of 16
Calibration of the Advanced LIGO detectors for the discovery of the binary
black-hole merger GW150914
B. P. Abbott,
1
R. Abbott,
1
T. D. Abbott,
2
M. R. Abernathy,
1
K. Ackley,
3
C. Adams,
4
P. Addesso,
5
R. X. Adhikari,
1
V. B. Adya,
6
C. Affeldt,
6
N. Aggarwal,
7
O. D. Aguiar,
8
A. Ain,
9
P. Ajith,
10
B. Allen,
6,11,12
P. A. Altin,
13
D. V. Amariutei,
3
S. B. Anderson,
1
W. G. Anderson,
11
K. Arai,
1
M. C. Araya,
1
C. C. Arceneaux,
14
J. S. Areeda,
15
K. G. Arun,
16
G. Ashton,
17
M. Ast,
18
S. M. Aston,
4
P. Aufmuth,
12
C. Aulbert,
6
S. Babak,
19
P. T. Baker,
20
S. W. Ballmer,
21
J. C. Barayoga,
1
S. E. Barclay,
22
B. C. Barish,
1
D. Barker,
23
B. Barr,
22
L. Barsotti,
7
J. Bartlett,
23
I. Bartos,
24
R. Bassiri,
25
J. C. Batch,
23
C. Baune,
6
B. Behnke,
19
A. S. Bell,
22
C. J. Bell,
22
B. K. Berger,
1
J. Bergman,
23
G. Bergmann,
6
C. P. L. Berry,
26
J. Betzwieser,
4
S. Bhagwat,
21
R. Bhandare,
27
I. A. Bilenko,
28
G. Billingsley,
1
J. Birch,
4
R. Birney,
29
S. Biscans,
7
A. Bisht,
6,12
C. Biwer,
21
J. K. Blackburn,
1
C. D. Blair,
30
D. Blair,
30
R. M. Blair,
23
O. Bock,
6
T. P. Bodiya,
7
C. Bogan,
6
A. Bohe,
19
P. Bojtos,
31
C. Bond,
26
R. Bork,
1
S. Bose,
32,9
P. R. Brady,
11
V. B. Braginsky,
28
J. E. Brau,
33
M. Brinkmann,
6
P. Brockill,
11
A. F. Brooks,
1
D. A. Brown,
21
D. D. Brown,
26
N. M. Brown,
7
C. C. Buchanan,
2
A. Buikema,
7
A. Buonanno,
19,34
R. L. Byer,
25
L. Cadonati,
35
C. Cahillane,
1
J. Calderón Bustillo,
36,35
T. Callister,
1
J. B. Camp,
37
K. C. Cannon,
38
J. Cao,
39
C. D. Capano,
6
S. Caride,
40
S. Caudill,
11
M. Cavaglià,
14
C. Cepeda,
1
R. Chakraborty,
1
T. Chalermsongsak,
1
S. J. Chamberlin,
11
M. Chan,
22
S. Chao,
41
P. Charlton,
42
H. Y. Chen,
43
Y. Chen,
44
C. Cheng,
41
H. S. Cho,
45
M. Cho,
34
J. H. Chow,
13
N. Christensen,
46
Q. Chu,
30
S. Chung,
30
G. Ciani,
3
F. Clara,
23
J. A. Clark,
35
C. G. Collette,
47
L. Cominsky,
48
M. Constancio, Jr.,
8
D. Cook,
23
T. R. Corbitt,
2
N. Cornish,
20
A. Corsi,
49
C. A. Costa,
8
M. W. Coughlin,
46
S. B. Coughlin,
50
S. T. Countryman,
24
P. Couvares,
1
D. M. Coward,
30
M. J. Cowart,
4
D. C. Coyne,
1
R. Coyne,
49
K. Craig,
22
J. D. E. Creighton,
11
J. Cripe,
2
S. G. Crowder,
51
A. Cumming,
22
L. Cunningham,
22
T. Dal Canton,
6
S. L. Danilishin,
22
K. Danzmann,
12,6
N. S. Darman,
52
I. Dave,
27
H. P. Daveloza,
53
G. S. Davies,
22
E. J. Daw,
54
D. DeBra,
25
W. Del Pozzo,
26
T. Denker,
6,12
T. Dent,
6
V. Dergachev,
1
R. DeRosa,
4
R. DeSalvo,
5
S. Dhurandhar,
9
M. C. Díaz,
53
I. Di Palma,
19,6
G. Dojcinoski,
55
F. Donovan,
7
K. L. Dooley,
14
S. Doravari,
4
R. Douglas,
22
T. P. Downes,
11
M. Drago,
6
R. W. P. Drever,
1
J. C. Driggers,
23
Z. Du,
39
S. E. Dwyer,
23
T. B. Edo,
54
M. C. Edwards,
46
A. Effler,
4
H.-B. Eggenstein,
6
P. Ehrens,
1
J. Eichholz,
3
S. S. Eikenberry,
3
W. Engels,
44
R. C. Essick,
7
T. Etzel,
1
M. Evans,
7
T. M. Evans,
4
R. Everett,
56
M. Factourovich,
24
H. Fair,
21
S. Fairhurst,
50
X. Fan,
39
Q. Fang,
30
B. Farr,
43
W. M. Farr,
26
M. Favata,
55
M. Fays,
50
H. Fehrmann,
6
M. M. Fejer,
25
E. C. Ferreira,
8
R. P. Fisher,
21
M. Fletcher,
22
Z. Frei,
31
A. Freise,
26
R. Frey,
33
T. T. Fricke,
6
P. Fritschel,
7
V. V. Frolov,
4
P. Fulda,
3
M. Fyffe,
4
H. A. G. Gabbard,
14
J. R. Gair,
57
S. G. Gaonkar,
9
G. Gaur,
58,59
N. Gehrels,
37
J. George,
27
L. Gergely,
60
A. Ghosh,
10
J. A. Giaime,
2,4
K. D. Giardina,
4
K. Gill,
61
A. Glaefke,
22
E. Goetz,
40
R. Goetz,
3
L. Gondan,
31
G. González,
2
A. Gopakumar,
62
N. A. Gordon,
22
M. L. Gorodetsky,
28
S. E. Gossan,
1
C. Graef,
22
P. B. Graff,
37,34
A. Grant,
22
S. Gras,
7
C. Gray,
23
A. C. Green,
26
H. Grote,
6
S. Grunewald,
19
X. Guo,
39
A. Gupta,
9
M. K. Gupta,
59
K. E. Gushwa,
1
E. K. Gustafson,
1
R. Gustafson,
40
J. J. Hacker,
15
B. R. Hall,
32
E. D. Hall,
1
G. Hammond,
22
M. Haney,
62
M. M. Hanke,
6
J. Hanks,
23
C. Hanna,
56
M. D. Hannam,
50
J. Hanson,
4
T. Hardwick,
2
G. M. Harry,
63
I. W. Harry,
19
M. J. Hart,
22
M. T. Hartman,
3
C.-J. Haster,
26
K. Haughian,
22
M. C. Heintze,
3,4
M. Hendry,
22
I. S. Heng,
22
J. Hennig,
22
A. W. Heptonstall,
1
M. Heurs,
6,12
S. Hild,
22
D. Hoak,
64
K. A. Hodge,
1
S. E. Hollitt,
65
K. Holt,
4
D. E. Holz,
43
P. Hopkins,
50
D. J. Hosken,
65
J. Hough,
22
E. A. Houston,
22
E. J. Howell,
30
Y. M. Hu,
22
S. Huang,
41
E. A. Huerta,
66
B. Hughey,
61
S. Husa,
36
S. H. Huttner,
22
T. Huynh-Dinh,
4
A. Idrisy,
56
N. Indik,
6
D. R. Ingram,
23
R. Inta,
49
H. N. Isa,
22
M. Isi,
1
G. Islas,
15
T. Isogai,
7
B. R. Iyer,
10
K. Izumi,
23
H. Jang,
45
K. Jani,
35
S. Jawahar,
67
F. Jiménez-Forteza,
36
W. W. Johnson,
2
D. I. Jones,
17
R. Jones,
22
L. Ju,
30
Haris K.,
68
C. V. Kalaghatgi,
16
V. Kalogera,
69
S. Kandhasamy,
14
G. Kang,
45
J. B. Kanner,
1
S. Karki,
33
M. Kasprzack,
2
E. Katsavounidis,
7
W. Katzman,
4
S. Kaufer,
12
T. Kaur,
30
K. Kawabe,
23
F. Kawazoe,
6
M. S. Kehl,
38
D. Keitel,
6
D. B. Kelley,
21
W. Kells,
1
R. Kennedy,
54
J. S. Key,
53
A. Khalaidovski,
6
F. Y. Khalili,
28
S. Khan,
50
Z. Khan,
59
E. A. Khazanov,
70
N. Kijbunchoo,
23
C. Kim,
45
J. Kim,
71
K. Kim,
72
N. Kim,
45
N. Kim,
25
Y.-M. Kim,
71
