Properties of the binary neutron star merger GW170817
The LIGO Scientific Collaboration and The Virgo Collaboration
(Compiled 17 October 2018)
On August 17, 2017, the Advanced LIGO and Advanced Virgo gravitational-wave detectors
observed a low-mass compact binary inspiral. The initial sky localization of the source of the
gravitational-wave signal, GW170817, allowed electromagnetic observatories to identify NGC 4993
as the host galaxy. In this work we improve initial estimates of the binary’s properties, including
component masses, spins, and tidal parameters, using the known source location, improved model-
ing, and re-calibrated Virgo data. We extend the range of gravitational-wave frequencies considered
down to 23 Hz, compared to 30 Hz in the initial analysis. We also compare results inferred using
several signal models, which are more accurate and incorporate additional physical effects as com-
pared to the initial analysis. We improve the localization of the gravitational-wave source to a 90%
credible region of 16 deg
2
. We find tighter constraints on the masses, spins, and tidal parameters,
and continue to find no evidence for non-zero component spins. The component masses are inferred
to lie between 1
.
00 and 1
.
89 M
when allowing for large component spins, and to lie between 1
.
16
and 1
.
60 M
(with a total mass 2
.
73
+0
.
04
−
0
.
01
M
) when the spins are restricted to be within the range
observed in Galactic binary neutron stars. Using a precessing model and allowing for large com-
ponent spins, we constrain the dimensionless spins of the components to be less than 0
.
50 for the
primary and 0
.
61 for the secondary. Under minimal assumptions about the nature of the compact
objects, our constraints for the tidal deformability parameter
̃
Λ are (0
,
630) when we allow for large
component spins, and 300
+420
−
230
(using a 90% highest posterior density interval) when restricting the
magnitude of the component spins, ruling out several equation of state models at the 90% credible
level. Finally, with LIGO and GEO600 data, we use a Bayesian analysis to place upper limits on
the amplitude and spectral energy density of a possible post-merger signal.
PACS numbers: 04.80.Nn, 97.60.Jd, 95.85.Sz, 97.80.–d
I. INTRODUCTION
On August 17, 2017 the advanced gravitational-wave
(GW) detector network, consisting of the two Advanced
LIGO detectors [1] and Advanced Virgo [2], observed the
compact binary inspiral event GW170817 [3] with a total
mass less than any previously observed binary coales-
cence and a matched-filter signal-to-noise ratio (SNR) of
32.4, louder than any signal to date. Followup Bayesian
parameter inference allowed GW170817 to be localized to
a relatively small sky area of 28 deg
2
and revealed com-
ponent masses consistent with those of binary neutron
star (BNS) systems. In addition, 1.7 s after the binary’s
coalescence time the Fermi and INTEGRAL gamma-ray
telescopes observed the gamma-ray burst GRB 170817A
with an inferred sky location consistent with that mea-
sured for GW170817 [4], providing initial evidence that
the binary system contained neutron star (NS) matter.
Astronomers followed up on the prompt alerts pro-
duced by this signal, and within 11 hours the transient
SSS17a/AT 2017gfo was discovered [5, 6] and indepen-
dently observed by multiple instruments [7–11], localiz-
ing the source of GW170817 to the galaxy NGC 4993.
The identification of the host galaxy drove an extensive
follow-up campaign [12], and analysis of the fast-evolving
optical, ultraviolet, and infrared emission was consistent
with that predicted for a kilonova [13–17] powered by
the radioactive decay of r-process nuclei synthesized in
the ejecta (see [18–28] for early analyses). The electro-
magnetic (EM) signature, observed throughout the entire
spectrum, provides further evidence that GW170817 was
produced by the merger of a BNS system (e.g. [29–31]).
