Draft version October 27, 2019Preprint typeset using LATEX style AASTeX6 v. 1.0

SUPPLEMENT: THE RATE OF BINARY BLACK HOLE MERGERS INFERRED FROM ADVANCED LIGO

OBSERVATIONS SURROUNDING GW150914

B. P. Abbott,1 R. Abbott,1 T. D. Abbott,2 M. R. Abernathy,1 F. Acernese,3,4 K. Ackley,5 C. Adams,6

T. Adams,7 P. Addesso,3 R. X. Adhikari,1 V. B. Adya,8 C. Affeldt,8 M. Agathos,9 K. Agatsuma,9 N. Aggarwal,10

O. D. Aguiar,11 L. Aiello,12,13 A. Ain,14 P. Ajith,15 B. Allen,8,16,17 A. Allocca,18,19 P. A. Altin,20 S. B. Anderson,1

W. G. Anderson,16 K. Arai,1 M. C. Araya,1 C. C. Arceneaux,21 J. S. Areeda,22 N. Arnaud,23 K. G. Arun,24

S. Ascenzi,25,13 G. Ashton,26 M. Ast,27 S. M. Aston,6 P. Astone,28 P. Aufmuth,8 C. Aulbert,8 S. Babak,29

P. Bacon,30 M. K. M. Bader,9 P. T. Baker,31 F. Baldaccini,32,33 G. Ballardin,34 S. W. Ballmer,35

J. C. Barayoga,1 S. E. Barclay,36 B. C. Barish,1 D. Barker,37 F. Barone,3,4 B. Barr,36 L. Barsotti,10

M. Barsuglia,30 D. Barta,38 J. Bartlett,37 I. Bartos,39 R. Bassiri,40 A. Basti,18,19 J. C. Batch,37 C. Baune,8

V. Bavigadda,34 M. Bazzan,41,42 B. Behnke,29 M. Bejger,43 A. S. Bell,36 C. J. Bell,36 B. K. Berger,1

J. Bergman,37 G. Bergmann,8 C. P. L. Berry,44 D. Bersanetti,45,46 A. Bertolini,9 J. Betzwieser,6 S. Bhagwat,35

R. Bhandare,47 I. A. Bilenko,48 G. Billingsley,1 J. Birch,6 R. Birney,49 S. Biscans,10 A. Bisht,8,17 M. Bitossi,34

C. Biwer,35 M. A. Bizouard,23 J. K. Blackburn,1 C. D. Blair,50 D. G. Blair,50 R. M. Blair,37 S. Bloemen,51

O. Bock,8 T. P. Bodiya,10 M. Boer,52 G. Bogaert,52 C. Bogan,8 A. Bohe,29 P. Bojtos,53 C. Bond,44 F. Bondu,54

R. Bonnand,7 B. A. Boom,9 R. Bork,1 V. Boschi,18,19 S. Bose,55,14 Y. Bouffanais,30 A. Bozzi,34 C. Bradaschia,19

P. R. Brady,16 V. B. Braginsky,48 M. Branchesi,56,57 J. E. Brau,58 T. Briant,59 A. Brillet,52 M. Brinkmann,8

V. Brisson,23 P. Brockill,16 A. F. Brooks,1 D. A. Brown,35 D. D. Brown,44 N. M. Brown,10 C. C. Buchanan,2

A. Buikema,10 T. Bulik,60 H. J. Bulten,61,9 A. Buonanno,29,62 D. Buskulic,7 C. Buy,30 R. L. Byer,40 L. Cadonati,63

G. Cagnoli,64,65 C. Cahillane,1 J. Calderon Bustillo,66,63 T. Callister,1 E. Calloni,67,4 J. B. Camp,68

K. C. Cannon,69 J. Cao,70 C. D. Capano,8 E. Capocasa,30 F. Carbognani,34 S. Caride,71 J. Casanueva Diaz,23

C. Casentini,25,13 S. Caudill,16 M. Cavaglia,21 F. Cavalier,23 R. Cavalieri,34 G. Cella,19 C. B. Cepeda,1

L. Cerboni Baiardi,56,57 G. Cerretani,18,19 E. Cesarini,25,13 R. Chakraborty,1 T. Chalermsongsak,1

S. J. Chamberlin,72 M. Chan,36 S. Chao,73 P. Charlton,74 E. Chassande-Mottin,30 H. Y. Chen,75 Y. Chen,76

C. Cheng,73 A. Chincarini,46 A. Chiummo,34 H. S. Cho,77 M. Cho,62 J. H. Chow,20 N. Christensen,78 Q. Chu,50

S. Chua,59 S. Chung,50 G. Ciani,5 F. Clara,37 J. A. Clark,63 F. Cleva,52 E. Coccia,25,12,13 P.-F. Cohadon,59

A. Colla,79,28 C. G. Collette,80 L. Cominsky,81 M. Constancio Jr.,11 A. Conte,79,28 L. Conti,42 D. Cook,37

T. R. Corbitt,2 N. Cornish,31 A. Corsi,71 S. Cortese,34 C. A. Costa,11 M. W. Coughlin,78 S. B. Coughlin,82

J.-P. Coulon,52 S. T. Countryman,39 P. Couvares,1 E. E. Cowan,63 D. M. Coward,50 M. J. Cowart,6 D. C. Coyne,1

R. Coyne,71 K. Craig,36 J. D. E. Creighton,16 J. Cripe,2 S. G. Crowder,83 A. Cumming,36 L. Cunningham,36

E. Cuoco,34 T. Dal Canton,8 S. L. Danilishin,36 S. D’Antonio,13 K. Danzmann,17,8 N. S. Darman,84 V. Dattilo,34

I. Dave,47 H. P. Daveloza,85 M. Davier,23 G. S. Davies,36 E. J. Daw,86 R. Day,34 S. De,35 D. DeBra,40

G. Debreczeni,38 J. Degallaix,65 M. De Laurentis,67,4 S. Deleglise,59 W. Del Pozzo,44 T. Denker,8,17 T. Dent,8

H. Dereli,52 V. Dergachev,1 R. De Rosa,67,4 R. T. DeRosa,6 R. DeSalvo,87 S. Dhurandhar,14 M. C. Dıaz,85

L. Di Fiore,4 M. Di Giovanni,79,28 A. Di Lieto,18,19 S. Di Pace,79,28 I. Di Palma,29,8 A. Di Virgilio,19 G. Dojcinoski,88

V. Dolique,65 F. Donovan,10 K. L. Dooley,21 S. Doravari,6,8 R. Douglas,36 T. P. Downes,16 M. Drago,8,89,90

R. W. P. Drever,1 J. C. Driggers,37 Z. Du,70 M. Ducrot,7 S. E. Dwyer,37 T. B. Edo,86 M. C. Edwards,78

A. Effler,6 H.-B. Eggenstein,8 P. Ehrens,1 J. Eichholz,5 S. S. Eikenberry,5 W. Engels,76 R. C. Essick,10

T. Etzel,1 M. Evans,10 T. M. Evans,6 R. Everett,72 M. Factourovich,39 V. Fafone,25,13,12 H. Fair,35 S. Fairhurst,91

X. Fan,70 Q. Fang,50 S. Farinon,46 B. Farr,75 W. M. Farr,44 M. Favata,88 M. Fays,91 H. Fehrmann,8 M. M. Fejer,40

I. Ferrante,18,19 E. C. Ferreira,11 F. Ferrini,34 F. Fidecaro,18,19 I. Fiori,34 D. Fiorucci,30 R. P. Fisher,35

R. Flaminio,65,92 M. Fletcher,36 H. Fong,69 J.-D. Fournier,52 S. Franco,23 S. Frasca,79,28 F. Frasconi,19 Z. Frei,53

A. Freise,44 R. Frey,58 V. Frey,23 T. T. Fricke,8 P. Fritschel,10 V. V. Frolov,6 P. Fulda,5 M. Fyffe,6

H. A. G. Gabbard,21 J. R. Gair,93 L. Gammaitoni,32,33 S. G. Gaonkar,14 F. Garufi,67,4 A. Gatto,30 G. Gaur,94,95

N. Gehrels,68 G. Gemme,46 B. Gendre,52 E. Genin,34 A. Gennai,19 J. George,47 L. Gergely,96 V. Germain,7

Archisman Ghosh,15 S. Ghosh,51,9 J. A. Giaime,2,6 K. D. Giardina,6 A. Giazotto,19 K. Gill,97 A. Glaefke,36

