First low-frequency Einstein@Home all-sky search for continuous
gravitational waves in Advanced LIGO data
B. P. Abbott
etal.
*
(LIGO Scientific Collaboration and Virgo Collaboration)
(Received 29 June 2017; revised manuscript received 14 September 2017; published 8 December 2017)
We report results of a deep all-sky search for periodic gravitational waves from isolated neutron stars in
data from the first Advanced LIGO observing run. This search investigates the low frequency range of
Advanced LIGO data, between 20 and 100 Hz, much of which was not explored in initial LIGO. The search
was made possible by the computing power provided by the volunteers of the Einstein@Home project.
We find no significant signal candidate and set the most stringent upper limits to date on the amplitude
of gravitational wave signals from the target population, corresponding to a sensitivity depth of
48
.
7
½
1
=
ffiffiffiffiffiffi
Hz
p
. At the frequency of best strain sensitivity, near 100 Hz, we set 90% confidence upper
limits of
1
.
8
×
10
−
25
. At the low end of our frequency range, 20 Hz, we achieve upper limits of
3
.
9
×
10
−
24
.
At 55 Hz we can exclude sources with ellipticities greater than
10
−
5
within 100 pc of Earth with fiducial
value of the principal moment of inertia of
10
38
kg m
2
.
DOI:
10.1103/PhysRevD.96.122004
I. INTRODUCTION
In this paper we report the results of a deep all-sky
Einstein@Home
[1]
search for continuous, nearly mono-
chromatic gravitational waves (GWs) in data from the first
Advanced LIGO observing run (O1). A number of all-sky
searches have been carried out on initial LIGO data,
[2
–
15]
,
of which
[2,3,7,9,14]
also ran on Einstein@Home.
Einstein@Home is a distributed computing project which
uses the idle time of computers volunteered by the general
public to search for GWs.
The search presented here covers frequencies from 20 Hz
through 100 Hz and frequency derivatives from
−
2
.
65
×
10
−
9
Hz
=
s through
2
.
64
×
10
−
10
Hz
=
s. A large portion of
this frequency range was not explored in initial LIGO due
to lack of sensitivity. By focusing the available computing
power on a subset of the detector frequency range, this
search achieves higher sensitivity at these low frequencies
than would be possible in a search over the full range of
LIGO frequencies. In this low-frequency range we establish
the most constraining gravitational wave amplitude upper
limits to date for the target signal population.
II. LIGO INTERFEROMETERS
AND THE DATA USED
The LIGO gravitational wave network consists of two
observatories, one in Hanford (Washington) and the other
in Livingston (Louisiana) separated by a 3000-km baseline
[16]
. The first observing run (O1)
[17]
of this network after
the upgrade towards the Advanced LIGO configuration
[18]
took place between September 2015 and January
2016. The Advanced LIGO detectors are significantly more
sensitive than the initial LIGO detectors. This increase in
sensitivity is especially significant in the low-frequency
range of 20 Hz through 100 Hz covered by this search: at
100 Hz the O1 Advanced LIGO detectors are about a factor
5 more sensitive than the Initial LIGO detectors during their
last run (S6
[19]
), and this factor becomes
≈
20
at 50 Hz.
For this reason all-sky searches did not include frequencies
below 50 Hz on initial LIGO data.
Since interferometers sporadically fall out of operation
(
“
lose lock
”
) due to environmental or instrumental disturb-
ances or for scheduled maintenance periods, the data set is
not contiguous and each detector has a duty factor of about
50%. To remove the effects of instrumental and environ-
mental spectral disturbances from the analysis, the data in
frequency bins known to contain such disturbances have
been substituted with Gaussian noise with the same average
power as that in the neighboring and undisturbed bands.
This is the same procedure as used in
[3]
. These bands are
identified in the Appendix.
III. THE SEARCH
The search described in this paper targets nearly mono-
chromatic gravitational wave signals as described for
example by Eqs. (1)
–
(4) of
[9]
. Various emission mech-
anisms could generate such a signal, as reviewed in
Sec.
IIA
of
[15]
. In interpreting our results we will consider
a spinning compact object with a fixed, nonaxisymmetric
l
¼
m
¼
2
mass quadrupole, described by an equatorial
ellipticity
ε
.