According to general relativity, the gravitational waves
emitted by inspiraling compact objects in a quasi-circular
orbit are characterized by a chirp-like time evolution in
their frequency that depends primarily on a combina-
tion of the component masses called the chirp mass [32]
and secondarily on the mass ratio and spins of the com-
ponents. In contrast to binary black hole (BBH) sys-
tems, the internal structure of the NS also impacts the
waveform, and needs to be included for a proper descrip-
tion of the binary evolution. The internal structure can
be probed primarily through attractive tidal interactions
that lead to an accelerated inspiral. These tidal inter-
actions are small at lower GW frequencies but increase
rapidly in strength during the final tens of GW cycles
before merger. Although tidal effects are small relative
to other effects, their distinct behavior make them poten-
tially measurable from the GW signal [33–37], providing
additional evidence for a BNS system and insight into
the internal structure of NSs.
In this work we present improved constraints on the
binary parameters first presented in [3]. These improve-
ments are enabled by (i) re-calibrated Virgo data [38], (ii)
a broader frequency band of 23–2048 Hz as compared to
the original 30–2048 Hz band used in [3], (iii) a wider
range of more sophisticated waveform models (see Ta-
ble I), and (iv) knowledge of the source location from
EM observations. By extending the bandwidth from 30–
2048 Hz to 23–2048 Hz, we gain access to an additional
∼
1500 waveform cycles compared to the
∼
2700 cy-
cles in the previous analysis. Overall, our results for
arXiv:1805.11579v2 [gr-qc] 15 Oct 2018
2
the parameters of GW170817 are consistent with, but
more precise than, those in the initial analysis [3]. The
main improvements are (i) improved 90% sky localiza-
tion from 28 deg
2
to 16 deg
2
without use of EM ob-
servations, (ii) improved constraint on inclination angle
enabled by independent measurements of the luminosity
distance to NGC 4993, (iii) limits on precession from a
new waveform model that includes both precession and
tidal effects, and (iv) evidence for a nonzero tidal de-
formability parameter that is seen in all waveform mod-
els. Finally, we analyze the potential post-merger sig-
nal with an unmodeled Bayesian inference method [39]
using data from the Advanced LIGO detectors and the
GEO600 detector [40]. This allows us to place improved
upper bounds on the amount of post-merger GW emis-
sion from GW170817 [41].
As in the initial analysis of GW170817 [3], we infer the
binary parameters from the inspiral signal while making
minimal assumptions about the nature of the compact
objects, in particular allowing the tidal deformability of
each object to vary independently. In a companion pa-
per [42], we present a complementary analysis assuming
that both compact objects are NSs obeying a common
equation of state. This results in stronger constraints on
the tidal deformabilities of the NSs than we can make un-
der our minimal assumptions, and allows us to constrain
the radii of the NSs and make for novel inferences about
the equation of state of cold matter at supranuclear den-
sities.
This paper is organized as follows. Section II details
the updated analysis, including improvements to the in-
strument calibration, improved waveform models, and
additional constraints on the source location. Section III
reports the improved constraints on the binary’s sky lo-
cation, inclination angle, masses, spins, and tidal param-
eters. Section IV provides upper limits on possible GW
emission after the binary merger. Finally, Sec. V summa-
rizes the results and highlights remaining work such as
inference of the NS radius and equation of state (EOS).
Additional results from a range of waveform models are
reported in Appendix A, and an injection and recovery
study investigating the systematic errors in our waveform
models is given in Appendix B.
II. METHODS
A. Bayesian method
All available information about the source parameters
~
θ
of GW170817 can be expressed as a posterior proba-
bility density function (PDF)
p
(
~
θ
|
d
(
t
)) given the data
d
(
t
) from the GW detectors. Through application of
Bayes’ theorem, this posterior PDF is proportional to
the product of the likelihood
L
(
d
(
t
)
|
~
θ
) of observing data
d
(
t
) given the waveform model described by
~
θ
and the
prior PDF
p
(
~
θ
) of observing those parameters. Marginal-
23
100
1000
2048
Frequency [Hz]
10
−
45
10
−
43
10
−
41
10
−
39
10
−
37
Power spectral density [1/Hz]
LIGO Livingston
LIGO Hanford
Virgo
FIG. 1. Power spectral densities (PSDs) of the Advanced
LIGO–Advanced Virgo network. Shown, for each detector,
is the median PSD computed from a posterior distribution
of PSDs as estimated by
BayesWave
[39, 47] using 128 s of
data containing the signal GW170817.
ized posteriors are computed through stochastic sam-
pling using an implementation of Markov-chain Monte
Carlo [43, 44] available in the
LALInference
pack-
age [45] as part of the LSC Algorithm Library (LAL) [46].