E. Goetz,98 R. Goetz,5 L. Gondan,53 G. Gonzalez,2 J. M. Gonzalez Castro,18,19 A. Gopakumar,99 N. A. Gordon,36

M. L. Gorodetsky,48 S. E. Gossan,1 M. Gosselin,34 R. Gouaty,7 C. Graef,36 P. B. Graff,62 M. Granata,65

A. Grant,36 S. Gras,10 C. Gray,37 G. Greco,56,57 A. C. Green,44 P. Groot,51 H. Grote,8 S. Grunewald,29

G. M. Guidi,56,57 X. Guo,70 A. Gupta,14 M. K. Gupta,95 K. E. Gushwa,1 E. K. Gustafson,1 R. Gustafson,98

J. J. Hacker,22 B. R. Hall,55 E. D. Hall,1 G. Hammond,36 M. Haney,99 M. M. Hanke,8 J. Hanks,37 C. Hanna,72

M. D. Hannam,91 J. Hanson,6 T. Hardwick,2 J. Harms,56,57 G. M. Harry,100 I. W. Harry,29 M. J. Hart,36

M. T. Hartman,5 C.-J. Haster,44 K. Haughian,36 A. Heidmann,59 M. C. Heintze,5,6 H. Heitmann,52 P. Hello,23

G. Hemming,34 M. Hendry,36 I. S. Heng,36 J. Hennig,36 A. W. Heptonstall,1 M. Heurs,8,17 S. Hild,36 D. Hoak,101

K. A. Hodge,1 D. Hofman,65 S. E. Hollitt,102 K. Holt,6 D. E. Holz,75 P. Hopkins,91 D. J. Hosken,102 J. Hough,36

E. A. Houston,36 E. J. Howell,50 Y. M. Hu,36 S. Huang,73 E. A. Huerta,103,82 D. Huet,23 B. Hughey,97 S. Husa,66

S. H. Huttner,36 T. Huynh-Dinh,6 A. Idrisy,72 N. Indik,8 D. R. Ingram,37 R. Inta,71 H. N. Isa,36 J.-M. Isac,59

M. Isi,1 G. Islas,22 T. Isogai,10 B. R. Iyer,15 K. Izumi,37 T. Jacqmin,59 H. Jang,77 K. Jani,63 P. Jaranowski,104

S. Jawahar,105 F. Jimenez-Forteza,66 W. W. Johnson,2 D. I. Jones,26 R. Jones,36 R. J. G. Jonker,9 L. Ju,50

Haris K,106 C. V. Kalaghatgi,24,91 V. Kalogera,82 S. Kandhasamy,21 G. Kang,77 J. B. Kanner,1 S. Karki,58

M. Kasprzack,2,23,34 E. Katsavounidis,10 W. Katzman,6 S. Kaufer,17 T. Kaur,50 K. Kawabe,37 F. Kawazoe,8,17

F. Kefelian,52 M. S. Kehl,69 D. Keitel,8,66 D. B. Kelley,35 W. Kells,1 R. Kennedy,86 J. S. Key,85

A. Khalaidovski,8 F. Y. Khalili,48 I. Khan,12 S. Khan,91 Z. Khan,95 E. A. Khazanov,107 N. Kijbunchoo,37 C. Kim,77

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J. Kim,108 K. Kim,109 Nam-Gyu Kim,77 Namjun Kim,40 Y.-M. Kim,108 E. J. King,102 P. J. King,37 D. L. Kinzel,6