E. J. King,
65
P. J. King,
23
D. L. Kinzel,
4
J. S. Kissel,
23
L. Kleybolte,
18
S. Klimenko,
3
S. M. Koehlenbeck,
6
K. Kokeyama,
2
V. Kondrashov,
1
A. Kontos,
7
M. Korobko,
18
W. Z. Korth,
1
D. B. Kozak,
1
V. Kringel,
6
C. Krueger,
12
G. Kuehn,
6
P. Kumar,
38
L. Kuo,
41
B. D. Lackey,
21
M. Landry,
23
J. Lange,
73
B. Lantz,
25
P. D. Lasky,
74
A. Lazzarini,
1
C. Lazzaro,
35
P. Leaci,
19
S. Leavey,
22
E. O. Lebigot,
39
C. H. Lee,
71
H. K. Lee,
72
H. M. Lee,
75
K. Lee,
22
A. Lenon,
21
J. R. Leong,
6
Y. Levin,
74
B. M. Levine,
23
T. G. F. Li,
1
A. Libson,
7
T. B. Littenberg,
76
N. A. Lockerbie,
67
J. Logue,
22
A. L. Lombardi,
64
J. E. Lord,
21
M. Lormand,
4
J. D. Lough,
6,12
H. Lück,
12,6
A. P. Lundgren,
6
J. Luo,
46
R. Lynch,
7
Y. Ma,
30
T. MacDonald,
25
B. Machenschalk,
6
M. MacInnis,
7
D. M. Macleod,
2
F. Magaña-Sandoval,
21
R. M. Magee,
32
M. Mageswaran,
1
I. Mandel,
26
V. Mandic,
51
V. Mangano,
22
G. L. Mansell,
13
M. Manske,
11
S. Márka,
24
Z. Márka,
24
A. S. Markosyan,
25
E. Maros,
1
I. W. Martin,
22
R. M. Martin,
3
D. V. Martynov,
1
J. N. Marx,
1
K. Mason,
7
T. J. Massinger,
21
M. Masso-Reid,
22
F. Matichard,
7
L. Matone,
24
N. Mavalvala,
7
N. Mazumder,
32
G. Mazzolo,
6
R. McCarthy,
23
D. E. McClelland,
13
S. McCormick,
4
S. C. McGuire,
77
G. McIntyre,
1
J. McIver,
64
D. J. McManus,
13
S. T. McWilliams,
66
G. D. Meadors,
19,6
A. Melatos,
52
G. Mendell,
23
D. Mendoza-Gandara,
6
R. A. Mercer,
11
E. Merilh,
23
S. Meshkov,
1
C. Messenger,
22
C. Messick,
56
P. M. Meyers,
51
H. Miao,
26
H. Middleton,
26
E. E. Mikhailov,
78
K. N. Mukund,
9
J. Miller,
7
PHYSICAL REVIEW D
95,
062003 (2017)
2470-0010
=
2017
=
95(6)
=
062003(16)
062003-1
© 2017 American Physical Society
M. Millhouse,
20
J. Ming,
19,6
S. Mirshekari,
79
C. Mishra,
10
S. Mitra,
9
V. P. Mitrofanov,
28
G. Mitselmakher,
3
R. Mittleman,
7
S. R. P. Mohapatra,
7
B. C. Moore,
55
C. J. Moore,
80
D. Moraru,
23
G. Moreno,
23
S. R. Morriss,
53
K. Mossavi,
6
C. M. Mow-Lowry,
26
C. L. Mueller,
3
G. Mueller,
3
A. W. Muir,
50
Arunava Mukherjee,
10
D. Mukherjee,
11
S. Mukherjee,
53
A. Mullavey,
4
J. Munch,
65
D. J. Murphy,
24
P. G. Murray,
22
A. Mytidis,
3
R. K. Nayak,
81
V. Necula,
3
K. Nedkova,
64
A. Neunzert,
40
G. Newton,
22
T. T. Nguyen,
13
A. B. Nielsen,
6
A. Nitz,
6
D. Nolting,
4
M. E. N. Normandin,
53
L. K. Nuttall,
21
J. Oberling,
23
E. Ochsner,
11
J. O
Dell,
82
E. Oelker,
7
G. H. Ogin,
83
J. J. Oh,
84
S. H. Oh,
84
F. Ohme,
50
M. Oliver,
36
P. Oppermann,
6
Richard J. Oram,
4
B. O
Reilly,
4
R. O
Shaughnessy,
73
C. D. Ott,
44
D. J. Ottaway,
65
R. S. Ottens,
3
H. Overmier,
4
B. J. Owen,
49
A. Pai,
68
S. A. Pai,
27
J. R. Palamos,
33
O. Palashov,
70
A. Pal-Singh,
18
H. Pan,
41
C. Pankow,
11,69
F. Pannarale,
50
B. C. Pant,
27
M. A. Papa,
19,11,6
H. R. Paris,
25
W. Parker,
4
D. Pascucci,
22
Z. Patrick,
25
B. L. Pearlstone,
22
M. Pedraza,
1
L. Pekowsky,
21
A. Pele,
4
S. Penn,
85
R. Pereira,
24
A. Perreca,
1
M. Phelps,
22
V. Pierro,
5
I. M. Pinto,
5
M. Pitkin,
22
A. Post,
6
J. Powell,
22
J. Prasad,
9
V. Predoi,
50
S. S. Premachandra,
74
T. Prestegard,
51
L. R. Price,
1
M. Principe,
5
S. Privitera,
19
L. Prokhorov,
28
O. Puncken,
6
M. Pürrer,
50
H. Qi,
11
J. Qin,
30
V. Quetschke,
53
E. A. Quintero,
1
R. Quitzow-James,
33
F. J. Raab,
23
D. S. Rabeling,
13
H. Radkins,
23
P. Raffai,
31
S. Raja,
27
M. Rakhmanov,
53
V. Raymond,
19
J. Read,
15
C. M. Reed,
23
S. Reid,
29
D. H. Reitze,
1,3
H. Rew,
78
K. Riles,
40
N. A. Robertson,
1,22
R. Robie,
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J. G. Rollins,
1
V. J. Roma,
33
G. Romanov,
78
J. H. Romie,
4
S. Rowan,
22
A. Rüdiger,
6
K. Ryan,
23
S. Sachdev,
1
T. Sadecki,
23
L. Sadeghian,
11
M. Saleem,
68
F. Salemi,
6
A. Samajdar,
81
L. Sammut,
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E. J. Sanchez,
1
V. Sandberg,
23
B. Sandeen,
69
J. R. Sanders,
40
B. S. Sathyaprakash,
50
P. R. Saulson,
21
O. Sauter,
40
R. L. Savage,
23
A. Sawadsky,
12
P. Schale,
33
R. Schilling,
6
,
J. Schmidt,
6
P. Schmidt,
1,44
R. Schnabel,
18
R. M. S. Schofield,
33
A. Schönbeck,
18
E. Schreiber,
6
D. Schuette,
6,12
B. F. Schutz,
50
J. Scott,
22
S. M. Scott,
13
D. Sellers,
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A. Sergeev,
70
G. Serna,
15
A. Sevigny,
23
D. A. Shaddock,
13
M. S. Shahriar,
69
M. Shaltev,
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Z. Shao,
1
B. Shapiro,
25
P. Shawhan,
34
A. Sheperd,
11
D. H. Shoemaker,
7
D. M. Shoemaker,
35
X. Siemens,
11
D. Sigg,
23
A. D. Silva,
8
D. Simakov,
6
A. Singer,
1
L. P. Singer,
37
A. Singh,
19,6
R. Singh,
2
A. M. Sintes,
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B. J. J. Slagmolen,
13
J. R. Smith,
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N. D. Smith,
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R. J. E. Smith,
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E. J. Son,
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B. Sorazu,
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T. Souradeep,
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A. K. Srivastava,
59
A. Staley,
