Determining the Hubble constant from gravitational wave observations of merging compact binaries
Samaya Nissanke, Daniel E. Holz, Neal Dalal, Scott A. Hughes, Jonathan L. Sievers, Christopher M. Hirata
DDraft version July 11, 2013
Preprint typeset using L A TEX style emulateapj v. 5/2/11
DETERMINING THE HUBBLE CONSTANT FROM GRAVITATIONAL WAVE OBSERVATIONS OF MERGINGCOMPACT BINARIES
Samaya Nissanke , Daniel E. Holz , Neal Dalal , Scott A. Hughes ,Jonathan L. Sievers , Christopher M. Hirata Draft version July 11, 2013
ABSTRACTRecent observations have accumulated compelling evidence that some short gamma-ray bursts(SGRBs) are associated with the mergers of neutron star (NS) binaries. This would indicate thatthe SGRB event is associated with a gravitational-wave (GW) signal corresponding to the final in-spiral of the compact binary. In addition, the radioactive decay of elements produced in NS binarymergers may result in transients visible in the optical and infrared with peak luminosities on hours-days timescales. Simultaneous observations of the inspiral GWs and signatures in the electromagneticband may allow us to directly and independently determine both the luminosity distance and redshiftto a binary. These standard sirens (the GW analog of standard candles) have the potential to providean accurate measurement of the low-redshift Hubble flow. In addition, these systems are absolutelycalibrated by general relativity, and therefore do not experience the same set of astrophysical sys-tematics found in traditional standard candles, nor do the measurements rely on a distance ladder.We show that 15 observable GW and EM events should allow the Hubble constant to be measuredwith 5% precision using a network of detectors that includes advanced LIGO and Virgo. Measuring30 beamed GW-SGRB events could constrain H to better than 1%. When comparing to standardGaussian likelihood analysis, we find that each event’s non-Gaussian posterior in H helps reduce theoverall measurement errors in H for an ensemble of NS binary mergers. Subject headings: cosmology: distance scale—cosmology: theory—gamma rays: bursts—gravitationalwaves INTRODUCTION
Gravitational wave (GW) standard sirens are the GWanalogs to traditional standard candles. They exem-plify multi-messenger astronomy (see Bloom et al. Caltech, Theoretical Astrophysics, California Institute ofTechnology, Pasadena, California 91125, USA Enrico Fermi Institute, Department of Physics, and Kavli In-stitute for Cosmological Physics, University of Chicago, Chicago,IL 60637, USA Department of Astronomy, University of Illinois, 1002 W.Green St., Urbana, IL 61801, USA Department of Physics and MIT Kavli Institute, MIT, 77Massachusetts Ave., Cambridge, MA 02139, USA Canadian Institute for Theoretical Astrohysics, University ofToronto, 60 St. George St., Toronto, ON M5S 3H8, Canada Perimeter Institute for Theoretical Physics, Waterloo, ONN2L 2Y5, Canada Jadwin Hall, Department of Physics, Princeton University,New Jersey, USA Astrophysics and Cosmology Research Unit, University ofKwazulu-Natal, Westville, Durban, 4000, South Africa
Universe’s expansion history, and thereby constrain cos-mological parameters such as the Hubble constant H ,the dark energy equation-of-state parameter w , and theaverage densities of matter Ω m and dark energy Ω λ .In this work we are interested in the constraints on theHubble constant that result from observations of GW-EM standard sirens at low redshift ( z < . H measurements at the few percent level, which have beendetermined using a combination of methods; see Suyu et al. (2012) for a brief review and references therein.All methods aim to i) provide independent H measure-ments, compared to derived constraints in the case ofthe Cosmic Microwave Background (CMB) and BaryonAcoustic Oscillations (BAO) measurements (see, for ex-ample, discussion of cosmological parameter constraintsby the Planck Collaboration et al. ∼
1% precision in random errors in the near future. Forinstance, the Planck satellite recently reported H of 67.4 ± − Mpc − , a low value compared to cosmicdistance ladder results of 73.8 ± − Mpc − (HSTCepheid–SNIa; Riess et al. et al. ± ± − Mpc − (Spitzer CHP; Freedman et al. ∼ σ discrepancy, suchdifferences suggest the necessity of additional indepen-dent measures. An independent ∼
