D
RAFT VERSION
O
CTOBER
8, 2018
Typeset using L
A
T
E
X
twocolumn
style in AASTeX62
Search for gravitational waves from a long-lived remnant
of the binary neutron star merger GW170817
T
HE
LIGO S
CIENTIFIC
C
OLLABORATION AND
T
HE
V
IRGO
C
OLLABORATION
(Dated: 2018-10-05; report no. LIGO-P1800195)
ABSTRACT
One unanswered question about the binary neutron star coalescence GW170817 is the nature of its post-
merger remnant. A previous search for post-merger gravitational waves targeted high-frequency signals from
a possible neutron star remnant with a maximum signal duration of 500 s. Here we revisit the neutron star
remnant scenario with a focus on longer signal durations up until the end of the second Advanced LIGO-Virgo
observing run, 8.5 days after the coalescence of GW170817. The main physical scenario for such emission is
the power-law spindown of a massive magnetar-like remnant. We use four independent search algorithms with
varying degrees of restrictiveness on the signal waveform and different ways of dealing with noise artefacts. In
agreement with theoretical estimates, we find no significant signal candidates. Through simulated signals, we
quantify that with the current detector sensitivity, nowhere in the studied parameter space are we sensitive to a
signal from more than
1 Mpc
away, compared to the actual distance of 40 Mpc. This study however serves as a
prototype for post-merger analyses in future observing runs with expected higher sensitivity.
Keywords:
gravitational waves – methods: data analysis – stars: neutron
1.
INTRODUCTION
The binary neutron star (BNS) observation GW170817 (Ab-
bott et al. 2017d) was the first multimessenger astronomy
event jointly detected in gravitational waves (GWs) and
at many electromagnetic (EM) wavelengths (Abbott et al.
2017e). It originated remarkably close to Earth, with a dis-
tance of
38
+8
−
18
Mpc
1
as measured by the LIGO and Virgo
GW detectors (Aasi et al. 2015a; Acernese et al. 2015) alone
and consistent EM distance estimates for the host galaxy
NGC4993 (Sakai et al. 2000; Freedman et al. 2001; Hjorth
et al. 2017; Lee et al. 2018).
A BNS merger is expected to leave behind a remnant com-
pact object, either a light stellar-mass black hole or a heavy
neutron star (NS), which can emit a variety of post-merger
GW signals. These are more difficult to detect than the pre-
merger inspiral signal, but the nearby origin of GW170817
has still generated interest in searching for a post-merger sig-
nal. Identifying the nature of the remnant would be highly
valuable for improving, among other things, constraints on
the nuclear equation of state (EoS) (Margalit & Metzger
2017; Bauswein et al. 2017; Rezzolla et al. 2018; Radice et al.
1
Updated distance estimate corresponding to Fig. 3 of Abbott et al.
(2018d), where the sky location of the counterpart is not assumed, hence
differing slightly from the one quoted in the text for fixed-location runs.
2018) over those obtained from the inspiral alone (e.g., Ab-
bott et al. 2017d, 2018d,c).
Abbott et al. (2017g) presented a first model-agnostic
search for short (
.
1
s) and intermediate-duration (
.
500
s)
GW signals. No signal candidates were found. The search
sensitivity was estimated for several GW emission mech-
anisms: oscillation modes of a short-lived hypermassive
NS, bar-mode instabilities, and rapid spindown powered by
magnetic-field induced ellipticities.
For all mechanisms,
a realistic signal from a NS remnant of GW170817 could
only have been detected with at least an order of magnitude
increase in detector strain sensitivity. A seconds-long post-
merger signal candidate was reported by van Putten & Della
Valle (2018) with an estimated GW energy lower than the
sensitivity estimates of Abbott et al. (2017g).
An additional analysis in Abbott et al. (2018d) used a
Bayesian wavelet-based method to put upper limits on the
energy and strain spectral densities over 1 s of data around
the coalescence. These strain upper limits are 3–10 times
above the numerical relativity expectations for post-merger
emission from a hypermassive NS at 40 Mpc.
In this paper, we focus on a long-lived NS remnant, cover-
ing possible signal durations which at the long end are lim-
ited by the end of the second observing run (O2) on 2017-
08-26, giving a total data set spanning 8.5 days from merger.
The shortest signal durations we cover are
∼
hundreds of
seconds after merger, so that the new search presented here
arXiv:1810.02581v1 [gr-qc] 5 Oct 2018
2
only partially overlaps with the intermediate-duration search
from Abbott et al. (2017g). We assume the sky location of the
EM counterpart (Coulter et al. 2017; Abbott et al. 2017e).
From considerations of realistic remnant NS properties,
detailed in Sec. 2, we do not expect to make a detection
with this search. Instead, the goal—as before in Abbott et al.
(2017g)—is mainly to make sure that no unexpected signal
is missed in the longer-duration part of the parameter space.
This study also serves as a rehearsal for future post-merger
searches with improved detectors. Hence, we use four search
methods with varying restrictiveness on the signal shape and
different ways of dealing with noise artefacts: two generic
unmodeled algorithms and two that use templates based on a
power-law spin-down waveform model.
The
Stochastic Transient Analysis Multidetector Pipeline
(STAMP, Thrane et al. 2011) is an unmodeled method us-
ing cross-power spectrograms. It was already used for the
intermediate-duration analysis in Abbott et al. (2017g), but
is employed here in a different configuration optimized for
much longer signal lengths.
The other three algorithms are derived from methods orig-
inally developed to search for continuous waves (CWs): per-
sistent, nearly-monochromatic GW signals from older NSs.
(For reviews, see Prix 2009; Riles 2017.). Some CW searches
have targeted relatively young NSs (Aasi et al. 2015b; Sun
et al. 2016; Zhu et al. 2016), and adaptations of CW search
methods to long-duration transient signals have been sug-
gested before (Prix et al. 2011; Keitel 2016). However, the
present search is the first time that any CW algorithms have
been modified in practice (on real data) to deal with transients
of rapid frequency evolution.
