2
P
1
1
P
1
2
df
O
2
fd
c
T
seg
1 day
2
(12)
with
f
the typical frequency and
d
the detector separation.
At the same time the interval length can be chosen such
that the detector noise is relatively stationary over one
interval. We use an interval length of 60 sec, which guar-
antees that the relative error
Y
err
T
seg
=Y
is less than 1%.
The cross correlation values are combined to produce a
final cross correlation estimator,
Y
opt
, that maximizes the
signal-to-noise ratio, and has variance
2
opt
:
Y
opt
X
t
2
Y
t
Y
t
=
2
opt
;
2
opt
X
t
2
Y
t
:
(13)
In practice the intervals are overlapping by 50% to avoid
the effective loss of half the data due to the required
windowing (Hanning). Thus Eq. (
13
) was modified slightly
to take the correlation of neighboring intervals into
account.
The data was downsampled to 4096 Hz and high-pass
filtered with a sixth order Butterworth filter with a cutoff
frequency at 40 Hz. Frequencies between 50 and 1800 Hz
were used for the analysis and the frequency bin width was
0.25 Hz. Frequency bins around multiples of 60 Hz up to
the tenth harmonic were removed, along with bins near a
set of nearly monochromatic injected signals used to simu-
late pulsars. These artificial pulsars proved useful in a
separate end-to-end check of this analysis pipeline, which
successfully recovered the sky locations, frequencies, and
strengths of three such pulsars listed in Table
I
. The result-
ing map for one of these pulsars is shown in Fig.
1
.
B. ABBOTT
PHYSICAL REVIEW D
76,
082003 (2007)
082003-4
III. LIGO S4 SCIENCE RUN
The LIGO S4 science run consisted of one month of
coincident data taking with all three LIGO interferometers
(22 February, 2005 noon to 23 March, 2005 midnight
CST). During that time all three interferometers where
roughly a factor of 2 in amplitude away from design
sensitivity over almost the whole frequency band. Also,
the Livingston interferometer was equipped with a hydrau-
lic external preisolation (HEPI) system, allowing it to stay
locked during daytime. This made S4 the first LIGO
science run with all-day coverage at both sites. A more
detailed description of the LIGO interferometers is given
in [
8
].
Since the radiometer analysis requires two spatially
separated sites we used only data from the two 4 km
interferometers (H1 in Hanford and L1 in Livingston).
For these two interferometers about 20 days of coincident
data was collected, corresponding to a duty factor of 69%.
The large spatial separation also reduces environmental
correlations between the two sites. Nevertheless we still
found a comb of 1 Hz harmonics that was coherent be-
tween H1 and L1. This correlation was found to be at least
in part due to an exactly 1-sec periodic signal in both
interferometers (Fig.
2
), which was caused by cross talk
from the GPS_RAMP signal. The GPS_RAMP signal
consists of a sawtooth signal that starts at every full second,
lasts for 1 msec, and is synchronized with the GPS re-
ceivers (see Fig.
2
). This ramp was used as an off-line
monitor of the analog-to-digital converter (ADC) card
timing and thus was connected to the same ADC card
that was used for the gravitational wave channel, which
resulted in a nonzero cross talk to the gravitational wave
channel.
To reduce the contamination from this signal a transient
template was subtracted in the time domain. This has
the advantage that effectively only a very narrow band
(
1
=
run time
1
10
6
Hz
) is removed around each
1 Hz harmonic, while the rest of the analysis is unaffected.
The waveform for subtraction from the raw (uncalibrated)
data was recovered by averaging the data from the whole
run in order to produce a typical second. Additionally,
since this typical second only showed significant features
in the first 80 msec, the transient subtraction template was
set to zero (with a smooth transition) after 120 msec. This
subtraction was done for only H1 since adding repetitive
data to both detectors can introduce an artificial correla-
tion. It eliminated the observed correlation. However, due
to an automatically adjusted gain between the ADC card
and the gravitational wave channel, the amplitude of the
transient waveform is affected by a residual systematic
error. Its effect on the cross correlation result was estimated
by comparing maps with the subtraction done on either H1
or L1. The systematic error is mostly concentrated around
the North and South Poles, with a maximum of about 50%
of the statistical error at the South Pole. In the declination
FIG. 1 (color).
Injected pulsar No. 3: The analysis was run
using the 108.625 Hz –109.125 Hz frequency band. The artificial
signal of pulsar No. 3 at 108.86 Hz stands out with a signal-to-
noise ratio of 9.2. The circle marks the position of the simulated
pulsar.