1% measurement in H is especially desirable when improving figure-of-meritconstraints on the dark energy equation of state (Hu2005; Weinberg et al. a r X i v : . [ a s t r o - ph . C O ] J u l Nissanke et al.is crucial in having a useful measurement of w , especiallyone that might involve the falsification of a cosmologicalconstant as the origin of the dark energy. Standard sirenapproaches offer a fundamentally different set of system-atics, and therefore provide a valuable counterpart to allother methods.For the next generation of advanced ground based GWobservatories, comprising LIGO , Virgo , and otherGW interferometers, inspiralling and merging NS-NS andNS-BH binaries are expected to be the most numerousand characterizable events. For an advanced LIGO–Virgo network, predicted event rates for NS–NS binarymergers range from 0 . − , and such sys-tems could be detectable to distances of several hun-dred Mpc (Abadie, the LIGO Scientific Collaboration& the Virgo Collaboration 2010; Aasi, the LIGO Sci-entific Collaboration & the Virgo Collaboration 2013).Similar rate estimates apply to NS-BH binaries. Thesenumbers are, however, far more uncertain (by several or-ders of magnitude) since they rely on population synthe-sis alone (Abadie, the LIGO Scientific Collaboration &the Virgo Collaboration 2010). NS-BH binaries are de-tectable to much larger distances ( > et al. et al. et al. underestimates thedistance errors that we expect.In Nissanke et al. (2010) (hereafter N10) we revis-ited the D06 analysis by following a Bayesian approachfor estimating parameter errors instead of a Fisher ma-trix analysis (Finn 1992). We implemented a Metropolis-Hastings Markov Chain Monte–Carlo (MCMC) methodto explore the posterior distribution of GW model pa-rameters, in particular for deriving measurement errorsin the luminosity distance. This was done for both NS–NS and NS–10 M (cid:12) BH binaries, using a careful selectionprocedure to decide which binaries to include in the anal-ysis. N10 also used a more accurate inspiral waveform than was used in D06, and considered various North-ern and Southern Hemisphere ground-based detector net-works. Specifically, in N10 we examined measurementaccuracies at different network combinations of LIGOHanford, LIGO Livingston, Virgo, KAGRA (formallyknown as the “Large-scale Cryogenic Gravitational-waveTelescope”) , and AIGO/LIGO Australia (Munch et al. γ -rayobservatories, reduces measurement errors in the lumi-nosity distance. In N10 we found that the distance to anindividual NS-NS binary is measured to within a frac-tional error of roughly 20–60%, with 20–30% being rep-resentative for the majority of events in our detecteddistribution. If the orientation of the orbital plane ofthe NS-NS binary is assumed to be face-on (as might beexpected for beamed SGRBs), we found that distancemeasurement errors improve by approximately a factorof two. If we instead assume that the EM counterpartsare NS–10M (cid:12) BH mergers, we found that the distribu-tion of fractional distance errors ranges from 15–50%,with most events clustered near 15–25%. Assuming thatthe EM counterpart is a beamed SGRB reduces the mea-surement errors by a factor of two.In what follows we update and refine the analysis ofD06 and N10. This paper focuses on the implicationssuch measurements will have when constraining cosmo-logical parameters such as H . Subtle and importantdifferences exist between our current analysis and thatused in N10. Primarily, we are interested here in H constraints for the ensemble of GW-EM events, and notin individual distance measures for GW-EM event as inN10. In addition to collimated SGRBs, we also con-sider more speculative transients, such as macronovae orkilonovae, associated with NS binary mergers that emitisotropically in the optical or near infrared. For consis-tency with N10, we assume that the BHs in our NS-BHpopulations have masses of 10M (cid:12) . We note, however,that recent numerical relativity simulations suggest thattidal disruption, and hence EM signatures, may only oc-cur for NS-BH mergers with much smaller BH masses( ∼ (cid:12) ); see e.g., Taniguchi et al. (2007), Shibata& Taniguchi (2008), Shibata et al. (2009), Kyutoku etal. (2011), Foucart et al. (2011), Foucart (2012). Inaddition, we examine measurement accuracies for net-works including LIGO India, an advanced interferome-ter whose construction is currently under consideration(Sathyaprakash 2012). Finally, in contrast to traditionalstandard candles such as Cepheid variables and TypeIa supernovae, we wish to emphasize that GW standardsirens are independent of the cosmological distance lad-der. Compared