Draft version May 21, 2019Preprint typeset using LATEX style AASTeX6 v. 1.0

MEASURING THE DELAY TIME DISTRIBUTION OF BINARY NEUTRON STARS. II. USING THEREDSHIFT DISTRIBUTION FROM THIRD-GENERATION GRAVITATIONAL WAVE DETECTORS

NETWORK

Mohammadtaher Safarzadeh1,2, Edo Berger1, Ken K. Y. Ng3,4, Hsin-Yu Chen5, Salvatore Vitale3,4, ChrisWhittle3,4, Evan Scannapieco2

1Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge MA 02138, USA msafarzadeh@cfa.harvard.edu2School of Earth and Space Exploration, Arizona State University Tempe AZ 85287, USA3LIGO, Massachusetts Institute of Technology, 185 Albany Street, Cambridge MA 02139, USA4Department of Physics and Kavli Institute for Astrophysics and Space Research, MIT, 77 Massachusetts Avenue, Cambridge MA 02139,

USA5Black Hole Initiative, Harvard University, 20 Garden Street, Cambridge MA 02138, USA

ABSTRACT

We investigate the ability of current and third-generation gravitational wave (GW) detectors to deter-mine the delay time distribution (DTD) of binary neutron stars (BNS) through a direct measurementof the BNS merger rate as a function of redshift. We assume that the DTD follows a power lawdistribution with a slope Γ and a minimum merger time tmin, and also allow the overall BNS formationefficiency per unit stellar mass to vary. By convolving the DTD and mass efficiency with the cosmicstar formation history, and then with the GW detector capabilities, we explore two relevant regimes.First, for the current generation of GW detectors, which are only sensitive to the local universe, butcan lead to precise redshift determinations via the identification of electromagnetic counterparts andhost galaxies, we show that the DTD parameters are strongly degenerate with the unknown massefficiency and therefore cannot be determined uniquely. Second, for third-generation detectors such asEinstein Telescope (ET) and Cosmic Explorer (CE), which will detect BNS mergers at cosmologicaldistances, but with a redshift uncertainty inherent to GW-only detections (δ(z)/z ≈ 0.1z), we showthat the DTD and mass efficiency can be well-constrained to better than 10% with a year of obser-vations. This long-term approach to determining the DTD through a direct mapping of the BNSmerger redshift distribution will be supplemented by more near term studies of the DTD through theproperties of BNS merger host galaxies at z ≈ 0 (Safarzadeh & Berger 2019).

1. INTRODUCTION

The joint gravitational wave (GW) and electromag-netic (EM) detections of the binary neutron star (BNS)merger, GW170817 (Abbott et al. 2017b), marked thedawn of multi-messenger astronomy. As the currentgeneration of GW detectors increases in sensitivity andnumber, the local rate of BNS mergers will soon be de-termined accurately for the first time, providing initialinsight into the formation channels of these binaries.Currently, the BNS merger rate is weakly constrainedby the single detection of GW170817 (1540+3200

−1220 Gpc−3

yr−1; Abbott et al . 2017), by the small known sample ofGalactic BNS systems (21+28

−14 Myr−1; Kim et al. 2015),and by the beaming-corrected rate of short gamma-raybursts (270+1580

−180 Gpc−3 yr−1; Fong et al. 2015). Theserates are in broad agreement, but the uncertainties fromall methods span at least two orders of magnitude.

Still, even when the local BNS merger rate is well de-termined, the more fundamental distribution of mergerdelay times may not be. The delay time distribution(DTD) encodes the time span between the formation of

the BNS system (or alternatively the time since the for-mation of the parent stars) until the two neutron starsmerge through the emission of gravitational waves. TheDTD therefore provides fundamental insight into theevolutionary processes that govern the initial separationof the binaries, including poorly-understood effects suchas common envelope evolution.

The DTD is usually parametrized as a power law dis-tribution above some minimum merger timescale, tmin,based on the following arguments: after the BNS for-mation, the binary’s orbit decays due to the emissionof gravitational waves on a timescale that depends onthe initial semi-major axis (a) as t ∝ a4. There-fore the resulting distribution of the merger times de-pends on the distribution of initial semi-major axes,dN/da ∝ a−β. The initial semi-major axis distributionof the O/B stellar progenitors is assumed to follow apower law dN/da ∝ a−1. If the binary experiences acommon envelope phase, then the distribution becomessteeper. Therefore, the expected merger times followdN/dtmerge ∝ t−β/4−3/4, where we define Γ ≡ −β/4 − 3/4(Belczynski et al. 2018).

