Cosmological inference from standard sirens without redshift measurements
Xuheng Ding, Marek Biesiada, Xiaogang Zheng, Kai Liao, Zhengxiang Li, Zong-Hong Zhu
PPrepared for submission to JCAP
Cosmological inference from standardsirens without redshift measurements
Xuheng Ding a,b,c , Marek Biesiada c,d , Xiaogang Zheng c,d , Kai Liao e ,Zhengxiang Li c , Zong-Hong Zhu a,c a School of Physics and Technology, Wuhan University, Wuhan 430072, China b Department of Physics and Astronomy, University of California, Los Angeles, CA, 90095-1547, USA c Department of Astronomy, Beijing Normal University, Beijing 100875, China d Department of Astrophysics and Cosmology, Institute of Physics, University of Silesia, 75Pu(cid:32)lku Piechoty 1, 41-500 Chorz´ow, Poland e School of Science, Wuhan University of Technology, Wuhan 430070, ChinaE-mail: [email protected]
Abstract.
The purpose of this work is to investigate the prospects of using the futurestandard siren data without redshift measurements to constrain cosmological parameters.With successful detections of gravitational wave (GW) signals an era of GW astronomy hasbegun. Unlike the electromagnetic domain, GW signals allow direct measurements of lumi-nosity distances to the sources, while their redshifts remain to be measured by identifyingelectromagnetic counterparts. This leads to significant technical problems for almost all pos-sible BH-BH systems. It is the major obstacle to cosmological applications of GW standardsirens. In this paper, we introduce the general framework of using luminosity distances alonefor cosmological inference. The idea is to use the prior knowledge of the redshift probabilitydistribution for coalescing sources from the intrinsic merger rates assessed with populationsynthesis codes. Then the posterior probability distributions for cosmological parameters canbe calculated. We demonstrate the performance of our method on the simulated mock dataand show that the luminosity distance measurement would enable an accurate determinationof cosmological parameters up to 20% uncertainty level. We also find that in order to infer H to 1% level with flat ΛCDM model, we need about 10 events. Keywords: gravitational waves / sources, gravitational waves / theory a r X i v : . [ a s t r o - ph . C O ] D ec ontents w CDM model 8
In the past decade, flat ΛCDM model has emerged as the standard cosmological model. Byassuming the existence of some form of dark energy with equation of state coefficient w = − H and the matterdensity parameter Ω m . The H is particularly noteworthy which represents the currentexpansion rate and is related to the age, size, and critical density of the Universe. However,the inferred value of H from the Planck satellite [1] is in tension with other low redshiftmeasurements [2–4]. Thus, a 1% accurate measurement of H is highly needed to understandwhether the tensions within the ΛCDM model are real and require new physics.Recent detections of gravitational waves (GW) by advanced LIGO/Virgo detectors haveopened a new window to the Universe [5–10] which will have a significant impact on physicsand astronomy. The detected GW signal came from the coalescence of double compactobjects (DCO) , i.e., BH-BH binary systems in all these cases, besides GW170817 [9] whichwas the first NS-NS coalescence ever detected and accompanied by successfully identifiedelectromagnetic (EM) counterpart. This category of GW sources (i.e., inspiralling DCOs)can be considered as standard sirens [11] — named so, in analogy to standard candles in theEM domain. However, contrary to EM probes, with GW inspiral signal one can measurethe luminosity distance D L to the source directly, without the need of taking into accountthe cosmic distance ladder. This idea has been widely discussed in the literature for bothground-based detectors [12–16] and space-based detectors [17, 18] — the above referencesbeing just indicative and by no means exhaustive.Encouraged by the development of new technologies leading to the Advanced LIGOdetectors, GW scientific community is designing and planning to build a new generationdetector called the Einstein Telescope (ET) which will broaden the accessible volume ofthe Universe by