of 34
Results of the deepest all-sky survey for continuous gravitational waves
on LIGO S6 data running on the Einstein@Home volunteer
distributed computing project
B. P. Abbott
etal.
*
(LIGO Scientific Collaboration and Virgo Collaboration)
(Received 6 July 2016; published 18 November 2016)
We report results of a deep all-sky search for periodic gravitational waves from isolated neutron stars in
data from the S6 LIGO science run. The search was possible thanks to the computing power provided by
the volunteers of the Einstein@Home distributed computing 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. At the frequency of best strain sensitivity, between 170.5 and 171 Hz we set a
90% confidence upper limit of
5
.
5
×
10
25
, while at the high end of our frequency range, around 505 Hz,
we achieve upper limits
10
24
. At 230 Hz we can exclude sources with ellipticities greater than
10
6
within 100 pc of Earth with fiducial value of the principal moment of inertia of
10
38
kg m
2
. If we assume a
higher (lower) gravitational wave spin-down we constrain farther (closer) objects to higher (lower)
ellipticities.
DOI:
10.1103/PhysRevD.94.102002
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 LIGO
s
sixth science (S6) run. A number of all-sky searches have
been carried out on LIGO data,
[2
11]
, of which
[5,7,10]
also ran on Einstein@Home. The search presented here
covers frequencies from 50 Hz through 510 Hz and
frequency derivatives from
3
.
39
×
10
10
Hz
=
s through
2
.
67
×
10
9
Hz
=
s. In this 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 (WA) and the other in
Livingston (LA) separated by a 3000-km baseline
[12]
.
The last science run (S6)
[13]
of this network before the
upgrade towards the advanced LIGO configuration
[14]
took place between July 2009 and October 2010. The
analysis in this paper uses a subset of this data: from GPS
949469977 (2010 Feb 6 05
39:22 UTC) through GPS
971529850 (2010 Oct 19 13
23:55 UTC), selected for good
strain sensitivity
[15]
. Since interferometers sporadically
fall out of operation (
lose lock
) due to environmental or
instrumental disturbances or for scheduled maintenance
periods, the data set is not contiguous and each detector has
a duty factor of about 50%
[16]
.
As done in
[7]
, frequency bands known to contain
spectral disturbances have been removed from the analysis.
Actually, the data has been substituted with Gaussian noise
with the same average power as that in the neighboring and
undisturbed bands. Table
III
identifies these bands.
III. THE SEARCH
The search described in this paper targets nearly mono-
chromatic gravitational wave signals as described for
example by Eqs. 1
4of
[7]
. Various emission mechanisms
could generate such a signal as reviewed in Sec. II A of
[11]
. In interpreting our results we will consider a spinning
compact object with a fixed, nonaxisymmetric mass quad-
rupole, described by an ellipticity
ε
.
We perform a stack-slide type of search using the GCT
(Global Correlation Transform) method
[17,18]
. In a stack-
slide search the data is partitioned in segments and each
segment is searched with a matched-filter method
[19]
. The
results from these coherent searches are combined by
summing (stacking) the detection statistic values from
the segments (sliding), 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
There are different ways to combine the single-segment
F
i
values, but independently of the way that this is done, this
type of search is usually referred to as a
semicoherent
*
Full author list given at end of the article.
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 3.0 License
. Further distri-
bution of this work must maintain attribution to the author(s) and
the published article
s title, journal citation, and DOI.
PHYSICAL REVIEW D
94,
102002 (2016)
2470-0010
=
2016
=
94(10)
=
102002(34)
102002-1
Published by the American Physical Society
search
. So stack-slide searches are a type of 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
χ
2
4
N
seg
chi-squared
distribution with
4
N
seg
degrees of freedom. These param-
eters 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 sum-
ming. 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
[20]
and in the GCT
methods papers cited above.
The sky grid is the union of two grids: one is uniform
over the projection of the celestial sphere onto the equa-
torial plane, and the tiling (in the equatorial plane) is
approximately square with sides of length
d
ð
m
sky
Þ¼
1
f
ffiffiffiffiffiffiffiffiffi
m
sky
p
πτ
E
;
ð
2
Þ
with
m
sky
¼
0
.
3
and
τ
E
0
.
021
s being half of the light
travel time across the Earth. As was done in
[7]
, the sky-
grids are constant over 10 Hz bands and the spacings are the
ones associated through Eq.
(2)
to the highest frequency
f
in the range. The other grid is limited to the equatorial
region (
0
α
2
π
and
0
.
5
δ
0
.
5
), with constant
right ascension
α
and declination
δ
spacings equal to
d
ð
0
.
3
Þ
see Fig.
1
. The reason for the equatorial
patch-
ing
with a denser sky grid is to improve the sensitivity of
the search: the sky resolution actually depends on the
ecliptic latitude and the uniform equatorial grid under-
resolves particularly in the equatorial region. The resulting
number of templates used to search 50 mHz bands as a
function of frequency is shown in Fig.
2
.
The search is split into work-units (WUs) sized to keep
the average Einstein@Home volunteer computer busy for
about six hours. Each WU searches a 50 mHz band, the
entire spin-down range and 13 points in the sky, corre-
sponding to
4
.
9
×
10
9
templates out of which it returns only
the top 3000. A total of 12.7 million WUs are necessary to
cover the entire parameter space. The total number of
templates searched is
6
.
3
×
10
16
.
A. The ranking statistic
The search was actually carried out in separate
Einstein@Home runs that used different ranking statistics
to define the top-candidate-list, reflecting different stages in
the development of a detection statistic robust with respect
TABLE I. Search parameters rounded to the first decimal
figure.
T
ref
is the reference time that defines the frequency
and frequency derivative values.
T
coh
60 hours
T
ref
960499913.5 GPS sec
N
seg
90
δ
f
1
.
6
×
10
6
Hz
δ
_
f
c
5
.
8
×
10
11
Hz
=
s
γ
230
m
sky
0
.
3
þ
equatorial patch
0.2
0.4
0.6
0.8
1
30
210
60
240
90
270
120
300
150
330
180
0
50-60 Hz band sky grid
0.2
0.4
0.6
0.8
1
30
210
60
240
90
270
120
300
150
330
180
0
110-120 Hz band sky grid
FIG. 1. Polar plots (
r
,
θ
plots with
θ
¼
α
and
r
¼
cos
δ
) of the
grid points in the northern equatorial hemisphere sky for the band
50
60 Hz (left panel) and for the band 110
120 Hz (right panel).
α
is the right ascension coordinate and
δ
the declination
coordinate. One can clearly see the higher density in the
0
.
5
δ
0
.
5
equatorial region and the higher density (
f
2
)
of grid points at higher frequencies. The southern hemispheres
looks practically identical to the respective northern ones.
FIG. 2. Number of searched templates in 50 mHz bands. The
variation with frequency is due to the increasing sky resolution.
N
f
×
N
_
f
3
.
7
×
10
8
, 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 50 and 510 Hz is
6
.
3
×
10
16
.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
94,
102002 (2016)
102002-2
to spectral lines in the data
[21]
. In particular, three ranking
statistics were used: the average
2
F
statistic over the
segments,
2
F
, which in essence at every template point
is the likelihood of having a signal with the shape given by
the template versus having Gaussian noise; the line-veto
statistic
ˆ
O
SL
which is the odds ratio of having a signal
versus having a spectral line; and a general line-robust
statistic,
ˆ
O
SGL
, that tests the signal hypothesis against a
Gaussian noise
þ
spectral line noise model. Such a statistic
can match the performance of both the standard average
2
F
statistic in Gaussian noise and the line-veto statistic in
presence of single-detector spectral disturbances and sta-
tistically outperforms them when the noise is a mixture of
both
[21]
.
We combine the
2
F
-ranked results with the
ˆ
O
SL
-ranked
results to produce a single list of candidates ranked
according to the general line-robust statistic
ˆ
O
SGL
.We
now explain how this is achieved. Alongside the detection
statistic value and the parameter space cell coordinates of
each candidate, the Einstein@Home application also
returns the single-detector
2
F
X
values (
X
indicates the
detector). These are used to compute, for every candidate of
any run, the
ˆ
O
SGL
through Eq. 61 of
[21]
ln
ˆ
O
SGL
¼
ln
ˆ
o
SL
þ
ˆ
F
ˆ
F
00
max
ln
ð
e
ˆ
F