J. S. Kissel,37 L. Kleybolte,27 S. Klimenko,5 S. M. Koehlenbeck,8 K. Kokeyama,2 S. Koley,9 V. Kondrashov,1

A. Kontos,10 M. Korobko,27 W. Z. Korth,1 I. Kowalska,60 D. B. Kozak,1 V. Kringel,8 B. Krishnan,8

A. Krolak,110,111 C. Krueger,17 G. Kuehn,8 P. Kumar,69 L. Kuo,73 A. Kutynia,110 B. D. Lackey,35 M. Landry,37

J. Lange,112 B. Lantz,40 P. D. Lasky,113 A. Lazzarini,1 C. Lazzaro,63,42 P. Leaci,29,79,28 S. Leavey,36

E. O. Lebigot,30,70 C. H. Lee,108 H. K. Lee,109 H. M. Lee,114 K. Lee,36 A. Lenon,35 M. Leonardi,89,90 J. R. Leong,8

N. Leroy,23 N. Letendre,7 Y. Levin,113 B. M. Levine,37 T. G. F. Li,1 A. Libson,10 T. B. Littenberg,115

N. A. Lockerbie,105 J. Logue,36 A. L. Lombardi,101 J. E. Lord,35 M. Lorenzini,12,13 V. Loriette,116 M. Lormand,6

G. Losurdo,57 J. D. Lough,8,17 H. Luck,17,8 A. P. Lundgren,8 J. Luo,78 R. Lynch,10 Y. Ma,50 T. MacDonald,40

B. Machenschalk,8 M. MacInnis,10 D. M. Macleod,2 F. Magana-Sandoval,35 R. M. Magee,55 M. Mageswaran,1

E. Majorana,28 I. Maksimovic,116 V. Malvezzi,25,13 N. Man,52 I. Mandel,44 V. Mandic,83 V. Mangano,36

G. L. Mansell,20 M. Manske,16 M. Mantovani,34 F. Marchesoni,117,33 F. Marion,7 S. Marka,39 Z. Marka,39

A. S. Markosyan,40 E. Maros,1 F. Martelli,56,57 L. Martellini,52 I. W. Martin,36 R. M. Martin,5

D. V. Martynov,1 J. N. Marx,1 K. Mason,10 A. Masserot,7 T. J. Massinger,35 M. Masso-Reid,36 F. Matichard,10

L. Matone,39 N. Mavalvala,10 N. Mazumder,55 G. Mazzolo,8 R. McCarthy,37 D. E. McClelland,20 S. McCormick,6

S. C. McGuire,118 G. McIntyre,1 J. McIver,1 D. J. McManus,20 S. T. McWilliams,103 D. Meacher,72

G. D. Meadors,29,8 J. Meidam,9 A. Melatos,84 G. Mendell,37 D. Mendoza-Gandara,8 R. A. Mercer,16 E. Merilh,37

M. Merzougui,52 S. Meshkov,1 C. Messenger,36 C. Messick,72 P. M. Meyers,83 F. Mezzani,28,79 H. Miao,44

C. Michel,65 H. Middleton,44 E. E. Mikhailov,119 L. Milano,67,4 J. Miller,10 M. Millhouse,31 Y. Minenkov,13

J. Ming,29,8 S. Mirshekari,120 C. Mishra,15 S. Mitra,14 V. P. Mitrofanov,48 G. Mitselmakher,5 R. Mittleman,10

A. Moggi,19 M. Mohan,34 S. R. P. Mohapatra,10 M. Montani,56,57 B. C. Moore,88 C. J. Moore,121 D. Moraru,37

G. Moreno,37 S. R. Morriss,85 K. Mossavi,8 B. Mours,7 C. M. Mow-Lowry,44 C. L. Mueller,5 G. Mueller,5

A. W. Muir,91 Arunava Mukherjee,15 D. Mukherjee,16 S. Mukherjee,85 N. Mukund,14 A. Mullavey,6 J. Munch,102

D. J. Murphy,39 P. G. Murray,36 A. Mytidis,5 I. Nardecchia,25,13 L. Naticchioni,79,28 R. K. Nayak,122 V. Necula,5

K. Nedkova,101 G. Nelemans,51,9 M. Neri,45,46 A. Neunzert,98 G. Newton,36 T. T. Nguyen,20 A. B. Nielsen,8

S. Nissanke,51,9 A. Nitz,8 F. Nocera,34 D. Nolting,6 M. E. Normandin,85 L. K. Nuttall,35 J. Oberling,37

E. Ochsner,16 J. O’Dell,123 E. Oelker,10 G. H. Ogin,124 J. J. Oh,125 S. H. Oh,125 F. Ohme,91 M. Oliver,66

P. Oppermann,8 Richard J. Oram,6 B. O’Reilly,6 R. O’Shaughnessy,112 D. J. Ottaway,102 R. S. Ottens,5

H. Overmier,6 B. J. Owen,71 A. Pai,106 S. A. Pai,47 J. R. Palamos,58 O. Palashov,107 C. Palomba,28 A. Pal-Singh,27

H. Pan,73 C. Pankow,82 F. Pannarale,91 B. C. Pant,47 F. Paoletti,34,19 A. Paoli,34 M. A. Papa,29,16,8 H. R. Paris,40

W. Parker,6 D. Pascucci,36 A. Pasqualetti,34 R. Passaquieti,18,19 D. Passuello,19 B. Patricelli,18,19 Z. Patrick,40

B. L. Pearlstone,36 M. Pedraza,1 R. Pedurand,65 L. Pekowsky,35 A. Pele,6 S. Penn,126 A. Perreca,1 M. Phelps,36

O. Piccinni,79,28 M. Pichot,52 F. Piergiovanni,56,57 V. Pierro,87 G. Pillant,34 L. Pinard,65 I. M. Pinto,87

M. Pitkin,36 R. Poggiani,18,19 P. Popolizio,34 E. K. Porter,30 A. Post,8 J. Powell,36 J. Prasad,14 V. Predoi,91

S. S. Premachandra,113 T. Prestegard,83 L. R. Price,1 M. Prijatelj,34 M. Principe,87 S. Privitera,29

G. A. Prodi,89,90 L. Prokhorov,48 O. Puncken,8 M. Punturo,33 P. Puppo,28 M. Purrer,29 H. Qi,16 J. Qin,50

V. Quetschke,85 E. A. Quintero,1 R. Quitzow-James,58 F. J. Raab,37 D. S. Rabeling,20 H. Radkins,37 P. Raffai,53

S. Raja,47 M. Rakhmanov,85 P. Rapagnani,79,28 V. Raymond,29 M. Razzano,18,19 V. Re,25 J. Read,22 C. M. Reed,37

T. Regimbau,52 L. Rei,46 S. Reid,49 D. H. Reitze,1,5 H. Rew,119 S. D. Reyes,35 F. Ricci,79,28 K. Riles,98

N. A. Robertson,1,36 R. Robie,36 F. Robinet,23 A. Rocchi,13 L. Rolland,7 J. G. Rollins,1 V. J. Roma,58

R. Romano,3,4 G. Romanov,119 J. H. Romie,6 D. Rosinska,127,43 S. Rowan,36 A. Rudiger,8 P. Ruggi,34 K. Ryan,37

S. Sachdev,1 T. Sadecki,37 L. Sadeghian,16 L. Salconi,34 M. Saleem,106 F. Salemi,8 A. Samajdar,122 L. Sammut,84,113

L. Sampson,82 E. J. Sanchez,1 V. Sandberg,37 B. Sandeen,82 J. R. Sanders,98,35 B. Sassolas,65

B. S. Sathyaprakash,91 P. R. Saulson,35 O. Sauter,98 R. L. Savage,37 A. Sawadsky,17 P. Schale,58 R. Schilling†,8

J. Schmidt,8 P. Schmidt,1,76 R. Schnabel,27 R. M. S. Schofield,58 A. Schonbeck,27 E. Schreiber,8 D. Schuette,8,17

B. F. Schutz,91,29 J. Scott,36 S. M. Scott,20 D. Sellers,6 A. S. Sengupta,94 D. Sentenac,34 V. Sequino,25,13

A. Sergeev,107 G. Serna,22 Y. Setyawati,51,9 A. Sevigny,37 D. A. Shaddock,20 S. Shah,51,9 M. S. Shahriar,82

M. Shaltev,8 Z. Shao,1 B. Shapiro,40 P. Shawhan,62 A. Sheperd,16 D. H. Shoemaker,10 D. M. Shoemaker,63

K. Siellez,52,63 X. Siemens,16 D. Sigg,37 A. D. Silva,11 D. Simakov,8 A. Singer,1 L. P. Singer,68 A. Singh,29,8

R. Singh,2 A. Singhal,12 A. M. Sintes,66 B. J. J. Slagmolen,20 J. R. Smith,22 N. D. Smith,1 R. J. E. Smith,1

E. J. Son,125 B. Sorazu,36 F. Sorrentino,46 T. Souradeep,14 A. K. Srivastava,95 A. Staley,39 M. Steinke,8

J. Steinlechner,36 S. Steinlechner,36 D. Steinmeyer,8,17 B. C. Stephens,16 S. Stevenson,44 R. Stone,85

K. A. Strain,36 N. Straniero,65 G. Stratta,56,57 N. A. Strauss,78 S. Strigin,48 R. Sturani,120 A. L. Stuver,6

T. Z. Summerscales,128 L. Sun,84 P. J. Sutton,91 B. L. Swinkels,34 M. J. Szczepanczyk,97 M. Tacca,30

D. Talukder,58 D. B. Tanner,5 M. Tapai,96 S. P. Tarabrin,8 A. Taracchini,29 R. Taylor,1 T. Theeg,8

M. P. Thirugnanasambandam,1 E. G. Thomas,44 M. Thomas,6 P. Thomas,37 K. A. Thorne,6 K. S. Thorne,76

E. Thrane,113 S. Tiwari,12 V. Tiwari,91 K. V. Tokmakov,105 C. Tomlinson,86 M. Tonelli,18,19 C. V. Torres‡,85

C. I. Torrie,1 D. Toyra,44 F. Travasso,32,33 G. Traylor,6 D. Trifiro,21 M. C. Tringali,89,90 L. Trozzo,129,19

M. Tse,10 M. Turconi,52 D. Tuyenbayev,85 D. Ugolini,130 C. S. Unnikrishnan,99 A. L. Urban,16 S. A. Usman,35

H. Vahlbruch,17 G. Vajente,1 G. Valdes,85 M. Vallisneri,76 N. van Bakel,9 M. van Beuzekom,9

J. F. J. van den Brand,61,9 C. Van Den Broeck,9 D. C. Vander-Hyde,35,22 L. van der Schaaf,9

J. V. van Heijningen,9 A. A. van Veggel,36 M. Vardaro,41,42 S. Vass,1 M. Vasuth,38 R. Vaulin,10 A. Vecchio,44

G. Vedovato,42 J. Veitch,44 P. J. Veitch,102 K. Venkateswara,131 D. Verkindt,7 F. Vetrano,56,57 A. Vicere,56,57

S. Vinciguerra,44 D. J. Vine,49 J.-Y. Vinet,52 S. Vitale,10 T. Vo,35 H. Vocca,32,33 C. Vorvick,37 D. Voss,5

W. D. Vousden,44 S. P. Vyatchanin,48 A. R. Wade,20 L. E. Wade,132 M. Wade,132 M. Walker,2 L. Wallace,1

S. Walsh,16,8,29 G. Wang,12 H. Wang,44 M. Wang,44 X. Wang,70 Y. Wang,50 R. L. Ward,20 J. Warner,37 M. Was,7

