GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at
Redshift 0.2 – Supplemental Material
The LIGO Scientific Collaboration and the Virgo Collaboration
I. NOISE PERFORMANCE OF THE
DETECTORS
Figure 1 shows a comparison of typical strain noise am-
plitude spectra during the first observing run and early
in the second for both of the LIGO detectors [1]. For the
Hanford detector, shot-noise limited performance was im-
proved above about 500 Hz by increasing the laser power.
There are new broad mechanical resonance features (e.g.,
at
∼
150 Hz, 320 Hz and 350 Hz) due to increased beam
pointing jitter from the laser, as well as the coupling
of the jitter to the detector’s gravitational-wave channel
that is larger than in the Livingston detector. The in-
crease in the noise between 40 Hz and 100 Hz is currently
under investigation. For the Livingston detector, signif-
icant reduction in the noise between 25 Hz and 100 Hz
was achieved mainly by the reduction of the scattered
light that re-enters the interferometer.
To date, the network duty factor of the LIGO detectors
in the second observing run is about 51% while it was
about 43% in the first observing run. The improvement
came from better seismic isolation at Hanford, and fine
tuning of the control of the optics at Livingston.
II. SEARCHES
The significance of a candidate event is calculated by
comparing its detection statistic value to an estimate of
the background noise [2–5, 7]. Figure 2 shows the back-
ground and candidate events from the offline searches for
compact binary coalescences obtained from 5.5 days of
coincident data. At the detection statistic value assigned
to GW170104, the false alarm rate is less than 1 in 70,000
years of coincident observing time.
III. PARAMETER INFERENCE
The source properties are estimated by exploring the
parameter space with stochastic sampling algorithms [8].
Calculating the posterior probability requires the like-
lihood of the data given a set of parameters, and the
parameters’ prior probabilities. The likelihood is deter-
mined from a noise-weighted inner product between the
data and a template waveform [9]. Possible calibration
error is incorporated using a frequency-dependent spline
model for each detector [10]. The analysis follows the
approach used for previous signals [11–13].
A preliminary analysis was performed to provide a
medium-latency source localization [14].
This analy-
sis used an initial calibration of the data and assumed
a (conservative) one-sigma calibration uncertainty of
10% in amplitude and 10
◦
in phase for both detec-
tors, a reduced-order quadrature model of the effective-
precession waveform [15–18] (the most computationally
expedient model), and a power spectral density cal-
culated using a parametrized model of the detector
noise [19, 20]. A stretch of 4 s of data, centered on
the event, was analysed across a frequency range of 20–
1024 Hz. We assumed uninformative prior probabili-
ties [11, 13]; technical restrictions of the reduced-order
quadrature required us to limit spin magnitudes to
<
0
.
8
and impose cuts on the masses (as measured in the de-
tector frame) such that
m
det
1
,
2
∈
[5
.
5
,
160]
M
,
M
det
∈
[12
.
3
,
45
.
0]
M
and mass ratio
q
=
m
2
/m
1
≥
1
/
8. The
bounds of the mass prior do not affect the posterior, but
the spin distributions were truncated. The source posi-
tion is not strongly coupled to the spin distribution, and
so should not have been biased by these limits [21, 22].
The final analysis used an updated calibration of the
data, with one-sigma uncertainties of 3
.
8% in ampli-
tude and 2
.
2
◦
in phase for Hanford, and 3
.
8% and
1
.
9
◦
for Livingston, and two waveform models, the
effective-precession model [16–18] and the full-precession
model [23–25]. The spin priors were extended up to
0
.
99. As a consequence of the computational cost of
the full-precession model, we approximate the likelihood
by marginalising over the time and phase at coalescence
as if the waveform contained only the dominant (2
,
±
2)
harmonics [8]. This marginalisation is not exact for pre-
cessing models, but should not significantly affect sig-
nals with binary inclinations that are nearly face on or
face off [12]. Comparisons with preliminary results from
an investigation using the full-precession waveform with-
out marginalisation confirm that this approximation does
not impact results. The two waveform models produce
broadly consistent parameter estimates, so the overall re-
sults are constructed by averaging the two distributions.
