A Standard Siren Cosmological Measurement from the Potential GW190521 Electromagnetic Counterpart ZTF19abanrhr
Hsin-Yu Chen, Carl-Johan Haster, Salvatore Vitale, Will M. Farr, Maximiliano Isi
AA Standard Siren Cosmological Measurement from the Potential GW190521Electromagnetic Counterpart ZTF19abanrhr
Hsin-Yu Chen,
1, 2, 3, ∗ Carl-Johan Haster,
2, 3, † Salvatore Vitale,
2, 3, ‡ Will M. Farr,
4, 5, § and Maximiliano Isi
2, 3, ¶ Black Hole Initiative, Harvard University, Cambridge, Massachusetts 02138, USA LIGO Laboratory, Massachusetts Institute of Technology, 185 Albany St, Cambridge, MA 02139, USA Department of Physics and Kavli Institute for Astrophysics and Space Research,Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA Department of Physics and Astronomy, Stony Brook University, Stony Brook NY 11794, USA (Dated: September 30, 2020)The identification of the electromagnetic counterpart candidate ZTF19abanrhr to the binaryblack hole merger GW190521 opens the possibility to infer cosmological parameters from thisstandard siren with a uniquely identified host galaxy. The distant merger allows for cosmologi-cal inference beyond the Hubble constant. Here we show that the three-dimensional spatial loca-tion of ZTF19abanrhr calculated from the electromagnetic data remains consistent with the lat-est sky localization of GW190521 provided by the LIGO-Virgo Collaboration. If ZTF19abanrhris associated with the GW190521 merger, and assuming a flat w CDM model, we find that H = 48 +23 − km s − Mpc − , Ω m = 0 . +0 . − . , and w = − . +0 . − . (median and 68% credible inter-val). If we use the Hubble constant value inferred from another gravitational-wave event, GW170817,as a prior for our analysis, together with assumption of a flat ΛCDM and the model-independentconstraint on the physical matter density ω m from Planck, we find H = 68 . +8 . − . km s − Mpc − . INTRODUCTION
Gravitational waves (GWs) emitted by compact objectbinaries are self-calibrating standard sirens [1], in thatthey yield a direct measurement of the source luminositydistance. If the redshift of the source can be estimated byother means, then GWs provide a way to measure cosmo-logical parameters that is entirely independent from clas-sic probes such as those based on standard candles [2, 3],the cosmic microwave background (CMB) [4, 5] and othermethods [6–9]. While a few different ways have been pro-posed to measure the redshift of the binary source, whichis not encoded in the GW signal [10–13], the two mostprominent are the identification of an electromagnetic(EM) counterpart, and a statistical analysis of all galax-ies included in the GW uncertainty volume [1, 14–16].To date, both approaches have been explored [17]. Thedetection of GWs from the binary neutron star (BNS)merger GW170817 [18] by the LIGO/Virgo Collabora-tion (LVC) [19, 20], together with the observation of EMcounterparts at multiple wavelengths [21] has allowed thefirst-ever standard siren measurement of the Hubble con-stant [22]. While the statistical standard siren methodhas the advantage that it can be applied to all typesof compact binary coalescences (CBCs), whether theyemit light or not, it is intrinsically less precise, as usuallymany galaxies are consistent with the GW uncertaintyvolume [16, 17, 23].The recent identification of an EM transient at non-negligible redshift ( z (cid:39) .
