Narrow-band search for gravitational waves from known pulsars
using the second LIGO observing run
B. P. Abbott
etal.
*
(LIGO Scientific Collaboration and Virgo Collaboration)
(Received 22 February 2019; published 27 June 2019)
Isolated spinning neutron stars, asymmetric with respect to their rotation axis, are expected to be sources
of continuous gravitational waves. The most sensitive searches for these sources are based on accurate
matched filtering techniques that assume the continuous wave to be phase locked with the pulsar beamed
emission. While matched filtering maximizes the search sensitivity, a significant signal-to-noise ratio loss
will happen in the case of a mismatch between the assumed and the true signal phase evolution. Narrow-
band algorithms allow for a small mismatch in the frequency and spin-down values of the pulsar while
coherently integrating the entire dataset. In this paper, we describe a narrow-band search using LIGO O2
data for the continuous wave emission of 33 pulsars. No evidence of a continuous wave signal is found, and
upper limits on the gravitational wave amplitude over the analyzed frequency and spin-down ranges
are computed for each of the targets. In this search, we surpass the spin-down limit, namely, the maximum
rotational energy loss due to gravitational waves emission for some of the pulsars already present in the
LIGO O1 narrow-band search, such as J
1400
−
6325
,J
1813
−
1246
,J
1833
−
1034
,J
1952
þ
3252
, and
for new targets such as J
0940
−
5428
and J
1747
−
2809
. For J
1400
−
6325
,J
1833
−
1034
, and
J
1747
−
2809
, this is the first time the spin-down limit is surpassed.
DOI:
10.1103/PhysRevD.99.122002
I. INTRODUCTION
Eleven gravitational wave (GW) signals have so far been
detected by the LIGO
[1,2]
and Virgo GW interferometers
[3]
in their first and second observing runs (O1 and O2,
respectively)
[4]
. All the signals detected so far come from
the coalescence of two compact objects. These signals
belong to the class of
transient signals
, since they are
observed only within a short time window during the
observing run. Ten detections of binary black hole mergers
[4
–
9]
and a detection from a binary neutron star (NS)
merger
[10]
have been accomplished during the first and
second observing runs.
Another class of GW signals potentially observable by
the LIGO and Virgo detectors is the so-called
continuous
wave
(CW). CWs could potentially be present during the
entire data-taking period of the GW detectors. Potential
sources of CWs are isolated spinning NSs asymmetric with
respect to their rotation axis. In the case of an oblate NS,
CWs are emitted at a frequency that is 2 times its rotational
frequency.
Different types of CW searches can be performed
according to the astrophysical scenario in which the NS
is observed. If the NS is a pulsar, an accurate ephemeris
may be available and matched filtering techniques can be
employed to reach, ideally, the best possible sensitivity by
using waveform templates that cover the entire observing
run. These types of searches are referred to as
targeted
searches
. The LIGO and Virgo Collaborations have already
searched for this type of emission from known pulsars
(both isolated and some in binaries)
[11
–
19]
, for which
accurate ephemerides were available. While for NSs
observed as a central compact object of a supernova
remnant or in a binary system, usually accurate ephemeri-
des are not available. In this case, we can pinpoint the
source and look for the CW signal over a wide frequency
range using semicoherent analysis, e.g., dividing the
observing run in several data chunks and looking for a
waveform template in each of them. Such searches are
called
“
directed
”
and offer the possibility to explore a large
number of templates at the price of a lower sensitivity with
respect to targeted searches
[20
–
24]
. Recently, there has
also been a study for a possible deviation of CW signals
from the general relativity model
[25]
by including non-
tensorial modes.
Between targeted and directed searches, we find the
narrow-band
searches. Such pipelines are based on algo-
rithms which allow us to make a full coherent search, and,
at the same time, we are able to deal with a frequency
mismatch between the CW signal and the electromagnetic
inferred value of the order of 500 mHz
[14,26,27]
. Usually,
this will correspond to the evaluation of millions of
waveform templates for each pulsar considered in the
analysis.
*
Full author list given at the end of the article.
PHYSICAL REVIEW D
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=
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=
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© 2019 American Physical Society
Hence, narrow-band searches offer a sensitivity compa-
rable to the one of targeted searches while relaxing the
phase-lock assumption of the CW signal with the NS
rotation. The CW phase locking is indeed a strong
assumption that may prevent the detection of a CW signal.
In fact, a coherent (or targeted) CW search that uses
one year of data has a frequency resolution of about
3
×
10
−
8
Hz. A mismatch between the rotational frequency
inferred from the ephemeris and the CW signal frequency
of this size or larger is enough to drastically reduce the
chance of detection.
A small frequency mismatch may arise for several
physical reasons that usually are parametrized in a fre-
quency mismatch of the form
Δ
f
gw
∼
f
gw
ð
1
þ
δ
Þ
[14]
.In
the case of a differential rotation between the GW engine
and the electromagnetic pulse engine, the factor
δ
will be
proportional to the timescale of some torque which enfor-
ces correlation between the two engines. Another possibil-
ity is that the NS is freely precessing. In this scenario, the
δ
factor will be proportional to the angle between the star
symmetry axis and the star rotation axis
[28]
. In some of the
previous narrow-band searches
[14,26]
, we used a value of
δ
∼
10
−
4
, which can accommodate the previous theoretical
models. However starting from the first narrow-band search
with advanced detector data
[27]
, we explore a frequency/
spin-down range corresponding to
δ
∼
10
−
3
.
Another possibility is that the pulsar ephemerides
provided are not accurate enough to carry on targeted
searches with the required resolution, or they are not
available during the observing time of our detectors.
That is the case for many low-frequency and energetic
pulsars observed in the x- and
γ
-ray bands, such as
J
1833
−
1034
and J
1813
−
1749
. For these reasons, along
with targeted searches, we search for CWs also with
narrow-band searches.
In this paper, we present the narrow-band search for
CWs from 33 known pulsars using LIGO O2 data. In
Sec.
II
, we provide a brief background on the CW signal
model and the algorithm used. In Sec.
III
, we summarize
the main features of the O2 narrow-band analysis, while in
Sec.
IV
, we introduce the pulsars that we have selected for
this search. The results of the search followed by the upper
limits on the signal strain amplitude are discussed in Sec.
V
.
Finally, in Sec.
VI
we draw the conclusion of this work.
II. BACKGROUND
A. The signal
The GW signal emitted by an asymmetric spinning NS
can be written at the detector frame using the formalism
introduced in
[29]
as the real part of
h
ð
t
Þ¼
H
0
ð
H
þ
ð
η
;
ψ
Þ
A
þ
ð
t
Þþ
H
×
ð
η
;
ψ
Þ
A
×
ð
t
ÞÞ
e
2
π
if
gw
ð
t
Þ
t
þ
i
φ
0
;
ð
1
Þ
where
f
gw
ð
t
Þ
is the GW frequency (which incorporates
all the modulation of the signal at the detector frame)
and
φ
0
an initial phase. The polarization amplitudes
H
þ
ð
η
;
ψ
Þ
;H
×
ð
η
;
ψ
Þ
are functions of the ratio of the
polarization ellipse semiminor to semimajor axis
η
and
the polarization angle
ψ
. The functions
A
þ
ð
t
Þ
;A
×
ð
t
Þ
are the
detector responses to the two wave polarizations. These two
functions depend on the detector geographical location and
the
0
;
1
;
2
harmonics of the sidereal rotational fre-
quency of Earth
F
sid
(the inverse of the sidereal day); see
[29]
for more details. In Eq.
