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Search for transient gravitational wave signals associated with magnetar bursts during Advanced LIGO’s second
observing run
The LIGO Scientific Collaboration and The Virgo Collaboration
ABSTRACT
We present the results of a search for short and intermediate-duration gravitational-wave signals
from four magnetar bursts in Advanced LIGO’s second observing run. We find no evidence of a signal
and set upper bounds on the root sum squared of the total dimensionless strain (
h
rss
) from incoming
intermediate-duration gravitational waves ranging from 1
.
1
×
10
−
22
at 150 Hz to 4
.
4
×
10
−
22
at 1550 Hz
at 50% detection efficiency. From the known distance to the magnetar SGR 1806-20 (8.7 kpc) we can
place upper bounds on the isotropic gravitational wave energy of 3
.
4
×
10
44
erg at 150 Hz assuming
optimal orientation. This represents an improvement of about a factor of 10 in strain sensitivity from
the previous search for such signals, conducted during initial LIGO’s sixth science run. The short
duration search yielded upper limits of 2
.
1
×
10
44
erg for short white noise bursts, and 2
.
3
×
10
47
erg
for 100 ms long ringdowns at 1500 Hz, both at 50% detection efficiency.
1.
INTRODUCTION
So far, the Laser Interferometer Gravitational-Wave Observatory (LIGO) (Aasi et al. 2015) and Virgo (Acernese
et al. 2015), have reported detections of a handful of gravitational-wave (GW) signals from coalescence of compact
binary systems (Abbott et al. 2016a,b, 2017b,c,d,e). Isolated compact objects may also emit detectable GWs, though
they are predicted to be much weaker than compact binary coalescences (Sathyaprakash & Schutz 2009). Because of
the high energies and mass densities required to generate detectable GWs, neutron stars and supernovae are among
the main targets of non-binary searches. This paper focuses on a type of neutron star: magnetars.
Magnetars are highly magnetized isolated neutron stars (Mereghetti et al. 2015; Woods & Thompson 2006). Orig-
inally classified as Anomalous X-ray Pulsars (AXPs), or Soft Gamma Repeaters (SGRs), some AXPs were observed
acting like SGRs and vice versa. They are now considered to be a single class of objects defined by their power source:
the star’s magnetic field, which, at 10
13
−
10
15
G, is about 100
×
stronger than a typical neutron star. Magnetars
occasionally emit short bursts of soft
γ
-rays, but the exact mechanism responsible for the bursts is unclear. There are
currently 23 known magnetars (and an additional 6 candidates) (Olausen & Kaspi 2014),
1
which were identified based
on observations across wavelengths of bursts, continuous pulsating emission, spin-down rates, and glitches in their
rotational frequency. The bursts last
∼
0
.
1s with luminosity of up to
∼
2
×
10
42
erg s
−
1
, and can usually be localized
well enough to allow identification of the source magnetar. Many magnetars also emit pulsed X-rays and some are
visible in radio.
The large energies involved originally led to the belief that magnetar bursts could be promising sources of detectable
gravitational waves, e.g. Ioka (2001); Corsi & Owen (2011). Further theoretical investigation indicates that most
mechanisms are likely too weak to be detectable by current detectors (Levin & van Hoven 2011; Zink et al. 2012).
Nevertheless, due to the large amount of energy stored in their magnetic fields and known transient activity, magnetars
remain a promising source of GW detections for ground-based detectors with rich underlying physics.
The search presented in this paper is triggered following identification of magnetar bursts by
γ
-ray telescopes. The
methodology is similar to one done during initial LIGO’s sixth science run (Quitzow-James et al. 2017; Quitzow-James
2016) with a few improvements and the use of an additional pipeline targeted toward shorter-duration signals (X-
Pipeline) (Sutton et al. 2010). This pipeline has been used to look for GWs coincident with
γ
-ray bursts (GRBs) (see
Abbott et al. (2017a) for such searches during advanced LIGO’s first observing run).