three orders of magnitude. Given this sensitivity, the service of ET would The DCO comprise of NS-NS, BH-NS and BH-BH binary systems. – 1 –ield 10 − detections per year up to redshift z = 17 [19], and thus could provide aconsiderable database of luminosity distances to these sources. Such rich statistics of D L measurements is very promising in the context of constraining cosmological parameters tomuch higher precision. In particular, it was demonstrated that with a dozen of lensed GWand EM signals, which would be quite realistic in the era of the ET, one could measure H with sub-percent accuracy [20].The standard way of cosmological inference is by using the D L − z diagram and con-fronting observed values with theoretical D L ( z ) function dependent on cosmological param-eters. Therefore one needs an independent determination of D L and the redshift z . In theEM domain redshift is an obvious observable, with D L being a tricky one. With GW sig-nals, the reverse is true: D L is a direct observable, while z should somehow be assessed. Ina vast majority of works discussing the standard siren approach, an optimistic assumptionwas made that an accompanying EM signal will be detected thus allowing for determinationof z . However it is not an easy task, first because the GW signals are detected with lowresolution of their sky location, typically ∼
10 deg accuracy [21]. Hence, identifying theEM counterpart from the extensive region is difficult. Fortunately, in the case of GW170817favorable location of the source with respect to three LIGO/Virgo detectors and the lumi-nosity distance inferred from the waveform considerably constrained candidate host galaxiesand eventually optical counterpart was observed in one of them. In several papers, it wasproposed to determine the ∆ z i range from the possible host galaxies for the GW source[14, 22, 23]. Similarly, the cross-correlation between spatial distributions of the DCOs andthe known-redshift galaxies was proposed to constrain the distance-redshift relation [24]. Theauthors of [25] suggested a self-calibrating iterative scheme to mitigate the misidentificationof DCO sources. However, some of the GW signals registered in the era of the ET wouldcome from very high redshifts (i.e., z >
4) where the EM counterparts and possible hosts areextremely faint and not readily observable. At last, the DCO signals are dominated by theBH-BH systems which most likely would not be accompanied by noticeable EM counterparts.This is what could be expected on theoretical grounds and what we indeed experienced withfive successful BH-BH detections so far.Attempting to overcome the difficulties with redshifts inherent to GW astrophysics, wepropose a new approach to construct the posterior probability distribution for the cosmo-logical parameters using the distribution of sources’ redshifts as a prior. In this paper, wecalculate the prior redshift distribution of sources based on the intrinsic merger rates of theDCOs together with the expected sensitivity of the ET. On the simulated data, we show thatone could achieve the precision of cosmological inference comparable to that achievable fromcurrent EM data and one would be able to measure H with 1% accuracy using the datagathered by the ET in one year of its operation.The paper is organized as follows. In Section 2, we outline the method of constructingthe posterior probability distribution for cosmological parameters and introduce an idea ofhow to set up the priors. In Section 3, we investigate the prospects of our approach bycarrying out the Monte Carlo probability maximization using simulated mock data for twopopular cosmological models. We discuss the results and conclude in Section 4.– 2 – Methodology
