A model-independent constraint on the Hubble constant with gravitational waves from the Einstein Telescope
Sixuan Zhang, Shuo Cao, Jia Zhang, Tonghua Liu, Yuting Liu, Shuaibo Geng, Yujie Lian
aa r X i v : . [ a s t r o - ph . C O ] S e p September 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2
A model-independent constraint on the Hubble constant withgravitational waves from the Einstein Telescope
Sixuan Zhang
Department of Astronomy, Beijing Normal University, Beijing 100875, China;
Shuo Cao ∗ Department of Astronomy, Beijing Normal University, Beijing 100875, China;[email protected]
Jia Zhang
School of Physics and Electrical Engineering, Weinan Normal University, Shanxi 714099,China;
Tonghua Liu
Department of Astronomy, Beijing Normal University, Beijing 100875, China;
Yuting Liu
Department of Astronomy, Beijing Normal University, Beijing 100875, China;
Shuaibo Geng
Department of Astronomy, Beijing Normal University, Beijing 100875, China;
Yujie Lian
Department of Astronomy, Beijing Normal University, Beijing 100875, China;
Received Day Month YearRevised Day Month YearIn this paper, we investigate the expected constraints on the Hubble constant fromthe gravitational-wave standard sirens, in a cosmological-model-independent way. In theframework of the well-known Hubble law, the GW signal from each detected binarymerger in the local universe ( z < .
10) provides a measurement of luminosity distance D L and thus the Hubble constant H . Focusing on the simulated data of gravitationalwaves from the third-generation gravitational wave detector (the Einstein Telescope,ET), combined with the redshifts determined from electromagnetic counter parts andhost galaxies, one can expect the Hubble constant to be constrained at the precision of ∼ − with 20 well-observed binary neutron star (BNS) mergers. Additional standard-siren measurements from other types of future gravitational-wave sources (NS-BH andBBH) will provide more precision constraints of this important cosmological parameter.Therefore, we obtain that future measurements of the luminosity distances of gravita-1 eptember 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2 Zhang, et al. tional waves sources will be much more competitive than the current analysis, whichmakes it expectable more vigorous and convincing constraints on the Hubble constantin a cosmological-model-independent way.
Keywords : cosmological parameters - gravitational wavesPACS numbers:
1. Introduction
The Hubble constant H , which illustrates the expansion rate of the universe today,plays a significant role in the deep understanding of fundamental physics ques-tions. Therefore, precise and accurate measurement of the Hubble constant is oneof the most fundamental issues influencing our understanding of the Universe. Al-though multiple paths to independent estimates of H have been accessed by manyastrophysical probes - in particular the observations of type Ia supernovae (SNeIa) and the first acoustic peak location in the pattern of anisotropies of the Cos-mic Microwave Background Radiation (CMBR) - two issue should be reminded.First of all, the Hubble constant cannot be constrained directly from CMB obser-vations, e.g. the latest Planck 2015 results, but must be inferred by assuming apre-assumed cosmological model (the standard ΛCDM model). It was found in that many parameters (i.e., the matter density parameter Ω m ) become degeneratewith the Hubble constant: a high value of Ω m will lead to a low value of H .When relaxing the ΛCDM assumption by introducing an exotic source of matterwith negative net pressure, the so-called dark energy to explain cosmic acceleration,the strong degeneracy between various cosmological parameters (such as the cos-mic equation of state w = p/ρ , or the interaction term between dark matter anddark energy) and the the Hubble constant was also noticed and discussed in. Second, alternative methods of deriving the Hubble constant from cosmological-model-independent probes, focus on the luminosity distance D L ( z ) using SNe Ia asstandard candles at lower redshifts and the time-delay distance D ∆ t using timedelays of strong lensing systems as standard rulers.