TABLE I. Injected pulsars: The table shows the level at which the three strongest injected pulsars were recovered.
Hdf
denotes the
rms strain power over the 0.5 Hz band that was used. The reported values for the injected
Hdf
include corrections that account for the
difference between the polarized pulsar injection and an unpolarized source that is expected by the analysis. The one sigma uncertainty
in the recovered
Hdf
is given in row two (Noise level), and the ratio between recovered
Hdf
and noise level is given in row five
(SNR). The significant underestimate of pulsar No. 4 is due to a known bias of the analysis method in the case of a signal strong enough
to affect the power spectrum estimation.
Injected pulsars
Quantity
Pulsar No. 3
Pulsar No. 4
Pulsar No. 8
Frequency during S4 run
108.86 Hz
1402.20 Hz
193.94 Hz
Noise level (
)
1
:
89
10
47
6
:
04
10
46
1
:
73
10
47
Injected
Hdf
(corrected for polarization)
1
:
74
10
46
4
:
28
10
44
1
:
54
10
46
Recovered
Hdf
on source
1
:
74
10
46
4
:
05
10
44
1
:
79
10
46
Signal-to-noise ratio (SNR)
9.2
67.1
10.3
Injected position
11 h 53 m 29.4 s
18 h 39 m 57.0 s
23 h 25 m 33.5 s
33
26
0
11
:
8
00
12
27
0
59
:
8
00
33
25
0
6
:
7
00
Recovered position (max SNR)
12 h 12 m
18 h 40 m
23 h 16 m
37
13
32
UPPER LIMIT MAP OF A BACKGROUND OF
...
PHYSICAL REVIEW D
76,
082003 (2007)
082003-5
range of
75
to
75
the error is less than 10% of the
statistical error. For upper limit calculations this systematic
error is added in quadrature to the statistical error. After the
S4 run the GPS_RAMP signal was replaced with a two-
tone signal at 900 and 901 Hz. The beat between the two is
now used to monitor the timing.
One postprocessing cut was required to deal with detec-
tor nonstationarity. To avoid a bias in the cross correlation
statistics the interval before and the interval after the one
being analyzed are used for the power spectral density
(PSD) estimate [
9
]. Therefore the analysis becomes vul-
nerable to large, short transients that happen in one instru-
ment in the middle interval — such transients cause a
significant underestimate of the PSD and thus of the theo-
retical standard deviation for this interval. This leads to a
contamination of the final estimate.
To eliminate this problem the standard deviation
is
estimated for both the middle interval and the two adjacent
intervals. The two estimates are then required to agree
within 20%:
1
1
:
2
<
middle
adjacent
<
1
:
2
:
(14)
The analysis is fairly insensitive to the threshold — the only
significant contamination comes from very large outliers
that are cut by any reasonable threshold [
10
]. The chosen
threshold of 20% eliminates less than 6% of the data.
IV. RESULTS FROM THE S4 RUN
A. Broadband results
In this analysis we searched for an
H
f
following a
power law with two different exponents
.
(i)
3
:
H
f
H
3
100 Hz
=f
3
.
This emphasizes low frequencies and is useful when
interpreting the result in a cosmological framework,
since it corresponds to a scale-invariant primordial
perturbation spectrum, i.e., the GW energy per loga-
rithmic frequency interval is constant.
(ii)
0
:
H
f
H
0
(constant strain power)
.
This emphasizes the frequencies for which the in-
terferometer strain sensitivity is highest.
The results are reported as point estimate
Y
^
and corre-
sponding standard deviation
^
for each pixel
^
. The
point estimate
Y
^
must be interpreted as best fit amplitude
H
for the pixel
^
[Eq. (
11
)].
Also we should note that the resulting maps have an
intrinsic spatial correlation, which is described by the point
spread function
A
^
;
^
0
h
Y
^
Y
^
0
i
h
Y
^
0
Y
^
0
i
:
(15)
It describes the spatial correlation in the following sense: if
either
Y
^
0
Y
due to random fluctuations, or if there is a
true source of strength
Y
at
^
0
, then the expectation value
at
^
is
h
Y
^
i
A
^
;
^
0
Y
. The shape of
A
^
;
^
0
depends
strongly on the declination. Figure
3
shows
A
^
;
^
0
for
different source declinations and both the
3
and
0
case, assuming continuous day coverage.
1. Scale-invariant case,
3
A histogram of the
SNR
Y=
(SNR, signal-to-noise
ratio) is plotted in Fig.