*
Full author list given at the end of the Letter.
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
’
s title, journal citation,
and DOI.
PHYSICAL REVIEW D
96,
122004 (2017)
2470-0010
=
2017
=
96(12)
=
122004(26)
122004-1
Published by the American Physical Society
We perform a stack-slide type of search using the global
correlation transform (GCT) method
[20
–
22]
. In a stack-
slide search the data is partitioned in segments, and each
segment is searched with a matched-filter method
[23]
. The
results from these coherent searches are combined by
summing the detection statistic values from the different
segments, one per segment (
F
i
), and this determines the
value of the core detection statistic:
̄
F
≔
1
N
seg
X
N
seg
i
¼
1
F
i
:
ð
1
Þ
The
“
stacking
”
part of the procedure is the summing, and
the
“
sliding
”
(in parameter space) refers to the fact that the
F
i
that are summed do not all come from the same
template.
Summing the detection statistic values is not the only
way to combine the results from the coherent searches; see
for instance
[4,24,25]
. Independently of the way that this is
done, this type of search is usually referred to as a
“
semicoherent search.
”
Important variables for this type
of search are the coherent time baseline of the segments
T
coh
, the number of segments used
N
seg
, the total time
spanned by the data
T
obs
, the grids in parameter space, and
the detection statistic used to rank the parameter space
cells. For a stack-slide search in Gaussian noise,
N
seg
×
2
̄
F
follows a chi-squared distribution with
4
N
seg
degrees of
freedom,
χ
2
4
N
seg
. These parameters are summarized in
Table
I
. The grids in frequency and spin-down are each
described by a single parameter, the grid spacing, which is
constant over the search range. The same frequency grid
spacings are used for the coherent searches over the
segments and for the incoherent summing. The spin-down
spacing for the incoherent summing,
δ
_
f
, is finer than that
used for the coherent searches,
δ
_
f
c
, by a factor
γ
. The
notation used here is consistent with that used in previous
observational papers
[2,3]
.
The sky grid is approximately uniform on the celestial
sphere projected on the ecliptic plane. The tiling is a
hexagonal covering of the unit circle with hexagons
’
edge
length
d
:
d
ð
m
sky
Þ¼
1
f
ffiffiffiffiffiffiffiffiffi
m
sky
p
πτ
E
;
ð
2
Þ
with
τ
E
≃
0
.
021
s being half of the light travel time across
the Earth and
m
sky
a constant which controls the resolution
of the sky grid. The sky grids are constant over 5 Hz bands
and the spacings are the ones associated through Eq.
(2)
to
the highest frequency in each 5 Hz. The resulting number of
templates used to search 50 mHz bands as a function of
frequency is shown in Fig.
1
.
This search leverages the computing power of the
Einstein@Home project, which is built upon the BOINC
(Berkeley Open Infrastructure for Network Computing)
architecture
[26
–
28]
: a system that exploits the idle time on
volunteer computers to solve scientific problems that
require large amounts of computer power. The search is
split into work units (WUs) sized to keep the average
Einstein@Home volunteer computer busy for about
8 CPU hours. Each WU performs
1
.
5
×
10
11
semicoherent
searches, one for each of the templates in 50 mHz band, the
entire spin-down range and 118 points in the sky. Out of
the semicoherent detection statistic values computed for
the
1
.
5
×
10
11
templates, it returns to the Einstein@Home
server only the highest 10000 values. A total of
1
.
9
×
10
6
WUs are necessary to cover the entire parameter space. The
total number of templates searched is
3
×
10
17
.
A. The ranking statistic
Two detection statistics are used in the search:
ˆ
β
S
=
GLtL
and
2
̄
F
.
ˆ
β
S
=
GLtL
is the ranking statistic which defines the
TABLE I. Search parameters rounded to the first decimal
figure.
T
ref
is the reference time that defines the frequency
and frequency derivative values.
Parameter
Value
T
coh
210 hr
T
ref
1132729647.5 GPS s
N
seg
12
δ
f
8
.
3
×
10
−
7
Hz
δ
_
f
c
1
.