with other recent standard siren studiesusing advanced GW interferometers (see e.g., Del Pozzo2012; Taylor, Gair & Mandel 2012; Messenger & Read2012), we consider the case where an EM observation ofthe NS binary inspiral is seen in conjunction with a GWmeasurement.In the next section we summarize the principles un-derlying GW standard sirens. We then outline how weselect our sample of binaries, and discuss the Bayesianmethod employed when estimating the luminosity dis-tance for each source. We conclude by discussing future http://gw.icrr.u-tokyo.ac.jp:8888/lcgt/ recision cosmology from gravitational waves 3constraints on H , the results of which are critically de-pendent on the assumed source population’s characteris-tics and the specific advanced detector network. STANDARD SIREN BINARIES
Many key observational methods employed in mappingout the expansion history of the universe rely on theluminosity distance-redshift relation: D L ( z ) = c (1 + z ) H √ Ω K sinh (cid:20)(cid:112) Ω K (cid:90) z H H ( z (cid:48) ) dz (cid:48) (cid:21) , (1)where the luminosity distance D L ( z ) is given as a red-shift integral of the Hubble parameter H ( z ), and theHubble constant H . For z (cid:38)
1, the evolution of H ( z )and D L ( z ) depends on cosmological parameters like Ω m and w , through the Friedmann equations. However, forlow redshifts z (cid:28)
1, the distance-redshift relation is welldescribed by D L ( z ) ≈ c z/H , independent of other cos-mological parameters. This is why measurements of thedistances to local sources, like Cepheids or GW standardsirens, can constrain the value of the Hubble constant.The inspiral signal of the GWs, modeled accuratelyusing the post-Newtonian (PN) approximation in gen-eral relativity, encodes geometrical and physical param-eters of the source (see e.g., Blanchet 2006). The sourceparameters include: the binary’s luminosity distance D L , its position on the sky n , its redshifted chirp mass M z = (1 + z ) m / m / / ( m + m ) / where m and m denote the mass of each compact object in the binary, itsredshifted reduced mass µ z = (1 + z ) m m / ( m + m ),its orientation on the sky given by its inclination angle ι , where cos ι = L · n / | L | and L is the binary’s orbitalangular momentum, and t c and Φ c , the time and GWphase at merger. A single detector a measures a linearcombination of the two GW polarizations: h a, meas ( θ ) = F + ( θ, φ, ψ ) h + + F × ( θ, φ, ψ ) h × . (2)The colatitude θ and longitude φ describe the binary’sposition on the sky n . The polarization angle ψ sets theinclination of the components of the unit vector ˆL or-thogonal to the unit vector ˆn . The components of thevector θ are all the various parameters (masses, angles,distance, etc.) upon which this measured waveform de-pends. For the two GW polarizations h + and h × , weuse the non-spinning restricted 3.5PN waveform in thefrequency domain (indicated by the ˜ h notation):˜ h + ( f ) = (cid:114) π − / M / z D L [1 + ( ˆL · ˆn ) ] f − / e i Ψ( f ) , (3)˜ h × ( f ) = (cid:114) π − / M / z D L ( ˆL · ˆn ) f − / e i Ψ( f ) − iπ/ , (4)which relies on the “stationary phase” approximation(Finn & Chernoff 1993), where the GW frequency f varies slowly over a single wave period. The GW phasein the frequency domain Ψ is computed to 3.5 PN order,where Ψ( f ) is given by:Ψ( f ) = 2 πf t c − Φ c − π π M z f ) − / × (cid:20) (cid:18) η (cid:19) ( πM z f ) / − π ( πM z f ) + 10 (cid:18) η + 617144 η (cid:19) ( πM z f ) / + π (cid:18) − η (cid:19) × (cid:20) πM z f ) / ln (cid:18) ff (cid:19)(cid:21) + (cid:20) − π − γ (cid:21) ( πM z f ) (cid:20)(cid:18) − π − − (cid:19) η + 760551728 η − η (cid:21) ( πM z f ) + π (cid:20) η − η (cid:21) ( πM z f ) / (cid:105) , (5)where M z = (1 + z )( m + m ) is the binary’s redshiftedtotal mass, η = µ z /M z is defined as the binary’s sym-metric mass parameter, γ is Euler’s constant, and f is aconstant frequency scale (Blanchet 2006). Central to theresults of this paper and N10, key geometrical source pa-rameters, such as D L and cos ι , appear in the amplitudeof each GW polarization ˜ h × ( f ) and ˜ h + ( f ). Therefore,measurement errors in D L and cos ι depend on the ex-tent of the degeneracy between these and other param-eters appearing only in the amplitude. We hence wishto assess how well we can disentangle each polarizationfrom the measured GW strain at a detector.Beyond the redshifting of masses (which is a sim-ple consequence of the cosmological redshift of