By marginalizing over all but one or two parameters, it
is then possible to generate credible intervals or credible
regions for those parameters.
B. Data
For each detector we assume that the noise is additive,
i.e., a data stream
d
(
t
) =
h
M
(
t
;
~
θ
)+
n
(
t
) where
h
M
(
t
;
~
θ
) is
the measured gravitational wave strain and
n
(
t
) is Gaus-
sian and stationary noise characterized by the one-sided
power spectral densities (PSDs) shown in Fig. 1. The
PSD is defined as
S
n
≡
(2
/T
)
〈|
̃
n
(
f
)
|
2
〉
where ̃
n
(
f
) is the
Fourier transform of
n
(
t
) and the angle brackets indicate
a time average over the duration of the analysis
T
, in this
case the 128 s containing the
d
(
t
) used for all results pre-
sented in Sec. III. This PSD is modeled as a cubic spline
for the broad-band structure and a sum of Lorentzians for
the line features, using the
BayesWave
package [39, 47],
which produces a posterior PDF of PSDs. Here we ap-
proximate the full structure and variation of these poste-
riors as a point estimate by using a median PSD, defined
separately at each frequency.
The analyses presented here use the same data and cal-
ibration model for the LIGO detectors as [3], including
subtraction of the instrumental glitch present in LIGO-
Livingston (cf. Fig. 2 of [3]) and of other independently
measured noise sources as described in [48–51]. The
method used for subtracting the instrumental glitch leads
to unbiased parameter recovery when applied to simu-
lated signals injected on top of similar glitches in detec-
tor data [52]. The data from Virgo has been re-calibrated
since the publication of [3], including the subtraction of
3
known noise sources during post-processing of the data,
following the procedure of [38] (the same as described
in Sec. II of [53]). While the assumption of stationary,
Gaussian noise in the detectors is not expected to hold
over long timescales, our subtraction of the glitch, known
noise sources, and recalibration of the Virgo data helps
to bring the data closer to this assumption. Applying the
Anderson-Darling test to the data whitened by the on-
source PSDs generated by
BayesWave
, we do not reject
the null hypothesis that the whitened data are consistent
with zero-mean, unit-variance Gaussian noise
N
(0
,
1).
The test returns
p
-values
>
0
.
1 for the LIGO detectors’
data. Meanwhile, the test is marginal when applied to
the Virgo data, with
p
∼
0
.
01. However, the information
content of the data is dominated by the LIGO detectors,
as they contained the large majority of the recovered sig-
nal power. The results of the Anderson-Darling tests
support the use of the likelihood function as described
in [45] for the signal characterization analyses reported
in this paper.
The measured strain
h
M
(
t
;
~
θ
) may differ from the true
GW strain
h
(
t
;
~
θ
) due to measured uncertainties in the
detector calibration [54, 55]. We relate the measured
strain to the true GW strain with the expression [56, 57]
̃
h
M
(
f
;
~
θ
) =
̃
h
(
f
;
~
θ
)
[
1 +
δA
(
f
;
~
θ
cal
)
]
exp
[
iδφ
(
f
;
~
θ
cal
)
]
,
(1)
where
̃
h
M
(
f
;
~
θ
) and
̃
h
(
f
;
~
θ
) are the Fourier transforms of
h
M
(
t
;
~
θ
) and
h
(
t
;
~
θ
) respectively. The terms
δA
(
f
;
~
θ
cal
)
and
δφ
(
f
;
~
θ
cal
) are the frequency-dependent amplitude
correction factor and phase correction factor respectively,
and are each modeled as cubic splines. For each detector,
the parameters are the values of
δA
and
δφ
at each of ten
spline nodes spaced uniformly in log
f
[58] between 23 Hz
and 2048 Hz.