In this section, we introduce the general framework of using luminosity distances alone forcosmological inference. Usually, the redshift is a key information in this context, but aswe already mentioned for most of GW events redshift will be unknown and unmeasurable.Therefore, we start with noticing that the redshift probability distribution for DCO coalescingsources could be calculated and then used to derive posterior probability distributions forcosmological parameters.Let us consider a number of n GW events detected by the ET, with luminosity distancesdirectly measured from their waveforms denoted collectively as (cid:126)D ≡ ( D , D , ..., D n ). Theredshifts of these events are unknown. Our goal is to construct the posterior probabilitydistribution of cosmological parameters (cid:126) Ω ≡ ( H , Ω m ) for ΛCDM model (or some biggercollection of parameters for other cosmological models). We will use Bayes theorem takingthe redshift distribution of the sources as a prior.Focusing on the i -th event one can write: P ( (cid:126) Ω , z i | D i , I ) = P ( (cid:126) Ω | z i , D i , I ) P obs ( z i | D i , I )= P ( D i | (cid:126) Ω , z i , I ) P ( (cid:126) Ω | z i , I ) P ( D i | z i , I ) P ( D i | z i , I ) P obs ( z i | I ) P ( D i | I )= P ( D i | (cid:126) Ω , z i , I ) P obs ( z i | (cid:126) Ω , I ) P ( D i | I ) P ( (cid:126) Ω | I ) (2.1)where P ( D i | (cid:126) Ω , z i , I ) is the likelihood function for the observed data. P obs ( z i | (cid:126) Ω , I ) and P ( (cid:126) Ω | I )are the priors on the redshift and the cosmological parameters, respectively. Let us emphasizethat the prior on redshifts is a prior of observed events and already includes detector selectioneffects. Hence we used the notation P obs ( z i | ... ) to make it clear to the reader that it doesnot represent intrinsic redshift distribution of sources. All the other background informationrelated to this study is denoted by I and all probabilities considered are conditional on it.As usual, P ( D i | I ) plays the role of normalization constant. Note that the expression of P obs ( z i | (cid:126) Ω , I ) means that the redshift probability distribution could be inferred invoking aspecific cosmological model. The likelihood P ( D i | (cid:126) Ω , z i , I ) can be taken in the form: P ( D i | (cid:126) Ω , z i , I ) ∝ e − χ ( D i | (cid:126) Ω ,z i ,I ) / (2.2)where χ ( D i | (cid:126) Ω , z i , I ) = (cid:16) D iL,obs − D iL,theo ( (cid:126) Ω , z i , I ) (cid:17) σ D L . (2.3)with D iL,theo ( (cid:126) Ω , z i , I ) denoting theoretical value of the luminosity distance corresponding tothe redshift z i calculated within a cosmological model with parameters (cid:126) Ω. One should notethat the GW amplitude h ( t ) measured in the detector is proportional to D − L . This meansthat if one refers to wave strain measurements one has to acknowledge this dependence informulating the likelihood. However, in our case we use the luminosity distances inferredfrom GW data (along with the precision of this inference) as observables. In such case thelikelihood (2.3) is justified and indeed such kind of expression was already used by otherauthors in the context of GW cosmography, e.g. in [23].– 3 –arginalizing P ( (cid:126) Ω , z i | D i , I ) over the redshift, we can write the posterior probability ofcosmological parameters as: P ( (cid:126) Ω | D i , I ) = (cid:90) z max P ( (cid:126) Ω , z (cid:48) i | D i , I ) dz (cid:48) i = P ( (cid:126) Ω | I ) (cid:90) z max P ( D i | (cid:126) Ω , z (cid:48) i , I ) P obs ( z (cid:48) i | (cid:126) Ω , I ) P ( D i | I ) dz (cid:48) i . (2.4)Given that one GW event is independent of the others, the combined posterior probabilityinferred from the entire set of events could be calculated. Note that the cosmological param-eter prior P ( (cid:126) Ω | I ) is common to all events, and accordingly this prior should be used onlyonce: P ( (cid:126) Ω | (cid:126)D, I ) = P ( (cid:126) Ω | I ) n (cid:89) i =1 (cid:90) z max P ( D i | (cid:126) Ω , z (cid:48) i , I ) P obs ( z (cid:48) i | (cid:126) Ω , I ) P ( D i | I ) dz (cid:48) i . (2.5)Once a set of measured luminosity distances from the GW events is obtained, theEq. (2.5) could be calculated, provided that the prior probability distributions P obs ( z i | (cid:126) Ω , I )and P ( (cid:126) Ω | I ) are given. As outlined above, in order to calculate the posterior, we need to set the priors concerningcosmological parameters P ( (cid:126) Ω | I ) and the redshifts of GW events (i.e., P obs ( z i | (cid:126) Ω , I )). Aim-ing to study the performance of cosmological inference from GW signals alone, we will setuniform priors on P ( (cid:126) Ω | I )) trying not to make