3
×
10
−
11
Hz
=
s
γ
100
m
sky
1
×
10
−
3
FIG. 1. Number of searched templates in 50 mHz band as a
function of frequency. The sky resolution increases with fre-
quency causing the variation in the number of templates.
N
f
×
N
_
f
∼
1
.
3
×
10
9
, where
N
f
and
N
_
f
are the number of
f
and
_
f
templates searched in 50 mHz bands. The total number of
templates searched between 20 and 100 Hz is
3
×
10
17
.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
96,
122004 (2017)
122004-2
top-candidate list; it is a line- and transient-robust statistic
that tests the signal hypothesis against a noise model which,
in addition to Gaussian noise, also includes single-detector
continuous or transient spectral lines. Since the distribution
of
ˆ
β
S
=
GLtL
is not known in closed form even in Gaussian
noise, when assessing the significance of a candidate
against Gaussian noise, we use the average
2
F
statistic
over the segments,
2
̄
F
[23]
; see Eq.
(1)
. This is in essence,
at every template point, the log-likelihood of having a
signal with the shape given by the template versus having
Gaussian noise.
Built from the multi- and single-detector
ˆ
F
statistics,
ˆ
β
S
=
GLtL
is the log
10
of
ˆ
B
S
=
GLtL
, the full definition of which is
given by Eq. (23) of
[29]
. This statistic depends on a few
tuning parameters that we describe in the remainder of the
paragraph for the reader interested in the technical details:
A transition-scale parameter
ˆ
F
ð
0
Þ
is used to tune the
behavior of the
ˆ
β
S
=
GLtL
statistic to match the performance
of the standard average
2
̄
F
statistic in Gaussian noise while
still statistically outperforming it in the presence of con-
tinuous or transient single-detector spectral disturbances.
Based on injection studies of fake signals in Gaussian-noise
data, we set an average
2
̄
F
transition scale of
ˆ
F
ð
0
Þ
¼
65
.
826
. According to Eq. (67) of
[30]
, with
N
seg
¼
12
this
2
̄
F
value corresponds to a Gaussian false-alarm probability
of
10
−
9
. Furthermore, we assume equal-odds priors
between the various noise hypotheses (
“
L
”
for line,
“
G
”
for Gaussian,
“
tL
”
for transient line).
B. Identification of undisturbed bands
Even after the removal of disturbed data caused by
spectral artifacts of known origin, the statistical properties
of the results are not uniform across the search band. In
what follows we concentrate on the subset of the signal-
frequency bands having reasonably uniform statistical
properties, or containing features that are not immediately
identifiable as detector artifacts. This comprises the large
majority of the search parameter space.
Our classification of
“
clean
”
versus
“
disturbed
”
bands
has no pretense of being strictly rigorous, because strict
rigor here is neither useful nor practical. The classification
serves the practical purpose of discarding from the analysis
regions in parameter space with evident disturbances and
must not dismiss detectable real signals. The classification
is carried out in two steps: an automated identification
of undisturbed bands and a visual inspection of the
remaining bands.
An automatic procedure, described in Sec. II F of
[31]
,
identifies as undisturbed the 50-mHz bands whose maxi-
mum density of outliers in the
f
−
_
f
plane and average
2
̄
F
are well within the bulk distribution of the values for
these quantities in the neighboring frequency bands. This
procedure identifies 1233 of the 1600 50-mHz bands as
undisturbed. The remaining 367 bands are marked as
potentially disturbed, and in need of visual inspection.
A scientist performs the visual inspection by looking at
various distributions of the
ˆ
β
S
=
GLtL
statistic over the entire
sky and spin-down parameter space in the 367 potentially
disturbed 50-mHz bands. She ranks each band with an
integer score 0,1,2 ranging from
“
undisturbed
”
(0) to
“
disturbed
”
(2). A band is considered
“
undisturbed
”
if
the distribution of detection statistic values does not show a
visible trend affecting a large portion of the
f
−
_
f
plane. A
band is considered
“
mildly disturbed
”
if there are outliers in
the band that are localized in a small region of the
f
−
_
f
plane. A band is considered
“
disturbed
”
if there are outliers
that are not well localized in the
f
−
_
f
plane.
