draft
GW170817: Measurements of Neutron Star Radii and Equation of State
The LIGO Scientific Collaboration and The Virgo Collaboration
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 hy-
pothesis 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
(
ρ
)
of the equation of state it-
self. From the LIGO and Virgo data alone and the first method, we measure the two neutron star radii as
R
1
= 10
.
8
+2
.
0
−
1
.
7
km for the heavier star and
R
2
= 10
.
7
+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 fur-
ther constrain
R
1
= 11
.
9
+1
.
4
−
1
.
4
km and
R
2
= 11
.
9
+1
.
4
−
1
.
4
km at the 90% credible level. Finally, we obtain
constraints on
p
(
ρ
)
at supranuclear densities, with pressure at twice nuclear saturation density measured
at
3
.
5
+2
.
7
−
1
.
7
×
10
34
dyn cm
−
2
at the 90% level.
INTRODUCTION
Since September 2015, the Advanced LIGO [1] and Ad-
vanced 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 Au-
gust 2017, when the GW signal emitted by a low-mass co-
alescing compact binary was observed [5]. This observa-
tion 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 obser-
vations that followed across the electromagnetic 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 study-
ing 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 ra-
dius, 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 sat-
uration density of nuclei
ρ
nuc
= 2
.
8
×
10
14
g cm
−
3
(see
e.g. [14–17] for a review), currently they cannot probe the
extreme conditions in the deep core of NSs. Astrophysi-
cal 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 deforma-
tion that each star’s gravitational field induces on its com-
panion. 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 dy-
namics [24–32], this effect causes the phase of the GW sig-
nal to differ from that of a binary BH (BBH) from the fifth
PN order onwards [21, 33, 34]. The leading-order con-
tribution is proportional to each star’s tidal deformability
parameter,
Λ = (2
/
3)
k
2
C
−
5
, an EOS-sensitive quantity
that describes how much a star is deformed in the pres-
ence of a tidal field. Here
k
2
is the
l
= 2
relativistic
Love number [35–39],
C
≡
Gm/
(
c
2
R
)
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 mod-
ifies 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 second order in the
GW phase [40, 41]. The EOS also affects the waveform
at merger, the merger outcome and its lifetime, as well as
the postmerger emission (see e.g. [42]). Finally, other stel-
lar 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 ef-
fect on the orbital evolution for systems with large NS spin
[46–49] but is also largely degenerate with the mass ra-
tio and the NS spins, making it difficult to measure inde-
pendently [50]. The postmerger signal, while rich in con-
tent, is also difficult to observe, with current detector sen-
sitivities being limited due to photon shot noise [1] at the
high frequencies of interest. The merger and postmerger
arXiv:1805.11581v2 [gr-qc] 15 Oct 2018
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2
signal make a negligible contribution to our inference for
GW170817 [51, 52].
In [5], we presented the first measurements of the prop-
erties 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 indepen-
dent analysis further exploring how well the gravitational-
wave data can be used to constrain the tidal deformabilites,
and, from that, the NS radii, has also been performed re-
cently [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 con-
straints on the EOS of NS matter. In a companion pa-
per [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 loca-
tion of the source in the sky to the one identified by the
electromagnetic observations.
Here we complement the analysis of [52], and work un-
der the hypothesis that GW170817 was the result of a coa-
lescence of two NSs whose masses and spins are consistent
with astrophysical observations and expectations. More-
over since NSs represent equilibrium ground-state con-
figurations, we assume that their properties are described
by the same EOS. By making these additional assump-
tions, we are able to further improve our measurements
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 pres-
sure of cold matter at supranuclear densities using GW ob-
servations. 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
J0348+0432 with
M
= 2
.
01
±
0
.
