of 32
Properties of the Binary Neutron Star Merger GW170817
B. P. Abbott
etal.
*
(LIGO Scientific Collaboration and Virgo Collaboration)
(Received 6 June 2018; revised manuscript received 20 September 2018; published 2 January 2019)
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 modeling, and recalibrated 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 compared 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 nonzero 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
þ
0
.
04
0
.
01
M
) when the spins are restricted to bewithin the range observed in
Galactic binary neutron stars. Using a precessing model and allowing for large component 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
þ
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 postmerger signal.
DOI:
10.1103/PhysRevX.9.011001
Subject Areas: Astrophysics, Gravitation
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 coalescence
and a matched-filter signal-to-noise ratio (SNR) of 32.4,
louder than any signal to date. Follow-up Bayesian param-
eter inference allowed GW170817 to be localized to a
relatively small sky area of
28
deg
2
and revealed compo-
nent 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 measured
for GW170817
[4]
, providing initial evidence that the
binary system contained neutron star (NS) matter.
Astronomers followed up on the prompt alerts produced
by this signal, and within 11 hours the transient SSS17a/AT
2017gfo was discovered
[5,6]
and independently observed
by multiple instruments
[7
11]
, localizing 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 Refs.
[18
28]
for early analyses). The electromagnetic (EM) signature,
observed throughout the entire spectrum, provides further
evidence that GW170817 was produced by the merger of a
BNS system (see, e.g., Refs.
[29
31]
).
According to general relativity, the gravitational waves
emitted by inspiraling compact objects in a quasicircular
orbit are characterized by a chirplike time evolution in their
frequency that depends primarily on a combination of the
component masses called the chirp mass
[32]
and second-
arily on the mass ratio and spins of the components. In
contrast to binary black hole (BBH) systems, the internal
structure of the NS also impacts the waveform and needs to
*
Full author list given at the end of the article.
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
s title, journal citation,
and DOI.
PHYSICAL REVIEW X
9,
011001 (2019)
2160-3308
=
19
=
9(1)
=
011001(32)
011001-1
Published by the American Physical Society
be included for a proper description of the binary evolution.
The internal structure can be probed primarily through
attractive tidal interactions that lead to an accelerated
inspiral. These tidal interactions 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 makes
them potentially 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 Ref.
[3]
. These
improvements are enabled by (i) recalibrated Virgo data
[38]
, (ii) a broader frequency band of 23
2048 Hz as
compared to the original 30
2048 Hz band used in Ref.
[3]
,
(iii) a wider range of more sophisticated waveform models
(see Table
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
about 1500 waveform cycles compared to the about 2700
cycles in the previous analysis. Overall, our results for 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 localization from
28
deg
2
to
16
deg
2
without use of EM observations,
(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 deformability parameter
that is seen in all waveform models. Finally, we analyze the
potential postmerger signal with an unmodeled Bayesian
inference method
[39]
using data from the Advanced LIGO
detectors and the GEO600 detector
[40]
. This method
allows us to place improved upper bounds on the amount of
postmerger GW emission 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. We allow for a large
range of tidal deformabilities in our analysis, including
zero, which means that our analysis includes the possibility
of phase transitions within the stars and allows for exotic
compact objects or even for black holes as binary compo-
nents. In a companion paper
[42]
, we present a comple-
mentary analysis assuming that both compact objects are
NSs obeying a common equation of state (EOS). This
analysis results in stronger constraints on the tidal deform-
abilities of the NSs than we can make under our minimal
assumptions, and it allows us to constrain the radii of the
NSs and make for novel inferences about the equation of
state of cold matter at supranuclear densities.
