of 13
Directional Limits on Persistent Gravitational Waves
from Advanced LIGO
s First Observing Run
B. P. Abbott
etal.
*
(LIGO Scientific Collaboration and Virgo Collaboration)
(Received 7 December 2016; published 24 March 2017)
We employ gravitational-wave radiometry to map the stochastic gravitational wave background
expected from a variety of contributing mechanisms and test the assumption of isotropy using data
from the Advanced Laser Interferometer Gravitational Wave Observatory
s (aLIGO) first observing run.
We also search for persistent gravitational waves from point sources with only minimal assumptions
over the 20
1726 Hz frequency band. Finding no evidence of gravitational waves from either point
sources or a stochastic background, we set limits at 90% confidence. For broadband point sources, we
report upper limits on the gravitational wave energy flux per unit frequency in the range
F
α
;
Θ
ð
f
Þ
<
ð
0
.
1
56
Þ
×
10
8
erg cm
2
s
1
Hz
1
ð
f=
25
Hz
Þ
α
1
depending on the sky location
Θ
and the spectral
power index
α
. For extended sources, we report upper limits on the fractional gravitational wave energy
density required to close the Universe of
Ω
ð
f;
Θ
Þ
<
ð
0
.
39
7
.
6
Þ
×
10
8
sr
1
ð
f=
25
Hz
Þ
α
depending on
Θ
and
α
. Directed searches for narrowband gravitational waves from astrophysically interesting objects
(Scorpius X-1, Supernova 1987 A, and the Galactic Center) yield median frequency-dependent limits on
strain amplitude of
h
0
<
ð
6
.
7
;
5
.
5
;
and
7
.
0
Þ
×
10
25
, respectively, at the most sensitive detector frequen-
cies between 130
175 Hz. This represents a mean improvement of a factor of 2 across the band compared
to previous searches of this kind for these sky locations, considering the different quantities of strain
constrained in each case.
DOI:
10.1103/PhysRevLett.118.121102
Introduction.
A stochastic gravitational-wave back-
ground (SGWB) is expected from a variety of mechanisms
[1
5]
. Given the recent observations of binary black hole
mergers GW150914 and GW151226
[6,7]
, we expect
the SGWB to be nearly isotropic
[8]
and dominated
[9]
by compact binary coalescences
[10
12]
. The Laser
Interferometer Gravitational Wave Observatory (LIGO)
and Virgo Collaborations have pursued the search for an
isotropic stochastic background from LIGO
s first obser-
vational run
[13]
. Here, we adopt an eyes-wide-open
philosophy and relax the assumption of isotropy in order
to allow for the greater range of possible signals. We search
for an anisotropic background, which could indicate a
richer, more interesting cosmology than current models.
We present the results of a generalized search for a
stochastic signal with an arbitrary angular distribution
mapped over all directions in the sky.
Our search has three components. First, we utilize a
broadband radiometer analysis
[14,15]
, optimized for
detecting a small number of resolvable point sources.
This method is not applicable to extended sources.
Second, we employ a spherical harmonic decomposition
[16,17]
, which can be employed for point sources but is
better suited to extended sources. Last, we carry out a
narrowband radiometer search directed at the sky position
of three astrophysically interesting objects: Scorpius X-1
(Sco X-1)
[18,19]
, Supernova 1987 A (SN 1987A)
[20,21]
,
and the Galactic Center
[22]
.
These three search methods are capable of detecting a
wide range of possible signals with only minimal assump-
tions about the signal morphology. We find no evidence of
persistent gravitational waves, and set limits on broadband
emission of gravitational waves as a function of sky
position. We also set narrowband limits as a function of
frequency for the three selected sky positions.
Data.
We analyze data from Advanced LIGO
s4km
detectors in Hanford, Washington (H1) and Livingston,
Louisana (L1) during the first observing run (O1), from
15
00 UTC, Sep 18, 2015
16:00 UTC, Jan 12, 2016.
During O1, the detectors reached an instantaneous strain
sensitivity of
7
×
10
24
Hz
1
=
2
in the most sensitive region
from 100
300 Hz, and collected 49.67 days of coincident
H1L1 data. The O1 observing run saw the first direct
detection of gravitational waves and the first direct obser-
vation of merging black holes
[6,7]
.
For our analysis, the time-series data are downsampled to
4096 Hz from 16 kHz, and divided into 192 s, 50%
overlapping, Hann-windowed segments, which are high-
pass filtered with a 16th order Butterworth digital filter
with knee frequency of 11 Hz (following
[13,23]
). We
apply data quality cuts in the time domain in order to
remove segments associated with instrumental artifacts and
*
Full author list given at the end of the article.
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=
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=
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=
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121102-1
Published by the American Physical Society
hardware injections used for signal validation
[24,25]
.We
also exclude segments containing known gravitational-
wave signals. Finally, we apply a standard nonstationarity
cut (see, e.g.,
[26]
), to eliminate segments that do not
behave as Gaussian noise. These cuts remove 35% of the
data. With all vetoes applied, the total live time for 192 s
segments is 29.85 days.
The data segments are Fourier transformed and coarse-
grained to produce power spectra with a resolution of
1
=
32
Hz. This is a finer frequency resolution than the
1
=
4
Hz used in previous LIGO-Virgo stochastic searches
[15,17]
in order to remove many finely spaced instrumental
lines occurring at low frequencies. Frequency bins asso-
ciated with known instrumental artifacts including suspen-
sion violin modes
[27]
, calibration lines, electronic lines,
correlated combs, and signal injections of persistent,
monochromatic, gravitational waves are not included in
the analysis. These frequency domain cuts remove 21% of
the observing band. For a detailed description of data
quality studies performed for this analysis, see the
Supplemental Material
[28]
of
[13]
.
