All-sky, all-frequency directional search for persistent gravitational waves
from Advanced LIGO
’
s and Advanced Virgo
’
s first three observing runs
R. Abbott
etal.
*
(LIGO Scientific Collaboration, the Virgo Collaboration, and the KAGRA Collaboration)
(Received 25 October 2021; revised 22 February 2022; accepted 17 May 2022; published 3 June 2022)
We present the first results from an all-sky all-frequency (ASAF) search for an anisotropic stochastic
gravitational-wave background using the data from the first three observing runs of the Advanced LIGO
and Advanced Virgo detectors. Upper limit maps on broadband anisotropies of a persistent stochastic
background were published for all observing runs of the LIGO-Virgo detectors. However, a broadband
analysis is likely to miss narrowband signals as the signal-to-noise ratio of a narrowband signal can be
significantly reduced when combined with detector output from other frequencies. Data folding and the
computationally efficient analysis pipeline,
PyStoch
, enable us to perform the radiometer map-making at
every frequency bin. We perform the search at 3072
HEALPix
equal area pixels uniformly tiling the sky
and in every frequency bin of width
1
=
32
Hz in the range 20
–
1726 Hz, except for bins that are likely to
contain instrumental artefacts and hence are notched. We do not find any statistically significant evidence
for the existence of narrowband gravitational-wave signals in the analyzed frequency bins. Therefore, we
place 95% confidence upper limits on the gravitational-wave strain for each pixel-frequency pair, the limits
are in the range
ð
0
.
030
−
9
.
6
Þ
×
10
−
24
. In addition, we outline a method to identify candidate pixel-
frequency pairs that could be followed up by a more sensitive (and potentially computationally expensive)
search, e.g., a matched-filtering-based analysis, to look for fainter nearly monochromatic coherent signals.
The ASAF analysis is inherently independent of models describing any spectral or spatial distribution of
power. We demonstrate that the ASAF results can be appropriately combined over frequencies and sky
directions to successfully recover the broadband directional and isotropic results.
DOI:
10.1103/PhysRevD.105.122001
I. INTRODUCTION
A stochastic gravitational-wave background (SGWB)
[1,2]
can be created by the superposition of a large number
of unresolved independent sources
[3
–
23]
. Improvements
in detector sensitivity suggest that the network of ground-
based gravitational-wave (GW) observatories may be able
to observe such a background in the coming years.
Astrophysical sources in the nearby universe can make
the background anisotropic
[24
–
31]
. Directional searches
were performed
[32
–
36]
on data from the Advanced LIGO-
Virgo detectors for two different source categories to probe
these anisotropies. One search focuses on a persistent
SGWB from a collection of pointlike and extended dis-
tributions of sources emitting GWs over a broad frequency
range
[36
–
40]
. The second search looks for narrowband
signals from known locations of potentially detectable
continuous wave sources
[36]
.
The past analyses, however, had limited prospects of
detecting an unknown narrowband anisotropic stochastic
background. The broadband searches are not optimized to
detect narrowband signals. While the broadband radiometer
search
[38,41]
is capable of coherently adding signals from
multiple narrowband sources, noise from thousands of
other frequency bins that do not contain any signal can
degrade the signal-to-noise ratio. Moreover, it is straight-
forward to combine results from a narrowband analysis
with appropriate frequency-dependent weight factors to
derive the broadband results for a variety of spectral shapes
including nonpower law predictions in astrophysical
[18,19,42]
and cosmological
[43
–
45]
scenarios. Such
efforts can be further extended to allow the spectral shape
to vary across the sky, which is expected when multiple
anisotropic backgrounds are simultaneously present in the
sensitive frequency bands of the detectors. Performing the
directional search at all the narrowband frequency bins
separately is thus well motivated. Recent developments on
data folding
[46]
and the introduction of a new analysis
pipeline
PyStoch
[47]
, that have made the radiometer
analysis hundreds of times faster, opens up the possibility of
performing an all-sky directed radiometer search for
unknownpersistent signals innarrow frequency bins
[48,49]
.
In addition, many unknown galactic and extra-galactic
continuous GW sources, primarily from neutron stars
[50
–
60]
and exotic scenarios like boson clouds around spinning
black holes
[61,62]
, may be detectable with the Advanced
*
Full author list given at the end of the article.
PHYSICAL REVIEW D
105,
122001 (2022)
2470-0010
=
2022
=
105(12)
=
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© 2022 American Physical Society
LIGO-Virgo detectors. Searching for such signals from
neutron stars across the sky using known models is always
computationally expensive
[63,64]
. Since these matched-
filtering-based searches depend on the signal model via
frequency evolution templates, the chances of detecting an
unknown or poorly modeled family of sources (e.g.,
accreting and/or short-period binary systems) are limited
and expanding the parameter space is computationally
challenging
[51,65]
. Also, the probability of detection in
such matched-filtering-based searches is reduced substan-
tially in the presence of glitches
[66]
. The all-sky all-
frequency (ASAF) analysis is robust with respect to
modeling and able to rapidly identify candidate fre-
quency
—
sky location pairs, that may warrant following
up with more sensitive matched filtering-based searches.