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M. Steinke,
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J. Steinlechner,
22
S. Steinlechner,
22
D. Steinmeyer,
6,12
B. C. Stephens,
11
R. Stone,
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K. A. Strain,
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N. A. Strauss,
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S. Strigin,
28
R. Sturani,
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A. L. Stuver,
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T. Z. Summerscales,
86
L. Sun,
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P. J. Sutton,
50
M. J. Szczepa
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czyk,
61
D. Talukder,
33
D. B. Tanner,
3
M. Tápai,
60
S. P. Tarabrin,
6
A. Taracchini,
19
R. Taylor,
1
T. Theeg,
6
M. P. Thirugnanasambandam,
1
E. G. Thomas,
26
M. Thomas,
4
P. Thomas,
23
K. A. Thorne,
4
K. S. Thorne,
44
E. Thrane,
74
V. Tiwari,
50
K. V. Tokmakov,
67
C. Tomlinson,
54
C. V. Torres,
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,
C. I. Torrie,
1
D. Töyrä,
26
G. Traylor,
4
D. Trifirò,
14
M. Tse,
7
D. Tuyenbayev,
53
D. Ugolini,
87
C. S. Unnikrishnan,
62
A. L. Urban,
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S. A. Usman,
21
H. Vahlbruch,
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G. Vajente,
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G. Valdes,
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D. C. Vander-Hyde,
21,15
A. A. van Veggel,
22
S. Vass,
1
R. Vaulin,
7
A. Vecchio,
26
J. Veitch,
26
P. J. Veitch,
65
K. Venkateswara,
88
S. Vinciguerra,
26
D. J. Vine,
29
S. Vitale,
7
T. Vo,
21
C. Vorvick,
23
W. D. Vousden,
26
S. P. Vyatchanin,
28
A. R. Wade,
13
L. E. Wade,
89
M. Wade,
89
M. Walker,
2
L. Wallace,
1
S. Walsh,
11
H. Wang,
26
M. Wang,
26
X. Wang,
39
Y. Wang,
30
R. L. Ward,
13
J. Warner,
23
B. Weaver,
23
M. Weinert,
6
A. J. Weinstein,
1
R. Weiss,
7
T. Welborn,
4
L. Wen,
30
P. Weßels,
6
T. Westphal,
6
K. Wette,
6
J. T. Whelan,
73,6
D. J. White,
54
B. F. Whiting,
3
R. D. Williams,
1
A. R. Williamson,
50
J. L. Willis,
90
B. Willke,
12,6
M. H. Wimmer,
6,12
W. Winkler,
6
C. C. Wipf,
1
H. Wittel,
6,12
G. Woan,
22
J. Worden,
23
J. L. Wright,
22
G. Wu,
4
J. Yablon,
69
W. Yam,
7
H. Yamamoto,
1
C. C. Yancey,
34
M. J. Yap,
13
H. Yu,
7
M. Zanolin,
61
M. Zevin,
69
F. Zhang,
7
L. Zhang,
1
M. Zhang,
78
Y. Zhang,
73
C. Zhao,
30
M. Zhou,
69
Z. Zhou,
69
X. J. Zhu,
30
M. E. Zucker,
1,7
S. E. Zuraw,
64
and J. Zweizig
1
(LIGO Scientific Collaboration)
*
1
LIGO, California Institute of Technology, Pasadena, California 91125, USA
2
Louisiana State University, Baton Rouge, Louisiana 70803, USA
3
University of Florida, Gainesville, Florida 32611, USA
4
LIGO Livingston Observatory, Livingston, Louisiana 70754, USA
5
University of Sannio at Benevento, I-82100 Benevento, Italy
and INFN, Sezione di Napoli, I-80100 Napoli, Italy
6
Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany
7
LIGO, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
8
Instituto Nacional de Pesquisas Espaciais, 12227-010 São José dos Campos, SP, Brazil
9
Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India
10
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research,
Bangalore 560012, India
11
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA
12
Leibniz Universität Hannover, D-30167 Hannover, Germany
13
Australian National University, Canberra, Australian Capital Territory 0200, Australia
14
The University of Mississippi, University, Mississippi 38677, USA
15
California State University Fullerton, Fullerton, California 92831, USA
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
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062003 (2017)
062003-2
16
Chennai Mathematical Institute, Chennai, Tamil Nadu 603103, India
17
University of Southampton, Southampton SO17 1BJ, United Kingdom
18
Universität Hamburg, D-22761 Hamburg, Germany
19
Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-14476 Potsdam-Golm, Germany
20
Montana State University, Bozeman, Montana 59717, USA
21
Syracuse University, Syracuse, New York 13244, USA
22
SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom
23
LIGO Hanford Observatory, Richland, Washington 99352, USA
24
Columbia University, New York, New York 10027, USA
25
Stanford University, Stanford, California 94305, USA
26
University of Birmingham, Birmingham B15 2TT, United Kingdom
27
RRCAT, Indore, Madhya Pradesh 452013, India
28
Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia
29
SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom
30
University of Western Australia, Crawley, Western Australia 6009, Australia
31
MTA Eötvös University,
Lendulet
Astrophysics Research Group, Budapest 1117, Hungary
32
Washington State University, Pullman, Washington 99164, USA
33
University of Oregon, Eugene, Oregon 97403, USA
34
University of Maryland, College Park, Maryland 20742, USA
35
Center for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology,
Atlanta, Georgia 30332, USA
36
Universitat de les Illes Balears, IAC3