ˆ
F
00
max
þh
ˆ
r
X
e
ˆ
F
X
ˆ
F
00
max
;
ð
3
Þ
with the angle-brackets indicating the average with respect
to detectors (
X
) and
ˆ
F
¼
N
seg
F
ð
4
Þ
ˆ
F
X
¼
N
seg
F
X
ð
5
Þ
ˆ
F
00
max
max
ð
ˆ
F

;
ˆ
F
X
þ
ln
ˆ
r
X
Þð
6
Þ
ˆ
F

ˆ
F
ð
0
Þ

ln
ˆ
o
LG
ð
7
Þ
ˆ
F
ð
0
Þ

ln
c
N
seg

with
c

set to
20
.
64
ð
8
Þ
ˆ
o
LG
¼
X
X
ˆ
o
X
LG
ð
9
Þ
ˆ
r
X
ˆ
o
X
LG
ˆ
o
LG
=N
det
ð
10
Þ
ˆ
p
L
ˆ
o
LG
1
þ
ˆ
o
LG
;
ð
11
Þ
where
ˆ
o
X
LG
is the assumed prior probability of a spectral line
occurring in any frequency bin of detector X,
ˆ
p
L
is the line
prior estimated from the data,
N
det
¼
2
is the number of
detectors, and
ˆ
o
SL
is an assumed prior probability of a line
being a signal (set arbitrarily to 1; its specific value does not
affect the ranking statistic). Following the reasoning of
Eq. 67 of
[21]
, with
N
seg
¼
90
we set
c

¼
20
.
64
corre-
sponding to a Gaussian false-alarm probability of
10
9
and
an average
2
F
transition scale of
6
(
F
ð
0
Þ

