First low-frequency Einstein@Home all-sky search for continuous gravitational waves
in Advanced LIGO data
The LIGO Scientific Collaboration and The Virgo Collaboration
et al.
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 [1/
√
Hz]. 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
.
I. INTRODUCTION
In this paper we report the results of a deep all-sky Ein-
stein@Home [1] search for continuous, nearly monochro-
matic 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 gen-
eral 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 (WA) and the other in Liv-
ingston (LA) 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 sen-
sitive 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 dis-
turbances or for scheduled maintenance periods, the data
set is not contiguous and each detector has a duty fac-
tor of about 50%. To remove the effects of instrumental
and environmental spectral disturbances from the anal-
ysis, 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 neighbouring
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
monochromatic gravitational wave signals as described
for example by Eqs. 1-4 of [9]. Various emission mecha-
nisms could generate such a signal, as reviewed in Section
IIA of [15]. In interpreting our results we will consider a
spinning compact object with a fixed, non-axisymmetric
`
=
m
= 2 mass quadrupole, described by an equatorial
ellipticity
.
We perform a stack-slide type of search using the GCT
(Global correlation transform) 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 com-
bined by summing the detection statistic values from the
different segments, one per segment (
F
i
), and this deter-
mines the value of the core detection statistic:
F
:=
1
N
seg
N
seg
∑
i
=1
F
i
.
(1)
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.
arXiv:1707.02669v2 [gr-qc] 14 Jul 2017
2
Parameter
Value
T
coh
210 hr
T
ref
1132729647.5 GPS s
N
seg
12
δf
8
.
3
×
10
−
7
Hz
δ
̇
f
c
1
.
3
×
10
−
11
Hz/s
γ
100
m
sky
1
×
10
−
3
TABLE I. Search parameters rounded to the first decimal
figure.
T
ref
is the reference time that defines the frequency
and frequency derivative values.
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
“semi-coherent 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 sum-
marised in Table I. The grids in frequency and spindown
are each described by a single parameter, the grid spac-
ing, 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
spindown 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 an
hexagonal covering of the unit circle with hexagons’ edge
length
d
:
d
(
m
sky
) =
1
f
√
m
sky
πτ
E
,
(2)
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 Ein-
stein@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 aver-
age Einstein@Home volunteer computer busy for about
8 CPU-hours. Each WU performs 1
.
5
×
10
11
semi-
FIG. 1. Number of searched templates in 50 mHz band as
a function of frequency. The sky resolution increases with
frequency 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
.
coherent searches, one for each of the templates in 50
mHz band, the entire spindown range and 118 points in
the sky. Out of the semicoherent detection statistic val-
ues 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 million 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
top-candidate-list; it is a line- and transient-robust statis-
tic 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
̂
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 re-
mainder of the paragraph for the reader interested in
3
the technical details: A transition-scale parameter
̂
F
(0)
∗
is used to tune the behaviour of the
ˆ
β
S
/
GLtL
statistic
to match the performance of the standard average 2
F
statistic in Gaussian noise while still statistically out-
performing it in the presence of continuous 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
̂
F
(0)
∗
= 65
.
826. According
to Eq. 67 of [30], with
N
seg
= 12 this 2
F
value cor-
responds 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 artefacts of known origin, the statistical prop-
erties 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 statis-
tical properties, or containing features that are not imme-
diately identifiable as detector artefacts. This comprises
the large majority of the search parameter space.
Our classification of “clean” vs. “disturbed” bands
has no pretence of being strictly rigorous, because strict
rigour here is neither useful nor practical. The classi-
fication serves the practical purpose of discarding from
the analysis regions in parameter space with evident dis-
turbances and must not dismiss detectable real signals.
The classification is carried out in two steps: an auto-
mated identification of undisturbed bands and a visual
inspection of the remaining bands.
An automatic procedure, described in Section IIF of
[31], identifies as undisturbed the 50-mHz bands whose
maximum density of outliers in the
f
−
̇
f
plane and aver-
age 2
F
are well within the bulk distribution of the values
for these quantities in the neighbouring 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 spindown parameter space in the 367 po-
tentially 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 “undis-
turbed” 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 dis-
turbed” if there are outliers in the band that are localised
in a small region of the
f
−
̇
f
plane. A band is consid-
ered “disturbed” if there are outliers that are not well
localised in the
f
−
̇
f
plane.
