Multimessenger Detection Rates and Distributions of Binary Neutron Star Mergers andTheir Cosmological Implications

Jiming Yu1,2 , Haoran Song3, Shunke Ai4, He Gao3 , Fayin Wang5,6, Yu Wang1,2, Youjun Lu7,8 , Wenjuan Fang1,2, andWen Zhao1,2

1 CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Chinese Academy ofSciences, Hefei, Anhui 230026, People’s Republic of China; yjm8012@mail.ustc.edu.cn, wzhao7@ustc.edu.cn

2 School of Astronomy and Space Science, University of Science and Technology of China, Hefei, 230026, People’s Republic of China3 Department of Astronomy, Beijing Normal University, Beijing 100875, People’s Republic of China

4 Department of Physics and Astronomy, University of Nevada Las Vegas, Las Vegas, NV 89154, USA5 School of Astronomy and Space Science, Nanjing University, Nanjing 210093, People’s Republic of China

6 Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, People’s Republic of China7 CAS Key Laboratory for Computational Astrophysics, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, People’s Republic of

China8 School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

Received 2020 December 15; revised 2021 May 26; accepted 2021 May 26; published 2021 July 27

Abstract

Gravitational wave (GW) events, produced by the coalescence of binary neutron stars (BNSs), can be treated as thestandard sirens to probe the expansion history of the universe, if their redshifts can be determined fromelectromagnetic (EM) observations. For the high-redshift (z 0.1) events, the short γ-ray bursts (sGRBs) and theafterglows are always considered as the primary EM counterparts. In this paper, by investigating various models ofsGRBs and afterglows, we discuss the rates and distributions of the multimessenger observations of BNS mergersusing GW detectors in the second-generation (2G), 2.5G, and 3G era with detectable sGRBs and afterglows. Forinstance, for the Cosmic Explorer GW detector, the rate is about 300–3500 yr−1 with a GECAM-like detector forγ-ray emissions and an LSST/WFST detector for optical afterglows. In addition, we find that these events haveredshifts z 2 and inclination angles ι 20°. These results justify the rough estimation in previous works.Considering these events as standard sirens to constrain the equation-of-state parameters of dark energy w0 and wa,we obtain the potential constraints of Δw0; 0.02–0.05 and Δwa; 0.1–0.4.

Unified Astronomy Thesaurus concepts: Gravitational waves (678); Gamma-ray bursts (629); Dark energy (351)

1. Introduction

The discoveries of dozens of gravitational wave (GW) signalsproduced by the inspiral and merger of compact binary systems(Abbott et al. 2016a, 2016b, 2016c, 2016d, 2017a, 2017b,2017c, 2017d, 2019, 2020a, 2020b, 2020c, 2021a) mark theopening of the era of GW astronomy. Since one can measure theluminosity distance of a GW source without any relying on acosmic distance ladder (Schutz 1986), if the source’s redshift canbe measured independently, this kind of GW event can be treatedas a standard siren to measure various cosmology parameters(Holz & Hughes 2005; Sathyaprakash et al. 2010; Zhao et al.2011; Yan et al. 2020; Wang et al. 2020). On 2017 August 17, aGW event (GW170817) produced by a binary neutron star (BNS)system, together with a γ-ray burst (GRB 170817A) wereobserved (Goldstein et al. 2017; Savchenko et al. 2017; Abbottet al. 2017a, 2017d). With the identification of their host galaxy,NGC 4993 (Arcavi 2017; Coulter et al. 2017; Soares-Santos et al.2017; Tanvir et al. 2017; Valenti et al. 2017), the Advanced LIGOand Virgo Collaborations gave the first constraint on the Hubbleconstant from standard sirens, = -

+ - -H 70 km s Mpc0 8.012.0 1 1 with

a 68.3% confidence level (LIGO Scientific Collaboration et al.2017). Many recent works also discussed the Hubble constantmeasurement through standard sirens (Chen et al. 2018; Fishbachet al. 2019; Mortlock et al. 2019; Soares-Santos et al. 2019; Grayet al. 2020; Yu et al. 2020).

Accurate measurement of the luminosity distance dL is crucialto standard sirens. Building more and more advanced detectorswill improve the dL measurement of the GW detection. At the

same time, this will increase the redshift detection limit of GWsignals, so that GW standard sirens can be used to study theevolution of the high-redshift universe. In the near future, theKamioka Gravitational Wave Detector (KAGRA; Abbott et al.2018) and LIGO-India (Unnikrishnan 2013), will be built.Together with Advanced LIGO (aLIGO) and Virgo, there willbe a network of five second-generation (2G) ground-based laserinterferometer GW detectors. In LIGO Scientific Collaboration(2016), a modest cost upgrade of aLIGO, named LIGO A+, isproposed. Looking forward, two 3G detectors, including theEinstein Telescope (ET; Punturo et al. 2010; Abernathy et al.2011) and Cosmic Explorer (CE; Dwyer et al. 2015; Abbott et al.2017e) are also under consideration.The measurement of redshift is also an important task for

standard sirens. One of the main methods is by observing theirelectromagnetic (EM) counterparts. The mergers of BNSs andneutron star black holes are believed to create short GRBs(Paczynski 1986; Eichler et al. 1989; Narayan et al. 1992;Rosswog 2013) and kilonovae (Li & Paczyński 1998; Rosswog2005; Tanvir et al. 2013; Metzger 2017). From the afterglow ofGRBs or kilonovae, one can measure the GW source’s redshift.Usually, the afterglows of GRB have a much narrowerobservation angle but a larger redshift range (Berger 2014).Therefore, the observations of afterglows are more important forstandard sirens with a high redshift.In this work, we assume the jet profile obtained from GRB

170817A is quasi-universal for all BNS mergers and performMonte Carlo simulations to estimate the magnitudes of theirGW signals, GRB counterparts, and afterglows. We select the

The Astrophysical Journal, 916:54 (13pp), 2021 July 20 https://doi.org/10.3847/1538-4357/ac0628© 2021. The American Astronomical Society. All rights reserved.