Specifically, these three are
Hidden Markov Model
(HMM)
tracking (Suvorova et al. 2016; Sun et al. 2018a)—a
template-free algorithm previously used to search for CWs
from the binary Scorpius X-1 (Abbott et al. 2017f)—and
two new model-dependent methods—
Adaptive Transient
Hough
(ATrHough, Oliver & Sintes 2018) and
General-
ized FrequencyHough
(Miller et al. 2018)—based on algo-
rithms (Krishnan et al. 2004; Sintes & Krishnan 2007; Aasi
et al. 2014; Palomba et al. 2005; Antonucci et al. 2008; As-
tone et al. 2014) previously used in CW all-sky searches (e.g.,
most recently in Abbott et al. 2017a, 2018b).
After discussing the astrophysical motivation and context
for this search in Sec. 2, presenting the analyzed data set in
Sec. 3 and the four search methods in Sec. 4, we discuss the
combined search results in Sec. 5 and conclude with remarks
on future applications in Sec. 6. Additional results and details
on the search methods are given in the appendices.
2.
ASTROPHYSICAL BACKGROUND AND
WAVEFORM MODEL
The probability for a long-lived NS remnant after a BNS
merger depends on the progenitor properties and on the nu-
clear EoS (Baiotti & Rezzolla 2017; Piro et al. 2017). Us-
ing the progenitor masses and spins as measured from the
inspiral (Abbott et al. 2017d, 2018d), for many EoS the pre-
ferred scenarios are prompt collapse to a black hole or the
formation of a hypermassive NS whose mass cannot be sup-
ported by uniform rotation and thus collapses in
.
1
s (Ab-
bott et al. 2017c). However, a supramassive NS—less mas-
sive, but above the maximum mass of a non-rotating NS and
stable for up to
∼
10
4
s (Ravi & Lasky 2014)—or even a long-
time stable NS could also be consistent with some physically-
motivated EoS which allow for high maximum masses.
From the EM observational side, circumstantial evidence
points towards a short-lived hypermassive NS (Kasen et al.
2017; Granot et al. 2017b,a; Pooley et al. 2018; Matsumoto
et al. 2018); though several authors (Yu et al. 2018; Ai et al.
2018; Geng et al. 2018; Li et al. 2018) consider continued en-
ergy injection from a long-lived remnant NS. Given this in-
conclusive observational situation, we agnostically consider
the possibility of GW emission from a long-lived remnant
NS and seek here to constrain it from the LIGO data.
In two of our search methods, and to estimate search sen-
sitivities with simulations, we use a waveform model (Lasky
et al. 2017; Sarin et al. 2018) originating from the general
torque equation for the spindown of a rotating NS:
̇
Ω =
−
k
Ω
n
.
(1)
Here,
Ω = 2
πf
and
̇
Ω
are the star’s angular frequency and
its time derivative, respectively, and
n
is the braking index.
A value of
n
≤
3
corresponds to spindown predominantly
through magnetic dipole radiation and
n
= 5
to pure GW
emission (Shapiro & Teukolsky 1983). A braking index of
n
= 7
is conventionally associated with spindown through
unstable
r
-modes (e.g., Owen et al. 1998), although the true
value can be less for different saturation mechanisms (Alford
& Schwenzer 2015, 2014). The value of
k
also depends on
these mechanisms; together with the starting frequency
Ω
0
it
defines a spin-down timescale parameter
τ
=
−
Ω
1
−
n
0
k
(1
−
n
)
.
(2)
Integrating Eq. (1) and solving for the GW frequency gives
the GW frequency evolution
f
gw
(
t
) =
f
gw0
(
1 +
t
τ
)
1
/
(1
−
n
)
,
(3)
where
f
gw
= 2
f
,
f
gw0
is the initial frequency at a starting
time
t
start
(e.g., coalescence time
t
c
of the BNS merger), and
t
is measured relative to
t
start
.
3
The dimensionless GW strain amplitude for a non-
axisymmetric rotating body following Eq. 3 is given by
h
0
(
t
) =
4
π
2
GI
zz
c
4
d
f
2
gw0
(
1 +
t
τ
)
2
/
(1
−
n
)
.
(4)
Here,
I
zz
is the principal moment of inertia,
is the ellipticity
of the rotating body,
d
is the distance to the source,
G
is
the gravitational constant, and
c
is the speed of light. This
model assumes that
n
,
and
I
zz
are constant throughout the
spin-down phase, while in reality the spindown could be e.g.
GW-dominated at early times and then transition into EM
dominance, and
I
zz
can decrease with
Ω
.
Our set of pipelines also allows for the power-law spin-
down model to be valid for only part of the observation
time. The FreqHough analysis only starts after delay times
of
∆
t
= 1
–7 hours after the merger (
t
c
≈
1187008882
.
443
in GPS seconds), to accomodate the possibility that the new-
born NS has not immediately settled into a state that obeys
the power-law model, and thus provides complementary con-
straints to the other searches. The unmodeled STAMP search
is also sensitive to signals starting at a later time, as it
does not impose a fixed starting time for any time-frequency
tracks. Moreover, neither STAMP nor HMM impose the spe-
cific waveform model for their initial candidate selection.
The detectability of long-lived post-merger GWs accord-
ing to the Eq. 3 spindown model has been explored by Sarin
et al. (2018). With aLIGO at design sensitivity (Abbott et al.
2018e) and an optimal matched-filter analysis, they found
that at
d
=
40 Mpc an ellipticity
∼
10
−
2
and timescale
τ
&
10
4
s would be required. However, such large
and long
τ
would imply more energy emitted than is available from
the remnant’s initial rotation. A detectable signal with cur-
rent pipelines and O2 detector sensitivities only seems pos-
sible for short
τ
but extremely large
≥
0
.