4
. The data points were weighted
with the corresponding sky area in square degrees. Because
neighboring points are correlated, the effective number of
independent points
N
eff
is reduced. Therefore the histo-
gram can exhibit statistical fluctuations that are signifi-
cantly larger than those naively expected from simply
counting the number of pixels in the map, while still being
consistent with (correlated) Gaussian noise. Indeed the
histogram in Fig.
4
features a slight bump around
SNR
2
, but is still consistent with
N
eff
100
— the dash-dotted
lines indicate the one sigma band around the ideal
0
50
100
150
−1
0
1
2
msec
uVolt Pentek
H1:DARM_ERR (1437280 averages)
0
2
4
6
8
10
12
14
−1
0
1
2
msec
uVolt Pentek
H1:DARM_ERR (1437280 averages)
0
50
100
150
−4
−2
0
2
4
msec
uVolt Pentek
L1:DARM_ERR (1447904 averages)
0
2
4
6
8
10
12
14
−4
−2
0
2
4
msec
uVolt Pentek
L1:DARM_ERR (1447904 averages)
FIG. 2.
Periodic timing transient in the gravitational wave channel (DARM_ERR), calibrated in
V
at the ADC (Pentek card) for H1
(left two graphs) and L1 (right two graphs) shown with a span of 200 and 14 msec in black. The
x
axis is the offset from a full GPS
second. About
1
:
4
10
6
sec
of DARM_ERR data was averaged to get this trace. Also shown in gray is the GPS_RAMP signal that
was used as a timing monitor. It was identified as a cause of the periodic timing transient in DARM_ERR. The H1 trace shows an
additional feature at 6 msec.
B. ABBOTT
PHYSICAL REVIEW D
76,
082003 (2007)
082003-6
Gaussian for
N
eff
100
. Additionally the SNR distribu-
tion also passes a Kolmogorov-Smirnov test for
N
eff
100
at the 90% significance level.
The number of independent points
N
eff
, which in effect
describes the diffraction limit of the LIGO detector pair,
was estimated by 2 heuristic methods.
(i)
Spherical harmonics decomposition
of the SNR
map. The resulting power versus
l
graph shows
structure up to roughly
l
9
and falls off steeply
above that — the
l
9
point corresponds to one
twentieth of the maximal power. The effective num-
ber of independent points then is
N
eff
l
1
2
100
.
(ii)
FWHM area
of a strong injected source, which is
latitude dependent but of the order of 800 square
degrees. To fill the sky we need about
N
eff
50
of
those patches. We used the higher estimate
N
eff
100
for this discussion.
Figure
4
suggests that the data are consistent with no
signal. Thus we calculated a Bayesian 90% upper limit for
each sky direction. The prior was assumed to be flat
between zero and an upper cutoff set to
5
10
45
Hz
1
at 100 Hz, the approximate limit that can be set from just
operating a single LIGO interferometer at the S4 sensitiv-
ity. Note, however, that this cutoff is so high that the upper
limit is completely insensitive to it. Additionally we margi-
nalized over the calibration uncertainty of 8% for H1 and
5% for L1 using a Gaussian probability distribution. The
resulting upper limit map is shown in Fig.
5
. The upper
limits on the strain power spectrum
H
f
vary between
1
:
2
10
48
Hz
1
100 Hz
=f
3
and
1
:
2
10
47
Hz
1
100 Hz
=f
3
, depending on the position in the sky. These
strain limits correspond to limits on the gravitational
wave energy flux per unit frequency
F
f
varying
between
3
:
8
10
6
erg cm
2
Hz
1
100 Hz
=f
and
3
:
8
10
5
erg cm
2
Hz
1
100 Hz
=f
.
FIG. 3 (color).
Point spread function
A
^
;
^
0
of the radiome-
ter for
3
(top two figures) and for
0
(bottom two
figures). Plotted is the relative expected signal strength assuming
a source at right ascension 12 h and declinations 20
and 60
.
Uniform day coverage was assumed, so the resulting shapes are
independent of right ascension. An Aitoff projection was used to
plot the whole sky.
−5
0
5
0
1000
2000
3000
4000
5000
6000
SNR
sky area (deg
2
)
S4, beta=−3 Histogram of SNR (40 bins)
Data
Ideal Gaussian (sigma=1 mean=0)
Max Likelihood: sigma=0.91836 mean=0.11816
1−sigma error for 100 indep. points
FIG. 4.