The broadband searches include frequencies from
20
500 Hz which more than cover the regions of 99%
sensitivity for each of the spectral bands (see Table 1 of
[13]
). The narrowband analysis covers the full 20
1726 Hz
band.
Method.
The main goal of a stochastic search is to
estimate the fractional contribution of the energy density in
gravitational waves
Ω
GW
to the total energy density needed
to close the Universe
ρ
c
. This is defined by
Ω
GW
ð
f
Þ¼
f
ρ
c
d
ρ
GW
df
;
ð
1
Þ
where
f
is frequency and
d
ρ
GW
represents the energy
density between
f
and
f
þ
df
[29]
. For a stationary and
unpolarized signal,
ρ
GW
can be factored into an angular
power
P
ð
Θ
Þ
and a spectral shape
H
ð
f
Þ
[30]
, such that
Ω
GW
ð
f
Þ¼
2
π
2
3
H
2
0
f
3
H
ð
f
Þ
Z
d
Θ
P
ð
Θ
Þ
;
ð
2
Þ
with Hubble constant
H
0
¼
68
km s
1
Mpc
1
from
[31]
.
The angular power
P
ð
Θ
Þ
represents the gravitational-
wave power at each point in the sky. To express this in
terms of the fractional energy density, we define the energy
density spectrum as a function of sky position
Ω
ð
f;
Θ
Þ¼
2
π
2
3
H
2
0
f
3
H
ð
f
Þ
P
ð
Θ
Þ
:
ð
3
Þ
We define a similar quantity for the energy flux, where
F
ð
f;
Θ
Þ¼
c
3
π
4
G
f
2
H
ð
f
Þ
P
ð
Θ
Þð
4
Þ
has units of erg s
1
Hz
1
sr
1
[15,16]
,
c
is the speed of light
and
G
is Newton
s gravitational constant.
Point sources versus extended sources.
We employ two
different methods to estimate
P
ð
Θ
Þ
based on the cross-
correlation of data streams from a pair of detectors
[17,29]
.
The radiometer method
[14,15]
assumes that the cross-
correlation signal is dominated by a small number of
resolvable point sources. The point source power is given
by
P
Θ
0
and the angular power spectrum is then
P
ð
Θ
Þ
P
Θ
0
δ
2
ð
Θ
;
Θ
0
Þ
:
ð
5
Þ
Although the radiometer method provides the optimal
method for detecting resolvable point sources, it is not well
suited for describing diffuse or extended sources, which
may have an arbitrary angular distribution. Hence, we
also implement a complementary spherical harmonic
decomposition (SHD) algorithm, in which the sky map
is decomposed into components
Y
lm
ð
Θ
Þ
with coefficients
P
lm
[16]
P
ð
Θ
Þ
X
lm
P
lm
Y
lm
ð
Θ
Þ
:
ð
6
Þ
Here, the sum over
l
runs from 0 to
l
max
and
l
m
l
.
We discuss the choice of
l
max
below. While the SHD
algorithm has comparably worse sensitivity to point
sources than the radiometer algorithm, it accounts for
the detector response, producing more accurate sky maps.
Spectral models.
In both the radiometer algorithm and
the spherical harmonic decomposition algorithm, we must
choose a spectral shape
H
ð
f
Þ
. We model the spectral
dependence of
Ω
GW
ð
f
Þ
as a power law
H
ð
f
Þ¼

f
f
ref

α
3
;
ð
7
Þ
where
f
ref
is an arbitrary reference frequency and
α
is the
spectral index (see, also,
[13]
). The spectral model will also
affect the angular power spectrum, so
P
ð
Θ
Þ
is implicitly a
function of
α
.
We can rewrite the energy density map
Ω
ð
f;
Θ
Þ
to
emphasize the spectral properties, such that
Ω
ð
f;
Θ
Þ¼
Ω
α
ð
Θ
Þ

f
f
ref

α
;
ð
8
Þ
where
Ω
α
ð
Θ
Þ¼
2
π
2
3
H
2
0
f
3
ref
P
ð
Θ
Þð
9
Þ
has units of fractional energy density per steradian
Ω
GW
sr
1
. The spherical harmonic analysis presents sky
maps of
Ω
α
ð
Θ
Þ
. Note that, when
P
ð
Θ
Þ¼
P
00
(the monop-
ole moment), we recover a measurement for the energy
density of the isotropic gravitational-wave background.
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Similarly, the gravitational-wave energy flux can be
expressed as
F
ð
f;
Θ
Þ¼
F
α
ð
Θ
Þ

f
f
ref

α
1
;
ð
10
Þ
where
F
α
ð
Θ
Þ¼
c
3
π
4
G
f
2
ref
P
ð
Θ
Þ
:
ð
11
Þ
In the radiometer case, we calculate the flux in each
direction
F
α
;
Θ
0
¼
c
3
π
4
G
f
2
ref
P
Θ
0
;
ð
12
Þ
which is obtained by integrating Eq.
(11)
over the sphere
for the point-source signal model described in Eq.
(5)
. This
quantity has units of erg cm
2
s
1
Hz
1
. Following
[13,32]
,
we choose
f
ref
¼
25
Hz, corresponding to the most sensi-
tive frequency in the spectral band for a stochastic search
with the Advanced LIGO network at design sensitivity.
We consider three spectral indices:
α
¼
0
, corresponding
to a flat energy density spectrum (expected from models of
a cosmological background),
α
¼
2
=
3
, corresponding to
the expected shape from a population of compact binary
coalescences, and
α
¼
3
, corresponding to a flat strain
power spectral density spectrum
[17,32]
. The different
spectral models are summarized in Table
I
.
Cross-correlation.
A stochastic background would
induce low-level correlation between the two LIGO detec-
tors. Although the signal is expected to be buried in the
detector noise, the cross-correlation signal-to-noise ratio
(SNR) grows with the square root of integration time
[29]
.
