Modelling Gravita/onal Waveforms
andElectromagne/c Signals
from Neutron Star Binary Mergers
Kyohei Kawaguchi
ICRR, The University of Tokyo
Collaborator: Kenta Kiuchi, Koutarou Kyutoku, Tatsuya Narikawa,
Yuichiro Sekiguchi, Masaru Shibata, Hideyuki Tagoshi,
Masaomi Tanaka, Keisuke Taniguchi, Nami Uchikata

2018.7.27 @ ICRR seminar

My Introduc,on•Kyohei Kawaguchi (川口 恭平）•

Ins,tute for Cosmic Ray Research (ICRR),
Gravita,onal waves / High energy astrophysics group•

YITP (Kyoto Uni.) → AEI (MPI, Postdam) → ICRR•

Compact binary merger (NS binary merger),numerical-rela,vity simula,on,
gravita,onal waves, electromagne,c counterparts

Today’s talk•Introduc,on•Gravita,onal waves from binary neutron stars•

Electromagne,c counterparts to binary neutron stars:
Kilonovae/Macronovae•

Future works

IntroducKon

Gravita,onal waves••

Gravita,onal waves (GW): the ripples of curvature that
propagate with the speed of lightadvanced LIGO
hPps://
www.ligo.caltech.edu/

GW sources for ground-based GW detectors•

Compact binary mergers•

Core collapse Super Novae•

Rota,ng Neutron stars•

Primordial GW (Inﬂa,on)•

Cosmic Strings

advanced Virgo
hPp://gwcenter.icrr.utokyo.ac.jp/

hPp://www.virgogw.eu/

Compact Binary: Binary composed ofblack-holes (BHs) and/or neutron stars (NSs)

KAGRA

Compact binary mergers•

Compact Binary—Binary composed of black-holes (BHs) and/or
neutron stars (NSs)— eﬃciently emit gravita,onal waves shrinking
their orbital separa,on, and the objects gradually merge•

Observa,on of gravita,onal waves provides us to the physicalinforma,on of the binary (masses, spins, inclina,on…)

Ref: K. Hotokezaka et al. 2013

Electromagne,c Counterparts
to NS binary mergers•

Various transient EM counterparts areproposed for NS binary mergers•

for example,•

short-hard gamma-ray-burst•A]erglow•

cocoon emission•

kilonovae/macronovae•

radio ﬂare, etc.

•

Host galaxy iden,ﬁca,on, remnant
proper,es, environment•

Possible synthesis site of r-process nuclei
Ref: B. Metzger and E. Berger 2012

Gravita,onal-wave
Astronomy

PHYSICAL REVIEW LETTERS

PRL 116, 061102 (2016)

properties of space-time in the strong-field, high-velocity
regime and confirm predictions of general relativity for the
nonlinear dynamics of highly disturbed black holes.
II. OBSERVATION

week en
12 FEBRUA

the coincident signal GW150914 shown in Fig. 1. Th
detection was made by low-latency searches forgravitational-wave transients [41] and was reported
three minutes of data acquisition [43]. Subsematched-filter analyses that use relativistic models
pact binary waveforms [44] recovered GW150914
most significant event from each detector for the otions reported here. Occurring within the 10-ms

・GW150914 (The ﬁrst GW event)On September 14, 2015 at 09:50:45 UTC, the LIGO
Hanford, WA, and Livingston, LA, observatories detected•

Since 14th of September 2015, many
GW events have been detected•

Binary BH (BBH; BH-BH)••

GW150914, GW151226,
GW170104, GW170608,
GW170814

Binary NS (BNS; NS-NS)
GW170817→Simultaneous detec,on ofelectromagne,c (EM) counterparts

BH-BH

Ref: B.P.Abbot et al. 2016

NS Binary
BH-NS

FIG. 1. The gravitational-wave event GW150914 observed by the LIGO Hanford (H1, left column panels) and Livingston (column panels) detectors. Times are shown relative to September 14, 2015 at 09:50:45 UTC. For visualization, all time series arwith a 35–350 Hz bandpass filter to suppress large fluctuations outside the detectors’ most sensitive frequency band, and ba
filters to remove the strong instrumental spectral lines seen in the Fig. 3 spectra. Top row, left: H1 strain. Top row, right: L
þ0.5
GW150914 arrived first at L1 and 6.9−0.4
ms later at H1; for a visual comparison, the H1 data are also shown, shifted in tim
amount and inverted (to account for the detectors’ relative orientations). Second row: Gravitational-wave strain projected odetector in the 35–350 Hz band. Solid lines show a numerical relativity waveform for a system with parameters consistent w
recovered from GW150914 [37,38] confirmed to 99.9% by an independent calculation based on [15]. Shaded areas show 90%

BNS (NS-NS)

the LIGO and Virgo detectors allowed a precise sky
2week ending
position localization to an area of 28 Pdeg
This
H
Y. S
I C measureAL REVIEW LETTERS
20 OCTOBER 2017
PRL
119,161101(2017)
ment enabled an electromagnetic follow-up campaign that
identified a counterpart near the galaxy NGC 4993, con∼100 s (calculated starting from 24 Hz) in the detectors’sistent with the localization and distance inferred fromsensitive band, the inspiral signal ended at 12∶41:04.4 UTC.
gravitational-wave data [46–50].
In addition, a γ-ray burst was observed 1.7 s after the
From the gravitational-wave signal, the best measured
coalescence time [39–45]. The combination of data fromcombination of the masses is the chirp mass [51]the LIGO þ0.004
and Virgo detectors allowed a precise sky
M
¼
1.188
. Fromtheof unionofcredibleweek ending
2 S90%−0.002 M ⊙ to
P
H
Y
I
C
A
L REVIEW LETTERS
position
localization
an
area
28
deg
.
This
measure20
OCTOBER
2017
PRL
119,
161101(2017)
intervals
obtainedusing differentfollow-up
waveformcampaign
models (seement enabled
an electromagnetic
that
Ref: B.P.Abbot et al. 2017
Sec.
IV
for
details),
the
total
mass
of
the
systemis
betweenidentified
a counterpart
nearfromthe 24galaxycon∼100s (calculated
startingHz) NGC
inare
thein4993,
detectors’
2.73
and
3.29
M
.
The
individual
massesthebroad⊙sistent withlocalizationanddistanceinferredUTC.
from
sensitive
band,thetheinspiralsignalended
at 12∶41:04.4
range
of
0.86
to
2.26
M
,
due
to
correlations
betweentheir⊙[46–50].
gravitational-wave
dataIn
addition,
a
γ-ray
burst
was
observed
1.7
s
after
the
uncertainties.
Thissuggests
a
BNS
as
the
sourceoftheFrom the time
gravitational-wave
signal, the best
measuredcoalescence[39–45].
The
combination
of
data
from
gravitational-wave
signal,as thetotalof known
combinationof Virgothemassesis allowed
the masses
chirp
mass
[51]
the
LIGO
and
detectorsaprecisesky
BNS systems
are between 2.57 and 2.88 M
withcompoþ0.004⊙
2
M ¼ 1.188
M . From
90%measurecredible
position
localization
areatheof union28 degof. This
−0.002
nents between
1.17⊙toandan∼1.6
M⊙ [52]. Neutron stars inintervals
obtainedusing differentfollow-up
waveformcampaign
models that
(see
ment
enabledan electromagnetic
generalhave precisely
measured masses as large
as 2.01 #Sec. IV for details),
the total massof the system
is between
identified
counterpart
the galaxy
NGC
4993,
con0.04 M⊙ a[53],whereas nearstellar-mass
black
holes
found
in
2.73 and
3.29theM⊙localization. The individual
masses arein the broad
sistent
withanddistanceinferredfrombinaries in our galaxy have masses substantially greater
range of 0.86 to 2.26
M⊙[46–50].
, due to correlations between theirgravitational-wave
data
than the components
of GW170817 [54–56].
uncertainties.
This suggests a BNS
asthe
thebestsourceof theFrom
the gravitational-wave
signal,measured
Gravitational-wave
observations
alone
are ableto meagravitational-wave
signal,as themassesmassof known
combinationof ofthemassesis total
the setchirp[51]sure the masses
thetwoobjects
and
a
lower
limit
on
þ0.004BNS
systemsarebetween
2.57
and
2.88
M
with
compo⊙
M
1.188−0.002 M⊙but. Fromthe unionof 90%
their¼ compactness,the resultspresented
herecredible
do Journal
not Letters, 848:L12 (59pp), 2017 October 20
The Astrophysicalnents
between
1.17
and
∼1.6
M
[52].
Neutronstars
in⊙Ref: P. S. Cowperthwaite 2017
intervals
obtainedmore
using
different
waveformmodels(see
exclude objects
compact
thanneutron stars
suchasgeneralhave
precisely
massesas largeisasbetween
2.01 #
Sec.
fordetails),
the measured
total
the system
FIG. 1. Time-frequency representations [65] of data containing
quarkIVstars,
black holes,
or mass
moreofexotic
objects [57–61].
0.04
M
[53],
whereasstellar-mass
black
holes
found
in
⊙ 3.29 M ⊙ . The individual masses are in the broad
2.73
the gravitational-wave event GW170817, observed by the LIGOThe anddetectionof GRB 170817A and subsequent electrobinaries
in ourgalaxysubstantiallygreater
Hanford (top), LIGO-Livingston (middle), and Virgo (bottom)
range
of 0.86
to 2.26
M⊙have, due masses
to correlations
betweentheirmagnetic
emission
demonstrates
the presenceof matter.
than the components
of GW170817[54–56].
detectors. Times are shown relative to August 17, 2017 12∶41:04
uncertainties.
This suggests
BNS astheMoreover, althougha neutronastar–black
holesource
systemofis thenotUTC. The amplitude scale in each detector is normalized to that
Gravitational-wave
observations
alonemasses
are able tomeagravitational-wave
signal,as
the mass
total
known
ruled out, the consistencyof theestimatesofwith
thedetector’s noise amplitude spectral density. In the LIGO data,
sure themassesthe two2.57
objects anda⊙lower
limit onBNS
systems
areofbetween
2.88setMneutron
with compodynamicallymeasuredmasses andof knownstars inindependently observable noise sources and a glitch that occurred
their compactness,but
resultspresented
herestars
do notnents
between
1.17inconsistency
andthe∼1.6
Mwith
Neutronin⊙ [52].
binaries,
and their
the massesof known
in the LIGO-Livingston detector have been subtracted, as
excludehave
objects
more measured
compact thanneutron
stars
such
as
generalprecisely
masses
as
large
as
2.01
#
black holes in galactic binary systems, suggests the source
described
in the text. Thisrepresentations
noise mitigation
same
as that
FIG. 1. Time-frequency[65]isofthe
data
containing
quarkMstars,
black
holes,stellar-mass
or more exotic
objects[57–61].
0.04[53],
whereas
black
holes
found
in
⊙
was composed of two neutron stars.
usedfor the results presented
in Sec. IV.observed by the LIGOthe gravitational-wave
event GW170817,
The detection
GRB have
170817A
andsubstantiallysubsequent electrobinaries
in ourofgalaxy
massesgreater
Hanford (top), LIGO-Livingston (middle), and Virgo (bottom)
magnetic
emission demonstrates
the[54–56].
presence of matter.
than
the componentsof II.
GW170817
DATA
detectors. Times are shown relative to August 17, 2017 12∶41:04
Moreover,althoughaneutronstar–blacksystemnot
Gravitational-wave observations
alonehole
are able
to ismeaUTC.
The
amplitudescaletointhe
eachdetector
is normalized
to that
in
the
Virgo
data
due
lower
BNS
horizon
and
the
At
the
time
of
GW170817,
theAdvanced
LIGO
detecruledout,theconsistency
of
the
mass
estimateswiththesure the masses of the two objects and set a lower limit on
detector’s noise amplitude spectral density. In the LIGO data,
directionof the source with respect to the detector’s antenna
tors
andthe measured
AdvancedVirgo
detector
wereinobserving
dynamicallyof known
neutron
stars
in
their
compactness,but masses
theresultspresented
here
do notindependently observable noise sources and a glitch that occurred
pattern.
mode.
maximum
distanceswhich
theofsuch
LIGObinaries,Theand their
inconsistency
withatneutron
themassesknown
in the LIGO-Livingston detector have been subtracted, as
excludeobjects
morecompact
than
stars
as
Figure 1. Localization of the gravitational-wave,
and optical signals.
Thedataleft panelasshows
an orthographic
projectionof the 90%
credible
Figure1 theillustrates
the
they
were
analyzed
toand
Livingston
LIGO-Hanford
detectors
couldthedetectblack holesingalacticbinarysystems,suggestssourceaFIG.
1. gamma-ray,Time-frequency
representations
ofthe
data
containing
described
in
text.
noise
same
as Fermi
thatquarkstars, andblackholes,or moreexoticobjects[57–61].
; light green), the initial
LIGO-Virgo
localization(31 deg This; darkgreen),
IPN mitigation
triangulation[65]fromis
the
time
delay
between
INT
LIGO (190 deg

GW170817:
Binary Neutron star Merger
・GW170817•

On 17th of August 2017, advanced
LIGO and advanced Virgo reported
the ﬁrst detec,on of gravita,onal
waves from a binary BNS merger•

Electromagne,c (EM)counterparts to GW170817 wereobserved over the en,rewavelength range (from radio togamma wavelengths)22Mul,-messenger Astronomy
PRL 119, 161101 (2017)•

week ending
20 OCTOBER 2017

NS Tidal deformability

Single event, but many physics• Constraints on NS masses, spins, and NS equa,on of state (NS ,dal deformability)• associa,on of kilonova/macronova (?): mass ejec,on, r-process synthesis• A]erglow, radio ﬂare: constraints on rela,vis,c jet
Ref: B.P.Abbot et al. 2017
, environment of the event• Host galaxy: Hubble parameter, GW propaga,on speed
Optical-IR EM counterparts of GW170817
• etc…
FIG. 5. Probability density for the tidal deformability parameters of the high and low mass components inferred from the detected
Data:
Villar +,1711.11576
~1042 erg/s
signals using the post-Newtonian model. Contours enclosing 90% and 50% of the probability
density areoverlaid(dashedlines). rTheidiagonal dashed line indicates the Λ1 ¼ Λ2 boundary. The Λ1 and-16
Λ2 parameters characterize the size of the tidally induced massz
deformations of each star and are proportional to k2 ðR=mÞ5 . Constraints are shown for the high-spin scenario≤ 0.89 (left panel)Jand
I jχjfraarset of representative
for the low-spin jχj ≤ 0.05 (right panel). As a comparison, we plot predictions for tidal deformability givennby
Hed-15 [161], all of which support stars of 2.01M
equations of state [156–160] (shaded filled regions), with labels following⊙ . Under
K theassumption that both components are neutron stars, we apply the function ΛðmÞ prescribed by that
equation
of
state
to
the
90%
most
Ne
probable region of the component mass posterior distributions shown in Fig. 4. EOS that produce lessar compact stars, such as MS1 and-14
MS1b, predict Λ values outside our 90% contour.
In…and many tasks and problems••

Today,
I will talkabout our recent works for modelling
accurate
GW templatefor BNSs
fra
accurate kilonovae/macronovae
lightcurve
predic/on
regravitaKonal
waves(brieﬂy)andd observations ofconsistent with the rate inferred fromcase and (1.0, 0.7) in the high-spin case. Further -13
Existence of rela,vis,c jetlow-spingalactic BNS systems [19,20,155,180].
analysis is required to establish the uncertainties of theseelectromagneKc
counterparts(mainly)
From this inferred rate, the stochastic background oftighter bounds,
and a detailed study of systematics is a subject
cocoon emission?, structured
jet?
gravitational
wave s produced by unresolved BNS mergers
of ongoing work.
ical
Opt

•

Absolute AB Maginitude•

PHYSICAL REVIEW LETTERS-12

from binary neutron star mergers
Preliminary comparisons with waveform models underdevelopment [171,173–177] also suggest the postNewtonian model used will systematically overestimate
the value of the tidal deformabilities. Therefore, based on

Ref: V. A. Villar 2017

throughout the history of the Universe should be comparable in magnitude to the stochastic background produced
0
2
4
6 As the8 advanced
10 detector
1214
by BBHmergers[181,182].
network improves in sensitivity
in themerger
coming years, thet (day) afterGravitaKonal waves frombinary neutron star mergers

Neutron star binaries
•

•

•

•

•

Gravita,onal waves contain the informa,on of thebinary (masses, spins, etc…) in its waveforms.
Tidal deformation

In par,cular, if the binary contains a NS, we canextract the informa,on of the internal structure of
the NSTidal
GW (t)
Modification in the GW phase

During the inspiral, a NS is deformed by the ,dalforce of the companion object. Deforma,on of a
NS (s) accelerates the orbital shrinking, andmodiﬁes gravita,onal waveforms
From the observed waveforms,
the %dal deformability of a NS can be extracted
The ,dal deformability reﬂects the internal
structure of a NS, and it can be used to constrain
the NS equa/on of state (EOS)⇤=G<latexit sha1_base64="WMkxFZNFyfHKVG9dtfYLPPbHpdY=">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</latexit>

✓

2

c
GMNS

◆5

✓ 2
◆5
c RNS
⇠
GMNS

（dimensionless）tidal deformability

Qij =<latexit sha1_base64="aJenXKMDU2QbNsOYCQJKLK6bQ0U=">AAACJnicbVDLSgMxFM3UV62vUZdugkVwY5kRQTeFogguW7AP6AzDnTRt02YeJBmhDPM1bvwVNy4qIu78FNMHYlsPBA7n3JPkHj/mTCrL+jJya+sbm1v57cLO7t7+gXl41JBRIgitk4hHouWDpJyFtK6Y4rQVCwqBz2nTH95N/OYTFZJF4aMaxdQNoBeyLiOgtOSZ5ZqXskFWvnC4DnUgdQhwfJ8tqk4MQjHgHvtlA6faZ55ZtErWFHiV2HNSRHNUPXPsdCKSBDRUhIOUbduKlZtO7iScZgUnkTQGMoQebWsaQkClm07XzPCZVjq4Gwl9QoWn6t9ECoGUo8DXkwGovlz2JuJ/XjtR3Rs3ZWGcKBqS2UPdhGMV4UlnuMMEJYqPNAEimP4rJn0QQJRutqBLsJdXXiWNy5JtlezaVbFyO68jj07QKTpHNrpGFfSAqqiOCHpGr2iM3o0X4834MD5nozljnjlGCzC+fwCx/KaJ</latexit>

Quadrupole

Eij =

@i @j

tidal fieldAccuracy of gravita,onal
waveform model•

Physical informa,on is extracted from observed gravita,onal
waves by the comparison with theore,cal templates→ an accurate waveform templates are crucial for parameteres/ma/on•Accurate waveform models for point-par,cle (no ,dal eﬀect)
waveforms calibrated by numerical simula,ons are welldeveloped (ex. EOB model, Phenom model)•

On the other hand, only a limited number of models (Dietrich
et al. 2017) exist for the waveforms including NS ,dal eﬀects
calibrated by numerical simula,ons

HB135-135
B135-135

HB125-125
B125-125

Analy,c GW models for BNS

FIG. 3. The relative deviation of the tidal-part phase of thehybrid waveforms from the 2.5 PN order tidal-part phase nor2.5PN
malized by ⇤2/3 , i.e., ( Hybrid
1)/⇤2/3 . Both vertitidal / tidal
cal and horizontal axes are plotted with logarithmic scales.