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Insight on the form of the DTD has been gained fromstudies of the small population of Galactic BNS sys-tems (Vigna-Gomez et al. 2018), from the propertiesof SGRB host galaxies (Zheng & Ramirez-Ruiz 2007;O’Shaughnessy et al. 2008; Leibler & Berger 2010; Fonget al. 2013; Behroozi et al. 2014; Berger 2014), and fromarguments related to r-process enrichment (Matteucciet al. 2014; Komiya et al. 2014; van de Voort et al. 2015;Shen et al. 2015; Cote et al. 2018; Hotokezaka et al. 2018;Safarzadeh et al. 2018, 2019). These results point to theneed for a fast merging channel if BNSs are assumedto be the primary source of r-process enrichment in theuniverse, which suggest that tmin may be rather small,. 0.1 Gyr or the slope of the DTD is steep. Populationsynthesis models have also made various predictions forthe values of Γ and tmin, but those are dependent on un-certain binary evolution processes (Dominik et al. 2012).We stress that the local BNS merger rate in itself can-not fully characterize the DTD since it also depends onan additional unknown parameter, the efficiency of BNSformation per unit stellar mass, λ.

In a recent paper, Safarzadeh & Berger (2019) (here-after, Paper I) showed that the mass distribution of BNSmerger host galaxies at z ≈ 0 can provide insight on Γand tmin with a sample size of O(102 − 103). This isbased on the fact that, on average, galaxy star forma-tion histories (SFH) depend on their mass, and hencethe convolution of the DTD and SFH leads to a specificprediction about the mass function of BNS merger hostgalaxies. Such an observational approach to determin-ing the DTD is only feasible in the local universe due tothe required detection of EM counterparts that will inturn lead to the identification of the host galaxies. It isanticipated that Advanced LIGO/Virgo, joined by KA-GRA1 and IndIGO2, can produce the required samplesize within the next two decades.

Here, we instead explore how the DTD can be deter-mined by directly observing the redshift distribution ofBNS mergers well beyond the local universe. Mappingthe rate of BNS mergers as a function of redshift canbreak the degeneracy between the shape of the DTD (Γand tmin) and the BNS mass efficiency (λ) when compar-ing to the cosmic star formation history. This approachrequires an order of magnitude increase in GW detec-tor sensitivity to detect BNS mergers at cosmologicaldistances. Such an improvement is expected for third-generation ground-based observatories such as EinsteinTelescope3 (ET; Punturo et al. 2010) and Cosmic Ex-plorer4 (CE; Abbott et al. 2017a). However, at thesedistances, it is unlikely that EM counterparts will bedetected for the majority of events, and therefore the

1 https://gwcenter.icrr.u-tokyo.ac.jp/en

2 http://www.gw-indigo.org/tiki-index.php

3 http://www.et-gw.eu

4 http://www.cosmicexplorer.org

distance (redshift) information will rely directly on theGW signal itself. We explore how the inherent distance-inclination degeneracy affects the ability to determinethe DTD.

The structure of the paper is as follows: In §2 wedelineate the method of estimating the observed red-shift distribution of BNS mergers as a function of DTDand mass efficiency, for different GW interferometer net-works; in §3 we show the results of DTD determinationfor existing GW detectors, which are only sensitive tothe local universe; in §4 we expand our analysis to a fu-ture network of ET and CE, including a determinationof the expected redshift uncertainties, and show the re-sulting constraints on the DTD and mass efficiency. Wediscuss some caveats and summarize the key results in§5. We adopt the Planck 2015 cosmological parameters(Planck Collaboration et al. 2016) where ΩM = 0.308,ΩΛ = 0.692, Ωb = 0.048 are total matter, vacuum, andbaryonic densities, in units of the critical density, ρc,H0 = 67.8 km s−1 Mpc−1 is the Hubble constant, andσ8 = 0.82 is the variance of linear fluctuations on the 8h−1 Mpc scale.

2. METHOD

The BNS merger rate as a function of redshift is aconvolution of the DTD with the cosmic star formationrate density:

Ûn(z) =∫ zb=z

zb=10λ

dPm

dt(t − tb − tmin)ψ(zb)

dtdz(zb)dzb, (1)

where dt/dz = −[(1 + z)E(z)H0]−1, and E(z) =√Ωm,0(1 + z)3 +Ωk,0(1 + z)2 +ΩΛ(z). Here, λ is the cur-

rently unknown BNS mass efficiency (assumed not toevolve5 with redshift) used as a free parameter that wetry to recover alongside the parameters governing theDTD; tb is the time corresponding to the redshift zb;dPm/dt is the DTD, parametrized to follow a power lawdistribution (∝ tΓ) with a minimum delay time, tmin thatrefers to the time since birth of the ZAMS stars andnot when the BNS system formed. Therefore, tmin cor-responds to the sum of the nuclear lifetime of the lowestmass component of the binary system and the minimalgravitational delay that is induced by the existence of aminimal separation between the two newly born neutronstars. We also impose a maximum delay time of 10 Gyrfor our fiducial case, although this does not affect ourresults, although we note that more than half of the ob-served BNS systems in the MW half merger times morethan 10 Gyr (Pol et al. 2019). We adopt the cosmicstar formation rate density6 from Madau & Dickinson

5 Although the DTD for binary black holes is likely highly de-pendent on the metallicity, the DTD for BNS systems has beenargued to be at most weakly dependent on metallicity (Dominiket al. 2012).