alltimescales), this waveform model does not encode any in-formation about source redshift. To investigate the D L – z relationship, in this work we require an independent mea-sure of the source’s z by observing an EM counterpart.Other methods of obtaining the source’s redshift includeusing statistical arguments regarding the underlying NSbinary merger distribution (e.g., Taylor, Gair & Man-del 2012), or adding information about the NSs’ (non-redshifted) tidal deformation in the GW phase (e.g., Mes-senger & Read 2012). Del Pozzo (2012) uses galaxy cat-alogs to infer probabilistically sources’ redshifts. In con-trast, an EM counterpart detected with a GW measure-ment may also advantageously indicate the source’s skyposition. As was shown in N10, localizing the binary withindependent EM observations reduces measurement er-rors in parameters D L and cos ι by breaking correlationswith other parameters, and by reducing the dimension-ality of the parameter space. An EM counterpart mayalso bring information about the time of merger for thebinary, which will increase the detection range of a co-herent network by a factor of ∼ . et al. METHOD
This section summarizes the methodology used to de-rive H measurements for an ensemble of NS-NS or NS-BH binary mergers. We first outline the schema of ourmethod. Based on Sections 3 and 4 of N10, we thendescribe technical aspects of simulating anticipated dis- Nissanke et al.tance measurements. Schema of our method
We detail below how we construct the posterior prob-ability density function (PDF) in H for a set of de-tected GW-EM standard siren measurements. Deriving H constraints for ensembles of NS binary mergers re-quires particular care, as we expect that the majority ofevents will be detected at low SNR. Consequently, PDFsin H for individual events will depend significantly onour prior knowledge of the events’ parameter distribu-tions. In this study, how we select for GW-EM eventsdetermines our choice in specific priors.We envision a scenario in which we have detected atotal of m GW-EM events. Each event has both a GWmeasurement of distance, D L,i , and an EM redshift, z i ,where the subscript i represents a particular binary andruns from 1 . . . m . When combined these produce a valuefor H [see Eq. (1)]. We assume a model that is de-scribed by: i) the event’s underlying redshift distributiondenoted by X , ii) each source’s true redshift ˆ z i , and iii)the vector set of source parameters θ R i for a single mea-sured GW binary. As shown in Eq. (1), the luminositydistance for a specific GW event depends on both H and ˆ z , and thus, we do not include D L,i in θ Ri . The set θ R differs from the set θ , which includes the parameter D L and was used in N10. In N10 we were interested inluminosity distance measurements for individual eventsand not for ensemble GW-EM standard sirens as in thiswork.The data matrix { s i , z i } comprises the measured GWtime streams s i , and the set of observed EM redshifts z i for a set of m binaries. If we assume that our modelparameters are independent of one another, the prior p prior ( H , X, ˆ z i , θ R i ) for m detected GW-EM coincidentevents is given by: p prior ( H , X, ˆ z i , θ R i ) = A p ( H ) p ( X ) × m (cid:89) i =1 p (ˆ z i | X ) p ( θ R i ) , (6)where p ( H ), p ( X ), and p ( θ R i ) are the individual pri-ors on H , X , and θ R i . The quantity p (ˆ z i | X ) is the priordistribution on a GW-EM event’s true redshift given theunderlying distribution X , and A is a normalization con-stant. We introduce the likelihood function L , whichmeasures the relative conditional probability of observ-ing the sources’ redshifts z i (via EM measurements), anda particular set of data s i (via GWs) given the source’sparameters θ R i . It assumes the form: L ( { s i , z i }| H , X, ˆ z i , θ R i ) = m (cid:89) i =1 L ( s i | H , z i , θ R i ) P ( z i | ˆ z i ) , (7)where the likelihood function for a single GW event is L ( s i | H , z i , θ R i ) = e − (cid:0) h a ( θ ) − s a (cid:12)(cid:12) h a ( θ ) − s a (cid:1) / . (8)The inner product ( g | h ) describes the noise-weightedcross correlation of g ( t ) and h ( t ) on the vector space of signals, and is defined as:( g | h ) = 2 (cid:90) ∞ df ˜ g ∗ ( f )˜ h ( f ) + ˜ g ( f )˜ h ∗ ( f ) S n ( f ) , (9)where S n ( f ) denotes the instrument’s power spectraldensity. The Fourier transform ˜ h ( f ) of h ( t ) is definedas: ˜ h ( f ) ≡ (cid:90) ∞−∞ e πift h ( t ) dt . (10)An important element of our analysis