The cross-correlation between two detectors, with (one-
sided strain) power spectral density
P
i
ð
f; t
Þ
for detector
i
,is
encoded in what is known as
the dirty map
[16]
X
ν
¼
X
ft
γ

ν
ð
f; t
Þ
H
ð
f
Þ
P
1
ð
f; t
Þ
P
2
ð
f; t
Þ
C
ð
f; t
Þ
:
ð
13
Þ
Here,
ν
is an index, which can refer to either individual
points on the sky (the pixel basis) or different
lm
indices
(the spherical harmonic basis). The variable
C
ð
f; t
Þ
is the
cross-power spectral density measured between the two
LIGO detectors at some segment time
t
. The sum runs over
all segment times and all frequency bins. The variable
γ
ν
ð
f; t
Þ
is a generalization of the overlap reduction func-
tion, which is a function of the separation and relative
orientation between the detectors, and characterizes the
frequency response of the detector pair
[33]
; see
[16]
for an
exact definition.
We can think of
X
ν
as a sky map representation of
the raw cross-correlation measurement before deconvolv-
ing the detector response. The associated uncertainty is
encoded in the Fisher matrix
Γ
μν
¼
X
ft
γ

μ
ð
f; t
Þ
H
2
ð
f
Þ
P
1
ð
f; t
Þ
P
2
ð
f; t
Þ
γ
ν
ð
f; t
Þ
;
ð
14
Þ
where the asterisk denotes complex conjugation.
Once
X
ν
and
Γ
μν
are calculated, we have the ingredients
to calculate both the radiometer map and the SHD map.
However, the inversion of
Γ
μν
is required to calculate the
maximum likelihood estimators of GW power
ˆ
P
μ
¼
Γ
1
μν
X
ν
[16]
. For the radiometer, the correlations between neigh-
boring pixels can be ignored. The radiometer map is
given by
ˆ
P
Θ
¼ð
Γ
ΘΘ
Þ
1
X
Θ
;
σ
rad
Θ
¼ð
Γ
ΘΘ
Þ
1
=
2
;
ð
15
Þ
where the standard deviation
σ
rad
Θ
is the uncertainty asso-
ciated with the point source amplitude estimator
ˆ
P
Θ
, and
Γ
ΘΘ
is a diagonal entry of the Fisher matrix for a pointlike
signal. For the SHD analysis, the full Fisher matrix
Γ
μν
must be taken into account, which includes singular
eigenvalues associated with modes to which the detector
pair is insensitive. The inversion of
Γ
μν
is simplified
by a singular value decomposition regularization. In this
decomposition, modes associated with the smallest eigen-
values contribute the least sensitivity to the detector net-
work. Removing a fraction of the lowest eigenmodes
regularizes
Γ
μν
without significantly affecting the
TABLE I. Values of the power-law index
α
investigated in this analysis, the shape of the energy density and strain power spectrum.
The characteristic frequency
f
α
, angular resolution
θ
[Eq.
(17)
], and corresponding harmonic order
l
max
[Eq.
(18)
] for each
α
are also
shown. The right hand section of the table shows the maximum SNR, associated significance (
p
value) and best upper limit values from
the broadband radiometer (BBR) and the SHD. The BBR sets upper limits on energy flux [erg cm
2
s
1
Hz
1
ð
f=
25
Hz
Þ
α
1
] while the
SHD sets upper limits on the normalized energy density [sr
1
ð
f=
25
Hz
Þ
α
] of the SGWB.
All-sky (broadband) Results
Max SNR (%
p
value)
Upper limit range
α
Ω
GW
H
ð
f
Þ
f
α
(Hz)
θ
ð
deg
Þ
l
max
BBR
SHD
BBR (×
10
8
) SHD (×
10
8
)
0
constant
f
3
52.50
55
3
3.32 (7)
2.69 (18)
10
56
2.5
7.6
2
=
3
f
2
=
3
f
7
=
3
65.75
44
4
3.31 (12)
3.06 (11)
5.1
33
2.0
5.9
3
f
3
constant
256.50
11
16
3.43 (47)
3.86 (11)
0.1
0.9
0.4
2.8
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sensitivity (see
[16]
). The estimator for the SHD and
corresponding standard deviation are given by
ˆ
P
lm
¼
X
l
0
m
0
ð
Γ
1
R
Þ
lm;l
0
m
0
X
l
0
m
0
;
σ
SHD
lm
¼½ð
Γ
1
R
Þ
lm;lm

1
=
2
;
ð
16
Þ
where
Γ
R
is the regularized Fisher matrix. We remove
1
=
3
of the lowest eigenvalues following
[16,17]
.
Angular scale.
In order to carry out the calculation in
Eq.
(16)
, we must determine a suitable angular scale, which
will depend on the angular resolution of the detector
network and vary with spectral index
α
. The diffraction-
limited spot size on the sky
θ
(in radians) is given by
θ
¼
c
2
df
50
Hz
f
α
;
ð
17
Þ
where
d
¼
3000
km is the separation of the LIGO detec-
tors. The frequency
f
α
corresponds to the most sensitive
frequency in the detector band for a power law with spectral
index
α
given the detector noise power spectra
[15]
.In
order to determine
f
α
, we find the frequency at which a
power law with index
α
is tangent to the single-detector
power-law integrated curve
[34]
. The angular resolution
scale is set by the maximum spherical harmonic order
l
max
,
which we can express as a function of
α
since
l
max
¼
π
θ
π
f
α
50
Hz
:
ð
18
Þ
The values of
f
α
,
θ
, and
l
max
for three different values of
α
are shown in Table
I
. As the spectral index increases, so
does
f
α
, decreasing the angular resolution limit, thus,
increasing
l
max
.
Angular power spectra.