The ASAF results could also be used to look for cross-
correlations between SGWB and the electromagnetic sky
[67
–
70]
, where each narrowband frequency bins may be
analysed independently for setting stronger constraints on
astrophysical and cosmological models.
In this paper we report the first ASAF upper limits on an
unmodeled anisotropic SGWB using data from the first
three observing runs (O
1
þ
O
2
þ
O
3
) of the Advanced
LIGO and Advanced Virgo detectors.
II. DATA
To perform the ASAF search, we analyze the data from
the first (O1) and second (O2) observing runs of the
Advanced LIGO
[71]
detectors located in Hanford (H)
and Livingston (L), and from the third observing runs (O3)
of both Advanced LIGO and Advanced Virgo
[72]
(V). The
recorded data have been processed and conditioned in the
same way as was done in the latest directional search
analysis by the LIGO-Virgo-KAGRA (LVK) collaboration
[36,73
–
75]
. To account for non-Gaussian features in the
data, we identify and remove segments containing detected
GW signals from compact binary coalescences
[36,76,77]
,
nonretracted events in the second half of O3
[78]
, transient
hardware injections (simulated signals injected by physi-
cally displacing the interferometer mirrors
[79]
) and seg-
ments associated with instrumental artifacts. We then
obtain the cross-spectral density (CSD) by combining
the short-term Fourier transforms (SFTs) of 192 second
data segments from pairs of detectors
[80]
, and the
corresponding variances, with a coarse-grained frequency
resolution of
Δ
f
¼
1
=
32
Hz
[81]
.
We apply the ASAF analysis in the frequency range 20
–
1726 Hz as in the past stochastic searches. In addition to the
time-domain data quality cuts, we also identify contami-
nated frequency bins using coherence studies
[82]
. These
frequency bins are mainly associated with known instru-
mental artifacts (calibration lines, power lines, and their
harmonics). We also remove the continuous-wave hard-
ware injections
[79]
in the final analysis, though we use
these for validating the analysis pipeline. Since we produce
results for each frequency bin separately, it is important to
have a stringent check on the noise contamination for all the
individual bins. In the broadband search
[36]
, while all
attempts are made to discard frequency bins with noise
contamination, to avoid too much loss of sensitivity,
notching is relatively less stringent, as noise in a few
frequency bins has an insignificant effect on the whole
integrated frequency band. We apply a more stringent
threshold on the permissible level of nonstationarity in
individual frequency bins compared to the broadband
search
[83]
. This choice results in removing approximately
21%, 34.8%, and 28.2% of the frequency bins from O3 data
for HL, HV, and LV baselines, respectively, compared to
14.8%, 25.2%, 21.9%, for the nonstringent notching used
in the broadband search
[36]
. After combining the baselines
and the three observing runs, the number of completely
notched frequencies reduces to 12.5% of the total.
III. ASAF ESTIMATORS
In contrast to the previous stochastic directional analy-
ses, which constrained either broadband integrated
anisotropy of the sky or narrowband sources in specific
directions, the ASAF analysis attempts to probe anisotropy
of the whole sky in each frequency bin. The anisotropies
can be characterized in terms of a dimensionless GW
energy density parameter
Ω
GW
ð
f;
ˆ
n
Þ
with units sr
−
1
,
defined as
[36,80,84]
,
Ω
GW
ð
f;
ˆ
n
Þ
≡
f
ρ
c
d
ρ
GW
df
¼
2
π
2
3
H
2
0
f
3
P
ð
f;
ˆ
n
Þ
;
ð
1
Þ
where
ρ
GW
is the energy density of incoming GWs from the
direction
ˆ
n
observed in the frequency range from
f
to
f
þ
df
,
ρ
c
¼
3
c
2
H
2
0
=
8
π
G
is the critical energy density
needed to close the universe,
c
denotes the speed of light,
G
is the gravitational constant, and
H
0
is the Hubble constant.
We are interested in estimating the sky-maps
P
ð
f;
ˆ
n
Þ
at
every frequency, which can be decomposed as,
P
ð
f;
ˆ
n
Þ¼
X
p
P
p
ð
f
Þ
e
p
ð
ˆ
n
Þ
;
ð
2
Þ
in terms of basis functions
e
p
ð
ˆ
n
Þ
. The choice of the basis
usually depends on the target source. For extended sources,
the spherical harmonic basis
Y
lm
ð
ˆ
n
Þ
is a common choice,
while for localized pointlike sources the pixel basis,
e
p
ð
ˆ
n
Þ¼
δ
2
ð
ˆ
n
−
ˆ
n
p
Þ
, where
ˆ
n
p
is the direction of pixel
number
p
, is an appropriate choice. Here we perform the
ASAF analysis in pixel basis only.