is that we expressthe joint posterior PDF in H given m observed GW-EMevents as: p joint ( H |{ s i , z i } ) ∝ p prior ( H , X, ˆ z i , θ R i ) × L ( { s i , z i }| H , X, ˆ z i , θ R ) , = N p ( H ) (cid:90) dX p ( X ) (cid:26) m (cid:89) i =1 (cid:90) d ˆ z i p (ˆ z i | X ) P ( z i | ˆ z i ) × (cid:20) (cid:90) d θ R i p ( θ R i ) ×L ( s i | H , X, z i , θ R ) (cid:21)(cid:27) , (11)where N is a normalization constant and we substituteEqs. (6) and (7) for the model’s prior and likelihood func-tions respectively. In the event where we have the preciseredshift measurement of binary i ’s EM counterpart [i.e., p ( z i | ˆ z i ) = δ ( z i − ˆ z i )], Eq. (11) then reduces to: p joint ( H |{ s i , z i } ) = N p ( H ) (cid:90) dX p ( X ) (cid:26) m (cid:89) i =1 p ( z i | X ) × (cid:20) (cid:90) d θ R i p ( θ R i ) ×L ( s i | H , X, z i , θ R i ) (cid:21)(cid:27) = N p ( H ) (cid:34) (cid:90) dXp ( X ) m (cid:89) i =1 p ( z i | X ) (cid:35) × m (cid:89) i =1 (cid:26) (cid:90) d θ R i p ( θ R i ) × L ( s i | H , X, z i , θ R i ) (cid:27) = N (cid:48) p ( H ) × m (cid:89) i =1 (cid:26) (cid:90) d θ R i p ( θ R i ) × L ( s i | H , X, z i , θ R i ) (cid:27) . (12)The normalization constant N (cid:48) absorbs the [ . . . ] part ap-pearing in the previous line, which is independent of H .We assume a uniform prior in H such that p ( H ) =constant. It is worth noting that our formalism could begeneralized to include the most precise current estimatesof H . We do not do this here, although we certainlyrecision cosmology from gravitational waves 5imagine that this would be done when one does an anal-ysis of this sort with actual GW detections.Since we take p to be constant, we need only computethe { . . . } term in the last part of Eq. (12), where θ R i does not include D L . Outlined below, our work relies oncomputing the key term: p ( θ R i ) L ( s i | H , X, z i , θ R i ) (13)for each GW-EM event. This contrasts with the meth-ods used in N10, where we instead computed the term p ( θ i ) L ( s i | θ i ) for each binary, where θ included D L . Summary of MCMC approach used
For each GW-EM event we explicitly map out theterm p ( θ R i ) L TOT ( s i | X, z i , θ i ) using MCMC methods(see N10 and Nissanke et al. θ now includes D L because of the one-to-one mappingbetween D L and H . The quantity L TOT ( s i | X, z i , θ i ) isthe likelihood function for an entire network. We as-sume that the instrument noise n is Gaussian, indepen-dent, and uncorrelated at each detector site. Therefore,the network likelihood function is the product of the in-dividual likelihoods at each detector. We generate thesignal s a at each detector a such that it comprises thepredicted GW signal h a ( ˆ θ ), which depends on the set oftrue source parameters ˆ θ , and the instrument noise n a .In our study we use the projected advanced LIGO sensi-tivity curve for S n ( f ) shown in N10 and denoted “Zero–Detuned, High–Power” in Harry & the LIGO ScientificCollaboration (2010) for all our GW interferometers.We generate predicted templates h a (and hence alsothe measured signals s a ) using the PN description ofthe binaries as the bodies inspiral about one anotherprior to their merger. Specifically, we use the restricted3.5PN waveform in the frequency domain, where the GWfrequency evolves with a characteristic chirp [see Eqns.(4)–(6)]. We note that the largest contribution to thesignal accumulates from the inspiral (and not the subse-quent merger and ringdown parts of the waveform) forNS binaries in the frequency band of ground based in-terferometers (Flanagan & Hughes 1998). When the skyposition n is assumed known from its EM counterpartobservation, the GW strain at each detector, h a ( t ; ˆ θ ), isdescribed using seven parameters, θ : the two redshiftedmass parameters ( M z and µ z ), two orientation angles (Ψand cos ι ), the GW merger’s time and phase ( t c and Φ c ),and the binary’s luminosity distance ( D L ). Apart fromexcluding D L , the six parameters in the reduced vectorset θ R are identical to those in θ .We use the MCMC algorithm discussed in Section 3.3of N10 to explore the likelihood function. For binarieswith an underlying population with isotropic orientation,we take prior distributions in the sources’ parameters tobe flat over the region of sample space that correspondsto our threshold SNR (described below). For the sub-set of beamed binaries, we assume a uniform prior onthe SGRB’s beaming angle distribution in the range of | cos ι | > .