For the SHD map, we calcu-
late the angular power spectra
C
l
, which describe the
angular scale of structure in the clean map, using an
unbiased estimator
[16,17]
ˆ
C
l
1
2
l
þ
1
X
m
½j
ˆ
P
lm
j
2
ð
Γ
1
R
Þ
lm;lm

:
ð
19
Þ
Narrowband radiometer.
The radiometer algorithm
can be applied to the detection of persistent gravitational
waves from narrowband point sources associated with a
given sky position
[15,17]
.We
point
the radiometer in the
direction of three interesting sky locations: Sco X-1, the
Galactic Center, and the remnant of supernova SN 1987A.
Scorpius X-1 is a low-mass x-ray binary believed to host
a neutron star that is potentially spun up through accretion,
in which gravitational-wave emission may provide a
balancing spin-down torque
[18,19,35,36]
. The frequency
of the gravitational-wave signal is expected to spread due to
the orbital motion of the neutron star. At frequencies below
930
Hz this Doppler line broadening effect is less than
1
=
4
Hz, the frequency bin width selected in past analyses
[15,17]
. At higher frequencies, the signal is certain to span
multiple bins. Therefore, we combine multiple
1
=
32
Hz
frequency bins to form optimally sized combined bins at
each frequency, accounting for the expected signal broad-
ening due to the combination of the motion of the Earth
around the Sun, the binary orbital motion, and any other
intrinsic modulation. For more detail on the method of
combining bins, see the Supplemental Material to this
Letter
[37]
.
The possibility of a young neutron star in SN 1987A
[20,21]
and the likelihood of many unknown, isolated
neutron stars in the Galactic Center region
[22]
indicate
potentially interesting candidates for persistent gravita-
tional-wave emission. We combine bins to include the
signal spread due to Earth
s modulation. For SN 1987A, we
choose a combined bin size of 0.09 Hz. We would be
sensitive to spin modulations up to
j
_
ν
j
<
9
×
10
9
Hz s
1
within our O1 observation time spanning 116 days. The
Galactic Center is at a lower declination with respect to the
orbital plane of the Earth. Therefore, the Earth modulation
term is more significant, so, for the Galactic Center, we
choose combined bins of 0.53 Hz across the band. In this
case, we are sensitive to a frequency modulation in the
range
j
_
ν
j
<
5
.
3
×
10
8
Hz s
1
.
Significance.
To assess the significance of the SNR in
the combined bins of the narrowband radiometer spectra,
we simulate many realizations of the strain power con-
sistent with Gaussian noise in each individual frequency
bin. Combining these in the same way as the actual analysis
leaves us with a distribution of maximum SNR values
across the whole frequency band for many simulations
of noise.
For a map of the whole sky, the distribution of maximum
SNR is complicated by the many dependent trials due
to covariances between different pixels (or patches)
on the sky. We calculate this distribution numerically by
simulating many realizations of the dirty map
X
ν
with
expected covariances described by the Fisher matrix
Γ
μν
[cf. Eqs.
(13)
and
(14)
, respectively]. This distribution is
then used to calculate the significance (or
p
value) of a
given SNR recovered from the sky maps
[17]
. We take a
p
value of 0.01 or less to indicate a significant result. The
absence of any significant events indicates the data are
consistent with no signal being detected, in which case we
quote Bayesian upper limits at 90% confidence
[15,17]
.
Results.
The search yields four data products: radiom-
eter sky maps, angular power spectra, and radiometer
spectra.
Radiometer sky maps optimized for broadband point
sources, are shown in Fig.
1
. The top row shows the SNR.
Each column corresponds to a different spectral index,
α
¼
0
;
2
=
3
, and 3, from left to right, respectively. The
maximum SNRs are, respectively, 3.32, 3.31, and 3.43
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corresponding to false-alarm probabilities typical of
what would be expected from Gaussian noise; see
Table
I
. We find no evidence of a signal and, so, set limits
on gravitational-wave energy flux, which are provided in
the bottom row of Fig.
1
and summarized in Table
I
.
SHD sky maps, suitable for characterizing an aniso-
tropic stochastic background, are shown in Fig.
2
. The
top row shows the SNR and each column corresponds to
a different spectral index (
α
¼
0
;
2
=
3
, and 3, respec-
tively). The maximum SNRs are 2.96, 3.06, and 3.86
corresponding to false-alarm probabilities typical of those
expected from Gaussian noise; see Table
I
. Failing
evidence of a signal, we set limits on energy density
per unit solid angle, which are provided in the bottom
row of Fig.
2
and summarized in Table
I
. Interactive
visualizations of the SNR and upper limit maps are also
available online
[38]
.
Angular power spectra are derived from the SHD sky
maps. We present upper limits at 90% confidence on the
angular power spectrum indices
C
l
from the spherical
harmonic analysis in Fig.
3
.
Radiometer spectra, suitable for the detection of a
narrowband point source associated with a given sky
position, are given in Fig.
4
, the main results of which
are summarized in Table
II
. For the three sky locations
(Sco X-1, SN 1987A, and the Galactic Center), we
FIG. 1. All-sky radiometer maps for pointlike sources showing SNR (top) and upper limits at 90% confidence on energy flux
F
α
;
Θ
0
[erg cm
2
s
1
Hz
1
] (bottom) for three different power-law indices,
α
¼
0
;
2
=
3
, and 3, from left to right, respectively. The
p
values
associated with the maximum SNR are (from left to right)
p
¼
7%
,
p
¼
12%
,
p
¼
47%
(see Table
I
).
FIG. 2. All-sky spherical harmonic decomposition maps for extended sources showing SNR (top) and upper limits at 90% confidence
on the energy density of the gravitational-wave background
Ω
α
(sr
1
) (bottom) for three different power-law indices
α
¼
0
;
2
=
3
, and 3,
from left to right, respectively. The
p
values associated with the maximum SNR are (from left to right)
p
¼
18%
,
p
¼
11%
,
p
¼
11%
(see Table
I
).