(1)
, the amplitude of the GW
H
0
is related to the canonical strain amplitude
h
0
given the
angle between the line of sight and the star rotation axis
ι
:
H
0
¼
h
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
þ
6
cos
2
ι
þ
cos
4
ι
4
r
ð
2
Þ
and
h
0
¼
1
d
4
π
2
G
c
4
I
zz
f
2
gw
ε
;
ð
3
Þ
with
d; I
zz
, and
ε
the star distance, moment of inertia with
respect to the rotation axis, and
ellipticity
. The ellipticity
measures the degree of asymmetry of the star with respect
to its rotation axis. In the detector reference frame, the
signal is modulated by several effects, the most important
being the
Römer delay
(also called the barycentric correc-
tion) due to the detector motion given by Earth
’
s orbital
motion and rotation, with respect to the GW source.
Moreover, the GW signal is also modulated by the source
’
s
intrinsic spin-down due to the rotational energy loss
from the source. Given a measure of the pulsar rotational
frequency
f
rot
, its derivative
_
f
rot
, and distance
d
, the GW
signal amplitude can be constrained, assuming that all the
star
’
s rotational energy is lost via gravitational radiation.
This theoretical value, which is an upper limit on the
rotational energy that can be emitted in GWs, is called the
spin-down limit
and is given by
[30]
h
sd
¼
8
.
06
×
10
−
19
I
1
=
2
38
1
kpc
d
_
f
rot
Hz
=
s
1
=
2
Hz
f
rot
1
=
2
;
ð
4
Þ
where
I
38
is the star
’
s moment of inertia in units of
10
38
kg m
2
. Different values of the moment of inertia are
possible according to the NS equation of state, mass, and
spin
[31]
; however, in this work we will assume its
canonical value to be
I
¼
10
38
kg m
2
. The corresponding
spin-down limit on the star
’
s equatorial fiducial ellipticity
can be obtained from Eq.
(3)
,
ε
sd
¼
1
.
91
×
10
5
I
−
1
=
2
38
_
f
rot
Hz
=
s
1
=
2
Hz
f
rot
5
=
2
;
ð
5
Þ
which does not depend on the star
’
s distance.
B. P. ABBOTT
et al.
PHYS. REV. D
99,
122002 (2019)
122002-2
B. The five-vector narrow-band pipeline
The narrow-band pipeline uses the five-vector method
[32]
and, in particular, its latest implementation for narrow-
band searches described in
[33]
.
The pipeline explores a range of frequency and spin-
down values by applying barycentric and spin-down
corrections to the data and then identifies the GW signal
using its characteristic frequency components.
The pipeline first removes the modulations given by
the barycentric corrections and intrinsic source spin-down.
The barycentric corrections are applied using a frequency-
independent nonuniform resampling
[33]
. The spin-down
is removed by applying a phase correction on the data time
series. Also, the Einstein delay is corrected in the time
domain.
Once we have removed the barycentric and spin-down
modulations of a possible signal, the GW signal power is
spread among five frequencies given by the coupling of
the signal frequency and the detector sidereal responses
A
þ
ð
t
Þ
;A
×
ð
t
Þ
. These frequency components are
f
gw
−
2
F
sid
;
f
gw
−
F
sid
;f
gw
;f
gw
þ
F
sid
, and
f
gw
þ
2
F
sid
, where
F
sid
is
the frequency corresponding to the Earth sidereal day.
Hence, a pair of matched filters, one for each sidereal
response function, is computed for each point of the explored
parameter space. This is done using a frequency grid which
allows us to compute the matched filters simultaneously over
the whole analyzed frequency band. These steps are done
separately for each detector. Then, the output of the matched
filters at each point of the parameter space are combined,
taking into account the phase shift
1
between the two datasets
in order to build a detection statistic.
The next step consists of selecting the maximum of
the detection statistic for every
10
−
4
Hz interval and over
the whole spin-down range. Within this set, points in the
parameter space with a p value below a 0.1% threshold
(taking into account the number of trials) are considered as
potentially interesting outliers and are subject to further
analysis steps; see Appendix
B
for more details.
III. THE ANALYSIS
The LIGO second observing run O2 started on
November 30, 2016 16
∶
00:00 UTC and ended on
August 25, 2017 22
∶
00:00 UTC, while Virgo joined the
run later on August 1, 2017 12
∶
00:00 UTC and ended on
August 25, 2017 22
∶
00:00 UTC. The narrow-band search
can be performed jointly between different detectors if the
datasets cover the same observing time. Since Virgo O2
data covered just approximately one month at the end of O2
and was characterized by a lower sensitivity with respect to
LIGO data, we have decided to exclude it from the analysis.
For this analysis, we have used the second version of
calibrated LIGO data (C02)
[34]
. We jointly analyzed
LIGO Hanford (LHO) and LIGO Livingston (LLO) data
over the period between January 4, 2017 00
∶
00:00 UTC
and August 25, 2017 22
∶
00:00 UTC. LLO data between the
beginning of the run and December 22, 2016 have been
excluded due to bad spectral contamination, while both
detectors underwent a commissioning break between
December 22, 2016 and January 4, 2017. The observing
time
T
obs
was
∼
232
days, implying frequency and spin-
down bins of, respectively,
δ
f
¼
5
×
10
−
8
Hz and
δ
_
f
¼
2
.
5
×
10
−
15
Hz
=
s. LHO and LLO duty cycles were
about 45% and 56% and corresponded to an effective
observing time of 104 and 129 days, respectively.
2
The
sensitivity of the O2 search is reported in Fig.
1
, where we
also show O1 sensitivity. While at lower frequency, only
O2 LLO seems to be much better than O1, at higher
frequencies the sensitivity is significantly better for both
detectors. In order to validate the analysis, we have looked
for four hardware injections in the data, checking if their
parameters were recovered correctly; see Appendix
A
.
The explored frequency and spin-down ranges were set
to 0.4% of the pulsar rotational frequency and spin-down
reported in the ephemeris. Since in this analysis we
subsampled data at 1 Hz, the explored frequency region
of some pulsars has been chosen manually in order to avoid
a possible signal aliasing.
We have decided to select as
outliers
for the follow-
up the points in the parameter space with a value of
the detection statistic corresponding to a p value of 0.1%
(taking into account the number of trials) or smaller. In the
previous O1 search, we used a threshold of 1% due to the
fact that the data quality of LHO and LLO was significantly
different at lower frequencies; see Appendix
B
for more
details.
IV. SELECTED TARGETS
In our O2 analysis, we have selected as an initial set of
targets all the pulsars present in the O1 narrow-band search
[27]
. Then, we have enlarged it, deciding to analyze all the
pulsars with rotation frequency of 10 and 350 Hz with spin-
down limit given in Eq.
(4)
within a factor of 10 from the
optimal sensitivity of the search of O2 LLO (in most cases).
This choice has been driven by the fact that available pulsar
distances can be affected by a large error. The spin-down
limit has been computed according to the most recent
estimation of the distance given in the ATNF catalog
[35]
(v1.58) and extrapolating the rotational frequency and spin-
down rate at the O2 epoch. For the pulsars J
1028
−
5819
,
J
1112
−
6103
,J
1813
−
1246
, and J
2043
þ
2740
,wehave
checked that the extrapolated rotational parameters together
with the ranges explored in the narrow-band search cover
1
This is given by the fact that the data sampling usually does
not begin at the exact same time for different detectors.
2
With the exception of pulsars that have glitched during the
analysis. For those, we have performed two independent analyses
before and after the glitch.
NARROW-BAND SEARCH FOR GRAVITATIONAL WAVES
...