6 We neglect the uncertainties in the cosmic SFRD since theGW source redshift uncertainty (see Appendix B) dominates theoverall error budget at cosmological redshifts.

3

(2014):

ψ(z) = 0.015(1 + z)2.7

1 + [(1 + z)/2.9]5.6M yr−1 Mpc−3. (2)

To determine the observed BNS merger rate as a func-tion of redshift we need to consider the matched filteringsignal-to-noise ratio as a function of GW detector sensi-tivity (Finn 1996):

ρ(z) = 8Θr0DL

(Mz

1.2M

)5/6√ζ( fmax), (3)

where Mz = (1 + z)M is the redshifted chirp mass, DL

is the luminosity distance, Θ is the orientation function,and

r20 ≡

5192π

(3G20

)5/3x7/3

M2

c3 ,

x7/3 ≡∫ ∞

0

df (πM)2

(π f M)7/3Sh( f ),

ζ( fmax) ≡1

x7/3

∫ 2 fmax

0

df (πM)2

(π f M)7/3Sh( f ), (4)

where 2 fmax is the wave frequency at which the inspi-ral detection template ends, r0 denotes the characteris-tic distance sensitivity, and Sh( f ) is the detector’s noisepower spectral density. The intrinsic chirp mass, M, isgiven in terms of the component masses by:

M =(

m1m2

(m1 + m2)2

)3/5(m1 + m2). (5)

Here we assume that both neutron stars have mass ofm1 = m2 = 1.4 M. The frequency at the end of theinspiral (taken to correspond to the innermost stablecircular orbit) is:

fmax =785 Hz

1 + z

(2.8M

M

), (6)

where M is the total mass of the binary. In Figure 1 weshow the sensitivity curves for Advanced LIGO, ET, andCE. The substantial reduction in noise amplitude forthe third-generation detectors with respect to AdvancedLIGO leads to an increase in the typical values of r0 from≈ 0.1 to ≈ 1.5 Gpc. Finally, the observed BNS mergerrate as a function of redshift is given by:

RD(z) =dVc

dzÛn(z)

1 + zPdet(z), (7)

where Pdet(z) is defined in Appendix A, and the redshiftderivative of the comoving volume is given by dVc/dz =(4πc/H0)[D2

L/(1 + z)2E(z)].In Figure 2 we show the intrinsic merger rate den-

sity, Ûn(z), for nine different choices of the DTD, withΓ = [−1.5,−1,−0.5] and tmin = [10, 100, 1000] Myr, and afixed mass efficiency value of λ = 10−5 M−1

. For compar-ison, we also show the curve corresponding to no delay(i.e., the cosmic star formation rate density). Clearly,

100 101 102 103 104

f/Hz

10-25

10-24

10-23

10-22

10-21

10-20

S1/

2h

(f)/

Hz−

1/2

adLIGOETCE

Figure 1. Comparison of the noise curves of different GW in-terferometers studied in this work. Red, black, and blue linescorrespond to Advanced LIGO, Einstein Telescope (ET), andCosmic Explorer (CE), respectively.

0 2 4 6 8 10redshift(z)

0.0e+00

2.0e-07

4.0e-07

6.0e-07

8.0e-07

1.0e-06

1.2e-06

1.4e-06

Mer

gerR

ateD

ensi

ty[M

pc−

3yea

r−1] NoDelay

tmin =10Myr,Γ= − 1/2

tmin =100Myr,Γ= − 1/2

tmin =1Gyr,Γ= − 1/2

tmin =10Myr,Γ= − 1

tmin =100Myr,Γ= − 1

tmin =1Gyr,Γ= − 1

tmin =10Myr,Γ= − 3/2

tmin =100Myr,Γ= − 3/2

tmin =1Gyr,Γ= − 3/2

Figure 2. The intrinsic redshift distribution of BNS mergersformed according to the cosmic star formation rate density,and with different DTDs spanning a range of Γ and tmin. Weassume a BNS mass efficiency of λ = 10−5 M−1

. For com-parison, the solid black line shows the merger rate density inthe absence of a delay.