3
). The
ˆ
o
X
LG
values are estimated from the data as described in Sec. VI.
Aof
[21]
in 50-mHz bands with a normalized-SFT-power
threshold
P
X
thr
¼
P
thr
ð
p
FA
¼
10
9
;N
X
SFT
6000
Þ
1
.
08
.
For every 50 mHz band the list of candidates from the
2
F
-ranked run is merged with the list from the
ˆ
O
SL
-ranked
run and duplicate candidates are considered only once. The
resulting list is ranked by the newly computed
ˆ
O
SGL
and the
top 3000 candidates are kept. This is our result-set, and it is
treated in a manner that is very similar to
[3]
.
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. This still leaves us with the majority of the
search parameter space while allowing us to use methods
that rely on theoretical modeling of the significance in the
statistical analysis of the results. Our classification of
clean
vs
disturbed
bands has no pretence 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 real signals.
The classification is carried out in two steps: a visual
inspection and a refinement on the visual inspection.
The visual inspection is performed by three scientists
who each look at various distributions of the detection
statistics over the entire sky and spin-down parameter space
in 50 mHz bands. They rank each band with an integer
score 0,1,2,3 ranging from
undisturbed
(0) to
disturbed
(3). A band is considered
undisturbed
if all three rankings
are 0. The criteria agreed upon for ranking are that the
distribution of detection statistic values should not show a
visible trend affecting a large portion of the
f
_
f
plane
and, if outliers exist in a small region, outside this region
the detection statistic values should be within the expected
ranges. Figure
3
shows the
ˆ
O
SGL
for three bands: two were
marked as undisturbed and the other as disturbed. One of
the bands contains the
f
_
f
parameter space that harbors a
fake signal injected in the data to verify the detection
pipelines. The detection statistic is elevated in a small
region around the signal parameters. The visual inspection
procedure does not mark as disturbed bands with such
features.
Based on this visual inspection 13% of the bands
between 50 and 510 Hz are marked as
disturbed
.Of
these, 34% were given by all visual inspectors rankings
RESULTS OF THE DEEPEST ALL-SKY SURVEY FOR
...
PHYSICAL REVIEW D
94,
102002 (2016)
102002-3
smaller than 3, i.e. they were only marginally disturbed.
Further inspection
rehabilitated
42% of these. As a result
of this refinement in the selection procedure we exclude
from the current analysis 11% of the searched frequencies
(Table
IV
).
Figure
4
shows the highest values of the detection
statistic in half-Hz signal-frequency bands compared to
the expectations. The set of candidates that the highest
detection statistic values are picked from, 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. In this paper we refer to the candidates with
the highest value of the detection statistic as the
loudest
candidates.
The loudest expected value 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
[20]
. For this search, we estimate that
N
trials
0
.
87
N
templ
, with
N
templ
being the number of
templates searched.
As a uniform measure of significance of the highest
2
F
value across bands that were searched with different values
of
N
trials
we introduce the critical ratio CR defined as the
deviation of the measured highest
2
F
from the expected
value, measured in units of the standard deviation
CR
2
F
meas
2
F
exped
σ
exped
:
ð
12
Þ
The highest and most significant detection statistic value
from our search is
2
F
¼
8
.
6
at a frequency of about
52.76 Hz with a CR
¼
29
. This is due to a fake signal. The
second highest value of the detection statistic is 7.04 at a
frequency of about 329.01 Hz corresponding to a CR of
4.6. The second highest-CR candidate has a
2
F
of 6.99, is
FIG. 3. On the z-axis and color-coded is the
ˆ
O
SGL
in three
50 mHz bands. The top band was marked as
undisturbed
. The
middle band is an example of a
disturbed band
. The bottom
band is an example of an
undisturbed band
but containing a
signal, a fake one, in this case.
FIG. 4. Highest values of
2
̄
F
in every half-Hz band as a
function of band frequency. Since the number of templates
increases with frequency so does the loudest
2
̄
F
.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
94,
102002 (2016)
102002-4
at 192.16 Hz and has a CR
¼
4
.
8
. The CR values are
plotted in Fig.
5
, and the distribution in Fig.
6
.
Sorting loudest candidates from half-Hz bands according
to detection statistic values is not the same as sorting them
according to CR. The reason for this is that the number of
templates is not the same for all half-Hz bands. This is due
to the grid spacings decreasing with frequency (Eq.
(2)
and
to the fact that, as previously explained, some 50 mHz
bands have been excluded from the current analysis and
hence some half-Hz bands comprise results from fewer
than ten 50 mHz bands. Figure
7
gives the fill-level of each
half-Hz band, i.e. how many 50 mHz bands have contrib-
uted candidates to the analysis out of ten. We use the CR as
a measure of the significance because it folds in correctly
the effect of varying number of templates in the half-
Hz bands.
After excluding the candidate due to the fake signal, in
this data we see no evidence of a signal: the distribution of p
values associated with every measured half-Hz band loud-
est is consistent with what we expect from noise-only
across the measured range (Fig.
8
). In particular we note
two things: 1) the two candidates at CR
¼
4
.
6
and
CR
¼
4
.
8
are not significant when we consider how many
half-Hz bands we have searched, and 2) there is no
population of low significance candidates deviating from
the expectation of the noise-only case. The p value for the
loudest measured in any half-Hz band searched with an
FIG. 5. Highest values of the significance (CR) in every half-Hz
band as a function of band frequency. Since the significance folds
in the expected value for the loudest
2
̄
F
and its standard
deviation, the significance of the loudest in noise does not
increase with frequency. Our results are consistent with this
expectation.
FIG. 6. Histogram of the highest values of the significance CR
in every half-Hz band.
FIG. 7. The fraction of 50 mHz bands (in signal frequency)
which contribute to the results in every half-Hz band. As
explained in the text, some bands are excluded because they
are all from fake data or because they are marked as disturbed by
the visual inspection. The list of excluded bands is given in
Table
IV
.
FIG. 8. p values for the loudest in half-Hz bands of our data
(histogram bars) and expected distribution of pure noise data for
reference (black markers).
RESULTS OF THE DEEPEST ALL-SKY SURVEY FOR
...
PHYSICAL REVIEW D
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effective number of independent trials
N
trials
¼
0
.
87
N
trials
is
obtained by integrating Eq. 6 of
[20]
between the observed
value and infinity.
IV. UPPER LIMITS
The search did not reveal any continuous gravitational
wave signal in the parameter volume that was searched. We
hence set frequentist upper limits on the maximum gravi-
tational wave amplitude consistent with this null result in
half-Hz bands:
h
90%
0
(f).
h
90%
0
(f) is the GW amplitude such
that 90% of a population of signals with parameter values in
our search range would have produced a candidate louder
than what was observed by our search. This is the criterion
hereafter referred to as
detection
.
Evaluating these upper limits with injection-and-
recovery Monte Carlo simulations in every half-Hz band
is too computationally intensive. So we perform them in a
subset of 50 bands and infer the upper limit values in the
other bands from these. The 50 bands are evenly spaced in
the search frequency range. For each band
j
¼
1
...
50
,we
measure the 90% upper limit value corresponding to
different detection criteria. The different detection criteria
are defined by different CR values for the assumed
measured loudest. The first CR bin, CR
0
, is for CR values
equal to or smaller than 0, the next bins are for
i<
CR
i
ð
i
þ
1
Þ
with
i
¼
1
...
5
. Correspondingly we have
h
90%
;j
0
;
CR
i
for
each band. For every detection criteria and every band we
determine the sensitivity depth
[22]
, and by averaging these
sensitivity depths over the bands we derive a sensitivity
depth for every detection criteria:
D
90%
CR
i
¼
1
=
50
P
j
D
90%
;j
CR
i
.
We use these to set upper limits in the bands