•

The waveforms including the linear- order ,dal eﬀects are derived
tidal part of the gravitational-wave
phase computedby post-Newtonian (PN) calcula,on (and thetheEﬀec,ve-One-Body
formalism)
HybridHybrid•Newtonian (Flanagan et al. 2008)•

1 PN (Vines et al. 2011)•

2.5 PN (Damour et al. 2012)•

Self force (Bernuzzi et al. 2015)

from the hybrid waveforms, tidal (=pp ,where Hybrid is the phase of the hybrid waveforms), forthe equal-mass binaries normalized by the 2.5 PN order・2.5 PN
Kdal phasecorrecKon(equal-mass)
tidal-partphase givenby2 [12]
✓
◆
3
39
2.5PN
⇤ x5/2
tidal =
322
3115
28024205 2 4283 5/2⇥ 1+
x ⇡x3/2 +
x
⇡x
,
1248
33022081092(2.14)⇣ ⌘v 2
2/3
x := (⇡m0 f ) 2/3⇡
: PN parametercwhere x = (⇡m0 f )is a dimensionless PN parame<latexit sha1_base64="jhRMUM93ar5JjVywPryCOp3/0Hk=">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</latexit>

ter, and ⇤ = ⇤1 = ⇤2 for the equal-mass cases. We
note that the tidal-part phase of the hybrid waveforms• Dynamical ,de (Hinderer et al. 2016)
in Fig. 2 is aligned with the 2.5 PN order tidal-partphase given by Eq. (2.14) for 10 Hz  f  50 Hz employing Eqs. (2.12) and (2.13). We find that the tidal-part• Tidal eﬀects become signiﬁcant in the lastphasehybridwaveforms
deviates significantly
frompartofofthethe
inspiral.
However,the
2.5 PN order tidal-part phase in the high-frequency
model based on PN calcula/on would not thebeaccurate
range,
f & 500 Hzjust(x &before
0.075 for them0 =merger.
2.7 M ), and thedeviation depends non-linearly on ⇤ (note the quantities
PredicKon by numerical simulaKons is importantmodelling.
shown infor
Fig.BNS
2 are waveform
already normalizedby ⇤). This indithat the non-linear contribution of ⇤ is appreciably(at least needed to cates
be checked)
present in Hybrid
for the high-frequency range.
tidal

Numerical Rela,vity Simula,on•Numerical-rela/vity (NR) simula/on is the unique method topredict dynamics and gravita,onal waves in the merger phase—aregime where the non-linear eﬀect of hydrodynamics should betaken into account in the framework of general rela,vity.

•

We performed the simula,ons by using a numerical-rela,vity
code, SACRA (Yamamoto et al. 2008),in which an adap,ve-meshreﬁnement (AMR) algorithm is implemented(Kiuchi et al. 2017, KK et al. 2018)

Einstein’s equaKon
Euler equaKon8⇡Gµµ⌫rµ (⇢u ) = 0 rµ T = 0
Gµ⌫ = 4 Tµ⌫c
EquaKon of state (EOS)(+MHD, viscosity, neutrino, etc., )P = P (⇢), P (⇢, T, Ye ), · · ·Results (1.35-1.35M_sun)!

•

Radius & Tidal deformability
15H>125H>H>HB>BComparison with EOB model•ΦNR-ΦTEOB (rad)These latest ,dal-EOB (SEOBNRv2T) waveforms
can be accurate even up to~3 ms before the onset of merger.

NR v.s. SEOBNRv2T 1.35-1.35 M_sun
1.5
1

15H
125H
H
HB
B

0.5
0
-0.5
0.15

h D /m0•

We compare our NR waveforms with the EOB
waveforms considering dynamical ,des(Hinderer et al. 2016)

0.1

15H, NR
15H, TEOB
125H, NR
125H, TEOB
H, NR

H, TEOB
HB, NR
HB, TEOB
B, NR
B, TEOB

0.05
The phase diﬀerence between the TEOB
0
waveforms and the numerical-rela,vity results is
0
10
20
30
40
50
60
tret (ms)s,ll larger than
~1 rad a]er two NSs come into contact for the FIG. 5. Comparison between NR and TEOB waveforms for thewith N = 150 (left) and 182 (right). The top and bottom panecase that the NS radii are larger than 13 km
A2,2 D/m0 of gravitational waves, respectively. Here, D denotes
peak amplitude time tpeak when the gravitational-wave amplitude(Λ~850)→Further improvement of the waveformmodel may be needed to suppress thesystema/c error in the measurement of the/dal deformability.

ΦNR-ΦTEOB (rad)

•

1.5
1
0.5
0

15H
125H
H
HB
B

15H135-135
125H135-135
H135-135
HB135-135
B135-135f [Hz]
0
200
400
600
800
15H125-125
TABLE I. Model name, mass of each neutron star, mi (i = 1, 2), equations of s
125H125-125
f [Hz]
2(m1 m2 )3/5H125-125/ (m1 + m2 )1/5 , symmetric mass ratio,
⌘
=
m
m
/
(m
+
m
)
, binary121
2
15H121-151
HB121-151
definition),
location of outer boundaries along125H121-151
each axis denoted by L, andfinest g
HB125-125
B121-151
radius andB125-125
the dimensionless tidal deformability H121-151
of neutron stars for each equation

Binary neutron star models

4 mass conﬁguraKon (m1+m2)
˜Model
m1 m2 EOS Mc
⌘
⇤
L [km
FIG.1.35
4. Thedi↵erence
betweenthetidal-part
phase
of
the
FIG.
5.
The
di↵erence
betweenthetidal-partphas
M_sun+1.35 M_sun
15H135-135
1.35 1.35 and15H the1.175240.25wave-phase1211
799
hybridwaveforms
andthetidal-partwave-phasemodel
dehybrid
waveforms
tidal-partmo1.21 M_sun+1.51 M_sun (q=m1/m2=0.8)
scribed in Eq. (2.15) for the equal-mass cases, where thescribed 1.35
in Eq.
the unequal-mass
125H135-135
1.35(2.21)
125Hfor1.17524
0.25 863 cases.
732
1.16areM_sun+1.58
phasesaligned forM_sun
10 Hz (q=m1/m2=0.73)
 f  1000 Hz employingH135-135
the phasesaligned
10 Hz f 6071000 Hz699em
1.35are1.35
H for
1.17524
0.25
M_sun+1.25
Eqs. 1.25(2.12)and (2.13). M_sun
Eqs. (2.12) and (2.13).
•

HB135-135 1.35 1.35 HB 1.17524 0.25 422
639
Mc = (m1 + m2 )⌘: chirp mass
B135-135 1.35 1.35 B 1.17524 0.25 289
586•
15H121-151 1.21 1.51 15H 1.17524 0.247 1198
m1 m
˜ as 782
the 1 PN ordertidal
2 correction to the wave phase can be
replacing
3/32
to3/128⌘
[48]
and
⇤
to
⇤
⌘ =in terms of the: symmetric
mass
ratio
125H121-151 1.21 1.51 125H 1.17524 0.247 856
732
writtensymmetricandanti-symmetricm1 + m2
 H 1.17524
contributions of neutron-star tidal deformation as [3] H121-151 1.21 1.51
0.247 ⌘604
682
3
39 ˜ ⇣
2/3
p
5/2
˜ x
=
⇤
1
+
a
⇤
x
• 5 equaKons of state (EOS)
tidal
HB121-151
1.21
1.51
HB
1.175240.247 422
632
✓
◆
128⌘
2
3
39 ˜ EOS
2 piecewise-polytropic
1PN
B121-151 1.21 1.51 B 1.17524 0.247 290599=⇤ x5/2
tidal
3115
28024205
4283
3/2
2(Read et128⌘
al. 2009,2Lackey et. al 2012)
⇥
1
+
x
⇡x
+
x
⇡x
15H116-158
1.16
1.58
15H
1.17524
0.244
1185
798
"
! #
1248
33022081092
˜
3115 6595 p⇤
125H116-158 1.16 1.58 125H 1.17524 0.244 848
749
⇥
1
+
1
4⌘
x
,
(2.18)
10.8 km  RNS
 13.7 km
˜1248 7098
H116-158 1.16 1.58 H 1.17524 0.244 601
699⇤
HB116-158
1.16
1.58 HB
1.17524di↵erence
0.244 421
649
Binary Kdal deformability (e.g. Wade et al.)Figure5 shows
the phase
between˜ and˜ ⇤˜ are defined by
B116-158
1.16 1.58andBthe1.17524
0.244
291
599
where
⇤
brid waveforms
tidal-partwave-phasemo300. ⇤ . 1900!
15H125-125
1.25(2.21)
15H for
1.08819
0.25 1875 cases782
scribed1.25
in Eq.
the unequal-mass
8 ⇥two phases
are aligned
for 10 Hz
f  1000˜
125H125-125
1.25 1.25
125H 1.08819
0.25 1352
732H⇤=
1 + 7⌘ 31⌘ 2 (⇤1 + ⇤2 )13
ploying1.25
Eqs.1.25(2.12)Hand1.08819(2.13). 0.25
Althoughfitti
H125-125
966 the 682!
prameters are determined only by employing the
1 4⌘ 1 + 9⌘ 11⌘ 2 (⇤1 ⇤2 )(2.19)HB125-125 1.25 1.25 HB 1.08819 0.25 683
632
waveform of 15H125-125, Eq. (2.17), we find th
B125-1251.25 1.25
B smaller
1.08819 than
0.25 ⇡ 476
phase error
is always
0.05 rad599
fo
•
3/5

<latexit sha1_base64="rCh/s5mU5eEQd8vbSjTnHXnlVHU=">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</latexit><latexit sha1_base64="VHjYFKsRQtMDIt0OIu49GiiWDPs=">AAACI3icbVBNSwMxEM3Wr1q/qh69BIsgWMquCIogFL14rGBV6JYlm85qaLK7JLNCWfpfvPhXvHhQihcP/heztQdtHQh5vDcvmXlhKoVB1/10SnPzC4tL5eXKyura+kZ1c+vGJJnm0OaJTPRdyAxIEUMbBUq4SzUwFUq4DfsXhX77CNqIJL7GQQpdxe5jEQnO0FJB9dQHZGd+pBnPVeCp4HBY3AcFOPXrua8VNQOlALXgfl0xY/y6LszDoFpzG+646CzwJqBGJtUKqiO/l/BMQYxc2oc6nptiN2caBZcwrPiZgZTxPruHjoUxU2C6+XjHId2zTI9GibYnRjpmfztypowdNLSdiuGDmdYK8j+tk2F00s1FnGYIMf/5KMokxYQWgdGe0MBRDixgXAs7K+UPzOaFNtaKDcGbXnkW3Bw2PLfhXR3VmueTOMpkh+ySfeKRY9Ikl6RF2oSTJ/JC3si78+y8OiPn46e15Ew82+RPOV/fM9CkoA==</latexit><latexit sha1_base64="AOc9Y9VrJZ6eCs2K4DJrK51H2RI=">AAACF3icbVDLSsNAFJ34rPUVdelmsAguJCQqtMuiG1dSH31AE8JkOm2HziRhZiKU0L9w46+4caGIW935N07SgNp6YODcc+7lzj1BzKhUtv1lLCwuLa+sltbK6xubW9vmzm5LRonApIkjFolOgCRhNCRNRRUjnVgQxANG2sHoIvPb90RIGoV3ahwTj6NBSPsUI6Ul37Qc26q5x6krOBzxicsIvPHz6uo2r5xTq/rj+2bFtuwccJ44BamAAg3f/HR7EU44CRVmSMquY8fKS5FQFDMyKbuJJDHCIzQgXU1DxIn00vyuCTzUSg/2I6FfqGCu/p5IEZdyzAPdyZEaylkvE//zuonq17yUhnGiSIini/oJgyqCWUiwRwXBio01QVhQ/VeIh0ggrHSUZR2CM3vyPGmdZNk612eV+nkRRwnsgwNwBBxQBXVwCRqgCTB4AE/gBbwaj8az8Wa8T1sXjGJmD/yB8fENi8qdrA==</latexit>andWe focus only on the case that the neutron-star
spins are absent.

✓
◆
p
1
13272
8944 2

unequal-mass cases.

Frequency-domain waveform model•

We derive a frequency domain waveform model based on the numerical-rela,vity
waveforms, in par,cular, in the frequency domain for convenience in data analysis•

NR waveforms are s,ll too short for the use of construc,ng an waveform model,
so we construct hybrid waveforms composed of our latest numerical-rela,vity
waveforms (>400 Hz) and the EOB (SEOBNRv2T) waveforms (<400 Hz) to calibrate thewaveform model.
•

Numerical-rela,vity waveforms ~30 gravita,onal cycles (>200 rad)0.1–0.6 rad phase error up to the peak of the amplitude(less than 0.1 rad un,l f < 1000Hz)•

The EOB (SEOBNRv2T) waveforms (Hinderer et al. 2016)
Ini,al frequency: 10 Hzalignment window˜ as the binary tidal deformait ising respectively. We refer to ⇤bility. For realistic cases, the tidal contributions to thetivityZ fmaxgravitational-wave
phase are dominated by the contribu9].
2
Hybrid00em
˜ terms
I =tions from the(f ) [19]. tidal(f ) that2⇡f tthe⇤Assuming
0 +contribu0 df,
tidal
fmin
˜ terms and those from the higher-ordertions
from the ⇤(2.16)
terms are always sub-dominant in the tidal part ofthephase, we extend the formula of Eq. (2.15) by replacing•whereThe
following
phenomenologicalform
of a ﬁwng
formula is
t
and
are
parametersthat
correspondto the˜00
el of
3/32 to 3/128⌘ [38] and ⇤ to ⇤ as
employed:of freedom for choosing the time and phase oriepen-degrees⌘
00
39 ˜ ⇣
tidalgins. Thus,=we 3minimize
parameters, a,
2/3 four˜the⇤ I 1 +fora⇤
xp x5/2tidal
2 is performed for fmin = 10 Hz
f thep, !t0 , and 0 . 128⌘
The fitting
tron-and fmax = 1000
3115Hz.
28024205 2 4283 5/2
3/2⇥ 1+x ⇡x +
x⇡x,
We
hybrid waveformof 15H125-125
3302208
1092 for deter! use the1248(2.21)
mining the fitting parameters because the non-linear
con-

tribution of ⇤ is most significant for this among the biwherethe values
indetermined
Eq. (2.17)byareused for thea andp.
• Modelparameters
employinghybridnary neutronstar aremodelsemployedin this work.
Then
Figureshows the phasebetweenhybridwaveformof 515H125-125,
whichdi↵erence
has the largest
valuetheof Λ,
weobtain2.18)
waveforms and the tidal-part phase model described in

0.1^Hybrid
tidal -^tidal [rad]em^Hybrid
tidal -^tidal[rad]Results

0.05
0
-0.05
-0.1
0

200

15H135-135
125H135-135
H135-135
HB135-135
B135-135

400
600
800
f [Hz]
15H125-125
125H125-125
H125-125
HB125-125
B125-1251000

7^Hybrid
tidal -^tidal [rad]em^Hybrid
tidal -^tidal[rad]and we obtained
0.1
0.1
Eq. (2.21) for the unequal-mass cases. Here, two phasesFIG. 4. The di↵erence between the tidal-part phase of the
waveforms and the tidal-part phase model described
are again aligned 0.05
fora =
10 12.55,
Hz  f  1000 Hz employinghybrid
0.05
in Eq. (2.15) for the equal-mass cases, where the phases! Eqs. (2.12) and (2.13).
are aligned for 10 Hz  f  1000 Hz employing Eqs. (2.12)
p =Although
4.240. the fitting parameters(2.17)and (2.13).
00
are determined only by employing the hybrid waveform
15H125-125,(2.17),
wewaveform
find thatthe phaseerroris as large as 0.08 rad for f ⇡ 1000 Hz. However, it is
4 shows
theEq.
phase
di↵erence
betweenthetidal•Figure
Theofphase
diﬀerence
between
our
model
and
the
-0.05
-0.05
is
alwayssmaller
than
⇡
0.07
rad
for
these
unequal-mass
smallerthan the phase error in the numerical-relativity
hybridwaveforms
is always
smallerandthan the0.1 rad.
of thehybridwaveforms
tidal-part phase2.19)part
waveforms associated with the finite-di↵erencing [29].
cases.
-0.1-0.1

model of Eq. (2.15) for 0the equal-masscases,
where
two
200
400
600
800
1000
0
200
400
600
800
1000
aligned
10 Hzbetween
 f 
1000
Hz employing
f [Hz]
f [Hz] cases
2. Unequal-mass•phases
There areis onlya small for
diﬀerence
thewaveform
15H135-135
15H125-125
15H121-151
15H116-158
Eqs.
(2.12)
and(2.13).
Althoughthefitting
parameters
125H135-135
125H125-125
125H121-151
125H116-158
D.determined
Tidal part
model
forBNSthe hybrid
gravitational-wave
modelsfrom
diﬀerent
waveforms
Next, we H121-151
extend the tidal-partH116-158
phase model of
H135-135
H125-125⇤2 ) are determined by employing
amplitude
only
thehybridwaveform
HB135-135
HB125-125
HB116-158 the depenEq. (2.15) to HB121-151
unequal-mass cases. Considering
B135-135
B125-125
B121-151
B116-158
dence on the symmetric mass ratio, the1 PN order tidalof 15H125-125 as a reference, we find that the error incorrection to the phase can be written in terms of the