FIG. 3. Upper limits on
C
l
at 90% confidence vs
l
for the
SHD analyses for
α
¼
0
(top, blue squares),
α
¼
2
=
3
(middle, red
circles) and
α
¼
3
(bottom, green triangles).
PRL
118,
121102 (2017)
PHYSICAL REVIEW LETTERS
week ending
24 MARCH 2017
121102-5
calculate the SNR in appropriately sized combined bins
across the LIGO band. For Sco X-1, the loudest observed
SNR is 4.58, which is consistent with Gaussian noise. For
SN 1987A and the Galactic Center, we observe maximum
SNRs of 4.07 and 3.92, respectively, corresponding to false
alarm probabilities consistent with noise; see Table
II
.
Since we observe no statistically significant signal, we
set 90% confidence limits on the peak strain amplitude
h
0
for each optimally sized frequency bin. Upper limits were
set using a Bayesian methodology with the constraint that
h
0
>
0
and validated with software injection studies. The
upper limit procedure is described in more detail in the
Supplemental Material
[37]
, while the subsequent software
injection validation is detailed in
[39]
.
The results of the narrowband radiometer search for the
three sky locations are shown in Fig.
4
. To avoid setting
limits associated with downward noise fluctuations, we
take the median upper limit from the most sensitive 1 Hz
band as our best strain upper limit. We obtain 90% con-
fidence upper limits on the gravitational-wave strain of
h
0
<
6
.
7
×
10
25
at 134 Hz,
h
0
<
7
.
0
×
10
25
at 172 Hz,
and
h
0
<
5
.
5
×
10
25
at 172 Hz from Sco X-1, SN 1987A,
and the Galactic Center, respectively, in the most sensitive
part of the LIGO band.
Conclusions.
We find no evidence to support the detec-
tion of either pointlike or extended sources and set upper
limits on the energy flux and energy density of the
anisotropic gravitational-wave sky. We assume three differ-
ent power law models for the gravitational-wave background
spectrum. Our mean upper limits present an improvement
over Initial LIGO results of a factor of 8 in flux for the
α
¼
3
broadband radiometer and factors of 60 and 4 for the
spherical harmonic decomposition method for
α
¼
0
and
3, respectively
[17,40]
. We present the first upper limits for
an anisotropic stochastic background dominated by compact
binary inspirals (with an
Ω
GW
f
2
=
3
spectrum) of
Ω
2
=
3
ð
Θ
Þ
<
ð
2
6
Þ
×
10
8
sr
1
depending on sky position. We can directly
compare the monopole moment of the spherical harmonic
decomposition to the isotropic search point estimate
Ω
2
=
3
¼ð
3
.
5

4
.
4
Þ
×
10
8
from
[13]
.Weobtain
Ω
2
=
3
¼
ð
2
π
2
=
3
H
2
0
Þ
f
3
ref
ffiffiffiffiffiffi
4
π
p
P
00
¼ð
4
.
4

6
.
4
Þ
×
10
8
.Thetwo
results are statistically consistent. Our spherical harmonic
estimate of
Ω
2
=
3
has a larger uncertainty than the dedicated
isotropic search because of the larger number of (covariant)
parameters estimated when
l
max
>
0
. We also set upper limits
on the gravitational-wave strain from point sources located in
the directions of Scorpius X-1, the Galactic Center, and
Supernova 1987A. The narrowband results improve on
previous limits of the same kind by more than a factor of
10instrainatfrequenciesbelow50Hzandabove300Hz,with
a mean improvement of a factor of 2 across the band
[17]
.
The authors gratefully acknowledge the support of the
United States National Science Foundation (NSF) for the
construction and operation of the LIGO Laboratory and
Advanced LIGO as well as the Science and Technology
Facilities Council (STFC) of the United Kingdom,
the Max-Planck Society (MPS), and the State of
Niedersachsen, Germany for support of the construction
of Advanced LIGO and construction and operation of the
GEO600 detector. Additional support for Advanced LIGO
was provided by the Australian Research Council. The
authors gratefully acknowledge the Italian Istituto
FIG. 4. Radiometer 90% upper limits (UL) on dimensionless strain amplitude (
h
0
) as a function of frequency for Sco X-1 (left),
SN1987A (middle), and the Galactic Center (right) for the O1 observing run (gray band) and standard deviation
σ
(black line). The large
spikes correspond to harmonics of the 60 Hz power mains, calibration lines, and suspension-wire resonances.
TABLE II. Results for the narrowband radiometer showing the
maximum SNR, corresponding
p
value and 1 Hz frequency band
as well as the 90% gravitational-wave strain upper limits, and
corresponding frequency band, for three sky locations of interest.
The best upper limits are taken as the median of the most sensitive
1 Hz band.
Direction
Max
SNR
p
value
(%)
Frequency
band (Hz)
Best UL
10
25
)
Frequency
band (Hz)
Sco X-1 4.58
10
616
617
6.7
134
135
SN1987A 4.07
63 195
196617 5.5
172
173
Galactic
Center
3.92
87 1347
1348617 7.0
172
173
PRL
118,
121102 (2017)
PHYSICAL REVIEW LETTERS
week ending
24 MARCH 2017
121102-6
Nazionale di Fisica Nucleare (INFN), the French Centre
National de la Recherche Scientifique (CNRS) and the
Foundation for Fundamental Research on Matter supported
by the Netherlands Organisation for Scientific Research, for
the construction and operation of the Virgo detector and the
creation and support of the EGO consortium. The authors
also gratefully acknowledge research support from these
agencies as well as by the Council of Scientific and
Industrial Research of India, Department of Science and
Technology, India, Science and Engineering Research
Board (SERB), India, Ministry of Human Resource
Development, India, the Spanish Ministerio de Economía
y Competitividad, the Conselleria d
Economia i
Competitivitat and Conselleria d
Educació, Cultura i
Universitats of the Govern de les Illes Balears, the
National Science Centre of Poland, the European
Commission, the Royal Society, the Scottish Funding
Council, the Scottish Universities Physics Alliance, the
Hungarian Scientific Research Fund (OTKA), the Lyon
Institute of Origins (LIO), the National Research
Foundation of Korea, Industry Canada and the Province
of Ontario through the Ministry of Economic Development
and Innovation, the Natural Science and Engineering
Research Council Canada, Canadian Institute for
Advanced Research, the Brazilian Ministry of Science,
Technology, and Innovation, Fundação de Amparo à
Pesquisa do Estado de São Paulo (FAPESP), Russian
Foundation for Basic Research, the Leverhulme Trust, the
Research Corporation, Ministry of Science and Technology
(MOST), Taiwan and the Kavli Foundation. The authors
gratefully acknowledge the support of the NSF, STFC, MPS,
INFN, CNRS and the State of Niedersachsen, Germany for
provision of computational resources. This article has been
assigned document number LIGO-P1600259.