1
. Such ellip-
ticities are physically unlikely (Johnson-McDaniel & Owen
2013) and would require internal magnetic fields greater than
∼
10
17
G (e.g., Cutler 2002), for which very rapid EM-
dominated dominated spindown would be expected.
For r-modes, the GW strain follows a different relation
than Eq. 4 (Owen et al. 1998), but the physically relevant
parameter (the saturation amplitude) is also expected to be
small (Arras et al. 2003; Bondarescu et al. 2009). They could
be an important emission channel especially at high frequen-
cies, and the search presented in this paper also covers brak-
ing indices up to
n
= 7
. The sensitivity estimates presented
in Sec. 5 however, for simplicity, will be for
n
= 5
only.
Still, with this first search for long-duration post-merger
signals, we demonstrate that available analysis methods can
comprehensively cover the relevant parameter space, and
thus will be ready once detector sensitivity has improved, or
in the case of a fortunate, very nearby BNS event.
3.
DETECTORS AND DATA SET
10
2
10
3
f
gw
[Hz]
10
−
23
10
−
22
10
−
21
ASD [1
/
√
Hz]
H1
L1
0
1
2
3
4
5
6
7
8
T from GPS=1187008882s [days]
H1
HL
L1
Figure 1.
Top panel: Noise strain amplitude spectral den-
sity (ASD) curves of LIGO Hanford (H1) and Livingston (L1)
on August 17, 2018. (Averaged over 1800 s stretches including
GW170817.) Lower panel: Analysable science mode data segments
for the remaining O2 run after the GW170817 event. Vertical dot-
ted lines mark the analysis end times, from left to right, for HMM,
FreqHough, ATrHough and STAMP.
In this analysis, we use data from the two Advanced LIGO
(aLIGO) detectors in Hanford, Washington (H1) and Liv-
ingston, Louisiana (L1). No data from Virgo (Acernese et al.
2015) or GEO600 (Dooley et al. 2016) was used because of
their lower sensitivity.
2
Three of the pipelines use data up to
2 kHz; STAMP also uses data up to 4 kHz. Both detectors in
their O2 configuration had their best sensitivity in the 100–
200 Hz range, with significantly less sensitivity in the kHz
range (e.g. a factor
∼
4 worse in strain at 2 kHz)—see Fig. 1.
For the lower analysis cutoff of each pipeline, see Sec. 4.
Starting from a rounded GW170817 coalescence time of
t
c
≈
1187008882
s, the HMM pipeline uses 9688 s of data
(until the first gap in H1 data), ATrHough uses 1 day of data
after
t
c
, FreqHough analyzes from 1 to about 18 hours af-
ter
t
c
in different configurations, and STAMP analyzes the
whole 8.5 days of data until the end of O2 on 2017-08-26.
The duty cycle of both detectors (H1, L1) was 100% during
the first 9688 s after the merger, (70%, 78%) for the first day
(62% in coincidence), and (83%, 85%) for the full data set
(75% in coincidence). The analysed data segments are also
illustrated in Fig. 1. STAMP processes the
h
(
t
)
strain data
into cross-power time-frequency maps (see Sec. 4.1 for de-
2
E.g. at 500 Hz the noise strain amplitude spectral density was about a
factor of
∼
5
for Virgo and
∼
20
for GEO600 worse than for L1 in late O2.
4
tails), while for the other pipelines the basic analysis units
are Short Fourier Transforms (SFTs) of 1–8 s duration.
Several known noise sources have been subtracted from
the strain data using a new automated procedure (Davis
et al. 2018) applied to the full O2 data set, processing a
much larger amount of time than the cleaning method (Drig-
gers et al. 2018) used for the shorter data sets analysed in
previous GW170817 publications.
Calibration uncertain-
ties (Cahillane et al. 2017) for this data set are estimated as
below 4.3% in amplitude and 2.3 degrees in phase for 20–
2000 Hz, and 4.5% in amplitude and 3.8 degrees in phase for
2–4 kHz; these are tighter than for the initial calibration ver-
sion used in Abbott et al. (2017g). These uncertainties are not
explicitly propagated into the sensitivity estimates presented
in this paper, since they are smaller than other uncertainty
contributions and the degeneracies in amplitude parameters.
4.
SEARCH METHODS AND CONFIGURATIONS
Here we briefly describe the four search methods, first the
unmodeled STAMP and HMM pipelines and then the two
Hough pipelines tailored to the power-law spindown model.
Additional details can be found in Appendix A.
Each analysis uses the known sky location of the counter-
part near RA = 13.1634 hrs, Dec. =
−
23
.
3815
◦
(Coulter et al.
2017; Abbott et al. 2017e), but makes different choices for
the analysed data span. The recovery efficiency of each algo-
rithm is studied with simulated signals under the waveform
model from Sec. 2, as described in Sec. 5 and Appendix B.
A summary of configurations for all four pipelines, both
for the main search and the sensitivity estimation simula-
tions, is given in Table 1.
4.1.
STAMP
STAMP (Thrane et al. 2011) is an unmodeled search
pipeline designed to detect gravitational wave transients.
Its basic unit is a spectrogram made from cross-correlated
data between two detectors. Narrowband transient gravita-
tional waves produce tracks of excess power within these
spectrograms, and can be detected by pattern recognition
algorithms. Each spectrogram pixel is normalized with the
noise to obtain a signal-to-noise ratio (SNR) for each pixel.
STAMP was used in the first GW170817 post-merger
search (Abbott et al. 2017g) in a configuration with 500 s
long spectrograms. To increase sensitivity to longer GW sig-
nals, here we use spectrogram maps of 15 000 s length. The
search is split into two frequency bands from 30–2000 Hz
and 2000–4000 Hz. The former uses pixels of
100 s
×
1 Hz
,
while the latter uses shorter-duration pixels of
50 s
×
1 Hz
to
limit SNR loss due to the Earth’s rotation changing the phase
difference between detectors.