04 M
[65], the heaviest
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 downloaded 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], as well as the 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
(
~
θ
|
d
)
, given the LIGO and Virgo data
d
. Given
a prior PDF
p
(
~
θ
)
on the parameter space (quantifying
our prior belief in observing a source with properties
~
θ
),
the posterior PDF is given by Bayes’s theorem
p
(
~
θ
|
d
)
∝
p
(
~
θ
)
p
(
d
|
~
θ
)
, where
p
(
d
|
~
θ
)
is the likelihood of obtaining
the data
d
given that a signal with parameters
~
θ
is present
in the data. Evaluating the multidimensional
p
(
~
θ
|
d
)
ana-
lytically is computationally prohibitive so we resort to sam-
pling techniques to efficiently draw samples from the un-
derlying distribution. We make use of the Markov-chain
Monte Carlo algorithm as implemented in the LALI
NFER
-
ENCE
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 den-
sity (PSD) of the noise is computed on source [52, 75, 76],
and we marginalize over the detectors’ calibration uncer-
tainties as described in [52, 73, 77].
In the analysis of a GW signal from a binary NS coales-
cence, the source parameters
~
θ
on which the signal depends
can be decomposed as
~
θ
= (
~
θ
PM
,
~
θ
EOS
)
, 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 param-
eters 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 con-
sider 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 spin measure-
ments [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 parameteri-
zations 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
(
ρ
)
from a 4-dimensional family of functions. In both cases,
the assumption that the binary consists of two NSs that are
draft
3
described by the same EOS is implicit in the parametriza-
tion 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 measurement process described above requires a
waveform model that maps the source parameters
~
θ
to a
signal
h
(
t
;
~
θ
)
that would be observed in the detector. The
publicly available LALS
IMULATION
software package of
LAL [74] contains several such waveform models obtained
with different theoretical approaches. The impact of vary-
ing 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 mat-
ter 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 base-
line model or using a post-Newtonian waveform prescrip-
tion and find results consistent with those presented in
Sec. III D and Table IV of [52]. Moreover, the study of Ap-
pendix A of [52] (Table V) suggests that varying the tidal
description in the waveforms also leads to broadly consis-
tent tidal measurements. 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 Tay-
lorF2 model that is based solely on post-Newtonian re-
sults for both the BBH baseline model [41, 90–94] and for
tidal effects [21, 22], as this model led to the conserva-
tively largest bounds. In this Letter, we use a more realis-
tic 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 com-
bining analytical information [22, 23, 86, 99] with results
from numerical-relativity simulations of binary NSs as de-
scribed in [84, 100], and matter effects in the spin-induced
quadrupole based post-Newtonian results [40, 41, 92–94].
The characteristic rotational quadrupole deformation pa-
rameters are computed from
Λ
through EOS-insensitive re-
lations [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 infer-
ence in [106]. It combines the mass ratio of the binary
q
≡
m
2
/m
1
≤
1
, the symmetric tidal deformability
Λ
s
≡
(Λ
2
+Λ
1
)
/
2
and the antisymmetric tidal deformabil-
ity
Λ
a
≡
(Λ
2
−
Λ
1
)
/
2
in a relation of the form
Λ
a
(Λ
s
,q
)
.
Fitting coefficients and an estimate of the relation’s intrin-
sic 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 mod-
els. We sample uniformly in the symmetric tidal deforma-
bility
Λ
s
∈
[0
,
5000]
, use the EOS-insensitive relation 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 intrin-
sic 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 de-
formability
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 stan-
dard deviation of
(6
.
5
/
3)%
and marginalize over it. We
verified that our results are not sensitive to this choice of er-
ror estimate by comparing to the more conservative choice
of a uniform distribution in
[
−
6
.
5%
,
6
.
5%]
.
Parametrized EOS
Instead of sampling macroscopic EOS-related parame-
ters such as tidal deformabilities, one may instead sample
the defining function
p
(
ρ
)
of the EOS directly. A num-
ber of parametrizations of different degrees of complex-
ity and fidelity to realistic EOS models have been pro-
posed (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(
p
;
γ
i
)
, as a polynomial of
the pressure
p
, where
γ
i
= (
γ
0
,γ
1
,γ
2
,γ
3
)
are the free
EOS parameters. The adiabatic index is then used to com-
pute the energy
(
p
;
γ
i
)
and rest-mass density
ρ
(
p
;
γ
i
)
,
which are inverted to give the EOS. The parameterized