To measure the anisotropy
P
ð
f;
ˆ
n
Þ
, the radiometer
search uses the maximum likelihood (ML) estimator
[38,85]
as the statistic,
ˆ
P
ð
f
Þ¼
Γ
ð
f
Þ
−
1
X
ð
f
Þ
;
ð
3
Þ
R. ABBOTT
et al.
PHYS. REV. D
105,
122001 (2022)
122001-2
where
X
is the
dirty map
[38]
and
Γ
is the Fisher
information matrix
[85]
in the weak signal limit, where
in the chosen basis, the signal is much smaller than the
standard deviation of the noise in each data segment.
The dirty map
X
represents the SGWB sky seen through
the response matrices of a baseline
I
formed by the
detectors
I
1
and
I
2
, defined as
X
p
ð
f
Þ¼
τ
Δ
f
Re
X
I
;t
γ
I
ft;p
C
I
ð
t
;
f
Þ
P
I
1
ð
t
;
f
Þ
P
I
2
ð
t
;
f
Þ
;
ð
4
Þ
where
τ
is the length (duration) of each time segment,
P
I
1
;
2
ð
t
;
f
Þ
denotes the one-sided noise power spectral
density (PSD) and
C
I
ð
t
;
f
Þ
≔
ð
2
=
τ
Þ
e
s
I
1
ð
t
;
f
Þ
e
s
I
2
ð
t
;
f
Þ
is
the CSD,
e
s
I
1
;
2
ð
t
;
f
Þ
are the SFTs of data from the detectors
at a time-segment marked by
t
. In practice, to prevent
spectral leakage without loss of data, overlapping Hanning
windows are applied to the time series data in each
segment, introducing additional normalization factors,
which have been accounted for in the analysis
[37,41,81,86,87]
. Also, the coarse-grained frequency bin
size
Δ
f
is greater than
1
=
τ
, i.e.,
τ
Δ
f
is not unity
[82,88]
.
The overlap reduction function (ORF),
γ
I
ft;p
, is defined as
γ
I
ft;p
∶
≡
X
A
F
A
I
1
ð
ˆ
n
p
;t
Þ
F
A
I
2
ð
ˆ
n
p
;t
Þ
e
2
π
if
ˆ
n
p
·
Δ
x
I
ð
t
Þ
=c
;
ð
5
Þ
where
Δ
x
I
ð
t
Þ
is the separation vector between the detec-
tors. In the above equation
A
¼þ
;
× denotes the polari-
zation (note that this analysis assumes statistically
equivalent
þ
and × polarizations). The ORF is necessary
to optimally
“
point
”
the radiometer
[38]
to a direction
ˆ
n
p
corresponding to the pixel index
p
by cross-correlating data
streams from pairs of detectors with time varying phase
delay, along with the sky modulation induced by the
antenna pattern functions of the detector (
F
A
I
1
;
2
)
[2,80]
.
The uncertainty in the dirty map measurement is encoded in
the Fisher information matrix defined as
Γ
pp
0
ð
f
Þ
≡
τ
Δ
f
2
Re
X
I
;t
γ
I
ft;p
γ
I
ft;p
0
P
I
1
ð
t
;
f
Þ
P
I
2
ð
t
;
f
Þ
:
ð
6
Þ
Since the estimators are obtained by summing over a large
number of
τ
¼
192
second time-segments, the central limit
theorem implies that the noise distribution is approximately
Gaussian as long as the total observation is longer than a
few hours.
Once
X
p
ð
f
Þ
and
Γ
pp
0
ð
f
Þ
are calculated using Eqs.
(4)
and
(6)
for all the baselines and observing runs, we
combine them to obtain the multi-baseline (HLV here
onward) dirty map and Fisher information matrix for all
the observing runs. From the combined Fisher matrix and
dirty map, we can construct the estimator in Eq.
(3)
. The
ML estimator
ˆ
P
involves inversion of the Fisher matrix
Γ
which has singular values associated with certain observed
modes on the sky where the baselines are insensitive. For
point sources, given the current sensitivity of detectors and
the pixel resolution used here, correcting for the pixel-to-
pixel correlation hardly makes any difference to the
analysis
[35]
. Therefore, to obtain the estimator of narrow-
band anisotropy and the corresponding signal-to-noise ratio
(SNR), instead of inverting the Fisher matrix we divide the
dirty map in a given pixel by the corresponding diagonal
element of the Fisher matrix. Then the estimator of
narrowband anisotropy and its uncertainty are given by,
ˆ
P
ð
f;
ˆ
n
Þ¼½
Γ
ˆ
n
;
ˆ
n
ð
f
Þ
−
1
X
ˆ
n
;
ð
7
Þ
σ
ˆ
n
ð
f
Þ¼½
Γ
ˆ
n
;
ˆ
n
ð
f
Þ
−
1
=
2
:
ð
8
Þ
Now one can write the observed (SNR) as
ρ
ð
f;
ˆ
n
Þ¼
ˆ
P
ð
f;
ˆ
n
Þ
=
σ
ˆ
n
ð
f
Þ
:
ð
9
Þ
In the absence of any signal,
ρ
ð
f;
ˆ
n
Þ
follows a Gaussian
distribution (Fig.