For the LIGO detectors, the calibration parameters
~
θ
cal
are informed by direct measurements of the calibra-
tion uncertainties [54], and are modeled in the same way
as in [3] with 1
σ
uncertainties of
<
7% in amplitude
and
<
3 deg in phase for LIGO Hanford and
<
5% in
amplitude and
<
2 deg in phase for LIGO Livingston,
all allowing for a non-zero mean offset. The correspond-
ing calibration parameters for Virgo follow [38], with
a 1
σ
amplitude uncertainty of 8% and a 1
σ
phase un-
certainty of 3 deg. This is supplemented with an addi-
tional uncertainty in the time stamping of the data of
20
μ
s (to be compared to the LIGO timing uncertainty
of
<
1
μ
s [59] already included in the phase correction
factor). At each of the spline nodes, a Gaussian prior is
used with these 1
σ
uncertainties and their correspond-
ing means. By sampling these calibration parameters
in addition to the waveform parameters, the calibration
uncertainty is marginalized over. This marginalization
broadens the localization posterior (Sec. III A), but does
not significantly affect the recovered masses, spins, or
tidal deformability parameters.
C. Waveform models for binary neutron stars
In this paper we use four different frequency-domain
waveform models which are fast enough to be used as
templates in
LALInference
. These waveforms incor-
porate point-particle, spin, and tidal effects in different
ways. We briefly describe them below. Each waveform’s
key features are stated in detail in Table I, and fur-
ther tests of the performance of the waveform models
can be found in [75]. In addition to these frequency
domain models, we employ two state-of-the-art time-
domain tidal EOB models that also include spin and tidal
effects [80, 81]. These tidal EOB models have shown good
agreement in comparison with NR simulations [80–83]
in the late inspiral and improve on the post-Newtonian
(PN) dynamics in the early inspiral. However, these im-
plementations are too slow for use in
LALInference
.
We describe these models in Sec. III D when we discuss
an alternative parameter-estimation code [84, 85].
The TaylorF2 model used in previous work is a
purely analytic PN model. It includes point-particle and
aligned-spin terms to 3.5PN order as well as leading-
order (5PN) and next-to-leading-order (6PN) tidal ef-
fects [34, 60–71, 86–88].
The other three waveform
models begin with point-particle models and add a fit
to the phase evolution from tidal effects, labeled NR-
Tidal [74, 75], that fit the high-frequency region to both
an analytic EOB model [80] and NR simulations [74, 89].
The SEOBNRT model is based on the aligned-spin point-
particle EOB model presented in [72] using methods pre-
sented in [73] to allow fast evaluation in the frequency
domain.
PhenomDNRT is based on an aligned-spin
point-particle model [76, 77] calibrated to untuned EOB
waveforms [90] and NR hybrids [76, 77]. Finally, Phe-
nomPNRT is based on the point-particle model presented
in [78] that includes an effective description of precession
effects. In addition to tidal effects, PhenomPNRT also
includes the spin-induced quadrupole moment that en-
ters in the phasing at the 2PN order [91]. For aligned-
spin systems, PhenomPNRT differs from PhenomDNRT
only in the inclusion of the spin-induced quadrupole mo-
ment. We include the EOS dependence of each NS’s spin-
induced quadrupole moment by relating it to the tidal
parameter of each NS using the quasi-universal relations
of [92]. Although this 2PN effect can have a large phase
contribution, even for small spins [37], it enters at similar
PN order as many other terms. We therefore expect it
to be degenerate with the mass ratio and spins.
These four waveform models have been compared to
waveforms constructed by hybridizing BNS EOB inspiral
waveforms [80, 83] with NR waveforms [74, 80, 82, 89] of
the late-inspiral and merger. Since only the PhenomP-
NRT model includes the spin-induced quadrupole mo-
ment, it was found that it has smaller mismatches than
PhenomDNRT and SEOBNRT [75]. In addition, because
PhenomPNRT is the only model that includes precession
effects, we use it as our reference model throughout this
paper.