DTDs that prefer longer delays result in a merger dis-tribution that is skewed to lower redshifts, with a higher

4

merger rate at z ≈ 0, but with some degeneracy betweenΓ and tmin. However, since the value of λ is not presentlyknown, all of the DTDs can reproduce the same localrate by simply scaling λ appropriately. This is essen-tially why a local measurement of the merger rate can-not by itself constrain the DTD.

To explore how well current and third-generation GWdetectors can determine the DTD, we inject a specificDTD model (Γ, tmin, λ), generate the resulting red-shift distribution with associated uncertainties, and thenfit this distribution using an interpolation table that isbased on the nine input DTDs. We fit for the inputparameters using Markov Chain Monte Carlo (MCMC)sampling with emcee, a python based affine invariantsampler (Foreman-Mackey et al. 2013). The likelihoodfunction is ln(L) = −χ2/2, with χ2 =

∑i=Ni=0 (RD,i −

ˆRD,i)/σ2t,i. Here the summation is over all of the red-

shift bins; RD,i and ˆRD,i are the constructed and simu-lated detection rates at redshift bin i, respectively; σt,i

is the total error on the detection rate at redshift bini, which is a combination of the Poisson error and theerror due to the distance-inclination degeneracy from

GW data, σ2t = σ

2p + σ

2z ; σp =

√N, where N is the ex-

pected number of events at a given redshift during theintegrated observation time of length Tobs; and the red-shift uncertainty (σz) for each redshift bin is estimatedbased on the vertical distance from the mean expecteddetection rate to the upper envelope corresponding towhen the detections’ redshift are all biased high. Wemodel the redshift uncertainty as δz/z = 0.1z based onre-scaled simulations of binary black hole redshift uncer-tainty estimates as detailed in Appendix B. We adopta flat prior distribution for all of our parameters in thelog λ ∈ [−7,−3], log tmin ∈ [1, 3], and Γ ∈ [−1.5,−0.5].

3. RESULTS FOR CURRENT GW DETECTORS

The predicted observed redshift distribution for thecurrent generation of detectors at design sensitivity isshown in Figure 3. As expected, because the detectiondistance is limited to only a few hundred Mpc, all of theDTDs predict the same shape of observed distribution,with a simple change in scaling that can be accommo-dated by varying the unknown value of λ.

For the purpose of assessing the resulting constraintson the DTD and λ we assume that BNS mergers fromthe current GW network will have precisely determinedredshifts through associated EM counterparts and hostgalaxies. Therefore, the error budget is dominated bythe Poisson error based on the detection rate. For theinput model we assume Γ = −0.6, tmin = 700 Myr, andλ = 10−5 M−1

. Using our MCMC approach we showthe resulting constraints on the DTD parameters for ayear of Advanced LIGO/Virgo operations at design sen-sitivity in Figure 4. The results indicate that the DTDremains largely unconstrained, with the posterior distri-butions strongly influenced by the flat priors. In partic-ular, tmin is unconstrained, while Γ and λ show a strongdegeneracy, with median values that are biased away

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16redshift[z]

0

50

100

150

200

250

300

350

400

Det

ection

rate

[yea

r−1]

adLIGO/Virgo

Γ = − 1/2, tmin = 10Myr

Γ = − 1/2, tmin = 100Myr

Γ = − 1/2, tmin = 1Gyr

Γ = − 1, tmin = 10Myr

Γ = − 1, tmin = 100Myr

Γ = − 1, tmin = 1Gyr

Γ = − 3/2, tmin = 10Myr

Γ = − 3/2, tmin = 100Myr

Γ = − 3/2, tmin = 1Gyr

Figure 3. The expected detection rate as a function of red-shift for Advanced LIGO, for the nine DTDs shown in Fig-ure 2. The detection PDFs are basically identical (moduloa scaling with the unknown value of λ) because AdvancedLIGO can only detect BNS mergers in the local universe.We consider a minimum signal-to-noise ratio of 8 for detec-tion.

log tmin = 1.97+0.69−0.64

adLIGO

Tobs =1year

4.95

4.80

4.65

logλ

logλ = −4.87+0.11−0.07

1.5 2.0 2.5 3.0

log tmin

1.25

1.00

0.75

0.50

Γ

4.95

4.80

4.65

logλ1.2

51.0

00.7

50.5

0

Γ

Γ = −0.77+0.17−0.27

Figure 4. Results of MCMC parameter estimation for a yearof Advanced LIGO/Virgo operations at design sensitivity.The red vertical lines and circles mark the input DTD model,while the green curves and contours show the posteriors ofthe model parameters. The black lines show the median andrange of 16th to 84th percentiles. Here we assume that theredshifts are known precisely thanks to EM counterparts andhost galaxy identifications.