1

samples that can be triggered by both GW interferometers andEM telescopes, and treat them as standard sirens. We get thestandard sirens’ rates and their distributions of redshiftsand inclination angles. Moreover, we use these samples ofstandard sirens, combined with the method mentioned inSathyaprakash et al. (2010) and Zhao et al. (2011), to discussthe implications for dark energy parameters constrained by 3GGW interferometers.

This paper is organized as follows. In Section 2, we show theBNS samples used in this work. We introduce the detectorresponse and the Fisher information matrix in Section 3. The jetprofile of GRB 170817A, the observation abilities of γ-raydetectors, and the optical telescopes we used are mentioned inSection 4. We estimate the detectability of BNS merger GWsignals, GRB counterparts, and afterglows, and discuss theimplications of dark energy parameter constraints in Section 5.As a supplement, we discuss several alternative models for theEM counterparts in Section 6. Finally, we summarize ourresults in Section 7.

Throughout the paper, we choose the unit with c=G= 1,where G is the Newtonian gravitational constant and c is thespeed of light in vacuum. We adopt the standard ΛCDM modelwith following parameters H0= 67.8 km s−1 Mpc−1, Ωm=0.308, ΩΛ= 0.682 (Planck Collaboration et al. 2016).

2. BNS Samples

The event rate of BNS mergers with redshift z can beestimated as

( )( ) ( ) ( )=

´

+N z dz

R f z

z

dV z

dzdz

1, 1

BNSmergers,0

where RBNSmergers,0 is the local BNS merger rate and we setit as RBNSmergers,0= 80–810 Gpc−3 yr−1 in this work (Abbottet al. 2021b). f (z) is the dimensionless redshift distributionfunction and dV(z)/dz is the differential comoving volume,which is given by

( )( )

( ) ( )pc=

dV z

dz

c

zz

4, 22

where χ(z) is the comoving distance and ( ) z is the conformalHubble parameter with redshift.

The dimensionless redshift distribution function f (z) dependson the initial distribution of the BNS system and the delay timefrom generation to merger. Here, we first assume the initialdistribution follows the star formation rate (SFR). We adopt themodel derived by Yuksel et al. (2008):

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥

( ) ( )

( )

µ + ++

++h

h h h- -z z

z zSFR 1

1

5000

1

9,

3

3.40.3 3.5 1

with η=−10 in units of MeGpc−3 yr−1. For delay time, weuse the log-normal distribution model (Wanderman & Piran2015). In this module, the delay time τ has the distribution

⎡⎣⎢

⎤⎦⎥

( ) ( ) ( )t tpts

ts

t= --

P dt

d1

2exp

ln ln

2, 4d

2

2

with td= 2.9 Gyr and σ= 0.2. The distribution of BNS mergersamples can be integrated from Equations (3) and (4):

( ( )) ( ( )) ( ) ( )ò t t t= -f z t z t P dSFR . 50 0

The range of the initial distribution is from 0 to 7, with thesame upper limit as Yuksel et al. (2008). So, we can get theBNS merger samples with the redshift distribution ofEquation (1). Their distribution is isotropic. As for thecomponent masses m1 and m2, we choose a normal distributionN (μ= 1.32Me, σ= 0.11Me) for simulation, which follows theobservationally deviating distribution of the galactic BNSsystem (Kiziltan et al. 2013). We denote the inclination anglebetween the binary’s orbital angular momentum and the line ofsight as ι. The distribution of ι is proportional to isin . All of theBNS mergers in our simulation are nonspinning systems.

3. GW Detection

In this work, we mainly consider the network with Nd GWdetectors and denote their spatial locations as rI with I= 1, 2,L , Nd. Here, we make an approximation that each detector hasa spatial size much smaller than the GW wavelength. It is worthnoting that for 3G detectors, this approximation wouldintroduce significant biases in localization (Essick et al.2017). Therefore, the frequency-dependent responses shouldbe considered in the real localization. In Figure 5 of Essicket al. (2017), the results of localization with/without thisapproximation are shown. Beyond the bias between them, theyhave similar errors. Since we concentrate on the error estimatein this work, this bias is unimportant and has little impact onour results. For the ith detector, the response to an incomingGW signal traveling in direction n could be written as a linearcombination of two wave polarizations,

( ) ( ) ( ) ( )= ++´

´´d t F h t F h t . 6I I I

The ith detector’s beam-pattern function is decided by thesource’s R.A. α, decl. δ, the polarization angle ψ, the position,and the orientation of the interferometer’s arms. In this paper,we consider the 2G interferometers LIGO (Livingston), LIGO(Handford), Virgo, KAGRA, LIGO-India, and the 3G inter-ferometers ET, CE, and an assumed CE-type detector. Ourprevious paper, Yu et al. (2020), listed the parameters of theseinterferometers, which are also mentioned in Jaranowski et al.(1998), Blair et al. (2015), and Vitale & Evans (2017).For LIGO (Livingston), LIGO (Handford), Virgo, KAGRA,and LIGO-India, we use the noise curve of the designedlevel for Advanced LIGO (aLIGO) and A+ (LIGO ScientificCollaboration 2016). For CE in the USA and the assumed CE-type detector in Australia, we use the proposed noise curve inAbbott et al. (2017e) and Dwyer et al. (2015). And for ET, weconsider the proposed ET-D project (Punturo et al. 2010;Abernathy et al. 2011). In this paper, we denote the network ofLIGO (Livingston), LIGO (Handford), and Virgo with anaLIGO-type noise curve as LHV, and the network of five 2Ginterferometers as LHVIK. LHV A+ and LHVIK A+ are usedto represent the networks with an A+-type noise curve. CEETrepresents the network of CE in the USA and ET in Europe,and CE2ET is the network of two CE-type interferometers inUSA and Australia, and one ET in Europe.

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The Astrophysical Journal, 916:54 (13pp), 2021 July 20 Yu et al.