We then use Stochtrack (Thrane & Coughlin 2013), a seed-
less clustering algorithm, to identify significant clusters of
pixels within these maps. The algorithm uses one million
quadratic B
́
ezier curves as templates for each map, and the
loudest cluster is picked for each map. More details about
the pixel size choice, the detection statistic and the search
results are in Appendix A.1.
The on-source data window is from just after the time of
the merger to the end of O2 (1187008942–1187733618). To
measure the background and estimate the significance of the
clusters found, we run the algorithm on time-shifted data
from June 24th to just before the merger.
4.2.
HMM tracking
Hidden Markov model (HMM) tracking provides a com-
putationally efficient strategy for detecting and estimating
a quasimonochromatic GW signal with unknown frequency
evolution and stochastic timing noise (Suvorova et al. 2016;
Sun et al. 2018a). It was applied to data from the first aLIGO
observing run to search for CWs from the low-mass X-ray bi-
nary Scorpius X-1 (Abbott et al. 2017f). The revision of the
algorithm in Sun et al. (2018a) is also well suited to search-
ing for a long-transient signal from a BNS merger remnant,
if the spin-down time-scale is in the range
10
2
s
.
τ
.
10
4
s.
A HMM is an automaton based on a Markov chain (a
stochastic process transitioning between discrete states at dis-
crete times), composed of a hidden (unmeasurable) state vari-
able and a measurement variable. A HMM is memoryless,
i.e., the hidden state at time
t
n
+1
only depends on the state at
time
t
n
, with a certain transition probability. The most prob-
able sequence of hidden states given the observations is com-
puted by the classic Viterbi algorithm (Viterbi 1967). Details
on the probabilistic model can be found in Appendix A.2.
In this analysis, we track the GW signal frequency as the
hidden variable, with its discrete states mapped one-to-one
to the frequency bins in the output of a frequency-domain es-
timator computed over an interval of length
T
drift
. We aim
at searching for signals with
10
2
s
.
τ
.
10
4
s, such that the
first time derivative
̇
f
gw
of the signal frequency
f
gw
satisfies
̇
f
gw
≈
f
gw
/τ
.
1
Hz s
−
1
, given
T
drift
= 1
s and a frequency
bin width of
∆
f
= 1
Hz. The motion of the Earth with re-
spect to the solar system barycenter (SSB) can be neglected
during a
T
drift
interval. Hence we use a running-mean nor-
malized power in SFTs with length
T
SFT
=
T
drift
= 1
s as
the estimator to calculate the HMM emission probability.
We analyze 9688 s of data (GPS times 1187008882–
1187018570) in a 100–2000 Hz frequency band with mul-
tiple configurations optimized for different
τ
. We do not an-
alyze longer data stretches because (i) several intervals in the
data after GPS time 1187018570 are not in analysable sci-
ence mode, and (ii) signals with
10
2
s
.
τ
.
10
4
s drop be-
low the algorithm’s sensitivity limit after
∼
10
4
s; observing
longer merely accumulates noise without improving SNR.
The 9688 SFTs are Hann-windowed. The detection statis-
5
Table 1.
Configurations of the four analysis pipelines used in this paper.
STAMP
HMM
ATrHough
FreqHough
search start
a
t
c
t
c
t
c
t
c
+
(1.5–7) hours
b
search duration [hours]
201.3
c
2.7
24
2–18
b
f
gw
data range [Hz]
30–4000
c
100–2000
187–2000
50–2000
n
coverage
unmodeled
unmodeled
2.5–7.0
2.5–7.0
f
start
coverage [Hz]
d
unmodeled
unmodeled
500–2000
500–2000
τ
coverage [s]
unmodeled
unmodeled
10
2
–
10
5
10
–
10
5
injection set for sensitivity estimation
e
signal start
a
random
t
c
t
c
t
c
+
(1.5–7) hours
b
n
coverage
5.0
2.5–7.0
5.0
5.0
f
start
coverage [Hz]
d
500–3000
500–2000
550–2000
390–2000
τ
coverage [s]
10
2
–
10
4
10
2
–
10
4
6
×
10
2
–
3
×
10
4
4
×
10
2
–
2
×
10
4
inclination
cos
ι
0.0, 1.0
random
0.0, 1.0
random
a
Coalescence time is
t
c
≈
1187008882
in GPS seconds.
b
FreqHough search start and duration vary across parameter space.
c
In separate maps of 15 000 s length and 20–2000 Hz and 2000–4000 Hz configurations.
d
f
start
=
f
gw
(
t
= 0)
for HMM and ATrHough;
f
start
=
f
gw
(
t
= ∆
t
)
for STAMP and FreqHough.
e
Discrete sets of injections within these ranges; not all combinations used. See Sec. 5 and the per-pipeline
tables in the appendix for details.
tic
P
is defined in Eqn. A11. The methodology and analysis
is fully described in Sun et al. (2018b).
4.3.
Adaptive Transient Hough
The Adaptive Transient Hough search method will be de-
scribed in detail by Oliver & Sintes (2018). It follows a semi-
coherent strategy similar to the SkyHough (Krishnan et al.
2004; Sintes & Krishnan 2007; Aasi et al. 2014) all-sky CW
searches, but adapted to rapid-spindown transient signals.
We start from data in the form of Hann-windowed SFTs
with lengths of [1,2,4,6,8] s, covering one day after merger
(GPS times 1187008882–1187095282). These are digitized
by setting a threshold of 1.6 on their normalized power, as
first derived by Krishnan et al. (2004), replacing each SFT
by a collection of zeros and ones called a peak-gram. For
each point in parameter space, the Hough number count is
the weighted sum of the peak-grams across a template track
accounting for Doppler shift and the spindown of the source.