The first searches for GW counterparts from magnetar activity targeted the 2004 hyperflare of SGR 1806-20. Initial
LIGO data was used to constrain the GW emission associated with the quasi-periodic seismic oscillations(QPOs) of
the magnetar following this catastrophic cosmic event (Matone & M ́arka 2007; Abbott et al. 2007) as well as the
1
See the catalog at http://www.physics.mcgill.ca/
∼
pulsar/magnetar/main.html
arXiv:1902.01557v2 [astro-ph.HE] 10 Feb 2019
2
instantaneous gravitational emission (Kalmus et al. 2007; Abbott et al. 2008). Abbott et al. (2008) and Abadie et al.
(2011) report on GW emission limits associated with additional magnetar activity observed during the initial detector
era until June 2009. LIGO data coinciding with the 2006 SGR 1900+14 storm was additionally analyzed by stacking
GW data Kalmus et al. (2009) corresponding to individual bursts in the storms EM light curve (Abbott et al. 2009).
Additionally, a magnetar was considered as a possible source for a GRB during initial LIGO (GRB 051103), and a
search using X-Pipeline and the Flare pipeline placed upper limits on GW emission from the star’s fundamental ringing
mode (Abadie et al. 2012).
The rest of this paper is laid out as follows: in Section 2, we provide a brief overview of the astrophysics of magnetars
as is relevant to gravitational-wave astronomy and the short bursts used in this analysis. Next, in Section 3 we describe
the methodology of the GW search. Section 4 describes the results and upper limits on possible gravitational radiation
from the studied bursts. The Appendix contains a discussion of the effect of GW polarization on the sensitivity of the
intermediate duration search.
2.
MAGNETAR BURSTS
Magnetars are currently not well understood. Their magnetic fields are strong and complex (Braithwaite & Spruit
2006), and power the star’s activity. Occasionally and unpredictably, magnetars give off short bursts of
γ
-rays whose
exact mechanism is unknown, but may be caused by seismic events, Alfv ́en waves in the star’s atmosphere, magnetic
reconnection events, or some combination of these, e.g. Thompson & Duncan (1995). After some of the brighter bursts
(giant flares, which have been seen only three times), there is a soft X-ray tail which lasts for hundreds of seconds.
Quasi-periodic oscillations (QPOs) have been observed in the tail of giant flares (Israel et al. 2005; Strohmayer &
Watts 2005) and some short bursts (Huppenkothen et al. 2014b,a), during which various frequencies appear, stay for
hundreds of seconds, and then disappear again, indicating a resonance within the magnetar. Many possible resonant
modes in the core and crust of the magnetar have been suggested to cause the QPOs, although it is unclear which
modes actually produce them. Some of these modes, such as f-modes and r-modes, couple well to GWs, and so, if
sufficiently excited, could produce detectable GWs, though current models indicate that they will be too weak (Levin
& van Hoven 2011; Zink et al. 2012). Other modes, such as the lowest order torsional mode, do not create the time
changing quadrupole moment needed for GW emission. None of these models provide precise predictions for emitted
GW waveforms.
This search was performed on data coincident with the four short bursts from magnetars during advanced LIGO’s
second observing run for which there was sufficient data (we require data from two detectors) for both short (less than
a second long) and intermediate (hundreds of seconds long) duration signals. Table 1 describes the four bursts. In
addition to the four studied bursts, there were five bursts that occurred during times when at least one detector was
offline. No GW analysis was done on them. All GW detector data comes from the two LIGO detectors because Virgo
was not taking data during any of these bursts.
Three bursts come from the magnetar SGR 1806-20. They were all identified by the Burst Alert Telescope (BAT)
aboard NASA’s Swift satellite (Gehrels et al. 2004). These were sub-threshold events that were found in BAT data (D.
M. Palmer, personal communication, June 6, 2017), an example of which is shown in Figure 1, with the data from the
other two found in Appendix C. The fourth was a short GRB with a soft spectrum observed by the
Fermi
Gamma-ray
Burst Monitor (Atwood et al. 2009), and named GRB 170304A.
Source
Date
Time
Duration
Fluence
Distance
(UTC)
(s)
(erg cm
−
2
)
(kpc)
SGR 1806-20
Feb 11, 2017
21:51:58
0.256
8
.
9
×
10
−
11
8.7
SGR 1806-20
Feb 25, 2017
06:15:07
0.016
1
.
2
×
10
−
11
8.7
GRB170304A March 4, 2017 00:04:26
0.16
3
.