We adopt the restricted post-Newtonian approximation of thewaveform for the nonspinning systems with only the waveformsin an inspiralling stage (Cutler et al. 1993; Damour et al. 2001;Itoh et al. 2001; Blanchet 2002; Blanchet et al. 2002,2004a, 2004b; Itoh & Futamase 2003; Itoh 2004a, 2004b;Blanchet & Iyer 2005; Sathyaprakash & Schutz 2009) in thispaper. The terms h+ and h× are dependent on the mass ratio

( )h º +m m m m1 2 1 22, the chirp mass ( )hº + m mc 1 2

3 5,dL, ι, the merging time tc, and the merging phase ψc. Therefore,for a BNS merger, the response of the interferometer dependson nine parameters, { ( )}q a d y i h f= M t d, , , , , , , , logc c L .Employing the nine-parameter Fisher matrix Γij and marginalizingover the other parameters, we derive the covariance matrix ( )G-

ij1

for the nine parameters (Wen & Chen 2010). The signal-to-noiseratio (S/N) for the GW signal can also be derived from theinterferometer’s response (see Zhao &Wen 2018 for details of theFisher matrix and S/N). In this work, we choose S/N> 12 as theGW signal’s threshold. Note that in realistic observation andanalysis, various selection effects might induce the bias for thesources’ parameters (Chen et al. 2018; Fishbach et al. 2019;Mortlock et al. 2019; Chen 2020), which is beyond the scope ofthe current paper with Fisher matrix analysis. We leave this to afuture work.

4. EM Counterpart Detection

4.1. γ-Ray Burst Detection

The observations of GW170817/GRB 170817A suggest ithas a Gaussian-shaped jet profile (Zhang & Mészáros 2002;Alexander et al. 2018; Lazzati et al. 2018; Mooley et al. 2018;Troja et al. 2018; Ghirlanda et al. 2019)

⎜ ⎟⎛⎝

⎞⎠

( ) ( )iii

= -E E exp2

7c

0

2

2

for ι� ιw, where E0 is the on-axis equivalent isotropic energy, ιcis the characteristic angle of the core, and ιw is the truncating angleof the jet. Troja et al. (2018) give the constraints on the jet withi = -

+0.057c 0.0230.025, = -

+Elog 52.7310 0 0.751.3 , and i = -

+0.62w 0.370.65. We

assume all BNS mergers in our simulation have a relativistic jetwith this profile. In order to convent this energy profile to theluminosity profile, we assume that the burst duration T90∼ 2 s(Abbott et al. 2017f) and that the spectrum is flat with time duringthe burst, where T90 is defined as the duration of the period inwhich 90% of the burst’s energy is emitted. So, the γ-ray flux ofeach BNS merger can be written as

⎜ ⎟⎛⎝

⎞⎠

( )h

pii

= -ggF

E

D T4exp

2, 8

L c

0

290

2

2

where ηγ is the radiative efficiency and we adopt it as 0.1 forthe bolometric energy flux in the 1–104 keV band. We assumethe spectrum for all BNS-associated GRBs follows the Bandfunction:

⎧

⎨

⎪⎪

⎩⎪⎪

⎛⎝

⎞⎠

⎡⎣⎢

⎤⎦⎥

⎛⎝

⎞⎠

⎡⎣⎢

⎤⎦⎥

( )

( )

( ) ( )

( )( )

( )

a a ba

a ba

a ba

=

-+ -

+

-

+

-

+

a

b a bb a

--

9

N E A

E E

EE

E

E Ee E

E

100 keVexp

2,

2,

100 keV 2 100 keV,

2.

p

p

p p

Here N(E) is in units of photons cm−2 s−1 keV−1 and Ep

corresponds to the peak energy in the ν Fν spectra. We adoptthe photon indices α and β as −1 and −2.3 (Preece et al. 2000)below and above the peak energy Ep, respectively. However,the bolometric isotropic luminosity L and the peak energy forGRB 170817A do not follow the Ep–L relation for short γ-raybursts (Zhang & Wang 2018) and long γ-ray bursts (Yonetokuet al. 2004), which can be written as

⎡⎣

⎤⎦

( )( )

( )= ´+

- -+ -

L E z

10 erg s2.34 10

1

1 keV. 10

p

52 1 1.762.59 5

2.0 0.2

There is a profile proposed by Ioka & Nakamura (2019) wherethe peak energy Ep changes with ι with the followingrelationship within the Gaussian structure jet framework:

( ) ( ) ( )i i i= ´ + -E E 1 , 11p p c,00.4

here Ep,0 is the peak energy that satisfies the Yonetoku relation.This profile makes the GRB 170817A observations consistentwith previous GRB observation.With the peak energy and the Band function we can get the

effective sensitivity limit for various γ-ray detectors. Here weadopt the same setting as Song et al. (2019), the sensitivity forFermi’s Gamma-ray Burst Monitor (Fermi-GBM) is adopted as∼2× 10−7 erg s−1 cm−2 in 50–300 keV (Meegan et al. 2009),the sensitivity for the Gravitational wave high-energy Electro-magnetic Counterpart All-sky Monitor (GECAM) is adopted as∼1× 10−7 erg s−1 cm−2 in 50–300 keV (Zhang et al. 2019),the sensitivity for Swift’s Burst Alert Telescope (BAT) and theSpace Variable Objects Monitor (SVOM)-ECLAIRS isadopted as ∼1.2× 10−8 erg s−1 cm−2 in 15–150 keV (Gehrelset al. 2004; Götz et al. 2014), and the sensitivity for theEinstein Probe (EP) is adopted as ∼3× 10−9 erg s−1 cm−2 in0.5–4 keV (Yuan et al. 2018). Another important factoraffecting the detection ability of the γ-ray detector is the sizeof its field of view (FOV). For Fermi-GBM, its FOV coversabout three-quarters of the whole sky and for GECAM, theFOV is about 4π. For other three detectors, their FOVs aremuch smaller. The factors are ∼1/9 for Swift-BAT, ∼1/5 forSVOM-ECLAIRS (Chu et al. 2016) and ∼1/11 for EPcompared with the whole sky. Note that these γ-ray detectorsmight have no overlapped observational time with future 3GGW detector networks. However, we expect similar or morepowerful detectors in the 3G era. So, for simplification,throughout this paper we only consider these γ-ray detectorsfor illustration.