PHYS. REV. D
99,
122002 (2019)
122002-3
the values reported by the updated ephemeris during the O2
epoch in
[19]
. For the pulsars J
0835
−
4510
,J
0940
−
5428
,J
1105
−
6107
,J
1410
−
6132
,J
1420
−
6048
,J
1531
−
5610
,J
1718
−
3825
,J
1809
−
1917
, and J
1838
−
0655
,
the extrapolated spin-down rate resulted off range with
respect to the one reported in
[19]
, and for this reason, the
searched parameter space has been adjusted in such a way
to cover the updated values. For the pulsars J
0205
þ
6449
,
J
0534
þ
2200
,J
1913
þ
1011
,J
1952
þ
3252
, and J
2229
þ
6114
, we have used updated ephemerides provided by
the telescopes at Jodrell Bank (UK). For the remaining
pulsars, no monitoring is present during the O2 run. Even
though we are aware that an extrapolation from outdated
ephemerides might bring a GW search which does not
cover the actual pulsar rotational parameters during O2,
we have decided to carry on the analysis in such a way to
exploit the possibility that the actual pulsar rotational
parameters were covered even partially by the narrow-band
search.
Table
I
reports the spin-down limit on amplitude
h
0
and
ellipticity
ε
for each target, given their distance estimation
and uncertainty. Hereafter, the distance uncertainties are
propagated to the derived quantities (such as the spin-down
limit) assuming normal distributions, namely,
σ
2
Y
¼
∂
Y
∂
d
2
σ
2
d
;
with
Y
being a function of the distance and
σ
2
the
distribution variance.
The spin-down limits are compared to the estimated
narrow-band search sensitivity in Fig.
1
. The analysis
covers the 11 targets that we have already analyzed for
O1 plus 22 new targets. Based on the estimated sensitivity,
we expected to surpass the spin-down limit in the O2
analysis for nine of the 11 O1 targets. The exceptions are
J
2043
þ
2740
and J
2229
þ
6114
, for which the current
distance estimation has been increased with respect to the
ATNF catalog v1.54 (the catalog used for O1
[27]
).
400
100
0
2
10
-26
10
-25
10
-24
O1 Upper limits
O2 Upper limits
O2 Upper-limits BG
O2 Upper-limits AG
LHO O2 sensitivity (232 days)
LLO O2 sensitivity (232 days)
LLO O1 sensitivity (141 days)
LHO O1 sensitivity (141 days)
Spin-down limit
FIG. 1. Vertical axis: CW amplitude. Horizontal axis: Searched GW frequencies. The different lines indicate the estimated search
sensitivity for O1 and O2 narrow-band searches, while the different markers indicate ULs. The labels
“
AG
”
and
“
BG
”
refer to a search
performed after or before the glitch of a given pulsar. The error bars correspond to the uncertainties on the pulsar distance and correspond
to
1
σ
confidence level.
B. P. ABBOTT
et al.
PHYS. REV. D
99,
122002 (2019)
122002-4
The new O2 targets mainly consist of pulsars with
rotational frequencies within 10 and 20 Hz with spin-down
rate
<
−
10
−
12
Hz
=
s, but there are also a few millisecond
pulsars, for which we can approach the spin-down limit.
Among these, there is the millisecond pulsar J
2124
þ
3358
,
for which we expect to barely approach the spin-down limit
with targeted searches. One of these millisecond pulsars
J
1300
þ
1240
is located in a binary system. However,
according to the orbital parameters in the ephemeris, the
intrinsic binary orbital modulation on a possible CW signal
would be of the order of
Δ
f
bin
≈
10
−
10
Hz, which is below
our frequency resolution and hence can be neglected.
3
Millisecond pulsars are characterized by a low rotational
spin-down value
_
f
rot
together with a high rotational
frequency
f
rot
; hence, according to Eq.
(4)
, their spin-
down limit will also be harder to surpass our search
sensitivities. Although the narrow-band search is currently
not sensitive enough for the millisecond pulsars, we have
decided to perform the search in order to test the capabil-
ities of the pipeline at higher frequencies. Furthermore,
pulsars J
0205
þ
6449
,J
0534
þ
2200
,J
0835
−
4510
,
J
1028
−
5819
, and J
1718
−
3825
had a glitch during O2.
J
0205
þ
6449
glitched on May 27, 2017, J
0534
þ
2200
glitched on March 27, 2017, J
0835
−
4510
had a glitch on
December 16, 2016
[51]
,J
1028
−
5819
glitched on May
29, 2017, and J
1718
−
3825
glitched on May 1 July 2017
[19]
. For these pulsars, we have performed two indepen-
dent analyses, one before and one after the glitch, excluding
TABLE I. Properties of analyzed pulsars. The second column reports the distance as provided by the ephemerides
based on the dispersion measure and the galactic electron density model of
[36]
. If the pulsar distance is estimated
according to an independent measure, we refer to it next to the name entry. The distance uncertainty refers to
1
σ
confidence level and is assumed to have a normal distribution. In the third and fourth columns, the spin-down limit
h
sd
and the corresponding ellipticity
ε
sd
are computed using Eqs.
(4)
and
(5)
.
Name
d
(kpc)
h
sd
ε
sd
J
0205
þ
6449
[37]
2
.
0
0
.
3
ð
6
.
9
1
.
1
Þ
×
10
−
25
1
.
42
×
10
−
3
J
0534
þ
2200
[38]
2
.
0
0
.
5
ð
1
.
4
0
.
4
Þ
×
10
−
24
7
.
56
×
10
−
4
J
0537
−
6910
[39]
49
.
7
0
.
2
ð
2
.
91
0
.
02
Þ
×
10
−
26
8
.
90
×
10
−
5
J
0540
−
6919
[39]
49
.
7
0
.
2
ð
4
.
99
0
.
02
Þ
×
10
−
26
1
.
50
×
10
−
3
J
0835
−
4510
[40]
0
.
28
0
.
02
ð
3
.
4
0
.
3
Þ
×
10
−
24
1
.
80
×
10
−
3
J
0940
−
5428
0
.
4
0
.
2
ð
1
.
3
0
.
5
Þ
×
10
−
24
8
.
97
×
10
−
4
J
1028
−
5819
1
.
4
0
.
6
ð
2
.
4
1
.
0
Þ
×
10
−
25
6
.
70
×
10
−
4
J
1105
−
6107
2
.
4
0
.
9
ð
1
.
7
0
.
7
Þ
×
10
−
25
3
.
82
×
10
−
4
J
1112
−
6103
4
.
5
1
.
8
ð
1
.
3
0
.
5
Þ
×
10
−
25
5
.
61
×
10
−
4
J
1300
þ
1240
[41]
0
.
7
0
.
2
ð
5
.
3
1
.
3
Þ
×
10
−
27
3
.
17
×
10
−
8
J
1302
−
6350
2
.
3
0
.
9
ð
7
.
6
3
.
0
Þ
×
10
−
26
9
.
52
×
10
−
5
J
1400
−
6325
[42]
0
.
9
0
.
3
ð
1
.
0
0
.
3
Þ
×
10
−
24
2
.
07
×
10
−
4
J
1410
−
6132
13
.
5
5
.
3
ð
4
.
8
1
.
9
Þ
×
10
−
26
3
.
83
×
10
−
4
J
1420
−
6048
5
.
6
2
.
2
ð
1
.
6
0
.
7
Þ
×
10
−
25
9
.
81
×
10
−
4
J
1524
−
5625
3
.
4
1
.
3
ð
1
.
7
0
.
7
Þ
×
10
−
25
8
.
25
×
10
−
4
J
1531
−
5610
2
.
8
1
.
1
ð
1
.
2
0
.
5
Þ
×
10
−
25
5
.
47
×
10
−
4
J
1617
−
5055
4
.
7
1
.
9
ð
2
.
4
1
.