1
×
10
−
10
–
SGR 1806-20
April 29, 2017 17:00:44
0.008
1
.
4
×
10
−
11
8.7
Table 1.
List of magnetar bursts considered in this GW search. GRB170304A is described in GCN circular 20813; data on
SGR 1806-20 burst activity is courtesy of David M. Palmer.
3.
METHOD
3
sw00737898000bevshto_uf.evt.gz
57.5
58.0
58.5
59.0
seconds of minute 2017-02-11 21:51
50
100
150
200
250
Counts per 8.0 ms
Figure 1.
SWIFT BAT’s data for the February 11 burst from SGR 1806-20. Image courtesy of David M. Palmer.
3.1.
Excess power searches
Fundamentally, all multi-detector GW searches seek to identify GW signals that are consistent with the data collected
at both detectors. Some searches identify candidate signals in each detector separately, then later consider only the
candidates that occur in all detectors within the light-travel time and with the same signal parameters. This approach
is disfavored in searches that do not rely on templates. We cannot perform a templated search here because there
is no current model which can produce templates for magnetar GW bursts. Instead, we first combine the two data
streams to create a time-frequency map where the value in each time-frequency pixel represents some measure of the
GWs (often energy) consistent with the observations from the detectors.
The next step is to identify GW signals in the time-frequency map. This is done by clustering together groups of
pixels, calculating the significance of each cluster with a metric, and searching for the most significant cluster. In order
to cover a broader range of frequencies and time scales, we use two different analysis pipelines which use different
clustering algorithms.
The short-duration search uses seed-based clustering implemented by X-Pipeline, which focuses on groups of bright
pixels (the seed) (Sutton et al. 2010). Specifically, the clusters considered by X-Pipeline are groups of neighboring
pixels that are all louder than a chosen threshold. This approach works well for short-duration searches, but fails for
longer-duration signals for two reasons: random noise will tend to break up the signal into multiple clusters, and each
pixel is closer to the background, so fewer of them will be above the threshold.
We rely on STAMP (Thrane et al. 2011) for the intermediate-duration search. STAMP offers a seedless method
whose clustering algorithm integrates over many, randomly chosen, B ́ezier curves (Thrane & Coughlin 2013, 2014).
Because of this, it can jump over gaps in clusters caused by noise, and thus it is better suited for longer-duration
signals. Additionally, it can build up signal-to-noise ratio (SNR) over many pixels of only slightly elevated SNR. This
method was previously used to search for signals from magnetars during initial LIGO (Quitzow-James et al. 2017;
Quitzow-James 2016).
3.2.
X-Pipeline
X-Pipeline is a software package designed to search for short-duration gravitational wave signals in multiple detectors,
and includes automatic glitch rejection, background calculation, and software injection processing (for details, see
Sutton et al. (2010)). It forms coherent combinations from multiple detectors, thus making it relatively insensitive to
4
non-GW signals, such as instrumental artifacts. X-Pipeline is used primarily to search for GWs coincident with
γ
-ray
bursts (GRBs), but is suitable for any short-duration coherent search.
X-Pipeline takes a likelihood approach to estimating the GW energy found in each time-frequency pixel. It models
the data collected at the detectors as a combination of signal and detector noise, then uses a maximum likelihood
technique to calculate the estimated GW signal power in each time-frequency pixel.
For clustering, X-Pipeline selects the loudest 1% of pixels and connects neighboring pixels. Each connected group is
a cluster, and the clusters are scored based on the likelihood described in the previous paragraph. We want to pick the
time length of the pixels in the time-frequency map so that signal is present in the smallest number of time-frequency
pixels, as this will recover the signal with the highest likelihood. Since we do not have a model for the waveform we
are searching for, we use multiple pixel lengths and run the clustering algorithm on all of them. After clusters are
identified, X-Pipeline identifies which candidate clusters are likely glitches by comparing three measurements of signal
energy: coherent energy consistent with GWs, coherent energy inconsistent with GWs, and sum of the signal energy
in all detectors (referred to as incoherent energy). GW signals can be differentiated from noise by the ratio of coherent
energy inconsistent with GWs to the incoherent energy (see Sections 2.6 and 3.4 of Sutton et al. (2010) for full details).