This paper is organized as follows. Section
II
details the
updated analysis, including improvements to the
instrument calibration, improved waveform models, and
additional constraints on the source location. Section
III
reports the improved constraints on the binary
s sky
location, 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 EOS. Additional results
from a range of waveform models are reported in
Appendix
A
, and an injection and recovery study inves-
tigating 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 probability
density function (PDF)
p
(
θ
j
d
ð
t
Þ
)
given the data
d
ð
t
Þ
from
the GW detectors. Through application of Bayes
theorem,
this posterior PDF is proportional to the product of the
likelihood
L
(
d
ð
t
Þj
θ
)
of observing data
d
ð
t
Þ
given the
waveform model described by
θ
and the prior PDF
p
ð
θ
Þ
of observing those parameters. Marginalized posteriors are
computed through stochastic sampling using an implemen-
tation of Markov-chain Monte Carlo algorithm
[43,44]
available in the LALI
NFERENCE
package
[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
ð
t
Þ¼
h
M
ð
t
;
θ
Þþ
n
ð
t
Þ
where
h
M
ð
t
;
θ
Þ
is
the measured gravitational-wave strain and
n
ð
t
Þ
is Gaussian
and stationary noise characterized by the one-sided power
spectral densities (PSDs) shown in Fig.
1
. The PSD is
defined as
S
n
ð
2
=T
Þhj
̃
n
ð
f
Þj
2
i
, where
̃
n
ð
f
Þ
is the Fourier
transform of
n
ð
t
Þ
and the angle brackets indicate a time
average over the duration of the analysis
T
, in this case the
128 s containing the
d
ð
t
Þ
used for all results presented in
Sec.
III
. This PSD is modeled as a cubic spline for the
broadband structure and a sum of Lorentzians for the line
features, using the B
AYES
W
AVE
package
[39,47]
, which
produces a posterior PDF of PSDs. Here, we approximate
the full structure and variation of these posteriors as a point
estimate by using a median PSD, defined separately at each
frequency.
The analyses presented here use the same data and
calibration model for the LIGO detectors as Ref.
[3]
,
including subtraction of the instrumental glitch present
in LIGO-Livingston (cf. Fig. 2 of Ref.
[3]
) and of other
independently measured noise sources as described in
B. P. ABBOTT
et al.
PHYS. REV. X
9,
011001 (2019)
011001-2
Refs.
[48
51]
. The method used for subtracting the
instrumental glitch leads to unbiased parameter recovery
when applied to simulated signals injected on top of similar
glitches in detector data
[52]
. The data from Virgo has been
recalibrated since the publication of Ref.
[3]
, including the
subtraction of known noise sources during postprocessing
of the data, following the procedure of Ref.
[38]
(the same
as described in Sec. II of Ref.
[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 bring the data closer to this assumption.
Applying the Anderson-Darling test to the data whitened
by the on-source PSDs generated by B
AYES
W
AVE
,wedo
not reject the null hypothesis that the whitened data are
consistent with zero-mean, unit-variance, Gaussian noise
N
ð
0
;
1
Þ
. 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 signal power. The results of the Anderson-
Darling tests support the use of the likelihood function
as described in Ref.
[45]
for the signal characterization
analyses reported in this paper.
The measured strain
h
M
ð
t
;
θ
Þ
may differ from the true
GW strain
h
ð
t
;
θ
Þ
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
ð
f
;
θ
Þ¼
̃
h
ð
f
;
θ
Þ½
1
þ
δ
A
ð
f
;
θ
cal
Þ
exp
½
i
δφ
ð
f
;
θ
cal
Þ
;
ð
1
Þ
where
̃
h
M
ð
f
;
θ
Þ
and
̃
h
ð
f
;
θ
Þ
are the Fourier transforms of
h
M
ð
t
;
θ
Þ
and
h
ð
t
;
θ
Þ
, respectively. The terms
δ
A
ð
f
;
θ
cal
Þ
and
δφ
ð
f
;
θ
cal
Þ
are the frequency-dependent amplitude correc-
tion factor and phase correction factor, respectively, and are
each modeled as cubic splines. For each detector, the
parameters are the values of
δ
A
and
δφ
at each of ten spline
nodes spaced uniformly in log
f
[58]
between 23 and
2048 Hz.
For the LIGO detectors, the calibration parameters
θ
cal
are informed by direct measurements of the calibration
uncertainties
[54]
and are modeled in the same way as in
Ref.