4) by the Zwicky Transient Fa-cility (ZTF) — ZTF19abanrhr [24] — consistent with be-ing a counterpart to the distant ( ∼ H , primarily due to theGW detectors’ current limit in their sensitive distanceto BNSs, and the number of galaxies consistent with thelarge BBH uncertainty volumes. Inference on other cos-mological parameters was expected to rely on future GWobservations at higher redshift [13, 30–32] ZTF19ABANRHR ASSOCIATION IN 3DLOCALIZATION
In Figure 1(a) we show the three-dimensional lo-calization uncertainty volume of GW190521 assuminga flat prior in luminosity volume ( ∝ D L ). Using aPlanck 2018 cosmology [5], we also mark the location ofZTF19abanrhr. We found that ZTF19abanrhr lies at a67% credible level of the GW190521 localization volume.The credible level at which the counterpart lies inthe localization of GW190521 depends on the assumedprior distribution of GW sources. Figure 1(b) showsthe posterior distribution of luminosity distance alongthe line of sight to ZTF19abanrhr for several differentchoices of prior; in all cases the luminosity distanceto ZTF19abanrhr computed from a reasonable cosmol- a r X i v : . [ a s t r o - ph . C O ] S e p (a) D L [Mpc] . . . . p ( D L ) [ M p c − ] Uniform inluminosity distanceUniform inluminosity volume Uniform in thecomoving frameTracing the starformation rate z=0.438(Planck 2018) (b)
FIG. 1. Panel (a): The 3D localization of GW190521 pre-sented in a Cartesian luminosity distance coordinates, cen-tered on the Earth marked with a black ⊕ . Here we use thelocalization inferred by the NRSur analysis from [27, 33] whichapplied a flat prior in luminosity volume. The size and hue ofeach point is weighted by the logarithm of its posterior prob-ability. The location of ZTF19abanrhr, assuming the Planck2018 cosmology [5], is shown by the orange star. Panel (b):The 1D D L posterior for GW190521 along the line of sight toZTF19abanrhr under four different prior assumptions for theluminosity distance D L [34]. The location of ZTF19abanrhr,assuming the Planck 2018 cosmology [5], is shown by the or-ange line. The priors are (solid blue line) uniform in lumi-nosity distance (i.e. proportional to the conditional distancelikelihood); uniform in luminosity volume (dashed blue line);uniform in the comoving frame (dotted blue line); and tracingthe star formation rate [35] (dash-dotted blue line). ogy [5] is found well within the bulk of this conditional distance distribution. For the primary estimate of thedistance marginal used in this study, we rely on a param-eter estimation analysis conditional on J1249 + 3449, thesky location of ZTF19abanrhr [34, 36], and otherwisematching the preferred analysis from [27, 33, 37]. COSMOLOGICAL INFERENCE
The mathematical and statistical background behind astandard siren measurement of the Hubble constant hasalready been presented in the literature [1, 14, 16, 22,38, 39]. In this letter, we follow the same framework toinfer the Hubble constant H , the matter density of theUniverse Ω m and the dark energy equation of state (EoS)parameter w .Given a set of GW data D GW and EM data D EM cor-responding to a common observation, the joint posteriorof ( H , Ω m , w ) can be written as: p ( H , Ω m , w |D GW , D EM ) = p ( H , Ω m , w ) β ( H , Ω m , w ) × (cid:90) p ( D GW | (cid:126) Θ) p ( D EM | (cid:126) Θ) p pop ( (cid:126) Θ | H , Ω m , w ) d(cid:126) Θ , (1)where (cid:126) Θ represents all the binary parameters, such as themasses, spins, luminosity distance, sky location, orbitalinclination etc. p ( H , Ω m , w ) denotes the prior prob-ability density function (PDF) on the cosmological pa-rameters. p pop ( (cid:126) Θ | H , Ω m , w ) is the distribution of thepopulation of binaries with parameters (cid:126) Θ in the Universe.The denominator, β , is the fraction of the population ofevents that would pass detection thresholds [22, 38, 40–43]: β ( H , Ω m , w ) ≡ (cid:90) P det ( (cid:126) Θ) p pop ( (cid:126) Θ | H , Ω m , w ) d (cid:126) Θ (2)where P det ( (cid:126) Θ) ≡ (cid:90) (cid:90) D GW > GW th , D EM > EM th p ( D GW | (cid:126) Θ) p ( D EM | (cid:126) Θ) d D GW d D EM , (3)is the probability of detecting a source with parameters (cid:126) Θin GW and EM emission. This latter integration shouldbe carried out over data above the GW and EM detec-tion thresholds, GW th and EM th . We assume that thecounterparts to systems like GW190521 can be observedby ZTF and other telescopes far beyond the distance atwhich GW observatories can detect them (ZTF19abanrhrwas ∼ . p pop ( (cid:126) Θ | H , Ω m , w ). Sincethe astrophysical rate of GW190521-like BBHs is still un-certain, we assume their redshift distribution follows thestar formation rate (SFR) as modeled by Ref. [35]. Weadopt the default assumptions of [27] that the populationis flat in the detector frame masses and spin magnitudesand isotropic over binary and spin orientations.Given the small uncertainty in the redshift and coun-terpart sky location measured in ZTF19abanrhr, we treatthe EM likelihood in Eq. (1) as a δ -function at thesemeasurements. Performing the integral over (cid:126) Θ, Eq. (1)becomes p ( H , Ω m , w | D GW , D EM ) ∝ p ( D GW | D L ( z EM | H , Ω m , w ) , α EM , δ EM ) × p pop ( z EM | H , Ω m , w ) β ( H , Ω m , w ) p ( H , Ω m , w ) . (4)The first term is the marginalized GW likelihood eval-uated at the right ascension α , declination δ , and lumi-nosity distance implied by the redshift of ZTF19abanrhrgiven cosmological parameters H , Ω m and w ; this func-tion is shown by the solid blue curve in Figure 1(b). Thenext term accounts for selection effects and the assumedGW source population and involves the ratio of the (nor-malized) population density at the ZTF19abanrhr red-shift and the fraction of the (normalized) population thatis jointly detectable in GW and EM emission as describedabove (in the local universe the effect of this term is to in-troduce a factor 1 /H [22, 38] but at z (cid:39) . H substantially [44]).The third term is the prior on cosmological parameters.We impose several different priors incorporating variousadditional cosmological measurements in the following.In our most generic analysis, we use flat priors in theranges H = [35 , − Mpc − , Ω m = [0 , w = [ − , − . H with a median and 68%credible interval of H = 48 +23 − km s − Mpc − , with apeak below the maximum likelihood Planck 2018 value [5](as well as the SH0ES estimate [2]), reported with ayellow (pink) solid line. The Planck and SH0ES es-timates are contained within the 90% credible regionsof our measurements. The posteriors for Ω m and w are nearly uninformative with Ω m = 0 . +0 . − . , and w = − . +0 . − . . Nevertheless, given the large inferreddistance of GW190521, they are mildly correlated with H , and must be included in the analysis.A joint GW190521–ZTF19abanrhr analysis furnishesa single measurement of the luminosity distance to The priors are uniform on the component masses in the detectorframe from [30, 200] M (cid:12) . The mass priors are further restrictedsuch that the total mass must be greater than 200 M (cid:12) , and thechirp mass to be between 70 and 150 M (cid:12) , both in the detectorframe. The mass ratio between the lighter and heavier objects isrestricted to be > . . . . Ω m
35 75 115 H [km s − Mpc − ] − . − . − . w . . . Ω m − . − . − . w w CDMPlanck 2018SH0ESPrior
FIG. 2. The joint posterior PDF of H , Ω m and w for theassociated GW190521–ZTF19abanrhr observations using uni-form priors (grey lines) for all parameters in a flat w CDMcosmology. The yellow (pink) solid lines report the Planck2018 [5] (SH0ES [2]) cosmology, with shaded regions repre-senting their respective 68% credible interval. For the 2Dplots, the contours are spaced 10 percentiles apart, from the10% (darkest) to 90% (lightest) credible regions.