IEEC, E-07122 Palma de Mallorca, Spain
37
NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
38
Canadian Institute for Theoretical Astrophysics, University of Toronto,
Toronto, Ontario M5S 3H8, Canada
39
Tsinghua University, Beijing 100084, China
40
University of Michigan, Ann Arbor, Michigan 48109, USA
41
National Tsing Hua University, Hsinchu City, Taiwan 30013, Republic of China
42
Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia
43
University of Chicago, Chicago, Illinois 60637, USA
44
Caltech CaRT, Pasadena, California 91125, USA
45
Korea Institute of Science and Technology Information, Daejeon 305-806, Korea
46
Carleton College, Northfield, Minnesota 55057, USA
47
University of Brussels, Brussels 1050, Belgium
48
Sonoma State University, Rohnert Park, California 94928, USA
49
Texas Tech University, Lubbock, Texas 79409, USA
50
Cardiff University, Cardiff CF24 3AA, United Kingdom
51
University of Minnesota, Minneapolis, Minnesota 55455, USA
52
The University of Melbourne, Parkville, Victoria 3010, Australia
53
The University of Texas Rio Grande Valley, Brownsville, Texas 78520, USA
54
The University of Sheffield, Sheffield S10 2TN, United Kingdom
55
Montclair State University, Montclair, New Jersey 07043, USA
56
The Pennsylvania State University, University Park, Pennsylvania 16802, USA
57
School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
58
Indian Institute of Technology, Gandhinagar Ahmedabad Gujarat 382424, India
59
Institute for Plasma Research, Bhat, Gandhinagar 382428, India
60
University of Szeged, Dóm tér 9, Szeged 6720, Hungary
61
Embry-Riddle Aeronautical University, Prescott, Arizona 86301, USA
62
Tata Institute of Fundamental Research, Mumbai 400005, India
63
American University, Washington, D.C. 20016, USA
64
University of Massachusetts-Amherst, Amherst, Massachusetts 01003, USA
65
University of Adelaide, Adelaide, South Australia 5005, Australia
66
West Virginia University, Morgantown, West Virginia 26506, USA
67
SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom
68
IISER-TVM, CET Campus, Trivandrum, Kerala 695016, India
69
Northwestern University, Evanston, Illinois 60208, USA
70
Institute of Applied Physics, Nizhny Novgorod 603950, Russia
71
Pusan National University, Busan 609-735, Korea
72
Hanyang University, Seoul 133-791, Korea
73
Rochester Institute of Technology, Rochester, New York 14623, USA
CALIBRATION OF THE ADVANCED LIGO DETECTORS
...
PHYSICAL REVIEW D
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74
Monash University, Victoria 3800, Australia
75
Seoul National University, Seoul 151-742, Korea
76
University of Alabama in Huntsville, Huntsville, Alabama 35899, USA
77
Southern University and A&M College, Baton Rouge, Louisiana 70813, USA
78
College of William and Mary, Williamsburg, Virginia 23187, USA
79
Instituto de Física Teórica, University Estadual Paulista/ICTP South American Institute for
Fundamental Research, São Paulo, São Paulo 01140-070, Brazil
80
University of Cambridge, Cambridge CB2 1TN, United Kingdom
81
IISER-Kolkata, Mohanpur, West Bengal 741252, India
82
Rutherford Appleton Laboratory, HSIC, Chilton, Didcot, Oxon OX11 0QX, United Kingdom
83
Whitman College, 280 Boyer Ave, Walla Walla, Washington 9936, USA
84
National Institute for Mathematical Sciences, Daejeon 305-390, Korea
85
Hobart and William Smith Colleges, Geneva, New York 14456, USA
86
Andrews University, Berrien Springs, Michigan 49104, USA
87
Trinity University, San Antonio, Texas 78212, USA
88
University of Washington, Seattle, Washington 98195, USA
89
Kenyon College, Gambier, Ohio 43022, USA
90
Abilene Christian University, Abilene, Texas 79699, USA
(Received 18 February 2016; published 28 March 2017)
In Advanced LIGO, detection and astrophysical source parameter estimation of the binary black hole
merger GW150914 requires a calibrated estimate of the gravitational-wave strain sensed by the detectors.
Producing an estimate from each detector
s differential arm length control loop readout signals requires
applying time domain filters, which are designed from a frequency domain model of the detector
s
gravitational-wave response. The gravitational-wave response model is determined by the detector
s opto-
mechanical response and the properties of its feedback control system. The measurements used to validate
the model and characterize its uncertainty are derived primarily from a dedicated photon radiation pressure
actuator, with cross-checks provided by optical and radio frequency references. We describe how the
gravitational-wave readout signal is calibrated into equivalent gravitational-wave-induced strain and how
the statistical uncertainties and systematic errors are assessed. Detector data collected over 38 calendar
days, from September 12 to October 20, 2015, contain the event GW150914 and approximately 16 days of
coincident data used to estimate the event false alarm probability. The calibration uncertainty is less than
10% in magnitude and 10° in phase across the relevant frequency band, 20 Hz to 1 kHz.