Fig. 2 shows the
ˆ
β
S
/
GLtL
for each type of band. Fig. 3
shows the
ˆ
β
S
/
GLtL
for a band that harbours 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 be-
tween 20 and 100 Hz are marked as “disturbed” and ex-
cluded from the current analysis. A further 6% of the
bands are marked as “mildly disturbed”. These bands
contain features that can not be classified as detector
disturbances without further study, therefore these are
included in the analysis.
Fig. 4 shows the highest values of the detection statis-
tic in half-Hz signal-frequency bands compared to the
expectations. The set of candidates from which the high-
est 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 numer-
ical 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 Section III) between the observed value and in-
finity, 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 can not exclude the presence of a
signal in this data based on this distribution alone, as
was done in [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 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 sub-threshold
candidates 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 harbours a real signal, the significance of the recov-
ered candidate will increase with respect to the signifi-
cance that it had in the previous stage. On the other
1
After a simple change of variable from 2
F
to
N
seg
×
2
F
.
4
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 “undis-
turbed”. The middle band is an example of a “mildly dis-
turbed band”. The bottom band is an example of a “dis-
turbed 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 frequency, so does the highest 2
F
. The highest expected
2
F ±
1
σ
(2
σ
) over
N
trials
independent trials is indicated by
the darker (faded) band. Two half-mHz bands have 2
F
val-
ues 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 dis-
turbed” in the visual inspection.
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
5
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.
T
coh
N
seg
δf
δ
̇
f
c
γ
m
sky
hr
Hz
Hz/s
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
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.
within which we could achieve a fully-coherent search,
given the available computing resources. Directly per-
forming a fully-coherent follow-up of all significant can-
didates from the all-sky search would have been compu-
tationally unfeasible.
A. Stage 0
We bundle together candidates from the all-sky search
that can be ascribed to the same root cause. This clus-
tering step is a standard step in a multi-stage approach
[2]: Both a loud signal and a loud disturbance produce
high values of the detection statistic at a number of dif-
ferent 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 to-
gether multiple candidates close to each other in param-
eter space, and assigns them the parameters of the loud-
est 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 cho-
sen such that only a handful of candidates per 50-mHz
would be selected if the data were consistent with Gaus-
sian noise. In this search, there are 15 million 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 clus-
ter. 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.
FIG. 6. Candidates that are followed-up in stage 1 : the
distribution of their detection statistic values
ˆ
β
S
/
GLtL
(left
plot) and their distribution as a function of frequency (right
plot).
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
∆
f
=
±
9
.
25
×
10
−
5
Hz
∆
̇
f
=
±
4
.
25
×
10
−
11
Hz/s
∆sky
'
4
.
5 sky grid points
(3)
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 uncertainty region decreases. For signals close to
the threshold used here, namely with
ˆ
β
S
/
GLtL
between
5.5 and 10, the detection confidence only drops by a few
2
We pick 99% confidence rather than, say, 100%, because to reach
the 100% confidence level would require an increase in contain-
ment region too large for the available computing resources.
6
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 hours on an average CPU of the
ATLAS computing cluster [35]. This yields an optimal
search set-up having a coherent baseline of 500 hours,
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
= 5.
For the population of simulated signals that passed the
previous stage, stage 0, 1268 of 1282 (99%) are recovered
within the uncertainty region
∆
f
=
±
1
.
76
×
10
−
5
Hz
∆
̇
f
=
±
9
.
6
×
10
−
12
Hz/s
∆sky
'
0
.
23 ∆sky
Stage-0
.
(4)
From each of the 35963 follow-up searches we record
the most significant candidate in
ˆ
β
S
/
GLtL
. The distribu-
tion of these is shown in Fig. 7. A threshold at
ˆ
β
S
/
GLtL
= 6.0, derived from Monte Carlo studies, is applied to se-
lect the candidates to consider in the next stage. There
are 14456 candidates above this threshold.