Figure
2
shows the
ˆ
β
S
=
GLtL
for each type of band.
Figure
3
shows the
ˆ
β
S
=
GLtL
for a band that harbors a fake
signal injected in the data to verify the detection pipelines.
In the latter case, the detection statistic is elevated in a small
region around the signal parameters.
Based on this visual inspection, 1% of the bands between
20 and 100 Hz are marked as
“
disturbed
”
and excluded
from the current analysis. A further 6% of the bands are
marked as
“
mildly disturbed.
”
These bands contain features
that cannot be classified as detector disturbances without
further study; therefore, these are included in the analysis.
Figure
4
shows the highest values of the detection
statistic in half-Hz signal-frequency bands compared to
the expectations. The set of candidates from which the
highest detection statistic values are picked does not
include the 50-mHz signal-frequency bands that stem
entirely from fake data, from the cleaning procedure, or
that were marked as disturbed. Two 50-mHz bands that
contained a hardware injection
[32]
were also excluded, as
the high amplitude of the injected signal caused it to
dominate the list of candidates recovered in those bands. In
this paper we refer to the candidates with the highest value
of the detection statistic as the
loudest
candidates.
The highest expected value from Gaussian noise over
N
trials
independent trials of
2
̄
F
is determined
1
by numerical
integration of the probability density function given, for
example, by Eq.
(7)
of
[33]
. Fitting to the distribution of the
highest
2
̄
F
values suggests that
N
trials
≃
N
templ
, with
N
templ
being the number of templates searched.
The
p
value for the highest
2
̄
F
measured in any half-Hz
band searched with
N
trials
independent trials is obtained by
integrating the expected noise distribution (
χ
2
4
N
seg
given in
Sec.
III
) between the observed value and infinity, as done in
Eq.
(6)
of
[33]
. The distribution of these
p
values is shown
in Fig.
5
and it is not consistent with what we expect from
Gaussian noise across the measured range. Therefore, we
cannot exclude the presence of a signal in this data based on
this distribution alone, as was done in
[3]
.
1
After a simple change of variable from
2
̄
F
to
N
seg
×
2
̄
F
.
FIRST LOW-FREQUENCY EINSTEIN@HOME ALL-SKY
...
PHYSICAL REVIEW D
96,
122004 (2017)
122004-3
IV. HIERARCHICAL FOLLOW UP
Since the significance of candidates is not consistent
with what we expect from Gaussian noise only, we must
investigate
“
significant
”
candidates to determine if they are
FIG. 2. On the vertical axis and color-coded is the
ˆ
β
S
=
GLtL
in
three 50-mHz bands. The top band was marked as
“
undisturbed.
”
The middle band is an example of a
“
mildly disturbed band.
”
The
bottom band is an example of a
“
disturbed band.
”
FIG. 3. This is an example of an
“
undisturbed band
”
but
containing a fake signal. On the
z
axis and color coded is the
ˆ
β
S
=
GLtL
.
FIG. 4. Highest
2
̄
F
value (also referred to as the
2
̄
F
of the
loudest candidate) in every half-Hz band as a function of band
frequency. Since the number of templates increases with fre-
quency, so does the highest
2
̄
F
. The highest expected
2
̄
F
1
σ
ð
2
σ
Þ
over
N
trials
independent trials is indicated by the darker
(faded) band. Two half-mHz bands have
2
̄
F
values greater than the
axes boundaries. The half-Hz bands beginning at 33.05 Hz and
35.55 Hz have loudest
2
̄
F
values of 159 and 500, respectively, due
to features in the 33.3 Hz and 35.75 Hz 50-mHz bands which were
marked
“
mildly disturbed
”
in the visual inspection.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
96,
122004 (2017)
122004-4
produced by a signal or by a detector disturbance. This is
done using a hierarchical approach similar to what was
used for the hierarchical follow-up of subthreshold candi-
dates from the Einstein@Home S6 all-sky search
[2]
.
At each stage of the hierarchical follow-up a semi-
coherent search is performed, the top ranking candidates
are marked and then searched in the next stage. If the data
harbors a real signal, the significance of the recovered
candidate will increase with respect to the significance that
it had in the previous stage. On the other hand, if the
candidate is not produced by a continuous-wave signal, the
significance is not expected to increase consistently over
the successive stages.