S4 Result: Histogram of the signal-to-noise ratio
(SNR) for
3
. The gray curve is a maximum likelihood
Gaussian fit to the data. The black solid line is an ideal Gaussian,
the two dash-dotted black lines indicate the expected one sigma
variations around this ideal Gaussian for 100 independent points
(
N
eff
100
).
UPPER LIMIT MAP OF A BACKGROUND OF
...
PHYSICAL REVIEW D
76,
082003 (2007)
082003-7
2. Constant strain power,
0
Similarly, Fig.
6
shows a histogram of the
SNR
Y=
for the constant strain power case. Structure in the spheri-
cal harmonics power spectrum goes up to
l
19
, thus
N
eff
was estimated to be
N
eff
l
1
2
400
. Alternatively
the FWHM area of a strong injection covers about
100
2
which also leads to
N
eff
400
. The dash-dotted lines in
the histogram (Fig.
6
) correspond to the expected one
sigma deviations from the ideal Gaussian for
N
eff
400
.
The histogram is thus consistent with (correlated) Gaussian
noise, indicating that there is no signal present. The SNR
distribution also passes a Kolmogorov-Smirnov test for
N
eff
400
at the 90% significance level.
Again we calculated a Bayesian 90% upper limit for
each sky direction, including the marginalization over the
calibration uncertainty. The prior was again assumed to be
flat between 0 and an upper cutoff of
5
10
45
Hz
1
at
100 Hz. The resulting upper limit map is shown in Fig.
7
.
The upper limits on the strain power spectrum
H
f
vary
between
8
:
5
10
49
Hz
1
and
6
:
1
10
48
Hz
1
de-
pending on the position in the sky. This corresponds to
limits on the gravitational wave energy flux per unit fre-
quency
F
f
varying between
2
:
7
10
6
erg cm
2
Hz
1
f=
100 Hz
2
and
1
:
9
10
5
erg cm
2
Hz
1
f=
100 Hz
2
.
3. Interpretation
The maps presented in Figs.
5
and
7
represent the first
directional upper limits on a stochastic gravitational wave
background ever obtained. They are consistent with no
gravitational wave background being present. This search
is optimized for well localized, broadband sources of
gravitational waves. As such it is best suited for unex-
pected, poorly modeled sources.
In order to compare the result to what could be expected
from known sources we also search for the gravitational
radiation from low-mass x-ray binaries (LMXBs). They
are accretion-driven spinning neutron stars, i.e., narrow
band sources and thus not ideal for this broadband search.
However they have the advantage that we can predict the
gravitational wave energy flux based on the known x-ray
flux. If gravitational radiation provides the torque balance
for LMXBs, then there is a simple relation between the
gravitational wave energy flux
F
GW
and x-ray flux
F
X
[
11
]:
F
GW
f
spin
f
Kepler
F
X
:
(16)
Here
f
Kepler
is final orbital frequency of the accreting
matter, about 2 kHz for a neutron star, and
f
spin
is the
spin frequency.
As an example we estimate the gravitational wave en-
ergy flux of all LMXBs within the Virgo galaxy cluster.
Their integrated x-ray flux is about
10
9
erg
=
sec
=
cm
2
(3000 galaxies at 15 Mpc,
10
40
erg
=
sec
=
galaxy
from
FIG. 5 (color).
S4 Result: Map of the 90% confidence level
Bayesian upper limit on
H
for
3
. The upper limit varies
between
1
:
2
10
48
Hz
1
100 Hz
=f
3
and
1
:
2
10
47
Hz
1
100 Hz
=f
3
, depending on the position in the sky. All fluctua-
tions are consistent with the expected noise.
−5
0
5
0
1000
2000
3000
4000
5000
SNR
sky area (deg
2
)
S4, beta=0 Histogram of SNR (40 bins)
Data
Ideal Gaussian (sigma=1 mean=0)
Max Likelihood: sigma=0.99738 mean=−0.025485
1−sigma error for 400 indep. points
FIG. 6.
S4 Result: Histogram of the SNR for
0
. The gray
curve is a maximum likelihood Gaussian fit to the data. The
black solid line is an ideal Gaussian, the two dash-dotted black
lines indicate the expected one sigma variations around this ideal
Gaussian for 400 independent points (
N
eff
400
).
FIG. 7 (color).
S4 Result: Map of the 90% confidence level
Bayesian upper limit on
H
for
0
. The upper limit varies
between
8
:
5
10
49
Hz
1
and
6
:
1
10
48
Hz
1
depending on
the position in the sky.
B. ABBOTT
PHYSICAL REVIEW D
76,
082003 (2007)
082003-8