4.2. Afterglows of γ-Ray Bursts

The measurement of the redshifts of GRBs relies on theidentification of their host galaxies and further opticalobservations. And the afterglow usually plays a crucial rolein the host galaxy identification. In this paper, we use thestandard afterglow models (Sari et al. 1998) to estimate theafterglows on the R band. The spectra and light curve aredetermined by the inclination angle ι, the half-opening angle θj,the total kinetic energy ( )q= -E E1 cosj j 0, the numberdensity of the interstellar medium (ISM) n0, the magnetic fieldenergy fraction òB, the accelerated electron energy fraction òe,the power-law index of shock-accelerated p, the luminositydistance dL, and the time after merger in days tj.

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The Astrophysical Journal, 916:54 (13pp), 2021 July 20 Yu et al.

Adopting the convention notion Q= 10xQ,x, for an adiabaticjet, the Lorentz factor is given by Sari et al. (1998)

( ) ( ) ( )g q= + --

- -t z E n t8.9 1 . 12j j d3 8

,511 8

01 8

, 11 4

,3 8

For an on-axis observer, a phenomenon named jet break isexpected when γ drops below θj and the jet’s material begin tospread sideways (Sari et al. 1998; Mészáros & Martin 1999).At the jet-break time, there will be a break in the afterglow’slight curve. From Equation (12), the jet-break time tj is givenby

( ) ( )q= + --t z E n0.82 1 days. 13j j j,51

1 30

1 3, 1

2

At the jet-break time, there are two cases for flux density, Fν, j,the slow-cooling case (νm< ν< νc) and the fast-cooling case(νc< ν), where νm is the typical synchrotron frequency ofthe accelerated electrons with the minimum Lorentz and νc is thecooling frequency. In the slow-cooling case, Fν, j∝ ν−(p −1)/2, weadopt p= 2.2 in this paper (the same as Nakar et al. 2002 and Zouet al. 2007). While in the fast-cooling case, Fν, j∝ ν−p/2 (Sariet al. 1998).

For an on-axis observer, the light curve is divided into twopower-law segments by the jet-break time (Sari et al. 1998).We denote the temporal decay index of the flux density Fν,0(t)in the time as α1 and α2, which represent the time before andafter tj, respectively. The index α1 is −(2–3p)/4 in the fast-cooling case and −3(1−p)/4 in the slow-cooling case. Whent> tj, the on-axis observer can only observe ∼q gj

2 2 of the fluxdensity in the isotropic fireball case. Since γ(t)∝ t−3/8, we haveα2= α1+ 3/4.

For a point source moving at angle ι, the observed flux is(Granot et al. 2002)

⎛⎝

⎞⎠

( )p

nn

=¢

¢n

n¢FL

d4, 14

A2

where dA is the angular distance, and ¢n ¢L and n¢ are the jetcomoving-frame spectral luminosity and frequency. For an off-axis observer (t, ν) at angle ι and an on-axis observer (t0, ν0) atangle 0, there are ( ) ( )n n b b i» = - - ºt t a1 1 cos0 0 ,

where b g= -1 1 2 . And we obtain

( ) ( ) ( )i =n nF t a F at, 0, . 15a3

Here we consider both the fast- and slow-cooling cases, whichare νc< ν and νm< ν< νc, respectively.

5. Multimessenger Observation of GWs and GRBs

5.1. Detection Rates

Since we found z= 3 exceeded the detection limit of theGRB detectors in the simulations, we set it as the upper redshift

limit and sampled 107 BNS mergers. However, this upper limitis too high for the 2G interferometers, and the simulationbecame very inefficient with them. The upper detection limitwith LHVIK A+ is about z∼ 0.2, so for the 2G interferom-eters, we resample them with zä (0, 0.3].There are ∼3.4× 105 BNS samples detected by ET and

5.1× 106 detected by CE. For the 2G GW detector networksLHV, LHVIK, LHV A+ and LHVIK A+, the numbers are2.7× 104, 5.7× 104, 1.6× 105, and 3.3× 105, respectively,with the samples’ z ä (0, 0.3]. The observational ability of theA+-type network will improve several times even with fewerdetectors. About 10%–12% of the GW signal from BNSmergers observed by 2G GW detector networks could betriggered by the γ-ray detector. For 3G detectors, the fractionsare much lower because more events have a high redshift.We convert these numbers into rates per year, after

multiplying the total rate of BNS mergers. IntegratingEquation (1), in z ä (0, 0.3], the total rate is 1159–11,736yr−1 and in z ä (0− 3], the total rate is 6.9× 104–7.0× 105

yr−1. Due to the limited observation area of the γ-ray detectors,not all GRB events whose flux reaches the threshold countcould be observed. So the multimessenger observation ratesrequire a discount, FOV/Ω0, where Ω0 is the solid angle ofwhole sky. Table 1 lists the observation rates per year with thisdiscount. For LHV, the rate is 0.042–0.425 per year with Swift-BAT and 0.072–0.731 per year with SVOM-ECLAIRS. ForGECAM and Fermi-GBM, the rates are a few times larger dueto their much larger observation areas. The result of EP isslightly worse than Swift-BAT due to its smaller observationarea, although it has better sensitivity. After adding two LIGO-type detectors, the rates of LHVIK are about twice as large asLHV. The future upgrade of aLIGO, A+, would increase thesenumbers by about five times. For the 3G interferometer ET,there will be about a 100∼ 800 multimessenger observationrate with different γ-ray detectors if we adopt 810 Gpc−3 yr−1

as the local merger rate. The rates are much larger even withonly one single interferometer. For another 3G interferometer,CE, the rates are a few times larger compared to ET, since it hasa further observation range. It is worth noting that we assumethat the GW and GRB detections are independent. However, inthe real situation, with a triggered GRB, one can search weakerGW signals. So the numbers in Table 1 might be slightlysmaller compared with the actual situation.We select the BNS samples that can be triggered by both

GW interferometers and γ-ray detectors. We choose theGECAM result as a representative one and show its redshiftdistribution in Figure 1. The redshift limits of LHV and LHVIKare around z∼ 0.1 and the detection limit of LHVIK A+ canreach ∼0.2 due to more sensitive detection capabilities. For the3G interferometers ET and CE, the redshift limits are z∼ 1.0and z∼ 2.5, respectively.