ZTF19abanrhr, which depends on H , Ω m and w .To compare with other GW measurements in the lo-cal universe, which depend only on H , we restrictto ΛCDM universes ( w = −
1) and apply the model-independent measurement of the physical matter den-sity [45] from Planck observations of the CMB [5], ω m ≡ Ω m h = 0 . ± . H = 48 . +21 . − . km s − Mpc − with this assumption.The model-independence of the ω m constraint meansthis measurement remains systematically independent ofearly-universe distance scales from CMB measurements.The best inference on H from gravitational wavestandard sirens comes from combining our measurementhere with GW170817. We apply the H likelihood fromGW170817 [17, 22] as a prior on H along with the Planck ω m constraint. These results are shown in the dark bluecurve in Fig. 3. The joint measurement is narrower thaneither measurement alone, with a median and 68% cred-ible interval of H = 68 . +8 . − . km s − Mpc − and a clearpeak consistent with estimates using observations fromboth the CMB [5] and the local distance ladders [2, 3, 6–9]; GW190521 rules out some large H values that arepermitted from GW170817.The choice of waveform models for GW data analy-
40 60 80 100 120 140 H [km s − Mpc − ] . . . . . . . p ( H ) [ k m − s M p c ] ΛCDM + ω m + GW170817ΛCDM + ω m Planck 2018SH0ESGW170817 (prior)
FIG. 3. The posterior PDF of H for the associatedGW190521–ZTF19abanrhr observations under the assump-tion of a flat ΛCDM cosmology and physical matter density ω m constraints from Planck 2018 [5]. The dark blue curveuses the inferred H posterior from GW170817 [17, 22] (greycurve) as a prior whereas the light blue curve assumes a flatprior on H . The yellow (pink) solid lines report the Planck2018 [5] (SH0ES [2]) H estimates, with shaded regions rep-resenting their respective 68% credible interval. sis can contribute to the systematic uncertainty of thestandard siren measurement via the luminosity distanceestimate. In Ref. [27], the LVC estimated the parame-ters of GW190521 with three different waveform mod-els [37, 46, 47]. We use a clustering decomposition fol-lowed by a kernel density estimate within clusters [44] toestimate the marginal posterior probability distributionof D L along the line-of-sight to ZTF19abanrhr [24] fromthese analyses. In Figure 4 we show the H inference withthe three waveform models using the GW170817 prior on H and Planck’s prior on ω m in a ΛCDM cosmology. Thestrong prior on H dominates over the slight differencebetween D L estimates from different models, and theyall yield a very similar posterior on H .Finally, in Figure 5 we present the measurements onΩ m and w with both of the GW events and Planck’sprior on ω m in a flat w CDM cosmology. We find that theΩ m posterior now shows a departure from its prior, andfeatures a peak with a median and 68% credible intervalof Ω m = 0 . +0 . − . . To a lesser extent, the same is truefor w now estimated as w = − . +0 . − . . DISCUSSION
The EM transient ZTF19abanrhr [24] could be as-sociated with the BBH merger GW190521. We findthat ZTF19abanrhr lies at the 67% credible level of theGW190521 three-dimensional localization volume undera default luminosity-distance prior and assuming thePlanck 2018 cosmology [5]. Assuming the GW–EM asso-ciation is true, we report a standard-siren measurement