[1] M. Maggiore,
Phys. Rep.
331
, 283 (2000)
.
[2] B. Allen, in
Relativistic Gravitation and Gravitational
Radiation: Proceedings of the Les Houches School of
Physics, Les Houches, Haute Savoie, 1995
, edited by
J.-A. Marck and J.-P. Lasota (Cambridge University Press,
Cambridge, England, 1997) p. 373.
[3] V. Mandic and A. Buonanno,
Phys. Rev. D
73
, 063008
(2006)
.
[4] X.-J. Zhu, X.-L. Fan, and Z.-H. Zhu,
Astrophys. J.
729
,59
(2011)
.
[5] X. Siemens, V. Mandic, and J. Creighton,
Phys. Rev. Lett.
98
, 111101 (2007)
.
[6] B. P. Abbott
et al.
(LIGO Scientific Collaboration and Virgo
Collaboration),
Phys. Rev. Lett.
116
, 061102 (2016)
.
[7] B. P. Abbott
et al.
(LIGO Scientific Collaboration and
Virgo Collaboration),
Phys. Rev. Lett.
116
, 241103 (2016)
.
[8] D. Meacher, E. Thrane, and T. Regimbau,
Phys. Rev. D
89
,
084063 (2014)
.
[9] T. Callister, L. Sammut, S. Qiu, I. Mandel, and E. Thrane,
Phys. Rev. X
6
, 031018 (2016)
.
[10] T. Regimbau and B. Chauvineau,
Classical Quantum
Gravity
24
, S627 (2007)
.
[11] C. Wu, V. Mandic, and T. Regimbau,
Phys. Rev. D
85
,
104024 (2012)
.
[12] X.-J. Zhu, E. J. Howell, D. G. Blair, and Z.-H. Zhu,
Mon.
Not. R. Astron. Soc.
431
, 882 (2013)
.
[13] B. Abbott
et al.
(LIGO Scientific Collaboration and Virgo
Collaboration), preceding Letter,
Phys. Rev. Lett.
118
,
121101 (2016)
.
[14] S. W. Ballmer,
Classical Quantum Gravity
23
, S179 (2006)
.
[15] B. Abbott
et al.
,
Phys. Rev. D
76
, 082003 (2007)
.
[16] E. Thrane, S. Ballmer, J. D. Romano, S. Mitra, D. Talukder,
S. Bose, and V. Mandic,
Phys. Rev. D
80
, 122002 (2009)
.
[17] J. Abadie
et al.
,
Phys. Rev. Lett.
107
, 271102 (2011)
.
[18] J. Aasi
et al.
,
Phys. Rev. D
91
, 062008 (2015)
.
[19] C. Messenger
et al.
,
Phys. Rev. D
92
, 023006 (2015)
.
[20] C. T. Y. Chung, A. Melatos, B. Krishnan, and J. T. Whelan,
Mon. Not. R. Astron. Soc.
414
, 2650 (2011)
.
[21] L. Sun, A. Melatos, P. D. Lasky, C. T. Y. Chung, and N. S.
Darman,
Phys. Rev. D
94
, 082004 (2016)
.
[22] J. Aasi
et al.
,
Phys. Rev. D
88
, 102002 (2013)
.
[23] B. Abbott
et al.
,
Astrophys. J.
659
, 918 (2007)
.
[24] B. P. Abbott
et al.
,
Classical Quantum Gravity
33
, 134001
(2016)
.
[25] C. Biwer
et al.
,
arXiv:1612.07864.
[26] B. P. Abbott
et al.
,
Nature (London)
460
, 990 (2009)
.
[27] S. M. Aston
et al.
,
Classical Quantum Gravity
29
, 235004
(2012)
.
[28] See Supplemental Material of [13] at
http://link.aps.org/
supplemental/10.1103/PhysRevLett.118.121102
for more
detail on the data quality studies performed during this
analysis.
[29] B. Allen and J. D. Romano,
Phys. Rev. D
59
, 102001
(1999)
.
[30] B. Allen and A. C. Ottewill,
Phys. Rev. D
56
, 545 (1997)
.
[31] P. A. R. Ade
et al.
(Planck Collaboration),
Astron.
Astrophys.
571
, A1 (2014)
.
[32] B. P. Abbott
et al.
,
Phys. Rev. Lett.
116
, 131102 (2016)
.
[33] N. Christensen,
Phys. Rev. D
46
, 5250 (1992)
.
[34] E.Thrane andJ. D.Romano,
Phys.Rev.D
88
,124032(2013)
.
[35] L. Bildsten,
Astrophys. J. Lett.
501
, L89 (1998)
.
[36] B. Abbott
et al.
,
Phys. Rev. D
76
, 082001 (2007)
.