DOI:
10.1103/PhysRevD.95.062003
I. INTRODUCTION
On September 14, 2015 09:50:45 UTC, the two
Advanced LIGO detectors observed a gravitational-wave
(GW) signal, GW150914, originating from the merging of
two stellar-mass black holes
[1]
. The event was observed in
coincident data from the two LIGO detectors between
September 12 to October 20, 2015. These detectors, H1
located on the Hanford Reservation in Richland,
Washington, and L1 located in Livingston Parish,
Louisiana, are laser interferometers
[2]
that use four mirrors
(referred to as test masses) suspended from multistage
pendulums to form two perpendicular optical cavities
(arms) in a modified Michelson configuration, as shown
in Fig.
1
. GW strain causes apparent differential variations
of the arm lengths which generate power fluctuations in the
interferometer
s GW readout port. These power fluctua-
tions, measured by photodiodes, serve as both the GW
readout signal and an error signal for controlling the
differential arm length
[3]
.
Feedback control of the differential arm length degree of
freedom (along with the interferometer
s other length and
angular degrees of freedom) is required for stable operation
of the instrument. This control is achieved by taking a
digitized version of the GW readout signal
d
err
ð
f
Þ
, apply-
ing a set of digital filters to produce a control signal
d
ctrl
ð
f
Þ
,
then sending the control signal to the test mass actuator
systems which displace the mirrors. Without this control
system, differential length variations arising from either
displacement noise or a passing GW would cause an
unsuppressed (free-running) change in differential length,
Δ
L
free
¼
L
x
L
y
¼
hL
, where
L
ð
L
x
þ
L
y
Þ
=
2
is the
average length of each detector
s arms, with lengths
L
x
and
L
y
, and
h
is the sensed strain,
h
Δ
L
free
=L
. In the
presence of feedback control, however, this free-running
displacement is suppressed to a smaller, residual length
change given by
Δ
L
res
¼
Δ
L
free
ð
f
Þ
=
½
1
þ
G
ð
f
Þ
, where
G
ð
f
Þ
is the open loop transfer function of the differential
arm length servo. Therefore, estimating the equivalent GW
*
Corresponding author.
lsc
spokesperson@ligo.org
Deceased.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
95,
062003 (2017)
062003-4
strain sensed by the interferometer requires detailed char-
acterization of, and correction for, the effect of this loop.
The effects of other feedback loops associated with other
degrees of freedom are negligible across the relevant
frequency band, from 20 Hz to 1 kHz.
The differential arm length feedback loop is character-
ized by a sensing function
C
ð
f
Þ
, a digital filter function
D
ð
f
Þ
, and an actuation function
A
ð
f
Þ
, which together give
the open loop transfer function
G
ð
f
Þ¼
A
ð
f
Þ
D
ð
f
Þ
C
ð
f
Þ
:
ð
1
Þ
The sensing function describes how residual arm length
displacements propagate to the digitized error signal,
d
err
ð
f
Þ
C
ð
f
Þ
Δ
L
res
ð
f
Þ
; the digital filter function describes
how the digital control signal is generated from the digital
error signal,
d
ctrl
ð
f
Þ
D
ð
f
Þ
d
err
ð
f
Þ
; and the actuation
function describes how the digital control signal produces
a differential displacement of the arm lengths,
Δ
L
ctrl
A
ð
f
Þ
d
ctrl
ð
f
Þ
. These relationships are shown schematically
in Fig.
2
.
Either the error signal, the control signal, or a combi-
nation of the two can be used estimate the strain sensed by
the detector
[4]
. For Advanced LIGO, a combination was
chosen that renders the estimate of the detector strain output
insensitive to changes in the digital filter function
D
, and
makes application of slow corrections to the sensing and
actuation functions convenient:
h
ð
t
Þ¼
1
L
½
C
1

d
err
ð
t
Þþ
A

d
ctrl
ð
t
Þ
;
ð
2
Þ
where
A
and
C
1
are time domain filters generated from
frequency domain models of
A
and
C
, and

denotes
convolution.
The accuracy and precision of this estimated strain rely
on characterizing the sensing and actuation functions of
each detector,
C
and
A
. Each function is represented by a
model, generated from measurements of control loop
parameters, each with associated statistical uncertainty
and systematic error. Uncertainty in the calibration model
parameters directly impacts the uncertainty in the recon-
structed detector strain signal. This uncertainty could limit
the signal-to-noise ratios of GW detection statistics, and
could dominate uncertainties in estimated astrophysical
parameters, e.g., luminosity distance, sky location, com-
ponent masses, and spin. Calibration uncertainty is thus
crucial for GW searches and parameter estimation.
This paper describes the accuracy and precision of the
model parameters and of the estimated detector strain
output over the course of the 38 calendar days of obser-
vation during which GW150914 was detected. Section
II
describes the actuation and sensing function models in
terms of their measured parameters. Section
III
defines the
treatment of uncertainty and error for each of these
parameters. In Sec.
IV
, a description of the radiation
pressure actuator is given. Sections
V
and
VI
discuss the
measurements used to determine the static statistical
uncertainties and systematic errors in the actuation and
sensing function models, respectively, and their results.
Section
VII
details the systematic errors in model param-
eters near the time of the GW150914 event resulting from
uncorrected, slow time variations. Section
VIII
discusses
each detector
s strain response function that is used to
estimate the overall amplitude and phase uncertainties
and systematic errors in the calibrated data stream
h
ð
t
Þ
.
FIG. 1. Simplified diagram of an Advanced LIGO interferom-
eter. Four highly reflective test masses form two Fabry
Pérot arm
cavities. At lower left, a power recycling mirror placed between
the laser and the beamsplitter increases the power stored in the
arms to 100 kW. A signal recycling mirror, placed between
the beamsplitter and the GW readout photodetector, alters the
frequency response of the interferometer to differential arm
length fluctuations. For clarity, only the lowest suspension stage
is shown for the optics. Inset: one of the dual-chain, quadruple
pendulum suspension systems is shown.
FIG. 2. Block diagram of the differential arm length feedback
control servo. The sensing function, digital filter function, and
actuation function combine to form the open loop transfer
function
G
ð
f
Þ¼
A
ð
f
Þ
D
ð
f
Þ
C
ð
f
Þ
. The signal
x
ð
PC
Þ
T
is the modu-
lated displacement of the test masses from the radiation pressure
actuator described in Sec.
IV
.
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...
PHYSICAL REVIEW D
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Section
IX
discusses the intersite uncertainty in the relative
timing of each detector
s data stream. In Sec.