B. Weaver,37 L.-W. Wei,52 M. Weinert,8 A. J. Weinstein,1 R. Weiss,10 T. Welborn,6 L. Wen,50 P. Weßels,8

T. Westphal,8 K. Wette,8 J. T. Whelan,112,8 D. J. White,86 B. F. Whiting,5 R. D. Williams,1 A. R. Williamson,91

J. L. Willis,133 B. Willke,17,8 M. H. Wimmer,8,17 W. Winkler,8 C. C. Wipf,1 H. Wittel,8,17 G. Woan,36 J. Worden,37

3

J. L. Wright,36 G. Wu,6 J. Yablon,82 W. Yam,10 H. Yamamoto,1 C. C. Yancey,62 M. J. Yap,20 H. Yu,10 M. Yvert,7

A. Zadrozny,110 L. Zangrando,42 M. Zanolin,97 J.-P. Zendri,42 M. Zevin,82 F. Zhang,10 L. Zhang,1 M. Zhang,119

Y. Zhang,112 C. Zhao,50 M. Zhou,82 Z. Zhou,82 X. J. Zhu,50 M. E. Zucker,1,10 S. E. Zuraw,101 and J. Zweizig1

†Deceased, May 2015. ‡Deceased, March 2015.

(LIGO Scientific Collaboration and Virgo Collaboration)

1LIGO, California Institute of Technology, Pasadena, CA 91125, USA2Louisiana State University, Baton Rouge, LA 70803, USA3Universita di Salerno, Fisciano, I-84084 Salerno, Italy4INFN, Sezione di Napoli, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy5University of Florida, Gainesville, FL 32611, USA6LIGO Livingston Observatory, Livingston, LA 70754, USA7Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Universite Savoie Mont Blanc, CNRS/IN2P3, F-74941 Annecy-le-Vieux,

France8Albert-Einstein-Institut, Max-Planck-Institut fur Gravitationsphysik, D-30167 Hannover, Germany9Nikhef, Science Park, 1098 XG Amsterdam, Netherlands

10LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA11Instituto Nacional de Pesquisas Espaciais, 12227-010 Sao Jose dos Campos, Sao Paulo, Brazil12INFN, Gran Sasso Science Institute, I-67100 L’Aquila, Italy13INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy14Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India15International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560012, India16University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA17Leibniz Universitat Hannover, D-30167 Hannover, Germany18Universita di Pisa, I-56127 Pisa, Italy19INFN, Sezione di Pisa, I-56127 Pisa, Italy20Australian National University, Canberra, Australian Capital Territory 0200, Australia21The University of Mississippi, University, MS 38677, USA22California State University Fullerton, Fullerton, CA 92831, USA23LAL, Universite Paris-Sud, CNRS/IN2P3, Universite Paris-Saclay, 91400 Orsay, France24Chennai Mathematical Institute, Chennai 603103, India25Universita di Roma Tor Vergata, I-00133 Roma, Italy26University of Southampton, Southampton SO17 1BJ, United Kingdom27Universitat Hamburg, D-22761 Hamburg, Germany28INFN, Sezione di Roma, I-00185 Roma, Italy29Albert-Einstein-Institut, Max-Planck-Institut fur Gravitationsphysik, D-14476 Potsdam-Golm, Germany30APC, AstroParticule et Cosmologie, Universite Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cite,

F-75205 Paris Cedex 13, France31Montana State University, Bozeman, MT 59717, USA32Universita di Perugia, I-06123 Perugia, Italy33INFN, Sezione di Perugia, I-06123 Perugia, Italy34European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy35Syracuse University, Syracuse, NY 13244, USA36SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom37LIGO Hanford Observatory, Richland, WA 99352, USA38Wigner RCP, RMKI, H-1121 Budapest, Konkoly Thege Miklos ut 29-33, Hungary39Columbia University, New York, NY 10027, USA40Stanford University, Stanford, CA 94305, USA41Universita di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova, Italy42INFN, Sezione di Padova, I-35131 Padova, Italy43CAMK-PAN, 00-716 Warsaw, Poland44University of Birmingham, Birmingham B15 2TT, United Kingdom

4

45Universita degli Studi di Genova, I-16146 Genova, Italy46INFN, Sezione di Genova, I-16146 Genova, Italy47RRCAT, Indore MP 452013, India48Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia49SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom50University of Western Australia, Crawley, Western Australia 6009, Australia51Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, Netherlands52Artemis, Universite Cote d’Azur, CNRS, Observatoire Cote d’Azur, CS 34229, Nice cedex 4, France53MTA Eotvos University, “Lendulet” Astrophysics Research Group, Budapest 1117, Hungary54Institut de Physique de Rennes, CNRS, Universite de Rennes 1, F-35042 Rennes, France55Washington State University, Pullman, WA 99164, USA56Universita degli Studi di Urbino “Carlo Bo,” I-61029 Urbino, Italy57INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy58University of Oregon, Eugene, OR 97403, USA59Laboratoire Kastler Brossel, UPMC-Sorbonne Universites, CNRS, ENS-PSL Research University, College de France, F-75005 Paris, France60Astronomical Observatory Warsaw University, 00-478 Warsaw, Poland61VU University Amsterdam, 1081 HV Amsterdam, Netherlands62University of Maryland, College Park, MD 20742, USA63Center for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA64Institut Lumiere Matiere, Universite de Lyon, Universite Claude Bernard Lyon 1, UMR CNRS 5306, 69622 Villeurbanne, France65Laboratoire des Materiaux Avances (LMA), IN2P3/CNRS, Universite de Lyon, F-69622 Villeurbanne, Lyon, France66Universitat de les Illes Balears, IAC3—IEEC, E-07122 Palma de Mallorca, Spain67Universita di Napoli “Federico II,” Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy68NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA69Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada70Tsinghua University, Beijing 100084, China71Texas Tech University, Lubbock, TX 79409, USA72The Pennsylvania State University, University Park, PA 16802, USA73National Tsing Hua University, Hsinchu City, 30013 Taiwan, Republic of China74Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia75University of Chicago, Chicago, IL 60637, USA76Caltech CaRT, Pasadena, CA 91125, USA77Korea Institute of Science and Technology Information, Daejeon 305-806, Korea78Carleton College, Northfield, MN 55057, USA79Universita di Roma “La Sapienza,” I-00185 Roma, Italy80University of Brussels, Brussels 1050, Belgium81Sonoma State University, Rohnert Park, CA 94928, USA82Northwestern University, Evanston, IL 60208, USA83University of Minnesota, Minneapolis, MN 55455, USA84The University of Melbourne, Parkville, Victoria 3010, Australia85The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA86The University of Sheffield, Sheffield S10 2TN, United Kingdom87University of Sannio at Benevento, I-82100 Benevento, Italy and INFN, Sezione di Napoli, I-80100 Napoli, Italy88Montclair State University, Montclair, NJ 07043, USA89Universita di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy90INFN, Trento Institute for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy91Cardiff University, Cardiff CF24 3AA, United Kingdom92National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan93School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom94Indian Institute of Technology, Gandhinagar Ahmedabad Gujarat 382424, India95Institute for Plasma Research, Bhat, Gandhinagar 382428, India