k
where we
have not performed injection-and-recovery simulations as
h
90%
0
ð
f
k
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
S
h
ð
f
k
Þ
p
D
90%
CR
i
ð
k
Þ
;
ð
13
Þ
where CR
i
ð
k
Þ
is the significance bin of the loudest
candidate of the
k
th band and
S
h
ð
f
k
Þ
the power spectral
density of the data (measured in
1
=
ffiffiffiffiffiffi
Hz
p
). The values of the
sensitivity depths range between
D
90%
CR
6
33
ð
1
=
ffiffiffiffiffiffi
Hz
p
Þ
and
D
90%
CR
0
37
ð
1
=
ffiffiffiffiffiffi
Hz
p
Þ
. The uncertainties on the upper limit
values introduced by this procedure are
10%
of the
nominal upper limit value. We represent this uncertainty as
a shaded region around the upper limit values in Fig.
9
. The
upper limit values are also provided in tabular form in the
FIG. 9. 90% confidence upper limits on the gravitational wave amplitude of signals with frequency within half-Hz bands, from the
entire sky and within the spin-down range of the search. The light red markers denote half-Hz bands where the upper limit value does not
hold for all frequencies in that interval. A list of the excluded frequencies is given in the Appendix. Although not obvious from the
figure, due to the quality of the data we were not able to analyze the data in some half-Hz bands, so there are some points missing in the
plot. For reference we also plot the upper limit results from two searches: one on the same data (Powerflux)
[2]
and on contemporary
data from the Virgo detector (frequency Hough)
[4]
. The Powerflux points are obtained by rescaling the best (crosses) and worst-case
(dots) upper limit values as explained in the text. It should be noted that the Powerflux upper limits are set at 95% rather than 90% but
refer to 0.25 Hz bands rather than half-Hz.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
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102002 (2016)
102002-6
Appendix in Table
II
. We do not set upper limits in half-Hz
bands where the results are entirely produced with fake data
inserted by the cleaning procedure described in Sec.
II
.
Upper limits for such bands will not appear in Table
II
nor
in Fig.
9
. There also exist 50 mHz bands that include
contributions from fake data as a result of the cleaning
procedure or that have been excluded from the analysis
because they were marked as disturbed by the visual
inspection procedure described in Sec.
III B
. We mark
the half-Hz bands which host these 50 mHz bands with a
different colour (light red) in Fig.
9
. In Table
IV
in the
Appendix we provide a complete list of such 50-mHz
bands because the upper limit values do not apply to those
50-mHz bands. Finally we note that, due to the cleaning
procedure, there exist signal frequency bands where the
search results
might
have contributions from fake data. We
list these signal-frequency ranges in Table
V
. For com-
pleteness this table also contains the cleaned bands of
Table
IV
, under the column header
all fake data
.
V. CONCLUSIONS
Our upper limits are the tightest ever placed for this set of
target signals. The smallest value of the GW amplitude
upper limit is
5
.
5
×
10
25
in the band 170.5
171 Hz.
Figure
9
shows the upper limit values as a function of
search frequency. We also show the upper limits from
[2]
,
another all-sky search on S6 data, rescaled according to
[23]
to enable a direct comparison with ours. Under the
assumption that the sources are uniformly distributed in
space, our search probes a volume in space a few times
larger than that of
[2]
. It should however be noted that
[2]
examines a much broader parameter space than the one
presented here. The Virgo VSR2 and VSR4 science runs
were contemporary to the S6 run and more sensitive at low
frequency with respect to LIGO. The Virgo data were
analyzed in search of continuous signals from the whole
sky in the frequency range 20
128 Hz and a narrower spin-
down range than that covered here, with
j
_
f
j
10
10
Hz
=
s
[4]
. Our sensitivity is comparable to that achieved by that
search and improves on it above 80 Hz.
Following
[24]
, we define the fraction
x
of the spin-down
rotational energy emitted in gravitational waves. The star
s
ellipticity necessary to sustain such emission is
ε
ð
f; x
_
f
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5
c
5
32
π
4
G
x
_
f
If
5
s
;
ð
14
Þ
where
c
is the speed of light,
G
is the gravitational constant,
f
is the GW frequency and
I
the principal moment of inertia
of the star. Correspondingly,
x
_
f
is the spin-down rate that
accounts for the emission of GWs, and this is why we refer
to it as the GW spin-down. The gravitational wave
amplitude
h
0
at the detector coming from a GW source
like that of Eq.
(14)
, at a distance
D
from Earth is
h
0
ð
f; x
_
f; D
Þ¼
1
D
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5
GI
2
c
3
x
_
f
f
s
:
ð
15
Þ
Based on this last equation, we can use our GW amplitude
upper limits to bound the minimum distance for compact
objects emitting continuous gravitational waves under
different assumptions on the object
s ellipticity (i.e.
FIG. 10. Gravitational wave amplitude upper limits recast as curves in the
f
x
j
_
f
j
plane for sources at given distances and having
assumed
I
¼
10
38
kg m
2
.
f
is the signal frequency and
x
j
_
f
j
is the gravitational-wave spin-down, i.e. the fraction of the actual spin-down
that accounts for the rotational energy loss due to GW emission. Superimposed are curves of constant ellipticity
ε
ð
f;
_
f
j
I
¼
10
38
kg m
2
).
The dotted line at
j
_
f
j
max
indicates the maximum magnitude of searched spin-down.
RESULTS OF THE DEEPEST ALL-SKY SURVEY FOR
...
PHYSICAL REVIEW D
94,
102002 (2016)
102002-7
gravitational wave spin-down). This is shown in Fig.
10
.
We find that for most frequencies above 230 Hz our upper
limits exclude compact objects with ellipticities of
10
6
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
10
38
kg m
2
I
q
(corresponding to GW spin-downs between
10
12
Hz
=
s and
10
11
Hz
=
s) within 100 pc of Earth. Both
the ellipticity and the distance ranges span absolutely
plausible values and could not have been excluded with
other measurements.
We expect the methodology used in this search to serve
as a template for the assessment of Einstein@Home run
results in the future, for example the next Einstein@Home
run, using advanced LIGO data that is being processed as
this paper is written. Results of searches for continuous
wave signals could also be mined further, probing sub-
threshold candidates with a hierarchical series of follow-up
searches. This is not the topic of this paper and might be
pursued in a forthcoming publication.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support
of the Einstein@Home volunteers, of the United States
National Science Foundation for the construction and
operation of the LIGO Laboratory, the Science and
Technology Facilities Council of the United Kingdom,
the Max-Planck-Society, and the State of Niedersachsen/
Germany for support of the construction and operation of
the GEO600 detector, and the Italian Istituto Nazionale di
Fisica Nucleare and the French Centre National de la
Recherche Scientifique for the construction and operation
of the Virgo detector. The authors also gratefully acknowl-
edge the support of the research by these agencies and by
the Australian Research Council, the International Science
Linkages program of the Commonwealth of Australia, the
Council of Scientific and Industrial Research of India, the
Istituto Nazionale di Fisica Nucleare of Italy, the Spanish
Ministerio de Educación y Ciencia, the Conselleria
d
Economia Hisenda i Innovació of the Govern de les
Illes Balears, the Foundation for Fundamental Research on
Matter supported by the Netherlands Organization for
Scientific Research, the Polish Ministry of Science and
Higher Education, the FOCUS Programme of Foundation
for Polish Science, the Royal Society, the Scottish Funding
Council, the Scottish Universities Physics Alliance, The
National Aeronautics and Space Administration, the
Carnegie Trust, the Leverhulme Trust, the David and
Lucile Packard Foundation, the Research Corporation,
and the Alfred P. Sloan Foundation.
This document has been assigned LIGO Laboratory
document No.
LIGO-P1600156-v22
.
APPENDIX: TABULAR DATA
1. Upper limit values
TABLE II. First frequency of each half Hz signal frequency band in which we set upper limits and upper limit value for that band.
f
(in Hz)
h
90%
0
×
10
25
f
(in Hz)
h
90%
0
×
10
25
f
(in Hz)
h
90%
0
×
10
25
f
(in Hz)
h
90%
0
×
10
25
50.063
70
.
3