The hierarchical approach used in this search consists of
four stages. This is the smallest number of stages within
which we could achieve a fully coherent search, given the
available computing resources. Directly performing a fully
coherent follow-up of all significant candidates from the
all-sky search would have been computationally unfeasible.
A. Stage 0
We bundle together candidates from the all-sky search
that can be ascribed to the same root cause. This clustering
step is a standard step in a multistage approach
[2]
: Both a
loud signal and a loud disturbance produce high values of
the detection statistic at a number of different template grid
points, and it is a waste of compute cycles to follow up each
of these independently.
We apply a clustering procedure that associates together
multiple candidates close to each other in parameter space,
and assigns them the parameters of the loudest among
them, the seed. We use a new procedure with respect to
[2]
that adapts the cluster size to the data and checks for
consistency of the cluster volume with what is expected
from a signal
[34]
. A candidate must have a
ˆ
β
S
=
GLtL
>
5
.
5
to be a cluster seed. This threshold is chosen such that only
a handful of candidates per 50 mHz would be selected if the
data were consistent with Gaussian noise. In this search,
there are
15
×
10
6
candidates with
ˆ
β
S
=
GLtL
>
5
.
5
. A lower
threshold of
ˆ
β
S
=
GLtL
>
4
.
0
is applied to candidates that can
be included in a cluster. If a cluster has at least two
occupants (including the seed), the seed is marked for
follow-up. In total, 35963 seeds are marked for follow-up.
The
ˆ
β
S
=
GLtL
values of these candidates are shown in Fig.
6
as well as their distribution in frequency.
Monte Carlo studies, using simulated signals added into
the data, are conducted to determine how far from the signal
parameters a signal candidate is recovered. These signals
are simulated at a fixed strain amplitude for which most
have
ˆ
β
S
=
GLtL
⪆
10
.
0
. We find that 1282 of 1294 signal
candidates recovered after clustering (99%) are recovered
within
8
<
:
Δ
f
¼
9
.
25
×
10
−
5
Hz
Δ
_
f
¼
4
.
25
×
10
−
11
Hz
=
s
Δ
sky
≃
4
.
5
sky grid points
ð
3
Þ
of the signal parameters. This confidence region
2
defines
the parameter space around each candidate which will be
searched in the first stage of the hierarchical follow-up. For
weaker signals the confidence associated with this uncer-
tainty region decreases. For signals close to the threshold
used here, namely with
ˆ
β
S
=
GLtL
between 5.5 and 10, the
FIG. 5. Distribution of
p
values, with binomial uncertainties,
for the highest detection statistic values measured in half-Hz
bands (circles) and expected from pure Gaussian noise (line). We
note that the measured
p
values for the highest
2
̄
F
in the
33.05 Hz and 35.55 Hz bands are not shown because they are
outside of the
x
axis boundaries.
FIG. 6. Candidates that are followed up in stage 1: the
distribution of their detection statistic values
ˆ
β
S
=
GLtL
(left) and
their distribution as a function of frequency (right).
2
We pick 99% confidence rather than, say, 100%, because to
reach the 100% confidence level would require an increase in
a containment region too large for the available computing
resources.
FIRST LOW-FREQUENCY EINSTEIN@HOME ALL-SKY
...
PHYSICAL REVIEW D
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122004 (2017)
122004-5
detection confidence only drops by a few percent (see
bottom panel of Fig.
7
and last row of Table II in
[34]
).
B. Stage 1
In this stage we search a volume of parameter space
[Eqs.
(3)
] around each cluster seed. We fix the run time per
candidate to be 4 hr on an average CPU of the ATLAS
computing cluster
[35]
. This yields an optimal search setup
having a coherent baseline of 500 hr, with 5 segments and
the grid spacings shown in Table
II
. We use the same
ranking statistic as the original search,
ˆ
β
S
=
GLtL
, with tunings
updated for
N
seg
¼
5
.
For the population of simulated signals that passed the
previous stage, stage 0, 1268 of 1282 (99%) are recovered
within the uncertainty region
8
<
:
Δ
f
¼
1
.