4
Model name
Name in LALSuite
BBH baseline
Tidal effects
Spin-induced
quadrupole effects
Precession
TaylorF2
TaylorF2
3.5PN (PP [60–65], SO [66]
SS [67–70])
6PN [71]
None
7
SEOBNRT
SEOBNRv4
ROM
NRTidal
SEOBNRv4
ROM [72, 73]
NRTidal [74, 75]
None
7
PhenomDNRT
IMRPhenomD
NRTidal
IMRPhenomD [76, 77]
NRTidal [74, 75]
None
7
PhenomPNRT
IMRPhenomPv2
NRTidal
IMRPhenomPv2 [78]
NRTidal [74, 75]
3PN [67–70, 79]
3
TABLE I. Waveform models employed to measure the source properties of GW170817. The models differ according to how
they treat the inspiral in the absence of tidal corrections, i.e. the BBH-baseline, in particular the point particle (PP), spin-orbit
(SO), and spin-spin (SS) terms, the manner in which tidal corrections are applied, whether the spin-induced quadrupole of
the neutron stars [67–70, 79] are incorporated, and whether the model allows for precession or only treats aligned spins. Our
standard model, PhenomPNRT, incorporates EOB- and NR-tuned tidal effects, the spin-induced quadrupole moment, and
precession.
In Fig. 2 we show differences in the amplitude and
phase evolution between the four models for an equal-
mass, non-spinning BNS system. The top panel shows
the fractional difference in the amplitude ∆
A/A
between
each model and PhenomPNRT, while the bottom panel
shows the absolute phase difference
|
∆Φ
|
between each
model and PhenomPNRT. Because none of the models
have amplitude corrections from tidal effects, the am-
plitude differences between the models are entirely due
to the underlying point-particle models. For the non-
precessing system shown here, PhenomPNRT and Phe-
nomDNRT agree by construction, and the difference with
SEOBNRT is also small. On the other hand, the purely
analytic TaylorF2 model that has not been tuned to NR
simulations deviates by up to 30% from the other mod-
els. For the phase evolution of non-spinning systems,
PhenomDNRT, PhenomPNRT, and SEOBNRT have the
same tidal prescription, so the small
<
∼
2 rad phase dif-
ferences are due to the underlying point-particle mod-
els. For non-spinning systems PhenomDNRT and Phe-
nomPNRT are the same, but for spinning systems, the
spin-induced quadrupole moment included in PhenomP-
NRT but not in PhenomDNRT will cause an additional
phase difference. For TaylorF2 the difference with re-
spect to PhenomPNRT is due to both the underlying
point-particle model and the tidal prescription, and is
∼
5 rad for non-spinning systems.
For reference, we also show in Fig. 2 the tidal contri-
bution to the phase for the NRTidal models (∆Φ
NRTidal
)
and the TaylorF2 model (∆Φ
TaylorF2
Tides
). For the system
here with tidal deformability
̃
Λ = 400 (Eq. (5)), the tidal
contribution is larger than the differences due to the un-
derlying point-particle models.
D. Source parameters and choice of priors
The signal model for the quasi-circular inspiral of com-
pact binaries is described by intrinsic parameters which
10
2
10
3
0
.
00
0
.
05
0
.
10
0
.
15
0
.
20
0
.
25
0
.
30
0
.
35
|
∆
A
|
/A
PhenomDNRT
−
PhenomPNRT
SEOBNRT
−
PhenomPNRT
TaylorF2
−
PhenomPNRT
10
2
10
3
Frequency [Hz]
0
5
10
15
|
∆Φ
|
[rad]
PhenomDNRT
−
PhenomPNRT
SEOBNRT
−
PhenomPNRT
TaylorF2
−
PhenomPNRT
∆Φ
NRTidal
∆Φ
TaylorF2
Tides
FIG. 2. Relative amplitude and phase of the employed wave-
form models starting at 23 Hz (see Table I) with respect to
PhenomPNRT after alignment within the frequency interval
[30
,
30
.