[37] See Supplemental Material at
http://link.aps.org/
supplemental/10.1103/PhysRevLett.118.121102
for details
on the directed narrowband radiometer procedure to calcu-
late upper limits on the gravitational-wave strain amplitude.
[38] B. P. Abbott
et al.
(LIGO Scientific Collaboration and Virgo
Collaboration), Mapping a stochastic gravitational-wave
background,
https://dcc.ligo.org/public/0139/G1602343/
003/index.html
.
[39] B. P. Abbott
et al.
(LIGO Scientific Collaboration and Virgo
Collaboration), O1 stochastic narrowband radiometer injec-
tion recovery summary,
https://dcc.ligo.org/LIGO
T1600563
.
[40] The spherical harmonic analysis uses different values of the
maximum order
l
max
than those used in
[17]
. The
l
max
value
was decreased by a factor of 2 for
α
¼
0
and increased by
30% for
α
¼
3
. The harmonic order resembles extra degrees
of freedom, meaning that larger values of
l
max
tend to reduce
the sensitivity in
Ω
GW
by increasing uncertainty in estimates
of the angular power
P
ð
Θ
Þ
.
PRL
118,
121102 (2017)
PHYSICAL REVIEW LETTERS
week ending
24 MARCH 2017
121102-7
B. P. Abbott,
1
R. Abbott,
1
T. D. Abbott,
2
M. R. Abernathy,
3
F. Acernese,
4,5
K. Ackley,
6
C. Adams,
7
T. Adams,
8
P. Addesso,
9
R. X. Adhikari,
1
V. B. Adya,
10
C. Affeldt,
10
M. Agathos,
11
K. Agatsuma,
11
N. Aggarwal,
12
O. D. Aguiar,
13
L. Aiello,
14,15
A. Ain,
16
P. Ajith,
17
B. Allen,
10,18,19
A. Allocca,
20,21
P. A. Altin,
22
A. Ananyeva,
1
S. B. Anderson,
1
W. G. Anderson,
18
S. Appert,
1
K. Arai,
1
M. C. Araya,
1
J. S. Areeda,
23
N. Arnaud,
24
K. G. Arun,
25
S. Ascenzi,
26,15
G. Ashton,
10
M. Ast,
27
S. M. Aston,
7
P. Astone,
28
P. Aufmuth,
19
C. Aulbert,
10
A. Avila-Alvarez,
23
S. Babak,
29
P. Bacon,
30
M. K. M. Bader,
11
P. T. Baker,
31
F. Baldaccini,
32,33
G. Ballardin,
34
S. W. Ballmer,
35
J. C. Barayoga,
1
S. E. Barclay,
36
B. C. Barish,
1
D. Barker,
37
F. Barone,
4,5
B. Barr,
36
L. Barsotti,
12
M. Barsuglia,
30
D. Barta,
38
J. Bartlett,
37
I. Bartos,
39
R. Bassiri,
40
A. Basti,
20,21
J. C. Batch,
37
C. Baune,
10
V. Bavigadda,
34
M. Bazzan,
41,42
C. Beer,
10
M. Bejger,
43
I. Belahcene,
24
M. Belgin,
44
A. S. Bell,
36
B. K. Berger,
1
G. Bergmann,
10
C. P. L. Berry,
45
D. Bersanetti,
46,47
A. Bertolini,
11
J. Betzwieser,
7
S. Bhagwat,
35
R. Bhandare,
48
I. A. Bilenko,
49
G. Billingsley,
1
C. R. Billman,
6
J. Birch,
7
R. Birney,
50
O. Birnholtz,
10
S. Biscans,
12,1
A. S. Biscoveanu,
74
A. Bisht,
19
M. Bitossi,
34
C. Biwer,
35
M. A. Bizouard,
24
J. K. Blackburn,
1
J. Blackman,
51
C. D. Blair,
52
D. G. Blair,
52
R. M. Blair,
37
S. Bloemen,
53
O. Bock,
10
M. Boer,
54
G. Bogaert,
54
A. Bohe,
29
F. Bondu,
55
R. Bonnand,
8
B. A. Boom,
11
R. Bork,
1
V. Boschi,
20,21
S. Bose,
56,16
Y. Bouffanais,
30
A. Bozzi,
34
C. Bradaschia,
21
P. R. Brady,
18
V. B. Braginsky,
49
,
M. Branchesi,
57,58
J. E. Brau,
59
T. Briant,
60
A. Brillet,
54
M. Brinkmann,
10
V. Brisson,
24
P. Brockill,
18
J. E. Broida,
61
A. F. Brooks,
1
D. A. Brown,
35
D. D. Brown,
45
N. M. Brown,
12
S. Brunett,
1
C. C. Buchanan,
2
A. Buikema,
12
T. Bulik,
62
H. J. Bulten,
63,11
A. Buonanno,
29,64
D. Buskulic,
8
C. Buy,
30
R. L. Byer,
40
M. Cabero,
10
L. Cadonati,
44
G. Cagnoli,
65,66
C. Cahillane,
1
J. Calderón Bustillo,
44
T. A. Callister,
1
E. Calloni,
67,5
J. B. Camp,
68
W. Campbell,
120
M. Canepa,
46,47
K. C. Cannon,
69
H. Cao,
70
J. Cao,
71
C. D. Capano,
10
E. Capocasa,
30
F. Carbognani,
34
S. Caride,
72
J. Casanueva Diaz,
24
C. Casentini,
26,15
S. Caudill,
18
M. Cavaglià,
73
F. Cavalier,
24
R. Cavalieri,