0
Þ
×
10
−
25
1
.
28
×
10
−
3
J
1718
−
3825
3
.
5
1
.
4
ð
9
.
7
3
.
8
Þ
×
10
−
26
4
.
48
×
10
−
4
J
1747
−
2809
8
.
2
3
.
2
ð
1
.
7
0
.
7
Þ
×
10
−
25
8
.
97
×
10
−
4
J
1747
−
2958
2
.
5
1
.
0
ð
2
.
5
1
.
0
Þ
×
10
−
25
1
.
47
×
10
−
3
J
1809
−
1917
3
.
3
1
.
3
ð
1
.
4
0
.
6
Þ
×
10
−
25
7
.
27
×
10
−
4
J
1811
−
1925
5
.
0
2
.
0
ð
1
.
3
0
.
6
Þ
×
10
−
25
6
.
59
×
10
−
4
J
1813
−
1246
[43]
>
2
.
5
<
1
.
9
×
10
−
25
2
.
67
×
10
−
4
J
1813
−
1749
[44]
4
.
7
0
.
8
ð
2
.
9
0
.
5
Þ
×
10
−
25
6
.
42
×
10
−
4
J
1831
−
0952
3
.
7
1
.
5
ð
7
.
7
3
.
0
Þ
×
10
−
26
3
.
04
×
10
−
4
J
1833
−
1034
[45]
4
.
1
0
.
3
ð
3
.
6
0
.
3
Þ
×
10
−
25
1
.
32
×
10
−
3
J
1838
−
0655
[46]
6
.
6
0
.
9
ð
1
.
0
0
.
2
Þ
×
10
−
25
7
.
94
×
10
−
4
J
1913
þ
1011
4
.
6
1
.
8
ð
5
.
4
2
.
1
Þ
×
10
−
26
7
.
54
×
10
−
5
J
1952
þ
3252
[41]
3
.
0
2
.
0
ð
1
.
0
0
.
7
Þ
×
10
−
25
1
.
15
×
10
−
4
J
2022
þ
3842
[47]
10
.
0
2
.
0
ð
1
.
1
0
.
3
Þ
×
10
−
25
6
.
00
×
10
−
4
J
2043
þ
2740
1
.
5
0
.
6
ð
6
.
3
2
.
5
Þ
×
10
−
26
2
.
03
×
10
−
4
J
2124
−
3358
[48]
0
.
4
0
.
1
ð
4
.
3
1
.
0
Þ
×
10
−
27
9
.
49
×
10
−
9
J
2229
þ
6114
[49]
3
.
0
2
.
0
ð
3
.
3
2
.
3
Þ
×
10
−
25
6
.
27
×
10
−
4
3
The frequency shift due to the binary motion has been
computed using
[50]
.
NARROW-BAND SEARCH FOR GRAVITATIONAL WAVES
...
PHYS. REV. D
99,
122002 (2019)
122002-5
the day in which the glitch was present. For J
0835
−
4510
and J
1718
−
3825
, only the analysis after or before the
glitch was done, since few data were available before or
after the two glitches.
Table
II
reports the frequency/spin-down regions that we
have analyzed for each of the 33 targets. The reference time
for the rotational parameters of the pulsars is December 1,
2016 00
∶
00:00 UTC.
V. RESULTS
The search has produced a total of 49 outliers for 15 of
the 33 targets. Every outlier underwent a chain of follow-up
steps aimed to test its nature, namely, (i) check for the
presence of known instrumental noise lines, (ii) compare
the SNR GW amplitude estimation among several detec-
tors, and (iii) study the outlier significance with software
injections. The outliers are given in Table
III
together with
the follow-up step where we excluded them.
The narrow-band search carried out in the past on O1
data
[27]
produced two interesting outliers for J
0835
−
4510
and
1833
−
1034
. In order to confirm or reject them,
the data from the first four months of O2 (available with
calibration version C01 at the time) were used and no
evidence of a signal was found. The full O2 analysis
discussed in this paper confirms those findings. No outlier
TABLE II. First column: Pulsar name. Second and third columns: Central frequency and frequency width
explored in the search. Fourth and fifth columns: Central spin-down and spin-down ranges explored in the search.
Sixth and seventh columns: Number of templates in frequency and spin-down. Frequency and spin-down
resolutions are, respectively,
δ
f
∼
5
×
10
−
8
Hz,
δ
_
f
∼
2
.
5
×
10
−
15
Hz
=
s. The labels
“
AG
”
and
“
BG
”
indicate,
respectively, after and before the glitch. Note that the frequency and spin-down resolution, and hence, the number of
templates, are lower in the case of pulsars with a glitch.
Name
f
(Hz)
Δ
f
(Hz)
_
f
(Hz/s)
Δ
_
f
(Hz/s)
n
f
ð
10
6
Þ
n
_
f
J
0205
þ
6449
AG
30.41
0.06
−
8
.
61
×
10
−
11
2
.
72
×
10
−
13
0.47
17
J
0205
þ
6449
BG
30.41
0.06
−
8
.
61
×
10
−
11
2
.
44
×
10
−
13
0.74
37
J
0534
þ
2200
AG
59.30
0.12
−
7
.
38
×
10
−
10
1
.
50
×
10
−
12
1.53
251
J
0534
þ
2200
BG
59.30
0.12
−
7
.
38
×
10
−
10
1
.
56
×
10
−
12
0.82
75
J
0537
−
6910
123.86
0.25
−
3
.
92
×
10
−
10
8
.
01
×
10
−
13
4.95
321
J
0540
−
6919
39.39
0.08
−
3
.
71
×
10
−
10
7
.
56
×
10
−
13
1.57
303
J
0835
−
4510
22.37
0.04
−
3
.
22
×
10
−
11
8
.
51
×
10
−
14
0.89
35
J
0940
−
5428
22.84
0.05
−
8
.
56
×
10
−
12
2
.
50
×
10
−
14
0.91
11
J
1028
−
5819
AG
21.88
0.04
−
3
.
86
×
10
−
12
3
.
56
×
10
−
14
0.33
3
J
1028
−
5819
BG
21.88
0.04
−
3
.
86
×
10
−
12
2
.
63
×
10
−
14
0.54
5
J
1105
−
6107
31.64
0.06
−
7
.
94
×
10
−
12
2
.
00
×
10
−
14
1.26
9
J
1112
−
6103
30.78
0.06
−
1
.
49
×
10
−
11
3
.
50
×
10
−
14
1.23
15
J
1300
þ
1240
321.62
0.64
−
5
.
91
×
10
−
15
5
.
00
×
10
−
15
12.86
3
J
1302
−
6350
41.87
0.08
−
2
.
00
×
10
−
12
5
.
00
×
10
−
15
1.67
3
J
1400
−
6325
64.12
0.13
−
8
.
00
×
10
−
11
1
.
65
×
10
−
13
2.56
67
J
1410
−
6132
39.95
0.08
−
2
.
52
×
10
−
11
7
.
01
×
10
−
14
1.60
29
J
1420
−
6048
29.32
0.06
−
3
.
57
×
10
−
11
1
.
00
×
10
−
13
1.17
41
J
1524
−
5625
25.56
0.05
−
1
.
27
×
10
−
11
3
.
00
×
10
−
14
1.02
13
J
1531
−
5610
23.75
0.05
−
3
.
88
×
10
−
12
1
.
50
×
10
−
14
0.95
7
J
1617
−
5055
28.80
0.06
−
5
.
62
×
10
−
11
1
.
15
×
10
−
13
1.15
47
J
1718
−
3825
BG
26.78
0.05
−
4
.
72
×
10
−
12
1
.
72
×
10
−
14
0.82
5
J
1747
−
2809
38.32
0.08
−
1
.