12
.
8
50.563
68
.
4

12
.
5
51.063
69
.
3

12
.
6
51.563
67
.
5

12
.
4
52.063
66
.
9

12
.
2
53.063
57
.
6

10
.
5
53.563
58
.
9

10
.
9
54.063
55
.
3

10
.
1
54.563
54
.
0

9
.
9
55.063
55
.
7

10
.
2
55.563
53
.
3

9
.
8
56.063
50
.
9

9
.
3
56.563
51
.
8

9
.
5
57.063
47
.
5

8
.
7
57.563
46
.
9

8
.
6
58.063
47
.
1

8
.
6
58.563
51
.
5

9
.
4
61.063
44
.
8

8
.
2
61.563
37
.
4

6
.
9
62.063
36
.
5

6
.
7
62.563
36
.
0

6
.
6
63.063
36
.
3

6
.
6
63.563
33
.
8

6
.
2
64.063
30
.
6

5
.
6
64.563
29
.
8

5
.
4
65.063
31
.
5

5
.
9
65.563
30
.
8

5
.
7
66.063
28
.
3

5
.
2
66.563
26
.
5

4
.
8
67.063
26
.
5

4
.
9
67.563
27
.
3

5
.
0
68.063
25
.
7

4
.
7
68.563
27
.
4

5
.
0
69.063
24
.
8

4
.
5
69.563
25
.
5

4
.
7
70.063
25
.
7

4
.
7
70.563
23
.
6

4
.
3
71.063
22
.
8

4
.
2
71.563
23
.
6

4
.
3
72.063
23
.
1

4
.
2
72.563
23
.
3

4
.
2
73.063
22
.
0

4
.
0
73.563
23
.
9

4
.
5
74.063
21
.
1

3
.
8
74.563
20
.
6

3
.
8
75.063
19
.
3

3
.
5
75.563
20
.
8

3
.
8
76.063
19
.
0

3
.
5
76.563
18
.
3

3
.
4
77.063
18
.
1

3
.
3
77.563
18
.
5

3
.
4
78.063
18
.
8

3
.
4
78.563
17
.
4

3
.
2
79.063
17
.
0

3
.
1
79.563
18
.
1

3
.
3
80.063
18
.
0

3
.
3
80.563
16
.
9

3
.
1
81.063
18
.
7

3
.
4
81.563
16
.
3

3
.
0
82.063
15
.
5

2
.
8
82.563
15
.
4

2
.
8
83.063
15
.
7

2
.
9
83.563
15
.
0

2
.
8
84.063
14
.
6

2
.
7
84.563
13
.
9

2
.
5
85.063
14
.
0

2
.
6
85.563
13
.
7

2
.
5
86.063
13
.
9

2
.
5
86.563
13
.
8

2
.
5
87.063
13
.
3

2
.
4
87.563
13
.
1

2
.
4
88.063
12
.
9

2
.
4
88.563
13
.
0

2
.
4
89.063
12
.
4

2
.
3
89.563
12
.
3

2
.
3
90.063
12
.
6

2
.
3
(Table continued)
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
94,
102002 (2016)
102002-8
TABLE II.
(Continued)
f
(in Hz)
h
90%
0
×
10
25
f
(in Hz)
h
90%
0
×
10
25
f
(in Hz)
h
90%
0
×
10
25
f
(in Hz)
h
90%
0
×
10
25
90.563
12
.
0