1
). This formalism is used to perform the
ASAF analysis.
IV. DATA FOLDING AND
PYSTOCH
Folding
[46]
makes use of the temporal symmetry in the
detector scan pattern to compress the data into a single
sidereal day. This reduces the computational cost of the
search by a factor equal to the total number of days in the
observation run.
PyStoch
[47]
is a fully python-based standalone pipe-
line for SGWB map-making that takes full advantage of the
compressed folded data and symmetries in the detector set-
up, further improving the efficiency of the search with
respect to the previous analysis pipeline. Both folding and
PyStoch
were recently adapted for the broadband direc-
tional search in LIGO-Virgo data
[36]
.
Since folding ensures that the data size is fixed (one
sidereal day long) and can be loaded entirely in the memory
of most computers,
PyStoch
is able to cast the segment-
wise radiometer analysis to a matrix multiplication problem
incorporating all the time segments together. This allows
interchangeable ordering of operations over time and
frequency, which is essential for the ASAF analysis, and
efficient parallel processing of data. Here we use
HEALPix
[89]
resolution of
N
side
¼
16
, which corresponds to the
number of pixels
N
pix
¼
12
N
2
side
¼
3072
. In principle, one
could use a lower (higher) pixel resolution at lower (higher)
frequencies. Since the chosen resolution is adequate for the
most sensitive frequency band of the baselines, we refrain
from introducing further complexities or using higher
resolution in the present analysis. With these search
ALL-SKY, ALL-FREQUENCY DIRECTIONAL SEARCH FOR
...
PHYS. REV. D
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parameters, it took less than an hour per baseline for each
dataset to run on a personal computer
[90]
.
V. DETECTION STATISTICS AND OUTLIERS
In order to search for significant outliers, which could
indicate the presence of a signal, we first need to determine
the noise background. Since it is impossible to directly
measure the detector noise in the absence of persistent GW
signals, the background is estimated by introducing a
constant unphysical
time-shift
[36,82]
of
∼
1
second
between the data streams from a pair of detectors which
is much greater than the light travel time between the
interferometers. If a signal is not correlated on longer time
scales, which is the case for black hole mergers or the
stochastic background created by such
“
popcorn
”
type
events
[91]
, these time-shifts are expected to remove the
signal correlation between detectors and the resultant
distribution of the detection statistic will then represent
the noise background
[92]
. However, these constant time-
shifts do not completely cancel out the types of sources that
have longer coherence. For example, if we consider an
isolated neutron star emitting GWs at a certain frequency,
the constant time-shift analysis will not remove the
presence of such a signal; the signal might appear to
originate from a different direction or with an unphysical
negative SNR. Since the ASAF results could be useful for
identifying potential locations and frequencies of previ-
ously unknown neutron stars, we apply an alternative
method to alleviate this problem.
We estimate the noise background by adding a
random
time-shift to each contiguous slice of data such that the net
shift (1
–
2 second) is always much greater than the
physically permissible time delay. In order to validate this
prescription, we use continuous hardware injections of
isolated pulsars with varying signal strength performed in
Advanced LIGO
’
s second (O2) observation run (the hard-
ware injections for O3 were too weak
[63]
for the ASAF
analysis to recover with enough SNR). To generate the
background we first apply the random time-shift for the
Hanford-Livingston pair and run the entire analysis. These
SNRs are plotted in the histogram shown in Fig.
1
along
with a Gaussian fit. It is clear that the time-shifted method is
successful in removing the effect of the injected signals (the
elevated tails of the distribution), leading to a Gaussian
noise background. Had the
“
foreground
”
distribution
—
the
distribution of SNR without any unphysical time-shifts
(zero-lag)
—
been inside the
2
σ
error bars around the
Gaussian, we would rule out the presence of any outliers
with 95% confidence. The presence of some large negative
SNR values for zero-lag in Fig.
1
is due to the mismatch
between the circular polarization model implicitly assumed
in Eq.
(5)
for a continuous-wave source and the elliptic
polarizations simulated by the hardware injections, some of
which are nearly linearly polarized. We searched for 9
injections that were in the analyzed frequency range and
with source-frame frequency variation less than
1
=
32
Hz
[93]
. Among these, 8 were detected as
“
outliers
”
and one as
a follow-up candidate (the ninth injection is slightly below
the detection threshold, but above the threshold for iden-
tifying as a follow-up candidate). The maximum dirty map
SNRs are plotted in Fig.
1
. In some cases, an injection is
recovered also in the previous or the next frequency bin due
to spectral leakage
[87]
, which appears as multiple circles
very close to one vertical line. The recovered locations of
the sources match the locations of injections within the
diffraction-limited resolution. Note that, as anticipated, the
broadband search did not detect any outliers
[83]
.
In order to identify potential candidates for follow-up,
we consider the distribution of maximum pixel SNR,
ρ
max
ð
f
Þ
≡
max
p
½
ρ
ð
f;
ˆ
n
p
Þ
. The maximum SNR are shown
as a scatter plot in the right panel of Fig.