14
×
10
−
10
2
.
35
×
10
−
13
1.53
95
J
1747
−
2958
20.23
0.04
−
1
.
25
×
10
−
11
3
.
00
×
10
−
14
0.81
13
J
1809
−
1917
24.17
0.05
−
7
.
44
×
10
−
12
2
.
00
×
10
−
14
0.97
9
J
1811
−
1925
30.91
0.06
−
2
.
10
×
10
−
11
4
.
50
×
10
−
14
1.23
19
J
1813
−
1246
41.60
0.08
−
1
.
52
×
10
−
11
3
.
50
×
10
−
14
1.66
15
J
1813
−
1749
44.71
0.09
−
1
.
27
×
10
−
10
2
.
60
×
10
−
13
1.79
105
J
1831
−
0952
29.73
0.06
−
3
.
67
×
10
−
12
1
.
00
×
10
−
14
1.19
5
J
1833
−
1034
32.29
0.06
−
1
.
05
×
10
−
10
2
.
15
×
10
−
13
1.29
87
J
1838
−
0655
28.36
0.06
−
1
.
99
×
10
−
11
5
.
51
×
10
−
14
1.13
23
J
1913
þ
1011
55.69
0.11
−
5
.
25
×
10
−
12
1
.
50
×
10
−
14
2.23
7
J
1952
þ
3252
50.59
0.10
−
7
.
48
×
10
−
12
2
.
00
×
10
−
14
2.02
9
J
2022
þ
3842
41.16
0.08
−
7
.
30
×
10
−
11
1
.
50
×
10
−
13
1.64
61
J
2043
þ
2740
20.80
0.04
−
2
.
75
×
10
−
13
5
.
00
×
10
−
15
0.83
3
J
2124
−
3358
405.59
0.81
−
16
.
92
×
10
−
16
5
.
00
×
10
−
15
16.21
3
J
2229
þ
6114
38.71
0.08
−
5
.
84
×
10
−
11
1
.
20
×
10
−
13
1.55
49
B. P. ABBOTT
et al.
PHYS. REV. D
99,
122002 (2019)
122002-6
has been found for J
0835
−
4510
, while an outlier has
been found for J
1833
−
1034
, at a slightly different
frequency which, however, as discussed in the next section,
has been vetoed.
A. Outliers follow-up
The first step of the follow-up was to check if a known
instrumental noise line was present in one of the two
detectors
[52]
. This ruled out most of the candidates for the
pulsars J
1105
−
6107
and J
2121
−
3358
; see Appendix
C
for more details.
The second step of the follow-up was to study the
evolution of the recovered signal-to-noise ratio (SNR) and
amplitude
h
0
with respect to the fraction of data samples
that we were integrating. We expect the SNR to increase as
the square root of the integration time and the amplitude
h
0
to be nearly constant. We have performed this type of test
in a LHO, LLO, and joint search for different integration
times, checking if the SNR and
h
0
estimation were
compatible across the different cases.
Many outliers at frequencies
<
100
Hz have been clas-
sified as LHO disturbances since they have been observed
only in LHO (see Appendix
C
). Some of these are in
proximity of unidentified noise lines (lines which are
confidently classified as detector disturbances but whose
origin is unknown). That is the case of the outliers from
J
1112
−
6103
,J
1302
−
6350
, and J
1813
−
1246
. Other
outliers at low frequency were not in proximity to uniden-
tified noise lines but have been vetoed as the signal-to-
noise ratio is bigger than 8 only in LHO data, which has
a sensitivity 2 to 3 times worse than LLO, thus, being
incompatible with a true CW signal.
Only three outliers survived up to the third step of the
follow-up, namely, from pulsars J
1300
þ
1240
,J
1617
−
5055
, and J
2124
−
3358
. For all these pulsars. we cannot
approach the theoretical spin-down limit with our current
search sensitivity, and this is a strong hint of the noise
origin of these outliers. The last step of the follow-up
consisted of studying the SNR and recovered CW ampli-
tude
h
0
with software injections with an amplitude
h
0
fixed
to that estimated for the outlier. The evolution of the SNR
and
h
0
for the outlier are then compared to the distributions
derived from the injections. If they are compatible among
the three different analyses, LHO, LLO, and joint combi-
nation, the outlier is subject to more dedicated studies. The
two remaining outliers for the millisecond pulsars were
ruled out since they were present in just one detector,
while the injections predicted that they would be visible in
both detectors. The J
1617
−
5055
remaining outlier was
also ruled out, as the injections show that they were likely
driven by a LHO disturbance. Refer to Appendix
C
for
more details on the last steps of this follow-up.
B. Upper limits
Since there was no evidence of the presence of a CW
signal, we have computed upper limits (ULs) on the CW
amplitude
h
0
. The ULs have been produced using the same
procedure as in the O1 narrow-band search
[27]
, which
consists of injecting nonoverlapping GW signals with fixed
amplitude
h
0
in data every
10
−
4
Hz intervals. When 95% of
injections produce a value of the detection statistic higher
than the one used for the outlier selection, we set the upper
limit to the injected amplitude value.
Figure
1
shows the median value of the UL for each of
the 33 targets. The ULs are driven at lower frequencies by
LLO sensitivity since it is the most sensitive detector in that
frequency region. On the other hand, at higher frequencies,
the ULs lie close to the sensitivity of the two detectors,
which are indeed similar.
Table
IV
summarizes our results of the O2 narrow-band
search. The table reports the median value of the UL on
the strain amplitude
h
0
and the corresponding ellipticity
computed using Eq.
(5)
. We consider the spin-down limit
surpassed for a given pulsar if the ULs are lower than the
spin-down limit over the entire frequency band.
The most stringent ULs have been set for the three
pulsars J
0537
−
6910
,J
1300
þ
1240
, and J
2124
−
3358
TABLE III. This table summarizes the outliers found in the O2
narrow-band search. The first column reports the name of the
pulsar for which we have found outliers. The second column
gives the central frequency of the pulsar search band, and the
third column the p value of the least significant outlier. The last
column reports the step of the follow-up in which we have vetoed
the outliers. For a description of the follow-up steps, refer to the
main text.
Name
f
Number of outliers p value Step
J
1105
−
6107
31.64
16
a
4
.
23
×
10
−
4
i,ii
J
1112
−
6103
30.78
1
b
1
.
83
×
10
−
4
ii
J1300
þ
1240
321.62
1
7
.
80
×
10
−
4
iii
J
1302
−
6350
41.87
4
c
7
.
79
×
10
−
4
ii
J
1420
−
6048
29.32
11
d
9
.
82
×
10
−
4
i,ii
J
1531
−
5610
23.75
1
4
.
65
×
10
−
4
ii
J
1617
−
5055
28.80
2
7
.
80
×
10
−
4
ii,iii
J
1747
−
2809
38.32
1
9
.
68
×
10
−
4
ii
J
1811
−
1925
30.91
1
3
.
30
×
10
−
4
ii
J
1813
−
1246
41.60
2
e
6
.
73
×
10
−
4
ii,iii
J
1831
−
0952
29.73
1
2
.
15
×
10
−
4
ii
J
1833
−
1034
32.29
1
9
.
33
×
10
−
4
ii
J
1952
þ
3252
50.59
4
f
4
.
48
×
10
−
4
i,ii
J
2124
−
3358
405.59
2
g
5
.
61
×
10
−
4
i,iii
J
2229
þ
6114
38.71
1
9
.