2
.
2
91.063
11
.
8

2
.
2
91.563
11
.
6

2
.
1
92.063
11
.
4

2
.
1
92.563
11
.
3

2
.
1
93.063
11
.
2

2
.
1
93.563
11
.
1

2
.
0
94.063
11
.
3

2
.
1
94.563
11
.
1

2
.
0
95.063
11
.
6

2
.
2
95.563
10
.
8

2
.
0
96.063
10
.
8

2
.
0
96.563
10
.
6

1
.
9
97.063
10
.
4

1
.
9
97.563
10
.
5

1
.
9
98.063
10
.
2

1
.
9
98.563
11
.
1

2
.
1
99.063
10
.
5

1
.
9
99.563
10
.
3

1
.
9
100.063
10
.
5

1
.
9
100.563
9
.
9

1
.
8
101.063
9
.
8

1
.
8
101.563
9
.
5

1
.
7
102.063
9
.
9

1
.
8
102.563
9
.
9

1
.
8
103.063
9
.
6

1
.
8
103.563
9
.
5

1
.
7
104.063
9
.
4

1
.
7
104.563
9
.
3

1
.
7
105.063
9
.
6

1
.
8
105.563
9
.
3

1
.
7
106.063
9
.
3

1
.
7
106.563
9
.
4

1
.
7
107.063
9
.
1

1
.
7
107.563
9
.
7

1
.
8
108.063
9
.
3

1
.
7
108.563
9
.
0

1
.
7
109.063
8
.
7

1
.
6
109.563
8
.
5

1
.
5
110.063
9
.
0

1
.
7
110.563
8
.
6

1
.
6
111.063
8
.
6

1
.
6
111.563
8
.
8

1
.
6
112.063
8
.
5

1
.
5
112.563
8
.
3

1
.
5
113.063
9
.
2

1
.
7
113.563
8
.
6

1
.
6
114.063
8
.
4

1
.
5
114.563
8
.
4

1
.
6
115.063
8
.
0

1
.
5
115.563
7
.
9

1
.
4
116.063
8
.
1

1
.
5
116.563
8
.
6

1
.
6
117.063
9
.
0

1
.
7
117.563
8
.
7

1
.
6
118.063
10
.
5

1
.
9
118.563
8
.
7

1
.
6
121.063
9
.
1

1
.
7
121.563
8
.
2

1
.
5
122.063
8
.
3

1
.
5
122.563
8
.
2

1
.
5
123.063
8
.
5

1
.
6
123.563
8
.
3

1
.
5
124.063
8
.
0

1
.
4
124.563
7
.
4

1
.
4
125.063
7
.
5

1
.
4
125.563
8
.
3

1
.
5
126.063
8
.
1

1
.
5
126.563
8
.
4

1
.
5
127.063
7
.
6

1
.
4
127.563
7
.
7

1
.
4
128.063
7
.
4

1
.
4
128.563
7
.
8

1
.
4
129.063
8
.
0

1
.
5
129.563
8
.
2

1
.
5
130.063
7
.
7

1
.
4
130.563
7
.
9

1
.
4
131.063
7
.
2

1
.
3
131.563
6
.
8

1
.
2
132.063
7
.
0

1
.
3
132.563
6
.
9

1
.
3
133.063
6
.
7

1
.
2
133.563
6
.
6

1
.
2
134.063
6
.
4

1
.
2
134.563
6
.
3

1
.
2
135.063
6
.
5

1
.
2
135.563
6
.
5

1
.
2
136.063
6
.
6

1
.
2
136.563
6
.
3

1
.
2
137.063
6
.
6

1
.
2
137.563
6
.
5

1
.
2
138.063
6
.
4

1
.
2
138.563
6
.
4

1
.
2
139.063
6
.
5

1
.
2
139.563
6
.
2

1
.
1
140.063
6
.
3

1
.
1
140.563
6
.
2

1
.
1
141.063
6
.
1

1
.
1
141.563
6
.
5

1
.
2
142.063
6
.
2

1
.
1
142.563
6
.
3

1
.
2
143.063
6
.
3

1
.
1
143.563
6
.
0

1
.
1
144.063
6
.
2

1
.
1
144.563
6
.
0

1
.
1
145.563
5
.
9

1
.
1
146.063
5
.
9

1
.
1
146.563
6
.
3

1
.
2
147.063
6
.
3

1
.
2
147.563
5
.
8

1
.
1
148.063
5
.
8

1
.
1
148.563
5
.
9

1
.
1
149.063
5
.
8

1
.
1
149.563
5
.
7

1
.
0
150.063
5
.
7

1
.
0
150.563
6
.
0

1
.
1
151.063
5
.
7

1
.
0
151.563
5
.
7

1
.
0
152.063
5
.
7

1
.
1
152.563
5
.
7

1
.
0
153.063
5
.
8

1
.
1
153.563
5
.
8

1
.
1
154.063
5
.
7

1
.
0
154.563
5
.
7

1
.
1
155.063
5
.
9

1
.
1
155.563
5
.
9

1
.
1
156.063
6
.
0

1
.
1
156.563
6
.
0

1
.
1
157.063
5
.
7

1
.
0
157.563
6
.
0

1
.
1
158.063
5
.
8

1
.
1
158.563
5
.
7

1
.
0
159.063
5
.
8

1
.
1
159.563
5
.
6

1
.
0
160.063
5
.
8

1
.
1
160.563
5
.
7

1
.
0
161.063
5
.
7

1
.
0
161.563
5
.
6

1
.
0
162.063
5
.
9

1
.
1
162.563
5
.
7

1
.
0
163.063
5
.
7

1
.
0
163.563
5
.
7

1
.
0
164.063
5
.
6

1
.
0
164.563
5
.
8

1
.
1
165.063
5
.
7

1
.
0
165.563
5
.
7

1
.
0
166.063
5
.
7

1
.
0
166.563
5
.
5

1
.
0
167.063
5
.
7

1
.
0
167.563
5
.
6

1
.
0
168.063
5
.
6

1
.
0
168.563
5
.
5

1
.
0
169.063
5
.
5

1
.
0
169.563
5
.
5

1
.
0
170.063
5
.
6

1
.
0
170.563
5
.
5

1
.
0
171.063
5
.
5

1
.
0
171.563
5
.
5

1
.
0
172.063
5
.
5

1
.
0
172.563
5
.
7

1
.
0
173.063
5
.
6

1
.
0
173.563
5
.
7

1
.
0
174.063
5
.
5

1
.
0
174.563
5
.
5

1
.
0
175.063
5
.
5

1
.
0
175.563
5
.
6

1
.
0
176.063
6
.
2

1
.
1
176.563
6
.
4

1
.
2
177.063
6
.
4

1
.
2
177.563
6
.
5

1
.
2
178.063
6
.
5

1
.
2
178.563
7
.
2

1
.
3
181.063
7
.
2

1
.
3
181.563
7
.
0

1
.
3
182.063
6
.
7

1
.
2
182.563
6
.
9

1
.
3
183.063
6
.
6

1
.
2
183.563
6
.
4

1
.
2
184.063
6
.
4

1
.
2
184.563
6
.
1

1
.
1
185.063
6
.
3

1
.
2
185.563
6
.
2

1
.
1
186.063
6
.
2

1
.
1
186.563
6
.
3

1
.
2
187.063
6
.
2

1
.
1
187.563
6
.
5

1
.
2
188.063
6
.
8

1
.
2
188.563
6
.
9

1
.
3
189.063
8
.
0

1
.
5
189.563
7
.
8

1
.
4
190.063
7
.
0

1
.
3
190.563
6
.
5

1
.
2
191.063
6
.
1

1
.
1
191.563
6
.
2

1
.
1
192.063
6
.
7

1
.
3
192.563