76
×
10
−
5
Hz
Δ
_
f
¼
9
.
6
×
10
−
12
Hz
=
s
Δ
sky
≃
0
.
23
Δ
sky
stage
0
:
ð
4
Þ
From each of the 35963 follow-up searches we record
the most significant candidate in
ˆ
β
S
=
GLtL
. The distribution
of these is shown in Fig.
7
. A threshold at
ˆ
β
S
=
GLtL
¼
6
.
0
,
derived from Monte Carlo studies, is applied to select the
candidates to consider in the next stage. There are 14456
candidates above this threshold.
C. Stage 2
In this stage we search a volume of parameter space
[Eqs.
(4)
] around each candidate from stage 1. We fix the
run time per candidate to be 4 hr on an average CPU of the
ATLAS computing cluster
[35]
. This yields an optimal
search setup having a coherent baseline of 1260 hr, with 2
segments and the grid spacings shown in Table
II
. We use a
different ranking statistic from the original search, because
with 2 segments the transient line veto is not useful. Instead
we use the ranking statistic
ˆ
β
S
=
GL
≔
log
10
ˆ
B
S
=
GL
, intro-
duced in
[30]
and previously used in
[3]
, with tunings
updated for
N
seg
¼
2
.
For the population of signals that passed the previous
stage, 1265 of 1268 (
>
99%
) are recovered within the
uncertainty region
8
<
:
Δ
f
¼
8
.
65
×
10
−
6
Hz
Δ
_
f
¼
7
.
8
×
10
−
12
Hz
=
s
Δ
sky
≃
0
.
81
Δ
sky
stage
1
:
ð
5
Þ
From each of the follow-up searches we record the most
significant candidate in
ˆ
β
S
=
GL
. The distribution of these is
shown in Fig.
8
. A threshold at
ˆ
β
S
=
GL
¼
6
.
0
is applied to
determine what candidates to consider in the next stage.
There are 8486 candidates above threshold.
TABLE II. Search parameters for each stage. The follow-up
stages are stages 1, 2, and 3. Also shown are the parameters for
stage 0, taken from Table
I
.
T
coh
hr
N
seg
δ
f
Hz
δ
_
f
c
Hz
=
s
γ
m
sky
Stage 0 210 12
8
.
3
×
10
−
7
1
.
3
×
10
−
11
100
1
×
10
−
3
Stage 1 500
5
6
.
7
×
10
−
7
2
.
9
×
10
−
12
80
8
×
10
−
6
Stage 2 1260
2
1
.
9
×
10
−
7
9
.
3
×
10
−
13
30
1
×
10
−
6
Stage 3 2512
1
6
.
7
×
10
−
8
9
.
3
×
10
−
14
1
4
×
10
−
7
FIG. 7. Detection statistic of the loudest candidate from each
stage 1 search: the distribution of their detection statistic values
ˆ
β
S
=
GLtL
(left) and their distribution as a function of frequency
(right). 411 candidates have
ˆ
β
S
=
GLtL
values lower than the axes
boundaries on the right plot. The red line marks
ˆ
β
S
=
GLtL
¼
6
.
0
which is the threshold at and above which candidates are passed
on to stage 2.
FIG. 8. Detection statistic of the loudest candidate from each
stage 2 search: the distribution of their detection statistic values
ˆ
β
S
=
GL
(left) and their distribution as a function of frequency
(right). The red line marks
ˆ
β
S
=
GL
¼
6
.
0
which is the threshold at
and above which candidates are passed on to stage 3.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
96,
122004 (2017)
122004-6
D. Stage 3
In this stage we search a volume of parameter space
[Eqs.
(5)
] around each candidate. We perform a fully
coherent search, with a coherent baseline of 2512 hr.
The grid spacings are shown in Table
II
. We use the same
ranking statistic as the previous stage,
ˆ
β
S
=
GL
, with tunings
updated for
N
seg
¼
1
.
For the population of signals that passed the previous
stage, 1265 of 1265 (
>
99%
) are recovered within the
uncertainty region
8
<
:
Δ
f
¼
7
.
5
×
10
−
6
Hz
Δ
_
f
¼
7
×
10
−
12
Hz
=
s
Δ
sky
≃
0
.