66
×
10
−
4
ii
a
Most vetoed since they are close to the comb line of the
0.987925 Hz comb in LLO and the comb line of 2.109223 Hz
in LHO.
b
Various unidentified lines around 35.51 Hz.
c
Unidentified noise disturbance in LHO at 41.8838 Hz.
d
Comb of 1.945501 Hz in LHO.
e
Unidentified broad line disturbance at 41.654
–
41.660 Hz.
f
Comb of 2.109223 Hz in LHO, comb of 1.9455045 Hz in
LHO, comb of 1.945437 Hz in LHO.
g
Comb of 0.9967943 Hz in LLO.
NARROW-BAND SEARCH FOR GRAVITATIONAL WAVES
...
PHYS. REV. D
99,
122002 (2019)
122002-7
and are of the order of
5
.
5
×
10
−
26
which, however, are
above the spin-down limit. The lowest ellipticity UL has
been set for J
1300
þ
1240
, of about
3
.
3
×
10
−
7
.Wehave
been able to surpass the spin-down limit for the pulsars
J
0205
þ
6449
,J
0534
þ
2200
(Crab), J
0835
−
4510
(Vela),
J
1400
−
6325
,J
1813
−
1246
(assuming the lower bound
for the distance), J
1813
−
1749
,J
1833
−
1034
, and
J
2229
þ
6114
. For J
0940
−
5428
, while the median value
TABLE IV. Upper limits summary table. First column: Pulsar name. Second and third columns: Median of the 95% confidence level
UL on the GWamplitude
h
0
and corresponding ellipticity
ε
. Fourth column: Surface deformation corresponding to the median ellipticity
for a NS with radius of 10 km
[53]
. Fifth column: Ratio between the median UL and the spin-down limit. Sixth column: Ratio between
the median UL on the GWand rotational energy losses. Last column: Minimum and maximum ratio between the ULs and the theoretical
spin-down limit over the analyzed frequency/spin-down region. All the entries that use information on the astrophysical distance also
include the corresponding uncertainty at
1
σ
confidence level.
Name
h
h
i
UL
h
ε
i
UL
r
ε
(cm)
h
h
i
UL
=h
sd
h
_
E
UL
i
=
_
E
sd
min
nb
ðh
h
i
UL
=h
sd
Þ
−
max
nb
ðh
h
i
UL
=h
sd
Þ
J
0205
þ
6449
AG
3
.
87
×
10
−
25
ð
7
.
9
1
.
2
Þ
×
10
−
4
197.9
0
.
56
0
.
08
0.3
0
.
48
þ
0
.
07
−
0
.
07
−
0
.
67
þ
0
.
10
−
0
.
10
J
0205
þ
6449
BG
3
.
19
×
10
−
25
ð
6
.
5
1
.
0
Þ
×
10
−
4
163.1
0
.
46
0
.
07
0.2
0
.
31
þ
0
.
05
−
0
.
05
−
0
.
58
þ
0
.
09
−
0
.
09
J
0534
þ
2200
AG
1
.
31
×
10
−
25
ð
7
.
1
1
.
8
Þ
×
10
−
5
17.4
0
.
09
0
.
02
0.008
0
.
07
þ
0
.
02
−
0
.
02
−
0
.
11
þ
0
.
03
−
0
.
03
J
0534
þ
2200
BG
1
.
64
×
10
−
25
ð
8
.
8
2
.
2
Þ
×
10
−
5
21.7
0
.
11
0
.
03
0.01
0
.
09
þ
0
.
02
−
0
.
02
−
0
.
14
þ
0
.
03
−
0
.
03
J
0537
−
6910
5
.
59
×
10
−
26
ð
1
.
7
0
.
01
Þ
×
10
−
4
1
.
92
0
.
01
1
.
13
þ
0
.
00
−
0
.
00
−
2
.
25
þ
0
.
01
−
0
.
01
J
0540
−
6919
1
.
47
×
10
−
25
ð
4
.
43
0
.
02
Þ
×
10
−
3
2
.
95
0
.
01
1
.
83
þ
0
.
01
−
0
.
01
−
3
.
47
þ
0
.
02
−
0
.
02
J
0835
−
4510
8
.
82
×
10
−
25
ð
4
.
7
0
.
4
Þ
×
10
−
4
116.8
0
.
26
0
.
02
0.07
0
.
14
þ
0
.
01
−
0
.
01
−
0
.
31
þ
0
.
02
−
0
.
02
J
0940
−
5428
8
.
55
×
10
−
25
ð
5
.
9
2
.
3
Þ
×
10
−
4
147.6
0
.
7
0
.
3
0.5
0
.
4
þ
0
.
2
−
0
.
2
−
0
.
8
þ
0
.
4
−
0
.
4
J
1028
−
5819
AG
1
.
18
×
10
−
24
ð
3
.
3
1
.
3
Þ
×
10
−
3
5
.
0
2
.
0
4
.
2
þ
1
.
7
−
1
.
7
−
6
.
0
þ
2
.
3
−
2
.
3
J
1028
−
5819
BG
1
.
37
×
10
−
24
ð
3
.
8
1
.
5
Þ
×
10
−
3
5
.
7
2
.
3
4
.
3
þ
1
.
7
−
1
.
7
−
7
.
0
þ
2
.
7
−
2
.
7
J
1105
−
6107
2
.
20
×
10
−
25
ð
5
.
0
2
.
0
Þ
×
10
−
4
123.0
1
.
3
0
.
6
1.7
0
.
68
þ
0
.
3
−
0
.
3
−
1
.
88
þ
0
.
8
−
0
.
8
J
1112
−
6103
2
.
48
×
10
−
25
ð
1
.
1
0
.
5
Þ
×
10
−
3
2
.
0
0
.
8
1
.
1
þ
0
.
5
−
0
.
5
−
2
.
5
þ
1
.
0
−
1
.
0
J
1300
þ
1240
5
.
60
×
10
−
26
ð
3
.
3
0
.
8
Þ
×
10
−
7
10
.
5
2
.
5
6
.
3
þ
1
.
5
−
1
.
5
−
13
.
1
þ
3
.
1
−
3
.
1
J
1302
−
6350
1
.
22
×
10
−
25
ð
1
.
5
0
.
6
Þ
×
10
−
4
38.0
1
.
6
0
.
7
2.6
0
.
7
þ
0
.
3
−
0
.
3
−
1
.
9
þ
0
.
8
−
0
.
8
J
1400
−
6325
8
.
57
×
10
−
26
ð
1
.
8
0
.
6
Þ
×
10
−
5
4.4
0
.
09
0
.
03
0.008
0
.
05
þ
0
.
01
−
0
.
01
−
0
.
10
þ
0
.
03
−
0
.
03
J
1410
−
6132
1
.
33
×
10
−
25
ð
1
.
1
0
.
5
Þ
×
10
−
3
2
.
8
1
.
1
1
.
4
þ
0
.
6
−
0
.
6
−
3
.
5
þ
1
.
4
−
1
.
4
J
1420
−
6048
2
.
75
×
10
−
25
ð
1
.
7
0
.
7
Þ
×
10
−
3
426.8
1
.
7
0
.
7
3.1
0
.
9
þ
0
.
4
−
0
.
4
−
2
.
2
þ
0
.
9
−
0
.
9
J
1524
−
5625
5
.
03
×
10
−
25
ð
2
.
5
1
.
0
Þ
×
10
−
3
3
.
0
1
.
2
1
.
7
þ
0
.
7
−
0
.
7
−
3
.
7
þ
1
.
5
−
1
.
5
J
1531
−
5610
7
.
51
×
10
−
25
ð
3
.
6
1
.
4
Þ
×
10
−
3
6
.
5
2
.
6
3
.
7
þ
1
.
5
−
1
.
5
−
7
.
7
þ
3
.
1
−
3
.
1
J
1617
−
5055
3
.