FIG.
hybr
in E
are a
and

resp
bilit
grav
tion
tion
term
phas
3/32

t

amplitude
model
is always
smaller than
10%. Therelainofourwaveform
model
assuming
the hybrid
wavei( ⇢+˜+lidity
TF2+
tidal )
the
case
that
⇤
&
850
and
&
35.
=
(A
A
)
e
.
TF2+
tidal
˜
and
tive aserror
is larger for ⇤850, and in particular, it is
ay- in particular, it isforms
signals.
to 15H121-151.
the binary (seeHow- larger than 15% for 15H135-135
and 15H121-151. Hownd
6
cle
This
waveform
model
has
6
parameters,
{✓
}
=
i
00
800
1000
n
o ever, such large errorsi=1
he higher
are only present for f & 900 Hz,
resentfor PNf &order900 Hz,he
B.
Systematic
error
˜
M
⇤, term. Inandthistheysectionwecheck
the vapolynomial
toc , ⌘,accuracy
0 , t0 , D
e↵our
have
only
minor
e↵ects on the accuracy of ourA.
Distinguishability
on
the
of
he
HB121-151
our waveform model
assuming
the hybrid
wave- in the next section.
waveformmodel
as is shown
elidity
nextofsection.
ayB121-151
Next, we estimate
the systematicof our waveforms
as signals.
To checkthe validity errorsof our waveformmodel derived in
5H125-125◆
model
in thethe previous
measurement
binaryparameters.
5H125-125
449 form
section,ofwecalculatethe distinguishability
6
q
TABLE
IV. The systematic error of our waveform model and the PN wave
isxEmploying+ bx , the hybrid
III.
VALIDITYOF
THE
ANALYTICAL
MODEL
H125-125
waveforms
ashypotheticalsignals,
betweenourwaveform
model
and
the
hybridwaveforms
64
ALYTICAL
MODEL
ofbinaryparameters
using the hybrid waveforms as hypothetical signals.
veHB125-125
A. Distinguishability
thesystematic
error
for
each
waveformparameter,✓,
supposing advanced LIGO as a fiducial detector.
For thisi(2.23)
SystemaKc
error:
hisB125-125
TT
our waveform model
is defined by ✓i purpose,✓i where✓iintroduce
is a parameter
ofproduct
the hybrid
We weconstructeda innerfrequency-domain
gravitationalan
andModelthenorm ofdomaingravitational˜˜To checkthe validity ofi our
waveformmodel
inneutron⇤
Mc [M ]⌘⇤
waveform
model
forderived
binarystars by employing
aswaveforms
iswaveforms
thecorresponding
best-fit
parameter
ters. We determineand ✓theTonby employing
6
thestarsprevious
section,fromdistinguishability✓we=calculate✓i⇣tidal-part
✓the
#
the
and"point-particle
part15H135-135
modelsof 1211
gravi- 1.1 ⇥ 10 7
1.8
⇥
10
2.1
determined
Z fmaxi⌘al-partwave-amplitude⇤
betweenourwaveform
model
and
the
hybridwaveformsh̃1 (f) h̃2 (f ) section and Appart
models of gravi- htational waves
76
i = 4Reh in theiprevious
derivedhe
125H135-135
863
1.9
⇥
10
7.7
⇥
10
2.8
the hybrid waveforms,
6
h̃
h̃
df
,
(3.2)
6
1
2
Tfiducial detector. For this
supposingadvanced
LIGO
as
a
vious
sectionandApminh̃Hybrid pendix✓i i=1h̃model fmin{✓asi }i=1Sn (f, ) (3.8)
A, respectively,nde of 2the hybrid
waveH135-135607
2.2 ⇥ 10 7
9.0 ⇥ 10 6
0.1
6

Validity check:
Systema,c error

purpose,wean inner product and the norm of{✓i }introduce
i=1
he
(f
)
df,
(2.24)⇣⌘l
HB135-135422
thewaveformsas˜ 0 , t0 , De↵
3].
and
h̃
=
h̃
f
;
M
,
⌘,
⇤,
(3.1)
⌘
model
model
c
"
#
where
h̃
is
the
Fourier
spectrumofthehybridwaveB135-135
289
deZ fmaxr⇣ ⌘⇣, De↵ ⌘ Hybrid
⇤
,
t
(3.1)
0
0
h̃
(f
)
h̃
(f
)
i( TF2++ tidal )1
2 = (Anly
forms.
error
does
not
de+
A
)
e
.(3.3)
15H121-151
1198
tidal
h̃
=We
4Renote that the systematicdf⇢ TF2+,(3.2)
mplitudehy1ofh̃2the
=
||
h̃||
=
h̃
h̃
,
Sn (f ratio.
)i( TF2+pend) the signal-to-noiseon
125H121-151 6 856
set
to be+ 10tidal
Hz.and fmin
Thiswaveform modelhas 6 parameters,{✓i }i=1=
hehybridwaveform
H121-151604no◆respectively,
whereS
denotesthe
one-sided
noise
specand
n
6
449
˜
tain
b
=
4251
and
5
6
M
,
⌘,
⇤,
,
t
,
D
we check the
HB121-151
422vadensity
The section
distinguishability
bec
0 detector.
e↵ . In thisxrameters,x {✓, i }(2.22)i=1 =trum r⇣ ⌘ of0 the64
tween
waveforms,h̃1 modelandh̃2assuming, is definedbyhybrid[17, 49]waveB121-151
290
lidityofour, waveformthe⇢ =va||h̃||
= two
h̃ h̃
(3.3)
ction
we
check
the
xtend Eq. S(2.23): noise spectrum
densityn (f ) to⌘forms as⇣signals.
15H116-158 1185
ming
theorderhybridwaveleading
coefceto thebinary(see⇢ one-sidedh̃1 ,POWER
h̃2 =noisemin ||spech̃1 h̃2 ( 0 , 125H116-158
t0 ) ||,
(3.4)
(the
ZERO
DETUNED
HIGH
configuration)
848
respectively,
whereS
denotes
the
n
˜
,t
0
0
and
respectively,the⇤,higherPN ordertrum density of the detector. The distinguishability beH116-158
601
Ref.tohPps://dcc.ligo.org/LIGO-T0900288/publica polynomial term
A.
Distinguishability
tween two waveforms, h̃1 and h̃2 , is defined by [17, 49]
HB116-158 421⇣⌘
291
bility ⇢ h̃ , h̃ = min ||h̃To checkof(3.4)
our waveformB116-158
model derivedinh̃2 ( 0the, t0 )validity
||,
1
2◆
1
0 ,t0
1875
449• 6 The
the previous
section,we calculate the15H125-125
distinguishability
5q systemaKc
error
in
the
x
x + bx ,1352
hybrid waveforms
eform64
model derived in between our waveform model and the125H125-125
measurementof Λ withrespect
966
supposingadvanced
LIGOtoas a fiducial H125-125
detector. For
this(2.23)
the distinguishability
we introduce
an inner productand the norm
hybrid purpose,
waveforms
is always
HB125-125
683 ofthe hybrid the
waveforms
the20
waveforms as
B125-125
476
meters.
We determine
ucial detector.
For thissmallerthan"Z#oduct and the norm offmax⇣⌘h̃1 (f ) h̃⇤2 (f )h̃1 h̃2 = 4Re
df ,
(3.2)
<latexit sha1_base64="ZD16OafcvC3mlA33SsqyDkkh778=">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</latexit>

2.3 ⇥ 10 7
1.7 ⇥ 10 7
4.1 ⇥ 10 7
2.2 ⇥ 10 7
1.8 ⇥ 10 7
2.6 ⇥ 10 7
1.6 ⇥ 10 7
6.3 ⇥ 10 7
5.0 ⇥ 10 7
2.8 ⇥ 10 7
2.3 ⇥ 10 7
1.4 ⇥ 10 7
1.7 ⇥ 10 7
7.1 ⇥ 10 8
5.3 ⇥ 10 8
4.5 ⇥ 10 8
4.3 ⇥ 10 8

8.8 ⇥ 10 6
6.0 ⇥ 10 6
2.4 ⇥ 10 5
1.0 ⇥ 10 5
6.9 ⇥ 10 6
1.1 ⇥ 10 5
5.3 ⇥ 10 6
4.0 ⇥ 10 5
2.9 ⇥ 10 5
1.4 ⇥ 10 5
9.7 ⇥ 10 6
2.9 ⇥ 10 6
4.3 ⇥ 10 6
3.2 ⇥ 10 6
5.3 ⇥ 10 6
6.9 ⇥ 10 6
1.5 ⇥ 10 5

-2.4
-3.7
9.8
2.1
-1.9
-2.6
-6.1
14.7
6.7
-1.5
-5.0
-9.1
1.5
-3.6
-8.1
-13
-20

*used for the model calibraKon

Sta,s,cal error

and dashed curves denote the cases with ⌘ = 0.25 and0.244, respectively. The black curves denote the results˜ = (2.7 M , 0) in which analysis thefor (Mc /(1/4)3/5 , ⇤)tides are not considered (note that the tides are consid˜ = 0 cases for which theered in the analysis for the ⇤results are shown with blue, green and light-blue curves
in Fig. 9).
The top panel in Fig. 9 shows that the statistical error
in the measurement of Mc depends only weakly on theupper-bound frequency of the analysis for fmax & 400 Hz.
The improvement of the statistical error by changing
fmax from 400 Hz to 1000 Hz is only ⇡ 25%. Figure 9
also shows that the statistical error becomes smaller forsmaller values of Mc , and depends only very weakly on
˜ The bottom panel in Fig. 9 shows that the⌘ and ⇤.
statistical error in the measurement of ⌘ depends morestrongly on the upper-bound frequency than that of Mc .
The statistical error is reduced by ⇡ 40% by changing
fmax from 400 Hz to 1000 Hz. On the other hand, thestatistical error of ⌘ depends only very weakly on the bi˜ The results ofnary parameters, such as Mc , ⌘, and ⇤.
the analysis without tides show that, if tides are considered, the statistical error of Mc increases by ⇡ 25–40%,
and that of ⌘ by a factor of 2. These results are consistent
with those found in Ref. [12].
Figure 10 shows the statistical error in the measure˜ The top panel of Fig. 10 shows that thement of ⇤.
˜ is significantly reduced if the upperstatistical error of ⇤bound frequency is increased. The statistical error de2
creases approximately in proportion to 1/fmax. On theother hand, the statistical error depends only weakly on
Mc and ⌘. This dependence on fmax and ⌘ is consistent
with Eq. (23) in Ref. [10]. The bottom panel of Fig. 10
˜ as a function of ⇤˜ forshows the statistical error of ⇤the case fmax = 1000 Hz. This indicates that the sta<latexit sha1_base64="xUE+jscwVMlr4/eMi0jXqsPSK60=">AAACF3icbZDLSsNAFIYn9VbrLerSzWAR6qYkIuiy6MaV1Esv0IQymU7boZOLMydiiXkLN76KGxeKuNWdb+O0DaitPwx8/OcczpzfiwRXYFlfRm5ufmFxKb9cWFldW98wN7fqKowlZTUailA2PaKY4AGrAQfBmpFkxPcEa3iD01G9ccuk4mFwDcOIuT7pBbzLKQFttc1y4kgfX51fpg6JIhneOepGQlJKHOCiw3A/vf/B/bRtFq2yNRaeBTuDIspUbZufTieksc8CoIIo1bKtCNyESOBUsLTgxIpFhA5Ij7U0BsRnyk3Gd6V4Tzsd3A2lfgHgsft7IiG+UkPf050+gb6aro3M/2qtGLrHbsKDKAYW0MmibiwwhHgUEu5wySiIoQZCJdd/xbRPJKGgoyzoEOzpk2ehflC2rbJ9cVisnGRx5NEO2kUlZKMjVEFnqIpqiKIH9IRe0KvxaDwbb8b7pDVnZDPb6I+Mj28Ux6B6</latexit>••

SystemaKcM /(1/4)
error~= 2.5 M , R = 0✓<latexit sha1_base64="gBK/ZIYqwO5zadNd3R5YUfVqic4=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBU0lE0GNRDx4r2A9oQtlsJ+3SzSbsToQS+je8eFDEq3/Gm//GbZuDtj4YeLw3w8y8MJXCoOt+O6W19Y3NrfJ2ZWd3b/+genjUNkmmObR4IhPdDZkBKRS0UKCEbqqBxaGETji+nfmdJ9BGJOoRJykEMRsqEQnO0Eq+fwcSmY8jQNav1ty6OwddJV5BaqRAs1/98gcJz2JQyCUzpue5KQY50yi4hGnFzwykjI/ZEHqWKhaDCfL5zVN6ZpUBjRJtSyGdq78nchYbM4lD2xkzHJllbyb+5/UyjK6DXKg0Q1B8sSjKJMWEzgKgA6GBo5xYwrgW9lbKR0wzjjamig3BW355lbQv6p5b9x4ua42bIo4yOSGn5Jx45Io0yD1pkhbhJCXP5JW8OZnz4rw7H4vWklPMHJM/cD5/AAcHkao=</latexit>

600
500

3/5

StaKsKcal error
1
/⇡
SNR
✓

400
300

Employing the standard Fisher-matrix
analysis, we es,mated the sta,s,cal error
in the measurement of binary parameterssun~
Mc/(1/4) = 2.7 Msun, R = 0~
Mc/(1/4)3/5 = 2.9 Msun, R = 0~
Mc/(1/4)3/5 = 2.5 Msun, R = 1200~
Mc/(1/4)3/5 = 2.7 Msun, R = 1200~
Mc/(1/4)3/5 = 2.9 Msun, R = 1200

700(h̃|h̃)<latexit sha1_base64="f2VlAFZWJY8jGNciqcowvtoAm+A=">AAACEnicbZDLSgMxFIYz3q23qks3wSLowjIjgi5FN67EW1Xo1HImTdvQZBKSM2IZ+gxufBU3LhRx68qdb2Nau/D2Q+DjP+dwcv7ESOEwDD+CkdGx8YnJqenCzOzc/EJxcenC6cwyXmFaanuVgONSpLyCAiW/MpaDSiS/TDoH/frlDbdO6PQcu4bXFLRS0RQM0Fv14kbsREtBPcY2R4iN1QZ1DMbDbR5bRc+OTnvX+WbUqxdLYTkciP6FaAglMtRxvfgeNzTLFE+RSXCuGoUGazlYFEzyXiHOHDfAOtDiVY8pKO5q+eCkHl3zToM2tfUvRTpwv0/koJzrqsR3KsC2+13rm//Vqhk2d2u5SE2GPGVfi5qZpKhpPx/aEJYzlF0PwKzwf6WsDRYY+hQLPoTo98l/4WKrHIXl6GS7tLc/jGOKrJBVsk4iskP2yCE5JhXCyB15IE/kObgPHoOX4PWrdSQYziyTHwrePgFJqJ5z</latexit>

200
100
0
400

Sta,s,cal error is always smaller than
the systema,c error of the model(against the hybrid waveforms)
if SNR is smaller than ~100

We obtained results consistent withthe previous studies(Hinderer et al. 2010, Damour et al. 2012,
Wade et al. 2014)

3/5

c

500

600

✓T
<latexit sha1_base64="H6tb2MH/NaioK856QauAZXtH34w=">AAAB9XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cK/YI2ls120y7dbMLuRCmh/8OLB0W8+l+8+W/ctjlo64OBx3szzMwLEikMuu63U1hb39jcKm6Xdnb39g/Kh0ctE6ea8SaLZaw7ATVcCsWbKFDyTqI5jQLJ28H4dua3H7k2IlYNnCTcj+hQiVAwilZ66OGII+1nPR2RxrRfrrhVdw6ySrycVCBHvV/+6g1ilkZcIZPUmK7nJuhnVKNgkk9LvdTwhLIxHfKupYpG3PjZ/OopObPKgISxtqWQzNXfExmNjJlEge2MKI7MsjcT//O6KYbXfiZUkiJXbLEoTCXBmMwiIAOhOUM5sYQyLeythI2opgxtUCUbgrf88ippXVQ9t+rdX1ZqN3kcRTiBUzgHD66gBndQhyYw0PAMr/DmPDkvzrvzsWgtOPnMMfyB8/kDf4uSfg==</latexit>

700
fmax [Hz]

✓best

800

900

1000

<latexit sha1_base64="WW1OcRf7Nvqwgbhp8qT9ag6nKHs=">AAAB+nicbVDLSgNBEJyNrxhfGz16GQyCp7Argh6DXjxGMA9IlmV20psMmX0w06uENZ/ixYMiXv0Sb/6Nk2QPmljQUFR1090VpFJodJxvq7S2vrG5Vd6u7Ozu7R/Y1cO2TjLFocUTmahuwDRIEUMLBUropgpYFEjoBOObmd95AKVFEt/jJAUvYsNYhIIzNJJvV/s4AmR+3lcRDUDj1LdrTt2Zg64StyA1UqDp21/9QcKzCGLkkmndc50UvZwpFFzCtNLPNKSMj9kQeobGLALt5fPTp/TUKAMaJspUjHSu/p7IWaT1JApMZ8RwpJe9mfif18swvPJyEacZQswXi8JMUkzoLAc6EAo4yokhjCthbqV8xBTjaNKqmBDc5ZdXSfu87jp19+6i1rgu4iiTY3JCzohLLkmD3JImaRFOHskzeSVv1pP1Yr1bH4vWklXMHJE/sD5/AITFlCc=</latexit>

fmin = 10Hz, fmax = 1000Hz, l = 50
180
160
140
120
mR~

•

fmin = 10Hz, l = 50

800

mR~

SNR ⇡

q

13

100
80
60

Mc/(1/4)3/5= 2.5 Msun, d = 0.25
3/5
Mc/(1/4)3/5 = 2.7 Msun, d = 0.25
Mc/(1/4) = 2.9 Msun, d = 0.25
Mc/(1/4)3/5= 2.5 Msun, d = 0.244
Mc/(1/4)3/5 = 2.7 Msun, d = 0.244
Mc/(1/4)3/5 = 2.9 Msun, d = 0.244

40
20
0
0

500

1000
~
R

1500

2000

Now-going works
e
r
P

y
r
a
n
i
lim

Ref: Narikawa et al. in prep.

•

We apply the BNS waveform model for the analysis of GW170817(with Narikawa-san, Uchikata-san, Tagoshi-san)•We are also working for upda,ng the waveform model by employing NR BNS
waveforms in a wider parameter range (lower mass, higher mass ra,o, NS spin)ElectromagneKc counterparts
to binary neutron stars:
Kilonovae/Macronovae

Mass Ejec,on•

A frac,on of NS material would be ejectedfrom the system during the merger•

Ejected material is neutron-rich→heavy radioac,ve nuclei would besynthesised in the ejecta by the so-called
r-process nucleosynthesis→EM emission in op,cal and NIR
wavelengths could occur by radioac,vedecays of heavy elements: kilonova/macronova (r-process nova)
Li & Paczyński 1998, Kulkarni 2005
, Metzger et al. 2010 …

Ref: K. Hotokezaka et al. 2013

Proper,es ofkilonovae / macronovae
Kilonova/macronova is expected to be nearly
isotropic emission.
(cf. ✓jet ⇠ 10 for sGRB)
The peak Kme of the emission will come in~1—10 days.
(cf. ~1 year for radio ﬂare)
The most of the emission occurs in around
op/cal and infrared.
The mass, velocity, morphology, and thecomposi/on(electron frac/on) of the ejecta
characterize the lightcurve of the kilonova/
macronova.