use of values suggested by other independentexperiments. On the other hand, the distribution of P obs ( z i | (cid:126) Ω , I ) is not straightforward andneeds to be considered prudently. In this work, we derive the P obs ( z i | (cid:126) Ω , I ) by considering theintrinsic merger rate and the expected sensitivity of the ET.In principle, the prior probability distribution of redshifts of GW sources is equivalent tothe number density of the detected events as a function of redshift which have been predictedmany times since the pioneering paper [26]. We refer the reader to more recent studies in[15, 27]. In particular, the detection rate of GWs has been calculated by [28, 29], takinginto account the intrinsic merger rates of the whole class of DCOs (i.e., NS-NS, BH-NS andBH-BH). These merger rates have been calculated by [30] as a function of redshift using StarTrack population synthesis evolutionary code.The general idea of such calculation is the following. The criterion, which defineswhether a DCO inspiral event is detectable, is that the value of its signal-to-noise ratio(SNR) is greater than the ET threshold (assumed as ρ =8). In general, the SNR ρ for asingle detector is: ρ = 8Θ r D L ( z s ) (cid:18) (1 + z ) M . M (cid:12) (cid:19) / (cid:112) ζ ( f max ) (2.6)where Θ is the orientation factor capturing part of sensitivity pattern due to (usually non-optimal) random relative orientation of a DCO system with respect to the detector. Fourangles describe this relative orientation: ( θ, φ ) describe the direction to the binary relative tothe detector, while ( ψ, ι ) describe the binary’s orientation relative to the line-of-sight betweenit and the detector. The quantity r is detector’s characteristic distance parameter. In thisstudy, we focus on the initial ET configuration for which r = 1527 Mpc. The dimensionlessfunction ζ ( f max ) depends only on detector’s noise, its argument is the orbital frequency when– 4 –he inspiral terminates, and its value is close to unity (see e.g. [15]). M is the intrinsic chirpmass of the DCO system. Following previous work ([27–29]), we assumed the chirp masses asaverage values for each category of DCO simulated by population synthesis: 1.2 M (cid:12) for NS-NS, 3.2 M (cid:12) for BH-NS and 6.7 M (cid:12) for BH-BH systems. Clearly, once all other parametersare fixed, ρ is a random quantity related to Θ. The probability distribution for Θ calculatedunder the assumption of uncorrelated orientation angles ( θ, φ, ψ, ι ) has the following form: P Θ (Θ) = 5Θ(4 − Θ) / , if 0 < Θ < P Θ (Θ) = 0 , otherwiseThe differential inspiral rate per redshift concerning events which exceed the threshold(i.e., ρ > ρ =8) can be expressed as: d ˙ N ( > ρ ) dz = 4 π (cid:18) cH (cid:19) ˙ n ( z s )1 + z s (cid:101) r ( z s ) E ( z s ) C Θ ( x ( z s )) (2.8)where ˙ n ( z s ) is the intrinsic coalescence rate of DCOs in the local Universe at redshift z s calculated by [30] from the population synthesis code, C Θ ( x ) = (cid:82) ∞ x P Θ (Θ) d Θ and x ( z, ρ ) = ρ (1 + z ) / cH ˜ r ( z ) r (cid:16) . M (cid:12) M (cid:17) / . Finally, the yearly detection rate of DCO sources extendingto the redshift z s can be calculated as:˙ N ( > ρ | z s ) = (cid:90) z s d ˙ N ( > ρ ) dz dz. (2.9)Eq. (2.9) was used by [28] to predict the yearly detection rate by the ET (see Table 1 and2 therein), showing that hundreds of thousand of DCOs can be detected per year. Thedifferential rate Eq. (2.8) describes the detected events distributed as a function of redshift.Therefore it could be used both to simulate the redshift distribution of the mock data andalso as the prior on the redshift P obs ( z i | (cid:126) Ω , I ). The purpose of this work is to investigate the prospects of using the future standard sirendata without redshift measurements to constrain cosmological parameters. To this end, wefirst randomly simulate the D L data representative of what could be observed by the ET,based on the redshift distribution of these events as described in Section 2.2. Then, we applyour approach, outlined in Section 2, to the simulated data and test its fidelity regarding thecosmological inference. We assume flat ΛCDM Universe with H = 70 km s − Mpc − , Ω m = 0 .
