of 16
GW170817: Measurements of Neutron Star Radii and Equation of State
B. P. Abbott
etal.
*
(The LIGO Scientific Collaboration and the Virgo Collaboration)
(Received 5 June 2018; revised manuscript received 25 July 2018; published 15 October 2018)
On 17 August 2017, the LIGO and Virgo observatories made the first direct detection of gravitational
waves from the coalescence of a neutron star binary system. The detection of this gravitational-wave signal,
GW170817, offers a novel opportunity to directly probe the properties of matter at the extreme conditions
found in the interior of these stars. The initial, minimal-assumption analysis of the LIGO and Virgo data
placed constraints on the tidal effects of the coalescing bodies, which were then translated to constraints on
neutron star radii. Here, we expand upon previous analyses by working under the hypothesis that both bodies
were neutron stars that are described by the same equation of state and have spins within the range observed in
Galactic binary neutron stars. Our analysis employs two methods: the use of equation-of-state-insensitive
relations between various macroscopic properties of the neutron stars and the use of an efficient
parametrization of the defining function
p
ð
ρ
Þ
of the equation of state itself. From the LIGO and Virgo
data alone and the first method, we measure the two neutron star radii as
R
1
¼
10
.
8
þ
2
.
0
1
.
7
km for the heavier
star and
R
2
¼
10
.
7
þ
2
.
1
1
.
5
km for the lighter star at the 90% credible level. If we additionally require that the
equation of state supports neutron stars with masses larger than
1
.
97
M
as required from electromagnetic
observations and employ the equation-of-state parametrization, we further constrain
R
1
¼
11
.
9
þ
1
.
4
1
.
4
km and
R
2
¼
11
.
9
þ
1
.
4
1
.
4
km at the 90% credible level. Finally, we obtain constraints on
p
ð
ρ
Þ
at supranuclear densities,
with pressure at twice nuclear saturation density measured at
3
.
5
þ
2
.
7
1
.
7
×
10
34
dyn cm
2
at the 90% level.
DOI:
10.1103/PhysRevLett.121.161101
Introduction.
Since September 2015, the Advanced
LIGO
[1]
and Advanced Virgo
[2]
observatories have
opened a window on the gravitational-wave (GW) universe
]
3,4 ]
. A new type of astrophysical source of GWs was
detected on 17 August 2017, when the GW signal emitted
by a low-mass coalescing compact binary was observed
[5]
. This observation coincided with the detection of a
γ
-ray
burst, GRB 170817A
[6,7]
, verifying that the source binary
contained matter, which was further corroborated by a
series of observations that followed across the electromag-
netic spectrum; see e.g.,
[8
12]
. The measured masses of
the bodies and the variety of electromagnetic observations
are consistent with neutron stars (NSs).
Neutron stars are unique natural laboratories for studying
the behavior of cold high-density nuclear matter. Such
behavior is governed by the equation of state (EOS), which
prescribes a relationship between pressure and density. This
determines the relation between NS mass and radius, as
well as other macroscopic properties such as the stellar
moment of inertia and the tidal deformability (see e.g.,
[13]
). While terrestrial experiments are able to test and
constrain the cold EOS at densities below and near the
saturation density of nuclei
ρ
nuc
¼
2
.
8
×
10
14
gcm
3
(see
e.g.,
[14
17]
for a review), currently they cannot probe the
extreme conditions in the deep core of NSs. Astrophysical
measurements of NS masses, radii, moments of inertia and
tidal effects, on the other hand, have the potential to offer
information about whether the EOS is soft or stiff and what
the pressure is at several times the nuclear saturation
density
[16,18
20]
.