10, 11
These results showed thatrecent determinations of H , from the Supernovae H for the Equation of Stateof Dark Energy (SH0ES) collaboration and a joint analysis of six gravitationallylensed quasars with measured time delays, are in strong tension with with thePlanck CMB measurements. The debate about the discrepancy between the Hub-ble constant measured locally and the value inferred from the Planck survey, haskept the discussion about a local underdensity alive. Therefore, such tension mayforce the rejection of the standard ΛCDM model or indicate new physics incorpo-rated into cosmology.However, it is worth noting that all of these H measurements performed throughelectromagnetic(EM) radiations. Gravitational wave offers an independent methodof determining H and resolving the H discrepancy. The inspiraling and merg-ing compact binaries consisting of neutron stars (NSs) and black holes (BHs), canbe considered analogously as the supernovae (SNe) standard candles, namely theeptember 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2
Model-independent constraint on the Hubble constant standard sirens. The most well-established method for measuring H is through theHubble law, based on the observations of the local Hubble flow velocity of a sourceand the distance to the source in the local Universe. Gravitational wave signalsfrom inspiraling binary systems are “standard sirens” in that the absolute value oftheir luminosity distances, and thus the distances to GWs in the Hubble flow canbe determined, and therefore can be used to infer H independent of any other dis-tance ladders: standard sirens are self-calibrating. The local Hubble flow velocity istypically obtained via the identification of electromagnetic counterpart and the hostgalaxy. The breakthrough took place with the first direct detection of GW170817 inboth gravitational waves and electromagnetic waves, which has opened an era ofgravitational-wave multi-messenger astronomy. determined the Hubble constantto be H = 70 . +12 . − . km/s/Mpc, which is well consistent with the currently existingmeasurements (CMB and SNe Ia). In the past years, many papers have studied thepossibility of the GW as standard sirens, and there are also estimations on theconstraint ability of the Hubble constant by the simulated GW data. Followingthis direction, extensive efforts have been made to use simulated GW data to placeconstraints on this important cosmological parameter, which showed that the con-straint ability of GWs is much better than the traditional probes (with a precisionof approximately two percent), if hundreds of GW events have been observed byLIGO and Virgo within five years.Inspired by the previous work, in this paper we explore the ability of the grav-itational wave detections of the Einstein Telescope(ET) to constrain the Hubbleconstant in a model-independent way based on the Hubble law. More importantly,the third-generation ground-based detector, i.e. Einstein telescope (ET), will beten times more sensitive in amplitude than the advanced ground-based detectors,covering the frequency range of 1-10 Hz. Therefore, 10 − GW events will bedetected by ET per year and about one of a thousand events will locate in thelow-redshift range of [0 , . This expected considerable number of low-redshiftGW events implies that it is possible to use these systems for estimating the Hubbleconstant, by combining the measurements of the sources’ redshitfs from, for exam-ple, the electromagnetic (EM) counterpart. In this paper, we explore the ability ofthe gravitational wave detections to constrain H . As result, we obtain that futureresults from GWs will be much more competitive with current limits from currentanalyses. The paper is organized as follows. In Section II we describe the method-ology used in our work. The simulated GW data and the error estimation of thestandard siren measurements are presented in Section III. In Section IV, we presentthe constraints these data put on the Hubble constant. Finally, the conclusions anddiscussions are presented in Section V. Throughout this paper, the Hubble constant H = 70 . Zhang, et al.
2. Methodology