Table 1Multimessenger Observation Rates (in Number per Year) for BNS Mergers with Different γ-ray Detectors and GW Detectors

Swift-BAT SVOM-ECLAIRS GECAM Fermi-GBM EP

LHV 0.042-0.425 0.072-0.731 0.278-2.820 0.198-2.001 0.029-0.297LHVIK 0.084-0.856 0.146-1.474 0.553-5.598 0.394-3.985 0.058-0.593LHV A+ 0.217-2.200 0.374-3.789 1.370-13.870 0.962-9.741 0.148-1.504LHVIK A+ 0.445-4.505 0.766-7.757 2.743-27.769 1.907-19.305 0.301-3.046ET 17.0-172.0 29.2-296.1 80.6-815.6 49.9-504.9 10.7-108.5CE 98.1-993.0 168.9-1710.0 342.1-3463.5 188.4-1907.4 58.1-587.9

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As an important method for measuring redshift information,we also calculate the magnitude of GRB afterglows with themethod mentioned in Section 4.2. We adopt n0= 1 cm−3,p= 2.2, òe= 0.1, òB= 0.01, and p= 2.2, where they are thesame as Nakar et al. (2002) and Zou et al. (2007). For the half-opening angle, we consider three cases, θj= 5°, 10°, 15°. Foroff-axis samples, we calculate the afterglows’ maximum R-band magnitude MR with different opening angles. However,for on-axis GRB samples, their afterglows’ power law decayswith time and we cannot get their maximumMR from their lightcurve, as shown in Figure 2, so we denote the R-band flux inthe jet-break time as MR in this case, as a comparison with theoff-axis samples.

Using the GECAM case and θj= 10° as an example, we showthe distributions of the redshift, inclination angle, and the flux ofafterglows for standard sirens in Figure 3. From this figure, wecan intuitively see the ι limit of multimessenger observation. Atvery low redshift, there is a BNS sample with ι∼ 22°. Thissample’s inclination angle and redshift are very similar toGW170817. For samples with z= 1, the ι limit becomes about15°. Zhao et al. (2011) and Sathyaprakash et al. (2010) have

roughly estimated the distribution of GW sources to an order ofmagnitude, which is about 1000 yr−1 with an inclination angleι< 20°. These predictions are based on a BNS rate of ∼several ×105 per year within the horizon of ET, and ∼10−3 of them willhave GRBs toward us. These estimates meet well with our result.We use the improved measurement of the BNS merger rate fromGW observations and give stronger constraints on the multi-messenger event rates. In the next subsection, we will use theirmethods, combined with the results of our simulations, to showthe potential for the determination of dark energy by the 3G GWinterferometers ET and CE. The design specification of theLegacy Survey of Space and Time (LSST) for 5σ depths for pointsources in the r band is ∼24.7 (Ivezić et al. 2019), whichcorresponds to point sources and fiducial zenith observations. Dueto the small inclination angles, all standard sirens have MR

brighter LSST r band depth. For the 2.5 m Wide Field SurveyTelescope (WFST), the single-visit depth with a 30 s exposuretime is r∼ 22.8. A small fraction of samples in the case of SVOMare fainter than r∼ 22.8. If we consider a 300 s exposure time, thesingle-visit depth of WFST will reach r∼ 24.1 and all sampleswill have an MR brighter than this. In the other two choices of θj,their afterglows all reach the design depth of LSST and WFST.Therefore, we have to assume all multimessenger BNS events inour simulation have a detectable afterglow and that we canmeasure their redshift from their afterglows.In Figure 4, we set the number density of the ISM as

n0= 0.1 cm−3 with n0= 0.01 cm−3 as a comparison. A GRBsurrounded by a thinner ISM usually has a weaker afterglow. Inthe case of n0= 0.01 cm−3, a small fraction of afterglows arefainter than the design depth of LSST and WFST. In that case,the measurement of the redshift of GRBs seems difficult andfurther discussion is needed.

5.2. Determination of Dark Energy

In the last two decades, the accelerated expansion of theuniverse has been confirmed by a series of observations(Eisenstein et al. 2005; Percival et al. 2007, 2010; Kowalskiet al. 2008; Hicken et al. 2009; Kilbinger et al. 2009; Komatsuet al. 2009; Schrabback et al. 2010). Various models of darkenergy have been proposed to explain it (see Copeland et al. 2006for a review). In order to understand the physical properties ofdark energy, it is important to measure its equation of state (EOS).In Sathyaprakash et al. (2010) and Zhao et al. (2011), a new

Figure 1. The number distribution of samples that can be triggered by both GW interferometers and GECAM.

Figure 2. The light curve of on-axis and off-axis afterglows. The red and blackcurves show the on-axis afterglow with ι = 5° and the off-axis afterglow withι = 15°, respectively. Other parameters in these two cases are same, withE0 = 1052.73 erg, z = 1, and θj = 10°.

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method of measuring the dark energy EOS using GW standardsirens was proposed. Zhao & Wen (2018) compared the results ofGW detector networks with 1000 face-on BNS sources, using the

baryon acoustic oscillations (BAO) and Type Ia supernovae (SNeIa) methods, and found that using the 3G GW detector networks,the constraints of dark energy parameters are similar with the SNe

Figure 3. The distributions of ι, the redshift of BNS samples, and their afterglow fluxes, which can be triggered by GW detectors and GECAM. The top and middlerows are the 2G GW detector networks LHV, LHVIK, LHV A+, and LHVIK A+, and show the results of 107 samples with z ä (0, 0.3]. The bottom row shows ETand CE and the results of 107 samples with z ä (0, 3]. The color bars show the R-band magnitude of afterglows with θj = 10°.

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Ia and BAO methods. In this subsection, we will discuss theseconstraints with a quantitatively distributed BNS sample.