5

0.0 0.5 1.0 1.5 2.0 2.5redshift[z]

0e+00

1e+04

2e+04

3e+04

4e+04

5e+04

6e+04

Det

ection

rate

[yea

r−1]

EinsteinTelescope

Γ = − 1/2, tmin = 10Myr

Γ = − 1/2, tmin = 100Myr

Γ = − 1/2, tmin = 1Gyr

Γ = − 1, tmin = 10Myr

Γ = − 1, tmin = 100Myr

Γ = − 1, tmin = 1Gyr

Γ = − 3/2, tmin = 10Myr

Γ = − 3/2, tmin = 100Myr

Γ = − 3/2, tmin = 1Gyr

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0redshift[z]

0.0e+00

2.0e+04

4.0e+04

6.0e+04

8.0e+04

1.0e+05

1.2e+05

1.4e+05

Det

ection

rate

[yea

r−1]

CosmicExplorer

0 1 2 3 4 5 6redshift[z]

0.0e+00

5.0e+04

1.0e+05

1.5e+05

2.0e+05

Det

ection

rate

[yea

r−1]

ET&CENetwork

Figure 5. The expected detection rate as a function of redshift for ET (left), CE (middle), and a network of ET+CE (right), forthe nine DTDs shown in Figure 2. Due to the ability of these third-generation detectors to detect BNS mergers at cosmologicaldistances, the resulting redshift distributions are no longer fully degenerate. The network of ET+CE not only leads to greatersensitivity, but also provides improved redshift determination compared to ET or CE alone (Appendix B). We consider aminimum signal-to-noise ratio of 8 for detection.

from the injected model. We find the same result for adecade of Advanced LIGO/Virgo operations.

Our results for Advanced LIGO/Virgo suggest thateven the proposed upgrades to the current facilities, suchas A+ (Miller et al. 2015) and Voyager (Lantz et al.2018) will not have a significant impact on the DTDsince these facilities will still only detect BNS mergersin the local universe (see e.g., Figure 1, right panel ofReitze et al. 2019). As argued in Paper I, a more robustconstraint on the DTD from the current generation ofGW detectors may be achieved through the mass distri-bution of BNS merger host galaxies. However, even thisapproach leaves a lingering degeneracy between Γ andtmin.

4. RESULTS FOR THIRD-GENERATIONDETECTORS

The situation is drastically different for the third-generation detectors, ET and CE. In Figure 5 we plotthe expected detection rate as a function of redshiftfor ET, CE, and a network of ET+CE. Two improve-ments are readily apparent. First, the expected detec-tion rate is about three orders of magnitude larger thanfor Advanced LIGO/Virgo. Second, the redshift rangefor BNS merger detections increases to z ∼ 5 in the caseof ET+CE. The latter improvement results in a clear dif-ference between the redshift distributions of the variousDTDs, while the former improvement provides the de-tection statistics needed to distinguish between the DTDmodels. The differences between the various DTDs canno longer be scaled away with a change in λ (as is thecase for Advanced LIGO/Virgo). In what follows wefocus on the case of ET+CE as a realistic version of athird-generation detector network.

In Figure 6 we show the result of MCMC fitting for thesame input model used in the previous section. We show

the results for 1 day, 1 week, 1 month, and 1 year of ob-servations. Unlike in the case of the current generationof GW detectors, we assume that the BNS mergers atcosmological distances will generally not have detectableEM counterparts (see Appendix C.). Instead we relyon distance information from the GW signal itself. Wemodel the resulting redshift uncertainty as δz/z = 0.1z; adetailed motivation for this parametrization is providedin Appendix B.

Our results show that an ET+CE network is able toconstrain the DTD parameters and overcome the intrin-sic degeneracy between Γ and tmin within a year of obser-vations. The values of Γ and tmin can be determined tobetter than 10% accuracy. We note that these numbersdepend on the overall event rate, which is determinedby λ; here we use an injected value of 10−5 M−1, butthe results can be rescaled for higher or lower values.

To assess the impact of our input model on the re-sults, in Figure 7 we repeat the same exercise, but fortwo different DTDs that favor short merger timescales:Γ = −1.2 with tmin = 30 Myr, and Γ = −1 withtmin = 100 Myr. Although the power law index is recov-ered with the same accuracy as before, we find that tminbecomes more challenging to determine when its value issmall. This is because the relative shift in the observedBNS merger redshift distribution becomes progressivelysmaller for small values of tmin, which is challenging todetect in the presence of realistic GW redshift uncer-tainties.