6
.
1

1
.
1
193.063
5
.
8

1
.
1
193.563
5
.
8

1
.
1
194.063
6
.
3

1
.
2
194.563
6
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1
.
1
195.063
6
.
1

1
.
1
195.563
6
.
2

1
.
1
196.063
6
.
5

1
.
2
196.563
6
.
3

1
.
2
197.063
6
.
4

1
.
2
197.563
6
.
9

1
.
3
198.063
6
.
8

1
.
2
198.563
6
.
8

1
.
2
199.063
7
.
9

1
.
4
199.563
8
.
5

1
.
6
200.063
7
.
1

1
.
3
200.563
7
.
3

1
.
3
201.063
7
.
5

1
.
4
201.563
7
.
0

1
.
3
202.063
6
.
7

1
.
2
202.563
6
.
8

1
.
2
203.063
6
.
4

1
.
2
203.563
5
.
7

1
.
1
204.063
5
.
8

1
.
1
204.563
6
.
0

1
.
1
(Table continued)
RESULTS OF THE DEEPEST ALL-SKY SURVEY FOR
...
PHYSICAL REVIEW D
94,
102002 (2016)
102002-9
TABLE II.
(Continued)
f
(in Hz)
h
90%
0
×
10
25
f
(in Hz)
h
90%
0
×
10
25
f
(in Hz)
h
90%
0
×
10
25
f
(in Hz)
h
90%
0
×
10
25
205.063
5
.
8

1
.
1
205.563
5
.
7

1
.
0
206.063
5
.
6

1
.
0
206.563
6
.
0

1
.
1
207.063
5
.
9

1
.
1
207.563
5
.
8

1
.
1
208.063
6
.
4

1
.
2
208.563
6
.
7

1
.
2
209.063
6
.
3

1
.
2
209.563
6
.
8

1
.
2
210.063
6
.
8

1
.
2
210.563
6
.
0

1
.
1
211.063
5
.
8

1
.
1
211.563
5
.
7

1
.
0
212.063
5
.
6

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0
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5
.
8

1
.
1
213.063
5
.
7

1
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0
213.563
5
.
9

1
.
1
214.063
5
.
5

1
.
0
214.563
5
.
8

1
.
1
215.063
5
.
9

1
.
1
215.563
5
.
8

1
.
1
216.063
5
.
5

1
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0
216.563
5
.
5

1
.
0
217.063
5
.
5

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.
0
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5
.
7

1
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0
218.063
5
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5

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218.563
5
.
8

1
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1
219.063
5
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5

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0
219.563
5
.
7

1
.
0
220.063
5
.
7

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0
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5
.
5

1
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221.063
5
.
6

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5
.
6

1
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0
222.063
5
.
7

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.
0
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5
.
8

1
.
1
223.063
6
.
2

1
.
1
223.563
6
.
2

1
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1
224.063
6
.
2

1
.
1
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5
.
8

1
.
1
225.063
5
.
8

1
.
1
225.563
5
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8

1
.
1
226.063
5
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7

1
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0
226.563
5
.
7

1
.
0
227.063
6
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0

1
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1
227.563
5
.
8

1
.
1
228.063
5
.
9

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.
1
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5
.
9

1
.
1
229.063
6
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1

1
.
1
229.563
5
.
9

1
.
1
230.063
6
.
2

1
.
1
230.563
5
.
8

1
.
1
231.063
5
.
9

1
.
1
231.563
5
.
8

1
.
1
232.063
5
.
7

1
.
1
232.563
5
.
9

1
.
1
233.063
6
.
2

1
.
1
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6
.
3

1
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1
234.063
6
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1

1
.
1
234.563
5
.
9

1
.
1
235.063
5
.
9

1
.
1
235.563
5
.
8

1
.
1
236.063
5
.
7

1
.
0
236.563
5
.
7

1
.
0
237.063
5
.
7

1
.
0
237.563
5
.
9

1
.
1
238.063
5
.
9

1
.
1
238.563
5
.
8

1
.
1
240.563
6
.
0

1
.
1
241.063
5
.
9

1
.
1
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5
.
9

1
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1
242.063
5
.
9

1
.
1
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6
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0