The primary target of this search are GWs produced from the excitation of the magnetar’s fundamental mode,
which are primarily dampened by the emission of GWs (Detweiler 1975; Andersson & Kokkotas 1998). We have
chosen parameters for X-Pipeline to search for signals a few hundred milliseconds long. The search window begins
4 seconds before the
γ
-rays arrive and ends 4 seconds after. The frequency range for the short-duration search is
64–4000 Hz, and the pixel lengths are every factor of 2 between 2 s and 1/128 s, inclusive.
3.3.
STAMP
STAMP is an unmodeled, coherent, directed excess power search suitable for longer-duration signals, described in
more detail in Thrane et al. (2011). In short, the pipeline calculates the cross power between the two detectors,
accounting for the time delay due to the light travel time between detectors. It then makes this into a time-frequency
SNR map, where pixel SNR is estimated from the variable
〈
ˆ
Y
(
t
;
f,
ˆ
Ω
,
)
〉
= Re
[
̃
Q
IJ
(
t
;
f,
ˆ
Ω)
(
2 ̃
s
∗
I
(
t
;
f
) ̃
s
J
(
t
;
f
)
)
]
. Here,
̃
s
I
(
t
;
f
) is the Fourier transformed data from detector
I
and
̃
Q
IJ
(
t
;
f,
ˆ
Ω) is the filter function required given the
locations and orientations of the pair of detectors and the sky position
ˆ
Ω of the source (Thrane et al. 2011). The ideal
filter function also depends on the polarization of the incoming GWs, which is unknown. Thus the best we can do is
to use the unpolarized filter function which causes a loss of signal power (though this loss is nearly zero for optimal
sky locations). This is more fully discussed in Appendix B.
To identify signals, STAMP uses seedless clustering and searches over a large number of clusters (30 million in this
search, see section III of (Thrane & Coughlin 2013)). For clusters, we use B ́ezier curves, which are parameterized by
three points (Thrane & Coughlin 2013).
GWs radiated through the mechanisms related to QPOs would be monochromatic, or close to it. So, it makes sense
to only search for such signals. Through STAMP, this is easily accomplished by restricting the search to clusters whose
frequencies change by only a small amount. This reduces the number of possible clusters, which means we get all of
the benefits of searching more clusters without the additional computational cost. Restricting the frequency change
too much may cause signals to be missed completely. Compromising between these, we restrict the searched clusters
to those with a frequency change less than 10%.
To estimate the background for this search, we use approximately 15 hours of data from each detector collected
around the time of each burst, excluding the data, coincident with the burst, that was searched for GWs (the on-
source). We can then perform background experiments free of any possible coincident GW signal by pairing data taken
at different times as if they were coincident. Then, breaking up the background data into 33 segments, we generate
1,056 background experiments. Each of these is run in exactly the same way as the on-source analysis, giving an
estimate of the SNR that can be expected from detector noise. The resulting distributions for the background data
for each burst, along with the on-source results, are shown in Figure 2.
With unlimited computing power, we would calculate upper limits by adding software injections of increasing am-
plitude until the desired fraction of injections are recovered. However, this is prohibitively expensive in computing
time. Instead, at each amplitude for the injection, we search only over a few previously identified clusters. To pick
those clusters we do a full run using the seedless algorithm with fifteen injections at varied amplitudes around the
expected recovery threshold and identify clusters which recover the injection, setting the same random seed as was
used for on-source recovery. We then analyze a large number of injections using only the pre-identified clusters, and
5
Figure 2.
SNR distribution of the background (lines) and on-source result (open circles) for each burst for the intermediate-
duration search. As expected, the background distributions are similar; since many background analyses give louder SNR than
the on-source, we conclude that no signal has been detected. Inset: a detailed view of the on-source results.
calculate the maximum SNR of those clusters. The injections at all amplitudes are done with the same time-frequency
parameters. This allows us to efficiently recover the injection by searching a small fraction of the total clusters. This
is shown to work as expected in Appendix A.
Since choosing a random seed also chooses which clusters will be searched, this value can affect the final upper limit
values. This effect is limited by using a large number of clusters (30 million), and analysis performed with different
seeds shows this effect leads to about a 10% uncertainty in the resulting upper limit value. However, this is not a
completely new source of error - changing the seed is functionally equivalent to changing the time-frequency location
of the software injections. In addition, the on-source analysis uses only one seed since it is only run once, so this
uncertainty is a manifestation of the random nature of the search.