GWs offer an opportunity for such astrophysical mea-
surements to be performed, as the GW signal emitted by
merging NS binaries differs from that of two merging
black holes (BHs). The most prominent effect of matter
during the observed binary inspiral comes from the tidal
deformation that each star
s gravitational field induces on
its companion. This deformation enhances GW emission
and thus accelerates the decay of the quasicircular inspiral
[21
23]
. In the post-Newtonian (PN) expansion of the
inspiral dynamics
[24
32]
, this effect causes the phase of
the GW signal to differ from that of a binary BH (BBH)
from the fifth PN order onwards
[21,33,34]
. The leading-
order contribution is proportional to each star
s tidal
deformability parameter,
Λ
¼ð
2
=
3
Þ
k
2
C
5
, an EOS-sensi-
tive quantity that describes how much a star is deformed in
the presence of a tidal field. Here
k
2
is the
l
¼
2
relativistic
Love number
[35
39]
,
C
Gm=
ð
c
2
R
Þ
is the compact-
ness,
R
is the areal radius, and
m
is the mass of the NS.
The deformation of each NS due to its own spin also
modifies the waveform and depends on the EOS.
This effect enters the post-Newtonian expansion as a
contribution to the (lowest order) spin-spin term at the
*
Full author list given at the end of the Letter.
PHYSICAL REVIEW LETTERS
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=
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=
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=
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161101-1
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second order in the GW phase
[40,41]
.TheEOSalso
affects the waveform at merger, the merger outcome and
its lifetime, as well as the postmerger emission (see e.g.,
[42]
). Finally, other stellar modes can couple to the tidal
field and affect the GW signal
[21,43
45]
.
Among the various EOS-dependent effects, the tidal
deformation is the one most readily measurable with
GW170817. The spin-induced quadrupole has a larger
effect on the orbital evolution for systems with large NS
spin
[46
49]
but is also largely degenerate with the mass
ratio and the NS spins, making it difficult to measure
independently
[50]
. The postmerger signal, while rich in
content, is also difficult to observe, with current detector
sensitivities being limited due to photon shot noise
[1]
at
the high frequencies of interest. The merger and postmerger
signal make a negligible contribution to our inference for
GW170817
[51,52]
.
In
[5]
, we presented the first measurements of the
properties of GW170817, including a first set of constraints
on the tidal deformabilities of the two compact objects,
from which inferences about the EOS can be made. An
independent analysis further exploring how well the gravi-
tational-wave data can be used to constrain the tidal
deformabilites, and, from that, the NS radii, has also been
performed recently
[53]
. Our initial bounds have facilitated
a large number of studies, e.g.,
[54
64]
, aiming to translate
the measurements of masses and tidal deformabilities into
constraints on the EOS of NS matter. In a companion paper
[52]
, we perform a more detailed analysis focusing on the
source properties, improving upon the original analysis of
[5]
by using Virgo data with reduced calibration uncer-
tainty, extending the analysis to lower frequencies, employ-
ing more accurate waveform models, and fixing the
location of the source in the sky to the one identified by
the electromagnetic observations.
Here we complement the analysis of
[52]
, and work
under the hypothesis that GW170817 was the result of a
coalescence of two NSs whose masses and spins are
consistent with astrophysical observations and expecta-
tions. Moreover, since NSs represent equilibrium ground-
state configurations, we assume that their properties are
described by the same EOS. By making these additional
assumptions, we are able to further improve our measure-
ments of the tidal deformabilities of GW170817, and
constrain the radii of the two NSs. Moreover, we use an
efficient parametrization of the EOS to place constraints on
the pressure of cold matter at supranuclear densities using
GW observations. This direct measurement of the pressure
takes into account physical and observational constraints
on the NS EOS, namely causality, thermodynamic stability,
and a lower limit on the maximum NS mass supported by
the EOS to be
M
max
>
1
.
97
M
. The latter is chosen as a
1
σ
conservative estimate, based on the observation of PSR
J
0348
þ
0432
with
M
¼
2
.
01

0
.
04
M
[65]
, the heavi-
est NS known to date.
The radii measurements presented here improve upon
existing results (e.g.,
[58,62]
) which had used the initial
tidal measurements reported in
[5]
. We also verify that our
radii measurements are consistent with the result of the
methodologies presented in these studies when applied to
our improved tidal measurements. Moreover, we obtain a
more precise estimate of the NS radius than
[53]
.