Rough EsKmaKon<latexit sha1_base64="PVbd6pqU4NRIq26Dbj5/l7OLM6A=">AAACkXicbVFdb9MwFHXCx0b5WPl448WiYhoSdEkLgvEwKnhBQpOGRNmkuoscx2mt2nFk30yrLP8g/s7e+De4JRJbx5UsHZ177ON7bl5LYSFJfkfxrdt37m5t3+vcf/Dw0U738ZOfVjeG8THTUpvTnFouRcXHIEDy09pwqnLJT/LFl1X/5JwbK3T1A5Y1nyo6q0QpGIVAZd1fkDliFK45XfhdQuva6As87A/J6zVf0KX1hHR2CQjFLZG8hD1SGsrckXdJPxkeZUQXGjwxYjaHV2cu3R/4q7rzlW7A/gnebCrIIvhS79LWlKmzwf7MX38y6/aC3brwTZC2oIfaOs66l6TQrFG8AiaptZM0qWHqqAHBJPcd0lheU7agMz4JsKJhvqlbR+rxy8AUuNQmnArwmr16w1Fl7VLlQakozO1mb0X+rzdpoPwwdaKqG+AV+2tUNhKDxqv94EIYzkAuA6DMiPBXzOY0pARhi50QQro58k0wHvQP+un3t73R5zaNbfQcvUB7KEXv0Qh9RcdojFi0E72LDqNP8bP4YzyKW20ctXeeomsVf/sDQ7THXg==</latexit>

tpeak ⇡ 3.3 days✓
◆1/2 ⇣
✓
◆1/2
⌘
1/2
M
v

⇥
0.03M
0.2c
1 cm2 /g

Lpeak ⇡ 2.0 ⇥ 1041 ergs/s✓◆✓◆1/2 ⇣✓◆ 1/2
f
Mv ⌘1/2

⇥
10 6
0.03M
0.2c
1 cm2 /g<latexit sha1_base64="EllYR/E4apuz2tXjhy/kGJS2IM8=">AAACvnicdVFda9swFJW9ry77ytbHvYiFlQ7WxPZKP6AP3fayhxU6WNZClBhZkR1hyTKSXBqE/uQeBvs3kxOPZe12QXA451zuvUdZzZk2UfQzCO/cvXf/wdbD3qPHT54+6z9/8U3LRhE6JpJLdZlhTTmr6Ngww+llrSgWGacXWfmx1S+uqNJMVl/NsqZTgYuK5Yxg46m0/+NzapESsKa4dDsI17WS1zAZRsgwQXUczex+7NDblYmqQo+0Q6i3s5YRp7nZRbnCxObOtu69A+eQYsXCvNlUz5yNhtG7sxTJuTS/HTMbjxK36btqfQn5vwGVfkfsZ3U7ETFLRoX707DXdqT9gR+3KngbxB0YgK7O0/53NJekEbQyhGOtJ3FUm6nFyjDCqeuhRtMakxIXdOJhhf31U7vK38HXnpnDXCr/KgNX7GaHxULrpci8U2Cz0De1lvyXNmlMfjS1rKobQyuyHpQ3HBoJ28+Ec6YoMXzpASaK+V0hWWAfk/Ff3vMhxDdPvg3GyfB4GH/ZH5x+6NLYAi/BK7ALYnAITsEncA7GgAQnQRaUAQ/fh0UoQrm2hkHXsw3+qvD6F8VO2Wo=</latexit>

Tpeak ⇡ 3.1 ⇥ 103 K✓◆1/4 ✓◆ 1/8 ⇣
✓
◆ 3/8
⌘
1/8
f
M
v

⇥
10 6
0.03M
0.2c
1 cm2 /g<latexit sha1_base64="yeCy6UGlSOnSX30jNK0+hKAqiLQ=">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</latexit>

Meje :ejecta mass

 :opacity

veje :expanding velocity

f : energy conversion rate
<latexit sha1_base64="K37+hlJ1NDdlUjKH83ENbdcQLNs=">AAACCnicbVDLSsNAFJ3UV62vqEs3g0VwUUoigo9V0Y3LCsYWmlIm05t26GQSZiZCCN278VfcuFBx6xe482+ctllo64GBwzn3cOeeIOFMacf5tkpLyyura+X1ysbm1vaOvbt3r+JUUvBozGPZDogCzgR4mmkO7UQCiQIOrWB0PfFbDyAVi8WdzhLoRmQgWMgo0Ubq2YfhpV/LfRlhECAHmV+jsSgCfk0SDeOeXXXqzhR4kbgFqaICzZ795fdjmkYgNOVEqY7rJLqbE6kZ5TCu+KmChNARGUDHUEEiUN18essYHxmlj8NYmic0nqq/EzmJlMqiwExGRA/VvDcR//M6qQ7PuzkTSapB0NmiMOVYx3hSDO4zCVTzzBBCJTN/xXRIJKHatFExJbjzJy8S76R+UXdvT6uNq6KNMjpAh+gYuegMNdANaiIPUfSIntErerOerBfr3fqYjZasIrOP/sD6/AFgeZra</latexit>

In parKcular the value of κ can vary signiﬁcantly depending on
the electron fracKon Ye of ejecta
κ = 0.1—10 cm^2/g

R-proc. nucleosynthesis depends on Ye (=[p]/([p]+[n]))(M. Tanaka et al., 1708.09101)

d

10

p

s
Se

0

f
Te Ba Nd

Ru
Ye = 0.30

Mass fraction

10

-1

Er

Ye = 0.25

Ye = 0.10-0.40

Black
Solarabundancelanthanide

10-2
10-3
10

-4

10

-5

30

40

ref) slide from Shibata-san’s talk

50

60
70
Atomic number

80

90

100

Heavy

Opacity depends strongly on abundance, i.e., Yedetermined by lanthanide

2−1

Opacity (cm g )

101

10

10

K-band102

Ye = 0.10−0.40
Ye = 0.25
Ye = 0.30

J-bandR-band10(M. Tanaka et al., 1708.09101)

3

10

0

0.1

−1

Presenceoflanthanide
enhances
opacitysignificantly
−2
10

0

5000

ref) slide from Shibata-san’s talk

10000
15000
Wavelength (Å)

20000

25000

(La + Ac)(+ actinide)
Mass fraction of X
Lanthanide

Fraction of lanthanides vs electron fraction
100

By M. Tanaka

10

-1

10

-2

10

-3

10

-4

10

-5

• Roughly speaking
1. For Ye < ~0.25
à κ~10 cm2/g
2. For ~0.25 < Ye < ~0.27 à κ=O(1) cm2/g
3. For Ye >~0.27
à κ=O(0.1) cm2/g

0.05

0.1

ref) slide from Shibata-san’s talk

0.15

0.2

0.25
Ye

0.3

0.35

0.4

Mass Ejec,on Mechanisms•

Merger process and evolu,on of themerger remnant have been studied by
NR simula,ons in the last decades
revealing the mass ejec,on process•

Dynamical mass ejec/on
mass ejec,on driven by ,dal interac,onorshock hea,ng during the collision•

Post-merger mass ejec/on
mass ejec,on from the merger remnant
driven by viscous and neutrino hea,ngDynamical mass ejecKon@merger

Post-merger mass ejecKon@arer merger

Dynamical mass ejec,on
SKﬀ EOS (DD2, R~13.2 km): 1.30-1.40 M_sun

Ref: Y. Sekiguchi et al. 2016

Tidal torque driven

Shock heaKng driven

NR simulaKon considering neutrino transport p + ν e → n + e−
and its eﬀect on the Yen +ν → p + e+eDynamical mass ejec,on

he Astrophysical Journal Letters, 778:L16 (5pp), 2013 November 20

Hotokezaka e

NS-NS models

1

10-2

APR4
SLy
ALF2
H4
MS1

1

10-4
10-5
10-6

0.14

N182
N150
N130
N110

-3

0.1

0.01
0.13

10

10
-2 sun]Meje/10
[M
M
Msun
ej

Mej/10-2Msun

10

BH-NS models
HB-135-135

0.15

0.16 0.17
Mtot/2R1.35

0.18

0.19

0.2

0.1

APR4
ALF2
H4
MS1
ALF2(7,0.5)
H4(7,0.5)
MS1(7,0.5)

10-70.01
0.10.130.2 0.14
0.3

0.4
0.7
0.15 0.50.160.6 0.17
vM/NS
c /R1.35

0.8
0.18 0.90.1

igure 2. Ejecta masses as a function of the compactness of the neutron star, which is defined by GMtot /2R1.35 c2 and GMNS /R1.35 c2 for NS–NS and BH–NS mo
Ref:denoteK. Hotokezaka
etofal.the2013
espectively. Left panel: NS–NS models. Each point shows the ejecta mass for the equal mass cases. Error bars
the dispersion
ejecta masses dhe various Q. Right panel: BH–NS models. The filled and open symbols correspond to the models with (Q, χ ) = (3–7, 0.75) and (7, 0.5), respectively. The
haded region in each panel shows the ejecta masses allowed in order to reproduce the observed near-infrared excess of GRB 130603B, 0.02 ! Mej /M⊙ ! 0.0.02 ! Mej /M⊙ ! 0.1 for the NS–NS and BH–NS models, respectively. The lower and upper bounds are imposed by hypothetical high- and low-heating mespectively.
A color version of this figure is available in the online journal.)

typical dynamical ejecta mass is 10^{-4}—10^{-2} M_sun
depends on the NS mass and EOS
typical velocity ~0.1-0.3 c (some fracKon has higher velocity)

the light curves of the radioactively powered emission f

10r-procwind-7
studie10
may
Figure 2. Color-coded distributions for density, temperature, Ye , and S/kB (from left to right) on the x–y (lower panels), x–z (positive sides of top panels), andy–zn-8(negative sides of top panels) planes at the end of simulation.
Mej ≈10(A color version of this figure is available in the online journal.)(a few
0
50
100
150 Ref: S.
200
250
Wanajoetal.
2014
mass number
with tY. Sekiguchi et al. 2015
discusfor0-2 the orbital and non-orbital planes, respectively (with higher
et al. 2
10
10

Ye distribu,onvalues for higher Ye ), which
are r-abundance
sizably greater than those in
solar
Goriely et al. (2011, S/kB ∼mass-averaged
1–3) with the Shen’s(x-y) EOS.
10-3
10-1
3. THE r-PROCESS
The
-4
10
subseqThe nucleosynthesis analysis makes use of the thermodyare fo
-5
-2
namic
trajectories
oftheejectaparticles
tracedontheorbitalheatin1010plane. A representative particle is chosen from each Ye -bin (from
tempo
-6
3).le
(top
10Ye = 0.09 to 0.44 with the interval of ∆Ye = 0.01 (Figure
For simplicity, we analyze only the x-y components because
the ofh
10-3
r-procthe-7 dominance of the ejecta masses close to the orbital plane.
10Each nucleosynthesis calculation is initiated when the temperaHotok
from
ture
-8 decreases to 10 GK, where the initial composition is given-4
10by Y and 1 − Y for the mass fractions of free protonsshort10
and0e
50 e 100
150
200
250
0.0
0.1
0.2
0.3
0.4
0.5
by
q̇an
neutrons.
mass
numberY
in dayeThereaction
network
consistsof
6300
speciesfromsingleFigure 4. Final nuclear abundances for selected trajectories (top) and that massheatin0neutrons
andprotonstheZ=
110
isotopes.
Experimental
averaged(bottom; comparedwith thetosolar
r-processabundances).
Ove10
rates,
when
are taken
fromjournal.)
the latest versions of REA(A color
versionof thisavailable,
figure is available
in the online
x-y
After
7
8
x-z
CLIB (Cyburt et al. 2010) and Nuclear Wallet Cards. Otherand by-z
9the theoreticalof fusion
rates (TALYS;
ejecta-1
slowswise,
the temperaturedrop estimatesaround 1 GK(e.g., Korobkin
et al. Goriely
10✓◆1/2 ⇣ 2012).⌘✓ is,and◆lessfindal.th
etThe
al. effect2008)
β-decay
half-livesthan(GT2;et
1/2 dramatic
however,thoseTachibana
found
1/2
M
v 1990)works
 wherea few
are adopted,
both arebased
on thesame nuclear
in previous
because of
the higher
ejectaentropies
in ourtpeak ⇡ 5.9 daysof T1/
et al. 2010). Theoretical fission propresult.masses (HFB-21;
0.01M
0.2c
10 cm2Goriely
/g
-2
10
(T1/2
(top)
displays
thefinal nuclear
erties4 adopted
are thoseestimatedon abundances
the basis offor
the HFB-14✓◆✓◆1/2 ⇣ Figure✓◆
1/2⌘trajectories.
for Yeselected
find fragments,
a variety of
nucleosyntheticdistribution
1/2
model. For
fissiona Gaussian-type
f
M
vmass
We
40
by theLpeak ⇡ 3.4 ⇥ 10 ergs/s
outcomes:iron-peakand2emission
A ∼ 90 abundances
made innuclear per event.
isassumed
with
of
four
promptneutrons
6
10
0.01M
0.2c
10forcm/g0.4, light r-process abundances for-3with q̇
quasi-equilibrium
Ye !
10
Neutrino
capturesare◆not included, which make only slight✓◆1/4 ✓◆ 1/8 ⇣Ye ∼ 0.2–0.4,✓
from t3/8abundances for Ye " 0.2.
heavy r-process⌘ 1/8of andshiftsY(typicallyanincrease
of
∼0.01
from
10
GK
to12
f
M
v

e
3
and
In
contrast
toprevious
works,
wefindnofissionrecycling;
Tpeak ⇡ 1.5 ⇥ 10 K
5 GK).flow 10
6
10
0.01M
0.2c
/g Ye (=0.09) trajectory reaches
Ourthenuclearforcm
the2lowest-4Thehydrodynamical
trajectories
end
with
temperatures
10(=0.0
A ∼ 280, the fissile point by neutron-induced fission, only at theabundancemass fraction

x-y
x-z
y-z

mass fraction

Ye par,ally becomes large due to shock hea,ng and neutrino irradia,on(cf. NS Ye~<0.1) Yet, lanthanide is synthesised→ κ~10cm^2/g<latexit sha1_base64="2gpTBwnflB+j4yBuUjkN+uW2akc=">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</latexit><latexit sha1_base64="PhH8f99oHmFlDjBwr+O1ZHyOV1Q=">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</latexit><latexit sha1_base64="TWeCyZquatWO4w8Qq2biBNnYoj4=">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</latexit>

Evolu,on a]er the merger•

For Mtot<2.8 M_sun, NS likely to remain a]er the merger.
→neutrino irradia,on from the remnant NS would be importantEvolu,on of Remnant NS &torus: Eﬀec,ve viscosity
1051

1.8e+

1050

1.6e+

1049

1.4e+

1048

Poloidal Toroidal

1047∆x = 12.5 m

1045

10

∆ x = 12.5 m∆ x = 70 m
∆ x = 110 m←KH instability∆ x = 12.5 m

6.0e+
0

5

10

15

< α >122530∆ x = 12.5 m∆ x = 70 m
∆ x = 110 m∆ x = 70 m
∆ x = 110 m< α >1320

t - tmerger [ms]

10

10-2

FIG. 4. Magnetic-field energy (left), rotational
the runs. The solid and dashed curves in the lef

10-3

-4

15

20

merger [ms]

25

10
Ref:
K.
Kiuchiet5al. 2015,
2017
30
10
15

20

t - tmerger [ms]

25

30

10-3

8.0e+

∆x = 110 m

-1

10-2

1.2e+

1.0e+

17

∆x = 70 m

1046

-1

Erot [erg]EB [erg]

Turbulence in the contact surface→ampliﬁcaKon of magneKc ﬁeld
→eﬀecKve viscosity would play a role

5

10

15

20

t - tmerger [ms]2530

Viscous-radia,ve simula,onNumerical-relaKvity simulaKon considering
the eﬀects of physical viscosity and neutrino radiaKon

Ref: S. Fujibayashi et al. 2018

Post-merger mass ejec,on
Ref: S. Fujibayashi et al.Neutrino/Viscosity-driven
2018
Ejecta from NS-NS Merger remnant

1

Mej [10-2 Msun]

4
3
2

Ek,ej [1050erg]

(a)

1
0
6

(b)

5
4
3
2
1
0
0.3

Vej [c]vis=0.04
vis=0.02
vis=0.01
vis=0.00

(c)

0.2
0.1
0

Ejecta mass ~10^{-2}—10^{-1} M_sun
Typical velocity 0.03 — 0.15 c
Figure 9. Same figures as panels (a)–(c) of Fig. 8 but for thejected toward thepolar angle 0  ✓  30 (solid curve
depending on the viscousmaterial
parameterandand 30 < ✓  90 (dotted curves) by the late-time viscosity-drive
mass ejection for t 0.5 s for the models DD2-135135-0.02-H (recurves) and DD2-135135-0.04-H(green curves), respectively.
the life,me of the remnant
NS

Figure 13. Same as Figure 12 but at t = 1 s. For ↵vis = 0.02 (top panels), the ejecta shown inviscosity-driven component toward the polar direction, while for ↵vis = 0.04 (bottom panels), it is com
component toward the equatorial direction.

Figuret =panels),
1 s. Forthe↵vis= 0.02(top
the isejecta
shownin thesepanels is
3. Same as Figure
12 13.
but atSame
t = 1ass. FigureFor ↵vis12= but
0.02 at(top
ejectashown
in panels),these panels
composedmainlyof theviscosity-driven
componentthepolar
direction,whilefor ↵itvis= 0.04 (bottom
panels),is composed primar
riven component
toward the polar
direction,toward
while for↵vis
= 0.04(bottom
panels),is composed
primarilyof the it
viscosity-driven
component
equatorial
direction.
Figure 12
at t = 1direction.
s. toward
For ↵visthe= 0.02(top panels),
the ejecta shown in these panels is composed mainly of thets toward
thebutequatorial
onent toward the polar direction, while for ↵vis = 0.04 (bottom panels), it is composed primarily of the viscosity-drivene equatorial direction.