34
G. Cella,
21
C. B. Cepeda,
1
L. Cerboni Baiardi,
57,58
G. Cerretani,
20,21
E. Cesarini,
26,15
S. J. Chamberlin,
74
M. Chan,
36
S. Chao,
75
P. Charlton,
76
E. Chassande-Mottin,
30
B. D. Cheeseboro,
31
H. Y. Chen,
77
Y. Chen,
51
H.-P. Cheng,
6
A. Chincarini,
47
A. Chiummo,
34
T. Chmiel,
78
H. S. Cho,
79
M. Cho,
64
J. H. Chow,
22
N. Christensen,
61
Q. Chu,
52
A. J. K. Chua,
80
S. Chua,
60
S. Chung,
52
G. Ciani,
6
F. Clara,
37
J. A. Clark,
44
F. Cleva,
54
C. Cocchieri,
73
E. Coccia,
14,15
P.-F. Cohadon,
60
A. Colla,
81,28
C. G. Collette,
82
L. Cominsky,
83
M. Constancio Jr.,
13
L. Conti,
42
S. J. Cooper,
45
T. R. Corbitt,
2
N. Cornish,
84
A. Corsi,
72
S. Cortese,
34
C. A. Costa,
13
E. Coughlin,
61
M. W. Coughlin,
61
S. B. Coughlin,
85
J.-P. Coulon,
54
S. T. Countryman,
39
P. Couvares,
1
P. B. Covas,
86
E. E. Cowan,
44
D. M. Coward,
52
M. J. Cowart,
7
D. C. Coyne,
1
R. Coyne,
72
J. D. E. Creighton,
18
T. D. Creighton,
87
J. Cripe,
2
S. G. Crowder,
88
T. J. Cullen,
23
A. Cumming,
36
L. Cunningham,
36
E. Cuoco,
34
T. Dal Canton,
68
S. L. Danilishin,
36
S. D
Antonio,
15
K. Danzmann,
19,10
A. Dasgupta,
89
C. F. Da Silva Costa,
6
V. Dattilo,
34
I. Dave,
48
M. Davier,
24
G. S. Davies,
36
D. Davis,
35
E. J. Daw,
90
B. Day,
44
R. Day,
34
S. De,
35
D. DeBra,
40
G. Debreczeni,
38
J. Degallaix,
65
M. De Laurentis,
67,5
S. Deléglise,
60
W. Del Pozzo,
45
T. Denker,
10
T. Dent,
10
V. Dergachev,
29
R. De Rosa,
67,5
R. T. DeRosa,
7
R. DeSalvo,
91
J. Devenson,
50
R. C. Devine,
31
S. Dhurandhar,
16
M. C. Díaz,
87
L. Di Fiore,
5
M. Di Giovanni,
92,93
T. Di Girolamo,
67,5
A. Di Lieto,
20,21
S. Di Pace,
81,28
I. Di Palma,
29,81,28
A. Di Virgilio,
21
Z. Doctor,
77
V. Dolique,
65
F. Donovan,
12
K. L. Dooley,
73
S. Doravari,
10
I. Dorrington,
94
R. Douglas,
36
M. Dovale Álvarez,
45
T. P. Downes,
18
M. Drago,
10
R. W. P. Drever,
1
J. C. Driggers,
37
Z. Du,
71
M. Ducrot,
8
S. E. Dwyer,
37
T. B. Edo,
90
M. C. Edwards,
61
A. Effler,
7
H.-B. Eggenstein,
10
P. Ehrens,
1
J. Eichholz,
1
S. S. Eikenberry,
6
R. C. Essick,
12
Z. Etienne,
31
T. Etzel,
1
M. Evans,
12
T. M. Evans,
7
R. Everett,
74
M. Factourovich,
39
V. Fafone,
26,15,14
H. Fair,
35
S. Fairhurst,
94
X. Fan,
71
S. Farinon,
47
B. Farr,
77
W. M. Farr,
45
E. J. Fauchon-Jones,
94
M. Favata,
95
M. Fays,
94
H. Fehrmann,
10
M. M. Fejer,
40
A. Fernández Galiana,
12
I. Ferrante,
20,21
E. C. Ferreira,
13
F. Ferrini,
34
F. Fidecaro,
20,21
I. Fiori,
34
D. Fiorucci,
30
R. P. Fisher,
35
R. Flaminio,
65,96
M. Fletcher,
36
H. Fong,
97
S. S. Forsyth,
44
J.-D. Fournier,
54
S. Frasca,
81,28
F. Frasconi,
21
Z. Frei,
98
A. Freise,
45
R. Frey,
59
V. Frey,
24
E. M. Fries,
1
P. Fritschel,
12
V. V. Frolov,
7
P. Fulda,
6,68
M. Fyffe,
7
H. Gabbard,
10
B. U. Gadre,
16
S. M. Gaebel,
45
J. R. Gair,
99
L. Gammaitoni,
32
S. G. Gaonkar,
16
F. Garufi,
67,5
G. Gaur,
100
V. Gayathri,
101
N. Gehrels,
68
G. Gemme,
47
E. Genin,
34
A. Gennai,
21
J. George,
48
L. Gergely,
102
V. Germain,
8
S. Ghonge,
17
Abhirup Ghosh,
17
Archisman Ghosh,
11,17
S. Ghosh,
53,11
J. A. Giaime,
2,7
K. D. Giardina,
7
A. Giazotto,
21
K. Gill,
103
A. Glaefke,
36
E. Goetz,
10
R. Goetz,
6
L. Gondan,
98
G. González,
2
J. M. Gonzalez Castro,
20,21
A. Gopakumar,
104
M. L. Gorodetsky,
49
S. E. Gossan,
1
M. Gosselin,
34
R. Gouaty,
8
A. Grado,
105,5
C. Graef,
36
M. Granata,
65
A. Grant,
36
S. Gras,
12
C. Gray,
37
G. Greco,
57,58
A. C. Green,
45
P. Groot,
53
H. Grote,
10
S. Grunewald,
29
G. M. Guidi,
57,58
X. Guo,
71
PRL
118,
121102 (2017)
PHYSICAL REVIEW LETTERS
week ending
24 MARCH 2017
121102-8