1
. We divide the
whole frequency range into 10 Hz bins, over which the
sensitivity of the radiometer does not vary significantly. We
make a histogram of
ρ
max
ð
f
Þ
for all the
1
=
32
Hz frequency
bins in each 10 Hz bin, separately for zero-lag and time-
shifted analysis. For each 10 Hz bin, we find the
ρ
max
ð
f
Þ
that corresponds to 99th percentile (1% false alarm rate) of
the maximum SNR distribution for time-shifted data and
average it over 3 neighboring 10 Hz bins (yellow line in the
right panel of Fig.
1
). Any
ρ
max
ð
f
Þ
that is above this
threshold in the zero-lag analysis is marked as a candidate
for follow-up studies. All hardware injections lie above this
threshold curve and qualify for detailed follow-up studies.
The
“
slope
”
of the maximum SNR distribution in the
scatter plot changes because the diffraction-limited reso-
lution is a function of frequency
[38,49]
. Therefore, at
lower frequencies, the number of independent sky patches
is smaller and, hence, the pixels are more correlated. The
correlation reduces at higher frequencies.
Reassured by the above hardware injection study, we
apply the same procedure to obtain the significance of the
results from the ASAF search. The distribution of SNR at
each frequency and pixel on the sky are shown in the
histogram in the bottom left panel of Fig.
1
along with the
SNR distributions from the time-shifted run. It is evident
from the figure that the distribution of SNRs follow the
noise background and in turn is consistent with Gaussian
noise within
2
σ
error bars. Since the number of frequency-
pixel pairs is
∼
10
8
, there is greater than a 5% probability of
at least one high SNR (
∼
6
) observation arising purely from
noise, as seen in the tails of the distributions. The ASAF
analysis thus rules out the existence of any significant
outliers in the O1+O2+O3 data.
We nevertheless apply the same procedure described for
hardware injections to identify potential candidates that can
be followed up by a more sensitive search. We again
consider the distribution of maximum SNRs in each 10 Hz
frequency bin. These maxima are plotted in the bottom
R. ABBOTT
et al.
PHYS. REV. D
105,
122001 (2022)
122001-4
right panel of Fig.
1
. Here, the noise background obtained
from the time-shifted method is shown in red in the scatter
plot whereas the results from zero-lag data are represented
in blue. The yellow line delimits the SNRs above the 99th
percentile of the background obtained from the unphysical
time-shifted data. The maximum SNRs in the zero-lag
analysis that pass this threshold can be classified as
candidates for follow-up using template-based searches,
even though they are not significant enough to be called
outliers by the ASAF analysis
—
SNR is below the thresh-
old of 6.05 that corresponds to the trials-factor-corrected,
one-sided global
p
-value of 5% shown by the orange dot-
dashed line.
VI. UPPER LIMITS
In the presence of a detectable signal of strength
P
ð
f;
ˆ
n
Þ
,
the ASAF point estimate,
ˆ
P
ð
f;
ˆ
n
Þ
, by construction would be
distributed with a mean
P
ð
f;
ˆ
n
Þ
and standard deviation
σ
ˆ
n
ð
f
Þ
. Since the distribution is found to be consistent with
the noise background obtained by unphysical time-shift
analysis, ruling out the detection of any significant signal,
we set upper limits on the strength of astrophysical signals.
Herewe take advantage of the fact that this noise background
is Gaussian. Since ASAF results are most relevant in the
context of nearly monochromatic continuous wave signals,
we set Bayesian upper limits on an equivalent strain
FIG. 1. Results from the O2 hardware injection study (top panel) and O
1
þ
O
2
þ
O
3
datasets (bottom panel) are shown. Left hand
side plots in the top and bottom panels show the histogram of SNR from both the (unphysical) time-shifted (TS) analysis (red) and zero-
lag (ZL) analysis (blue). Time-shifted and zero-lag datasets are both consistent with Gaussian noise (gray histogram) within the
2
σ
error
bars. The outliers cause the extended tails in blue histogram representing the distribution of foreground SNRs. These extended tails in
the top left plot depicts the recovery of the hardware injections performed on the O2 data. The point with SNR
∼−
6
in bottom left plot is
not astrophysically motivated and most likely caused by low number statistics (the two-tailed
p
-value is more than 5%). The right hand
side plots in the top and bottom panels show the distribution of maximum SNRs from both the time-shifted and zero-lag analyses. In
these plots, the gray solid vertical lines represent the frequencies notched in the data, the yellow curve shows the 99th percentile of
maximum SNR for every 10 Hz frequency bin in the time-shifted analysis, smoothed over 3 neighboring 10 Hz bins, and the orange dot-
dashed line delineates the trials-factor-corrected, one-sided global
p
-value of 5%. The points above the yellow curve, the identified
candidates for follow-up studies, are marked with teal circles. For the O2 hardware injection study, one can see that (top right) all the
injections are recovered which appear as multiple teal circles very close to the black vertical lines indicating the frequency bin where the
injections were made. On the other hand, for O
1
þ
O
2
þ
O
3
dataset, we do not find any outliers significantly above the noise
background (bottom right). In this case, the teal circles represent 515 candidates which may be followed up by a more sensitive matched-
filtering-based analysis.