5. SUMMARY AND DISCUSSION

We investigated how well the DTD and mass efficiencyof BNS systems can be determined through the redshiftdistribution of BNS mergers detected by current andfuture GW networks. We model the DTD as a powerlaw with a minimum merger timescale, and leave the

6

log tmin = 2.88+0.06−0.09

δ(z)/z= 0.1 z

CE&ETNetwork

Tobs =1day

0.07

0.04

0.01

0.02

logλ

5 logλ = −5.01+0.02−0.02

2.55

2.70

2.85

3.00

log tmin

1.05

0.90

0.75

0.60

Γ

0.07

0.04

0.01

0.02

logλ 51.0

50.9

00.7

50.6

0

Γ

Γ = −0.64+0.10−0.17

log tmin = 2.85+0.07−0.08

δ(z)/z= 0.1 z

CE&ETNetwork

Tobs =1week

0.04

0.02

0.00

0.02

logλ

5 logλ = −5.00+0.01−0.01

2.6 2.7 2.8 2.9 3.0

log tmin

0.90

0.75

0.60

Γ

0.04

0.02

0.00

0.02

logλ 50.9

00.7

50.6

0

Γ

Γ = −0.59+0.07−0.13

log tmin = 2.85+0.06−0.08

δ(z)/z= 0.1 z

CE&ETNetwork

Tobs =1month

0.025

0.010

0.005

0.020

logλ

5 logλ = −5.00+0.01−0.01

2.7 2.8 2.9 3.0

log tmin

0.90

0.75

0.60

Γ

0.025

0.010

0.005

0.020

logλ 5

0.90

0.75

0.60

Γ

Γ = −0.61+0.08−0.11

log tmin = 2.84+0.05−0.07

δ(z)/z= 0.1 z

CE&ETNetwork

Tobs =1year

0.008

0.000

0.008

logλ

5 logλ = −5.00+0.00−0.00

2.7 2.8 2.9

log tmin

0.72

0.64

0.56

Γ

0.008

0.000

0.008

logλ 5

0.72

0.64

0.56

Γ

Γ = −0.60+0.06−0.06

Figure 6. Results of MCMC parameter estimation for a network of CE+ET with a range of operating timescales, spanning 1day to 1 year. The red vertical lines and circles mark the input DTD model, while the green curves and contours show theposteriors of the model parameters. The black lines show the median and range of 16th to 84th percentiles. In this case theredshift uncertainty is modeled as δz/z = 0.1z. With a year of observations all of the DTD parameters can be determinedaccurately to high precision.

mass efficiency as a free parameter. While other DTDshave been proposed (e.g., Simonetti et al. 2019), ourprimary conclusions should not be affected by the exactform of the DTD.

We find that current GW detectors, which can onlydetect BNS mergers in the local universe, cannot di-rectly constrain the DTD due to their limited sensitivity.In effect, the various DTDs, with an appropriate scal-ing of λ, predict the same BNS merger detection rateat z ≈ 0. However, the situation is dramatically differ-ent for the anticipated third-generation detectors, whichwill be able to detect BNS mergers to z ≈ 5. For thiscosmological population, even in the presence of red-

shift uncertainties of δz/z ≈ 0.1z from the GW data, thelarge detection rate and broad redshift range will pre-cisely determine the DTD within about a year of opera-tions. It has been previously argued that the cosmologi-cal merger population uncovered by third-generation de-tectors will be able to constrain cosmological parameters(Sathyaprakash et al. 2009; Taylor et al. 2011; Taylor &Gair 2012; Vitale & Farr 2018); our results for the DTDfurther bolster the science case for third-generation GWdetectors.

This work was supported by the National ScienceFoundation under grant AST14-07835 and by NASA

7

log tmin = 1.51+0.32−0.34

δ(z)/z= 0.1 z

CE&ETNetwork

Tobs =1year

0.04

0.02

0.00

0.02

logλ

5 logλ = −5.01+0.01−0.01

1.2 1.6 2.0 2.4

log tmin

1.35

1.20

1.05

Γ

0.04

0.02

0.00

0.02

logλ 51.3

51.2

01.0

5

Γ

Γ = −1.21+0.11−0.13

log tmin = 2.18+0.33−0.34

δ(z)/z= 0.1 z

CE&ETNetwork

Tobs =1year

0.02

0.00

0.02

0.04

logλ

5 logλ = −5.00+0.01−0.01

1.5 2.0 2.5

log tmin

1.4

1.2

1.0

0.8

Γ

0.02

0.00

0.02

0.04

logλ 5

1.4 1.2 1.0 0.8

Γ

Γ = −1.06+0.12−0.15

Figure 7. The same as in Figure 6 but for two DTD models that favor short merger timescales. Left: An injected DTD withΓ = −1.2 and tmin = 30 Myr. Right: An injected DTD with Γ = −1. and tmin = 100 Myr. Although the value of Γ is still recoveredwith about 10% uncertainty in both cases, tmin becomes more challenging to accurately determine when its value is small.

under theory grant NNX15AK82G. The Berger Time-Domain Group at Harvard is supported in part byNSF under grant AST-1714498 and by NASA undergrant NNX15AE50G. MTS is thankful to Harvard-Smithsonian Center for Astrophysics for hospitalitywhich made this work possible. SV, KKYN and CW

acknowledge support of the National Science Founda-tion, the LIGO Laboratory and the LIGO Data Gridclusters. LIGO was constructed by the California Insti-tute of Technology and Massachusetts Institute of Tech-nology with funding from the National Science Founda-tion and operates under cooperative agreement PHY-0757058.