40 60 80 100 120 140 H [km s − Mpc − ] . . . . . . . p ( H ) [ k m − s M p c ] NRSur – ΛCDM + ω m + GW170817Phenom – ΛCDM + ω m + GW170817SEOBNR – ΛCDM + ω m + GW170817Planck 2018SH0ESGW170817 (prior) FIG. 4. The posterior PDF of H for the associatedGW190521–ZTF19abanrhr observations using ω m constraintsfrom Planck 2018 [5], a prior on H from GW170817 [17, 22](shown in grey) and a flat ΛCDM cosmology. We showestimates on H using all three waveform analyses fromRef. [27, 33]. The yellow (pink) solid lines report the Planck2018 [5] (SH0ES [2]) cosmology, with shaded regions repre-senting their respective 68% credible interval. of cosmological parameters from these transients. Thelarge inferred distance of GW190521 enables probing H and additional cosmological parameters Ω m and the darkenergy EoS parameter w . We note that other indepen-dent analyses also conduct the standard-siren measure-ment assuming the association between GW190521 andZTF19abanrhr [48, 49].We find H = 68 . +8 . − . km s − Mpc − from the as-sociated ZTF19abanrhr–GW190521 and the kilonovaAT 2017gfo–GW170817 observations assuming a model-independent constraints on the physical matter density ω m from the Planck observations [5] in a flat ΛCDM cos-mology. The same measurement yields Ω m = 0 . +0 . − . and w = − . +0 . − . in a flat w CDM cosmology. Sincethere is only one standard siren measurement at higherredshift, the inference on Ω m mainly relies on the priorfrom GW170817 and Planck. The strong prior on H from GW170817 dominates the H measurement. WhenGW170817 is combined with the Planck prior on ω m ,Ω m is constrained to ∼ w is only marginally confinedeven when both GW170817 and GW190521 are includedin the analysis.We find that the choice of GW waveform for the es-timation of luminosity distance and the assumption ofBBH population for the evaluation of selection effect donot introduce noticeable difference in our results. How-ever, when more events are combined in the future andthe cosmological parameters are confined more precisely,the systematic uncertainties arising from waveform andselection effect will have to be investigated more care-fully. For example, a joint inference of the BBH popu-lation and the cosmological parameters will help reduce . . . Ω m
35 75 115 H [km s − Mpc − ] − . − . − . w . . . Ω m − . − . − . w
60 75 90 H [kms − Mpc − ] Ω m w CDM + ω m +GW170817Planck 2018SH0ESPrior FIG. 5. The joint posterior PDF of H , Ω m and w forthe associated GW190521–ZTF19abanrhr observations usinga prior for H equal to the posterior of GW170817 [17, 22], andadditional constraints on ω m from Planck 2018 [5], shown asgrey curves. The yellow (pink) solid lines report the Planck2018 [5] (SH0ES [2]) cosmology, with shaded regions repre-senting their respective 68% credible interval. For the 2Dplots, the contours are spaced 10 percentiles apart, from the10% (darkest) to 90% (lightest) credible regions. bias from unrealistic population assumptions.In the next five years, LIGO, Virgo and KAGRA arepredicted to detect hundreds of BBHs per year [50]. If in-deed ZTF19abanrhr is the counterpart of GW190521, weshould see more BBHs accompanied by EM counterparts.Owing to their generally larger distances, compared tostandard BNS bright sirens, these have a significant po-tential of yielding an interesting GW measurement of Ω m and w .The authors would like to thank Jonathan Gair his re-view and suggestions to this work. HYC was supportedby the Black Hole Initiative at Harvard University, whichis funded by grants from the John Templeton Foundationand the Gordon and Betty Moore Foundation to Har-vard University. HYC and MI are supported by NASAthrough NASA Hubble Fellowship grants No. HST-HF2-51452.001-A and No. HST-HF2-51410.001-A awarded bythe Space Telescope Science Institute, which is operatedby the Association of Universities for Research in As-tronomy, Inc., for NASA, under contract NAS5-26555.CJH and SV acknowledge support of the National Sci-ence Foundation, and the LIGO Laboratory. LIGO wasconstructed by the California Institute of Technologyand Massachusetts Institute of Technology with funding from the National Science Foundation and operates un-der cooperative agreement