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 plot) and their distribution as a function
of frequency (right plot). 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
= 6.0 which is the threshold at and above
which candidates are passed on to stage-2.
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 hours on an average CPU
of the ATLAS computing cluster [35]. This yields an op-
timal search set-up having a coherent baseline of 1260
hours, 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 statis-
tic
ˆ
β
S
/
GL
:= log
10
ˆ
B
S
/
GL
, introduced in [30] and previ-
ously used in [3], with tunings updated for
N
seg
= 2.
For the population of signals that passed the previous
stage, 1265 of 1268 (
>
99%) are recovered within the
uncertainty region
∆
f
=
±
8
.
65
×
10
−
6
Hz
∆
̇
f
=
±
7
.
8
×
10
−
12
Hz/s
∆sky
'
0
.
81 ∆sky
Stage-1
.
(5)
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
= 6.0 is applied
to determine what candidates to consider in the next
stage. There are 8486 candidates above threshold.
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 plot) and their distribution as a function of
frequency (right plot). The red line marks
ˆ
β
S
/
GL
= 6.0 which
is the threshold at and above which candidates are passed on
to stage-3.
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 hours.
The grid spacings are shown in Table II. We use the same
ranking statistic as the previous stage,
ˆ
β
S
/
GL
, with tun-
ings updated for
N
seg
= 1.
For the population of signals that passed the previous
stage, 1265 of 1265 (
>
99%) are recovered within the
7
uncertainty region
∆
f
=
±
7
.
5
×
10
−
6
Hz
∆
̇
f
=
±
7
×
10
−
12
Hz/s
∆sky
'
0
.
99 ∆sky
Stage-2
.
(6)
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 re-
gions would be significant enough to pass through all
follow-up stages. In this case the uncertainty on the sig-
nal parameters would be larger than the uncertainty re-
gion 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
= 6.0 is ap-
plied to determine what candidates require further study.
There are 6349 candidates above threshold. Many candi-
dates appear to be from the same feature at a specific fre-
quency. There are 57 distinct narrow frequency regions
at which these 6349 candidates have been recovered.
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 plot) and their distribution as a function of
frequency (right plot). The red line marks
ˆ
β
S
/
GL
= 6.0 which
is the threshold below which candidates are discarded.
E. Doppler Modulation off veto
We employ a newly developed Doppler modulation off
(DM-off) veto [36] to determine if the surviving candi-
dates 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 de-modulation is disabled, a candidate of as-
trophysical origin would not be recovered with the same
significance. In contrast, a candidate of terrestrial origin
could potentially become more significant. This is the
basis of the DM-off veto.
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 orig-
inal all-sky search range, and extends into large positive
values of
̇
f
to allow for a wider range of detector artefact
behaviour.
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 par-
ticular we find that for candidates with 2
F
DM-on
<
500,
after the third follow-up, 2
F
thr
DM-off
= 62. The threshold
increases for candidates with 2
F
DM-on
>
500, scaling lin-
early with the candidates 2
F
DM-on
(see Figure 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. 653 of
the 6349 candidates pass the 2
F
thr
DM-off
threshold. These
surviving candidates undergo another similar search, ex-
cept that the search is performed separately on the data
from each of the LIGO detectors. 101 candidates survive,
and undergo a final DM-off search stage. This search
uses the fine grid parameters of the stage-3 search (Ta-
ble II), covers the parameter space which resulted in the
largest 2
F
DM-off
from the previous DM-off steps, and is
performed using both detectors jointly and each detector
separately. For a candidate to survive this stage it has
to pass
all
three stage-3 searches.
Four candidates survive the full DM-off veto. The pa-
rameters 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.
F. Follow-up in LIGO O2 data
If the signal candidates surviving the O1 search are
standard continuous-wave signals, i.e. continuous wave
signals arising from sources that radiate steadily over
many years, they should be present in data from the Ad-
vanced LIGO’s second observing run (O2) with the same
parameters. We perform a follow-up search using three
months of O2 data, collected from November 30 2016 to
February 28 2017.
The candidate parameters in Table III are translated
to the O2 midtime, which is the reference time of the
new search. The parameter space covered by the search
is determined by the uncertainty on the candidate pa-
rameters in Eq. 5. The frequency region is widened to
account for the spindown uncertainty. The O2 follow-up
covers a frequency range of
±
5
.