Similar to Zhao et al. (2011), we adopt a phenomenologicalform for the EOS parameter w as a function of redshift(Chevallier & Polarski 2001)

( ) ( )r

º = ++

w zp

w wz

z1, 16a

de

de0

where pde is the pressure of dark energy and ρde is the energydensity. The parameter w0 represents the present EOS and wa

represents the evolution with redshift. In the ΛCDM model, therelation of dL−z is determined by (H0, Ωm, Ωk, w0, wa) together(Weinberg 2008). So if the dL−z relation can be measured froma series of multimessenger observations, the set of these fiveparameters would have a chance to be constrained. However,due to the strong degeneracy between the backgroundparameters (H0, Ωm, Ωk) and the dark energy parameter (w0,wa), the constraints of the full parameter set seems unrealistic(Zhao et al. 2011). The same problem also happens in othermethods for dark energy detection (e.g., the SNe Ia and BAOmethods) (Albrecht et al. 2006; Zhao et al. 2011). A generalway to break this degeneracy is to combine the result with

the cosmic microwave background (CMB) data, which aresensitive to the background parameters (H0, Ωm, Ωk), andprovide the necessary complement to the GW data. It has alsobeen discovered in Zhao et al. (2011) that taking the PlanckCMB observation as a prior is nearly equivalent to treating theparameters (H0, Ωm, Ωk) as known in the data analysis. Thus,similar to Zhao & Wen (2018), Zhao et al. (2018), and Yanet al. (2020), in this article we use the GW data to constrain thedark energy parameters (w0, wa) only.To estimate the errors of (w0, wa), we consider a Fisher

matrix Fij for a collection of N BNS mergers that follows oursample distribution, which is given by Zhao et al. (2011)

( ) ( )( ( ˆ ))

( )å d g=

¶¶

¶¶=

Fd z

p

d z

p d d z

ln ln 1

,, 17ij

k

NL k

i

L k

j L L k k12

where i and j run from 1 to 2, denoting the free parameters w0

and wa, and g represents the angle (α, δ, ι, ψ). The distanceerror δ dL/dL has two components: the instrument error ΔdL/dL,and the error caused by the effects of weak lensing. We denotethis error as Dd dL L and assume it satisfies D =d d z0.05L L

(Sathyaprakash et al. 2010; Zhao et al. 2011; Zhao & Wen 2018;

Figure 4. Same a Figure 3, but with a different number density for the ISM. The left panels show the results with n0 = 0.1 cm−3 and the right panels show the resultswith n0 = 0.01 cm−3.

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Wang & Wang 2019). So the total distance error could be

estimated from ( ) ( ˜ )d = D + Dd d d d d dL L L L L L2 2 . Similar

to Gupta et al. (2019) and Zhao & Santos (2019), in order toevaluate ΔdL for each event, we assume the parameters (α, δ, ι)are well constrained by γ-ray observation and consider a Fishermatrix of six parameters, ( )y h fM t d, , , , , logc c L . This scenariois possible as we have already seen in the case of GW170817. Thesky position of GW170817 was constrained by finding the hostgalaxy NGC 4993 (Abbott et al. 2017f), whereas the inclinationangle was constrained from the X-ray and ultraviolet observations(Evans et al. 2017). In Figure 5 we use the case of GECAM as arepresentative case and show the distribution of sample instrumenterrors with two single 3G GW detectors, ET and CE. There is anobvious cutoff around D ~dlog 0.08L , since the instrumenterrors have a strong correlation with S/N and we choose S/N=12 as the threshold.

The uncertainties of cosmological parameters are given by( )D = -w F0

111 and ( )D = -w Fa

122. If the statistical errors are

dominant, the uncertainties of w0 and wa are proportional toN1 . In Table 2, we list the 68.3% (1σ) uncertainties of w0

and wa marginalized over other parameters with one year’s

multimessenger observation. Also, to estimate the goodness-of-parameter constraints, we list the figure of merit (FoM)(Albrecht et al. 2006) in Table 2, which is written as

[ ( )] ( )= -C w wFoM Det , , 18a01 2

where C(w0, wa) is the covariance matrix of w0 and wa. Withthe large observation area, the results of GECAM and Fermi-GBM are much better than other γ-ray detectors. For GECAMand CE, we obtain Δw0= 0.016–0.051, Δwa= 0.119–0.380,and FoM= 171–1732; for Fermi-GBM, the results areΔw0= 0.020–0.065, Δwa= 0.157–0.499, and FoM= 100–1009. These results are comparable with the detection abilitiesof the SNe Ia and BAO methods in the long-term (Stage IV)projects, which are FoM∼ 300 for the SNe Ia method andFoM∼ 100 for the BAO method (Albrecht et al. 2006; Zhaoet al. 2011). We can also see that the ET has a poor ability toconstrain w0 and wa due to its low observation limit. InFigure 6, the corresponding two-dimensional 1σ uncertaintycontours of these multimessenger observations with ET and CEare plotted, and in each panel, we compare the results with fourdifferent γ-ray detectors. For the GECAM case, the constraints

Figure 5. The distribution of the samples’ ΔdL and redshift, which can be triggered by 3G interferometers and GECAM. The left and right panels are the results withET and CE, respectively. The color bar shows the samples’ inclination angle.

Table 2The 1σ Constraints of Δw0 and Δwa with One Year’s Joint Observations with Different GW Interferometers and γ-Ray Detectors

Swift-BATSVOM-ECLAIRS GECAM Fermi-GBM EP

Δw0 ET 0.129-0.412 0.099-0.314 0.057-0.181 0.070-0.222 0.160-0.510CE 0.034-0.107 0.026-0.082 0.016-0.051 0.020-0.065 0.043-0.136

CEET 0.032-0.104 0.025-0.079 0.015-0.049 0.020-0.062 0.041-0.131CE2ET 0.028-0.090 0.022-0.069 0.013-0.042 0.017-0.054 0.036-0.114

Δwa ET 1.173-3.734 0.894-2.846 0.531-1.690 0.664-2.114 1.462-4.652CE 0.237-0.754 0.181-0.575 0.119-0.380 0.157-0.499 0.303-0.966

CEET 0.228-0.727 0.174-0.554 0.115-0.366 0.151-0.481 0.293-0.931CE2ET 0.198-0.630 0.151-0.480 0.100-0.318 0.132-0.419 0.254-0.808

FoM ET 2.4-24.4 4.1-42.0 11.7-118.8 7.4-75.0 1.5-15.6CE 42.5-430.5 73.2-741.3 171.1-1732.0 99.7-1009.1 26.0-263.0

CEET 45.7-462.9 78.7-797.1 183.7-1860.3 107.0-1083.8 27.2-282.7CE2ET 60.4-611.4 104.0-1052.9 241.7-2447.6 140.6-1423.9 36.8-372.9

Note. The FoM in each case is also shown.