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APPENDIX

A. DETECTION PROBABILITY OF A NETWORK OF DETECTORS

The strain measured by a GW interferometer in frequency domain is given by

h( f ) = F+ h+( f ) + F× h×( f ), (A1)

where h+,× are the +,×-polarization bases and F+,× are the corresponding beam pattern functions,

F+ = g

[12

(1 + cos2 θ

)cos 2φ cos 2ψ − cos θ sin 2φ sin 2ψ

], (A2)

F× = g

[12

(1 + cos2 θ

)cos 2φ sin 2ψ + cos θ sin 2φ cos 2ψ

], (A3)

where θ, φ and ψ are the zenith, azimuth and polarization angles respectively, and g is a dimensionless coefficientdetermined by the geometry of an interferometer (Sathyaprakash & Schutz 2009; Chen et al. 2017; Schutz 2011).CE is a single interferometer with the angle between two arms equal to 90 deg, hence gCE = 1 (Chen et al. 2017;Schutz 2011). ET consists of 3 identical interferometers with the angle between two arms equal to 60 deg, forming an

equilateral triangle, hence gET =√

3/2 for each interferometer in ET (Sathyaprakash et al. 2012; Punturo et al. 2010).The detection probability, Pdet, is defined as the probability of a detection with ρnet ≥ ρT , where ρT is the SNR

threshold of detection and ρ2net =

∑i ρ

2i is the network SNR as a geometric sum of SNR of each inteferometer. Assuming

isotropic sky locations, orbital orientation and polarization, (i.e., uniform distribution of (cos θ, φ, ψ, cos ι)), Pdet is givenanalytically as

Pdet (θint, z) =∫

H(

ρTρnet(θint, z, θ, φ, ψ, ι)

)d cos θdφdψd cos ι, (A4)

where θint is the set of intrinsic parameters, which are fixed at 1.4 − 1.4 M and zero-spin for BNS systems, andH(w(z, θ, φ, ψ, ι)) is the unitary step function defined in w ∈ (0, 1] for w(z, θ, φ, ψ, ι) = ρT /ρnet(z, θ, φ, ψ, ι).

For single CE or ET, we follow the inspiral approximation in Finn (1996). The SNR of a single interferometer isapproximately

ρ2 = 64Θ2(

r0dL

)2 (Mz

1.2 M

)5/3ζ2( fmax), (A5)

where Θ2 = 4[F2+

(1 + cos2 ι

)2+ 4F2

× cos2 ι], and proportional to Θ2

net =∑

i Θ2i for the same signal strain observed by

homogeneous detectors such as single CE or ET. Hence Pdet(w(z)) is equivalently the survival function of Θnet/Θmaxnet ,

where Θmaxnet is the maximum response of a particular network of detectors (Sathyaprakash & Schutz 2009; Chen et al.

2017; Schutz 2011; Finn 1996). Here w(z) = ρT /ρopt(z) where ρopt is the optimal SNR at maximum Θ2. CE has

ΘmaxCE = 4 and ET has Θmax

ET = 4×√

3 × 3/4 = 6. We have verified that the inspiral approximation in 3G detectors onlyresults in few-percent difference in SNR for z . 6, which is the region we are interested in.

Approximated forms of Pdet assuming uniformly distributed (cos θ, φ, ψ, cos ι) provided in Finn (1996) or Dominiket al. (2015) are only suitable for a single CE or a second-generation network. Therefore, we generate the survivalfunction of Θ by drawing 106 points of uniformly distributed (cos θ, φ, ψ, cos ι) in single CE or ET and fit the survival

9

0.0 0.2 0.4 0.6 0.8 1.0w

0.0

0.2

0.4

0.6

0.8

1.0

Pd

et(w

)

CE

ET

CE-ET

Figure A1. Comparison of simulation and parametric fits for CE, ET and CE+ET network. Solid lines show the simulation,dashed-dotted lines show the fit using Equation A6 and dashed lines show the fit using Equation A7. Here w = ρT /ρopt(z) forsingle CE or ET and w = z/zhorizon for CE+ET network.

function with the following parametric form:

Pdet(w; A, B) = erf (A − Bw) − erf (A − B)erf (A) − erf (A − B) , (A6)

where erf(x) = 2√π

∫ x

0 e−t2dt is the error function. We also employ a 10th order polynomial,

Pdet(w; ak) =9∑

k=1ak(1 − w)k +

(1 −

9∑k=1

ak

)(1 − w)10 , (A7)

where ak are the polynomial coefficients.The above simplification using distribution of Θ breaks down for a CE+ET network due to the different sensitivity

and heterogeneous geometry of each interferometer. Instead, we calculate the integral in Equation A4 by simulatingwaveform and network SNR for each redshift. Then w(z) = z/zhorizon and zhorizon is the redshift of the detector horizon,which is ∼ 12.5 in CE+ET for a BNS merger. Again we fit the simulation result for CE+ET using Equations A6 andA7.