94, which corresponds to a beamed popula-tion with an opening jet angle of approximately 20 ◦ (e.g.,Burrows et al. et al. et al. Binary Selection
We follow the approach of N10 in generating a sampleof detectable GW-EM events. Having assumed a con-stant comoving density of GW-EM events in a ΛCDMuniverse (Komatsu et al. bina-ries uniformly in volume with random sky positions andorientations to redshift z = 1 ( D L (cid:39) . detected sample of binary eventsfor every network under consideration. Assuming priorknowledge of merger time and source position allows usto set the threshold for the network to SNR = 7 .
5, a valuelower than that used in the absence of an EM counter-part (see D06 and N10). Figure 2 in N10 shows howthe detectable GW-EM events for each detector networkincreases as the number of detectors in a network in-creases. Notice that N10 included an advanced detectorin Australia and not LIGO India as in this work (thecoordinates assumed for LIGO India are given in Nis-sanke, Kasliwal & Georgieva 2013). The term “total de-tectable binaries” refers to binaries which are detectableby a network of all five detectors — both LIGO sites,Virgo, LIGO India, and KAGRA.Finally we obtain our subsample of beamed SGRBsfrom our original sample of total detected GW-EM bina-ries by assuming that the SGRB has a uniform beamingangle distribution of | cos ι | > . RESULTS AND DISCUSSION
We now present our results for H constraints usingGW standard sirens observed by advanced ground-basedGW detector networks. As discussed in Sec. 2, we areinterested in the joint PDF of H given an ensemble ofGW-EM observations.Figure 1 shows the normalized joint posterior PDFs in H , indicated by the thick blue line, for a sample of 15isotropically oriented NS-NS binaries observed with thebaseline LIGO-Virgo network. The thin lines in Figure 1represent each individual binary’s measurement of H .We note that each binary gives a relatively poor con-straint on H with 68% confidence level (c.l.) fractionalerrors of ∼ H for an ensemble of 15mergers detected by LIGO-Virgo network being peakedaround our assumed true value of 70 . / s / Mpc with astandard deviation of 5 km / s / Mpc. This result would becompetitive with current H constraints using either thecosmological distance ladder (e.g., see Freedman et al. et al. et al. et al. et al. et al. et al. H as a func-tion of the number of binaries detectable by a particularnetwork. We randomly select 30 NS-NS and 30 NS-BHmergers detected in GWs using a five detector network.The actual number of detectable binaries is a functionof the detector network, on whether the EM counter-part is collimated (as we expect in the case of SGRBs),and on whether the progenitor model is a NS-NS or NS-BH binary (N10, Chen & Holz 2012). Specifically, Ta-ble 1 shows the measurement errors in H for samples Nissanke et al. Fig. 1.—
The joint posterior PDF in H for a sample of 15isotropically-oriented NS-NS binaries observed using a three de-tector network (LIGO Livingston, LIGO Hanford, Virgo). Thelight coloured lines mark the normalized posterior PDF for H foreach event, whereas the thicker blue line denotes the joint posteriorPDF in H given all the observed events. The vertical dashed blackline denotes the value of H of 70 . / s / Mpc used in generatingthe simulations. As the number of detections increases, the jointposterior PDF gets progressively narrower, and its center comescloser to the true value of H . of unbeamed and beamed NS-NS or NS-BH binaries us-ing different detector networks. The percentage errorsare quoted as the fraction of measured standard devia-tions over an assumed true value ( H = 70 . / s / Mpc).Such a measurement corresponds to a range of observa-tion times because of the wide range of uncertainties inNS binary merger rates. The general trends seen in Ta-ble 1 can be summarized as: • As expected, the errors in H decrease with an in-crease in the number of GW detectors in a network.Table 1 shows that a five detector network will re-sult in an improvement of up to a factor of 2 com-pared to a three detector network. Such a feature isa consequence of the increase in the detected num-ber of binaries, rather than due to the decrease inmeasurement error in D L for each individual event(see discussion in N10). Due to differences in theinstruments’ antenna response functions, the addi-tion of LIGO India has a greater impact than thatof KAGRA. • The errors in H reduce by a factor from two tofive when the EM counterpart