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of dark energy parameters are significantly more stringent thanthe other three γ-ray detectors due to its large observation areas.The results of CEET and CE2ET are not shown here becausetheir shapes are very similar to the results with CE. Our resultsroughly match the results of Zhao & Wen (2018), but using amuch more quantitative BNS sample.

6. Effect of Alternative Models for EM Counterparts

In previous discussions, we assume all BNS mergers have aGaussian-shaped jet profile in the form of Equation (7). In Abbottet al. (2017d), other two jet profiles are used to explain the

observed properties of GRB 170817A, uniform top-hat jets and a‘cocoon’ emission model (Lazzati et al. 2017). However, thecompact radio emission observations favor the structured-jet profile(Ghirlanda et al. 2019). Therefore, we will not consider these twomodels here. Meanwhile, to combine GRB 170817A and otherSGRBs, Tan & Yu (2018) produced a two-Gaussian profile,

⎜ ⎟ ⎜ ⎟⎡

⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤

⎦⎥( ) ( )i

iq

iq

= - + -L L exp2

exp2

, 19iso on

2

in2

2

out2

where Lon, θin, θout, and are four free parameters, with θout=10θin and 1. We choose the θin= 2° model as example. The

Figure 6. The two-dimensional uncertainty contours of the dark energy parameters w0 and wa for one year’s multimessenger observation of ET, CE, and five γ-raydetectors with a 68.3% (1σ) confidence interval. The left panels are the results with RBNSmergers, 0 = 80 Gpc−3 yr−1 and the right panels are the results with RBNSmergers,

0 = 810 Gpc−3 yr−1.

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multimessenger observation rates with this model are shown inTable 3. For the 2G GW network, the rates with Swift-BAT andSVOM are a few times larger than the case of the Gaussian jetprofile, since more GRBs in this model have a peak energy inSwift-BAT and SVOM’s flux band. Other rates are roughly same.For 3G interferometers, the rates are about half compared withTable 1, since the two-Gaussian profile is narrower around ι= 0°.

In Section 2, we used the log-normal distribution time-delaymodel. In Sun et al. (2015), the authors investigated it withanother two models, i.e., the Gaussian delay model (Virgiliet al. 2011) and power-law delay model (Sun et al. 2015). Thepower-law delay model is not supported by the SGRB data(Virgili et al. 2011). Therefore, we concentrate on thedifference between the Gaussian and log-normal delay modelshere. The f (z) mentioned in Equation (5) with these two modelsare shown in Figure 7 and the observation rates with theGaussian delay model are shown in Table 4. While z< 0.5, thedistributions are almost same. So for 2G GW networks, themultimessenger observation rates are also almost the same.However, for high redshift, the Gaussian delay model predicts alarger number of BNS samples, so the rates in the case of 3Ginterferometers, especially for CE, are larger than the log-normal delay model.

Another issue that needs to be discussed is GW190425(Abbott et al. 2020a), a compact binary coalescence with totalmass ∼3.4 Me that does not fit in with the BNS massdistribution assumed in our work. In order to cover theGW190425-type systems, we follow the same setting as Abbottet al. (2021b), assuming a uniform mass distribution between1 Me and 2.5 Me. In Figure 8, we compare the redshiftdistributions of BNS samples, which can both be triggered byGW interferometers and GECAM, with two BNS massdistributions. Due to the heavy BNS mergers considered, for2G GW interferometers and ET, more and further BNS mergerscan be observed. However, for CE, since the maximumobservation distance exceeds that of the γ-ray detectors and thedominant constraints come from GRB observations, the resultsof the two mass distributions are very similar. In Table 5, themerger rates with other γ-ray detectors and GW interferometersare listed.

In this section, we discuss three different models as acomparison and predict the multimessenger observation rateswith each model. Following the method mentioned inSection 5.2, we calculate the constraints on w0 and wa throughmultimessenger observations with these three models. InFigure 9, we compare the corresponding two-dimensional 1σuncertainty contours of Δw0 and Δwa with four models. Thecontours with a two-Gaussian jet profile are larger than theothers due to smaller observation rates. In the case of ET andflat mass distribution, the constraints are better. And in othercases, the results are roughly similar.

7. Conclusions

The detection of GW170817 opens a door to multimessengerobservation. From the GW waveforms of these events we couldmeasure the sources’ luminosity distances independently,without the comic distance ladder. If the redshifts of thesources could also be measured from other methods, these kindof GW events could be treated as standard sirens to constrain

Table 3The Same as Table 1, but with the Jet Profile from Tan & Yu (2018)

Swift-BAT SVOM-ECLAIRS GECAM Fermi-GBM EP

LHV 0.185-1.873 0.318-3.226 0.384-3.884 0.183-1.850 0.055-0.561LHVIK 0.315-3.194 0.543-5.500 0.580-5.868 0.290-2.936 0.075-0.760LHV A+ 0.529-5.356 0.911-9.222 1.007-10.198 0.563-5.702 0.117-1.186LHVIK A+ 0.791-8.011 1.362-13.795 1.680-17.015 1.018-10.305 0.187-1.890ET 8.8-89.3 15.2-153.7 44.3-448.3 29.2-259.2 5.3-54.0CE 45.1-457.0 77.7-786.9 202.1-2046.3 122.3-1228.7 29.5-298.4

Table 4The Same as Table 1, but with a Gaussian Time-delay Model

Swift-BAT SVOM-ECLAIRS GECAM Fermi-GBM EP

LHV 0.041-0.411 0.070-0.708 0.265-2.681 0.189-1.916 0.028-0.287LHVIK 0.082-0.829 0.141-1.427 0.528-5.347 0.376-3.810 0.057-0.577LHV A+ 0.208-2.112 0.359-3.636 1.308-13.239 0.922-9.332 0.142-1.444LHVIK A+ 0.432-4.373 0.744-7.531 2.660-26.938 1.856-18.797 0.293-2.970ET 18.3-185.7 31.5-319.8 85.0-860.7 53.0-536.2 11.5-116.4CE 130.2-1318.2 224.2-2268.9 423.4-4287.4 227.4-2303.0 75.9-768.9

Figure 7. The redshift distribution of BNS merger samples. The green and bluelines represent the log-normal and Gaussian time-delay models, respectively.