4.
RESULTS AND DISCUSSION
No signals were found by either the short- or intermediate-duration searches. We present the results and upper
limits on GW strain and energy for each analysis below.
4.1.
Short-Duration Search Upper Limits
No significant signal was found by X-Pipeline. After glitch rejection, the most significant cluster for the February
25 burst had a p-value of 0.63.
Following the previous f-mode search (Abadie et al. 2012), we injected white noise bursts (frequencies: 100–200 Hz
and 100–1000 Hz; durations: 11 ms and 100 ms), and ringdowns (damped sinusoids, at frequencies: 1500 Hz and
2500 Hz; time constants: 100 ms and 200 ms), and chirplets (chirping sine-Gaussians; this differs from the prior search,
which used sine-Gaussians). The best limits for the white noise bursts were for the 11 ms long bursts in the 100–200
Hz band, at 2
.
1
×
10
44
erg in total isotropic energy and
h
rss
of 5
.
6
×
10
−
23
at the detectors. We are most sensitive to
ringdowns at 1500 Hz and a time constant of 100 ms, with an upper limit of 2
.
3
×
10
47
erg and
h
rss
of 1
.
9
×
10
−
22
.
Directly comparing the
h
rss
limits to Abadie et al. (2011), we see that limits have improved by roughly a factor of 10,
though the ringdowns we used had slightly different parameters. Comparing to Abadie et al. (2012), which provided
only energy upper limits assuming a distance of 3.6 Mpc, we see an improvement of factor of 60 after correcting for
6
the larger distance. This corresponds to roughly a factor of 8 improvement in
h
rss
limits. A full list of upper limits
for the waveforms tested is found in Table 2.
Injection Type
Frequency (Hz) Duration/
h
rss
Energy (erg)
τ
(ms)
chirplet
100
10
5
.
42
×
10
−
23
8
.
49
×
10
43
chirplet
150
6.667
4
.
93
×
10
−
23
1
.
58
×
10
44
chirplet
300
3.333
5
.
29
×
10
−
23
7
.
27
×
10
44
chirplet
1000
1
1
.
15
×
10
−
22
3
.
82
×
10
46
chirplet
1500
0.6667
1
.
69
×
10
−
22
1
.
81
×
10
47
chirplet
2000
0.5
2
.
32
×
10
−
22
5
.
92
×
10
47
chirplet
2500
0.4
3
.
06
×
10
−
22
1
.
56
×
10
48
chirplet
3000
0.3333
3
.
96
×
10
−
22
3
.
65
×
10
48
chirplet
3500
0.2857
5
.
30
×
10
−
22
8
.
51
×
10
48
white noise burst
100–200
11
5
.
57
×
10
−
23
2
.
09
×
10
44
white noise burst
100–200
100
7
.
88
×
10
−
23
4
.
15
×
10
44
white noise burst
100–1000
11
1
.
00
×
10
−
22
1
.
04
×
10
46
white noise burst
100–1000
100
1
.
83
×
10
−
22
3
.
55
×
10
46
ringdown
1500
200
1
.
89
×
10
−
22
2
.
25
×
10
47
ringdown
2500
200
2
.
87
×
10
−
22
1
.
37
×
10
48
ringdown
1500
100
1
.
89
×
10
−
22
2
.
25
×
10
47
ringdown
2500
100
2
.
80
×
10
−
22
1
.
30
×
10
48
Table 2.
Upper limits on isotropic energy from the short-duration search for the February 25 burst from SGR 1806-20. For
white noise bursts, we give the duration of the injection; for the other waveforms, the characteristic time. All limits are given
at 50% detection efficiency, meaning that a signal with the given parameters would be detected 50% of the time.
4.2.
Intermediate Duration Search Upper Limits
To calculate upper limits, we add software injections of two waveforms (half-sine Gaussians and exponentially
decaying sinusoids) at five frequencies (55, 150, 450, 750, and 1550 Hz) and at two timescales (150 seconds and 400
seconds). Reported upper limits are for 50% recovery efficiency, where recovery is defined as finding a cluster, at the
same time and frequency as the injection, with SNR greater than that of the on-source (for the February 25 event, it
was 6.09). Full results are shown in Table 3.