1
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1
243.063
6
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2

1
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1
243.563
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0

1
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1
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5
.
9

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244.563
5
.
9

1
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1
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0

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5
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8

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1
246.063
5
.
8

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1
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5
.
8

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1
247.063
5
.
9

1
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1
247.563
6
.
0

1
.
1
248.063
5
.
9

1
.
1
248.563
6
.
2

1
.
1
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6
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1

1
.
1
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6
.
4

1
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2
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5
.
9

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1
250.563
6
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0

1
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1
251.063
5
.
8

1
.
1
251.563
5
.
9

1
.
1
252.063
5
.
9

1
.
1
252.563
5
.
8

1
.
1
253.063
5
.
8

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.
1
253.563
5
.
8

1
.
1
254.063
5
.
9

1
.
1
254.563
6
.
1

1
.
1
255.063
5
.
9

1
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1
255.563
6
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1

1
.
1
256.063
6
.
0

1
.
1
256.563
6
.
0

1
.
1
257.063
6
.
6

1
.
2
257.563
6
.
0

1
.
1
258.063
6
.
4

1
.
2
258.563
6
.
2

1
.
1
259.063
6
.
1

1
.
1
259.563
6
.
1

1
.
1
260.063
6
.
0

1
.
1
260.563
6
.
0

1
.
1
261.063
6
.
0

1
.
1
261.563
6
.
0

1
.
1
262.063
6
.
3

1
.
1
262.563
6
.
1

1
.
1
263.063
6
.
2

1
.
1
263.563
6
.
2

1
.
1
264.063
6
.
3

1
.
2
264.563
6
.
1

1
.
1
265.063
6
.
1

1
.
1
265.563
6
.
3

1
.
1
266.063
6
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1

1
.
1
266.563
6
.
4

1
.
2
267.063
6
.
6

1
.
2
267.563
6
.
3

1
.
2
268.063
6
.
4

1
.
2
268.563
6
.
3

1
.
2
269.063
6
.
2

1
.
1
269.563
6
.
2

1
.
1
270.063
7
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0

1
.
3
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6
.
6

1
.
2
271.063
6
.
4

1
.
2
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6
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3

1
.
2
272.063
6
.
6

1
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2
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6
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5

1
.
2
273.063
6
.
7

1
.
2
273.563
6
.
5

1
.
2
274.063
6
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2

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1
274.563
6
.
3

1
.
1
275.063
6
.
3

1
.
1
275.563
6
.
3

1
.
2
276.063
6
.
7

1
.
2
276.563
6
.
5

1
.
2
277.063
6
.
6

1
.
2
277.563
7
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0

1
.
3
278.063
6
.
6

1
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2
278.563
6
.
7

1
.
2
279.063
6
.
8

1
.
3
279.563
7
.
2

1
.
3
280.063
7
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1

1
.
3
280.563
6
.
8

1
.
2
281.063
6
.
9

1
.
3
281.563
7
.
3

1
.
3
282.063
6
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8

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3
282.563
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9

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.
3
283.063
6
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7

1
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2
283.563
6
.
9

1
.
3
284.063
6
.
6

1
.
2
284.563
6
.
6

1
.
2
285.063
6
.
8

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.
3
285.563
6
.
5

1
.
2
286.063
6
.
7

1
.
2
286.563
6
.
6

1
.
2
287.063
6
.
7

1
.
2
287.563
6
.
5

1
.
2
288.063
6
.
6

1
.
2
288.563
6
.
8

1
.
2
289.063
6
.
6

1
.
2
289.563
6
.
7

1
.
2
290.063
6
.
6

1
.
2
290.563
6
.
6

1
.
2
291.063
6
.
7

1
.
2
291.563
6
.
6

1
.
2
292.063
6
.
7

1
.
2
292.563
6
.
6

1
.
2
293.063
6
.
6

1
.
2
293.563
6
.
8

1
.
2
294.063
6
.
9

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.
3
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6
.
6

1
.
2
295.063
6
.
6

1
.
2
295.563
6
.
9

1
.
3
296.063
6
.
9

1
.
3
296.563
6
.
7

1
.
2
297.063
6
.
9

1
.
3
297.563
6
.
7

1
.
2
298.063
6
.
9

1
.
3
298.563
6
.
9

1
.
3
300.563
7
.
1

1
.
3
301.063
7
.
2

1
.
3
301.563
6
.
9

1
.
3
302.063
6
.
9

1
.
3
302.563
7
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1

1
.
3
303.063
7
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1

1
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3
303.563
7
.
3

1
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3
304.063
7
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2

1
.
3
304.563
6
.
9

1
.
3
305.063
7
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0

1
.
3
305.563
7
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2

1
.
3
306.063
7
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1

1
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3
306.563
7
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1

1
.
3
307.063
7
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2

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3
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7
.
2

1
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3
308.063
7
.
2

1
.
3
308.563
7
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3

1
.
3
309.063
7
.
2

1
.
3
309.563
7
.
3

1
.
3
310.063
7
.
4

1
.
4
310.563
7
.
2

1
.
3
311.063
7
.
5

1
.
4
311.563
7
.
6

1
.
4
312.063
7
.
4

1
.
4
312.563
7
.
3

1
.
3
313.063
7
.
3

1
.
3
313.563
7
.
3

1
.
3
314.063
7
.
3

1
.
3
314.563
7
.
5

1
.
4
315.063
7
.
3

1
.
3
315.563
7
.
4

1
.
4
316.063
7
.
8

1
.
4
316.563
7
.
7

1
.
4
317.063
8
.
2

1
.
5
317.563
7
.
8

1
.
4
(Table continued)
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
94,
102002 (2016)
102002-10