[3]
with
1
σ
uncertainties of less than 7% in amplitude
and less than 3 degrees in phase for LIGO Hanford and less
than 5% in amplitude and less than 2 degrees in phase for
LIGO Livingston, all allowing for a nonzero mean offset.
The corresponding calibration parameters for Virgo follow
Ref.
[38]
, with a
1
σ
amplitude uncertainty of 8% and a
1
σ
phase uncertainty of 3 degrees. This is supplemented with
an additional uncertainty in the time stamping of the data
of
20
μ
s (to be compared to the LIGO timing uncertainty
of less than
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 corresponding means. By sampling these calibration
parameters in addition to the waveform parameters, the
calibration uncertainty is marginalized over. This margin-
alization broadens the localization posterior (Sec.
III A
)but
FIG. 1. PSDs of the Advanced LIGO
Advanced Virgo net-
work. Shown, for each detector, is the median PSD computed
from a posterior distribution of PSDs as estimated by B
AYES-
W
AVE
[39,47]
using 128 s of data containing the signal
GW170817.
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 quadrupoles 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 effective-one-body (EOB)- and numerical relativity (NR)-tuned tidal effects, the spin-induced quadrupole moment, and
precession.
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
SEOBNRT
SEOBNRv4_ROM_NRTidal SEOBNRv4_ROM
[72,73]
NRTidal
[74,75]
None
PhenomDNRT
IMRPhenomD_NRTidal
IMRPhenomD
[76,77]
NRTidal
[74,75]
None
PhenomPNRT
IMRPhenomPv2_NRTidal
IMRPhenomPv2
[78]
NRTidal
[74,75]
3PN
[67
70,79]
PROPERTIES OF THE BINARY NEUTRON STAR MERGER
...
PHYS. REV. X
9,
011001 (2019)
011001-3
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 LALI
NFERENCE
. These waveforms incorpo-
rate point-particle, spin, and tidal effects in different ways.
We briefly describe them below. Each waveform
skey
features are stated in detail in Table
I
, and further tests of
the performance of the waveform models can be found in
Ref.
[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 implementations are
too slow for use in LALI
NFERENCE
. 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 effects
[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 NRTidal
[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 Ref.
[72]
using
methods presented in Ref.
[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, PhenomPNRT is based on the point-particle model
presented in Ref.
[78]
, which includes an effective descrip-
tion of precession effects. In addition to tidal effects,
PhenomPNRT also includes the spin-induced quadru-
pole moment that enters in the phasing at 2PN order
[91]
.
For aligned-spin systems, PhenomPNRT differs from
PhenomDNRT only in the inclusion of the spin-induced
quadrupole moment. 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 Ref.
[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 PhenomPNRT model
includes the spin-induced quadrupole moment, 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.
In Fig.
2
, we show differences in the amplitude and phase
evolution between the four models for an equal-mass,
nonspinning BNS system. The top panel shows the frac-
tional difference in the amplitude
Δ
A=A
between each
model and PhenomPNRT, while the bottom panel shows
the absolute phase difference
j
ΔΦ
j
between each model and
PhenomPNRT. Because none of the models have amplitude
corrections from tidal effects, the amplitude differences
between the models are entirely due to the underlying point-
particle models. For the nonprecessing system shown here,
PhenomPNRT and PhenomDNRT 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 models. For the phase evolution of nonspinning
systems, PhenomDNRT, PhenomPNRT, and SEOBNRT
have the same tidal prescription, so the small
2
rad phase
differences are due to the underlying point-particle models.
For nonspinning systems PhenomDNRTand PhenomPNRT
are the same, but for spinning systems, the spin-induced
quadrupole moment included in PhenomPNRT but not in
PhenomDNRTwill cause anadditionalphasedifference. For
TaylorF2 the difference with respect to PhenomPNRT is due
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
½
30
;
30
.