Due to the improved sensitivity of Advanced LIGO, we are able to set strain upper limits about a factor of 10 lower
than the previous search during initial LIGO (Quitzow-James et al. 2017), see Figure 3. Unlike the previous search,
this search showed little difference in
h
rss
sensitivity between the two injection lengths. STAMP has been refined to
improve PSD estimation, which explains the small gap between the injection timescales for this search.
4.3.
Discussion
This search has set the strongest upper limits on short- and intermediate-duration GW emission associated with
magnetar bursts. The energy limits, which are as low as 10
44
–10
47
erg, are now well below the EM energy scale of
magnetar giant flares (10
46
erg). The short bursts analyzed here were much weaker than a giant flare (see Table
1), so for these bursts the limit is much larger than the observed electromagnetic energy. In addition, these limits
assume ideal orientation of the magnetar (both sky position and polarization of produced GWs). The impact of other
polarizations on the intermediate-duration search are discussed in Appendix B, and plotted in Figure 4.
The upper limits set by this search are still far above the GW energy from f-mode excitation during a giant flare
according to Zink et al. (2012), unless the magnetic field strength is far higher than currently accepted value of
2
×
10
15
G (Olausen & Kaspi 2014). Using Equation 2 from Zink et al. (2012), f-mode GW emission from a giant flare
would be about 1
.
4
×
10
38
erg. A surface magnetic field of 1
.
8
×
10
16
G would be required to reach the best upper
limit found with the short-duration source.
7
Frequency (Hz) Tau (sec)
h
rss
Energy (erg)
Half Sine-Gaussian
Ringdown
Half Sine-Gaussian
Ringdown
55
400
2
.
29
×
10
−
22
2
.
43
×
10
−
22
1
.
82
×
10
44
2
.
06
×
10
44
55
150
1
.
97
×
10
−
22
2
.
11
×
10
−
22
1
.
35
×
10
44
1
.
55
×
10
44
150
400
1
.
32
×
10
−
22
1
.
37
×
10
−
22
4
.
52
×
10
44
4
.
86
×
10
44
150
150
1
.
14
×
10
−
22
1
.
22
×
10
−
22
3
.
37
×
10
44
3
.
89
×
10
44
450
400
1
.
69
×
10
−
22
1
.
79
×
10
−
22
6
.
62
×
10
45
7
.
47
×
10
45
450
150
1
.
78
×
10
−
22
1
.
83
×
10
−
22
7
.
43
×
10
45
7
.
83
×
10
45
750
400
2
.
56
×
10
−
22
2
.
70
×
10
−
22
4
.
21
×
10
46
4
.
69
×
10
46
750
150
2
.
11
×
10
−
22
2
.
37
×
10
−
22
2
.
87
×
10
46
3
.
61
×
10
46
1550
400
5
.
86
×
10
−
22
6
.
22
×
10
−
22
9
.
21
×
10
47
1
.
03
×
10
48
1550
150
4
.
38
×
10
−
22
4
.
58
×
10
−
22
5
.
16
×
10
47
5
.
62
×
10
47
Table 3.
Upper limits on GW strain and energy from the intermediate-duration search for the February 25 burst from SGR
1806-20. All limits are at 50% detection efficiency.
As the LIGO detectors increase in sensitivity, these upper limits will improve, and will be well-positioned to place
meaningful limits on emitted GW energy in the event of a future nearby magnetar giant flare.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the con-
struction 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 LIGO and construction and operation of the GEO600 detector. Additional
support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge 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 Re-
search, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium.
The authors also gratefully 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 (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investi-
gaci ́on, the Vicepresid`encia i Conselleria d’Innovaci ́o, Recerca i Turisme and the Conselleria d’Educaci ́o i Universitat
del Govern de les Illes Balears, the Conselleria d’Educaci ́o, Investigaci ́o, Cultura i Esport de la Generalitat Valenciana,
the National Science Centre of Poland, the Swiss National Science Foundation (SNSF), the Russian Foundation for
Basic Research, the Russian Science Foundation, the European Commission, the European Regional Development
Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hun-
garian Scientific Research Fund (OTKA), the Lyon Institute of Origins (LIO), the Paris
ˆ
Ile-de-France Region, the
National Research, Development and Innovation Office Hungary (NKFIH), the National Research Foundation of Ko-
rea, 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 (ICTP-SAIFR), the Research Grants Council of Hong
Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation,
the Ministry of Science and Technology (MOST), Taiwan and the Kavli Foundation. The authors gratefully acknowl-
edge the support of the NSF, STFC, MPS, INFN, CNRS and the State of Niedersachsen/Germany for provision of
computational resources.
