Results of the deepest all-sky survey for continuous gravitational waves on LIGO S6
data running on the Einstein@Home volunteer distributed computing project
The LIGO Scientific Collaboration and The Virgo Collaboration
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
spindown we constrain farther (closer) objects to higher (lower) ellipticities.
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 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 pre-
sented 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 spec-
tral 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 neigh-
bouring and undisturbed bands. Table A 2 identifies
these bands.
III. THE SEARCH
The search described in this paper targets nearly
monochromatic gravitational wave signals as described
for example by Eqs. 1-4 of [7]. Various emission mecha-
nisms could generate such a signal as reviewed in Section
IIA of [11]. In interpreting our results we will consider a
spinning compact object with a fixed, non-axisymmetric
mass quadrupole, described by an ellipticity
.
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 com-
bined by summing (stacking) the detection statistic val-
ues from the segments (sliding), one per segment (
F
i
),
and this determines the value of the core detection statis-
tic:
F
:=
1
N
seg
N
seg
∑
i
=1
F
i
.
(1)
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 “semi-
coherent search”. So stack-slide searches are a type of
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
χ
2
4
N
seg
chi-squared distribution with
4
N
seg
degrees of freedom. 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
arXiv:1606.09619v2 [gr-qc] 2 Jul 2016
2
,
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
=
α
and
r
= cos
δ
)
of the grid points in the northern equatorial hemisphere sky
for the band 50-60Hz (left panel) and for the band 110-120Hz
(right panel).
α
is the right ascension coordinate and
δ
the
declination coordinate. One can clearly see the higher den-
sity in the
−
0
.
5
≤
δ
≤
0
.
5 equatorial region and the higher
density (
∝
f
2
) of grid points at higher frequencies. The south-
ern hemispheres looks practically identical to the respective
northern ones.
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
(
m
sky
) =
1
f
√
m
sky
πτ
E
,
(2)
with
m
sky
= 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 spac-
ings 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
spac-
ings equal to
d
(0
.
3) – see Fig.1. The reason for the equa-
torial “patching” with a denser sky grid is to improve the
sensitivity of the search: the sky resolution actually de-
pends 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 6 hours. Each WU searches a 50 mHz band, the
entire spindown 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 nec-
essary to cover the entire parameter space. The total
number of templates searched is 6
.
3
×
10
16
.
FIG. 2. Number of searched templates in 50-mHz bands. The
variation with frequency is due to the increasing sky resolu-
tion.
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 to-
tal number of templates searched between 50 and 510 Hz is
6
.
3
×
10
16
.
1. The ranking statistic
The search was actually carried out in separate Ein-
stein@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 to spectral lines in the data [21]. In particu-
lar, 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
̂
O
SL
which is the odds ratio
of having a signal versus having a spectral line; and a
general line-robust statistic,
̂
O
SGL
, that tests the signal
hypothesis against a Gaussian noise + 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 statistically outperforms them
when the noise is a mixture of both [21].
We combine the 2
F
-ranked results with the
̂
O
SL
-
ranked results to produce a single list of candidates
ranked according to the general line-robust statistic
̂
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 appli-
cation 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
̂
O
SGL
through Eq. 61 of
3
T
coh
60 hrs
T
ref
960499913.5 GPS sec
N
seg
90
δf
1
.
6
×
10
−
6
Hz
δ
̇
f
c
5
.
8
×
10
−
11
Hz/s
γ
230
m
sky
0.3 + equatorial patch
TABLE I. Search parameters rounded to the first decimal figure.
T
ref
is the reference time that defines the frequency and
frequency derivative values.
[21]:
ln
̂
O
SGL
= ln
̂
o
SL
+
̂
F −
̂
F
′′
max
−
ln
(
e
̂
F
∗
−
̂
F
′′
max
+
〈
̂
r
X
e
̂
F
X
−
̂
F
′′
max
〉)
,
(3)
with the angle-brackets indicating the average with re-
spect to detectors (
X
) and
̂
F
=
N
seg
F
(4)
̂
F
X
=
N
seg
F
X
(5)
̂
F
′′
max
≡
max
(
̂
F
∗
,
̂
F
X
+ ln
̂
r
X
)
(6)
̂
F
∗
≡
̂
F
(0)
∗
−
ln
̂
o
LG
(7)
̂
F
(0)
∗
≡
ln
c
N
seg
∗
with
c
∗
set to 20
.
64
(8)
̂
o
LG
=
∑
X
̂
o
X
LG
(9)
̂
r
X
≡
̂
o
X
LG
̂
o
LG
/N
det
(10)
̂
p
L
≡
̂
o
LG
1 +
̂
o
LG
(11)
where
̂
o
X
LG
is the assumed prior probability of a spectral
line occuring in any frequency bin of detector X,
̂
p
L
is
the line prior estimated from the data,
N
det
= 2 is the
number of detectors, and
̂
o
SL
is an assumed prior prob-
ability of a line being a signal (set arbitrarily to 1; its
specific value does not affect the ranking statistic). Fol-
lowing the reasoning of Eq. 67 of [21], with
N
seg
= 90
we set
c
∗
= 20
.
64 corresponding to a Gaussian false-
alarm probability of 10
−
9
and an average 2
F
transition
scale of
∼
6 (
F
(0)
∗
∼
3). The
̂
o
X
LG
values are estimated
from the data as described in Section VI.A of [21] in
50-mHz bands with a normalized-SFT-power threshold
P
X
thr
=
P
thr
(
p
FA
= 10
−
9
,N
X
SFT
∼
6000)
≈
1
.
08. For
every 50-mHz band the list of candidates from the 2
F
-
ranked run is merged with the list from the
̂
O
SL
-ranked
run and duplicate candidates are considered only once.
The resulting list is ranked by the newly-computed
̂
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].
2. 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. This still leaves us with the majority of
the search parameter space while allowing us to use meth-
ods that rely on theoretical modelling of the significance
in the statistical analysis of the results. Our classifica-
tion of “clean” vs. “disturbed” bands has no pretence
of being strictly rigorous, because strict rigour 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 spindown 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. Fig. 3 shows the
̂
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 harbours 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 be-
tween 50 and 510 Hz are marked as “disturbed”. Of
these, 34% were given by all visual inspectors rankings
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.