30 as a fiducial modelin our simulation. By setting this model, the redshift distribution of the mock data can becalculated using Eq. (2.8). We adopt the values of intrinsic inspiral rates ˙ n ( z s ) reported byDominik et al. [30] for the whole class of DCO. Our fiducial model is the same they used.For simplicity, we only considered the standard scenario with “low-end” case of metallicityevolution. It has been tested (see e.g. Fig 2 in [28]) that different choices of evolutionaryscenarios would not strongly affect the final distribution.– 5 – z o f D C O e v e n t s p e r z DCO events distribution detected by ETTotalBHBHNSBHNSNS . . . . . . . . z o f D C O e v e n t s Histogram of simulated DCO events
Figure 1 : Redshift distribution of DCO inspiral events predicted for the ET (left) and oneexample of the histogram of simulated DCO events (right).The calculated redshift distribution of DCO inspiral events predicted to be detected bythe ET including NS-NS, BH-NS, BH-BH is shown in Fig. 1. This distribution can serve as asampling distribution to generate the simulated redshifts of DCO systems. Overwhelmingly,the distribution is dominated by the BH-BH systems. This is because the BH-BH systems arethe predominant population of DCOs and typically have stronger signals than NS-NS, BH-NS. As an example, the histogram of redshifts obtained with 10,000 simulations is shown inFig. 1. This sample size is sufficient for our purpose, and at the same time, it is representativeof what would be achieved very soon when the ET is put into service. In previous work [28],it has been estimated that the ET would register about 10 − inspiral DCO events peryear. Following the common practice, mock luminosity distance is generated as D L,sim ( z s ) = D L,fid ( z s ) + N(0 , σ ), where D L,sim ( z s ) and D L,fid ( z s ) are simulated and fiducial values ofthe luminosity distance at a given redshift, respectively. Assuming the fiducial cosmologicalmodel as a ‘true’ one, the values of luminosity distance ( D L,fid ( z )) at the corresponding red-shift can be calculated within such model. The N(0 , σ ) term is the Gaussian random variablewith zero mean and variance σ corresponding to the uncertainty regarding the luminositydistance measurement. Given the values of uncertainty level, one can randomly generate thesimulated values of D L,sim . In this section, we investigate posterior distributions of cosmological parameters from theanalysis of simulated mock data using our approach based on Eq. (2.4) and (2.5). Weconsider two simplest cosmological models with the following expansion rates: H ( z ) = H (cid:112) Ω m (1 + z ) + (1 − Ω m ) , (3.1) H ( z ) = H (cid:113) Ω m (1 + z ) + (1 − Ω m )(1 + z ) w ) , (3.2)– 6 – = 0.30 +0.050.04 . . . . m H H H = 70.20 +3.923.73 (a) D L uncertainty distributed as U (5% , m = 0.29 +0.050.04 . . . . m H H H = 70.95 +4.313.98 (b) D L uncertainty distributedas U (5% , m = 0.28 +0.050.04 . . . . m H H H = 71.67 +4.434.24 (c) D L uncertainty distributed as U (5% , Figure 2 : Illustration of the maximization distribution of the posterior for the cosmologicalparameters in the flat ΛCDM model based on ∼ ,
000 realizations of datasets, each datasetcontain 10,000 events. In the simulation, D L are set with different uncertainty levels. Inthe fitting, the universal prior of uncertainty level is adopted as U([5%,20%]). The contourregions denote the 68% (1- σ ) and 95% (2- σ ) confidence region. The blue lines denote thetrue values of the fiducial model parameters.which phenomenologically describe the dark energy modeled as a perfect fluid with barotropicequation of state: p = − ρ and p = wρ , respectively .We performed the Monte Carlo probability maximization by first repeatedly simulatinga set of mock data realizations with random noise and of a sufficiently big size; each realizationcontained 10,000 events. We then derived the maximization distribution of the posterior forthe parameters (cid:126) Ω using Eq. (2.5) based on the realizations. The simulation process continueduntil the maximization distribution was stable. In this study, only the luminosity distanceis considered as the observed data, and the uncertainty of this distance would not usuallybe perfectly known. Thus, in our analysis, the uncertainty is assumed as a parameter whichshould be marginalized over in the final result. Moreover, these uncertainty levels wouldaffect the posterior; thus we adopt three different uncertainty levels randomly distributedas U ([5% , U ([5% , U ([5% , Λ CDM model
We assume uniform priors: H ∼ U ([45 , m ∼ U ([0 . , . D L is not perfectly unknown, we adopt theuncertainty level as U ([5% , D L at different levels as mentionedabove. We see that the reliable inference (within 68% (1- σ ) confidence) could be achievedat each uncertainty level. Not surprisingly, with the higher uncertainty level, the scatter of These models are known as flat ΛCDM and w CDM, respectively. – 7 – = 0.29 +0.050.05 . . . . . w w = 1.03 +0.250.32 .
20 0 .
25 0 .
30 0 .