The use of weights minimizes the influence of time-varying
detector antenna patterns and noise levels (Sintes & Krishnan
2007). For this post-merger search it also accounts for the
amplitude modulation related to the transient nature of the
signal.
The search parameter space for the model from Sec. 2
covers a band of 500–2000 Hz in starting frequencies
f
gw0
,
braking indices of
2
.
5
≤
n
≤
7
and spindown timescales of
10
2
≤
τ
≤
10
5
s. The search runs over 16042 subgroups,
each containing a range of 150 Hz in
f
0
, 0.25 in
n
and
1000
s
in
τ
. Each subgroup is analyzed with the longest possible
SFTs according to the criterion (Oliver & Sintes 2018)
T
SFT
≤
√
(
n
−
1)
τ
√
f
gw
,
(5)
and for each template the observation time is selected as
T
obs
= min(4
τ,
24 hours)
. Over the whole template bank,
the search uses data from 187–2000 Hz.
Each template is ranked based on the deviation of its
weighted number count from the theoretical expectation
for Gaussian noise (the critical ratio) as described in ap-
pendix A.3. The detection threshold corresponds to a two-
detector
5
σ
false alarm probability for the entire template
bank. A per-detector critical ratio threshold was also set to
check the consistency of a signal between H1 and L1.
4.4.
Generalized FrequencyHough
The FrequencyHough is a pattern-recognition technique
originally developed to search for CWs by mapping points
in time-frequency space of the detector to lines in frequency-
spindown space (Antonucci et al. 2008; Astone et al. 2014).
This only works if the signal frequency varies in time very
slowly. Miller et al. (2018) have generalized the Frequency-
Hough for postmerger signals, where we expect much higher
spindowns.
The search starts at a time offset
∆
t
=
t
start
−
t
c
after
coalescence time
t
c
, so that the waveform model is inter-
preted with starting frequency
f
start
=
f
gw
(
t
= ∆
t
)
taking
6
the place of
f
gw0
in Eq. 3. In this way, assuming that the
model can be extrapolated backwards to
t
c
, we would be
probing higher initial frequencies and spindowns through a
less challenging parameter space during the search window,
while also allowing for the case that the remnant does not ac-
tually follow the power-law model immediately after merger.
Furthermore, the source parameters
(
n,f
start
,τ
) are trans-
formed to new coordinates such that in the new space the
behavior of the signal is linear. See appendix A.4 for the
transformation relations.
We search across the parameter space with a fine, nonuni-
form grid: For each braking index
n
, we do a Hough trans-
form and then record the most significant candidates over the
parameter range of the resulting map. This is done separately
on the data from each detector, and then we check candi-
dates for coincidence between detectors according to their
Euclidean distance in parameter space.
The search is run in three configurations using varying
T
SFT
= 2
,
4
,
8
s, covering different observing times, start-
ing
∆
t
= 1
–
7
hours after merger. It covers
n
= [2
.
5
,
7]
,
f
start
= [500
,
2000]
Hz and
τ
= [10
,
10
5
]
s, analyzing detec-
tor data from 50 to 2000 Hz.
Candidates are also ranked by critical ratio (deviation from
the theoretical expectation for Gaussian noise) in this analy-
sis. Most can be vetoed by the coincidence step or by con-
sidering detector noise properties; a follow-up procedure for
surviving candidates is also described in appendix A.4.
5.
SEARCH RESULTS AND SENSITIVITY ESTIMATES
5.1.
Absence of significant candidates
The four search methods all either found no significant
candidates in the aLIGO data after GW170817; or those that
were found, were clearly vetoed as instrumental artifacts.
For the unmodeled STAMP search, the most significant
triggers in the low- and high-frequency bands have SNRs of
3.18 and 3.07 respectively. These correspond to insignificant
false alarm probabilities of 0.81 and 0.80 as measured from
the time-shifted background. (Seen Fig. 4 in appendix.)
For HMM, the loudest trigger has a detection statistic
P
= 2
.
6749
(as defined in Eq. A11), corresponding to a
false-alarm probability of 0.01, right below the threshold
set beforehand as significant enough for further study. The
trigger is found with observing time
T
obs
= 200
s starting
from
t
=
t
c
. Monte-Carlo simulations show that for signals
that this setup is sensitive to, higher
P
should be obtained
with longer
T
obs
. Follow-up analysis of the trigger with
300 s
≤
T
obs
≤
1000 s
confirms that it does not follow this
expectation; hence it is discarded as spurious.
ATrHough found 51 initial candidates over the covered part
of
(
n,f
gw0
,τ
)
parameter space. All of these were excluded
with the follow-up procedure described in appendix A.3 as
inconsistent with the expected spindown model and more
likely to be caused by monochromatic detector artifacts
(lines) contaminating the search templates.
The FreqHough search returned 521 candidates over the
covered part of
(
n,f
start
,τ
)
parameter space. We vetoed 10
of them because they were within frequency bands contami-
nated with known noise lines (Covas et al. 2018). 510 of the
remaining candidates had much higher (
>
4
times) critical
ratios in H1 than in L1, which is inconsistent with true astro-
physical signals when considering the relative sensitivities,
duty factors and antenna patterns. There was one remain-
ing candidate, with a critical ratio of 5.21 in H1 and 4.88 in
L1, which was followed up and excluded with the procedure
described in appendix A.4.
5.2.