99
Δ
sky
stage
2
:
ð
6
Þ
This uncertainty region assumes candidates are within
the uncertainty regions shown in Eqs.
(3)
,
(4)
, and
(5)
for
each of the corresponding follow-up stages. It is possible
that a strong candidate which is outside these uncertainty
regions would be significant enough to pass through all
follow-up stages. In this case the uncertainty on the signal
parameters would be larger than the uncertainty region
defined in Eq.
(6)
.
From each of the follow-up searches we record the
most significant candidate in
ˆ
β
S
=
GL
. The distribution of
these is shown in Fig.
9
. A threshold at
ˆ
β
S
=
GL
¼
6
.
0
is
applied to determine what candidates require further
study. There are 6349 candidates above threshold. Many
candidates appear to be from the same feature at a
specific frequency. There are 57 distinct narrow frequency
regions at which these 6349 candidates have been
recovered.
E. Doppler modulation off veto
We employ a newly developed Doppler modulation off
(DM-off) veto
[36]
to determine if the surviving candidates
are of terrestrial origin. When searching for CW signals, the
frequency of the signal template at any point in time is
demodulated for the Doppler effect from the motion of the
detectors around the Earth and around the Sun. If this
demodulation is disabled, a candidate of astrophysical origin
would not be recovered with the same significance. In
contrast, a candidate of terrestrial origin could potentially
becomemoresignificant. Thisisthe basisofthe DM-offveto.
For each candidate, the search range of the DM-off
searches includes all detector frequencies that could have
contributed to the original candidate, accounting for
_
f
and
Doppler corrections. The
_
f
range includes the original
all-sky search range, and extends into large positive values
of
_
f
to allow for a wider range of detector artifact behavior.
For a candidate to pass the DM-off veto it must be that its
2
F
DM-off
≤
2
F
thr
DM-off
. The
2
F
thr
DM-off
is picked to be safe,
i.e. to not veto any signal candidate with
2
F
DM-on
in the
range of the candidates under consideration. In particular
we find that for candidates with
2
F
DM-on
<
500
, after the
third follow-up,
2
F
thr
DM-off
¼
62
. The threshold increases for
candidates with
2
F
DM-on
>
500
, scaling linearly with the
candidates
2
F
DM-on
(see Fig. 4 of
[36]
).
As described in
[36]
, the DM-off search is first run using
data from both detectors and a search grid which is ten
times coarser in
f
and
_
f
than the stage 3 search. The coarser
search grid is used to minimize computational cost. 653 of
the 6349 candidates pass the
2
F
thr
DM-off
threshold. These
surviving candidates undergo another similar search,
except that the search is performed separately on the data
from each of the LIGO detectors. We search each detector
separately because a detector artifact present only in one
detector may still pass the previous, multidetector search, as
its significance is
“
diluted
”
by the clean data of the other
detector. 101 candidates survive, and undergo a final DM-
off search stage. This search uses the fine grid parameters of
the stage 3 search (Table
II
), covers the parameter space
which resulted in the largest
2
F
DM-off
from the previous
DM-off steps, and is performed three times, once using
both detectors jointly and once for each of the two LIGO
detectors. For a candidate to survive this stage it has to pass
all
three stage 3 searches.
Four candidates survive the full DM-off veto. Such veto
is designed to be safe, i.e. not falsely dismiss real signals.
However, its false alarm rate for noise disturbances is not
fully characterized because very little is known about such
weak and rare spectral disturbances, which this type of deep
search unveils. This means that we cannot exclude that the
four surviving candidates are in fact noise disturbances.
The parameters of the candidates, after the third follow-up,
are given in Table
III
. The
2
F
DM-off
values are also given in
this table.
FIG. 9. Detection statistic of the loudest candidate from each
stage 3 search: the distribution of their detection statistic values
ˆ
β
S
=
GL
(left) and their distribution as a function of frequency
(right). The red line marks
ˆ
β
S
=
GL
¼
6
.
0
which is the threshold
below which candidates are discarded.
FIRST LOW-FREQUENCY EINSTEIN@HOME ALL-SKY
...
PHYSICAL REVIEW D
96,
122004 (2017)
122004-7