41
×
10
−
25
ð
1
.
8
0
.
6
Þ
×
10
−
3
461.0
1
.
5
0
.
6
2.1
0
.
8
þ
0
.
3
−
0
.
3
−
1
.
8
þ
0
.
8
−
0
.
8
J
1718
−
3825
BG
3
.
88
×
10
−
25
ð
1
.
8
0
.
7
Þ
×
10
−
3
4
.
0
1
.
6
2
.
5
þ
1
.
0
−
1
.
0
−
4
.
8
þ
2
.
0
−
2
.
0
J
1747
−
2809
1
.
43
×
10
−
25
ð
7
.
5
2
.
9
Þ
×
10
−
4
188.1
0
.
8
0
.
4
0.6
0
.
5
þ
0
.
2
−
0
.
2
−
1
.
0
þ
0
.
4
−
0
.
4
J
1747
−
2958
1
.
35
×
10
−
24
ð
7
.
9
3
.
1
Þ
×
10
−
3
5
.
4
2
.
1
3
.
2
þ
1
.
3
−
1
.
3
−
6
.
7
þ
2
.
6
−
2
.
6
J
1809
−
1917
6
.
95
×
10
−
25
ð
3
.
7
1
.
5
Þ
×
10
−
3
5
.
1
2
.
0
3
.
1
þ
1
.
2
−
1
.
2
−
6
.
2
þ
2
.
4
−
2
.
4
J
1811
−
1925
2
.
53
×
10
−
25
ð
1
.
2
0
.
5
Þ
×
10
−
3
1
.
9
0
.
8
1
.
3
þ
0
.
5
−
0
.
5
−
2
.
3
þ
0
.
9
−
0
.
9
J
1813
−
1246
1
.
23
×
10
−
25
≤
7
×
10
−
4
42.0
≥
0
.
7
≥
0
.
5
≥
ð
0
.
4
−
0
.
8
Þ
J
1813
−
1749
1
.
16
×
10
−
25
ð
2
.
6
0
.
5
Þ
×
10
−
4
64.5
0
.
40
0
.
07
0.2
0
.
25
þ
0
.
04
−
0
.
04
−
0
.
49
þ
0
.
08
−
0
.
08
J
1831
−
0952
2
.
56
×
10
−
25
ð
1
.
0
0
.
4
Þ
×
10
−
3
3
.
3
1
.
3
2
.
1
þ
0
.
9
−
0
.
9
−
4
.
2
þ
1
.
7
−
1
.
7
J
1833
−
1034
1
.
96
×
10
−
25
ð
7
.
3
0
.
6
Þ
×
10
−
4
182.5
0
.
55
0
.
04
0.3
0
.
35
þ
0
.
03
−
0
.
03
−
0
.
71
þ
0
.
05
−
0
.
05
J
1838
−
0655
3
.
03
×
10
−
25
ð
2
.
4
0
.
4
Þ
×
10
−
3
3
.
0
0
.
4
1
.
8
þ
0
.
3
−
0
.
3
−
3
.
6
þ
0
.
5
−
0
.
5
J
1913
þ
1011
1
.
02
×
10
−
25
ð
1
.
4
0
.
6
Þ
×
10
−
4
1
.
9
0
.
8
1
.
1
þ
0
.
5
−
0
.
5
−
2
.
31
þ
0
.
9
−
0
.
9
J
1952
þ
3252
9
.
09
×
10
−
26
ð
1
.
0
0
.
7
Þ
×
10
−
4
25.2
0
.
9
0
.
6
0.8
0
.
5
þ
0
.
4
−
0
.
4
−
1
.
1
þ
0
.
8
−
0
.
8
J
2022
þ
3842
1
.
32
×
10
−
25
ð
7
.
4
1
.
5
Þ
×
10
−
4
184.0
1
.
2
0
.
3
1.4
0
.
7
þ
0
.
2
−
0
.
2
−
1
.
5
þ
0
.
3
−
0
.
3
J
2043
þ
2740
1
.
12
×
10
−
24
ð
3
.
6
1
.
4
Þ
×
10
−
3
17
.
8
7
.
0
10
.
3
þ
4
.
0
−
4
.
0
−
21
.
42
þ
9
.
0
−
9
.
0
J
2124
−
3358
5
.
97
×
10
−
26
ð
1
.
3
0
.
3
Þ
×
10
−
7
14
.
0
3
.
3
7
.
3
þ
1
.
8
−
1
.
8
−
17
.
4
þ
4
.
2
−
4
.
2
J
2229
þ
6114
1
.
39
×
10
−
25
ð
2
.
7
1
.
8
Þ
×
10
−
4
65.8
0
.
4
0
.
3
0.2
0
.
3
þ
0
.
2
−
0
.
2
−
0
.
5
þ
0
.
4
−
0
.
4
B. P. ABBOTT
et al.
PHYS. REV. D
99,
122002 (2019)
122002-8
of the UL is below the spin-down limit, a small fraction of
the individual results are above. For J
1747
−
2809
and
J
1952
þ
3252
, we are close to surpassing the spin-down
limit
4
; see Table
IV
. For all the pulsars for which we have
surpassed the spin-down limit, we have computed the upper
limit on the ratio of the GW to the rotational energy loss.
The lower ULs on the GW energy loss are for J
0534
þ
2200
and J
1400
−
6325
corresponding to a fraction of
about 0.8%. The lowest ULs on the GW amplitude and
ellipticity among the pulsars for which we have surpassed
the spin-down limit are, respectively,
8
.
29
×
10
−
26
and
1
.
78
×
10
−
5
for J
1400
−
6325
. For a canonical pulsar with
a radius of about 10 km, this number would correspond to a
maximum surface deformation of about 5 cm.
For the remaining 22 targets, we were not able to surpass
the spin-down limit. Table
IV
roughly suggests to us that an
improvement in sensitivity of a factor 3 is needed for most
of the low-frequency pulsars. It must be considered,
however, that the spin-down limits have been computed
assuming a canonical value for the moment of inertia of
10
38
kg m
2
. In fact, it could be significantly larger, depend-
ing on the NS equation of state, up to
∼
3
×
10
38
kg m
2
,
implying a spin-down limit
∼
ffiffi
ð
p
3
Þ
times larger.
VI. CONCLUSION
Overall, the narrow-band search over O2 data has
brought an improvement with respect to previous searches
in terms of ULs. On the other hand, ULs are similar to
those found in O1 for pulsars with rotation frequency below
30 Hz. For instance, the UL on the Vela Pulsar (around
22 Hz) has improved by 10%, while the UL on
J
0205
þ
6449
5
has improved by about 22%. On the other
hand, for pulsars with expected GW frequencies
>
30
Hz,
the UL is improved even by a factor 2. The UL on J
0534
þ
2200
did not improve, since in O2 we split the analysis
in two different chunks due to the presence of the
glitch. For this reason, the UL both before and after the
glitch is comparable to the one found in the O1 analysis.
We have also been able to surpass the spin-down limit
for two pulsars that were not analyzed in O1,
J
0940
−
5428
,J
1747
−
2809
.
We still have not been able to surpass the spin-down limit
for the millisecond pulsars and for low-frequency pulsars
with spin-down below
∼
10
−
12
Hz
=
s. However, we have
been able to surpass the spin-down limit for low-frequency
and high energetic pulsars (such as Crab or J
1833
−
1034
)
or for low-frequency pulsars that are close to Earth.
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 Technology
Facilities Council (STFC) of the United Kingdom, the Max-
Planck-Society (MPS), and the State of Niedersachsen/
Germany for support of the construction of Advanced
LIGOandconstruction andoperationof theGEO600detector.