Methods.
In this section we describe the details of the
analysis. We use the same LIGO and Virgo data and
calibration model analyzed in
[52]
. The data can be down-
loaded from the Gravitational Wave Open Science Center
(GWOSC)
[66]
. The data include the subtraction of an
instrumental artifact occurring at LIGO-Livingston within
2 s of the GW170817 merger
[5,67]
,aswellasthe
subtraction of independently measurable noise sources
[68
71]
.
Bayesian methods.
We employ a coherent Bayesian
analysis to estimate the source parameters
θ
as described
in
[72,73]
. The goal is to determine the posterior probability
density function (PDF),
p
ð
θ
j
d
Þ
,giventheLIGOandVirgo
data
d
. Given a prior PDF
p
ð
θ
Þ
on the parameter space
(quantifying our prior belief in observing a source with
properties
θ
), the posterior PDF is given by Bayes
stheorem
p
ð
θ
j
d
Þ
p
ð
θ
Þ
p
ð
d
j
θ
Þ
,where
p
ð
d
j
θ
Þ
is the likelihood of
obtaining the data
d
given that a signal with parameters
θ
is
present in the data. Evaluating the multidimensional
p
ð
θ
j
d
Þ
analytically is computationally prohibitive so we resort
to sampling techniques to efficiently draw samples from
the underlying distribution. We make use of the Markov-
chain Monte Carlo algorithm as implemented in the
LALI
NFERENCE
package
[72]
, which is part of the publicly
available LSC Algorithm Library (LAL)
[74]
. For the
likelihood calculation, we use 128 s of data around
GW170817 over a frequency range of 23
2048 Hz, covering
both the time and frequency ranges where there was
appreciable signal above the detector noise, and we estimate
the likelihood for our waveform templates up to merger. The
power spectral density (PSD) of the noise is computed on
source
[52,75,76]
, and we marginalize over the detectors
calibration uncertainties as described in
[52,73,77]
.
In the analysis of a GW signal from a binary NS
coalescence, the source parameters
θ
on which the signal
depends can be decomposed as
θ
¼ð
θ
PM
;
θ
EOS
Þ
, into
parameters that would be present if the two bodies behaved
like point masses
θ
PM
, and EOS-sensitive parameters
θ
EOS
that arise due to matter effects of the two finite-sized
bodies (e.g., tidal deformabilities). The priors on the point-
mass parameters that we use are described in Sec. II D
of
[52]
and we do not repeat them here. We also use the
same convention for the component masses, i.e.,
m
1
m
2
.
We only consider the
low-spin
prior of
[52]
where the
dimensionless NS spin parameter is restricted to
χ
0
.
05
,
in agreement with expectations from Galactic binary NS
PHYSICAL REVIEW LETTERS
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161101 (2018)
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spin measurements
[78]
, and we fix the location of the
source in the sky to the one given by EM observations.
Regarding the EOS-related part of the parameter space and
the corresponding priors, we consider two physically
motivated parameterizations of different dimensionalities,
which we describe in detail in the following sections. The
first method requires the sampling of tidal deformability
parameters, whereas the second method directly samples
the EOS function
p
ð
ρ
Þ
from a 4-dimensional family of
functions. In both cases, the assumption that the binary
consists of two NSs that are described by the same EOS is
implicit in the parametrization of matter effects (in contrast
with the analysis of
[52]
, where minimal assumptions are
made about the nature of the source).
Waveform models and matter effects.
The measure-
ment process described above requires a waveform model
that maps the source parameters
θ
to a signal
h
ð
t
;
θ
Þ
that
would be observed in the detector. The publicly available
LAL Simulation software package of LAL
[74]
contains
several such waveform models obtained with different
theoretical approaches. The impact of varying the models
among several choices
[21,22,41,79
94]
is analyzed in
detail in
[52]
, showing that for GW170817 the systematic
uncertainties due to the modeling of matter effects or the
underlying point-particle description are smaller than the
statistical errors in the measurement. We perform a similar
analysis here by varying the BBH baseline model or using a
post-Newtonian waveform prescription and find results
consistent with those presented in Sec. III D and Table
IV of
[52]
. Moreover, the study of Appendix A of
[52]
(Table V) suggests that varying the tidal description in the
waveforms also leads to broadly consistent tidal measure-
ments. Since the net effect of varying waveform models is
very different for each of the source properties, we refer to
the tables and figures in
[52]
for quantitative statements to
assess the impact of modeling uncertainties.