We assume that in the homogeneous and isotropic universe, its geometry can bedescribed by the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric ds = − dt + a ( t ) − kr dr + a ( t ) r d Ω , (1)where t is the cosmic time, a ( t ) is the scale factor whose evolution depends onthe matter and energy contents of the universe, while k represents the spatial cur-vature. k = +1 , − , k = − k/H . In the framework ofFLRW metric, at nearby distances the mean expansion rate of the Universe is wellapproximated by the expression v H = H D H , (2)where v H is the local Hubble flow velocity of a source, D H is the Hubble distanceto the source (all cosmological distance measures, such as luminosity distance, co-moving distance and angular diameter distance can not be distinguished at lowredshifts). In this case, the exact value of other cosmological parameters (such asthe matter density parameter Ω m , the cosmic equation of state w ) is not our con-cern, since they are similarly insensitive to the distance measurements. In a closeduniverse, this linear redshift-distance relation is usually expressed in the form of cz = H D H , where c is the speed of light and z is the redshift of the galaxy. However, with the exception of cosmological models in which the Hubble parameteris a constant at higher redshifts, the Hubble law is linear only for low redshifts( z ≪ When it comes to a higher redshift, such approximation can lead to asignificant error in measuring D H and one needs to consider the relativistic cor-rection of the approximation, z = p (1 + vc / − vc ). With this correction, thecorrected Hubble law can be rewritten as (1 + z ) − z ) + 1 c = H D H . (3)More specifically, by taking the relativistic correction form, it is estimated thatthe observable distances (luminosity distance, angular diameter distance, etc.) willdiffer from the Hubble distance by less than 5% (when z ≤ In this paper, wechoose to implement a stringent redshift criterion when the relativistic correctionis considered (i.e, z < . H , on the one hand, we must also measure a redshift foreach binary merger. Throughout, we take the redshift z to be the peculiar-velocity-corrected redshift, i.e., the redshift of the source if it is located in the Hubble flow. Itshould be noted two different cases will be considered in this work: the GW sourcesare caused by binary merger of a neutron star with either a neutron star or blackhole, which can generate an intense burst of γ -rays (SGRB) with measurable sourceredshift, or the redshift information either comes from a statistical analysis over aeptember 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2 Model-independent constraint on the Hubble constant catalogue of potential host galaxies, when the GW sources are caused by compactbinaries consisting of black holes (BHs).On the other hand, different from the luminosity distance measurements fromEM observations, the GW signal from a compact binary system can provide themeasurement in another way: through its dependency on the amplitude of the GWevent and the so-called chirp mass of the binary system, which can be measuredfrom the GW signals phasing. More specifically, the GW signal from each detectedbinary merger (with component masses m and m ) provides a measurement of D L ,which can be directly inferred from the amplitude A = 1 D L q F (1 + cos ( ι )) + 4 F × cos ( ι ) × p π/ π − / M / c (4)of Fourier transform of the strain h ( t ) of GW signal H ( f ) = A f − / exp[ i (2 πf t − π/ ψ ( f / − φ . )] , (5)where t is the epoch of the merger, while the definitions of the functions ψ and ϕ (2 . can be found in. In the transverse-traceless (TT) gauge, the strain can bewritten as the linear combination of the two polarization states h ( t ) = F × ( θ, φ, ψ ) h × ( t ) + F + ( θ, φ, ψ ) h + ( t ) (6)where h × and h + are the two independent components of the GW tensor, F × and F + are the beam pattern functions, ψ denotes the polarization angle, and ( θ, φ )are the location angles of the source in the sky, which describes the location of thesource relative to the detector. The exact forms of pattern functions for ET aregiven by F (1)+ ( θ, φ, ψ ) = √
32 [ 12 (1 + cos ( θ )) cos(2 φ ) cos(2 ψ ) − cos( θ ) sin(2 φ ) sin(2 ψ )] ,F (1) × ( θ, φ, ψ ) = √
32 [ 12 (1 + cos ( θ )) cos(2 φ ) sin(2 ψ )+ cos( θ ) sin(2 φ ) cos(2 ψ )] . (7)and the other two interferometer’s antenna pattern functions are F (2)+ , × ( θ, φ, ψ ) = F (1)+ , × ( θ, φ + 2 π/ , ψ ) and F (3)+ , × ( θ, φ, ψ ) = F (1)+ , × ( θ, φ + 4 π/ , ψ ), since the three in-terferometers of the ET are arranged in an equilateral triangle. We can definethe chirp mass M c = M η / and its corresponding observational counterpart as M c, obs = (1 + z ) M c, phys (with the total mass M = m + m and the symmetricmass ratio η = m m /M ). ι is the angle between inclination of the binary system’sorbital angular momentum and line of sight. One should note that, from observa-tional point of view, the maximal inclination is about ι = 20 ◦ and averaging theeptember 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2 Zhang, et al.
Fisher matrix over the inclination ι with the limit ι < ◦ is approximately equiv-alent to taking ι = 0. Therefore, one can take ι = 0 for simplicity, as argued in and references therein. M p c M p c Fig. 1. The luminosity distance measurements from 200 low-redshift GW events generated frominspiraling binary neutron stars (left panel). Details of the measurements in the redshift range0 .
06 0 .
07 are also shown for comparison (right panel).