25] Hz. Note that in particular the alignment between
SEOBNRT and PhenomPNRT is sensitive to the chosen in-
terval due to the difference in the underlying BBH-baseline
models at early frequencies. We show as an example an equal-
mass, non-spinning system with a total mass of 2
.
706
M
and
a tidal deformability of
̃
Λ = 400. In the bottom panel, we
also show the tidal contribution to the phasing for the Tay-
lorF2 and the NRTidal models. This contribution can be in-
terpreted as the phase difference between the tidal waveform
models and the corresponding BBH models. The TaylorF2
waveform terminates at the frequency of the innermost stable
circular orbit, which is marked by a small dot.
describe the components of the binary, and extrinsic pa-
rameters which determine the location and orientation
of the binary with respect to the observer. The intrinsic
parameters include the component masses
m
1
and
m
2
,
where we take as convention that
m
1
≥
m
2
. The best
measured parameter for systems displaying a long inspi-
5
ral is the chirp mass [32, 61, 93, 94],
M
=
(
m
1
m
2
)
3
/
5
(
m
1
+
m
2
)
1
/
5
.
(2)
Meanwhile, ground-based GW detectors actually mea-
sure redshifted (detector-frame) masses, and these are
the quantities we state our prior assumptions on.
Detector-frame masses are related to the astrophysically
relevant source-frame masses by
m
det
= (1 +
z
)
m
, were
z
is the redshift of the binary [93, 95]. Dimensionless quan-
tities such as the ratio of the two masses,
q
=
m
2
/m
1
≤
1,
are thus the same in the detector frame and the source
frame. When exploring the parameter space
~
θ
we assume
a prior PDF
p
(
~
θ
) uniform in the detector-frame masses,
with the constraint that 0
.
5 M
≤
m
det
1
,m
det
2
≤
7
.
7 M
where
m
det
1
≥
m
det
2
, and with an additional constraint on
the chirp mass, 1
.
184 M
≤ M
det
≤
2
.
168 M
. These
limits were chosen to mimic the settings in [3] to allow
for easier comparisons, and were selected originally for
technical reasons. The posterior does not have support
near those limits. Despite correlations with the prior on
the distance to the source, the source masses also have
an effectively uniform prior in the region of parameter
space relevant to this analysis.
When converting from detector-frame to source-frame
quantities, we use the MUSE/VLT measurement of the
heliocentric redshift of NGC 4993,
z
helio
= 0
.
0098 re-
ported in [96, 97]. We convert this into a geocentric
redshift using the known time of the event, yielding
z
= 0
.
0099.
The spin angular momenta of the two binary com-
ponents
S
i
represent six additional intrinsic parameters,
and are usually represented in their dimensionless forms
χ
i
=
c
S
i
/
(
Gm
2
i
). For these parameters we have, fol-
lowing [3], implemented two separate priors for the mag-
nitudes of the dimensionless spins,
|
χ
|
=
χ
, of the two
objects. In both cases, we assume isotropic and uncor-
related orientations for the spins, and we use a uniform
prior for the spin magnitudes, up to a maximum mag-
nitude. In the first case we enforce
χ
≤
0
.
89 to be con-
sistent with the value used in [3]. This allows us to ex-
plore the possibility of exotic binary systems. The exact
value of this upper limit does not significantly affect re-
sults. Meanwhile, observations of pulsars indicate that
while the fastest-spinning neutron star has an observed
χ
<
∼
0
.
4 [98], the fastest-spinning BNSs capable of merg-
ing within a Hubble time, PSR J0737–3039A [99] and
PSR J1946+2052 [100], will at most have dimensionless
spins of
χ
∼
0
.
04 or
χ
∼
0
.
05 when they merge. Consis-
tent with this population of BNS systems, in the second
case we restrict
χ
≤
0
.
05.