5

96University of Szeged, Dom ter 9, Szeged 6720, Hungary97Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA98University of Michigan, Ann Arbor, MI 48109, USA99Tata Institute of Fundamental Research, Mumbai 400005, India

100American University, Washington, D.C. 20016, USA101University of Massachusetts-Amherst, Amherst, MA 01003, USA102University of Adelaide, Adelaide, South Australia 5005, Australia103West Virginia University, Morgantown, WV 26506, USA104University of Bia lystok, 15-424 Bia lystok, Poland105SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom106IISER-TVM, CET Campus, Trivandrum Kerala 695016, India107Institute of Applied Physics, Nizhny Novgorod, 603950, Russia108Pusan National University, Busan 609-735, Korea109Hanyang University, Seoul 133-791, Korea110NCBJ, 05-400 Swierk-Otwock, Poland111IM-PAN, 00-956 Warsaw, Poland112Rochester Institute of Technology, Rochester, NY 14623, USA113Monash University, Victoria 3800, Australia114Seoul National University, Seoul 151-742, Korea115University of Alabama in Huntsville, Huntsville, AL 35899, USA116ESPCI, CNRS, F-75005 Paris, France117Universita di Camerino, Dipartimento di Fisica, I-62032 Camerino, Italy118Southern University and A&M College, Baton Rouge, LA 70813, USA119College of William and Mary, Williamsburg, VA 23187, USA120Instituto de Fısica Teorica, University Estadual Paulista/ICTP South American Institute for Fundamental Research, Sao Paulo SP 01140-

070, Brazil121University of Cambridge, Cambridge CB2 1TN, United Kingdom122IISER-Kolkata, Mohanpur, West Bengal 741252, India123Rutherford Appleton Laboratory, HSIC, Chilton, Didcot, Oxon OX11 0QX, United Kingdom124Whitman College, 345 Boyer Avenue, Walla Walla, WA 99362 USA125National Institute for Mathematical Sciences, Daejeon 305-390, Korea126Hobart and William Smith Colleges, Geneva, NY 14456, USA127Janusz Gil Institute of Astronomy, University of Zielona Gora, 65-265 Zielona Gora, Poland128Andrews University, Berrien Springs, MI 49104, USA129Universita di Siena, I-53100 Siena, Italy130Trinity University, San Antonio, TX 78212, USA131University of Washington, Seattle, WA 98195, USA132Kenyon College, Gambier, OH 43022, USA133Abilene Christian University, Abilene, TX 79699, USA

ABSTRACT

Supplemental information for a Letter reporting the rate of binary black hole (BBH) coalescences in-

ferred from 16 days of coincident Advanced LIGO observations surrounding the transient gravitational

wave (GW) signal GW150914. In that work we reported various rate estimates whose 90% credible

intervals (CIs) fell in the range 2–600 Gpc−3 yr−1. Here we give details of our method and com-

putations, including information about our search pipelines, a derivation of our likelihood function

for the analysis, a description of the astrophysical search trigger distribution expected from merging

BBHs, details on our computational methods, a description of the effects and our model for calibration

uncertainty, and an analytic method of estimating our detector sensitivity that is calibrated to our

measurements.

The first detection of a gravitational wave (GW) sig- nal from a merging binary black hole (BBH) system is

6

described in Abbott et al. (2016d). Abbott et al. (2016g)

reports on inference of the local BBH merger rate from

surrounding Advanced LIGO observations. This Supple-

ment provides supporting material and methodological

details for Abbott et al. (2016g), hereafter referred to as

the Letter.

1. SEARCH PIPELINES

Both the pycbc and gstlal pipelines are based on

matched filtering against a bank of template waveforms.

See Abbott et al. (2016c) for a detailed description of

the pipelines in operation around the time of GW150914;

here we provide an abbreviated description.

In the pycbc pipeline, the single-detector signal-to-

noise ratio (SNR) is re-weighted by a chi-squared fac-

tor (Allen 2005) to account for template-data mismatch

(Babak et al. 2013); the re-weighted single-detector SNRs

are combined in quadrature to produce a detection statis-

tic for search triggers.

The gstlal pipeline’s detection statistic, however, is

based on a likelihood ratio (Cannon et al. 2013, 2015)

constructed from the single-detector SNRs and a signal-

consistency statistic. An analytic estimate of the distri-

bution of astrophysical signals in multiple-detector SNR

and signal consistency statistic space is compared to a

measured distribution of single-detector triggers without

a coincident counterpart in the other detector to form a

multiple-detector likelihood ratio.

Both pipelines rely on an empirical estimate of the

search background, making the assumption that triggers

of terrestrial origin occur independently in the two detec-

tors. The background estimate is built from observations

of single-detector triggers over a long time (gstlal) or

through searching over a data stream with one detector’s

output shifted in time relative to the other’s by an in-

terval that is longer than the light travel time between

detectors, ensuring that no coincident astrophysical sig-

nals remain in the data (pycbc). For both pipelines it

is not possible to produce an instantaneous background

estimate at a particular time; this drives our choice of

likelihood function as described in Section 2.

The gstlal pipeline natively determines the functions

p0(x) and p1(x) for its detection statistic x. For this anal-

ysis a threshold of xmin = 5 was applied, which is suffi-

ciently low that the trigger density is dominated by ter-

restrial triggers near threshold. There were M = 15 848

triggers observed above this threshold in the 17 days of

observation time analyzed by gstlal.

For pycbc, the quantity x′ is the re-weighted SNR de-

tection statistic.1 We set a threshold x′min = 8, above

which M ′ = 270 triggers remain in the search. We use a

histogram of triggers collected from time-shifted data to

1 When quoting pipeline-specific values we distinguish pycbcquantities with a prime.

8.0 8.5 9.0 9.5 10.0

x′

10−3

10−2

10−1

100

101

p(x′ )

p0

p1

Figure 1. Inferred terrestrial (p0; blue) and astrophysical

(p1; green) trigger densities for the pycbc pipeline as

described in Section 1.

estimate the terrestrial trigger density, p0 (x′), and a his-

togram of the recovered triggers from the injection sets

described in Section 2.2 of the Letter to estimate the as-

trophysical trigger density, p1 (x′). These estimates are

shown in Figure 1. The uncertainty in the distribution of

triggers from this estimation procedure is much smaller

than the uncertainty in overall rate from the finite num-

ber statistics (see, for example, Figure 4). The empirical

estimate is necessary to properly account for the inter-

action of the various single- and double-interferometer

thresholds in the pycbc search (Abbott et al. 2016c).

At high SNR, where these thresholds are irrelevant, the

astrophysical triggers follow an approximately flat-space

volumetric density (see Section 3) of

p1(x′) ' 3x′3min

x′4, (1)

but they deviate from this at smaller SNR due to thresh-

old effects in the search.