Tables A1 and A2 show the definitions of w and fitting parameters of Equations A6 and A7 in each network.Figure A1 shows the comparison of actual simulation and the parametric fits.

Network w(z) A BCE ρT /ρopt(z) 1.63 3.93

ET ρT /ρopt(z) 1.05 3.33

CE-ET z/zhorizon -0.58 -5.05

Table A1. Definition of w and fitting parameters of Equation A6 for CE, ET and CE+ET.

Network w(z) a1 a2 a3 a4 a5 a6 a7 a8 a9CE ρT /ρopt(z) 0.038 -0.57 10.76 -49.76 143.1 -293.1 470.5 -512.4 307.5

ET ρT /ρopt(z) 0.22 -8.51 132.8 -966.0 3995.7 -9923.5 15017.1 -13487.3 6581.3

CE-ET z/zhorizon -0.15 5.48 -71.4 455.0 -1613.3 3363.5 -4119.8 2794.4 -877.2

Table A2. Definition of w and fitting parameters of Equation A7 for CE, ET and CE+ET.

10

B. REDSHIFT UNCERTAINTY FROM GW DETECTIONS

In the case of third-generation detectors it is unlikely that most BNS merger detections will have EM counterparts.Instead, the distance information will need to be gleaned from the GW signal itself. There is an inherent degeneracybetween a binary’s inclination angle in the sky with respect to a GW detector, θJN, and the luminosity distance (Schutz2011; Abbott et al . et al. 2016; Chen et al. 2018; Usman et al. 2018). Messenger & Read (2012) show that for arange of representative neutron star equations of state the redshift of such systems can be determined to an accuracyof ∼ 8− 40% for z < 1 and ∼ 9− 65% for 1 < z < 4. However, for a binary to be detectable at high redshifts, we expectthe inclination to be close to face-on (θJN = 0o), or face-off (θJN = 180o), as most of the energy in gravitational wavesis released along the angular momentum vector of the binary. Schutz (2011) derived an analytic formulation for thedistribution of inclination angle of sources detectable by advanced detectors; . 7% (. 3%) of detectable events willhave viewing angles of > 70o (> 80o) (Chen et al. 2018).

In principle, we can simulate BNS signals at different redshifts in third-generation detectors and estimate the evolu-tion of uncertainty in parameter estimation using MCMC. Since BNS are a low-mass system with a long coalescencetime, the parameter estimation is computationally expensive. To approximate the redshift uncertainty of a BNS atfixed redshift, we may extrapolate a binary black hole (BBH) signal by lowering the starting frequency from 10 Hz to5 Hz, as to mimic the long inspiral phase in a BNS merger. We obtain the redshift uncertainties of BBHs in CE+ETfrom the simulations of Vitale & Whittle (2018), and fit the mean redshift uncertainties as a function of true redshifts.Then we calibrate our fit using the above extrapolation scheme to approximate the redshift uncertainties as δz/z ≈ 0.1z.

Alternatively, Chen et al. (2018) developed a rapid algorithm that provides a luminosity distance uncertainty esti-mate for a large population of BNS merger detections. We use this algorithm to simulate 2000 BNS detections in athird-generation network, and compare the results to the extrapolation procedure above. We find that the two ap-proaches yield consistent results. We therefore use the extrapolated distance uncertainty, and convert it to the redshiftuncertainty through the adopted cosmology in this work.

C. REDSHIFT UNCERTAINTY FROM JOINT SGRB DETECTIONS

The distance-inclination degeneracy can be broken through the detections of an associated SGRB. The relativefraction of on-axis mergers is only a few percent. A γ-ray detection alone will thereby reduce the overall redshiftuncertainty of at most a few percent of BNS merger detections, and likely over a restricted redshift range (perhaps toz ∼ 2). An afterglow detection can further lead to a precise redshift determination through an associated host galaxy,but to date such detections have mainly been limited to z . 1 (Berger 2014), which is generally not high enough tomake an impact on the DTD determination (see Figure 5). Thus, it seems unlikely that associated SGRBs and theirafterglows will substantially improve the constraints on the DTD.