As a proxy for the theoretical error from waveform mod-
eling, we use the difference between the results from the
two approximants [11]. A detailed summary of results is
given in Table I, and the final sky localization is shown
in Fig. 3.
Figure 4 illustrates the distance, and the angle be-
tween the total angular momentum and the line of sight
θ
JN
. The latter is approximately constant throughout
the inspiral and serves as a proxy for the binary inclina-
tion [26, 27]. The full-precessing model shows a greater
preference (after accounting for the prior) for face-on or
face-off orientations with
θ
JN
'
0
◦
or 180
◦
. This leads to
the tail of the
D
L
distribution extending to farther dis-
III PARAMETER INFERENCE
10
2
10
3
10
−
23
10
−
22
10
−
21
Strain Noise [Hz
−
1
/
2
]
O1
O2
10
2
10
3
O1
O2
Frequency [Hz]
Hanford
Livingston
FIG. 1. Comparison of typical noise amplitude spectra of the LIGO detectors in the first observing run (O1) and the early
stages of the second observing run (O2). The noise is expressed in terms of equivalent gravitational-wave strain amplitude.
Some narrow features are calibration lines (22–24 Hz for L1, 35–38 Hz for H1, 330 Hz and 1080 Hz for both), suspension fibers’
resonances (500 Hz and harmonics) and 60 Hz power line harmonics.
8
.
0
8
.
5
9
.
0
9
.
5
10
.
0
10
.
5
Detection statistic
ρ
10
−
6
10
−
4
10
−
2
10
0
Number of events
GW170104
Search Result
Background
10
15
20
25
30
Detection statistic
ln
L
10
−
6
10
−
4
10
−
2
10
0
Number of events
GW170104
Search Result
Background
FIG. 2.
Left:
Search results from the binary coalescence search described in [2–4]. The histogram shows the number of
candidate events (orange markers) in the 5.5 days of coincident data and the expected background (black lines) as a function
of the search detection statistic. The reweighted SNR detection statistic
%
is defined in [3]. GW170104 has a larger detection
statistic value than all of the background events in this period. At the detection statistic value assigned to GW170104, the
search’s false alarm rate is less than 1 in 70,000 years of coincident observing time. No other significant candidate events
are observed in this time interval.
Right:
Search results from an independently-implemented analysis [5], where the detection
statistic ln
L
is an approximate log likelihood ratio statistic that is an extension of [6]. The two search algorithms give consistent
results.
tances. There is a preference towards face-on or face-off
inclinations over those which are edge on; the probabil-
ity that
|
cos
θ
JN
|
>
1
/
√
2 is 0
.
62, compared to a prior
probability of 0
.
29. These inclinations produce louder
signals and so are expected to be most commonly de-
tected [28, 29]. Viewing the binary near face-on or face-
off minimises the impact (if present) of precession [11, 30].
For GW170104, we obtain weak constraints on the
spins. The amount of information we learn from the
signal may be quantified by the Kullback–Leibler diver-
gence, or relative entropy, from the prior to the pos-
terior [31, 32]. For
χ
eff
we gain 0
.
36 nat of informa-
tion, and for
χ
p
we only gain 0
.
03 nat. As compari-
son, the Kullback–Leibler divergence between two equal-
width normal distributions with means one standard de-
viation apart is 0
.
5 nat = 0
.
72 bit. We cannot gain much
insight from these spin measurements, but this may be-
come possible by considering the population of binary
black holes [33]. Figure 5 shows the inferred
χ
eff
dis-
tributions for GW170104, GW150914, LVT151012 and
GW151226 [13]. Only GW151226 has a
χ
eff
(and hence
at least one component spin) inconsistent with zero. The
others are consistent with positive or negative effective
inspiral spin parameters; the probabilities that
χ
eff
>
0
are 0
.
18, 0
.
23 and 0
.
59 for GW170104, GW150914 and
LVT151012, respectively. Future analysis may reveal if
2