In the GW170817 discovery paper
[5]
the results for
the inferred tidal deformabilities were obtained with the
TaylorF2 model that is based solely on post-Newtonian
results for both the BBH baseline model
[41,90
94]
and for
tidal effects
[21,22]
, as this model led to the conservatively
largest bounds. In this Letter, we use a more realistic
waveform model, PhenomPNRT
[79
83]
, which is also
used as the reference model in our detailed analysis of the
properties of GW170817
[52]
. The BBH baseline in this
model, constructed based on
[24,91
93,95
98]
, is cali-
brated to numerical relativity data and describes relativistic
point-mass, spin, and the dominant precession effects.
The model further includes tidal effects in the phase from
combining analytical information
[22,23,86,99]
with
results from numerical-relativity simulations of binary
NSs as described in
[84,100]
, and matter effects in the
spin-induced quadrupole based post-Newtonian results
[40,41,92
94]
. The characteristic rotational quadrupole
deformation parameters are computed from
Λ
through
EOS-insensitive relations
[101,102]
as described in
[48,103]
. Other matter effects with nonzero spins are not
taken into account in our analysis.
EOS-insensitive relations.
Despite the microscopic
complexity of NSs, some of their macroscopic properties
are linked by EOS-insensitive relations that depend only
weakly on the EOS
[104]
. We use two such relations to
ensure that the two NSs obey the same EOS and to translate
NS tidal deformabilities to NS radii.
The first such relation we employ was constructed in
[105]
and studied in the context of realistic GW inference
in
[106]
. It combines the mass ratio of the binary
q
m
2
=m
1
1
, the symmetric tidal deformability
Λ
s
ð
Λ
2
þ
Λ
1
Þ
=
2
and the antisymmetric tidal deformability
Λ
a
ð
Λ
2
Λ
1
Þ
=
2
in a relation of the form
Λ
a
ð
Λ
s
;q
Þ
.
Fitting coefficients and an estimate of the relation
s
intrinsic error were obtained by tuning to a large set of
EOS models
[104,106]
, ensuring that the relation gives
pairs of tidal deformabilities that correspond to realistic
EOS models. We sample uniformly in the symmetric tidal
deformability
Λ
s
½
0
;
5000

, use the EOS-insensitive rela-
tion to compute
Λ
a
, and then obtain
Λ
1
and
Λ
2
, which are
used to generate a waveform template. The sampling of
tidal parameters also involves a marginalization over the
intrinsic error in the relation, which is also a function of
Λ
s
and
q
. This procedure leads to unbiased estimation of the
tidal parameters for a wide range of EOSs and mass
ratios
[106]
.
The second relation we employ is between NS tidal
deformability
Λ
and NS compactness
C
[107,108]
.We
employ this
Λ
-
C
relation with the coefficients given in
Sec. (4.4) of
[104]
to compute the posterior for the radius
and the mass of each binary component. Reference
[104]
reports a maximum 6.5% relative error in the relation when
compared to a large set of EOS models. We assume that the
relative error is constant across the parameter space and
distributed according to a zero-mean Gaussian with a
standard deviation of
ð
6
.
5
=
3
Þ
%
and marginalize over it.
We verified that our results are not sensitive to this choice
of error estimate by comparing to the more conservative
choice of a uniform distribution in
½
6
.
5%
;
6
.
5%

.
Parametrized EOS.