Sensitivity estimates with simulated signals
Starting from this non-detection result, we use simulated
signals according to Eq. 3 to quantify the sensitivity of each
analysis given the data set around the time of GW170817
and its known sky location. The sets of injected parameters
are different for each pipeline, and there are also some dif-
ferences in procedure: STAMP performs injections on the
same data as the main search but with a non-physical time
shift between the detectors (as in Abbott et al. 2017g, 2018a);
HMM injects signals into the original set of SFTs but with
randomly permuted timestamps; and the other two pipelines
inject signals into exactly the same data as analysed in the
main search. HMM and ATrHough perform all injections
starting at merger time
t
c
, with
f
gw0
in Eq. 3 interpreted
as the frequency at
t
c
, while injections for FreqHough are
done at
∆
t
= 1
,
2
or
5
hours after
t
c
, chosen as representative
starting times for each search configuration, and
f
gw0
cor-
respondingly set at
t
c
+ ∆
t
. Similarly, STAMP treats
f
gw0
as the starting frequency of each injection, which have
∆
t
distributed through the whole search range, yielding a time-
averaged sensitivity. In the following, we use
f
start
to refer
to any of these choices.
These differences in injection procedure, and different
choices of detection threshold, mean that any comparison of
the following results does not correspond to a representative
evaluation of general pipeline performance, but is solely in
the interest of estimating how much sensitivity is missing for
a GW170817-like post-merger detection based on the spe-
cific configurations as used in the present search.
We focus here on results for a braking index of
n
= 5
, as
expected for spin-down dominated by GW emission from a
static quadrupole deformation. The signal amplitude
h
0
(as
given in Eq. 4) is degenerate between the ellipticity
, mo-
ment of inertia
I
zz
and distance
d
. We choose a fiducial
value of
I
zz
= 100
M
3
G
2
/c
4
≈
4
.
34
×
10
38
kg m
2
, consis-
tent with EoS yielding a supramassive or stable remnant. For
a given set of model parameters
{
n
= 5
,f
start
,τ
}
we con-
sider the maximum
allowed by energy conservation (Sarin
7
500
1000
1500
2000
f
start
[Hz]
0.1
1
10
d
90%
[Mpc]
GW170817
random cos
ι
cos
ι
= 0
cos
ι
= 1
STAMP
HMM
ATrHough
FreqHough
500
1000
1500
2000
f
start
[Hz]
10
53
10
55
10
57
10
59
E
90%
GW
[erg]
E
rot
(
f
start
)
E
tot
= 3
.
265
M
c
2
10
−
2
10
−
1
10
0
10
1
10
2
10
3
10
4
E
90%
GW
[
M
c
2
]
random cos
ι
cos
ι
= 0
cos
ι
= 1
STAMP
HMM
ATrHough
FreqHough
Figure 2.
A sample of search sensitivities achieved for the power-law spindown signal model with braking index
n
= 5
. Results are shown as
sensitive distance
d
90%
(left panel) for otherwise physical parameters, or as required emitted energy
E
90%
gw
at a fixed distance
d
=
40 Mpc (right
panel), both as a function of reference starting frequency
f
start
used for the injections of each pipeline. (
f
start
=
f
gw
(
t
= ∆
t
)
for STAMP and
FreqHough and
f
start
=
f
gw0
=
f
gw
(
t
= 0)
for the others.)
See Fig. 3 for the parameter ranges covered by each injection set. This figure shows the subset with highest sensitivity for each analysis; this
corresponds to the shortest (
τ
= 100
s) injections for STAMP and HMM, while for ATrHough and FreqHough
τ
(
f
start
)
is variable, depending
on the search coherence length, as also listed in Tables 4 and 5. Note that detection thresholds are also different between pipelines.
The NS ellipticity
is always chosen as the maximum allowed by energy conservation (
E
gw
=
E
rot
) at each (
n,f
start
,τ
)
parameter point,
assuming a NS moment of inertia of
I
zz
= 100
M
3
G
2
/c
4
≈
4
.
34
×
10
38
kg m
2
. Injections were randomized over source inclination
cos
ι
for HMM and FreqHough, while for STAMP and ATrHough injections for the best case (
cos
ι
= 1
) and worst case (
cos
ι
= 0
) are shown
separately.
For comparison, the known distance to the source of GW170817 is indicated by a horizontal dashed line in the left panel, as well as two
(optimistic) energy upper limits in the right panel: the total system energy (dotted line, using a fiducial value of
E
tot
= 3
.
265
M
c
2
as in
Abbott et al. 2017g) and the initial rotational energy
E
rot
as a function of
f
start
(dashed line).
500
1000
1500
2000
2500
3000
f
start
[Hz]
10
0
10
1
10
2
10
3
10
4
10
5
τ
[s]
= 10
−
4
= 10
−
3
= 10
−
2
= 10
−
1
STAMP
HMM
ATrHough
FreqHough
500
1000
1500
2000
2500
3000
f
start
[Hz]
10
−
4
10
−
3
10
−
2
10
−
1
τ
= 10
1
s
τ
= 10
2
s
τ
= 10
3
s
τ
= 10
4
s
τ
= 10
5
s
STAMP
HMM
ATrHough
FreqHough
Figure 3.
Parameter coverage in
f
start
,
τ
and
of the injection sets used for the
n
= 5
sensitivity estimates, as listed in Tables 2–5. As
shown in the left panel, the HMM and STAMP injections are at fixed
τ
∈
[10
2
,
10
3
,
10
4
]
s, while for ATrHough and FreqHough different
τ
(
f
start
)
curves are covered for different choices of
T
SFT
(and, in the case of FreqHough,
∆
t
) in the search setup. At each
(
n,f
start
,τ
)
parameter space point, the maximum
allowed by energy conservation (
E
gw
=
E
rot
) is chosen (right panel), assuming a NS moment of inertia
of
I
zz
= 100
M
3
G
2
/c
4
≈
4
.