ALL-SKY, ALL-FREQUENCY DIRECTIONAL SEARCH FOR
...
PHYS. REV. D
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122001 (2022)
122001-5
amplitude of a circularly polarized signal,
h
ð
f;
ˆ
n
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
ð
f;
ˆ
n
Þ
df
p
, without correcting for Doppler modulation
caused by the Earth
’
s motion. We assume a Gaussian
likelihood for
ˆ
P
ð
f;
ˆ
n
Þ
which is then marginalized over
calibration uncertainty
[94]
with a uniform prior on
h
ð
f;
ˆ
n
Þ
in the wide range
0
−
10
σ
of the point estimate. The
95% confidence Bayesian upper limits placed on the
HLV dataset are shown in Fig.
2
. The matrix plot shown
in Fig.
2
is a qualitative representation of the upper limit,
where the upper limit sky maps are plotted as a function of
HEALPix
pixel index on the horizontal axis and frequency
on the vertical axis. The color bar represents the respective
upper limit range. The horizontal
“
bands
”
and gaps in the
matrix plot correspond to the notched frequencies in one or
more detector pairs. These upper limits are in the range
ð
0
.
030
–
9
.
6
Þ
×
10
−
24
. Note that, when interpreting upper
limits on particular frequency-pixel pairs for potential
point sources, such as neutron stars, allowance should
be made forthe angular distancebetween thesource and the
center of the pixel containing it; at the highest frequency
searched (1726 Hz) the SNR for a point source at the edge
of a pixel (at the chosen resolution) is
∼
20%
less than for
the same source at the center of the pixel for most parts of
the sky. The inclusion of different baselines and observing
run data reduces the fraction of completely notched
frequencies and also improves the upper limits (more
details are provided in
[83]
). In order to compare the
relative sensitiv
ities of different baselines and observing
runs, one could define an effective strain sensitivity
averaged over all pixels and frequencies (including the
notched ones) as
½h
σ
−
2
ˆ
n
ð
f
Þi
=df
2
−
1
=
4
, which turns out to be
10
−
25
times 1.5, 0.89, 2.5, 2.2, 0.88 and 0.86 respectively,
for O
1
þ
O
2
, HL, LV, HV and HLV for O3 and HLV
for O
1
þ
O
2
þ
O
3
.
We also include the plots of the upper limit sky maps for
three example frequencies 34.0 Hz, 200.0 Hz and, 910.0 Hz
in Fig.
2
. The sky map on the top right corner shows how
the uncertainty in the ASAF estimator is nearly axially
symmetric and varies in latitude. This pattern is also
reflected in the upper limit matrix plot.
FIG. 2. The 95% confidence Bayesian upper limit on the strain amplitude
h
for all-sky directions and all-frequency bins. These upper
limits are set using the ASAF search performed on the O
1
þ
O
2
þ
O
3
dataset. The color bar here denotes the range of upper limit
variations. A vertical cross section in this diagram shows the frequency dependent upper limit in a particular direction, 3 such plots are
shown in the subplots on top. Whereas horizontal cross sections form a map of upper limits in a particular frequency, 3 such maps are
shown on the right. An example of the standard deviation in these maps is also shown in the top right corner.
R. ABBOTT
et al.
PHYS. REV. D
105,
122001 (2022)
122001-6
VII. DERIVATION OF BROADBAND
RADIOMETER (BBR) ESTIMATES FROM
ASAF SEARCH
One can integrate the ASAF results over frequency to
obtain the broadband radiometer point estimate, variance
and upper limits. We invoke the standard assumption used
in the BBR search
[36]
, that the anisotropic PSD can be
decomposed as a product of frequency and direction
dependent terms,
P
ð
f;
ˆ
n
Þ¼
P
ð
ˆ
n
Þ
H
ð
f
Þ
;
ð
10
Þ
where
H
ð
f
Þ
is assumed to be a power law model propor-
tional to
ð
f=f
ref
Þ
α
where
α
is the spectral index and
f
ref
is a
reference frequency set to 25 Hz. The BBR estimator and
its standard deviation
[95
–
97]
can then be written in terms
of ASAF estimators as
ˆ
P
ð
ˆ
n
Þ¼
P
f
ˆ
P
ð
f;
ˆ
n
Þ
σ
−
2
ˆ
n
ð
f
Þ
H
ð
f
Þ
P
f
σ
−
2
ˆ
n
ð
f
Þ
H
2
ð
f
Þ
;
σ
ˆ
n
¼
X
f
σ
−
2
ˆ
n
ð
f
Þ
H
2
ð
f
Þ
−
1
=
2
:
ð
11
Þ
Now one can calculate the GW flux in each direction
ˆ
n
,
F
ð
ˆ
n
Þ¼
c
3
π
4
G
f
2
ref
P
ð
ˆ
n
Þ
:
ð
12
Þ
This quantity has the units of erg cm
−
2
s
−
1
Hz
−
1
sr
−
1
measured with respect to the reference frequency. Using
the methods outlined in
[36]
, we place upper limits on the
amount of GW flux in each pixel. The upper limit range
obtained from this analysis is reported in Table
I
.Wehave
also shown in Fig.