3. Simulation and error estimation
In this section we simulate GW events based on the Einstein Telescope, the thirdgeneration of the ground-based GW detector. In the simulation, the mass distribu-tion of NS is chosen to be uniform in the in interval of [1.4, 2.4] M J . We adopt theredshift distribution of the GW sources observed on Earth, which can be writtenas P ( z ) ∝ πD c ( z ) R ( z ) H ( z )(1 + z ) , (8)where H ( z ) is the Hubble parameter of the fiducial cosmological model, D c ( z ) isthe co-moving distance at redshift z , and R ( z ) represents the time evolution of theburst rate taken as R ( z ) = z, z ≤ (5 − z ) , < z < , z ≥ . (9)For the network of three independent ET interferometers, the combined signal-to-noise ratio (SNR) of the GW waveform is ρ = vuut X i =1 (cid:10) H ( i ) , H ( i ) (cid:11) . (10)eptember 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2 Model-independent constraint on the Hubble constant Here the inner product is defined as h a, b i = 4 Z f upper f lower ˜ a ( f )˜ b ∗ ( f ) + ˜ a ∗ ( f )˜ b ( f )2 dfS h ( f ) , (11)where ˜ a ( f ) and ˜ b ( f ) are the Fourier transforms of the functions a ( t ) and b ( t ). S h ( f )is the one-side noise power spectral density (PSD) characterizing the performanceof a GW detector S h ( f ) = S [ x p + a x p + a f ( x )] (12)where f ( x ) takes the form as f ( x ) = 1 + b x + b x + b x + b x + b x + b x c x + c x + c x + c x (13)with the definition of x = f / The lower cutoff frequency f lower is fixed at 1Hz. The upper cutoff frequency, f upper , is decided by the last stable orbit (LSO), f upper = 2 f LSO , where f LSO = 1 / (6 / πM obs ) is the orbit frequency at the LSO,and M obs = (1 + z ) M phys is the observed total mass. Meanwhile, the signal isidentified as a GW event only if the ET interferometers have a network SNR of ρ > . Different sources of uncertainties are included in our simulation of luminositydistance D L . Firstly, in the standard framework of GW data analysis, the infor-mation of relevant parameters ( θ ) are derived by fitting the frequent-domain wavemodel h ( f ; θ ) to the frequent-domain detector data d ( f ). If the noise is stationaryand Gaussian, the likelihood function L is defined as ln L = − Z ∞ df | d ( f ) − h ( f ; θ ) | S h ( f ) (14)where d ( f ) and S h ( f ) are implicit functions of the calibration parameters ( λ ). Spe-cially, considering the difference between the detector’s true calibration parameters( λ t ) and the calibration parameters used to produce the strain data from powerfluctuations ( λ ), the generated calibration error should be included into the param-eter estimation pipeline, described by the parameter ∆ λ = λ − λ t . In this analysis,we take the calibration-induced error ( σ cal D L ) as one third of the noise-induced er-ror ( σ noi D L ), when the parameter estimation is dominated by systematics induced bydetector noise. Secondly, when the error on luminosity distance is uncorrelated with errors onthe remaining GW parameters, the noise-induced error can be estimated with Fishermatrix by σ noi D L ≃ s(cid:28) ∂ H ∂D L , ∂ H ∂D L (cid:29) − , (15)It should be pointed out that in the ET era, we will be confronted with a family ofphenomenological waveforms that incorporates the dynamics of the inspiral, merger,and ringdown phases of the coalescence,
37, 38 while the inclusion of the merger phaseeptember 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2 Zhang, et al. and ringdown phase may helpfully break the degeneracy between the luminositydistance D L and inclination angle ι .
39, 40
In this paper, following the procedureextensively applied in the literature,
20, 25 we focus only on the inspiral phase of theGW signal, with the corresponding instrumental error written as σ inst D L = q ( σ noi D L ) + ( σ cal D L ) (16)where σ noi D L is the noise-induced error that can be estimated as σ noi D L ≃ D L ρ .