Ye distribu,onMass per bin [M sun]

10-2

Ref: S. Fujibayashi et al. 20180-1

 vis=0.02
 vis=0.04

Lanthanides-2
Log XZ10-30o-15o15o-30o30o-45o45o-60o60o-90o

10-4

-3
-4
-5
-6

10-5

-7

0

30

40

50

60

70

80

90

100

0.1 0.2 0.3 0.4 0.5
Atomic number Z
Ye
Figure 15. Mass fraction of the nuclei as a function of atomic
4.
Mass histogram of the ejecta as a function of Ye
for the ofmodel
DD2-135135-0.04-H
t = fraction
1.3 s. The
s/kB (right)att ejecta= 1.314.
s.as The
distributions
elec-ejecta
15. numberMassfractionas a function
atomichistogramof thea functionof Ye of the
Figure15. ofatMassof difthe nuclei
FigureMasshistogramofFiguretheasa function
ofthe
Yenuclei
ferentcolor
curvescorrespondtotheresults
indi↵erent
angularon(leftpanel)
andthespecific
entropy(rightpanel)
arethe model DD2-135135-0.04-H
at t = 1.3
dif- DD2-135135-0.04-Ht) at t = 1.3 s.(left)
The distributionsof the at
elecnumber
for s.theThemodel
and s/kB (right)
t = 1.3number
s. The for
distributionsof the elecregions.correspond
The shaded
regionindicates
the rangeof the lanthanide
he redcurvesdenotethepanel)resultsare
for the models
ferentcolor
curvestoare
theresults
in di↵erent
angularnel)andandthegreen
specific
entropy(rightferentcolor
curvescorrespond to the restron
fraction(leftpanel)
andthespecific
entropy(rightpanel)
elements(Z =57–71).
35-0.04-H
andDD2-135135-0.02-H,
respectively.
regions.
The
shadedregionindicates
the
range
of
the
lanthanide
green curves
denotetheresultsforthemodelsregions. The shaded region indicates theplotted. The red and green curves denote the results for the models
elements(Z=57–71).
nd DD2-135135-0.02-H,
respectively.
elements (Z =57–71).
DD2-135135-0.04-H and DD2-135135-0.02-H, respectively.

post-merger ejecta typically has Ye>0.25 due to neutrino irradia,on→ lanthanide-free → κ~0.1-1 cm^2/g
✓
◆1/2
⌘
1/2
M
v

tpeak ⇡ 1.5 days
0.03M
0.1c
0.1 cm2 /g✓◆✓◆1/2 ⇣
✓
◆ 1/2
⌘
1/2
f
M
v

Lpeak ⇡ 4.3 ⇥ 1041 ergs/s
10 6
0.03M
0.1c
0.1 cm2 /g✓◆1/4 ✓◆ 1/8 ⇣
✓
◆ 3/8
⌘
1/8
f
M
v

3
Tpeak ⇡ 8.0 ⇥ 10 K
10 6
0.03M
0.1c
0.1 cm2 /g<latexit sha1_base64="S4vsKp/xTu/8q2VgAtEOuYD4PYU=">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</latexit><latexit sha1_base64="EhI39HBXI975De8gt+GdptNtWkw=">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</latexit><latexit sha1_base64="T53U8dxkwvlJLFyRlGh8yXk0ygc=">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</latexit>✓◆1/2 ⇣

kilonova/macronova fordiﬀerent components

Dynamical ejectaYe ~ 0.05− 0.5••

Dynamical mass ejec/on
M~0.0001—0.01 M_sun, v~0.1-0.2 c
Ye~0.1-0.5→t_peak~10 days,
long-las,ng, dim, low temperature(red) kilonova→ κ ~ 10 cm 2 / g
M ~ 10−2 M ⊙v / c = 0.1− 0.9

Remnant
MNS

Post-merger mass ejec/on
M~0.01—0.1 M_sun, v~0.03-0.15 c
Ye~0.3-0.4
Post-merger ejecta
→t_peak~1 days,
Ye ~ 0.3− 0.4
short-las,ng, bright, high temperature→ κ ~ 0.1 cm 2 / g(blue) kilonova
M ~ 3×10−2 M⊙

v / c = 0.03− 0.1

SSS17a (GW170817)
Ref: Waxman et al. 2017

~1042 erg/s

-16

Data: Villar +, 1711.11576r

Infr
ared

-15
Ne
ar

-14
-13

In



i
z
J
H
K

fra

re
d

ical
Opt

Absolute AB Maginitude

Optical-IR EM counterparts of GW170817

-12
0

2

ref) Shibata-san’s slide

4

6
8
10
t (day) after merger

1214Mul,ple component
Kilonovae/macronovaeaWhatis the origin of blue/red kilonova?
Blue (lanthanide-free)relative flux F¸ (plus offset)

3.5

10
light r-process component
heavy r-process component
composite

See e.g., Metzger+18,
Waxman+18

t = 1.5 day

3.0

Blue kilonova
M ~2.50.02 Msunt = 2.5 dayv ~ 0.25c
Too fast
2.0
as post-merger ejecta??
1.5

t = 4.5 day

Red kilonova
t = 7.5 day
Red (lanthanide-rich) M ~0.50.03 Msunv ~ 0.1 c
Too massive as
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ref) Tanaka-san’s slide
dynamical
ejecta??
wavelength (microns)
1.0

1g (0.40-0.55 ¹m)bbroadband luminosity, ¸F¸ (1041 ergs s¡1 )

4.0

r (0.55-0.70 ¹m)
i (0.70-0.85 ¹m)
j (1.10-1.35 ¹m)
h (1.50-1.80 ¹m)
k (2.00-2.30 ¹m)

10

10

0

-1

0

2

4

6

8

10

12

days since merger

Ref: D. Kasen
et al. 2017
Figure 5 | A unified kilonova model explaining the optical/infrared
counterpartof••

GW170817.
The model is the superposition of the emission from two spatially distinct
Post-merger
ejectacomponents:a ‘blue’ kilonova
r-process ejecta
withobservaKon
M = 0.025M!, vk = 0.3c
InsidemulKple
ofejecta
dynamical
ejectaKilonova/macronova model withcomponentswell(lightinterpretsthedynamical
and Xlan = ejecta10"4) plusa ‘red’kilonovaKasen(heavy r-processwith M= 0.04Met,(see e.g., Kasliwal et al. 2017,(lessCowperthwaite
et al.
2017,
et al.ejecta
2017,
Villar! al. 2017)"1.5
near thevpole)). a, Optical–infrared spectral time series, where the black line
k = 0.15c, and Xlan = 10

early-blue component (~1day) from islanthanide-freeejectathe sum of the light r-process(blue line) and heavy r-process (red line) contributions.

b, Composite broadband light curves. The light r-process component produces the rapidly
evolving
opticallanthanide-richemission while the heavy ejecta
r-process component produces the extended+ long-lasKng red component (~10days)
frominfrared emission. The composite model predicts a distinctive colour evolution, spectral
continuum shape and infrared spectral peaks, all of which resemble the properties of14

broadbt = 7.5 day
0.5

Problems
10

0.0

•

0.5

1.0

1.5

2.0

2.5

-1

3.0

wavelength (microns)

0

2

4

6

8

10

12

14

days since merger

MulKple component ejecta kilonova/macronova seem to work, but…

!
!
!
•

Figure 5 | A unified kilonova model explaining the optical/infrared counterpart of
GW170817. The model is the superposition of the emission from two spatially distinct
ejecta components: a ‘blue’ kilonova (light r-process ejecta with M = 0.025M!, vk = 0.3c
and Xlan = 10"4) plus a ‘red’ kilonova (heavy r-process ejecta with M = 0.04M!,
vk = 0.15c, and Xlan = 10"1.5). a, Optical–infrared spectral time series, where the black lineis the sum of the light r-process (blue line) and heavy r-process (red line) contributions.
b, Composite broadband light curves. The light r-process component produces the rapidly
lanthanide-richejecta with >0.01 M_sun seems too large
Ref: D. Kasen et al. 2017
evolving
opticalemission
while
the
heavy
r-processcomponent
producestheextendedfor the dynamical ejecta (typically less than 0.01 M_sun)
infrared emission. The composite model predicts a distinctive colour evolution, spectral•

continuum
shape and infrared
spectralpeaks,
of highwhichforresemblethe properties ofvelocityof lanthanide-freeejecta~0.3
c isalltoo
the post-merger
ejectaAT 2017gfo.
(typically~0.05 c)&•Tension between observed properKes and predicKons of numerical relaKvity?
(e.g., Waxman et al. 2017, Metzger et al. 2018, Matsumoto et al. 2018)•

Photon interplay between the dynamical & post-merger ejecta is not taken into
account ( only simple composi/on of each lightcurve)
Page 16 of 29

Radia,ve transfer between
diﬀerent ejecta components
soabMain energy sourceReprocessed Emission:

ion
rpt


Remnant
MNS
Post-merger ejecta

Dynamical ejectaYe ~ 0.3− 0.4

Ye ~ 0.05− 0.5→ κ ~ 0.1 cm 2 / g→ κ ~ 10 cm 2 / g

M ~ 3×10−2 M ⊙M ~ 10−2 M ⊙v / c = 0.03− 0.1

v / c = 0.1− 0.9

We perform an axisymmetric radiaKve transfer simulaKon for kilonovae/macronovae
taking the interplay of mulKple ejecta components of non-spherical morphology into account.
(KK. et al 2018, see also Perego et al. 2017 for studies with similar setups)

Radia,ve transfer simula,on•

We employ a wavelength-dependent Monte-Carlo radiaKve transfer simulaKon code
(M. Tanaka et al. 2013, 2014, 2017)•

The density, velocity, and Ye proﬁles of ejecta are employed within the range of predicKonsby numerical-relaKvity simulaKons.
(e.g., Dietrich et al. 2016, Hotokezaka et al. 2018, Metzger&Fernandez et al. 2014
, Fujibayashi et al. 2018)•

The abundance paPern and nuclear heaKng rate are given based on r-process nucleosynthesis
calculaKons by (Wanajo et al. 2014)•

RealisKc opacity table constructed by the atomic structure calculaKons(Se, Ru, Te, Nd, and Er)! improved points•

The grid resoluKon of the simulaKon is also improved by an oder ofmagnitude from our previous works by imposing axisymmetry.

•special-relaKvisKc eﬀects on photon transfer are fully taken into accountDensity distribu,on

Axisymmetricw thatlog10 (t = 1[day]) [g/cm3 ]typi0.9
-13
0.9
0.9
10
-15
pend0.8
0.8
0.8
here c
-16
✓<
⇡/4
0.7
0.7
0.7
nami-13.5
5
3
⇢⇥
10
0.6
large
0.6
0.6
-17
ejec0.5
0.5
0.5
-14
0-18
driven
θ
0.4
0.4
0.4
et al.
0.3
0.3
0.3
-19
Siegel
-14.5
-5
0.2
0.2
0.2
et al.
-20a.) It0.1
0.1
0.1
dering
0
-15
00
-10
-21
iation00.40.5
0.1
0.2
0.3
0.4
0.5
00 0.1
0.10.20.20.30.30.40.50.6
0.60.70.70.80.80.90.90.6 0.7 0.8 0.9
p
from
⇢
/
r
vx/c
v
v
/c
/c
x
x
. The
0.1 Dynamicalc.
Fig. ejecta:
1.— Density
distributionsof0.08—0.9
the ejecta employedin the0.009
M_sun,
c,
Ye=0.1-0.4,
p=-6
radiative-transfer simulation. Red and blue regions denote the reremgions of the
dynamical
the post-merger
ejecta, respectively.
Post-merger
ejecta:0.02and
M_sun,
0.025—0.08
c, Ye=0.3-0.4, p=-3
ejectapost-merger ejecta

vz/c

vvz/c
z/c

<latexit sha1_base64="Jzdx6YDThc5ttJ5A4jGApoqfDd0=">AAAB+XicdVDLSsNAFJ3UV62vVJduBovgxpDUQu2u6MZlBaOFJpbJdNIOncyEmYlSYj/FjQsVt/6JO//G6UPweeDC4Zx7ufeeKGVUadd9twoLi0vLK8XV0tr6xuaWXd6+VCKTmPhYMCHbEVKEUU58TTUj7VQSlESMXEXD04l/dUOkooJf6FFKwgT1OY0pRtpIXbscyIEINE2I8tzr/PBo3LUrrlNz61WvAX8Tz3GnqIA5Wl37LegJnCWEa8yQUh3PTXWYI6kpZmRcCjJFUoSHqE86hnJkdoX59PQx3DdKD8ZCmuIaTtWvEzlKlBolkelMkB6on95E/MvrZDo+DnPK00wTjmeL4oxBLeAkB9ijkmDNRoYgLKm5FeIBkghrk1bJhPD5Kfyf+FWn4XjntUrzZJ5GEeyCPXAAPFAHTXAGWsAHGNyCe/AInqw768F6tl5mrQVrPrMDvsF6/QBo15Oo</latexit>l(t=1 [day])[g/cm3]<latexit sha1_base64="nSWtR9bkLOdjH9pxges75AbKf0E=">AAAB8nicdVDLSgNBEJz1GeMr6tHLYBA8rbshEAMegl48RnBNILuE2clsMmR2dpjpFULIb3jxoOLVr/Hm3zh5CD4LGoqqbrq7YiW4Ac97d5aWV1bX1gsbxc2t7Z3d0t7+rclyTVlAM5HpdkwME1yyADgI1laakTQWrBUPL6d+645pwzN5AyPFopT0JU84JWClMIQBA3IeKn5a7ZbKnlv1ahW/jn8T3/VmKKMFmt3SW9jLaJ4yCVQQYzq+pyAaEw2cCjYphrlhitAh6bOOpZKkzETj2c0TfGyVHk4ybUsCnqlfJ8YkNWaUxrYzJTAwP72p+JfXySE5i8ZcqhyYpPNFSS4wZHgaAO5xzSiIkSWEam5vxXRANKFgYyraED4/xf+ToOLWXf+6Wm5cLNIooEN0hE6Qj2qoga5QEwWIIoXu0SN6cnLnwXl2XuatS85i5gB9g/P6ASjQkVE=</latexit>

dynamical ejecta<latexit sha1_base64="WB1UY3EYgZ0JfOKGYqyZl+BpzMM=">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</latexit><latexit sha1_base64="tBXOORdfZlzW9rFcfjIBcOdNQQk=">AAAB+nicbVBNSwMxEM36WevXWo9egkXwYtkVQb0VvXis4NpCdy3ZNNuGZpOQZMWy7F/x4kHFq7/Em//GtN2Dtj4YeLw3w8y8WDKqjed9O0vLK6tr65WN6ubW9s6uu1e71yJTmARYMKE6MdKEUU4CQw0jHakISmNG2vHoeuK3H4nSVPA7M5YkStGA04RiZKzUc2uhGopQKiGNgOohP5FFz617DW8KuEj8ktRBiVbP/Qr7Amcp4QYzpHXX96SJcqQMxYwU1TDTRCI8QgPStZSjlOgon95ewCOr9GEilC1u4FT9PZGjVOtxGtvOFJmhnvcm4n9eNzPJRZRTLjNDOJ4tSjIG7Z+TIGCfKoING1uCsKL2VoiHSCFsbFxVG4I///IiCU4blw3/9qzevCrTqIADcAiOgQ/OQRPcgBYIAAZP4Bm8gjencF6cd+dj1rrklDP74A+czx+eYpRd</latexit>Results:Light curves

4

D=40 Mpc, 20°)e<28°

AB absolute magnitude

-16
-15

g
r
i
z
J
H
K

-14
-13

-17
-16
AB absolute magnitude

-17

-15
-14
-13

-12

-12

-11

-11
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
t [day]0 1 2 3Fig. 2.— Optical and near-infrared lightcurves of SSS17a compared with the kilono
and 86  ✓  90 (right panel). The optical and near-infrared data points are taken
at a distance of 40 Mpc. All the magnitudes are given in AB magnitudes.

Host galaxy NGC4993: ~40 Mpc
GW data analysis gave a constraint on θ <~30°

-11

-11

Spectra0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
t [day]0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
t [day]Fig. 2.— Optical and NIR lightcurves of SSS17a compared with the kilonova model observed from 20  ✓  28 (left panel) and86  ✓  90 (right panel). The optical and NIR data points are taken from Villar et al. (2017). We assume that SSSa17 is at a distanceof 40 Mpc. All the magnitudes are given in AB magnitudes. Note that the large deviation of the model lightcurves in H-band may be dueto the incompleteness of the line list for the opacity estimation.

Our model

ObservaKon

2012; Wanajo et al. 2014). Figure 5 compares the elemental abundance in our model with the solar abundance. Though some abundance peaks are smaller thanthose of the solar abundance, broadly speaking, the massaveraged element abundance of our model reproduces thetrend of the solar abundance for a wide range of r-processelements, in particular, including the first (Z = 34) abundance peaks.