35 0 . m H . . . . . w
66 72 78 84 90 H H = 70.88 +6.195.08 (a) D L uncertainty distributed as U (5% , m = 0.29 +0.050.04 . . . . . w w = 1.03 +0.250.31 .
20 0 .
25 0 .
30 0 .
35 0 . m H . . . . . w
66 72 78 84 90 H H = 71.38 +6.085.23 (b) D L uncertainty distributedas U (5% , m = 0.28 +0.040.04 . . . . . w w = 1.02 +0.270.33 .
20 0 .
25 0 .
30 0 .
35 0 . m H . . . . . w
66 72 78 84 90 H H = 71.56 +5.894.51 (c) D L uncertainty distributed as U (5% , Figure 3 : Results for w CDM model. H realizations is slightly larger with the central value slightly shifted from the true point.We tested that adopting the prior uncertainty levels other than U ([5% , σ .Besides the uncertainty level, the confidence regions for the inferred cosmological pa-rameters are related to the number of detected events. As we discussed in Section 1, theprecision of H measurement is necessary to shed light on the tension between Planck andlocal probes. Therefore we also investigated how many data on D L would be required toachieve a percent precision for the Hubble constant. Fixing the uncertainty level for D L at10%, we increased the size of the data gradually from 2 × to 1 × and obtained thecorresponding 1- σ confidence region of inferred H , as listed in Tab. 1. The result shows thatthe inference of H with ∼
1% precision requires 1 × samples of D L . w CDM model
In the w CDM model, the equation of state coefficient w is a free parameter. Therefore,besides H and Ω m , for which we assume the same uniform priors as in ΛCDM case, weshould set a prior on w as well. To calculate the posterior, we assumed a uniform prior w ∼ U ([ − . , − . w -parameter) and other parameters, the posterior distributions are supposed to be wider when w is set free. Indeed, in Fig. 3 we present the results for the w CDM model with widerconfidence contours. Despite of this degeneracy, the results indicate that our approach is stillable to recover cosmological parameters within 1- σ for the distance uncertainty level up to20%. Concerning the uncertainty of cosmological inference as a function of sample size (Tab. 1),one can see that with the biggest sample of 100 × measurements, the 1 − σ confidenceregion for H would be as big as ± .
2, which corresponds to ∼
7% precision.
In this paper, we investigated the prospects of using gravitational waves from inspirallingcompact binaries as standard sirens for the cosmological inference. Though the redshift z – 8 – able 1 : The 1- σ confidence region for H as a function of sample size. The D L uncertaintyis distributed as U (5% , H N ( × ) 2 5 10 50 100ΛCDM ± . ± . ± . ± . ± . w CDM ± . ± . ± . ± . ± . P obs ( z i | (cid:126) Ω , I ),is predictable, given the intrinsic merger rate of DCO events and the expected sensitivity ofthe detector, i.e. the ET in the case we discussed. Then, we estimated the precision andaccuracy of our approach using simulated mock data generated by combining the P obs ( z i | (cid:126) Ω , I )with the fiducial cosmological model.We repeatedly generated the realizations of datasets each containing 10,000 systems andcomputed the maximization distribution for the parameters ( (cid:126) Ω) of ΛCDM and w CDM model.Because the data were simulated from the fiducial model, the true values of cosmologicalparameters were assumed as known. Therefore the inferred values of these parameters allowedto study both the precision and accuracy of the inference as well as their changes as afunction of data quality, i.e. the uncertainty of D L measurements. We stress again that inthis work, we assumed that the only observable quantity was the luminosity distance whoseuncertainty level was not perfectly known. Consequently, the luminosity distance uncertaintywas assumed as a free parameter in the analysis and marginalized over. Our results showthat one can obtain the non-biased cosmological inference at different D L uncertainty levelsup to 20% in agreement with pre-assumed true values within 1- σ level.We also investigated the confidence regions for the inferred cosmological parameters asa function of sample size. We found that if one aims at the H measurement contributingto the resolution of the tension in ΛCDM model between Planck and other low redshiftmeasurements, one needs a sample size of ∼ × events (see. Tab. 1). Even though itseems large, such a sample size could be provided in one year of successful operation of theET. In the literature concerning LIGO/Virgo or LISA detectors, different concepts concern-ing redshift priors have been discussed. Mostly, the idea there was to incorporate all potentialhost galaxies [14, 23, 31] or clusters [22] from wide-field sky surveys such as the SDSS. Re-cently, [32] performed a statistical standard siren analysis of GW170817 