25

Hz. Note that, in particular, the alignment between
SEOBNRT and PhenomPNRT is sensitive to the chosen interval
due to the difference in the underlying BBH-baseline models at
early frequencies. We show, as an example, an equal-mass,
nonspinning system with a total mass of
2
.
706
M
and a tidal
deformability of
̃
Λ
¼
400
. In the bottom panel, we also show the
tidal contribution to the phasing for the TaylorF2 and the NRTidal
models. This contribution can be interpreted as the phase
difference between the tidal waveform models and the corre-
sponding BBH models. The TaylorF2 waveform terminates at the
frequency of the innermost stable circular orbit, which is marked
by a small dot.
B. P. ABBOTT
et al.
PHYS. REV. X
9,
011001 (2019)
011001-4
to both the underlying point-particle model and the tidal
prescription, and it is about 5 rad for nonspinning systems.
For reference, we also show in Fig.
2
the tidal contri-
bution to the phase for the NRTidal models (
ΔΦ
NRTidal
) and
the TaylorF2 model (
ΔΦ
TaylorF
2
Tides
). For the system here with
tidal deformability
̃
Λ
¼
400
[Eq.
(5)
], the tidal contribution
is larger than the differences due to the underlying point-
particle models.
D. Source parameters and choice of priors
The signal model for the quasicircular inspiral of
compact binaries is described by intrinsic parameters that
describe the components of the binary and extrinsic
parameters that 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 choose the convention
m
1
m
2
. The best
measured parameter for systems displaying a long inspiral
is the chirp mass
[32,61,93,94]
,
M
¼
ð
m
1
m
2
Þ
3
=
5
ð
m
1
þ
m
2
Þ
1
=
5
:
ð
2
Þ
Meanwhile, ground-based GW detectors actually measure
redshifted (detector-frame) masses, and these are the quan-
tities we state our prior assumptions on. Detector-frame
masses are related to the astrophysically relevant source-
frame masses by
m
det
¼ð
1
þ
z
Þ
m
, where
z
is the redshift of
the binary
[93,95]
. Dimensionless quantities such as the ratio
of the two masses,
q
¼
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
ð
θ
Þ
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
.Theselimitswerechosen
to mimic the settings in Ref.
[3]
to allow for easier
comparisons and were originally selected for technical
reasons. The posterior does not have support near those
limits. Despite correlations with the prior on the distance to
thesource,thesource masses also have an effectivelyuniform
prior in the regionofparameter 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
¼
0
.
0098
, reported
in Refs.
[96,97]
. We convert this into a geocentric redshift
using the known time of the event, yielding
z
¼
0
.
0099
.
The spin angular momenta of the two binary components
S
i
represent six additional intrinsic parameters and are
usually represented in their dimensionless forms
χ
i
¼
c
S
i
=
ð
Gm
2
i
Þ
. For these parameters, we have, following Ref.
[3]
,
implemented two separate priors for the magnitudes of the
dimensionless spins,
j
χ
χ
, of the two objects. In both
cases, we assume isotropic and uncorrelated orientations
for the spins, and we use a uniform prior for the spin
magnitudes, up to a maximum magnitude. In the first case,
we enforce
χ
0
.
89
to be consistent with the value used in
Ref.
[3]
. This allows us to explore the possibility of exotic
binary systems. The exact value of this upper limit does not
significantly affect the results. 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 merging within a Hubble time, PSR J0737
3039A
[99]
and PSR J1946+2052
[100]
, will have at most
dimensionless spins of
χ
0
.
04
or
χ
0
.
05
when they
merge. Consistent 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 unsup-
ported 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 detail
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 constraint is discussed in a
companion paper that focuses on the NS EOS
[42]
.
The remaining signal parameters in
θ
are extrinsic
parameters that give the localization and orientation of
the binary. When we infer the location of the binary from
GW information alone (in Sec.