2

Flux [erg/cm /s/Å]t = 1.4 [day], D = 40 Mpc, 20°)e<28°
4×10ï16

T = 5150 [K]

ï16

3×10

2×10ï16
1×10ï16005000

10000 15000
h[Å]

20000

25000

2

Flux [erg/cm /s/Å]t = 3.7 [day], D = 40 Mpc, 20°)e<28°
1×10ï16
8×10ï17
6×10ï17
4×10ï17
2×10ï170

T = 3060 [K]

0

5000

10000 15000
h[Å]

20000

°

25000

°

2

Flux [erg/cm /s/Å]t = 6.8 [day], D = 40 Mpc, 20 )e<28
5×10ï17
4×10ï17
3×10ï17
2×10ï17
ï17
1×10
0

T = 2190 [K]

0

5000

10000 15000
h[Å]

20000

25000

4. DISCUSSION & CONCLUSION

In this letter, we showed a result of an axisymmetricradiative transfer simulation for a kilonova/macronova
with a setup indicated by numerical-relativity simulations. In particular, the interplay of multiple nonspherical ejecta components via photon transfer are consistently taken into account in the lightcurve prediction.
We found that the optical and NIR lightcurves of
SSS17a are reproduced naturally by the numericalrelativity-simulation-motivated model observed from20  ✓  28 . In particular, we showed that the observed NIR lightcurves can be interpreted by the emission from the dynamical ejecta of which mass is consistent with the prediction of numerical relativity. The
observed lightcurves are reproduced by a smaller massof the post-merger ejecta than that estimated by previous studies (e.g., Kasliwal et al. 2017; Cowperthwaite
et al. 2017; Kasen et al. 2017; Perego et al. 2017; Villar et al. 2017) because the e↵ect of the photon di↵usion
preferentially to the polar direction is taken into account.
The observed blue optical lightcurves as well as the photospheric velocity of ⇡ 0.3 c can be interpreted by thephoton-reprocessing in the low-density dynamical ejecta,
which locates in the polar region above the post-merger
ejecta.
Our results indicate that there is no tension
between et al. 2017
Ref: Waxman

Photospheric quan,,es
Lbol = 4⇡<latexit sha1_base64="6WFzuXnCEBGOjAFE+ZmwuUtmTTQ=">AAACGnicbVDLSgMxFM34rPU16tJNsAiuykwp6EYounHhokpf0BmHTJq2oclMSDJCGeY73Pgrblwo4k7c+Dem7YDaeuDCyTn3kntPKBhV2nG+rKXlldW19cJGcXNre2fX3ttvqTiRmDRxzGLZCZEijEakqalmpCMkQTxkpB2OLid++55IReOooceC+BwNItqnGGkjBbZ7HaSe5DCMWXZe9QSFnqIDjuDtTBfD7K4CGz+PamCXnLIzBVwkbk5KIEc9sD+8XowTTiKNGVKq6zpC+ymSmmJGsqKXKCIQHqEB6RoaIU6Un05Py+CxUXqwH0tTkYZT9fdEirhSYx6aTo70UM17E/E/r5vo/pmf0kgkmkR49lE/YVDHcJIT7FFJsGZjQxCW1OwK8RBJhLVJs2hCcOdPXiStStl1yu5NtVS7yOMogENwBE6AC05BDVyBOmgCDB7AE3gBr9aj9Wy9We+z1iUrnzkAf2B9fgPeoaAf</latexit>

2
4
Rph Tphvph = Rph /t
5<latexit sha1_base64="+uXpmdg+OYnsJoFDcrK+BNlXreo=">AAACAXicbZDLSsNAFIYnXmu9Rd0IbgaL4KomIuhGKLpxWcVeoA1hMp20Q2eSMHNSKKFufBU3LhRx61u4822ctkG09YeBj/+cw5nzB4ngGhzny1pYXFpeWS2sFdc3Nre27Z3duo5TRVmNxiJWzYBoJnjEasBBsGaiGJGBYI2gfz2uNwZMaR5H9zBMmCdJN+IhpwSM5dv7Az9rK4mT3ugS3/3wCfh2ySk7E+F5cHMooVxV3/5sd2KaShYBFUTrlusk4GVEAaeCjYrtVLOE0D7pspbBiEimvWxywQgfGaeDw1iZFwGeuL8nMiK1HsrAdEoCPT1bG5v/1VophBdexqMkBRbR6aIwFRhiPI4Dd7hiFMTQAKGKm79i2iOKUDChFU0I7uzJ81A/LbtO2b09K1Wu8jgK6AAdomPkonNUQTeoimqIogf0hF7Qq/VoPVtv1vu0dcHKZ/bQH1kf39G4lnc=</latexit>

1x1042Temperature [K]Bolometric Luminosity [erg/s]

12000

41

1x10

1x1040

5000
3000
2000

20°°)e<28°°

1x103986 )e<90
Waxman et al. 2017
1

1000
10

20°°)e<28°°86 )e<90
Waxman et al. 20171

t [day]10t [day]1
0.3
Velocity [c]Radius [cm]2x1015
15

1x10

14

5x10

14

3x10

2x10140.003
10

t [day]0.03
0.0120°°)e<28°°86 )e<90
Waxman et al. 201710.120°°)e<28°°86 )e<90
Waxman et al. 20171

10
t [day]Important points•

A smaller value of mass the dynamical ejecta (<~0.01 M_sun) thanprevious studies is needed for reproducing the observed
lightcurves, which agrees with the predicKon of NR simulaKons•

Velocity of the post-merger ejecta (~<0.05 c) also agrees within
the predicKon of NR simulaKons•

The total mass of ejecta is smaller than the predicKon of previous
studies (~0.04-0.08 M_sun -> ~0.03 M_sun)log10 (t = 1[day]) [g/cm3 ]D=40 Mpc, 20°)e<28°<latexit sha1_base64="WB1UY3EYgZ0JfOKGYqyZl+BpzMM=">AAACJHicbVBNS8NAEN3Ur1q/qh69LBZBLzVRQQWFohePClaFJpbNdpMu3c2G3YlQQv6MF/+KFw8qHrz4W9x+HLT6YODx3gwz88JUcAOu++mUpqZnZufK85WFxaXllerq2o1RmaasSZVQ+i4khgmesCZwEOwu1YzIULDbsHc+8G8fmDZcJdfQT1kgSZzwiFMCVmpXT3JfSyxUXLRzzy183VW+YBFsw6nXGnod0i8CX/O4Czt4JMW7VN7vF0G7WnPr7hD4L/HGpIbGuGxX3/yOoplkCVBBjGl5bgpBTjRwKlhR8TPDUkJ7JGYtSxMimQny4ZcF3rJKB0dK20oAD9WfEzmRxvRlaDslga6Z9Abif14rg+goyHmSZsASOloUZQKDwoPIcIdrRkH0LSFUc3srpl2iCQUbbMWG4E2+/Jc09+rHde/qoNY4G6dRRhtoE20jDx2iBrpAl6iJKHpEz+gVvTlPzovz7nyMWkvOeGYd/YLz9Q2gUqTo</latexit>

0.6
0.6

0.6

0.5
0.5

0.5

0.4
0.4

0.4

0.3
0.3

0.3

0.2
0.2

0.2

0.1
0.1

0.1

-16
5
-17
0-18

-19
-5
-20

-16
-15

-13
-13.5
-14

-14
-13

g
r
i
z
J
H
K

post-merger ejecta

0.7
0.7

0.7

dynamical ejecta

0.8

-17

3
l(t=1
[day])[g/cm
AB absolute magnitude ]

0.8
0.8

10
-15

Photospheric velocity [c]

0.9

vz/c

0.9
0.9

-14.5

-12 -15-10-2100.40.5
0.1
0.2
0.3
0.4
0.5
00 0.1
0.10.20.20.30.30.40.50.6
0.60.70.70.80.80.90.90.6 0.7 0.8 0.9
-11
v
/c
vxv/c
/c
x
x
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
t [day]Fig. 1.— Density distributions of the ejecta employed in theradiative-transfer simulation. Red and blue regions
denote the reFig. 2.— Optical and near-infrared lightcurves of SSS17a compar
gionsof the dynamical
and irradiate
the post-merger
ejecta,• post-merger
ejectaand heattheejecta,andup
86respectively.
 ✓ dynamical
90 (right panel).
The optical and near-infrared data
Axisymmetry with respect to the z-axis is imposedinthesimulaat a distance of 40 Mpc. All the magnitudes are given in AB magnit
tion.and help the long-lasKng NIR lightcurves to be reproduced by less massive00

0dynamical ejecta (see also Perego et al. 2017)

t = 1.4 [day], D = 40 Mpc, 20°)e<28°

lightcurves by less massive dynamical ejecta. Numericalï16
Å]

ud-

4

vvz/c
z/c

hat
pindec
mirge
ecven
al.
gel
al.
It
ng
on
om
The
1 c.
mcta
of
of

Hea,ng up of dynamicalejecta by post-merger ejecta

4×10Comparison with composited
dynamical
lightcurves+ post-merger ejecta

w thatlog10 (t = 1[day]) [g/cm3 ]e typi0.9
-13
0.9
0.9
10
-15
epend0.8
0.8
0.8
where c
-16
0.7
0.7
0.7
ynami-13.5
5
0.6
a large
0.6
0.6-17
l ejec0.5
0.5
0.5
-14
0-18
driven0.4
0.4
0.4
et al.
0.3
0.3
0.3
-19
Siegel
-14.5
-5
0.2
0.2
0.2
et al.
-20ta.) It0.1
0.1
0.1
dering
0
-15
00
-10
-21
4
diation00.40.5
0.1
0.2
0.3
0.4
0.5
00 0.1
0.10.20.20.30.30.40.50.6
0.60.70.70.80.80.90.90.6 0.7 0.8 0.9
d from
vx/c
vxv/c
x/c
r. The°0.1 c.
Fig. 1.— Density distributions of the ejecta
employedin the
D=40 Mpc, 20°)e<28
radiative-transfer simulation. Red and blue regions denote the rem remejecta -17 gions of the dynamical and the post-merger ejecta, respectively.
Axisymmetry with respect to the z-axis is imposed in the simulahat ofgtion.
alue ofr

Only post-merger ejecta
0.6

0.6

0

0.3
0.2

-5

0.1

+

5

0.5
vz/c

0.4

10

0.7

5

0.5
vz/c

Only dynamical ejecta

10

0.7

post-merger ejecta

dynamical ejectal(t=1 [day])[g/cm3]

vz/c

vvz/c
z/c

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0.4

0

0.3
0.2

-5

0.1

0

-10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
vx/c

0

-10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
vx/c
°

°°
D=40
)e<28
D=40 Mpc,
86°Mpc,)e<9020

-17

-17

g
r
i
z
J
H
K

g
r
i
z
J
H
K

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AB absolute magnitude

AB absolute magnitude

AB absolute magnitude

i
-16
-16
stud- -16 lightcurves by less massive dynamical ejecta. Numericalrelativity simulations also show that most of the dynamz
h kiloicalejectaisconcentrated
in
✓
⇡/4,
and
only
a
part
J of
more)
the
dynamical
ejecta,
which
has
high
velocity,ispresentynam- -15
H
-15
-15
in the polar region (✓  ⇡/4), where ✓ is the inclinationnd theangle measured from the orbital axis of the binaryK(Hoity (⇡tokezaka et al. 2013; Bauswein et al. 2013; Sekiguchi et al.
et al.
-14-14-14
2016; Radice et al. 2016; Dietrich et al. 2017). However,r et al.
such low-density dynamical ejecta in the polar region canservedstill significantly modify the spectrum due to large opacsts for-13-13
e post- -13 ity determined by lanthinides (this is known as the lanthanide
curtaine↵ect(Kasenetal.
2015;
Wollaegeretal.
ly. On2017).) Since the dynamical ejecta have high velocity, weta paon are -12 can expect that the reprocess of photon in the dynami-12-12
cal ejecta helps the photospheric velocity to be enhanced.
ativity
The gravitational-wave data analysis of GW170817 infers
mass ofthat the event was observed from ✓ . 28 (Abbott et al.
e near-11-11
nitude -11 2017a). Therefore, photon-reprocessing in both the lowdensity
andhigh-density
dynamical
ejectawould
be
ima large
0 1 for2 the
3 lightcurve
4 5 6 prediction.
7 8 9 A10
11 12
13 14 15 16
0 1 2 0 3 1 4 2 5 3 6 4 7 5 8 6 9 710811912101311141215131614 15 16
portant
similarpictureor thewas proposed and studied semi-analyticallyby Peregot [day]t [day] t [day]photoetal.
(2017)
toexplaintheinconsistency
betweentheaxman
estimated mass of the dynamical ejecta and that of theealizedFig. 2.—
Optical and near-infrared lightcurves of SSS17a compared with the kilonova model observed from 20  ✓  28 (left panel)
theoretical prediction.
i et al.

F
M =1!
⇡ 2.5 (F : energy flux)Fud-

log10 (t = 1[day]) [g/cm3 ]

0.8
0.8

0.8

0.7
0.7

0.7

0.6
0.6

0.6

0.5
0.5

0.5
0.4

0.3
0.3

0.3

0.2
0.2

0.2 Photosphere

0.1
0.1

0.1

00

0

l(t=1 [day])[g/cm3]

0.4
0.4

-16
5
-17

0.3

0-18

-13
-13.5

0.1

-19
-5
0.03
-20

20°)<e<28°

post-merger ejecta

10-15

dynamical ejecta

0.9

vz/c

0.9
0.9

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Enhancement ofphotospheric velocity
-14

-14.5

Dynamical + Post-merger ejecta
-15
-10
-21
Only Post-merger ejecta
Waxman et al. 201700.40.5
0.1
0.2
0.3
0.4
0.5
00 0.1
0.10.20.20.30.30.40.50.6
0.60.70.70.80.80.90.90.6 0.7 0.8 0.9
0.01
vx/c
vxv/c
1
x/c
t [day]Fig. 1.— Density distributions of the ejecta employed in theradiative-transfer simulation. Red and blue regions denote the regions of the dynamical and the post-merger ejecta, respectively.
• The reprocess
Axisymmetry
with respect
the z-axisin
is imposedin the simulaof tophotons
the dynamicalejecta helps thetion.

photospheric velocity to be enhanced

lightcurves by less massive dynamical ejecta. Numerical-

10

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0.8

0.7
0.7

0.7

0.6
0.6

0.6

0.5
0.5

0.5

0.4
0.4

0.4

0.3
0.3

0.3

0.2
0.2

0.2

0.1
0.1

0.1

00

0

-16
5
-17

43

1x10

0-18
-19
-5
-20

post-merger ejecta

0.8
0.8

-13

10
-15
Bolometric
Luminosity[erg/s]dynamical
ejecta0.9l(t=1 [day])[g/cm3]

0.9
0.9

vz/c

vvz/c
z/c

hat
pindec
mirge
ecven
al.
gel
al.
It
ng
on
om
The
1 c.
mcta
of
of

Angular dependence ofphoton diﬀusion
-13.5
42

1x10

-14
41

1x10

-14.5
40

1x10

-15
-10
-21
1x103900.40.5
0.1
0.2
0.3
0.4
0.5
00 0.1
0.10.20.20.30.30.40.50.6
0.60.70.70.80.80.90.90.6 0.7 0.8 0.9
vx/c
vxv/c
x/c

20°°)e<28°°86 )e<90
All directions
1

10
t [day]Fig. 1.— Density distributions of the ejecta employed in the•radiative-transfer
Photons diﬀuse
preferenKally
to thepolar
direcKon
simulation.
Red and blue
regions
denotethe re- in the presence of thegions of the dynamical and the post-merger ejecta, respectively.
opKcally withthickrespect
dynamical
ejectais in
the equatorial
plane,
Axisymmetry
to the z-axis
imposedin the simulation.and then, luminosity is eﬀecKvely enhanced in the polar direcKon→less ejecta mass is required to reproduced observed (isotropic) luminosity

lightcurves by less massive dynamical ejecta. Numerical-

Summary•

We found that the opKcal and NIR lightcurves as well as
photospheric velocity of SSS17a are reproduced by
the ejecta model which is within the predicKon of NR simulaKons, and thus, there is no tension between the predicKon ofnumerical-relaKvity simulaKons and the observaKon of SSS17a•

The interplay of the mulKple non-spherical ejecta components via
photons plays a key role for the kilonova/macronova lightcurves.

•

Our model can be examined by the kilonova/macronova
observed from diﬀerent inclinaKon(for BNS with similar total mass)Future work•

Varia,ons of Kilonova/macronova lightcurves should be studied:ex) smaller total mass system, a black hole-neutron star merger•

Systema,c & Quan,ta,ve study varying ejecta parameters, such as themasses and velocity of dynamical and post merger ejecta, are needed
to understand the variety of kilonova/macronova lightcurves•

Combined analysis of gravita,onal waves and kilonova/macronova
lightcurves (+other EMs)
Gravita,onal waves → total mass, inclina,on, ,dal deformability
Electromagne,c waves → ejecta mass, Ye (weak process) informa,onToward rapid KN predic,on:
Gaussian Process Regression
RT Data sets (3×3×3=27 models)Mpost = 0.01, 0.03, 0.05 M
Mdyn,eq = 0.001, 0.005, 0.01(⇥0.72) M
Mdyn,polar = 10 5 , 5 ⇥ 10 5 , 10 4 (⇥0.28) M<latexit sha1_base64="6XQMvZaCeyaPy3sWKjO8BvafJSE=">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</latexit>•

Gaussian process regression model may be useful tointerpolate the data point of RT simula,ons•

RT simula,on ~1 Day /model → ~1 -10Minutes/modelAppendix

Contamina,on of
Post-merger waveforms
9000
8000

8000
t = 0.00 ms
t = 1.02 ms
t = 2.04 ms
t = 3.06 ms
t = 20.9 ms

0.1•

The post-merger waveforms can be modiﬁed
by detailed physical eﬀects that are nottaken into account for our current
numerical-rela,vity simula,ons(e.g., M. Shibata et al. 2017)
we restrict the frequency range ofgravita,onal-wave modelling to 10–1000 Hz
to avoid the contamina,on from the postmerger waveforms

15H135-135
125H135-135
H135-135
15H125-125
125H125-125
H125-125

6000
5000
4000
3000
2000
1000

0.01
5

7000
6000
5000
4000
3000
2000
1000

10

15
500

20

25
700

R [km]

35
40 1500 20005
1000
f [Hz]

30

15
3000

10

20

R [km]

Dependence of the post-merger
onstars
physicalwaves waveforms
from binary neutron
with threeviscositydi↵erent equations

FIG. 3. Averaged angular-velocity profiles along the cylindrical radicolumn),column),and 0.02spectra(right column).
FIG. 1. 0.01
The(middle
comparisonof Fourierof gravitationalof state and with two values of total mass. A(f ), De↵ , and m0
denote the amplitude of the spectrum, thedistance
αv =e↵ective
0.00
0.004
α
=
0.01
to the gravitational-wave source, and thev total mass of the
α = 0.02
binary, respectively. The vertical dashedv line denotes f =1000 Hz.
0.002

D m0 Re[Ψ4]•Ω [rad/s]

7000

0

D m0 |Ψ4|

The gravita,onal-wave spectra for f>1000 Hz
would be aﬀected by the post-merger
waveforms.