which did not utilizeknowledge of NGC 4993 as the unique host galaxy. By weighting the host galaxies by stellarmass or star-formation rate they obtained consistent results with potentially tighter con-straints. Admittedly, such statistical methods were claimed to be able to constrain H toseveral percent levels. Such approaches are only applicable whenever the redshift proxies canbe well assessed or the electromagnetic counterpart is detected. However, the GW eventsregistered in the era of the ET would come from very high redshifts at which host galax-ies would not be available to the wide-field surveys. Hence, it is possible that the approach– 9 – H P D F Figure 4 : The inferred H when the standard scenario for ˙ n ( z s ) was used to generate themock data, but the delayed SN scenario was used as a prior. The red line is the true valueof H used in mock data generation.proposed in this paper might be the only option in the era of 3rd generation of GW detectors.In this work, we used only one particular population synthesis model of inspiral rates˙ n ( z s ) (i.e. the “low-metallicity” standard scenario) to generate the mock data, and thento derive cosmological parameters. Even though the difference between each populationmodel by Dominik et al. [30] is small, it can be expected that this small difference could beamplified by the selection effects inherent to the GW observations. This means that if thewrong prior of ˙ n ( z s ) is assumed, an extra bias would be introduced. To test this, we tookfor the simulations the ˙ n ( z s ) according to the standard scenario, but used the delayed SNscenario as a prior for the inference. In order to directly observe the bias on H induced bysuch mismatch in assumptions, we fixed the value of Ω m during the fitting. The result shownin Fig. 4 from which the bias on the inferred H is clearly seen. In the future, we hope that˙ n ( z s ) would be known better, following better understanding of the DCO evolution andrefinement of population synthesis models based on existing and forthcoming GW detectionsby LIGO/Virgo. Let us also remark that our approach could also be applied to constrainingthe right scenario for ˙ n ( z s ) — one can set the cosmology as prior and select the best scenario.Assessment of this idea would require more extensive tests and simulations, and is left forthe future work.Furthermore, the ˙ n ( z s ) also depends on the cosmological model, since the populationsynthesis models predict coalescence rate as a function of time, and time-redshift relationshould be used. We have avoided this problem by adopting the same cosmology as Dominiket al. [30] have used. This issue deserves more comprehensive studies in the future usingparametrized models for the DCO population.Even though it was shown in [20] that with about 10 lensed GW and EM signals (realisticnumber for the ET) one would achieve a sub-percent accuracy of the H measurement, yet– 10 –t would not be fast and easy to gather such a sample. In particular, the pipelines to identifylensed GW events are still under development. On the contrary, luminosity distances inferredfrom the inspiral waveforms would be routinely measured in significant numbers quicklybuilding up the samples we discussed. Therefore it would be promising to develop furtherthe method we proposed. Acknowledgments
We would like to express our deep gratitude to the referee for a thorough reading of ourpaper at each stage of revisions and his/her time devoted to constructively discuss the issuesthat needed improvements. Especially the comments regarding the bias associated with as-sumptions of uncertainty level and population synthesis scenario are gratefully acknowledged.This constructive engagement of the referee allowed to improve the paper substantially. Wethank Xi-Long Fan for contributing to the formulation of this work; unfortunately, he did notwish to be an author because of restrictions required by the LIGO Scientific Collaborationpolicies. This work was supported by the National Basic Science Program (Project 973) ofChina under (Grant No. 2014CB845800), the National Natural Science Foundation of Chinaunder Grants Nos. 11633001 and 11373014, the Strategic Priority Research Program of theChinese Academy of Sciences, Grant No. XDB23000000 and the Interdiscipline ResearchFunds of Beijing Normal University. X. Ding acknowledges support by China PostdoctoralScience Foundation Funded Project (No. 2017M622501). M.B. was supported by the KeyForeign Expert Program for the Central Universities No. X2018002 K. Liao was supportedby the National Natural Science Foundation of China (NSFC) No. 11603015. Z. Li wassupported by NSFC under Grants Nos. 11505008.
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