15
×
10
−
4
Hz around the
candidates.
The search parameters of the O2 follow-up are given in
Table IV. The expected loudest 2
F
per follow-up search
8
ID
f
[Hz]
α
[rad]
δ
[rad]
̇
f
[Hz/s]
2
F
2
F
H1
2
F
L1
2
F
DM-off
1
58
.
970435900
1
.
87245
−
0
.
51971
−
1
.
081102
×
10
−
9
81
.
4
48
.
5
33
.
4
55
2
62
.
081409292
4
.
98020
0
.
58542
−
2
.
326246
×
10
−
9
81
.
9
45
.
5
39
.
0
52
3
97
.
197674733
5
.
88374
−
0
.
76773
2
.
28614
×
10
−
10
86
.
5
55
.
0
31
.
8
58
4
99
.
220728369
2
.
842702
−
0
.
469603
−
2
.
498113
×
10
−
9
80
.
2
41
.
4
45
.
8
55
TABLE III. Stage-3 follow-up results for each of the 4 candidates that survive the DM-off veto. For illustration purposes
in the 7th and 8th column we show the values of the average single-detector detection statistics. Typically, for signals, the
single-detector values do not exceed the multi-detector 2
F
.
Parameter
Value
T
coh
2160 hrs
T
ref
1168447494.5 GPS sec
N
seg
1
δf
9
.
0
×
10
−
8
Hz
δ
̇
f
c
1
.
1
×
10
−
13
Hz/s
γ
1
m
sky
4
×
10
−
7
TABLE IV. Search parameters, rounded to the first decimal
place, for the follow-up of surviving LIGO O1 candidates in
LIGO O2 data.
T
ref
is the reference time that defines the
frequency and frequency derivative values.
Candidate
Expected 2
F ±
1
σ
Loudest 2
F
recovered
1
85
±
18
44
2
90
±
19
52
3
84
±
18
49
4
77
±
17
47
TABLE V. Highest 2
F
expected after the follow-up in O2
data, if the candidates were due to a signal, compared with the
highest 2
F
recovered from the follow-up. The 2
F
expected
in Gaussian noise data is 52
±
3.
due to Gaussian noise alone, is 52
±
3, assuming indepen-
dent search templates.
If a candidate in Table III were due to a signal, the
loudest 2
F
expected after the follow-up would be the
value given in the second column of Table V. This ex-
pected value is obtained by scaling the 2
F
in Table III
according to the different duration and the different noise
levels between the data set used for the third follow-up
and the O2 data set. The expected 2
F
also folds-in a
conservative factor of 0.9 due to a different mismatch of
the O2 template grid with respect to the template grid
used for the third follow-up. Thus the expected 2
F
in
Table V is a conservative estimate for the minimum 2
F
that we would expect from a signal candidate.
The loudest 2
F
after the follow-up in O2 data is also
given in Table V. The loudest 2
F
recovered for each can-
didate are
≈
2
σ
below the expected 2
F
for a signal can-
didate. The recovered 2
F
are consistent with what is
expected from Gaussian data. We conclude that it is un-
likely that any of the candidates in Table III arises from
a long-lived astronomical source of continuous gravita-
tional waves.
V. RESULTS
The search did not reveal any continuous gravitational
wave signal in the parameter volume that was searched.
We hence set frequentist 90% confidence upper limits on
the maximum gravitational wave amplitude consistent
with this null result in 0
.
5 Hz bands,
h
90%
0
(
f
). Specif-
ically,
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 been detected by our search. We de-
termined the upper limits in bands that were marked as
undisturbed in Section III B. These upper limits may not
hold for frequency bands that were marked as mildly dis-
turbed, which we now consider disturbed as they were
excluded by the analysis. These bands, as well as bands
which were excluded from further analysis, are identified
in Appendix A 3.
Since an actual full scale fake-signal injection-and-
recovery Monte Carlo for the entire set of follow-ups in
every 0
.