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cosmological parameters, such as the Hubble constant, darkenergy, and so on. A direct way to measure the sources’redshifts is to find their host galaxies by the observation of theirEM counterparts, such as GW170817.

In this paper, we discuss the multimessenger observations ofBNS merger GW and GRB signals. We find that the 2.5Gnetwork LHVIK A+ expect to detect more than threemultimessenger signals per year with different γ-ray detectors,if the local merger rate is 810 Gpc−3 yr−1. And the upperredshift limit of these BNS samples is about 0.2. For LHVIK orLHV A+, the numbers are roughly several times smaller thanLHVIK A+. Most of the BNS mergers that could be triggeredby both GW interferometers and γ-ray detectors have ι� 22°,because of the limitation of the Gaussian jet profile. Only a fewsamples with a very low redshift and inclination angles largerthan 22° can be triggered by γ-ray detectors. For the 3Ginterferometers ET and CE, the number of multimessengerobservations are tens to hundreds of times larger compared toLHVIK A+. The upper redshift limit for ET is ∼1 and ∼2.5 forCE. We also find for the samples with z larger than 1.0 that therequirements on the inclination angles become stricter,becoming less than 15°. The results for 3G GW detectors arein line with the magnitude estimates of BNS event rates inSathyaprakash et al. (2010) and Zhao et al. (2011). BNS mergerinclinations that can be observed by both GW and γ-raydetectors are almost all less than 22°, which also matches theestimate of these works.

Due to the high-redshift upper limit of the multimessengerobservations with the 3G GW detectors, we discuss theimplication for constraining dark energy parameters with BNSmergers. One year’s observations by ET can constrain the EOS

of cosmic dark energy with accuracies Δw0∼ 0.06–0.18 andΔwa∼ 0.5–1.7 with GECAM. For CE, the results are muchbetter, being Δw0∼ 0.02–0.05 and Δwa∼ 0.1–0.4. We alsodiscuss the cases of 3G GW detector networks, CEET andCE2ET. The results of CEET are quite close to CE and those ofCE2ET have a ∼10%–20% progress. Therefore, the usage of3G interferometers will provide an important method to help usunderstand the high-redshift universe.Note that Belgacem et al. (2019) also estimated the multi-

messenger observation rates, where they used a Gaussianstructured-jet profile with specific parameters and selectedobserved GRBs through their peak flux. In this work, we makea much more comprehensive and integrated analysis of this issue.We discuss a more general structured-jet profile and calculate thespectrum of each GRB sample from the Band function, integratingit in the band of different GRB detectors, instead of peak flux.Because of this, our predictions are several times larger than theresults in Belgacem et al. (2019). In order to study theuncertainties in BNS samples and GRB jets, we discuss severalalternative models. The two-Gaussian jet profile, the Gaussiantime-delay model and a flat BNS mass distribution are discussedas a supplement. The two-Gaussian jet profile model brings anarrower jet and a softer spectrum. The results of 2G GWnetworks and Swift-BAT/SVOM are ∼4 times larger and therates with 3G GW detectors are about half as large. The Gaussiantime-delay model predicts a larger BNS merger rate in highredshift and the results of 3G interferometers are slightly larger.The flat BNS mass distribution takes heavy BNS mergers intoaccount, and the rates with ET have a∼50% increase. But for CE,due to the constraints from GRB observations, the rates are almostsame. We also estimate the magnitudes of optical afterglows and

Figure 8. The comparison of two distributions of BNS mass. Model A represents the normal distribution mentioned in Section 2 and model B represents the flat massdistribution mentioned in Abbott et al. (2021b).

Table 5The Same as Table 1, but with a Flat Mass Distribution between 1 Me and 2.5Me

Swift-BAT SVOM-ECLAIRS GECAM Fermi-GBM EP

LHV 0.079-0.800 0.136-1.378 0.506-5.129 0.360-3.648 0.054-0.551LHVIK 0.163-1.655 0.281-2.850 1.033-10.461 0.730-7.392 0.112-1.132LHV A+ 0.421-4.264 0.725-7.343 2.563-25.944 1.780-18.021 0.284-2.878LHVIK A+ 0.881-8.917 1.516-15.354 5.260-53.257 3.612-36.572 0.590-5.978ET 27.3-276.2 47.0-475.7 119.4-1209.0 71.6-724.6 16.9-171.4CE 90.6-917.4 156.0-1579.7 318.2-3221.7 176.0-1781.6 53.8-545.0

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find that if the number density of the ISM is 1 cm−3, due to thelow inclination angle, all multimessenger events have opticalafterglows that reach the single-visit depth of LSST and/or WFST.

This work is supported by the National Natural ScienceFoundation of China under grant Nos. 11903030, 11773028,11603020, 11633001, and 11653002, the Strategic PriorityProgram of the Chinese Academy of Sciences (grant No. XDB23040100), and the Fundamental Research Funds for theCentral Universities under grant Nos. WK2030000036 andWK3440000004.

ORCID iDs

Jiming Yu https://orcid.org/0000-0001-6319-0866He Gao https://orcid.org/0000-0002-3100-6558Youjun Lu https://orcid.org/0000-0002-1310-4664Wen Zhao https://orcid.org/0000-0002-1330-2329

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Figure 9. The two-dimensional 1σ uncertainty contours of the dark energy parameters w0 and wa with different BNS samples. Case A represents the results of delaytime with a log-normal distribution and a Gaussian BNS mass distribution, case B uses a flat mass distribution, case C uses a Gaussian time-delay model, and case Duses a two-Gaussian jet profile.

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