For the pycbc pipeline, a detection statistic x′ ≥ 10.1

corresponds to an estimated search false alarm rate

(FAR) of one per century.

2. DERIVATION OF POISSON MIXTURE MODEL

LIKELIHOOD

In this section we derive the likelihood function in Eq.

(3) of the Letter. Consider first a search of the type de-

scribed in Section 1 over NT intervals of time of width

δi, {i = 1, . . . , NT }. Triggers above some fixed threshold

occur with an instantaneous rate in time and detection

statistic x given by the sum of the terrestrial and astro-

physical rates:

dN

dtdx(t, x) = R0(t)p0(x; t) +R1(t)V (t)p1(x; t), (2)

7

where R0(t) is the instantaneous rate (number per unit

time) of terrestrial triggers, R1(t) is the instantaneous

rate density (number per unit time per unit comoving

volume) of astrophysical triggers, p0 is the instantaneous

density in detection statistic of terrestrial triggers, p1 is

the instantaneous density in detection statistic of astro-

physical triggers, and V (t) is the instantaneous sensitive

comoving redshifted volume (Abbott et al. 2016a, see also

Eq. (15) of the Letter) of the detectors to the assumed

source population. The astrophysical rate R1 is to any

reasonable approximation constant over our observations

so we will drop the time dependence of this term from

here on.2 Note that R0 and R1 have different units in

this expression; the former is a rate (per time), while the

latter is a rate density (per time-volume). The density

p1 is independent of source parameters as described in

Section 3. Let

dN

dt≡∫

dxdN

dtdx= R0(t) +R1V (t). (3)

If the search intervals δi are sufficiently short, they

will contain at most one trigger and the time-dependent

terms in Eq. (2) will be approximately constant. Then

the likelihood for a set of times and detection statistics

of triggers, {(tj , xj)|j = 1, . . . ,M}, is a product over in-

tervals containing a trigger (indexed by j) and intervals

that do not contain a trigger (indexed by k) of the cor-

responding Poisson likelihoods

L =

M∏

j=1

dN

dtdx(tj , xj) exp

[−δj

dN

dt(tj)

]

×{

NT−M∏

k=1

exp

[−δk

dN

dt(tk)

]}(4)

(cf. Farr et al. (2015, Eq. (21)) or Loredo & Wasserman

(1995, Eq. (2.8))).3 Now let the width of the observation

intervals δi go to zero uniformly as the number of inter-

vals goes to infinity. Then the products of exponentials

in Eq. (4) become an exponential of an integral, and we

have

L =

M∏

j=1

[dN

dtdx(tj , xj)

]exp [−N ] , (5)

where

N =

∫dt

dN

dt(6)

is the expected number of triggers of both types in the

total observation time T .

2 The astrophysical rate can, in principle, also depend on red-shift, but in this paper we assume that the BBH coalescence rateis constant in the comoving frame.

3 There is a typo in Eq. (2.8) of Loredo & Wasserman (1995).The second term in the final bracket is missing a factor of δt.

As discussed in Section 1, in our search we observe

that R0 remains approximately constant and that p0 re-

tains its shape over the observation time discussed here;

this assumption is used in our search background esti-

mation procedure (Abbott et al. 2016c). The astrophys-

ical distribution of triggers is universal (Section 3) and

also time-independent. Finally, the detector sensitivity

is observed to be stable over our 16 days of coincident

observations, so V (t) ' const (Abbott et al. 2016b). We

therefore choose to simply ignore the time dimension in

our trigger set. This generates an estimate of the rate

that is sub-optimal (i.e. has larger uncertainty) but con-

sistent with using the full data set to the extent that the

detector sensitivity varies in time; since this variation is

small, the loss of information about the rate will be cor-

respondingly small. We do capture any variation in the

sensitivity with time in our Monte-Carlo procedure for

estimating 〈V T 〉 that is described in Section 2.2 of the

Letter.

If we ignore the trigger time, then the appropriate like-

lihood to use is a marginalization of Eq. (5) over the tj .

Let

L ≡∫ ∏

j

dtj

L

=∏

j

[Λ0p0 (xj) + Λ1p1 (xj)] exp [−Λ0 − Λ1] , (7)

where

Λ0p0(x) =

∫dtR0(t)p0 (x; t) , (8)

and

Λ1p1(x) =

∫dtR1V (t)p1 (x; t) , (9)

with ∫dx p0(x) =

∫dx p1(x) = 1. (10)

If we assume that R1 is constant in (comoving) time,

and measure p1(x) by accumulating recovered injections

throughout the run as we have done, then this expression

reduces to the likelihood in Eq. (3) of the Letter. A sim-

ilar argument with an additional population of triggers

produces Eq. (10) of the Letter.

Our procedure for estimating p0(x) in the pipelines also

provides an estimate of the mean number of background

events per experiment Λ0 (Abbott et al. 2016c). We,

however, have chosen to leave Λ0 as a free parameter in

our analysis with a broad prior and infer it from the ob-

served data, rather than using the background estimate

to constrain the prior, which would result in a much nar-

rower posterior on Λ0. This is conservative in the sense

that it will broaden the posterior on Λ1 from which we

infer rates. However, because there are so many more

8

200 220 240 260 280 300 320 340

Λ0

0

2

4

6

8

10

12Λ

1

Figure 2. The two-dimensional posterior on terrestrial

and astrophysical trigger counts (Λ0 and Λ1 in Eq. (5) of

the Letter) for the pycbc search. Countours are drawn

at the 10%, 20%, . . . , 90%, and 99% credible levels.

There is no meaningful correlation between the two vari-

ables. The Poisson uncertainty in the terrestrial count

is ∼√

270, or 16, which is also very nearly the Poisson

uncertainty in the total count. Because this uncertainty

is much larger than the astrophysical count, changes in

the astrophysical count do not force the terrestrial count

to adjust in a meaningful way and the variables are un-

correlated in the posterior.

triggers in both searches of terrestrial origin than astro-

physical there is little correlation between Λ0 and Λ1,

and so there is little difference between the posterior we

obtain on Λ1 and the posterior we would have obtained

had we implemented the tight prior on Λ0. Figure 2shows the two-dimensional posterior we obtain from Eq.

(5) of the Letter on Λ0 and Λ1.

3. UNIVERSAL ASTROPHYSICAL TRIGGER

DISTRIBUTION

Both the pycbc and gstlal pipelines rely on the SNR

as part of their detection statistic, x. The SNR of an

astrophysical trigger is a function of the detector noise

at the time of detection and the parameters of the trigger.

Schutz (2011) and Chen & Holz (2014) demonstrate that

the distribution of SNR ρ in a simple model of a detection

pipeline that simply thresholds on SNR, ρ ≥ ρth, with

sources in the local universe is universal, following

p (ρ) =3ρ3thρ4

. (11)

This result follows from the fact that the expected value

of the SNR in a matched-filter search for compact binary

coalescence (CBC) signals scales inversely with trans-

verse comoving distance (Hogg 1999):

ρ =A (m1,m2,~a1,~a2, S(f), z)B (angles)

DM, (12)

where A is an amplitude factor that depends on the in-

trinsic properties (source-frame masses and spins) of the

source, the detector sensitivity expressed as a noise power

spectral density S(f) as a function of observer frequency

and redshift z, and B is an angular factor depending

on the location of the source in the sky and the relative

orientations of binary orbit and detector. The redshift

enters A only through shifting the source waveform to

lower frequency at higher redshift, changing A because

the sensitivity varies with observer frequency f . For the

redshifts to which we are sensitive to BBH in this obser-

vation period this effect on A is small.