5 Hz band is prohibitive, in the same spirit as
[2, 5, 31], we perform such a study in a limited set of trial
bands. We choose 20 half-Hz bands to measure the up-
per limits. If these half-Hz bands include 50-mHz bands
which were not marked undisturbed, no upper limit in-
jections are made in those 50-mHz bands.
The amplitudes of the fake signals bracket the 90%
confidence region typically between 70% and 100%. The
h
0
versus confidence data is fit in this region with a sig-
moid of the form
C
(
h
0
) =
1
1 + exp(
a
−
h
0
b
)
(7)
and the
h
90%
0
value is read-off of this curve. The fitting
procedure
3
yields the best-fit a and b values and the co-
variance matrix. Given the binomial confidence values
uncertainties, using the covariance matrix we estimate
the
h
90%
0
uncertainty.
For each of these frequency bands we determine the
sensitivity depth
D
90%
[39] of the search corresponding
to
h
90%
0
(
f
):
D
90%
:=
√
S
h
(
f
)
h
90%
0
(
f
)
[1
/
√
Hz]
,
(8)
3
We used the
linfit
Matlab routine.
9
20
30
40
50
60
70
80
90
100
search frequency (Hz)
10
25
10
24
h
90%
0
Powerflux O1 search
Time-domain F-stat O1 search
Sky Hough O1 search
Frequency Hough O1 search
Results from this search
FIG. 10. 90% confidence upper limits on the gravitational wave amplitude of continuous gravitational wave signals with
frequency in 0.5 Hz bands and with spindown values within the searched range. The lowest set of points (black circles) are the
results of this search. The empty circles 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. The lighter grey region around the upper limit
points shows the 11% relative difference bracket between upper limits inferred with the procedure described in Section V and
upper limits that would have been derived (at great computational expense) with direct measurements in all half-Hz bands. We
estimate that less than
∼
0
.
5% of the upper limit points would fall outside of this bracket if they were derived with the direct-
measurement method in Gaussian noise. For comparison we also plot the most recent upper limits results in this frequency
range from O1 data obtained with various search pipelines [40]. We note that these searches cover a broader frequency and
spindown range than the search presented here. All upper limits presented here are population-averaged limits over the full
sky and source polarisation.
where
√
S
h
(
f
) is the noise level of the data as a function
of frequency.
As representative of the sensitivity depth of this hi-
erarchical search, we take the average of the measured
depths at different frequencies: 48
.
7 [1
/
√
Hz]. We then
determine the 90% upper limits by substituting this value
in Eq. 8 for
D
90%
.
The upper limit that we get with this procedure, in
general yields a different number compared to the up-
per limit directly measured as done in the twenty test
bands. An 11% relative error bracket comprises the range
of variation observed on the measured sensitivity depths,
including the uncertainties on the single measurements.
So we take this as a generous estimate of the range of
variability of the upper limit values introduced by the
estimation procedure. If the data were Gaussian this
bracket would yield a
∼
0
.
5% probability of a
measured
upper limit falling outside of this bracket.
Figure 10 shows these upper limits as a function of fre-
quency. They are also presented in tabular form in the
Appendix with the uncertainties indicating the range of
variability introduced by the estimation procedure. The
associated uncertainties amount to
∼
20% when also in-
cluding 10% amplitude calibration uncertainty. The most
constraining upper limit in the band 98.5-99 Hz, close to
the highest frequency, where the detector is most sensi-
tive, is 1
.
8
×
10
−
25
. At the lowest end of the frequency
range, at 20 Hz, the upper limit rises to 3
.
9
×
10
−
24
.
In general not all the rotational kinetic energy lost is
due to GW emission. Following [38], we define
x
to be
the fraction of the spindown rotational energy emitted
in gravitational waves. The star’s ellipticity necessary to
10
20
30
40
50
60
70
80
90
100
search frequency (Hz)
10
8
10
7
10
6
10
5
10
4
10
3
10
2
10
1
source ellipticity
1 pc
10 pc
100 pc
1 kpc
10 kpc
max
FIG. 11. Ellipticity
of a source at a distance d emitting con-
tinuous gravitational waves that would have been detected
by this search. The dashed line shows the spin-down ellip-
ticity for the highest magnitude spindown parameter value
searched: 2.6
×
10
−
9
Hz/s. The spin-down ellipticity is the
ellipticity necessary for all the lost rotational kinetic energy
to be emitted in gravitational waves. If we assume that the
observed spin-down is all actual spin-down of the object, then
no ellipticities could be possible above the dashed curve. In
reality the observed and actual spindown could differ due to
radial acceleration of the source. In this case the actual spin-
down of the object may even be larger than the apparent one.