Ω [rad/s]•

f A Deff /m0

1

the analysis in Sec. III and Sec. IV. The details and thederivation of these point-particle models are presented in-0.002
AppendixA.
We note that, in this work, we focus only on gravitational-0.004
waves forRef:
f M.
1000
Hz. Thefor thisis thatShibata& reason
K. Kiuchi
2017
the gravitational-wave spectra for f > 1000 Hz would be
0
5
10
15
20
a↵ected by the post-mergert waveforms:
In
Fig.
1, we
ret [ms]show the amplitude of the gravitational-wave spectra forSet up•

Binary neutron stars in quasi-circular orbits are numerically derived for the ini,alcondi,ons of the simula,ons using a spectral-method library, LORENE(h•p://www.lorene.obspm.fr/)•

small orbital eccentricity ∼10^{−3} is realized by an eccentricity-reduc,onprocedure described in Kyutoku et al. 2014•

The ini,al gravita,onal-wave frequency is ~400 Hz, and~30 gravita9onal cycles (>200 rad) are obtained before the onset of merger•

Grid resolu,on of the simula,on is improved by more than a factor of 2.
(With these grid spacing, the semi-major diameter of the neutron stars is coveredby about ~260 grid points.)•

540-650k core hours (~2 month for 16 core x 32 nodes) for the highest resolu,onmodelsError and Convergencephase at the Kme of the peak amplitude
1.21-1.51 M_sun•

The phase error is dominated by thatcomes from the ﬁnite grid spacing
We es,mate the error of the phase at the,me of the peak amplitude by data with 6
diﬀerent grid-resolu,ons

as

3.5

15H: 6q(N=182)= 0.16 [rad], p= 3.5
125H: 6q(N=182)= 0.31 [rad], p= 2.7
H: 6q(N=182)= 0.44 [rad], p= 2.4
HB: 6q(N=182)= 0.47 [rad], p= 2.2
B: 6q(N=182)= 0.21 [rad], p= 3.3

3
qpeak(182)-qpeak(N)

•

2.5

p

2
1.5
1
0.5
0
0

0.2 0.4 0.6 0.81

1.2 1.4 1.6 1.8

2

whe
pha
for N
conv
peak
tion

p

•

The nearly convergent result is likely to beachieved for all the cases, and the order ofthe convergence is likely to be about 2-4.
The error of the peak phase for highestresolu,on run due to the ﬁnite grid spacing
is conserva,vely about 0.1–0.6 rad.

pe

3.5

qpeak(182)-q
qpeak(182)-q
peak(N)peak(N)•6q(N=182)[(182/N) -1]

3
3.5
2.5
3
2
2.5
1.5
2
1
1.5
0.5
1
0
0.5 0

0
0
3.5

1.16-1.58 M_sun

15H: 6q(N=182)= 0.54 [rad], p= 2.2
125H: 6q(N=182)= 0.34 [rad], p= 2.7
H: 6q(N=182)= 0.31 [rad], p= 2.6
15H:
6q(N=182)= 0.28
0.54 [rad],[rad], p=
p= 2.8
2.2
HB: 6q(N=182)=
125H:
6q(N=182)= 0.37
0.34 [rad],[rad], p=
p= 2.52.7
B: 6q(N=182)=
H: 6q(N=182)= 0.31 [rad], p= 2.6
HB: 6q(N=182)= 0.28 [rad], p= 2.8
B: 6q(N=182)= 0.37 [rad], p= 2.5

0.5

1

1.5

6q(N=182)[(182/N)p-1]
0.5
1
1.5
p

6q(N=182)[(182/N) -1]
15H: 6q(N=182)= 0.09 [rad], p= 4.2
125H: 6q(N=182)= 0.03 [rad], p= 5.9

and
obta

2

2.5

2

2.5

F
twee
func
ues
neut
verg
the
How
fitti
the
rad

V.

SUMMARY

found that the systematic error of our waveform˜ is always smaller thanin the measurement of ⇤also showed that it is smaller than or at most com
˜ with respect to the mass rato the variation of ⇤˜ can be overestim
the other hand, we found that ⇤˜ & 600 when employing P
the order of 100 for ⇤formulas of Eqs. (3.6) and (3.7).

In this paper, we derived a frequency-domain model for
gravitational waves from inspiraling binary neutron stars
employing the hybrid waveforms composed of the latest numerical-relativity waveforms and the SEOBNRv2T
waveforms. In this work, we restrict the frequency rangeof gravitational waves from 10 Hz to 1000 Hz to focus onthewaveforms. We obtained the tidal cor- 30• inspiral-stage
Tidal part amplitude
model for theAssuming that the approximate correlation
rection to the gravitational-wave phase as
25 ⇤˜ and the value of distingusihability found in
hybrid waveforms
isalso
derivedbased⌘
20 holds for the SEOBNRv2T waveforms, we found3
39 ˜ ⇣
2/3
4.240
5/2
˜
on
1PN
order
,dal
part
formula
=
⇤
1
+
12.55
⇤
x
x
tidal
128⌘
2
15 systematic error of the SEOBNRv2T waveform˜ is as large as ⇠ 50 for ⇤˜ & 12✓◆
measurement of ⇤3115
28024205 2 4283 5/2
˜ ⇡ 1900. This indicates⇥ 1+
x ⇡x3/2 +x⇡x, 10 in particular, ⇠ 100 for ⇤• The same,dal1248 hybrid waveform
3302208 as the1092
5 improvement of the TEOB formalism is neededpart phase model is used for parameter (5.1) 0 large values of ⇤˜ to constrain ⇤˜ accurately. We athat, while we restrict our analysis up to f = 1000
determina,on(15H125-125)
and tothe gravitational-wave
amplitude as-5di↵erenceand the0200between
400 the hybrid
600 waveforms
800
1000
r
NRv2T waveforms would be more significant in
5⇡⌘ m20 ˜ 7/4
f [Hz]Atidal =⇤xfrequency
range
[29]
(the
gravitational-wave fr
15H135-135
15H116-158
24 De↵✓◆
at125H135-135
the time of the maximum
amplitude is ⇡ 150
125H116-158
27
449!
H135-135or 15H135-135,
H116-158
15H125-125
and much higher fo⇥
x5
x6 4251 x7.890 .
(5.2)
HB135-135
HB116-158
1664
equations of state).
Hybrid(AHybrid
-A
)
/
A
[%]
tidal
tidal

Tidal part amplitude model

B135-135
B116-158
15H121-151
15H125-125
125H121-151
125H125-125
WeH121-151
estimated the statisticalerror in the meas
H125-125
HB121-151
HB125-125the standardof binaryparameters employing
B121-151
B125-125

We showed that our waveform model reproduces the
The rela,ve error of the ,dal-partphase of the hybrid waveforms in the frequency domain
˜ withinamplitudemodel
always% for thewithin0.1 rad error
for is300. ⇤. 19005andmassforratio
between
1. Wewithinnote thatthe modelf<~900Hz,0.73
andandalways10 %matrix analysis. We obtained results consistent w
parameters are determined using the hybrid waveform ofprevious studies [10, 12, 19]: We reconfirmed thatfor Λ<=850 at 1000 Hz
a specific equal-mass binary. The relative error ofFIG.
the 6. The di↵erence between the tidal-part amplitude
˜ depends
tistical
error
in
the
measurementof⇤tidal-part amplitude model is always within 5% for given
f . by Eq. (2.25), Atidal , and that of the hybrid waveforms,on the upper-boundfrequency of the analysis,˜Hybridby the amplitude of the hybrid wave900 Hz, and in particular, is always within 10% for A⇤
 , normalized
tidal
stronglyon ⌘. We also reconfirmed that the valu
forms, AHybrid.
850 at 1000 Hz.
•

Frequency-domain
GW model for inspiralling BNS
of each neutron star is below 0.05 [3].
V.

SUMMARY

In this paper, we derived a frequency-domain model for
gravitational waves from inspiraling binary neutron stars
employing the hybrid waveforms composed of the latof each neutron star is below 0.05 [3].
We also computed the systematic error of ourestnumerical-relativity
waveforms
andbinary
the SEOBNRv2T• We constructed a frequency-domain gravita,onal-waveform
model
for
form model in themeasurementof binary paramete
waveforms. In this work, we restrict the frequency range
ploying
the hybrid waveforms as hypothetical signa
neutron stars by employing the ,dal-part models
and point-par,cleof gravitational
waves from 10 Hzmodelto 1000 Hz to focus on
V. SUMMARY
found
that
the
systematic
error ofwaveformthe inspiral-stage waveforms. We˜obtained
theourtidal
corinthethegravitational-wave
measurement of ⇤phaseis always
smaller than 2
rection toas• We
showedthatandit is calibrated
smaller thanto
or at most compemployphenom-likepoint-par,cle
modelalso
derivedby
In thispaper, thewe derived
a frequency-domain
model for

⇣ of ⇤
˜ with respect ⌘tothevariation
3
39
gravitational
waves
from
inspiraling
binaryneutronstars
2/3 4.240 to the
5/2 mass rat
˜˜
the SEOBNRv2 waveforms
=
⇤
1
+
12.55
⇤
x
x
tidal
˜ can be overestimatheother hand,
we found that ⇤employing the hybrid waveforms composed of the lat128⌘2◆
˜ & 600 when employing⌘ of 100 for
order
⇤
PN
est numerical-relativity waveforms and the⇣SEOBNRv2T✓ the
3115
28024205
4283
3/2
2
5/2
˜⇥ 01, +x⇡x,= h̃model
f ; Mc , ⌘,⇤,t0formulas, De↵x of⇡xEqs.+(3.6) and (3.7).
waveforms.
In this work,h̃model
we restrict
the frequency
range
!
1248
33022081092
of gravitational waves from 10 Hz to 1000 Hz to focusi( on +(5.1)
= obtained(App + Athe) e cor-pp tidal ) .
tidal
the inspiral-stage waveforms. We
tidal
that theamplitude
approximate!
rectionto the gravitational-wave phase as
and to the ˜Assuming
gravitational-wave
as correlation b⇤ andthe value of distingusihability found in

r
⇣
⌘
2
holds forthemSEOBNRv2T
waveforms, we found th
3
39 ˜
5⇡⌘
2/3 4.240
5/2
˜
0
7/4
˜
⇤ 1 + 12.55 ⇤ x
xAtidal=! tidal = 128⌘
systematic
error⇤xof the SEOBNRv2T waveforms
2
24 D
e↵
˜ is as large as ⇠◆50 for ⇤˜ & 120✓◆✓
measurementof ⇤27 5 449 63115
28024205 2 4283 5/2
˜4251⇥
x ⇠ 100xfor ⇤x7.890This. indicates(5.2) th⇥ 1+
x ⇡x3/2 +x⇡x,
in particular,⇡ 1900.
16
64
!
1248
33022081092
improvement of the TEOB formalism is needed(5.1)
˜ to constrain
˜ reproduces
large valuesof ⇤⇤accurately. theWe als
We showedthat ourwaveform
model
while werestrictouranalysis
up todomain
f = 1000 H
phasebyof that,the hybrid
waveforms
in thefrequency• andWetocheckthe validity ofamplitude
our waveformmodel
comparing
our waveformmodel
the gravitational-waveas˜ . 1900
di↵erence
waveforms
within 0.1
rad errorbetween
for 300the. hybrid⇤
and forand
thethe Sr
and the hybrid
waveforms
supposingadvanced
LIGO
asaﬁducialdetector.
NRv2T
waveforms
moremass ratio
between0.73 andwould
1. Webenote
thatsignificant
the modelin a
5⇡⌘ m20 ˜ 7/4
Atidal =⇤x
parameters
are determinedusing
the gravitational-wave
hybrid waveform of freq
frequency
range [29](the24 De↵✓◆
a specificatequal-mass
The relative
error isof ⇡the1500
the time ofbinary.
the maximum
amplitude27
449<latexit sha1_base64="NdO99WWtY3HSR2yfVX5V42U09HE=">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</latexit>

Tidal part phase model<latexit sha1_base64="QHcVLUYD2cX+iYUaNrgZ8Ls34+E=">AAACSHicdZDNS8MwGMbT+TXnV9Wjl+AQvDjaoehx6GUnmeA+YJ0lTdMtLGlLkgqj9M/z4tGbf4MXD4p4M+t60G0+EHh4nvclyc+LGZXKsl6N0srq2vpGebOytb2zu2fuH3RklAhM2jhikeh5SBJGQ9JWVDHSiwVB3GOk641vpn33kQhJo/BeTWIy4GgY0oBipHTkmq4TCIRTpyWpmzqCQ0V9xLKH3DcnnqB+dra8rdcuWrdZ9s9u0bpm1apZueCisQtTBYVarvni+BFOOAkVZkjKvm3FapAioShmJKs4iSQxwmM0JH1tQ8SJHKQ5iAye6MSHQST0CRXM098bKeJSTrinJzlSIznfTcNlXT9RwdUgpWGcKBLi2UVBwqCK4JQq9KkgWLGJNggLqt8K8Qhpskqzr2gI9vyXF02nXrOtmn13Xm1cFzjK4Agcg1Ngg0vQAE3QAm2AwRN4Ax/g03g23o0v43s2WjKKnUPwR6XSD1G2syo=</latexit>

2.5PN
2/3
(^Hybrid
/
^
-1)/R
tidal
tidal

determined by employing several hyb
the tidal part of the gravitational-wave phase computed
Hybridfitting parameters, a and p, are deter
from the hybrid waveforms, Hybrid
(=
pp ,
tidal
ing
where Hybrid is the phase of the hybrid waveforms), forequal-mass
binaries
normalized
by the
2.5 PN order•theThe
2.5 PN order(equal-mass),dal-partphase
Z fmax2
Hybrid(equal-mass)
tidal-partem(Damour et al.
2012) phase given by [12]I 00 =(f)
tidal (f )
tidal
✓
◆
fmin
3
39
2.5PN
=
⇤ x5/2
tidal
!
32
2
6

where t0 and 0 are parameters that
3115
28024205 2 4283 5/2
3/2⇥ 1+x ⇡x +x⇡x,(m0 = 2.7)
degrees off [Hz]freedom
forMsunchoosing
the t
2/3
1248
3302208
1092
PN! order tidal-part phase normalized by ⇤ , i.e.,00
300 Thus,
400we 500
600 700gins.
minimizeI
for900theHybrid
2.5PN
2/3
900
(2.14) 0.002( tidal / tidal1)/⇤ . Figure 3 clearly shows thatp, t0 , and 0 . The fitting is performe
therelativedeviation
can
be
well
approximatedbyaThe
relaKve deviaKonof the Kdal-part phase!
2/30.001and fmax = 1000 Hz.
power xlaw=in(⇡mx. Furthermore,
it
shows
that
the
relative
where
f
)
is
a
dimensionlessPN
parame00.0007 We use the hybrid waveform of 15H
2/3
deviationisapproximately
proportional
to⇤becauseHybrid
2.5PNter, and ⇤ = ⇤1 = ⇤2 for tidal
the equal-mass cases. We0.0005
tidal
mining the fitting parameters because
all thecurvesalign.
We notethat,ofexceptionally,therelanotethatthetidal-part
phase
the
hybridwaveforms
2.5PN!
tive deviation for the B equationof state shows a slightly 0.0003tribution of ⇤ is most significant for
tidal
in Fig. 2 is aligned with the 2.5 PN order tidal-part0.0002
di↵erent trend from the other cases. The reason for this isnary neutron star models employed
phase
given
by
Eq.
(2.14)
for
10
Hz

f

50
Hz
employthe
is so small
its e↵ect can- 0.0001we obtain• thatWeﬁndtidalthat deformability
therela,vedevia,oncanthatbe welling
Eqs.
(2.12)
and(2.13).
We
find
that
the tidal-part
0.06
0.07 0.08 0.09 0.1 0.11
not be accurately extracted from the numerical-relativity
approximatedby aawaveforms
power
law indeviates
thePN parameter,x fromphaseof theforhybrid
significantly
x = (/m0 f)2/3 a = 12.55,
waveformsuch
soft
equationofstate.
11
the To2.5correct
PN ordertidal-partwe
phasein thethe 2.5
high-frequency
this deviation,
extendPN order15H135-135
15H125-125
p = 4.240.
range,
f
&
500
Hz
(x
&
0.075
for
m
=
2.7
M
),
and
the
tidal-part
phaseformulaofEq.
(2.14)
bymultiplyinga0
125H135-135
125H125-125• Furthermore, we ﬁnd that the rela,ve devia,on is
H135-135
H125-125
deviation
non-linearlyon ⇤ (note the quantities
non-lineardepends
correctionto ⇤ as
HB135-135
HB125-125
approximately propor,onal to Λ^{2/3}
Figure4 shows the phasedi↵erence
shown in Fig. 2 are alreadynormalized
by
⇤).
This
indiB135-135
B125-125⌘3
39 ⇣part of the hybrid waveforms and th
em
2/3
p
5/2
catestidal
thatthenon-linear
contributionof⇤isappreciably=⇤ 1 + a⇤ x
x
Eq. (2.15)
for the phaseequal-ma
32Hybrid2
FIG. 3. Themodel
relative ofdeviationof the tidal-partof thepresentinforthehigh-frequency
range.
tidal
arem2.5
alignedfortidal-part10
Hz f nor
waveforms
from
order
phase
mphases
total
mass
of2/3
binary3115
28024205
4283 5/2ofhybrid
02/3= m1 +the2 :PN
3/2 the2 deviation
Hybrid
In
Fig.
3,
we
plot
relative
the
2.5PN
⇥ 1+
x ⇡x +
x⇡x,malized by ⇤Eqs.
, i.e.,( tidaland/ tidal1)/⇤. Both verti(2.12)(2.13).
Althoughthe1248
33022081092
2/3
cal2.5and horizontal
axes
withparameter
logarithmic scales.
of the tidal-part phase of the hybrid waveforms from thex :=determined(⇡mare): PN
0 f plottedarebyemploying
only th(2.15)
se norof 15H125-125 as a reference, we fin
verti<latexit sha1_base64="J9RT1Ftlvr2wEwn2aC36fJdiQu8=">AAACFHicbVDLSgMxFM34rPU16tJNsAiCpcwUQRGEohuXFewDOsOQSTNtaJIZkowwDP0IN/6KGxeKuHXhzr8xbWehrRdCDufce5NzwoRRpR3n21paXlldWy9tlDe3tnd27b39topTiUkLxyyW3RApwqggLU01I91EEsRDRjrh6Gaidx6IVDQW9zpLiM/RQNCIYqQNFdinPHCueOCau+5VL3NPcqhjjZhX5UgprxpHXjWkAslsHNgVp+ZMCy4CtwAVUFQzsL+8foxTToTGzGzruU6i/RxJTTEj47KXKpIgPEID0jNQIE6Un09NjeGxYfowiqU5QsMp+3siR1ypjIemkyM9VPPahPxP66U6uvBzKpJUE4FnD0UpM7bhJCHYp5JgzTIDEJbU/BXiIZIIa5Nj2YTgzlteBO16zXVq7t1ZpXFdxFECh+AInAAXnIMGuAVN0AIYPIJn8ArerCfrxXq3PmatS1YxcwD+lPX5AwoCnZc=</latexit><latexit sha1_base64="inWf0ZOLy0c27MX0cKKvnR49suQ=">AAACEXicbVDLSsNAFJ3UV62vqEs3g0WoIDWpglIQim5cSQX7gCaWyXTSDp1JwsxELCG/4MZfceNCEbfu3Pk3TtsstHrgwuGce7n3Hi9iVCrL+jJyc/MLi0v55cLK6tr6hrm51ZRhLDBp4JCFou0hSRgNSENRxUg7EgRxj5GWN7wY+607IiQNgxs1iojLUT+gPsVIaalrlu6rZyUnopB3Lejv3yaVw6O0mjiCw/qVcxAhgThRRKRds2iVrQngX2JnpAgy1Lvmp9MLccxJoDBDUnZsK1JugoSimJG04MSSRAgPUZ90NA30Hukmk49SuKeVHvRDoStQcKL+nEgQl3LEPd3JkRrIWW8s/ud1YuWfugkNoliRAE8X+TGDKoTjeGCPCoIVG2mCsKD6VogHOgSsM5AFHYI9+/Jf0qyUbatsXx8Xa+dZHHmwA3ZBCdjgBNTAJaiDBsDgATyBF/BqPBrPxpvxPm3NGdnMNvgF4+MbqpibsA==</latexit>