Additional support for Advanced LIGO was provided by the
Australian Research Council. The authorsgratefully acknowl-
edge the Italian Istituto Nazionale di Fisica Nucleare (INFN),
the French Centre National de la Recherche Scientifique
(CNRS), and the Foundation for Fundamental Research on
Matter supported by the Netherlands Organisation for
Scientific Research for the construction and operation of the
Virgo detector and the creation and support of the European
GravitationalObservatoryconsortium.The authors also grate-
fully acknowledge research support from these agencies as
well as by the Council of Scientific and Industrial Research of
India, the Department of Science and Technology, India, the
Science & Engineering Research Board, India, the Ministry of
Human Resource Development, India, the Spanish Agencia
Estatal de Investigación, the Vicepresid`
encia i Conselleria
d
’
Innovació, Recerca i Turisme and the Conselleria
d
’
Educació i Universitat del Govern de les Illes Balears, the
Conselleria d
’
Educació, Investigació, Cultura i Esport de la
Generalitat Valenciana, the National Science Centre of
Poland, the Swiss National Science Foundation, the
Russian Foundation for Basic Research, the Russian
Science Foundation, the European Commission, the
European Regional Developm
ent Funds, the Royal Society,
the Scottish Funding Council, the Scottish Universities
Physics Alliance, the Hungarian Scientific Research Fund,
the Lyon Institute of Origins, the National Research,
Development and Innovation Office Hungary, the National
Research Foundation of Korea, Industry Canada and the
Province of Ontario through the Ministry of Economic
Development and Innovation, the Natural Science and
Engineering Research Council Canada, the Canadian
Institute for Advanced Research, the Brazilian Ministry of
Science, Technology, Innovations, and Communications, the
International Center for Theoretical Physics South American
Institute for Fundamental Research, the Research Grants
Council of Hong Kong, the National Natural Science
Foundation of China, the Leverhulme Trust, the Research
Corporation,theMinistryofScienceandTechnology,Taiwan,
andtheKavliFoundation.Theauthorsgratefullyacknowledge
the support of the NSF, STFC, MPS, INFN, CNRS, and the
State of Niedersachsen/Germany for provision of computa-
tional resources. Work at Naval Research Laboratory is
supported by the National Aeronautics and Space
Administration. The authors acknowledge the anonymous
referees for helping improve this paper. This work has been
assigned LIGO Document No. LIGO-P1800391.
4
Excluding a frequency band heavily contaminated by noise.
5
Please note that the spin-down limit of this pulsar has been
computed using two different distances in O1 and O2. For O1, we
used 2.0 kpc
[54]
, while for O2 the nominal ephemeris value was
3.2 kpc.
NARROW-BAND SEARCH FOR GRAVITATIONAL WAVES
...
PHYS. REV. D
99,
122002 (2019)
122002-9
APPENDIX A: VALIDATION WITH
HARDWARE INJECTIONS
Hardware injections are simulated signals in LIGO-Virgo
data for testing purposes. These artificial signals are injected
by a control system which acts on the mirror and simulates
a CW signal. The hardware injections are continuously
monitored and their injected parameters are known. In order
to validate the efficiency of the pipeline used in this paper,
we have looked for four hardware injections in LIGO data
studying the accuracy of the recovered parameters. We
define the relative error on the CW amplitude recovery as
ε
h
0
¼
1
−
h
esti
0
=h
inj
0
,where
h
inj
0
is the injected CW amplitude
and
h
esti
0
is the recovered value, whereas we define the
relative error on the angular parameters
ψ
,
η
as
ε
ψ
¼j
ψ
inj
−
ψ
esti
j
=
90
deg and
ε
η
¼j
η
inj
−
η
esti
j
=
2
.Table
V
reports the
errors on the parameter estimation for the validation tests
performed with the O2 hardware injections.
APPENDIX B: VALIDATION OF
THE THRESHOLD
The narrow-band search is based on the five-vector
method
[29]
that was implemented originally for
targeted
searches
. In that context, just one template is explored
for each detector, and an overall threshold on the p value
of, say, 1% for the candidate selection, is sufficient to
efficiently recover 95% of injected signals with SNR
¼
8
.
However, in narrow-band searches, we are exploring a large
number of templates in a frequency region of about 0.04 Hz
or more, using two detectors that have different data
quality, i.e., different level of noise and duty cycle. The
threshold in this case is computed by using as noise
background the values of the statistic excluded from the
local maxima selection and then extrapolating the long tails
of the distribution. By definition, these excluded points are
representative of the noise level in the given frequency
bands. This means that, if the noise level in the
10
−
4
Hz
wide frequency subband that we are analyzing is slightly
higher than the noise level in the overall frequency region
from which we are generating noise backgrounds, then
close-to-threshold outliers will occur. These close-to-
threshold outliers may not be completely distinguishable
from the actual noise. As an example, we have generated
200 software injections with amplitude
h
0
fixed to the one
that generated a 1% p-value outlier in the postglitch
analysis of pulsar J
0534
þ
2200
. We have estimated the
recovered signal-to-noise ratio of the injections by inte-
grating coherently more and more data from LHO and
LLO. If the injections are distinguishable from the noise,
we expect 95% of the injections to have a recovered signal-
to-noise ratio greater than 8. However, it is shown by Fig.
2
that this is not the case. For a full coherent LHO-LLO
search, the distribution of the recovered SNR is below 8.
We have also performed the same test by injecting fake
signals with an amplitude
h
0
that would correspond to a
0.1% outlier. In this case, as shown in Fig.
2
, the recovered
SNR of the injections is higher than 8, confirming that the
0.1% p-value threshold represents a more conservative
choice while recovering CW signals.
APPENDIX C: FOLLOW-UP TEST CASES
We report in this Appendix some explanatory plots of the
analysis steps used for outliers follow-up. The first step
TABLE V. Accuracy of the parameter estimation for the O2
hardware injections. The first three columns report the name,
frequency, and spin-down of the hardware injections (reference
time of December 1, 2017 UTC 00
∶
00:00). The last three columns
report the relative errors in percentage for the parameter estima-
tion. The relative errors are defined in the text.
Name
f
gw
(Hz)
_
f
gw
(Hz/s)
ε
h
0
ε
η
ε
ψ
Pulsar 2 575.16
−
1
.
37
×
10
−
13
6% 0.3%
Pulsar 3 108.86
−
1
.
46
×
10
−
17
0.01% 0.3%
2%
Pulsar 5 52.81
−
4
.
03
×
10
−
18
3% 0.07% 1%
Pulsar 8 190.46
−
8
.
65
×
10
−
9
8% 0.03% 0.07%
0
5
10
15
20
0.2
0.4
0.6
0.8
1
Fraction of injections
FIG. 2. Vertical axis: Fraction of injections recovered with a
SNR equal to or higher than the one indicated on the
horizontal
axis
. The different line colors indicate a set of software injections
that would produce an outlier at 1% and 0.1% according to the
evaluation of the noise-only distribution of the detection statistic.
The red dashed vertical line indicates the SNR
¼
8
threshold that
is commonly used to distinguish the signal from the noise.
0.85
4
6
8
10
0.610 0.615 0.620 0.625 0.630 0.635 0.64 0.645 0.65 0.655 0.66
2
2.5
3
3.5
4
FIG. 3. Top: LHO spectrum around the expected frequency of
J
1105
−
6107
. Bottom: LLO spectrum around the expected signal
frequency of J
1105
−
6107
. In both detectors, we see the contribu-
tion of various noise lines which are known combs with fundamental
frequency0.987925HzinLLOand2.109223HzinLHO.
B. P. ABBOTT
et al.
PHYS. REV. D
99,
122002 (2019)
122002-10