X
the
implications of these uncertainties on the detection and
astrophysical parameter estimation of GW150914 are
summarized. Finally, in Sec.
XI
we give an outlook on
future calibration and its role in GW detection and
astrophysical parameter estimation.
II. MODEL DESCRIPTION
We divide the differential arm length feedback loop into
two main functions, sensing and actuation. In this section,
these functions are described in detail. The interferometer
response function is also introduced; it is composed of
these functions and the digital control filter function (which
is precisely known and carries no uncertainty), and is useful
for estimating the overall uncertainty in the estimated
strain.
A. Sensing function
The sensing function
C
converts residual test mass
differential displacement
Δ
L
res
to a digitized signal repre-
senting the laser power fluctuation at the GW readout
port,
d
err
, sampled at a rate of 16 384 Hz. It includes the
interferometric response converting displacement to laser
power fluctuation at the GW readout port, the response of
the photodiodes and their analog readout electronics, and
effects from the digitization process.
The complete interferometric response is determined by
the arm cavity mirror (test mass) reflectivities, the reflec-
tivity of the signal recycling mirror (see Fig.
1
), the length
of the arm cavities and the length of the signal recycling
cavity
[5,6]
. The response is approximated by a single-pole
low-pass filter with a gain and an additional time delay.
The sensing function is thus given by
C
ð
model
Þ
ð
f
Þ¼
K
C
1
þ
i
f=f
C
C
R
ð
f
Þ
exp
ð
2
π
i
f
τ
C
Þ
;
ð
3
Þ
where
K
C
is combined gain of the interferometric response
and analog-to-digital converter (see Fig.
3
). It describes, at
a reference time, how many digital counts are produced in
d
err
in response to differential arm length displacement. The
pole frequency,
f
C
, is the characteristic frequency that
describes the attenuation of the interferometer response to
high-frequency length perturbations
[5,7]
. Though each
interferometer is designed to have the same pole frequency,
the exact value differs as result of discrepant losses in their
optical cavities: 341 Hz and 388 Hz for H1 and L1,
respectively. The time delay
τ
C
includes the light travel
time
L=c
along the length of the arms (
L
¼
3994
.
5
m),
computational delay in the digital acquisition system, and
the delay introduced to approximate the complete inter-
ferometric response as a single pole. Finally, the dimen-
sionless quantity
C
R
ð
f
Þ
accounts for additional frequency
dependence of the sensing function above 1 kHz, arising
from the properties of the photodiode electronics, as well
as analog and digital signal processing filters.
B. Actuation function
The interferometer differential arm length can be
controlled by actuating on the quadruple suspension system
for any of the four arm cavity test masses. Each of these
systems consists of four stages, suspended as cascading
pendulums
[8,9]
, which isolate the test mass from residual
motion of the supporting active isolation system
[10]
. Each
suspension system also includes an adjacent, nearly-
identical, cascaded reaction mass pendulum chain which
can be used to independently generate reaction forces on
each mass of the test mass pendulum chain. A diagram of
one of these suspension systems is shown in Fig.
1
.
For each of the three lowest stages of the suspension
system
the upper intermediate mass (
U
), the penultimate
mass (
P
), and thetest mass (
T
)
digital-to-analog converters
and associated electronics drive a set of four actuators that
work in concert to displace each stage, and consequently the
test mass suspended at the bottom. The digital control signal
d
ctrl
is distributed to each stage and multiplied by a set of
dimensionless digital filters
F
i
ð
f
Þ
, where
i
¼
U
,
P
,or
T
,so
that the lower stages are used for the highest frequency signal
content and the upper stages are used for high-range, low-
frequency signal content.
While the differential arm length can be controlled using
any combination of the four test mass suspension systems,
FIG. 3. The magnitude and phase of the sensing function model
C
ð
f
Þ
for the L1 detector. Below 1 kHz the frequency dependence
is determined by
f
C
, while above 1 kHz it is determined by the
analog-to-digital conversion process.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
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062003 (2017)
062003-6
only one, the
Y
-arm end test mass, is used to create
Δ
L
ctrl
.
Actuating a single test mass affects both the common and
the differential arm lengths. The common arm length
change is compensated, however, by high-bandwidth
(
14
kHz) feedback to the laser frequency.
The model of the actuation function
A
of the suspension
system comprises the mechanical dynamics, electronics,
and digital filtering, and is written as
A
ð
model
Þ
ð
f
Þ¼½
F
T
ð
f
Þ
K
T
A
T
ð
f
Þþ
F
P
ð
f
Þ
K
P
A
P
ð
f
Þ
þ
F
U
ð
f
Þ
K
U
A
U
ð
f
Þ
exp
ð
2
π
i
f
τ
A
Þ
:
ð
4
Þ
Here
K
i
and
A
i
ð
f
Þ
are the gain and the normalized
frequency dependence of the
i
th suspension stage actuator,
measured at a reference time, that define the actuation
transfer function for each suspension stage;
τ
A
is the
computational delay in the digital-to-analog conversion.
The overall and individual stage actuation functions are
plotted as a function of frequency in Fig.
4
. The gain
converts voltage applied at suspension stage
i
to test mass
displacement. The frequency response is primarily deter-
mined by the mechanical dynamics of the suspension, but
also includes minor frequency dependent terms from
digital-to-analog signal processing, analog electronics,
and mechanical interaction with the locally-controlled
suspension stage for the top mass (see Fig.
1
). While
opto-mechanical interaction from radiation pressure can
affect the actuation function
[11]
, the laser power resonat-
ing in the arm cavities during the observation period was
low enough that radiation pressure effects can be ignored.
The H1 and L1 suspensions and electronics are identical by
design, but there are slight differences, mostly due to the
digital filtering for each stage
F
i
, which are precisely
known and carry no uncertainty.
C. Response function
For uncertainty estimation, it is convenient to introduce
the response function
R
ð
f
Þ
that relates the differential arm
length servo error signal to strain sensed by the interfer-
ometer:
h
ð
f
Þ¼ð
1
=L
Þ
R
ð
f
Þ
d
err
ð
f
Þ
. As shown schemati-
cally in Fig.
2
, the response function is given by
R
ð
f
Þ¼
1
þ
A
ð
f
Þ
D
ð
f
Þ
C
ð
f
Þ
C
ð
f
Þ
¼
1
þ
G
ð
f
Þ
C
ð
f
Þ
:
ð
5
Þ
We will use this response function to evaluate the overall
accuracy and precision of the calibrated detector strain
output. The actuation function dominates at frequencies
below the differential arm length servo unity gain fre-
quency, 40 Hz and 56 Hz for H1 and L1, respectively.
Above the unity gain frequency, the sensing function
dominates (see Figs.
3
and
4
).
III. DEFINITIONS OF PARAMETER
UNCERTAINTY
From Eqs.