other hand,
the SEOBNRv2T
largerboundsof the[51].
integrationin Eq. (3.2) are ratio
set to andbe 10theHz distinlarger
than 0.5 forwaveforms
the caseshavewiththeval15H equation ofthe
1 levelThe signal-to-noiseues of the distinguishability and the mismatch than ourand 1000 Hz, respectively. We note that ⇢ corresponds
state, and in
the distinguishability is ⇡ 0.8
guishability are proportional to the inverse of the e↵ective
˜ &particular,
waveform model for ⇤700 with respect to the hyto the signal-to-noise ratio [50], and ⇢ = 1 indicatesfor 15H125-125.
Theseresults are consistent
with thedistance,De↵ .
brid waveforms.
The value ofthe distinguishability
isthat two waveforms
are distinguishable approximately at
resultsof theRefs.
[26,with29]thein 15Hwhichlarger ofphase di↵erence
III, Thewe signal-to-noisesummarize thedistinguishability
be- thanlarger
0.5 forcases
equationtheIn1 Tablelevel [51].
ratio
and the distinbetween
the SEOBNRv2T
waveforms
state, andin particular,
the distinguishability
is ⇡and
0.8 the numericaltween
our arewaveformmodel
andthe ofhybrid
waveforms.
guishability
proportional
to theinversethe e↵ective
˜
results are
consistentthedistance,
relativityThese
waveforms
is foundfor withthe larger
values of ⇤.
Here,theDsignal-to-noise
ratio is always fixed to be 50for
by15H125-125.
e↵ .
results ofWe
Refs.
[26, that29] in thewhichlarger phase di↵erence
In Table DIII,
we summarize the distinguishability benote
SEOBNRv2T
formalism is a timeadjustinge↵ because the tidal deformability is clearly
˜
Model
⇤
our
waveformmodel 2.5PNtidal
betweenthe
SEOBNRv2T
waveforms
andthenumericaltween
ourwaveform
model
and
the
hybridwaveforms.
domain approximant, and thus, the computational
costs
measurable only for events with a high signal-to-noise ra˜6⇤.
relativity
waveforms
is
found
for
the
largervaluesofHere, the signal-to-noise ratio is always fixed to be 50 by
15H135-135
1211 would be 4.1⇥ 10
1.4 ⇥ 10 3
fordata
analysishigherthan
ourfrequencytio.
Forcomparison,we
also
computethe
distinguishaWe note that the SEOBNRv2T formalism is a timeadjusting De↵ because the tidal deformability is clearly6domain waveformmodel.
125H135-135863
4.0 ⇥ 10costs
5.6 ⇥ 10 4
2.5
PNKdal:
same
as
our
model
bilityofthe
SEOBNRv2T
waveforms
andPN
waveform
domainapproximant,
and
thus,
the computational
measurable only for events with a high signal-to-noise raPN waveformmodels,PNtidal(TF2)
and1.8
PNtimodels
withrespect
thecomputehybridthe
waveforms.
theH135-135
data analysis
would607
be higher
than 2.9
ourfrequencytio.
Forcomparison,
weto
also
distinguisha-For for⇥
10 6⇥ 10 4
but
with
2.5
PN
Kdal
phase
formulawaveformmodel.
dal(TF2+),
show poor agreements with6the hybrid wavebility partof theof
SEOBNRv2T
waveformsmodels,and PN waveformtidal
the PN waveform
we employ domain
the5
HB135-135
422
2.6
⇥
10
5.0
⇥
10
PN waveformmodels,PNtidal(TF2)
and PNti(and1PN
Kdal
formula)modelsrespect
the hybrid
waveforms.
For theforms. For
PNtidal(TF2),
the distinguishability
and the
2.5
PN withorderphasetoamplitude
andthe 1 PN
order amplitudefor6
5
3 1.3 ⇥ 10
B135-135
289
2.0
⇥
10
dal(TF2+),
show
poor
agreements
withthehybridwavetidal
part
of
the
PN
waveformmodels,we
employthemismatch are always larger than 2 and 8 ⇥ 10 , respecmulas given by [11, 12, 36]
63
forms.
For
PNtidal(TF2),
the distinguishability
and the
2.5 PN order phase and the 1 PN order amplitude for15H121-151
1198
⇥
⇥
10
tively,and inparticular,
the6.3
distinguishability
is1.6
larger3 10✓◆
mismatch are always larger than 2 and 8 ⇥ 10 , respecmulas givenby [11, 12, 36]
6
4
3
39 ˜
than
4 for 15H121-151,
15H116-158,
and 15H125-125.
2.5PN05/2
125H121-151
856
2.5
⇥
10
5.7
⇥
10
tively,andinparticular,the
distinguishability
islarger
=
⇤
x
✓
◆
tidal
This
large distinguishability
is15H125-125.
notonly6 due to the
lack
4
128⌘
3
39 ˜2 5/2
thanH121-151
4 for
15H121-151,
15H116-158, and
2.5PN0
604
2.8
⇥
10
1.8
⇥
10
=
⇤
x

tidal
of distinguishability
higher-order terms
thedue
tidal
part
large
is notin
only
to thelack
128⌘
2
6 but also due to5
3115
28024205 2 4283 5/2 This
3/2
HB121-151
422
3.0
⇥
10
5.2
⇥
10

⇥ 1 +3115 x ⇡x 28024205+
x4283⇡x of, higher-orderterms
in theterms
tidal part
butpoint-particle
also due tothe lackof thosein thepart of PN3/2
2
5/2
6
1248
33022081092⇥ 1+
x ⇡x +x⇡x,
the B121-151
lack ofthose termswaveforms.
in the point-particle
ofdistinguishability
PN290
3.1part⇥1.1 ⇥of10 5
tidal(TF2)
Indeed,the10
1248
3302208
1092(3.6)
tidal(TF2)
waveforms. Indeed,
5of which purely
3
PNtidal(TF2+)
fromthethedistinguishability
hybridwaveforms,
15H116-158
1185
1.1
⇥
10
1.7
⇥
10
(3.6)
PNtidal(TF2+) from the hybrid waveforms, which purely
6 from the hybrid
4
reflects
the ofdi↵erenceof PNtidal(TF2+)
125H116-158
848
3.8
⇥
10
6.3
⇥
10
and
reflects
thedi↵erence
PNtidal(TF2+)
fromthehybridandwaveforms
in parts
the tidal
parts of waves,gravitational
waves, is6is4
waveforms
in the tidalof gravitationalr✓◆
H116-158
601
3.0
⇥
10
1.9
⇥
10
r
✓
◆
2
alwaysthansmallerthat of PNtidal(TF2),
and in partic0
always smallerthat ofthan
PNtidal(TF2),
and in partic5⇡⌘m20m˜0 ˜7/4 7/4 27 527 449
449 6
5⇡⌘
1PN
5
6
5
1PN0
6
A
=
⇤x
x
x
,
(3.7)
Atidal⇤xxx , (3.7)
HB116-158
421
4.0
⇥
10
5.1
⇥
10
tidal =
ular,
is
as
small
as
⇠
0.3
for
the
cases
with
the
B
equation
ular,
is
as
small
as
⇠
0.3
for
the
cases
with
the
B
equation
2424De↵16 16 64
De↵
64
6of state.
for PNtidal(TF2+),
the10
distinB116-158
291⇥⇥ 10 5ofHowever,state. even
However,even for 5.0
PNtidal(TF2+),
the1.0
distin˜ 850. This ˜indiguishability
is larger than
⇡ 1.4 for⇤3
respectively.3 3“PNtidal(TF2)”
and and“PNtidal(TF2+)”
inguishability
is larger
than
⇡1.6
1.4⇥
for10⇤ 6 850. This
indirespectively.
“PNtidal(TF2)”“PNtidal(TF2+)”
in
15H125-125
1875
2.5
⇥
10
cates that PN tidal formulas of Eqs. (3.6) and (3.7) are
Table
IIIVines
denotewaveformmodelsemploying
TayRef.
et PN
al.PN
2011,
Damouret al.
2012,
6
cates that PN
tidal formulasof ⇥Eqs. (3.6)
and (3.7)
are10 3Table
III
denotewaveformmodelsemploying
Tay˜ & 850
125H125-125
1352
1.6
1.1 ⇥
lorF2 and TF2+ (see Appendix A) as the point-particle
notsuitable for the data
analysis if ⇤and10
⇢˜& 35
Hotokezaka
et al. 2016
lorF2 and TF2+
(see AppendixA) as the point-particle
notsuitablefor the data
analysis⇢ & 35no matterhow
the point-particle
model
is3.5
accurate.
H125-125966⇥if10⇤ &6 850 and3.7⇥ 10 4no matter how the point-particle model6 is accurate.
4

Mismatch (2.5 PN ,dal)

HB125-125
B125-125 B.

3 We note that, for Eqs. (3.6) and (3.7), the dependence on themass ratio is considered only up to the leading order for simplic3 We
note that, for Eqs. (3.6) and (3.7), the dependence on the683
5.0 ⇥ 10
6
Systematic
error 8.3 ⇥ 10476

1.0 ⇥ 10
1.8 ⇥ 10 5

B. Systematic error
ity. This can be justified by the fact that the asymmetric-tidal
mass
ratioisisexpectedconsidered
up to the[19].
leadingorder
for simplicNext, we estimate the systematic error of our wavecorrection
to beonly
sub-dominant
Indeed,we findity.
can beby thefact
the asymmetric-tidal
that This
employing
PNjustified
tidal formulas
withfullthatdependence
on the 3 form model4 in the measurement of binary parameters.
Next,
we waveforms
estimate asthesystematic
error of our wavecorrection
is expected
to bein sub-dominant
Indeed, we Employing
findmass ratio changes
the results
Table III only by[19].
. 10%.
thehybridhypotheticalsignals,<latexit sha1_base64="+AvXPC0zPS9KqHJPU4AxSL5AGmQ=">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</latexit>•

Large values of mismatch ( 10<latexit sha1_base64="B042exhB/vX75ZYoW6bmxs1w3XI=">AAAB+HicbVDLSsNAFL2pr1ofjbp0M1gENy2JFnRZdOOygn1AG8tkOm2HTiZhZiLUkC9x40IRt36KO//GaZqFth643MM59zJ3jh9xprTjfFuFtfWNza3idmlnd2+/bB8ctlUYS0JbJOSh7PpYUc4EbWmmOe1GkuLA57TjT2/mfueRSsVCca9nEfUCPBZsxAjWRhrYZdd5SKoXaTXr9XRgV5yakwGtEjcnFcjRHNhf/WFI4oAKTThWquc6kfYSLDUjnKalfqxohMkUj2nPUIEDqrwkOzxFp0YZolEoTQmNMvX3RoIDpWaBbyYDrCdq2ZuL/3m9WI+uvISJKNZUkMVDo5gjHaJ5CmjIJCWazwzBRDJzKyITLDHRJquSCcFd/vIqaZ/XXKfm3tUrjes8jiIcwwmcgQuX0IBbaEILCMTwDK/wZj1ZL9a79bEYLVj5zhH8gfX5A2Y/kZ0=</latexit>that employing PN tidal formulas with full dependence on themass ratio changes the results in Table III only by . 10%.

10) between the 2.5 PN ,dal waveforms

form model in the measurement of binary parameters.
Employing the hybrid waveforms as hypothetical signals,

Systema,c error (2.5 PN ,dal)
Model

•

We found that Λ can beoveres,mated by the order of 100
for Λ~600 when employing PN,dal formulas (assuming the hybrid
waveforms are the realis,c signal)•

In the absence of the non-linear
enhancement of Λ , spuriously
larger values of Λ are needed for
2.5PN ,dal formula to complement
the enhancement of ,dal eﬀects.

15H135-135
125H135-135
H135-135
HB135-135
B135-135
15H121-151
125H121-151
H121-151
HB121-151
B121-151
15H116-158
125H116-158
H116-158
HB116-158
B116-158
15H125-125
125H125-125
H125-125
HB125-125
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˜
⇤
1211
863
607
422
289
1198
856
604
422
290
1185
848
601
421
291
1875
1352
966
683
476

Mc [M ]
3.8 ⇥ 10 6
3.1 ⇥ 10 6
2.0 ⇥ 10 6
1.4 ⇥ 10 6
7.8 ⇥ 10 7
3.9 ⇥ 10 6
3.2 ⇥ 10 6
2.1 ⇥ 10 6
1.4 ⇥ 10 6
8.0 ⇥ 10 7
4.0 ⇥ 10 6
3.4 ⇥ 10 6
2.2 ⇥ 10 6
1.5 ⇥ 10 6
7.7 ⇥ 10 7
2.5 ⇥ 10 6
3.2 ⇥ 10 6
2.2 ⇥ 10 6
1.4 ⇥ 10 6
7.8 ⇥ 10 7

2.5PNtidal⌘
3.1 ⇥ 10 4
2.5 ⇥ 10 4
1.6 ⇥ 10 4
1.0 ⇥ 10 4
5.5 ⇥ 10 5
3.1 ⇥ 10 4
2.5 ⇥ 10 4
1.6 ⇥ 10 4
1.0 ⇥ 10 4
5.5 ⇥ 10 5
3.2 ⇥ 10 4
2.6 ⇥ 10 4
1.6 ⇥ 10 4
1.0 ⇥ 10 4
5.2 ⇥ 10 5
2.6 ⇥ 10 4
3.0 ⇥ 10 4
2.0 ⇥ 10 4
1.2 ⇥ 10 4
6.2 ⇥ 10 5

˜
⇤
250
177
105
59
28.9
245
175
105
59
27
243
176
105
58
24
296
265
168
93
39

dependence on the mass ratio contained in the
coupling constants, Eq. (2). Also, while we use
data up to !
ˆ = 0.17, the model remains accu
also for BNSs with smaller T2 that merge at hi
frequencies. Let us stress that the model performa
are independent of the BBH baseline, provided the laComparison
Dietrich+17
is a faithful representationof withBBH
waveforms.

Comparison with Dietrich et al. 2017

obtained following Eq. (3). The panels show the realof the GW signals (NR data – gray, tidal approximantnge).• WeDiﬀerent
also include
the phase between
NR data
NS waveforms
and thethe
TidalEOB
respect towaveforms
our tidal approximant(5), calibra,on
to Taylor T2
used for theEq.
model
approximant Eq. (4) (cyan), and for some cases to EOB
n dashed [8], green dot dashed [7]. We also indicate theated •uncertaintyof the NR
datais(blueshaded)andthe
The waveformmodel
derivedin the,mement region (gray shaded). Simulations use the sameand forthen,transformed
to aof [13]on as in domain,
Fig. 1 except
theisunequalmass casefrequency-domain
waveform model
EOSMA +M
B.

Frequency-domain tidal approximant. In thei (f )
quency domain h̃(f ) = f 7/6 Ã(f )eOriginal . The exp
0.001000sion of the tidal phase is computed using the statio
phase approximation (SPA) [29]MismatchA BNS GW model is also derived in
Dietrichofet NR
al. 2017
based onNRmodelwaveforms
2. Comparison
simulations
with
wave•

fmin = 10 Hz, fmax = 1000 Hz,
m1 = m2 = 1.35 Msun

0.000100d2 SPA
Q! (!f )
T
=
,
2
2
d!f
!f

where 0.000010!f is the Fourier domain circular frequency !
2⇡M f , and Q! (!) = d /d log !. The integratio
Eq. (6) with (5) is performed numerically; the const
0.000001
fixed by demanding continuity with
employing the sta,onary-phase approxima,on of integration 0are200
400 600 800 1000 1200 1400 1600 1800
TaylorF21PN in the limit f ! 0. The resulting expres
R
NR
can
be
approximatedbyaPadé function:T. Although the final fit depends only weakly on• While their model and our model agrees quite
xact number of points of the interpolating grid, usNRP
T c̃Newt
5/2
=

x
⇥
well,
some
diﬀerence
is
found
for
a
large
value
T
2
more points at lower frequencies helps constraining
XA XBof regime.
Λ due to Our
the tothe diﬀerence
that theby
nont in thatapproximant
is defined
1 + ñ1 x + ñ3/2 x3/2 + ñ2 x2 + ñ5/2 x5/2
linear,dal coefficients
correc,on n = (-17.941,57.983,5) with thefitting
1 + d˜1 x + d˜3/2 x3/2
76,964.192,-936.844), and d3/2 = 43.446.
Ref: dDietrich2017
˜1 = ñ1et al.3115/1248,
time-domain approximant of a generic spin-alignedwith c̃Newt = 39/16 andthe oconfiguration is computed by prescribing T2 andparameters read: ñ = (-17.428,31.867,-26.414,62.

Radia,ve transfer simula,on•

We employ a wavelength-dependent radiaKve transfer simulaKon code
(M. Tanaka et al. 2013, 2014, 2017)•

The density, velocity, and Ye proﬁles of ejecta are employed within the range of predicKonsby numerical-relaKvity simulaKons.
(e.g., Dietrich et al. 2016, Hotokezaka et al. 2018, Metzger&Fernandez et al. 2014, Fujibayashi et al. 2018)•

The abundance paPern and nuclear heaKng rate are given based on r-process nucleosynthesis
calculaKons by (Wanajo et al. 2014)•

RealisKc opacity table constructed by the atomic structure calculaKons(Se, Ru, Te, Nd, and Er)! improved points•

The grid resoluKon of the simulaKon is also improved by an oder ofmagnitude from our previous works by imposing axisymmetry.

•special-relaKvisKc eﬀects on photon transfer are fully taken into accountR-process nucleosynthesis
Credit) Sho FujibayashiZβdecayN captureN⌧Ncap ⌧ ⌧ dec<latexit sha1_base64="Rs9wf7gaMtiaGEU5pLuIublymYI=">AAACD3icbVDLSgNBEJz1bXxFPXoZDKKnsCuCehO9eJIIrglkQ+iddJLB2QczvUJY8gle/BUvHlS8evXm3zh5gJpY0FBUddPdFaZKGnLdL2dmdm5+YXFpubCyura+UdzcujVJpgX6IlGJroVgUMkYfZKksJZqhChUWA3vLgZ+9R61kUl8Q70UGxF0YtmWAshKzeI+DwiyZh7oiF8JSPuBUj9KECIBb6HoN4slt+wOwaeJNyYlNkalWfwMWonIIoxJKDCm7rkpNXLQJIXCfiHIDKYg7qCDdUtjiNA08uFDfb5nlRZvJ9pWTHyo/p7IITKmF4W2MwLqmklvIP7n1TNqnzRyGacZYSxGi9qZ4pTwQTq8JTUKUj1LQGhpb+WiCxoE2QwLNgRv8uVp4h+WT8ve9VHp7HycxhLbYbvsgHnsmJ2xS1ZhPhPsgT2xF/bqPDrPzpvzPmqdccYz2+wPnI9vDUeczw==</latexit>