34
×
10
38
kg m
2
. Lines of constant
(left panel) or
τ
(right panel) are shown for comparison. STAMP injections
include
f
start
up to 3000 Hz for longer
τ
, with those above 2000 Hz covered by the high-frequency search configuration. But for
τ
= 100
s,
we limit
f
start
to 2000 Hz because injections at higher frequencies would leave the high-frequency band too rapidly to be recoverable.
et al. 2018): the total emitted GW energy as
t
→∞
,
E
gw
=
−
∞
∫
t
=
t
start
d
t
32
G
5
c
5
I
2
zz
2
Ω
6
(
t
)
,
(6)
must not exceed the remnant’s initial rotational energy
E
rot
= 0
.
5
I
zz
f
2
start
π
2
.
Given each pipeline’s detection threshold, we can rescale
the amplitude of simulated signals until 90% of them are re-
covered above threshold, while randomising over nuisance
parameters (polarization angle and initial phase of the sig-
nal; also source inclination
ι
and signal start time for some
of the pipelines). We can then either interpret this amplitude
scaling as a need to lower the distance of simulated sources,
8
i.e. estimating the sensitive distance
d
90%
of the search. Or
we can fix the true distance to the source of GW170817 and
interpret the square of the required amplitude change as the
factor by which we exceed the expected energy output of a
remnant NS with the given parameters, yielding
E
90%
gw
. Both
interpretations are shown in Fig. 2, with coverage of the in-
jection sets illustrated in Fig. 3. Full results are listed in the
Appendix in Tables 2–5.
The highest sensitivities are achieved at low
f
start
and for
rapid spindown (low
τ
). This is mostly due to the energy
conservation constraint enforced on
: while higher
f
start
would yield higher initial amplitudes and longer
τ
would al-
low to accumulate SNR over longer observation times, in turn
must be lower and hence actual detectability is reduced.
The four pipelines perform differently across
τ
regimes:
The unmodeled STAMP and HMM are most sensitive at
the shortest
τ
= 10
2
s, but lose up to an order of magnitude
in
d
90%
when going to
τ
= 10
4
s. On the other hand, the
model-based semi-coherent ATrHough and FreqHough have
focused on longer
τ
of
4
×
10
2
s to
3
×
10
4
s, with only up to
a factor of 2 loss in
d
90%
for the longest
τ
at fixed
f
start
. See
Figs. 6–9 in the Appendix for sensitivity estimates over each
pipeline’s full injection set.
As this parameter dependence is shaped by the
E
gw
=
E
rot
constraint, and also influenced by some practical tradeoffs
in pipeline configuration, in this paper we do not attempt to
provide a general evaluation of pipeline performance on fully
equivalent injection sets, nor for generic GW signals. Such
a comparison would require a detailed mock data challenge
similar to Messenger et al. (2015) and Walsh et al. (2016).
Instead, Fig. 2 shows results from each pipeline for the parts
of parameter space where it achieved its highest sensitivity.
In summary, in no part of the
n
= 5
parameter space cov-
ered by the four search ranges and injection sets do we reach
90% sensitive distances of
1 Mpc
or further. This corre-
sponds to a lowest 90% detectable energy of
E
gw
.
8
M
c
2
at
f
start
= 500
Hz and
τ
= 100
s. At higher
f
start
, the sen-
sitive distances for any
τ
are even lower due to the energy
constraint. Note again that this covers power-law spindown
signals both starting right at coalescence time
t
c
and signals
starting with some time delay, with a delay time of 1–7 hours
for FreqHough and any possible delay time until the end of
O2 for STAMP.
At the shortest
τ
, the parameter space covered here over-
laps with the magnetar injections in the shorter-duration
search of Abbott et al. (2017g)
3
, though results in that paper
were quoted as recoverable at 50% confidence, and hence are
more optimistic than the new results at 90%.
4
6.
CONCLUSION
We have searched for GW emission from a putative rem-
nant neutron star of the BNS merger GW170817, concen-
trating on signals lasting from hundreds of seconds upwards
and described by a power-law spin-down model. Two of the
four employed analysis methods however were designed to
be sensitive to any generic signal morphology in the covered
observation time. In keeping with the available energy bud-
get and theoretical sensitivity estimates, we have not found
any significant signal candidates. Studies with simulated sig-
nals confirm that we would have only been sensitive to a sig-
nal from GW-dominated spin-down (at the time and sky lo-
cation of GW170817) for distances of less than
1 Mpc
, or
equivalently for unphysical amounts of emitted GW energy.
The four analysis pipelines used in this work have comple-
mentary strengths in parameter space coverage and in their
response to noise artifacts and gaps in the data. While further
development of these methods is expected, improvements are
also needed—and already in progress—on the instrumenta-
tion side. Ongoing instrumental enhancements of Advanced
LIGO and Virgo towards their design sensitivies (Abbott
et al. 2018e), and further upgrades like LIGO A+ (Barsotti
et al. 2018) in the next decade, will improve strain sensitivity
across the detector band. Improved high-frequency perfor-
mance is of particular importance for post-merger searches,
as the highest signal amplitudes are emitted in the early, high-
frequency part of the spin-down, where the detectors are cur-
rently much less sensitive than around a few hundred Hz.
Searches for long-duration post-merger signals from supra-
massive or stable NSs could then enter into the astrophysi-
cally constraining regime. However, from scaling the sensi-
tivies obtained in this analysis (or even those estimated for
an optimal matched-filter analysis by Sarin et al. 2018) with
the expected improvements of 2–4 in strain, they will still be
limited to the most nearby BNS events.
Third generation detectors, such as the Einstein Telescope
(Hild et al. 2011; Sathyaprakash et al. 2012) and Cosmic Ex-
3
We note here a mistake in Abbott et al. (2017g): In section 3.2.4,
the equivalent energies for the best STAMP results should have read
E
gw
≈
0
.
6
M
c
2
for bar modes and
E
gw
≈
10
M
c
2
for the magnetar
model, instead of the quoted 2 and 4
M
c
2
. The corresponding
h
rss
val-
ues in the text of Abbott et al. (2017g) and in its Tables 2 and 3, as well as
Figure 1, are correct as published.