34, 36
Note that the maximal effect of the inclination on the SNR is a factor of 2 (between ι = 0 ◦ and ι = 90 ◦ ), for a conservative estimation of the correction between D L and ι . Thirdly, following the strategy described by, weak lensing has been estimatedas a major source of error on D L ( z ) for standard sirens. For the ET we estimatethe uncertainty from weak lensing according to the fitting formula of σ lensD L /D L =0 . z . Therefore, the distance precision per GW is taken to be σ sta D L = q ( σ inst D L ) + ( σ lens D L ) . (17)Finally, precise redshift measurements are the crucial point of our idea. Weconsider two cases: In the with-counterpart case, with the observation of the EMcounter parts, the redshift of a GW event can be determined. We assume thatthe EM counterpart is close enough to its host galaxy that the host can be un-ambiguously identified, and we can measure its sky position and redshift. In theprevious works, the uncertainty of the redshift measurement is always ig-nored, because it is ignorable compared to the uncertainty of the luminosity dis-tance. However, in the case of local universe, the redshift uncertainty caused bythe uncertainty of peculiar velocity should be taken into account. Throughout thispaper, we take the redshift z to be the peculiar-velocity-corrected redshift (i.e., theredshift of the source located in the Hubble flow) and apply two different cases.Following the procedure performed in the recent analysis, a standard deviationof cσ z =200 km/s is assumed for each BNS and BH-NS system (with a direct EMcounterpart), which is a typical uncertainty for the peculiar velocity correction wellconsistent with the peculiar velocity measurement of NGC 4993 (the host galaxy ofGW170817). Note that the recent analysis of BH-NS mergers has discussed thepossibility that the neutron star can be tidally disrupted and emit electromagneticradiation, depending on the mass ratio and the black hole spin. For the latter casein which the host galaxy of a GW event can not be identified (BBH), the redshiftcomes from a statistical analysis over a catalogue of potential host galaxies, whichwill be discussed later.Now the final key question required to be answered is: how many low-redshiftGW events can be detected per year for the ET? Focusing on the GW sources causedby binary merger of neutron stars (with detectable EM counterpart measurablesource redshift), it is revealed that the third generation ground-based GW detectorcan detect up to 10 - 10 events, with the upper detection limit of z ∼ . eptember 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2 Model-independent constraint on the Hubble constant Following our detailed calculation that only 0.1% of the total GW events will belocated in the redshift range of [0 , . low-redshiftevents could be used in our analysis. In addition, recent analysis revealed thatthe five-detector network including LIGO, Virgo, KAGRA and LIGO-India plansto detect ∼
40 events per year, if the designed sensitivity of the network could beachieved in the future. Therefore, assuming the luminosity distance measurementsobey the Gaussian distribution, we simulate 200 GW events of BNS merging usedfor statistical analysis in the next section, the redshift distribution of which is shownin Fig. 1.We summarize the main route of our method as follows: • Simulate 200 GW events according to the redshift distribution in Eq. (8).The angles describing the position of each BNS system are randomly sam-pled within the interval of θ ∈ [0 , π ) and φ ∈ [0 , π ). • Calculate the luminosity distance D L ( z ) according to Eq. (2)-(3). Randomlysample the mass of neutron star within [1 . , . M J . Evaluate the signal-to-noise ratio (SNR) and the error σ D L when the SNR of the detector networkreaches above 8. • For a confirmed GW event, the statistical error of luminosity distance σ D L sta can be figured out through Eq. (17). Moreover, considering the effectof the peculiar velocity of the host galaxy v pec , we also add the systematicaluncertainty of observed redshift z obs to project uncertainties σ sys D L onto thefinal uncertainty of distance estimation (Eq. (3)). We therefore take thetotal uncertainty on the luminosity distance as σ tot D L = q ( σ sta D L ) + ( σ sys D L ) .The observed luminosity distance D L follows a Gaussian distribution whosemean is D fidL and variance is σ tot D L , i.e., D L ∼ N ( D fidL , σ tot D L ).
25 50 75 100 125 150 175 200N68.569.069.570.070.5 H ( k m / s / M p c )
25 50 75 100 125 150 175 200N0.0020.0040.0060.0080.0100.012 Δ H Δ H Fig. 2. Left: Inferred Hubble constant as a function of the number of GW events (BNS mergerswith EM counterparts). Right: The corresponding precision of the Hubble constant constraints fora variable number of GW events (BNS mergers with EM counterparts). eptember 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2 Zhang, et al.