III A
), we use an isotropic
prior PDF for the location of the source on the sky. For most
of the results presented here, we restrict the sky location to
the known position of SSS17a/AT 2017gfo as determined
by electromagnetic 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
corrections applied; in other words, the distance prior
grows with the square of the luminosity distance.
Meanwhile, we use EM observations to reweight our
distance posteriors 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
¼
ˆ
J
·
ˆ
N
, defined for the total angular
momentum
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 analyses
with the nonprecessing waveform models implement a
likelihood function where the phase at coalescence is
analytically marginalized out
[45]
.
PROPERTIES OF THE BINARY NEUTRON STAR MERGER
...
PHYS. REV. X
9,
011001 (2019)
011001-5
III. PROPERTIES INFERRED FROM
INSPIRAL AND MERGER
A. Localization
For most of the analyses in this work, we assume
a priori
that the source of GW170817 is in NGC 4993. However,
the improved calibration of Virgo data enables better
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
[105]
),
shown in Fig.
3
. We find a reduction in the 90% localization
region from
28
deg
2
[3]
to
16
deg
2
. This improved locali-
zation is still consistent with the associated counterpart
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 distance
from the GW data alone. Using the PhenomPNRT wave-
form model, we find that the luminosity distance is
D
L
¼
41
þ
6
12
Mpc in the high-spin case and
D
L
¼
39
þ
7
14
Mpc in
the low-spin case. Combining this distance information
with the redshift associated with the Hubble flow at NGC
4993, we measure the Hubble parameter as in Ref.
[106]
.
We find that
H
0
¼
70
þ
13
7
km s
1
Mpc
1
(we use maximum
a posteriori
and 68.3% credible interval for only
H
0
in this
work) in the high-spin case and
H
0
¼
70
þ
19
8
km s
1
Mpc
1
in the low-spin case; both measurements are within the
uncertainties seen in Extended Data Table I and Extended
Data Fig. 2 of Ref.
[106]
. As noted in Refs.
[106,107]
,
when only measuring one polarization of GW radiation
from a binary merger, in the absence of strong precession
there is a degeneracy between distance and inclination of
the binary. When using GW170817 to measure the Hubble
constant, this degeneracy is the main source of uncertainty.
The slightly stronger constraints on
H
0
in the high-spin
case arise because, under that prior, our weak constraint on
precession (see Sec.
III C
) helps to rule out binary incli-
nations that are closer to edge-on (i.e.,
θ
JN
¼
90
deg) and
where precession 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
¼
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
PhenomPNRT,
θ
JN
¼
152
þ
21
27
deg, as compared to the
low-spin case. The inclination measurement in the low-spin
case,
θ
JN
¼
146
þ
25
27
deg, agrees with the inferred values for
both the high- and low-spin cases of our three waveform
models that treat only aligned spins (see Table
IV
in
Appendix
A
). This agreement gives further evidence
that it is the absence of strong precession effects in
the signal, which can only occur in the high-spin
case of the precessing model, that leads to tighter constraints
on
θ
JN
. These tighter constraints are absent for systems
restricted to the lower spins expected from Galactic NS
binaries.
Conversely, EM measurements of the distance to the host
galaxy can be used to reduce the effect of this degeneracy,
improving constraints on the luminosity distance of the
binary and its inclination, which may be useful for
constraining emission mechanisms. Figure
4
compares
our posterior estimates for distance and inclination with
no
a priori
assumptions regarding the distance to the binary
(i.e., using a uniform-in-volume prior) to the improved
constraints from an EM-informed prior for the distance to
the binary. For the EM-informed results, we have
reweighted the posterior distribution to use a prior in
distance following a normal distribution with mean
40.7 Mpc and standard deviation 2.36 Mpc
[108]
. This
leads to improved measurements of the inclination angle
θ
JN
¼
151
þ
15
11
deg (low spin) and
θ
JN
¼
153
þ
15
11
deg (high
spin). This measurement is consistent for both the high-spin
and low-spin cases since the EM measurements constrain
the source of GW170817 to higher luminosity distances
and correspondingly more face-off inclination values. They
are also consistent with the limits reported in previous
studies using afterglow measurements
[109]
and combined
FIG. 3. The improved localization of GW170817, with the
location 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 presented in
Ref.