is assumed to bebeamed. In the case of beamed NS binary mergerswhose PDFs are more Gaussian in shape, we findthat the error in H decreases as 1 / √ N , where N is the number of GW-EM events detected. We ex-pect this trend in the joint PDF of H as we fixthe inclination angle of each binary in the ensem-ble, since individual H constraints are Gaussianin distribution due to the absence of the D L -cos ι degeneracy.Fig. 2 shows the 68% c.l. measurement error in H as afunction of the number of GW-EM detectable NS binarymerger events. We assume that our detectable sample comprises 26 GW-EM binary mergers observed with aLIGO-Virgo network; we expect that the errors in H willdecrease with 1 / √ N in the limit of large N , where N isthe number of detectable GW-EM events. We computethe posterior PDF in H for each NS-NS binary mergerin our sample averaged over 100 noise realizations. Thesolid bars indicate the measurement error in H for the joint PDF of some i binary mergers; at low i , we selectthe i -th merger with the mean value in the H error of theremaining (26 − ( i − i GW-EM events will have on the convergenceof measurement errors in H . For an identically-orderedensemble of NS-NS mergers, the dashed line indicatesthe measurement error in H derived assuming Gaussianerrors for each GW-EM independent merger. In the limitof large i events, the difference in H error constraintsdecreases between the two methods. Furthermore, forlow i events in particular, we find that the non-Gaussianshapes of the individual H distributions improve thecombined H distribution. For example and as discussedin N10, after observing 15 NS-NS mergers in GWs andEM, we find that H may be measured to within 5% usingthe combined posterior PDF method (or to within 8%assuming Gaussian posterior PDFs for each individualevent). Without an EM counterpart and based solelyon statistical cross-correlations of GW sky errors withwide-field galaxy surveys, Del Pozzo (2012) finds a 14% H measurement error (with a 95% confidence interval)using ten GW merger events with a LIGO-Virgo network(and assuming a SNR ∼ − Myr − . Weexpect 15 (30) isotropically-oriented NS-NS mergers tobe detectable in GWs over a ∼ three month period usinga three (five) GW detector network and an EM precur-sor trigger (Nissanke, Kasliwal & Georgieva 2013). If weinstead consider beamed NS binary mergers and use theSGRB rate of 10 Gpc − yr − , we expect ∼
30 GW-SGRBevents per year (Berger 2011; Chen & Holz 2012; EnricoPetrillo, Dietz & Cavagli`a 2013).In the case of isotropically-oriented NS-10 M (cid:12)
BHmergers, we use a merger rate of 0.03 Mpc − Myr − (Abadie, the LIGO Scientific Collaboration & the VirgoCollaboration 2010). We then expect 15 (30) GW-detectable events in GWs over a six month period (wescale the results given in Table 1 of Nissanke, Kasliwal& Georgieva 2013 by a factor M / c to account for thedifference between the NS-10 M (cid:12) BH and the NS-5 M (cid:12)
BH mergers used there and here respectively). Due to anabsence of observed systems, we emphasize that NS-BHmerger rates based on population sythesis results vary byseveral orders of magnitude. In the case of beamed NS-BH mergers, we use the SGRB rate of 10 Gpc − yr − andfind 1 GW-SGRB event per year (e.g., Chen & Holz 2012;Enrico Petrillo, Dietz & Cavagli`a 2013; Kelley, Mandel& Ramirez-Ruiz 2013; Dietz et al. H , it is unclear how long this will take given the rangerecision cosmology from gravitational waves 7 TABLE 1Measurement errors in H for a sample of GW-EM events. Results are presented for unbeamed and beamedsources, for both NS-NS and NS-BH mergers, and for a range of detector networks. The % values are the68% c.l. fractional errors, and the number of binaries detected by each network is given in parentheses. Network LIGO+Virgo (LLV) LLV+LIGO India LLV+KAGRA LLV+LIGO India+KAGRANS-NS Isotropic 5.0% (15) 3.3% (20) 3.2% (20) 2.1% (30)NS-NS Beamed 1.1% (19) 1.0% (26) 1.0% (25) 0.9% (30)NS-BH Isotropic 4.9% (16) 3.5% (21) 3.6% (19) 2.0% (30)NS-BH Beamed 1.2% (18) 1.0% (25) 1.1% (24) 0.9% (30)
Number of GW−EM NS−NS mergers M ea s u r e m en t e rr o r i n H Fig. 2.— H measurement error as a function of the number ofmulti-messenger (GW+EM) NS-NS merger events observed by aLIGO-Virgo network. The solid bars indicate the 68% c.l. mea-surement error in H for the joint PDF of the independent binarymergers; the dashed line shows the 68% c.l. measurement errorin H derived assuming Gaussian errors for each GW-EM merger.When specifying the particular order of events shown, we choosethe GW-EM merger in the remaining ensemble with the mean mea-surement error in H . of uncertainty in binary merger rates. Current estimatessuggest that the median timescale to achieve this numberof events is likely about one year, but could be as shortas a few months, or as long as a decade. IMPLICATIONS FOR COSMOLOGY