4. Results and discussions
In order to place constraints on the Hubble constant with MCMC method, thelikelihood estimator is determined by χ statistics χ = N X i =1 (cid:18) D obsL ( z i ) − D thL ( z i ) σ D L,i (cid:19) (18)where N denotes the number of data sets, D thL is the predicted luminosity distancevalue in the Hubble law and D obsL is the measured value with a uncertainty of σ D L in the simulated data.Fig. 2 shows the precision of the curvature parameter assessment as a function ofGW sample size for future ET detector, where weighted means and correspondingstandard deviations are illustrated for comparison. Projected fractional error forthe standard siren H measurement for BNSs is also shown. One can see that, evenwith about 20 well-observed GW events due to BNS mergers one can expect theHubble constant to be estimated with the precision of δH ∼
1% (at the 1 σ level),if it is possible to independently measure a unique redshift for all BNS events. Moreimportantly, we find that in this counterpart case, the fractional H uncertaintywill be proportional to 1 / √ N , where N is the number of BNS mergers detected bythe ET. Still, there are several remarks that remain to be clarified as follows.Firstly, in the above analysis we assume that EM counterparts are detectablefor all BNS systems. For example, as GW170817 demonstrated, for the mergerof a BNS system it is possible to identify a kilonova counterpart independentlyof the short γ ray burst (SGRB). These EM counter parts can help us in locat-ing the events in the sky and identifying the host galaxy of the event, and thelocating ability can be improved with the use of H.E.S.S.Imaging Air CherenkovTelescopes(IACTs). However, from observational point of view we don’t expect toobserve EM counter parts for all GW events.For example, the optical counter parts,kilo-nova, are too dim to be observed in large luminosity distance. More specifi-cally, SGRBs are strongly beamed phenomena which carry a great deal of energyand we can only detect them when they are almost face-on, which will significantlydecline the number of detectable GW events with SGRB. Following the recentdiscussion of GW170817, the eject matter from a merging BNS system can cause asecondary radiation which might power the radiation for a longer time and a widerradiation angle. This suggest that only 10% of the 200 BNS events detected byET will be available with the EM counterparts such as SGRB. Therefore, afterabout 20 gravitational-wave standard sirens, the fractional uncertainty on H couldstill reach 1% (at the 1 σ level) by the end of five years of ET at design sensitivity,sufficient to arbitrate the current tension between local and high- z measurementsof H .Secondly, besides BNS with EM counterparts, there are other types of GWsources that can contribute in providing precise determination of the Hubble con-stant. Theoretically, ET could also detect a large number of GW signals for blackeptember 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2 Model-independent constraint on the Hubble constant hole - neutron star (BH-NS) merger systems and binary black hole (BBH) mergersystems. These two types of GW sources are also of concern to our investigationin this paper. While we do not expect EM counter parts from BBH systems, wedo want to observe the EM counter parts from BN-BH systems, and the opticalluminosity depends on the property of the neutron star. For the former type, theelectromagnetic (EM) signals are emitted during the merger processes, allowing usto determine the redshift of sources. In the framework of ET project, the expecteddetect rate of BNS and BHNS are in the same order of 10 to 10 . Therefore, it isreasonable to assume that with the observation of 20 BNS merger with EM counterparts, one can detect 20 BHNS merger with their EM counterparts as well. For thelatter case where a unique counterpart cannot be identified for a BBH merger, itis possible to carry out a measurement of the Hubble constant using the statisti-cal approach, i.e., there are still other methods to identify their host galaxies, in astatistical way which might not be that accurate. More specifically, we will applythe methodology proposed in, in which a galaxy catalogue is used to describe allpotential host galaxies in the case that the EM counterpart is absent. The simulatedgalaxy catalogue is constructed by distributing galaxies uniformly in the co-movingvolume of 10000 Mpc with a number density of 0.02 Mpc − , each of which has thesame probability to be the host galaxy of the GW event. Given the detailed calcula-tion presented in, after two years of full operation, the LIGO and Virgo network isexpected to detect ∼
16 BBH events (with well estimation of statistical redshifts) inthe local universe (located in the volume of 10000 Mpc ), which will lead to a 10%of Hubble constant measurement. Therefore, it is reasonable to simulate 30 BBHsystems with redshift determination in the framework of ET configurations (in thesimulation, the mass distribution of BH is chosen to be uniform in the in interval of[3,10] M ⊙
21, 22 ). A representative result is illustrated in Fig. 3, in order to comparethe H constraints with different types of gravitational-wave events. The left panelis obtained using BNS GW sirens, while the middle panel is obtained using BHNSsirens, which is compared in the right panel with BBH GW sirens, respectively. Aswas noted in the previous work based on the LIGO and Virgo network, the errorbars will be greatly reduced when different types of GW events are included. Mean-while, benefit from a larger total mass and smaller merger frequency than BNS, onecould also expect a larger SNR for a specific BHNS event, which will greatly con-tribute to the distance measurements and thus the standard siren H constraint. However, constraints from BBH systems without counterparts are inferior, due tothe larger number of potential host galaxies compared with other two types of GWevents. Now it is worthwhile to compare our forecast results with some previous H tests fitting the Hubble constant in different cosmological models, based on the“ D L − z ” relation at higher redshifts in the GW domain. Using the informationof luminosity distances and redshifted chirp masses for a catalog of BNSs detectedby an advanced era network, studied a technique to obtain constraints on theeptember 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2 Zhang, et al.