[3]
.
B. P. ABBOTT
et al.
PHYS. REV. X
9,
011001 (2019)
011001-6
GW and EM constraints
[108,110,111]
to infer the incli-
nation of the binary.
B. Masses
Owing to its low mass, most of the SNR for GW170817
comes from the inspiral phase, while the merger and
postmerger phases happen at frequencies above 1 kHz,
where LIGO and Virgo are less sensitive (Fig.
1
). This case
is different than the BBH systems detected so far, e.g.,
GW150914
[112
115]
or GW170814
[53]
. The inspiral
phase evolution of a compact binary coalescence can be
written as a PN expansion, a power series in
v=c
, where
v
is
the characteristic velocity within the system
[65]
. The
intrinsic parameters on which the system depends enter the
expansion at different PN orders. Generally speaking,
parameters that enter at lower orders have a large impact
on the phase evolution and are thus easier to measure using
the inspiral portion of the signal.
The chirp mass
M
enters the phase evolution at the
lowest order; thus, we expect it to be the best constrained
among the source parameters
[32,61,93,94]
. The mass ratio
q
, and consequently the component masses, are instead
harder to measure due to two main factors: (1) They are
higher-order corrections in the phase evolution, and (2) the
mass ratio is partially degenerate with the component of the
spins aligned with the orbital angular momentum
[93,94,116]
, as discussed further below.
In Fig.
5
, we show one-sided 90% credible intervals of
the joint posterior distribution of the two component masses
in the source frame. We obtain
m
1
ð
1
.
36
;
1
.
89
Þ
M
and
m
2
ð
1
.
00
;
1
.
36
Þ
M
in the high-spin case, and
FIG. 4. Marginalized posteriors for the binary inclination (
θ
JN
)
and luminosity distance (
D
L
) using a uniform-in-volume prior
(blue) and EM-constrained luminosity distance prior (purple)
[108]
. The dashed and solid contours enclose the 50% and 90%
credible regions, respectively. Both analyses use a low-spin prior
and make use of the known location of SSS17a. The 1D marginal
distributions have been renormalized to have equal maxima to
facilitate comparison, and the vertical and horizontal lines mark
90% credible intervals.
FIG. 5. The 90% credible regions for component masses using
the four waveform models for the high-spin prior (top panel) and
low-spin prior (bottom panel). The true thickness of the contour,
determined by the uncertainty in the chirp mass, is too small to
show. The points mark the edge of the 90% credible regions. The
1D marginal distributions have been renormalized to have equal
maxima, and the vertical and horizontal lines give the 90% upper
and lower limits on
m
1
and
m
2
, respectively.
PROPERTIES OF THE BINARY NEUTRON STAR MERGER
...
PHYS. REV. X
9,
011001 (2019)
011001-7
tighter constraints of
m
1
ð
1
.
36
;
1
.
60
Þ
M
and
m
2
ð
1
.
16
;
1
.
36
Þ
M
in the low-spin case. These estimates are
consistent with, and generally more precise than, those
presented in Ref.
[3]
. The inferred masses for the compo-
nents are also broadly consistent with the known masses of
Galactic neutron stars observed in BNS systems (see, e.g.,
Ref.
[117]
).
As expected, the detector-frame chirp mass is measured
with muchhigherprecision, with
M
det
¼
1
.
1976
þ
0
.
0004
0
.
0002
M
(high spin) and
M
det
¼
1
.
1975
þ
0
.
0001
0
.
0001
M
(low spin). These
uncertainties are decreased by nearly a factor of 2 as
compared to the value reported in Ref.
[3]
for the detec-
tor-frame chirp mass, while the median remains consistent
with the 90% credible intervals previously reported. The
main source of uncertainty in the source-frame chirp mass
comes from the unknownvelocity of the source: The line-of-
sight velocity dispersion
σ
v
¼
170
km s
1
of NGC 4993
reported in Ref.