Assuming GR accurately describes the inspiral dynam-ics and GW emission, GW standard sirens should pro-vide a measure of H based on absolutely-calibrated GWdistances that are independent of the cosmological dis-tance ladder. Given that we anticipate a network of ad-vanced GW interferometers reaching their design sensi-tivity within the next decade, this physics-based tech-nique could play a large role in precision determination ofthe Hubble constant, especially in conjuction with otherapproaches (see Suyu et al. H to anaccuracy of ∼ H constraintsusing solely GW observations, and are based on statisti-cal arguments or galaxy catalogs to infer the mergers’redshifts. We emphasize that an individual standardsiren may only constrain H to a precision ranging from5 to 50%. We have shown that the error in H dependscritically on the number of GW-EM mergers observed ,which in turn depends on the NS binary progenitor, onwhether the NS binary is face-on (due to GRB beaming), and on the number and sensitivy of GW interferometersin a network. We find that the critical limitation whenprojecting the timescale for this measurement (once theGW detectors are operational) is the few orders of magni-tude uncertainty in NS binary merger rates, independentof GW detections. Using mean NS merger rates derivedfrom population synthesis or the observed Galactic bi-nary pulsar distribution, we estimate that percent-levelmeasurements of H are possible within ∼ H at the per-cent level, when combined with precision CMB measure-ments of the absolute distance to the last scattering sur-face, would constrain the dark energy equation of stateparameter w to ∼
10% (D06). The power of such a result(e.g., to falsify the cosmological constant model for darkenergy) depends critically on understanding the system-atic errors associated with the measurement of H . Itis for this reason that GW standard sirens may have animportant role to play in constraining cosmology in thenear future. ACKNOWLEDGEMENTS
We thank Curt Cutler, Phil Marshall, and Michele Val-lisneri for very useful discussions on selection effects andbiases. We thank Vicky Scowcroft for discussion on H measurements, Edo Berger, Josh Bloom and Brian Met-zger for discussions on GW-SGRB measurements, andFrancois Foucart for discussion on the status of numeri-cal relativity simulations. Some of the simulations wereperformed using the Sunnyvale cluster at Canadian In-stitute for Theoretical Astrophysics (CITA), which isfunded by NSERC and CIAR. Part of this work wasperformed at the Jet Propulsion Laboratory, CaliforniaInstitute of Technology, under contract with the Na-tional Aeronautics and Space Administration. SMN issupported by the David & Lucile Packard Foundation.ND is supported by NASA under grants NNX12AD02Gand NNX12AC99G, and by a Sloan Research Fellowshipfrom the Alfred P. Sloan Foundation. DEH acknowledgessupport from National Science Foundation CAREERgrant PHY-1151836. SAH is supported by NSF GrantPHY-1068720. SAH also gratefully acknowledges fellow-ship support by the John Simon Guggenheim MemorialFoundation, and sabbatical support from CITA and thePerimeter Institute for Theoretical Physics. CH is sup-ported by the Simons Foundation, the David & LucilePackard Foundation, and the US Department of Energy(award de-sc0006624). Nissanke et al. REFERENCESAasi, J., the LIGO Scientific Collaboration, and the VirgoCollaboration 2013, ArXiv e-prints, 1304.0670.Abadie, J., the LIGO Scientific Collaboration, and the VirgoCollaboration 2010, Classical and Quantum Gravity, 27(17),173001, 1003.2480.Barnes, J. and Kasen, D. 2013, ArXiv e-prints, 1303.5787.Berger, E. 2011, New Ast. Rev., 55, 1, 1005.1068.Blanchet, L. 2006, Living Reviews in Relativity, 9, 4.Bloom, J. S. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al . 2012. https://dcc.ligo.org/cgi-bin/DocDB/ShowDocument?docid=91470.Schutz, B. F. 1986, Nature, 323, 310.Shibata, M., Kyutoku, K., Yamamoto, T., and Taniguchi, K.2009, Phys. Rev. D, 79(4), 044030, 0902.0416.Shibata, M. and Taniguchi, K. 2008, Phys. Rev. D, 77(8), 084015,0711.1410.Soderberg, A. M. et al. et al.et al.