3
an example upper limit sky map
derived from the ASAF search, performed on the O3 HL
folded data for a specific spectral shape (
α
¼
0
). These
limits are consistent with the results we directly obtained
from the broadband analysis of O3 data
[36]
.
VIII. DERIVATION OF ISOTROPIC ESTIMATES
FROM ASAF SEARCH
From the ASAF search, it is also possible to derive the
isotropic search results. One can integrate over all the sky
directions to obtain the isotropic estimator
[98,99]
.
ˆ
P
iso
ð
f
Þ
σ
−
2
iso
ð
f
Þ¼
5
4
π
Z
d
ˆ
n
ˆ
P
ð
f;
ˆ
n
Þ
σ
−
2
ˆ
n
ð
f
Þ
;
σ
−
2
iso
ð
f
Þ¼
5
4
π
2
Z
d
ˆ
n
Z
d
ˆ
n
0
Γ
ˆ
n
;
ˆ
n
0
ð
f
Þ
:
ð
13
Þ
To compute the standard deviation of the isotropic esti-
mator
σ
iso
ð
f
Þ
, one requires the full narrowband pixel-to-
pixel covariance matrix
Γ
ˆ
n
;
ˆ
n
0
ð
f
Þ
which became computa-
tionally realistic with
PyStoch
and folded data.
Conventionally isotropic searches report the SGWB in
terms of
Ω
GW
;
iso
ð
f
Þ
. Equation
(1)
can be used for the
corresponding conversion of units. Since the
HEALPix
grid is a discrete grid, we perform discrete integration
with
d
ˆ
n
¼
4
π
=N
pix
.
The broadband isotropic estimators can then be derived
using Eq.
(11)
by replacing ASAF estimators with iso-
tropic-all-frequency estimators given in Eq.
(13)
. The
cross-correlation spectra from the O3 HL dataset are shown
(up to 100 Hz) in Fig.
4
and summarized in Table
I
. The
narrowband uncertainty obtained from the ASAF estima-
tors and the one obtained using the analytically derived
isotropic ORF are consistent up to few hundred Hz (which
contributes more than 99% sensitivity for the HL baseline
[82]
). On the other hand, the point estimate and standard
deviation of the isotropic broadband estimate shown in
Table
I
differ from our recently published results
[36,82]
due to different time and frequency domain data quality
cuts and a different pixel resolution used in the ASAF
analysis.
FIG. 3. The broadband 95% confidence GW flux upper limit
sky map for
α
¼
0
spectral shape. The sky map is represented as a
color bar plot on a Mollweide projection of the sky in ecliptic
coordinates.
TABLE I. Broadband directional radiometer upper limit and the
isotropic search results derived from the ASAF search (with the
stricter notching) using the O3 HL dataset. Results from
the previous LVK O3 analyses are also added in the table for
a direct comparison. The three spectral indices used in the search
are denoted by
α
.
Broadband results: HL baseline
α
Upper limit on
F
ð
ˆ
n
Þ
(×
10
−
8
)
ˆ
Ω
GW
;
iso
ð
×
10
−
9
Þ
ASAF
BBR
[36]
ASAF
ISO
[82]
0
2.1
–
9.6
1.8
–
9.2
2
.
5
8
.
6
−
2
.
1
8
.
2
2
=
3
1.1
–
5.6
0.92
–
4.9
0
.
61
6
.
5
−
3
.
4
6
.
1
3
0.016
–
0.13
0.014
–
0.12
−
0
.
49
0
.
99
−
1
.
3
0
.
9
ALL-SKY, ALL-FREQUENCY DIRECTIONAL SEARCH FOR
...
PHYS. REV. D
105,
122001 (2022)
122001-7
IX. CONCLUSIONS
We present the first all-sky all-frequency radiometer
search results for data from ground-based laser interfero-
metric detectors. No GW signal is detected by our analysis
in the first three observing runs from the Advanced LIGO
and Advanced Virgo detectors. We set 95% confidence
upper limits on the gravitational-wave strain at every
frequency bin and sky location searched.
Note that, while the matched-filtering-based analyses
can search for neutron stars at narrower frequency bins and
are more sensitive
[63]
when such template-based searches
are computationally feasible, ASAF analysis can rapidly
search for such monochromatic signals with very little
computation power and set upper limits at all frequency and
sky-locations for the resolutions used here. The candidate
frequency-pixel pairs identified by ASAF can be followed
up by matched-filtering-based searches. We employed a
heuristic prescription for identifying these follow-up can-
didates. We recognize that alternative approaches may be
explored in this regard. Future analyses will also examine
the potential gains from refining frequency and pixel
resolution. Since the total number of frequency-pixel pairs
is very large (
∼
10
8
), the prescription must be robust enough
not to miss feasibly detectable signals, yet should limit
follow-up candidates to a computationally viable number.