[96]
translates into an uncertainty on the
geocentric redshift of the source
z
¼
0
.
0099

0
.
0009
, and
thereby onto the chirp mass. This uncertainty dominates
over the statistical uncertainty in
M
and over the subpercent
level uncertainty in the redshift measurement of NGC 4993
reported in Ref.
[97]
. The use of the velocity dispersion to
estimate the uncertainty in the radial velocity of the source is
consistent with the impact of the second supernova on the
center-of-mass velocity of the progenitor of GW170817
being relatively small
[96,118]
, especially given that the
probable delay time of GW170817 is much longer than the
dynamical time of its host galaxy. Both sources of uncer-
tainty are incorporated into the values reported in Table
II
,
which still correspond to a subpercent level of precision on
the measurement of
M
. This method of determining
M
from the detector-frame chirp mass differs from the original
method used in Ref.
[3]
, and the resulting median value of
M
lies at the edge of the 90% credible interval reported
there, with uncertainties reduced by a factor of 2 or more.
The fact that chirp mass is estimated much better than the
individual masses is the reason why, in Fig.
5
, the two-
dimensional posteriors are so narrow in one direction
[119]
.
Meanwhile, the unknown velocity of the progenitor of
GW170817 impacts the component masses at a subpercent
level and is neglected in the bounds reported above and in
Table
II
.
C. Spins
The spins of compact objects directly impact the phasing
and amplitude of the GW signal through gravitomagnetic
interactions (e.g., Refs.
[120
122]
) and through additional
contributions to the mass- and current-multipole moments,
which are the sources of GWs (e.g., Ref.
[65]
). This result
allows for the measurement of the spins of the compact
objects from their GW emission. The spins produce two
qualitatively different effects on the waveform.
First, the components of spins along the orbital angular
momentum
L
have the effect of slowing down or speeding
up the overall rate of inspiral, for aligned-spin components
and anti-aligned spin components, respectively
[123]
. The
most important combination of spin components along
L
is
a mass-weighted combination called the effective spin,
χ
eff
[124
126]
, defined as
TABLE II. Properties for GW170817 inferred using the PhenomPNRTwaveform model. All properties are source
properties except for the detector-frame chirp mass
M
det
¼
M
ð
1
þ
z
Þ
. Errors quoted as
x
þ
z
y
represent the median,
5% lower limit, and 95% upper limit. Errors quoted as
ð
x; y
Þ
are one-sided 90% lower or upper limits, and they 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.
Low-spin prior (
χ
0
.
05
Þ
High-spin prior (
χ
0
.
89
Þ
Binary inclination
θ
JN
146
þ
25
27
deg
152
þ
21
27
deg
Binary inclination
θ
JN
using EM
distance constraint
[108]
151
þ
15
11
deg
153
þ
15
11
deg
Detector-frame chirp mass
M
det
1
.
1975
þ
0
.
0001
0
.
0001
M
1
.
1976
þ
0
.
0004
0
.
0002
M
Chirp mass
M
1
.
186
þ
0
.
001
0
.
001
M
1
.
186
þ
0
.
001
0
.
001
M
Primary mass
m
1
ð
1
.
36
;
1
.
60
Þ
M
ð
1
.
36
;
1
.
89
Þ
M
Secondary mass
m
2
(1.16, 1.36) M
ð
1
.
00
;
1
.
36
Þ
M
Total mass
m
2
.
73
þ
0
.
04
0
.
01
M
2
.
77
þ
0
.
22
0
.
05
M
Mass ratio
q
(0.73, 1.00)
(0.53, 1.00)
Effective spin
χ
eff
0
.
00
þ
0
.
02
0
.
01
0
.
02
þ
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
þ
500
190
ð
symmetric
Þ
=
300
þ
420
230
ð
HPD
Þ
(0, 630)
B. P. ABBOTT
et al.
PHYS. REV. X
9,
011001 (2019)
011001-8