In this case our search would be sensitive to objects with el-
lipticities above the dashed line.
sustain such emission is
(
f,x
̇
f
) =
√
5
c
5
32
π
4
G
x
|
̇
f
|
If
5
,
(9)
where
c
is the speed of light,
G
is the gravitational con-
stant,
f
is the GW frequency and
I
the principal mo-
ment of inertia of the star. Correspondingly,
x
̇
f
is the
spindown rate that accounts for the emission of GWs and
this is why we refer to it as the GW spindown. The grav-
itational wave amplitude
h
0
at the detector coming from
a GW source like that of Eq. 9, at a distance
D
from
Earth is
h
0
(
f,x
̇
f,D
) =
1
D
√
5
GI
2
c
3
x
|
̇
f
|
f
.
(10)
Based on this last equation, we can use the GW ampli-
tude upper limits to bound the minimum distance for
compact objects emitting continuous gravitational waves
under different assumptions on the object’s ellipticity
(i.e. gravitational wave spindown). This is shown in
Fig. 11. Above 55 Hz we can exclude sources with ellip-
ticities larger than 10
−
5
within 100 pc of Earth. Rough
estimates are that there should be of order 10
4
neutron
stars within this volume.
VI. CONCLUSIONS
This search concentrates the computing power of Ein-
stein@Home in a relatively small frequency range at
low frequencies where all-sky searches are significantly
“cheaper” than at higher frequencies. For this reason,
the initial search could be set-up with a very long coher-
ent observation time of 210 hours and this yields a record
sensitivity depth of 48.7 [1/
√
Hz].
The O1 data set in the low frequency range investi-
gated with this search is significantly more polluted by
coherent spectral artefacts than most of the data sets
from the Initial-LIGO science runs. Because of this, even
a relatively high threshold on the detection statistic of the
first search yields tens of thousands of candidates, rather
than just
O
(100). We follow each of them up through
a hierarchy of three further stages at the end of which
O
(7000) survive. After the application of a newly de-
veloped Doppler-modulation-off veto, 4 survive. These
are finally followed up with a fully coherent search using
three months of O2 data, which produces results com-
pletely consistent with Gaussian noise and falls short of
the predictions under the signal hypothesis. We hence
proceed to set upper limits on the intrinsic GW ampli-
tude
h
0
. The hierarchical follow-up procedure presented
here has also been used to follow-up outliers from other
all-sky searches in O1 data with various search pipelines
[40].
The smallest value of the GW amplitude upper limit
is 1
.
8
×
10
−
25
in the band 98.5-99 Hz. Fig. 10 shows the
upper limit values as a function of search frequency. Our
upper limits are the tightest ever placed for this popu-
lation of signals, and are a factor 1.5-2 smaller than the
most recent upper limits [40]. We note that [40] presents
results from four different all-sky search pipelines cover-
ing a broader frequency and spindown range than the one
explored here. The coherent time-baseline for all these
pipelines is significantly shorter than the 210 hours used
by the very first stage of this search. This limits the
sensitivity of those searches but it makes them more ro-
bust to deviations in the signal waveform from the target
waveform, with respect this search.
Translating the upper limits on the GW amplitude
in upper limits on the ellipticity of the GW source,
we find that for frequencies above 55 Hz our results
exclude isolated compact objects with ellipticities of
10
−
5
√
10
38
kg m
2
/I
(corresponding to GW spindowns be-
tween 10
−
14
Hz/s and 10
−
13
Hz/s) or higher, within 100
pc of Earth.
VII. ACKNOWLEDGMENTS
The authors gratefully acknowledge the support of the
United States National Science Foundation (NSF) for
the construction and operation of the LIGO Laboratory
and Advanced LIGO as well as the Science and Tech-