Instead of sampling macroscopic
EOS-related parameters such as tidal deformabilities, one
may instead sample the defining function
p
ð
ρ
Þ
of the EOS
directly. A number of parametrizations of different degrees
of complexity and fidelity to realistic EOS models have
been proposed (see
[109]
for a review), and here we employ
the spectral parametrization constructed and validated in
[110
112]
. This parametrization expresses the logarithm of
the adiabatic index of the EOS
Γ
ð
p
;
γ
i
Þ
, as a polynomial of
the pressure
p
, where
γ
i
¼ð
γ
0
;
γ
1
;
γ
2
;
γ
3
Þ
are the free EOS
parameters. The adiabatic index is then used to compute the
energy
ε
ð
p
;
γ
i
Þ
and rest-mass density
ρ
ð
p
;
γ
i
Þ
, which are
inverted to give the EOS. The parameterized high-density
EOS is then stitched to the SLy EOS
[113]
below about half
PHYSICAL REVIEW LETTERS
121,
161101 (2018)
161101-3
the nuclear saturation density. This is chosen because such
low densities do not significantly impact the global proper-
ties of the NS
[114]
. Different low density EOSs can
produce a difference in radius, for a given
m
, of order
0.1 km. Though use of a specific parametrization makes our
results model-dependent, we have checked that they are
consistent with another common EOS parametrization, the
piecewise polytropic one
[115,116]
, as also found in
[117]
.
In this analysis, we follow the methodology detailed in
[117]
, developed from the work of
[118]
, to sample directly
in an EOS parameter space. We sample uniformly in all
EOS parameters within the following ranges:
γ
0
½
0
.
2
;
2

,
γ
1
½
1
.
6
;
1
.
7

,
γ
2
½
0
.
6
;
0
.
6

, and
γ
3
½
0
.
02
;
0
.
02

and additionally impose that the adiabatic index
Γ
ð
p
Þ
½
0
.
6
;
4
.
5

. This choice of prior ranges for the
EOS parameters was chosen such that our parametrization
encompasses a wide range of candidate EOSs
[110]
and
leads to NSs with a compactness below 0.33 and a tidal
deformability above about 10. Then for each sample, the
four EOS parameters and the masses are mapped to a
ð
Λ
1
;
Λ
2
Þ
pair through the Tolman-Oppenheimer-Volkoff
(TOV) equations describing the equilibrium configuration
of a spherical star
[119]
. The two tidal deformabilities are
then used to compute the waveform template.
Sampling directly in the EOS parameter space allows for
certain prior constraints to be conveniently incorporated
in the analysis. In our analysis, we impose the following
criteria on all EOS and mass samples: (i) causality, the
speed of sound in the NS (
ffiffiffiffiffiffiffiffiffiffiffiffiffi
dp=d
ε
p
) must be less than the
speed of light (plus 10% to allow for imperfect para-
metrization) up to the central pressure of the heaviest star
supported by the EOS; (ii) internal consistency, the EOS
must support the proposed masses of each component; and
(iii) observational consistency, the EOS must have a
maximum mass at least as high as previously observed
NS masses, specifically
1
.
97
M
. Another condition the
EOS must obey is that of thermodynamic stability; the EOS
must be monotonically increasing (
d
ε
=dp >
0
). This con-
dition is built into the parametrization
[110]
, so we do not
need to explicitly impose it.
Results.
We begin by demonstrating the improvement
in the measurement of the tidal deformability parameters
due to imposing a common but unknown EOS for the two
NSs. In Fig.
1
we show the marginalized joint posterior
PDF for the individual tidal deformabilities. We show
results from our analysis using the
Λ
a
ð
Λ
s
;q
Þ
relation in
green and the parametrized EOS without a maximum mass
constraint in blue. These are compared to results from
[52]
,
where the two tidal deformability parameters are sampled
independently, in orange. The shaded region marks the
Λ
2
<
Λ
1
region that is naturally excluded when a common
realistic EOS is assumed, but is not excluded from the
analysis of
[52]
. In both cases imposing a common EOS
leads to a smaller uncertainty in the tidal deformability
measurement. The area of the 90% credible region for the
Λ
1
-
Λ
2
posterior shrinks by a factor of
3
,whichis
consistent with the results of
[106]
for soft EOSs and
NSs with similar masses. The tidal deformability of a
1
.