For the waveforms in Table I that do not support spin-
precession, the components of the spins aligned with the
orbital angular momentum
χ
1
z
and
χ
2
z
still follow the
same prior distributions, which are marginalized over the
unsupported spin components. We use the labels “high-
spin” and “low-spin” to refer to analyses that use the
prior
χ
≤
0
.
89 and
χ
≤
0
.
05, respectively.
The dimensionless parameters Λ
i
governing the tidal
deformability of each component, discussed in greater de-
tail in Sec. III D, are given a prior distribution uniform
within 0
≤
Λ
i
≤
5000 where no correlation between Λ
1
,
Λ
2
, and the mass parameters is assumed. If we assume
the two components are NSs that obey the same EOS,
then Λ
1
and Λ
2
must have similar values when
m
1
and
m
2
have similar values [101–103]. This additional con-
straint is discussed in a companion paper that focuses on
the NS EOS [42].
The remaining signal parameters in
~
θ
are extrinsic
parameters which give the localization and orientation
of the binary. When we infer the location of the bi-
nary from GW information alone (in the Localization
section), we use an isotropic prior PDF for the location
of the source on the sky. For most of the results pre-
sented here, we restrict the sky location to the known
position of SSS17a/AT 2017gfo as determined by elec-
tromagnetic observations [12]. In every case, we use a
prior on the distance which assumes a homogeneous rate
density in the nearby Universe, with no cosmological cor-
rections applied; in other words, the distance prior grows
with the square of the luminosity distance. Meanwhile
we use EM observations to reweight our distance poste-
riors when investigating the inclination of the binary in
Sec. III A, and we use the measured redshift factor to
the host galaxy NGC 4993 in order to infer source-frame
masses from detector frame masses in Sec. III B. For the
angle cos
θ
JN
=
ˆ
J
·
ˆ
N
, defined for the total angular mo-
mentum
J
and the line of sight
N
, we assume a prior
distribution uniform in cos
θ
JN
[104]. To improve the
convergence rate of the stochastic samplers, the analy-
ses with the non-precessing waveform models implement
a likelihood function where the phase at coalescence is
analytically marginalized out [45].
III. PROPERTIES INFERRED FROM
INSPIRAL AND MERGER
A. Localization
For most of the analyses in this work we assume
a pri-
ori
that the source of GW170817 is in NGC 4993. How-
ever, the improved calibration of Virgo data enables bet-
ter localization of the source of GW170817 from GW data
alone. To demonstrate the improved localization we use
results from the updated TaylorF2 analysis (the choice
of model does not meaningfully affect localization [106]),
shown in Fig. 3. We find a reduction in the 90% localiza-
tion region from 28 deg
2
[3] to 16 deg
2
. This improved
localization is still consistent with the associated coun-
terpart SSS17a/AT 2017gfo (see Fig. 3).
For the remainder of this work we incorporate our
knowledge of the location of the event.
While fixing the position of the event to the known
location within NGC 4993, we infer the luminosity dis-
tance from the GW data alone. Using the PhenomPNRT
6
Low-spin prior (
χ
≤
0
.
05)
High-spin prior (
χ
≤
0
.
89)
Binary inclination
θ
JN
146
+25
−
27
deg
152
+21
−
27
deg
Binary inclination
θ
JN
using EM distance constraint [105]
151
+15
−
11
deg
153
+15
−
11
deg
Detector frame chirp mass
M
det
1
.
1975
+0
.
0001
−
0
.
0001
M
1
.
1976
+0
.
0004
−
0
.
0002
M
Chirp mass
M
1
.
186
+0
.
001
−
0
.
001
M
1
.
186
+0
.
001
−
0
.
001
M
Primary mass
m
1
(1
.
36, 1
.
60) M
(1
.
36, 1
.
89) M
Secondary mass
m
2
(1
.
16, 1
.
36) M
(1
.
00, 1
.
36) M
Total mass
m
2
.
73
+0
.
04
−
0
.
01
M
2
.
77
+0
.