PHY-1764464. This researchhas made use of data, software and/or web tools ob-tained from the Gravitational Wave Open Science Center( ), a service of LIGOLaboratory, the LIGO Scientific Collaboration and theVirgo Collaboration. LIGO is funded by the U.S. Na-tional Science Foundation. Virgo is funded by the FrenchCentre National de Recherche Scientifique (CNRS), theItalian Istituto Nazionale della Fisica Nucleare (INFN)and the Dutch Nikhef, with contributions by Polish andHungarian institutes. This analysis was made possi-ble by the numpy [51, 52], SciPy [53], matplotlib [54], emcee [55], pandas [56, 57], pymc3 [58], seaborn [59] and astropy [60, 61] software packages. This is LIGO Doc-ument Number LIGO-P2000233. The data behind Fig-ures 1(b), 2, 3, 4 and 5 are publicly available at [62]. ∗ [email protected]; NHFP Einstein fellow † [email protected] ‡ [email protected] § [email protected] ¶ [email protected]; NHFP Einstein fellow[1] B. F. Schutz, Nature , 310 (1986).[2] A. G. Riess et al., Astrophys. J. , 56 (2016),1604.01424.[3] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, andD. Scolnic, Astrophys. J. , 85 (2019), 1903.07603.[4] P. Ade et al. (Planck), Astron. Astrophys. , A13(2016), 1502.01589.[5] N. Aghanim et al. (Planck), Astron. Astrophys. , A6(2020), 1807.06209.[6] E. Macaulay et al. (DES), Mon. Not. Roy. Astron. Soc. , 2184 (2019), 1811.02376.[7] W. Yuan, A. G. Riess, L. M. Macri, S. Casertano, andD. Scolnic, Astrophys. J. , 61 (2019), 1908.00993.[8] W. L. Freedman et al., Astrophys. J. , 34 (2019),1907.05922.[9] D. Pesce et al., Astrophys. J. Lett. , L1 (2020),2001.09213.[10] C. Messenger, K. Takami, S. Gossan, L. Rezzolla, andB. S. Sathyaprakash, Phys. Rev. X , 041004 (2014),1312.1862.[11] W. M. Farr, M. Fishbach, J. Ye, and D. Holz, Astrophys.J. Lett. , L42 (2019), 1908.09084.[12] S. R. Taylor, J. R. Gair, and I. Mandel, Phys. Rev. D , 023535 (2012), 1108.5161.[13] W. Del Pozzo, T. G. F. Li, and C. Messenger, Phys. Rev.D , 043502 (2017), 1506.06590.[14] D. E. Holz and S. A. Hughes, Astrophys. J. , 15(2005), astro-ph/0504616.[15] W. Del Pozzo, Phys. Rev. D , 043011 (2012),1108.1317.[16] H.-Y. Chen, M. Fishbach, and D. E. Holz, Nature ,545 (2018), 1712.06531.[17] B. Abbott et al. (LIGO Scientific, Virgo), ArXiv e-prints(2019), 1908.06060.[18] B. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. , 161101 (2017), 1710.05832.[19] G. M. Harry (LIGO Scientific Collaboration), Class.Quant. Grav. , 084006 (2010).[20] F. Acernese et al. (Virgo Collaboration), Class. Quant.Grav. , 024001 (2015), 1408.3978.[21] B. Abbott et al. (LIGO Scientific, Virgo, Fermi GBM,INTEGRAL, IceCube, AstroSat Cadmium Zinc Tel-luride Imager Team, IPN, Insight-Hxmt, ANTARES,Swift, AGILE Team, 1M2H Team, Dark Energy CameraGW-EM, DES, DLT40, GRAWITA, Fermi-LAT, ATCA,ASKAP, Las Cumbres Observatory Group, OzGrav,DWF (Deeper Wider Faster Program), AST3, CAAS-TRO, VINROUGE, MASTER, J-GEM, GROWTH,JAGWAR, CaltechNRAO, TTU-NRAO, NuSTAR, Pan-STARRS, MAXI Team, TZAC Consortium, KU, NordicOptical Telescope, ePESSTO, GROND, Texas TechUniversity, SALT Group, TOROS, BOOTES, MWA,CALET, IKI-GW Follow-up, H.E.S.S., LOFAR, LWA,HAWC, Pierre Auger, ALMA, Euro VLBI Team, Pi ofSky, Chandra Team at McGill University, DFN, AT-LAS Telescopes, High Time Resolution Universe Survey,RIMAS, RATIR, SKA South Africa/MeerKAT), Astro-phys. J. Lett. , L12 (2017), 1710.05833.[22] B. Abbott et al. (LIGO Scientific, Virgo, 1M2H, DarkEnergy Camera GW-E, DES, DLT40, Las CumbresObservatory, VINROUGE, MASTER), Nature , 85(2017), 1710.05835.[23] M. Soares-Santos et al. (DES, LIGO Scientific, Virgo),Astrophys. J. Lett. , L7 (2019), 1901.01540.[24] M. Graham et al., Phys. Rev. Lett. , 251102 (2020),2006.14122.