Also, it may be worth exploring the possibility and
effectiveness of incorporating the full ASAF Fisher matrix
in the analysis, especially for frequencies below 100 Hz
where the pixel correlations may affect the upper limits.
The ASAF maps and Fisher information matrices were
also appropriately integrated over frequency, for different
choices of spectral models of the background, to recover
the previously published search results for the broadband
isotropic and anisotropic stochastic backgrounds. The
ASAF results could be useful for many other future studies.
ACKNOWLEDGMENTS
This material is based upon work supported by NSF
’
s
LIGO Laboratory which is a major facility fully funded by
the National Science Foundation. The authors also grate-
fully acknowledge the support of 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
Nazionale di Fisica Nucleare (INFN), the French Centre
National de la Recherche Scientifique (CNRS) and the
Netherlands Organization for Scientific Research (NWO),
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, the Department of Science and
Technology, India, the Science & Engineering Research
Board (SERB), India, the Ministry of Human Resource
Development, India, the Spanish Agencia Estatal de
Investigación (AEI), the Spanish Ministerio de Ciencia e
Innovación and Ministerio de Universidades, the
Conselleria de Fons Europeus, Universitat i Cultura and
the Direcció General de Política Universitaria i Recerca del
Govern de les Illes Balears, the Conselleria d
’
Innovació,
Universitats, Ci`
encia i Societat Digital de la Generalitat
Valenciana and the CERCA Programme Generalitat de
Catalunya, Spain, the National Science Centre of Poland
and the European Union
—
European Regional
Development Fund; Foundation for Polish Science
(FNP), the Swiss National Science Foundation (SNSF),
the Russian Foundation for Basic Research, the Russian
Science Foundation, the European Commission, the
European Social Funds (ESF), the European Regional
Development Funds (ERDF), the Royal Society, the
Scottish Funding Council, the Scottish Universities
Physics Alliance, the Hungarian Scientific Research Fund
(OTKA), the French Lyon Institute of Origins (LIO), the
Belgian Fonds de la Recherche Scientifique (FRS-FNRS),
Actions de Recherche Concert ́
ees (ARC) and Fonds
Wetenschappelijk Onderzoek
—
Vlaanderen (FWO),
Belgium, the Paris Île-de-France Region, the National
Research, Development and Innovation Office Hungary
(NKFIH), the National Research Foundation of Korea, the
Natural Science and Engineering Research Council Canada,
Canadian Foundation for Innovation (CFI), the Brazilian
Ministry of Science, Technology, and Innovations, the
FIG. 4. The isotropic cross-correlation spectra derived from the
ASAF analysis using the O3 HL baseline data. The red line
depicts the estimated value of standard deviation with the
isotropic ORF, while the yellow points show the uncertainty in
the cross-power estimator obtained from the ASAF search. The
green vertical line, which fluctuates around zero mean represents
the point estimates from the ASAF search. The uncertainty
calculated from the ASAF maps is not available at certain
frequencies due to the more stringent notching.
R. ABBOTT
et al.
PHYS. REV. D
105,
122001 (2022)
122001-8
International Center for Theoretical Physics South American
Institute for Fundamental Research (ICTP-SAIFR), the
Research Grants Council of Hong Kong, the National
Natural Science Foundation of China (NSFC), the
Leverhulme Trust, the Research Corporation, the Ministry
of Science and Technology (MOST), Taiwan, the United
States Department of Energy, and the Kavli Foundation. The
authors gratefully acknowledge the support of the NSF,
STFC, INFN and CNRS for provision of computational
resources. This work was supported by MEXT, JSPS
Leading-edge Research Infrastructure Program, JSPS
Grant-in-Aid for Specially Promoted Research 26000005,
JSPS Grant-in-Aid for Scientific Research on Innovative
Areas 2905: No. JP17H06358, No. JP17H06361 and
No. JP17H06364, JSPS Core-to-Core Program A.
Advanced Research Networks, JSPS Grant-in-Aid for
Scientific Research (S) No. 17H06133 and
No. 20H05639, JSPS Grant-in-Aid for Transformative
Research Areas (A) 20A203: No. JP20H05854, the joint
research program of the Institute for Cosmic Ray Research,
University of Tokyo, National Research Foundation (NRF)
and Computing Infrastructure Project of KISTI-GSDC in
Korea, Academia Sinica (AS), AS Grid Center (ASGC) and
the Ministry of Science and Technology (MoST) in Taiwan
under grants including AS-CDA-105-M06, Advanced
Technology Center (ATC) of NAOJ, Mechanical
Engineering Center of KEK. We would like to thank all
of the essential workers who put their health at risk during the
COVID-19 pandemic, without whom we would not have
been able to complete this work.
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