BH-NS BNS BNS+BHNS BNS+BHNS+BBHType of source69.669.870.070.270.4 H ( k m / s / M p c ) Fig. 3. Constraints on the Hubble constant with three types of GW sources, 20 BNS with EMcounterparts, 20 BH-NS with EM counterparts, and 30 BBH without EM counterparts but withwell-estimated statistical redshifts.
Hubble constant and NS mass-distribution parameters simultaneously. It was foundthat H could be estimated at the precision of 10% with ∼
100 such kind of GWevents. Meanwhile, in the framework of a range of ground-based detector networks, examined how well distances (and thus cosmological parameters) can be measuredfrom BNSs with electromagnetic counterparts (such as the associated SGRB). Theanalysis results revealed that H could be measured with a fractional error of ∼
13% with 4 GW-SGRB events, which could be improved to ∼
5% with 15 eventsdetected by the advanced LIGO-Virgo detector network. Focusing on constraintability of the third-generation gravitational wave detector (the Einstein Telescope),the recent analysis showed that with the simulated data of 500 standard sirens, onecan constrain the Hubble constant with an accuracy comparable to the most recentPlanck results. By considering our results and those from, our results showthat strong constraints on the Hubble constant can be obtained in a cosmological-model-independent fashion. Such conclusion agrees very well with that obtained inthe framework of LIGO-Virgo detector network. Measuring the Hubble constant ( H ) independent of CMB observations is oneof the most important complementary probes for understanding the nature of theUniverse. Therefore, the fractional uncertainty on H will reach 1% by the end offive years of ET at design sensitivity, which furthermore strengthens the probativepower of the third generation ground-based GW detectors to inspire new observingprograms or theoretical work in the moderate future. Finally, one should note that,in order to achieve this goal, dedicated observations of the sky position of each hostgalaxy (that is, with negligible measurement error) would be necessary. Althoughthe GW distance posterior changes slowly over the sky and therefore is not sensi-tive to the precise location of the counterpart, obtaining such measurements for asample of different types of GW events would require substantial follow-up efforts,which can lead to significant improvements in the distance, and hence H measure-eptember 10, 2020 0:32 WSPC/INSTRUCTION FILE ms2 Model-independent constraint on the Hubble constant ments. We also hope future observational data such as strongly lensed gravitationalwaves (GWs) from compact binary coalescence and their electromagnetic (EM)counterparts systems,
47, 48 precise measurements of the Hubble parameter obtainedby cosmic chronometer and radial BAO size methods, and VLBI observationsof compact radio quasars with higher sensitivity and angular resolution mayimprove remarkably the constraints on this key cosmological parameter.
Acknowledgments
We are grateful to Jingzhao Qi for helpful discussions. This work was supported byNational Key R&D Program of China No. 2017YFA0402600, the National NaturalScience Foundation of China under Grants Nos. 11690023, 11373014, and 11633001,the Strategic Priority Research Program of the Chinese Academy of Sciences, GrantNo. XDB23000000, the Interdiscipline Research Funds of Beijing Normal Univer-sity, and the Opening Project of Key Laboratory of Computational Astrophysics,National Astronomical Observatories, Chinese Academy of Sciences.