If we assume that the distribution of source parameters

is constant over the range of distances to which we are

sensitive, and ignore the small redshift-dependent sensi-

tivity correction mentioned above, then the distribution

of SNR will be governed entirely by the distribution of

distances of the sources, which, in the local universe is

approximately

p (DM ) ∝ D2M , (13)

yielding the distribution of SNR given in Eq. (11).

Both the pycbc and gstlal pipelines use a more com-

plicated system of thresholds than this simple model,

but the distribution of detection statistics remains, to

an excellent approximation, independent of the source

parameters. Figure 3 shows the distribution of recovered

detection statistics for the various injection campaigns

with varying source distribution used to estimate sensi-

tive time-volumes in the pycbc pipeline. For loud signals,

the detection statistic is proportional to SNR, and the

distribution is not sensitive to the complicated thresh-

olding in the pipeline, so we recover Eq. (13); for quiet

signals the interaction of various single-detector thresolds

in the pipeline causes the distribution to deviate from

this analytic approximation, but it remains independent

of the distribution of sources.

Because the distribution of detection statistics is uni-

versal, we cannot learn anything about the source popu-

lation from the detection statistic alone; we must instead

resort to parameter estimation (PE) followup (Veitch

et al. 2015; Abbott et al. 2016e) of triggers to determine

their parameters. The parameters of the waveform tem-

plate that produced the trigger can be used to guess the

parameters of the source that generated that trigger, but

the bias and uncertainty in this estimate are very large

compared to the PE estimate. We therefore ignore the

parameters of the waveform template that generated the

trigger in the assignment of triggers to BBH classes.

Strictly speaking, the fact that we have performed

PE means that we have examined the full interferom-

9

101 102

x

10−3

10−2

10−1

100

p(x

)Analytic

GW150914

LVT151012

Flat in Log

Power Law

All

Figure 3. The distribution of detection statistics in the

pycbc pipeline for the signals recovered in the injection

campaigns used to estimate sensitive time-volumes for

various BBH population assumptions. The solid line

gives the analytic approximation to the distribution from

Eq. (11), which agrees well with the recovered statistics

for loud signals; for quieter signals the interaction of var-

ious thresholds in the pipeline causes the distribution to

deviate from the analytic approximation, but it remains

independent of the source distribution.

eter data surrounding GW150914 and LVT151012, so we

should modify the background density: p0 for these trig-

gers should be the density of terrestrial triggers near the

corresponding x that can produce data giving the asso-

ciated PE posteriors, and pi should be the density of

astrophysical triggers near the corresponding x that can

produce data giving the associated PE posteriors. This

will modify the density ratio pi/p0, and therefore adjust

the probability that LVT151012 is of astrophysical origin;

the probability of astrophysical origin for GW150914 will

remain 1 to an excellent approximation. This approach

to rate estimation is under active development and re-

sults will appear in the future.

4. COUNT POSTERIOR

We impose a prior on the Λ parameters of:

p (Λ1,Λ0) ∝ 1√Λ1

1√Λ0

. (14)

The posterior on counts is proportional to the product

of the likelihood from Eq. (3) of the Letter and the prior

from Eq. (14):

p (Λ1,Λ0| {xj |j = 1, . . . ,M})

∝

M∏

j=1

[Λ1p1 (xj) + Λ0p0 (xj)]

× exp [−Λ1 − Λ0]1√

Λ1Λ0

. (15)

For estimation of the Poisson rate parameter in a simple

Poisson model, the Jeffreys prior is 1/√

Λ. With this

prior, the posterior mean on Λ is N+1/2 for N observed

counts. With a prior proportional to 1/Λ the mean is

N for N > 0, but the posterior is improper when N =

0. For a flat prior, the mean is N + 1. Though the

behaviour of the mean is not identical with our mixture

model posterior, it is similar; because we find 〈Λ1〉 �1/2, the choice of prior among these three reasonable

options has little influence on our results here.

For the pycbc data set we find the posterior median

and 90% credible range Λ1 = 3.2+4.9−2.4 above our thresh-

old. For the gstlal set we find the posterior median and

90% credible range Λ1 = 4.8+7.9−3.8. Though we have only

one event (GW150914) at exceptionally high significance,

and one other at marginal significance (LVT151012), the

counting analysis shows these to be consistent with the

possible presence of several more events of astrophysical

origin at lower detection statistic in both pipelines.

The thresholds applied to the pycbc and gstlal trig-

gers for this analysis are not equivalent to each other

in terms of either SNR or false alarm rate; instead, both

thresholds have been chosen so that the rate of triggers of

terrestrial origin (Λ0p0) dominates near threshold. Since

the threshold is set at different values for each pipeline,

we do not expect the counts to be the same between

pipelines.

The estimated astrophysical and terrestrial trigger rate

densities (Eq. (1) of the Letter) for pycbc are plotted in

Figure 4. We select triggers from a subset of the search

parameter space (i.e. our bank of template waveforms)

that contains GW150914 as well as the mass range con-

sidered for possible alternative populations of BBH bina-

ries in Section 3 of the Letter. There are M ′ = 270 two-

detector coincident triggers in this range in the pycbc

search (Abbott et al. 2016c). Figure 4 also shows an es-

timate of the density of triggers that comprise our data

set which agrees well with our inference of the trigger

rate.

Based on the probability of astrophysical origin in-

ferred for LVT151012 from the two-component mixture

model in Eq. (15) and shown in Figure 5, we introduce a

third class of signals and use a three-component mixture

model with counts Λ0 (terrestrial), Λ1 (GW150914-like),

and Λ2 (LVT151012-like) to infer rates in Sections 2.1 of

10

8 9 10 20 30

x′

10−2

10−1

100

101

102

103

Λipi(x′ )

Figure 4. The inferred number density of astrophysical (green), terrestrial (blue), and all (red) triggers as a function

of x′ for the pycbc search (cf. Eq. (1) of the Letter), using the models for each population described in Section 2.1 of

the Letter. The solid lines give the posterior median and the shaded regions give the symmetric 90% credible interval

from the posterior in Eq. (5) of the Letter. We also show a binned estimate of the trigger number density from the

search (black); bars indicate the 68% confidence Poisson uncertainty on the number of triggers in the vertical-direction

and bin width in the horizontal-direction.

5 10 15 20

x

10−5

10−4

10−3

10−2

10−1

100

P1(x|{xj|j

=1,...,M})

8.0 8.5 9.0 9.5 10.0

x′

10−4

10−3

10−2

10−1

100

P1

( x′ |{

x′ j|j

=1,...,M})

Figure 5. The posterior probability that coincident triggers in our analysis come from an astrophysical source (see

Eq. (7) of the Letter), taking into account the astrophysical and terrestrial counts estimated in Section 2.1 of the

Letter. Left: the gstlal triggers with x > 5; right: pycbc triggers with x′ > 8. GW150914 is not shown in the plot

because its probability of astrophysical origin is effectively 100%. The only two triggers with P1 & 50% are GW150914

and LVT151012. For GW150914, we find P1 = 1 to very high precision; for LVT151012, the gstlal pipeline finds

P1 = 0.84 and the pycbc pipeline finds P1 = 0.91.

11

the Letter and 2.2 of the Letter.

We use the Stan and emcee Markov-Chain Monte Carlo

samplers (Foreman-Mackey et al. 2013; Stan Develop-

ment Team 2015b,a) to draw samples from the posterior

in Eq. (5) of the Letter for the two pipelines. We have

assessed the convergence and mixing of our chains us-

ing empirical estimates of the autocorrelation length in

each parameter (Sokal 1996), the Gelman-Rubin R con-

vergence statistic (Gelman & Rubin 1992), and through

visual inspection of chain plots. By all measures, the

chains appear well-converged to the posterior distribu-

tion.