4
M
NS can be estimated through a linear expansion
of
Λ
ð
m
Þ
m
5
around
1
.
4
M
as in
[5,48,120]
to be
Λ
1
.
4
¼
190
þ
390
120
at the 90% level when a common EOS is imposed
(here and throughout this paper we quote symmetric credible
intervals). Our results suggest that
soft
EOSs such as
APR4, which predict smaller values of the tidal deform-
ability parameter, are favored over
stiff
EOSs such as H4
or MS1, which predict larger values of the tidal deform-
ability parameter and lie outside the 90% credible region.
We next explore what inferences we can make about
the structure of NSs. We do this using the spectral EOS
parametrization described above in combination with the
requirement that the EOS must support NSs up to at least
1
.
97
M
, a conservative estimate based on the heaviest
known pulsar
[65]
. From this we obtain a posterior for the
NS interior pressure as a function of rest-mass density. The
result is shown in Fig.
2
, along with marginalized posteriors
for central densities and central pressures and predictions of
the pressure-density relationship from various EOS models.
The pressure posterior is shifted from the 90% credible
prior region (marked by the purple dashed lines) and
towards the soft floor of the parameterized family of
FIG. 1. Marginalized posterior for the tidal deformabilities of
the two binary components of GW170817. The green shading
shows the posterior obtained using the
Λ
a
ð
Λ
s
;q
Þ
EOS-insensitive
relation to impose a common EOS for the two bodies, while the
green, blue, and orange lines denote 50% (dashed) and 90%
(solid) credible levels for the posteriors obtained using EOS-
insensitive relations, a parametrized EOS without a maximum
mass requirement, and independent EOSs (taken from
[52]
),
respectively. The gray shading corresponds to the unphysical
region
Λ
2
<
Λ
1
while the seven black scatter regions give the
tidal parameters predicted by characteristic EOS models for this
event
[113,115,121
125]
.
PHYSICAL REVIEW LETTERS
121,
161101 (2018)
161101-4
EOS. This means that the posterior is indicating more
support for softer EOS than the prior. The solid vertical
lines denote the nuclear saturation density and two
more rest-mass density values that are known to approx-
imately correlate with bulk macroscopic properties
of NSs
[19]
. The pressure at twice (six times) the nuclear
saturation density is measured to be
3
.
5
þ
2
.
7
1
.
7
×
10
34
ð
9
.
0
þ
7
.
9
2
.
6
×
10
35
Þ
dyn
=
cm
2
at the 90% level.
The pressure posterior appears to show minor signs of a
bend above a density of
5
ρ
nuc
. Evidence of such behavior
at high densities would be an indication of extra degrees of
freedom, though this is not an outcome of the GW data
alone. Indeed in the top (right) panel, the vertical (hori-
zontal) lines denote the 90% confidence intervals for the
central densities (pressures) of the two stars, suggesting that
our data are not informative for densities (pressures) above
those intervals. The bend is an outcome of two competing
effects: the GW data point toward a lower pressure, while
the requirement that the EOS supports masses above
1
.
97
M
demands a high pressure at large densities. The
result is a precise pressure estimate at around
5
ρ
nuc
and a
broadening above that, giving the impression of a bend in
the pressure. We have verified that the bend is absent if we
remove the maximum mass constraint from our analysis.
Finally we place constraints in the 2-dimensional param-
eter space of the NS mass and areal radius for each binary
component. This posterior is shown in Fig.
3
. The left panel
is obtained by first using the
Λ
a
ð
Λ
s
;q
Þ
relation to obtain
tidal deformability samples assuming a common EOS and
then using the
Λ
-
C
relation to compute the NS radii. The
right panel is computed by integrating the TOVequation to
compute the radius for each sample in the spectral EOS
parametrization after imposing a maximum mass of at least
1
.