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APPENDIX
A.
VALIDATION OF THE INJECTION RECOVERY METHOD
In Section 3, we described the way we obtain the precise detectability threshold of injected waveforms. Given that
this method is an approximation of the full search protocol, we check to see how closely the results match that of the
full search. Figure 5 shows the result of this test, run over 100 injection trials. A point falling below the diagonal
indicates that the noise and injected signal conspire together in such a way that the cluster with the highest SNR is
one that had not been previously identified. This test indicates that this happens infrequently. On average, we recover
98% of the SNR that is found by the full search.
Figure 5.
Direct comparison the SNR recovered by a full run of 30 million clusters and a run with only a few previously
selected clusters. This shows that this method is highly effective in recovering the injected signal, at a small fraction of the
computational cost.
B.
STAMP POLARIZATION
The cross power of a pixel in the time-frequency map is calculated with
ˆ
Y
≡
Re
[
̃
Q
IJ
(
t
;
f,
ˆ
Ω)
C
IJ
(
t
;
f
)
]
, where
̃
Q
IJ
(
t
;
f,
ˆ
Ω) is the filter function which depends on the polarization of the GWs. Since we do not know the polarization
of the GWs that we are looking for, we use an approximation of the ideal filter function. We choose the unpolarized
filter function
̃
Q
IJ
(
t
;
f,
ˆ
Ω) =
1
IJ
(
t
;
ˆ
Ω)
e
2
πif
ˆ
Ω
·
∆
~x
IJ
/c
(Thrane et al. 2011). The second term in this equation corrects for
a difference in phase due to different travel distances from the source to the two detectors. For this discussion, we will
set it to 1 for simplicity.
If the GWs emitted from magnetars are due to a pure quadrupole, we expect them to be polarized elliptically
according to standard linearized gravity, with the polarization depending on the orientation of the source’s time
changing quadrupole moment with respect to our detectors.
Using Equation A48 from Thrane et al. (2011), we can see how the cross power of an individual pixel will change
with changing polarization. In Fig. 6, we plot this parametrically in the complex plane for two of the bursts analyzed.
The main effect of the filter function is to pick a phase (the cross power is multiplied by the filter function, then the
real part is taken). The unpolarized filter function sets that phase to zero, so the imaginary part of the cross power
is ignored, which limits the sensitivity to nearly all elliptically polarized waveforms. Because the time-frequency map
is normalized by the average value of the power spectral density nearby in time, the magnitude of the filter function
only matters if it changes over the course of a time-frequency map.
In addition, for some polarizations, the cross power is negative, which will result in pixels in the SNR map being
negative. For this reason these signals, no matter how large in amplitude, would be missed by this search. However,
Figure 6.
Parametric plots of the complex valued cross power due to elliptically polarized signals of varying polarizations from
two different sky locations (left: SGR 1806-20 during the event on April 29, right: SGR 1806-20 during the event on February
25). The polarization of incoming GWs is defined by two angles,
ι
and
ψ
.
ι
is the angle between the vector from Earth to the
source and the source’s rotation vector, while
ψ
indicates the orientation of the source’s rotation vector when projected into a
plane perpendicular to the propagation vector. The ends of the boomerang are at
ι
= 0
,π
; changing
ψ
changes the real part
only. For ideal sky positions, the boomerang collapses to the real axis and reaches about 0.95. It does not reach 1 because the
detectors are not aligned.
these signals produce little cross power (as evidenced by how close they are to zero) such that the likelihood of detecting
them even with the ideal filter function is low. It is possible to recover these by taking the absolute value of each
cluster, but doing this is roughly equivalent to running the search twice with opposite phase filter functions. Thus, it
will double the background.
C.
SUPPLEMENTARY DATA