Table 1 contains the full results on counts and associ-

ated sensitive time-volumes for both pipelines.

5. CALIBRATION UNCERTAINTY

The LIGO detectors are subject to uncertainty in their

calibration, in both the measured amplitude and phase of

the gravitational-wave strain. Abbott et al. (2016b) dis-

cusses the methods used to calibrate the strain output

of the detector during the 16 days of coincident obser-

vations discussed here. Abbott et al. (2016b) estimates

that the reported strain is accurate to within 10% in am-

plitude and 10 degrees in phase between 20 Hz and 1 kHz

throughout the observations.

The SNRs reported by our searches are quadratically

sensitive to calibration errors because they are maxi-

mized over arrival time, waveform phase, and a template

bank of waveforms (Allen 1996; Brown & LIGO Scien-

tific Collaboration 2004). Abbott et al. (2016c) demon-

strates that the other search pipeline outputs are also

not affected to a significant degree by the calibration un-

certainty present during our observing run. Therefore,

we ignore effects of calibration on the pipeline detection

statistics x and x′ we use here to estimate rates from the

pycbc and gstlal pipelines.

The amplitude calibration uncertainty in the detector

results at leading order in a corresponding uncertainty

between the luminosity distances of sources measured

from real detector outputs (Abbott et al. 2016e) and the

luminosity distances used to produce injected waveforms

used to estimate sensitive time-volumes in this work. A

10% uncertainty in dL at these redshifts corresponds to

an approximately 30% uncertainty in volume. We model

this uncertainty by treating 〈V T 〉 as a parameter in our

analysis, and imposing a log-normal prior:

p (log 〈V T 〉) ∝ N(

logµ,σ

µ

), (16)

where µ is the Monte-Carlo estimate of sensitive time-

volume produced from the injection campaigns described

in Section 2.2 of the Letter and

σ2 = σ2cal + σ2

stat, (17)

with σcal = 0.3µ and σstat is the estimate of the Monte-

Carlo uncertainty from the finite number of recovered

injections reported above. In all cases σcal � σstat.

Since the likelihood in Eqs. (3) of the Letter or (10)

of the Letter does not constrain 〈V T 〉 independently of

R, sampling over 〈V T 〉 at the same time as Λ and R

has the effect of convolving the log-normal distribution

of 〈V T 〉 with the posterior on Λ in the inference of R.

In spite of the 30% relative uncertainty in 〈V T 〉 from

calibration uncertainty, the counting uncertainty on R

from the small number of detected events dominates the

width of the posterior on R.

6. ANALYTIC SENSITIVITY ESTIMATE

As a rough check on our 〈V T 〉 estimates and the in-

tegrand d 〈V T 〉 /dz, we find that the following approxi-

mate, analytic procedure also produces a good approxi-

mation to the pycbc Monte-Carlo estimate in Table 1.

1. Generate inspiral–merger–ringdown waveforms in a

single detector at various redshifts from the source

distribution s(θ) with random orientations and sky

positions.

2. Using the high-sensitivity early Advanced LIGO

noise power spectral density from Abbott et al.

(2016f), compute the SNR in a single detector.

3. Consider a signal found if the SNR is greater than

8.

Employed with the source distributions described

above, this approximate procedure yields 〈V T 〉1 '0.107 Gpc3 yr and 〈V T 〉2 ' 0.0225 Gpc3 yr for the sensi-

tivity to the two classes of merging BBH system. Figure

6 shows the sensitive time-volume integrand,

d〈V T 〉dz

≡ T 1

1 + z

dVcdz

∫dθ s(θ)f(z, θ) (18)

estimated from this procedure for systems with various

parameters superimposed on the Monte-Carlo estimates

from the injection campaign described above.

The authors gratefully acknowledge the support of the

United States National Science Foundation (NSF) for

the construction and operation of the LIGO Laboratory

and Advanced LIGO as well as the Science and Tech-

nology Facilities Council (STFC) of the United King-

dom, the Max-Planck-Society (MPS), and the State of

Niedersachsen/Germany for support of the construction

of Advanced LIGO and construction and operation of

the GEO600 detector. Additional support for Advanced

LIGO was provided by the Australian Research Coun-

cil. The authors gratefully acknowledge the Italian Isti-

tuto Nazionale di Fisica Nucleare (INFN), the French

12

Table 1. Counts and sensitive time-volumes to BBH mergers

estimated under various assumptions. See Sections 2.1 of the

Letter, 2.2 of the Letter, 3 of the Letter and 4.

Λ 〈V T 〉 /Gpc3 yr

pycbc gstlal pycbc gstlal

GW150914 2.1+4.1−1.7 3.6+6.9

−2.9 0.130+0.084−0.051 0.21+0.14

−0.08

LVT151012 2.0+4.0−1.7 3.0+6.8

−2.7 0.032+0.020−0.012 0.048+0.031

−0.019

Both 4.5+5.5−3.1 7.4+9.2

−5.1 · · · · · ·

Astrophysical

Flat in log mass3.2+4.9−2.4 4.8+7.9

−3.8

0.050+0.032−0.019 0.080+0.051

−0.031

Power Law (-2.35) 0.0154+0.0098−0.0060 0.024+0.015

−0.009

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

z

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

d〈VT〉/

dz

(Gp

c3yr)

GW150914

LVT151012

10–10

20–20

40–40

Figure 6. The rate at which sensitive time-volume ac-

cumulates with redshift. Curves labeled by component

masses in M� are computed using the approximate pre-

scription described in Section 6, assuming sources with

fixed masses in the comoving frame and without spin; the

GW150914 and LVT151012 curves are determined from

the Monte-Carlo injection campaign described in Section

2.2 of the Letter.

Centre National de la Recherche Scientifique (CNRS)

and the Foundation for Fundamental Research on Mat-

ter supported by the Netherlands Organisation for Sci-

entific Research, for the construction and operation of

the Virgo detector and the creation and support of the

EGO consortium. The authors also gratefully acknowl-

edge research support from these agencies as well as by

the Council of Scientific and Industrial Research of India,

Department of Science and Technology, India, Science &

Engineering Research Board (SERB), India, Ministry of

Human Resource Development, India, the Spanish Min-

isterio de Economıa y Competitividad, the Conselleria

d’Economia i Competitivitat and Conselleria d’Educacio,

Cultura i Universitats of the Govern de les Illes Balears,

the National Science Centre of Poland, the European

Commission, the Royal Society, the Scottish Funding

Council, the Scottish Universities Physics Alliance, the

Hungarian Scientific Research Fund (OTKA), the Lyon

Institute of Origins (LIO), the National Research Foun-

dation of Korea, Industry Canada and the Province of

Ontario through the Ministry of Economic Development

and Innovation, the Natural Science and Engineering Re-

search Council Canada, Canadian Institute for Advanced

Research, the Brazilian Ministry of Science, Technology,

and Innovation, Russian Foundation for Basic Research,

the Leverhulme Trust, the Research Corporation, Min-

istry of Science and Technology (MOST), Taiwan and

the Kavli Foundation. The authors gratefully acknowl-

edge the support of the NSF, STFC, MPS, INFN, CNRS

and the State of Niedersachsen/Germany for provision of

computational resources. This article has been assigned

the document number LIGO-P1500217.

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