97
M
. At the 90% level, the radii of the two NSs are
R
1
¼
10
.
8
þ
2
.
0
1
.
7
km and
R
2
¼
10
.
7
þ
2
.
1
1
.
5
km from the left
panel and
R
1
¼
11
.
9
þ
1
.
4
1
.
4
km and
R
2
¼
11
.
9
þ
1
.
4
1
.
4
km from
the right panel. The one-sided 90% lower [upper] limit on
m
2
ð
m
1
Þ
is
ð
1
.
15
;
1
.
36
Þ
M
½ð
1
.
36
;
1
.
62
Þ
M

from the left
panel and
ð
1
.
18
;
1
.
36
Þ
M
½ð
1
.
36
;
1
.
58
Þ
M

from the right
panel, consistent with the results of Ref.
[52]
. We note
that the
Λ
-
C
relation has not been established to values
of
Λ
less than 20
[104]
. In order to check the validity of our
EoS-insensitive results in this regime, we first verify that
the parametrized-EoS results without a maximum mass
constraint satisfy the
Λ
-
C
relation to the required accuracy,
even for
Λ
1
<
20
. Furthermore, we find that our radius and
mass estimates are unaffected if we discard all
Λ
1
<
10
samples.
The difference between the two radius estimates is
mainly due to different physical information included in
each analysis. The EOS-insensitive-relation analysis (left
panel) is based on GW data alone, while the parametrized-
EOS analysis (right panel) imposes an additional observa-
tional constraint, namely that the EOS must support NSs of
at least
1
.
97
M
. This has a large effect on the radii priors
as shown in the 1-dimensional plots of Fig.
3
, since small
radii are typically predicted by soft EOSs, which cannot
support large NS masses. In the case of EOS-insensitive
relations (left panel), the prior allows for smaller values of
the radius than in the parametrized-EOS case (right panel),
something that is reflected in the posteriors since the GW
data alone cannot rule out radii below
10
km. Therefore
the lower radius limit in the EOS-insensitive-relations
analysis is determined by the GW measurement, while
in the case of the parametrized-EOS analysis it is deter-
mined by the mass of the heaviest observed pulsar and its
implications for NS radii
[65]
. Additionally, we verified
that the parametrized-EOS analysis without the maximum
mass constraint leads to similar results to the EOS-insen-
sitive-relations analysis.
To quantify the improvement from assuming that both
NSs obey the same EOS, we apply the
Λ
-
C
relation to
tidal deformability samples calculated without assuming
the
Λ
a
ð
Λ
s
;q
Þ
relation (the orange posterior of Fig.
1
) and
obtain
R
1
¼
11
.
8
þ
2
.
7
3
.
3
km and
R
2
¼
10
.
8
þ
2
.
9
3
.
0
km at the 90%
level. This suggests that imposing a common EOS for the
two binary components leads to a reduction of the 90%
FIG. 2. Marginalized posterior (green bands) and prior (purple
dashed) for the pressure
p
as a function of the rest-mass density
ρ
of the NS interior using the spectral EOS parametrization and
imposing a lower limit on the maximum NS mass supported by
the EOS of
1
.
97
M
. The dark (light) shaded region corresponds
to the 50% (90%) posterior credible level and the purple dashed
lines show the 90% prior credible interval. Vertical lines
correspond to once, twice, and six times the nuclear saturation
density. Overplotted in gray are representative EOS models
[121,122,124]
, using data taken from
[19]
; from top to bottom
at
2
ρ
nuc
we show H4, APR4, and WFF1. The corner plots show
cumulative posteriors of central densities
ρ
c
(top) and central
pressures
p
c
(right) for the two NSs (blue and orange), as well as
for the heaviest NS that the EOS supports (black). The 90%
credible intervals for
ρ
c
and
p
c
are denoted by vertical and
horizontal lines respectively for the heavier (blue dashed) and
lighter (orange dot-dashed) NS.
PHYSICAL REVIEW LETTERS
121,
161101 (2018)
161101-5