(3)
and
(4)
, we identify the set
Q
ð
model
Þ
of
parameters shown in Table
I
that define the model for each
detector
s sensing and actuation functions. These model
parameters have both statistical uncertainty and systematic
error. In this section, we outline how the uncertainty and
error for each parameter are treated. Discussion of how
these are propagated to inform the total uncertainty and
error in final estimated strain
h
ð
t
Þ
is left to Sec.
VIII
.
Combinations of the model
s scalar parameters (
K
C
,
K
T
,
K
P
,
K
U
,
f
C
, and
τ
C
) and frequency-dependent functions
(
A
T
ð
f
Þ
,
A
P
ð
f
Þ
,
A
U
ð
f
Þ
, and
C
R
ð
f
Þ
) are constrained by a set
of directly measurable properties of the detector
Q
ð
meas
Þ
:
Q
ð
meas
Þ
ð
f
Þ¼f
K
T
A
T
ð
f
Þ
;
K
P
A
P
ð
f
Þ
;
K
U
A
U
ð
f
Þ
;
K
C
C
R
ð
f
Þ
=
ð
1
þ
i
f=f
C
Þ
exp
ð
2
π
if
τ
C
Þg
:
ð
6
Þ
The parameters in
Q
ð
model
Þ
not included in Table
I
,
F
i
ð
f
Þ
and
τ
A
, are part of the digital control system, known with
FIG. 4. Overall actuation transfer function
A
ð
f
Þ
and actuation
functions for each suspension stage
F
i
ð
f
Þ
K
i
A
i
ð
f
Þ
for the L1
detector. The mechanical response of the pendulums and
F
i
dictate the characteristics of each stage. The strongest actuator,
that for the upper intermediate mass, is used below a few Hz.
Above
30
Hz, only the test mass actuator is used. At certain
frequencies (e.g., 10, 14, and 500 Hz), digital notch filters are
implemented for high quality factor features of the pendulum
responses in order to avoid mechanical instabilities. The H1
actuation function differs slightly in scale, frequency dependence,
and digital filter choice.
CALIBRATION OF THE ADVANCED LIGO DETECTORS
...
PHYSICAL REVIEW D
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negligible uncertainty, and are thus removed from the
measured quantities without consequence. Each quantity
q
ð
meas
Þ
i
Q
ð
meas
Þ
is measured using sinusoidal excitations
injected at various points in the control loop while the
detector is in its lowest noise state. The measurements
consist of excitations that are injected consecutively at
discrete frequencies,
f
k
. Only measurements made at a
reference time
t
0
are used to determine the corresponding
model parameters
q
ð
model
Þ
i
, however the measurements are
repeated periodically to inform and reduce uncertainty.
The frequency-dependent model parameters
Q
ð
model
Þ
described in Table
I
do not completely describe the
frequency-dependent quantities in
Q
ð
meas
Þ
at the reference
time. In addition, the scalar quantities in
Q
ð
meas
Þ
vary with
time after the reference measurement. Both discrepancies
are systematic errors,
δ
q
i
. Albeit small, they are carried
with each parameter
Q
ð
model
Þ
through to inform the known
systematic error in the response function, and quantified in
the following fashion.
Any discrepancy between
A
i
ð
f
Þ
and
C
R
ð
f
Þ
and the
measurements exposes poorly modeled properties of the
detector, and thus are systematic errors in Eqs.
(3)
and
(4)
;
δ
q
i
¼
q
ð
meas
Þ
i
q
ð
model
Þ
i
. We find it convenient to quantify
this systematic error in terms of a multiplicative correction
factor to Eqs.
(3)
and
(4)
,
ζ
ð
fd
Þ
i
q
ð
meas
Þ
i
=q
ð
model
Þ
i
1
þð
δ
q
i
=q
ð
model
Þ
i
Þ
, instead of dealing directly with the
systematic error
δ
q
i
. These frequency-dependent discrep-
ancies are confirmed with repeated measurements beyond
the reference time.
The scalar parameters,
K
i
and
f
C
, are monitored con-
tinuously during data taking to track small, slow temporal
variations beyond the reference measurement time
t
0
.
Tracking is achieved using a set of sinusoidal excitations
at select frequencies, typically referred to as
calibration
lines
. The observed time dependence is treated as an
additional systematic error,
δ
q
i
ð
t
Þ
, also implemented as a
correction factor,
ζ
ð
td
Þ
i
δ
q
i
ð
t
Þ
=q
ð
model
Þ
i
.
In order to quantify the statistical uncertainties in the
frequency-dependent parameters in
Q
ð
model
Þ
, we divide the
measurements
Q
ð
meas
Þ
by the appropriate combination of
reference model parameters
q
ð
model
Þ
i
, time-dependent scalar
correction factors,
ζ
ð
td
Þ
i
, and a fit to any frequency-dependent
correction factors,
ζ
ð
fd
;
fit
Þ
i
to form a statistical residual,
ξ
ð
stat
Þ
i
¼
q
ð
meas
Þ
i
=
ð
q
ð
model
Þ
i
ζ
ð
td
Þ
i
ζ
ð
fd
;
fit
Þ
i
Þ
1
:
ð
7
Þ
We assume this remaining residual reflects an estimate of
the complex, scalar (i.e. frequency
independent
), statistical
uncertainty,
σ
q
i
q
j
, randomly sampled over the measurement
frequency vector
f
k
, and may be covariant between
parameter
q
ð
meas
Þ
i
and
q
ð
meas
Þ
j
. Thus, we estimate
σ
q
i
q
j
by
computing the standard deviation of the statistical residual,
ξ
ð
stat
Þ
i
, across the frequency band,
σ
q
i
q
j
¼
X
N
k
¼
1
ð
ξ
ð
stat
Þ
i
ð
f
k
Þ
ξ
ð
stat
Þ
i
Þð
ξ
ð
stat
Þ
j
ð
f
k
Þ
ξ
ð
stat
Þ
j
Þ
ð
N
1
Þ
ð
8
Þ
where
ξ
ð
stat
Þ
i
¼
P
k
ξ
ð
stat
Þ
i
ð
f
k
Þ
=N
is the mean across the
N
points in the frequency vector
f
k
.
The time-dependent correction factor,
ζ
ð
td
Þ
i
, has associ-
ated statistical uncertainty
σ
ζ
ð
td
Þ
i
that is governed by the
signal-to-noise ratio of the continuous excitation. Only a
limited set of lines were used to determine these time-
dependent systematic errors, so their estimated statistical
uncertainty is also, in general covariant.
In Secs.
V
,
VI
, and
VII
, we describe the techniques for
measuring
Q
ð
meas
Þ
at the reference time
t
0
, and discuss
resulting estimates of statistical uncertainty
σ
q
i
q
j
and
systematic error
δ
q
i
, via correction factors
ζ
i
, for each
detector. In Sec.