[25] LIGO Scientific Collaboration, Virgo Collaboration,Gamma-ray Coordinates Network (2017), URL https://gcn.gsfc.nasa.gov/gcn3/24621.gcn3 .[26] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev.Lett. , 101102 (2020), 2009.01075.[27] R. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J.Lett. , L13 (2020), 2009.01190.[28] G. Ashton, K. Ackley, I. Maga˜na Hernand ez, andB. Piotrzkowski, arXiv e-prints arXiv:2009.12346 (2020),2009.12346.[29] B. McKernan, K. Ford, I. Bartos, M. Graham, W. Lyra,S. Marka, Z. Marka, N. Ross, D. Stern, and Y. Yang,Astrophys. J. Lett. , L50 (2019), 1907.03746.[30] B. Sathyaprakash, B. Schutz, and C. Van Den Broeck,Class. Quant. Grav. , 215006 (2010), 0906.4151.[31] S. R. Taylor and J. R. Gair, Phys. Rev. D , 023502(2012), 1204.6739.[32] S.-J. Jin, D.-Z. He, Y. Xu, J.-F. Zhang, and X. Zhang,JCAP , 051 (2020), 2001.05393.[33] LIGO Scientific Collaboration and Virgo Collaboration, GW190521 parameter estimation samples and figure data (2020), URL https://dcc.ligo.org/LIGO-P2000158/public .[34] M. Isi,
GW190521: posterior samples conditional onAGN J1249+3449 (2020), URL https://doi.org/10.5281/zenodo.4057131 .[35] P. Madau and M. Dickinson, Ann. Rev. Astron. Astro- phys. , 415 (2014), 1403.0007.[36] R. Abbott et al. (LIGO Scientific, Virgo), ArXiv e-prints(2019), 1912.11716.[37] V. Varma, S. E. Field, M. A. Scheel, J. Blackman,D. Gerosa, L. C. Stein, L. E. Kidder, and H. P. Pfeif-fer, Phys. Rev. Research. , 033015 (2019), 1905.09300.[38] M. Fishbach et al. (LIGO Scientific, Virgo), Astrophys.J. Lett. , L13 (2019), 1807.05667.[39] R. Gray et al., Phys. Rev. D , 122001 (2020),1908.06050.[40] T. J. Loredo, AIP Conf. Proc. , 195 (2004), astro-ph/0409387.[41] I. Mandel, W. M. Farr, and J. R. Gair, Mon. Not. Roy.Astron. Soc. , 1086 (2019), 1809.02063.[42] S. Vitale, ArXiv e-prints (2020), 2007.05579.[43] W. M. Farr and J. R. Gair, A derivation of the likeli-hood function for a statistical h measurement , https://github.com/farr/H0StatisticalLikelihood (2020).[44] W. M. Farr, Gw190521sky , https://github.com/farr/GW190521Sky (2020).[45] W. Hu and S. Dodelson, Ann. Rev. Astron. Astrophys. , 171 (2002), astro-ph/0110414.[46] S. Khan, F. Ohme, K. Chatziioannou, and M. Hannam,Phys. Rev. D , 024056 (2020), 1911.06050.[47] S. Ossokine et al., Phys. Rev. D , 044055 (2020),2004.09442.[48] S. Mukherjee et al. (2020), in prep.[49] V. Gayathri et al. (2020), in prep.[50] B. Abbott et al. (KAGRA, LIGO Scientific, VIRGO),Living Rev. Rel. , 3 (2018), 1304.0670.[51] T. Oliphant, NumPy: A guide to NumPy , USA: TrelgolPublishing (2006–), URL .[52] C. R. Harris et al., Nature , 357 (2020), URL https://doi.org/10.1038/s41586-020-2649-2 .[53] P. Virtanen et al., Nature Meth. , 261 (2020),1907.10121.[54] J. D. Hunter, Comput. Sci. Eng. , 90 (2007).[55] D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Good-man, Publ. Astron. Soc. Pac. , 306 (2013), 1202.3665.[56] The pandas development team, pandas-dev/pandas:Pandas (2020), URL https://doi.org/10.5281/zenodo.3509134 .[57] Wes McKinney, in Proceedings of the 9th Python in Sci-ence Conference , edited by St´efan van der Walt and Jar-rod Millman (2010), pp. 56 – 61.[58] J. Salvatier, T. V. Wiecki, and C. Fonnesbeck, PeerJComputer Science , e55 (2016), 1507.08050.[59] M. Waskom et al., mwaskom/seaborn: v0.10.1 (april2020) (2020), URL https://doi.org/10.5281/zenodo.3767070 .[60] T. P. Robitaille et al. (Astropy Collaboration), Astron.Astrophys. , A33 (2013), 1307.6212.[61] A. Price-Whelan et al., Astron. J. , 123 (2018),1801.02634.[62] H.-Y. Chen, C.-J. Haster, S. Vitale, W. M. Farr, andM. Isi, Data release - A Standard Siren CosmologicalMeasurement from the Potential GW190521 Electromag-netic Counterpart ZTF19abanrhr (2020), URL https://doi.org/10.5281/zenodo.4057311https://doi.org/10.5281/zenodo.4057311