4
For example, at
f
start
= 1000
Hz and
τ
= 100
s, the STAMP analysis
in the previous paper found
E
50%
gw
≈
24
M
c
2
while the new STAMP and
HMM analyses presented here obtain
E
90%
gw
≈
100
M
c
2
at these param-
eters. 50% and 90% detectability amplitudes typically differ by factors of
2–4 for the pipelines in this paper. The main goal of the new analysis is to
extend coverage to longer
τ
.
9
plorer (Abbott et al. 2017b), promise a strain sensitivity in-
crease of
∼
20
–
30
over aLIGO at design sensitivity. GWs
from a long-lived remnant of another BNS at the same dis-
tance as GW170817 should then become observable.
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 Niedersach-
sen/Germany for support of the construction of Advanced
LIGO and construction and operation of the GEO600 detec-
tor. Additional support for Advanced LIGO was provided
by the Australian Research Council. The authors gratefully
acknowledge the Italian Istituto Nazionale di Fisica Nucle-
are (INFN), the French Centre National de la Recherche
Scientifique (CNRS) and the Foundation for Fundamental
Research on Matter supported by the Netherlands Organi-
sation for Scientific Research, for the construction and op-
eration of the Virgo detector and the creation and support
of the EGO consortium. The authors also gratefully ac-
knowledge research support from these agencies as well as
by the Council of Scientific and Industrial Research of In-
dia, the Department of Science and Technology, India, the
Science & Engineering Research Board (SERB), India, the
Ministry of Human Resource Development, India, the Span-
ish Agencia Estatal de Investigaci
́
on, the Vicepresid
`
encia i
Conselleria d’Innovaci
́
o, Recerca i Turisme and the Consel-
leria 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 Re-
gional Development Funds (ERDF), the Royal Society, the
Scottish Funding Council, the Scottish Universities Physics
Alliance, the Hungarian Scientific Research Fund (OTKA),
the Lyon Institute of Origins (LIO), the Paris
ˆ
Ile-de-France
Region, the National Research, Development and Innova-
tion Office Hungary (NKFI), the National Research Founda-
tion of Korea, Industry Canada and the Province of Ontario
through the Ministry of Economic Development and Innova-
tion, 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 Min-
istry of Science and Technology (MOST), Taiwan and the
Kavli Foundation. The authors gratefully acknowledge the
support of the NSF, STFC, MPS, INFN, CNRS and the State
of Niedersachsen/Germany for provision of computational
resources. This article has been assigned document number
LIGO-P1800195.
10
0
1
2
3
4
5
6
7
8
SNR
10
−
3
10
−
2
10
−
1
10
0
p
FA
loudest trigger
30 – 2000 Hz band
loudest trigger
2 – 4 kHz band
30 – 2000 Hz
2 – 4 kHz
Figure 4.
STAMP background distributions, (in terms of false alarm probability
p
FA
as a function of detection statistic (SNR), for the low-
and high-frequency searches, and the corresponding loudest foreground triggers (dot and diamond symbols).
APPENDIX
A.
ADDITIONAL DETAILS ON SEARCH METHODS
A.1.
STAMP
Spectrogram pixel sizes
—
The low frequency band from 30–2000 Hz uses pixels of
100 s
×
1 Hz
, while the high frequency band
from 2000–4000 Hz uses pixels of smaller durations of
50 s
×
1 Hz
. Smaller pixels at higher frequency are necessary to account
for the rotation of the Earth, which causes the GW phase difference between detectors to change with time. If the pixel durations
are too large, this results in a loss of SNR which increases with frequency. The durations are thus chosen to limit the maximum
possible SNR loss in a pixel (at the highest frequencies) from this effect to about
10%
(Thrane et al. 2015).
Detection statistic
—
Each spectrogram in the STAMP search (Thrane et al. 2011; Thrane & Coughlin 2013; Thrane et al. 2015) is
analyzed with many randomly chosen quadratic B
́
ezier curves. The SNR of each track
ρ
Γ
is a weighted sum of the SNR of the
pixels covered by the track. The quantity
ρ
Γ
also serves as the detection statistic and is calculated as:
ρ
Γ
=
1
N
3
/
4
∑
i
ρ
i
,
(A1)
where
i
runs over all the pixels in a track and N is the total number of pixels in it. These are then ranked and the track with largest
ρ
Γ
is picked as the trigger for a map. This is done for both the main on-source search and for the background estimation over
time-shifted data.
Background triggers and loudest events
—
Fig. 4 shows the distribution of false alarm probabilities
p
FA
for the SNRs of triggers
collected in background data, for both high- and low-frequency spectrograms. The loudest on-source event in each frequency
range is also shown.
A.2.
HMM tracking
A general description of the HMM method is given by Suvorova et al. (2016) and Sun et al. (2018a). The following summary
is intended to clarify the configuration used for the search presented in this paper.
Probabilistic model
—
A Markov chain is a stochastic process transitioning between discrete states at discrete times
{
t
0
,
···
,t
N
T
}
.
A HMM is an automaton based on a Markov chain, composed of the hidden (unmeasurable) state variable
q
(
t
)
∈{
q
1
,
···
,q
N
Q
}
and the measurement variable
o
(
t
)
∈{
o
1
,
···
,o
N
O
}
. A HMM is memoryless, i.e., the hidden state at time
t
n
+1
only depends on
the state at time
t
n
, with transition probability
A
q
j
q
i
=
P
[
q
(
t
n
+1
) =
q
j
|
q
(
t
n
) =
q
i
]
.
(A2)
The hidden state
q
i
is in observed state
o
j
at time
t
n
with emission probability
L
o
j
q
i
=
P
[
o
(
t
n
) =
o
j
|
q
(
t
n
) =
q
i
]
.
(A3)