22
−
0
.
05
M
Mass ratio
q
(0
.
73, 1
.
00)
(0
.
53, 1
.
00)
Effective spin
χ
eff
0
.
00
+0
.
02
−
0
.
01
0
.
02
+0
.
08
−
0
.
02
Primary dimensionless spin
χ
1
(0
.
00, 0
.
04)
(0
.
00, 0
.
50)
Secondary dimensionless spin
χ
2
(0
.
00, 0
.
04)
(0
.
00, 0
.
61)
Tidal deformability
̃
Λ with flat prior
300
+500
−
190
(symmetric)/ 300
+420
−
230
(HPD)
(0
,
630)
TABLE II. Properties for GW170817 inferred using the PhenomPNRT waveform model. All properties are source properties
except for the detector frame chirp mass
M
det
=
M
(1 +
z
). Errors quoted as
x
+
z
−
y
represent the median, 5% lower limit, and
95% upper limit. Errors quoted as (
x,y
) are one-sided 90% lower or upper limits, and are used when one side is bounded by
a prior. For the masses,
m
1
is bounded from below and
m
2
is bounded from above by the equal mass line. The mass ratio
is bounded by
q
≤
1. For the tidal parameter
̃
Λ, we quote results using a constant (flat) prior in
̃
Λ. In the high-spin case we
quote a 90% upper limit for
̃
Λ, while in the low-spin case we report both the symmetric 90% credible interval and the 90%
highest posterior density (HPD) interval, which is the smallest interval that contains 90% of the probability.
13h40m
20m
00m
12h40m
20m
-15
◦
-20
◦
-25
◦
-30
◦
Right Ascension
Declination
SSS17a
FIG. 3. The improved localization of GW170817, with the lo-
cation of the associated counterpart SSS17a/AT 2017gfo. The
darker and lighter green shaded regions correspond to 50%
and 90% credible regions respectively, and the gray dashed
line encloses the previously-derived 90% credible region pre-
sented in [3].
waveform model, we find that the luminosity distance is
D
L
= 41
+6
−
12
Mpc in the high-spin case and
D
L
= 39
+7
−
14
Mpc in the low-spin case. Combining this distance in-
formation with the redshift associated with the Hubble
flow at NGC 4993, we measure the Hubble parameter
as in [107]. We find that
H
0
= 70
+13
−
7
km s
−
1
Mpc
−
1
(we use maximum
a posteriori
and 68.3% credible inter-
val for only
H
0
in this work) in the high-spin case and
H
0
= 70
+19
−
8
km s
−
1
Mpc
−
1
in the low-spin case; both
measurements are within the uncertainties seen in Ex-
tended Data Table 1 and Extended Data Fig. 2 of [107].
As noted in [107, 108], when only measuring one polar-
ization of GW radiation from a binary merger, in the
absence of strong precession there is a degeneracy be-
tween distance and inclination of the binary. When us-
ing GW170817 to measure the Hubble constant this de-
generacy is the main source of uncertainty. The slightly
stronger constraints on
H
0
in the high-spin case arise be-
cause under that prior our weak constraint on precession
(see Sec. III C) helps to rule out binary inclinations which
are closer to edge-on (i.e.,
θ
JN
= 90 deg) and where pre-
cession effects would be measurable, and hence increases
the lower bound on the luminosity distance. Meanwhile,
the upper bound on the luminosity distance is achieved
with face-off (i.e.,
θ
JN
= 180 deg) binary inclinations,
and is nearly the same for both high-spin and low-spin
cases.
This same weak constraint on precession leads to a
tighter constraint on the inclination angle in the high-
spin case when using the precessing signal model Phe-
nomPNRT,
θ
JN
= 152
+21
−
27
deg, as compared to the low-
spin case. The inclination measurement in the low-spin
case,
θ
JN
= 146
+25
−
27
deg, agrees with the inferred values
for both the high- and low-spin cases of our three wave-
form models that treat only aligned-spins (see Table IV
in Appendix A). This gives further evidence that it is the