VIII
, we describe how the uncertainty and
error estimates for these parameters are combined to
estimate the overall accuracy and precision of the calibrated
detector strain output
h
ð
t
Þ
.
IV. RADIATION PRESSURE ACTUATOR
The primary method for calibrating the actuation func-
tion
A
and sensing function
C
is an independent radiation
pressure actuator called the
photon calibrator
(PC)
[12]
.
A similar system was also used for calibration of the initial
LIGO detectors
[13]
.
Each detector is equipped with two photon calibrator
systems, one for each end test mass, positioned outside the
vacuum enclosure at the ends of the interferometer arms.
For each system, 1047 nm light from an auxiliary, power-
modulated, Nd
3
þ
:YLF laser is directed into the vacuum
envelope and reflects from the front surface of the mirror
(test mass). The reflected light is directed to a power sensor
located outside the vacuum enclosure. This sensor is an
TABLE I. The set of differential arm length control loop
parameters,
Q
ð
model
Þ
that must be characterized to define the
sensing and actuation functions.
Parameter
Description
A
T
ð
f
Þ
Normalized test mass actuation function
A
P
ð
f
Þ
Normalized penultimate mass actuation function
A
U
ð
f
Þ
Normalized upper intermediate mass actuation function
C
R
ð
f
Þ
Residual sensing function frequency dependence
K
C
Sensing function gain
K
T
Test mass actuation function gain
K
P
Penultimate mass actuation function gain
K
U
Upper intermediate mass actuation function gain
f
C
Cavity pole frequency
τ
C
Sensing function time delay
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
95,
062003 (2017)
062003-8
InGaAs photodetector mounted on an integrating sphere
and is calibrated using a standard that is traceable to the
National Institute of Standards and Technology (NIST).
Power modulation is accomplished via an acousto-optic
modulator that is part of an optical follower servo that
ensures that the power modulation follows the requested
waveform. After modulation, the laser beam is divided
optically and projected onto the mirror in two diametrically
opposed positions. The spots are separated vertically,

11
.
6
cm from the center of the optical surface, on the
nodal ring of the drumhead elastic body mode, to minimize
errors at high-frequency caused by bulk deformation
[13
16]
.
The laser power modulation induces a modulated dis-
placement of the test mass that is given by
[13]
x
ð
PC
Þ
T
ð
f
Þ¼
2
P
ð
f
Þ
c
s
ð
f
Þ
cos
θ

1
þ
M
T
I
T
~
a
·
~
b

:
ð
9
Þ
This modulated displacement is shown schematically on
the left of Fig.
2
. The terms entering this formula are as
follows:
f
is the frequency of the power modulation,
P
ð
f
Þ
is the power modulation amplitude,
c
is the speed of light,
s
ð
f
Þ
is the mechanical compliance of the suspended mirror,
θ
8
.
8
° is the angle of incidence on the mirror,
M
T
¼
39
.
6
kg and
I
T
¼
0
.
415
kg m
2
are the mass and rotational
moment of inertia of the mirror, and
~
a
and
~
b
are displace-
ment vectors from the center of the optical surface to the
photon calibrator center of force and the main interferom-
eter beam, respectively. These displacements determine the
amount of unwanted induced rotation of the mirror.
The compliance
s
ð
f
Þ
of the suspended mirror can be
approximated by treating the mirror as rigid body that is
free to move along the optical axis of the arm cavity:
s
ð
f
Þ
1
=
½
M
T
ð
2
π
f
Þ
2

. Cross-couplings between other
degrees of freedom of the multistage suspension system,
however, require that
s
ð
f
Þ
be computed with a full, rigid-
body model of the quadruple suspension. This model has
been validated by previous measurements
[9,17]
and is
assumed to have negligible uncertainty.
Significant sources of photon calibrator uncertainty
include the NIST calibration of the reference standard
(0.5%), self-induced test mass rotation uncertainty (0.4%),
and uncertainty of the optical losses along the projection
and reflection paths (0.4%). The overall
1
σ
uncertainty
in the displacement induced by the photon calibrator,
x
ð
PC
Þ
T
ð
f
Þ
,is
0
.
8%
.
V. ACTUATION FUNCTION CALIBRATION
The actuation strength for the
i
th suspension stage,
½
K
i
A
i
ð
f
Þ
ð
meas
Þ
, can be determined by comparing the
interferometer
s response,
d
err
ð
f
Þ
, to an excitation from
that suspension stage
s actuator, exc
i
ð
f
Þ
, with one from the
photon calibrator,
x
ð
PC
Þ
T
ð
f
Þ
,
½
K
i
A
i
ð
f
Þ
ð
meas
Þ
¼
x
ð
PC
Þ
T
ð
f
Þ
d
err
ð
f
Þ
×
d
err
ð
f
Þ
exc
i
ð
f
Þ
:
ð
10
Þ
Figures
5
and
6
show the collection of these measurements
for the H1 and L1 interferometers in the form of correction
factors,
ζ
ð
fd
Þ
i
¼½
K
i
A
i
ð
f
Þ
ð
meas
Þ
=
½
K
i
A
i
ð
f
Þ
ð
model
Þ
. The col-
lection includes the reference measurement and subsequent
measurements normalized by any scalar, time-dependent
correction factors,
ζ
ð
td
Þ
i
. These data are used to create the
fit,
ζ
ð
fd
;
fit
Þ
i
, and estimate the actuation components of the
statistical uncertainty
σ
q
i
q
j
.
As described in Sec.
II
, the actuation function, and
therefore its uncertainty and error, only contribute signifi-
cantly to the uncertainty estimate for
h
below
45
Hz,
which is the unity gain frequency for the differential arm
length servo. While there are no data at frequencies above
100 Hz for H1, the L1 high-frequency data confirm that
above 100 Hz, frequency-dependent deviations from the
model are small.
There are larger frequency-dependent errors in the
models for the upper intermediate stages
K
U
A
U
for both
detectors. Additional measurements, not explicitly included
in this paper, have shown that these result from unmodeled
mechanical resonances as well as the non-negligible
inductance of the electromagnetic coil actuators. As shown
FIG. 5. Measured frequency-dependent correction factors,
ζ
ð
fd
Þ
i
,
for the actuators of the lower three stages of the H1 suspension
(symbols) and corresponding fits,
ζ
ð
fd
;
fit
Þ
i
(solid lines). Only data
up to 100 Hz for the bottom two stages were collected because the
sensing function dominates the actuation function above
45
Hz.
Data for the upper intermediate mass is presented only up to
30 Hz because the actuation function for this stage is attenuated
sharply above
5
Hz.
CALIBRATION OF THE ADVANCED LIGO DETECTORS
...
PHYSICAL REVIEW D
95,
062003 (2017)
062003-9