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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. A
ff
eldt,
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. Calder
́
on 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. Cavagli
`
a,
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. D
́
ıaz,
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. E
ffl
er,
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. Fy
ff
e,
4
H. A. G. Gabbard,
14
J. R. Gair,
57
S. G. Gaonkar,
9
G. Gaur,
58
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59
N. Gehrels,
37
J. George,
27
L. Gergely,
60
A. Ghosh,
10
J. A. Giaime,
2
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4
K. D. Giardina,
4
K. Gill,
61
A. Glaefke,
22
E. Goetz,
40
R. Goetz,
3
L. Gondan,
31
G. Gonz
́
alez,
2
A. Gopakumar,
62
N. A. Gordon,
22
M. L. Gorodetsky,
28
S. E. Gossan,
1
C. Graef,
22
P. B. Gra
ff
,
37
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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
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4
M. Hendry,
22
I. S. Heng,
22
J. Hennig,
22
A. W. Heptonstall,
1
M. Heurs,
6
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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. Jim
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enez-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
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12
H. L
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12
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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. Maga
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R. M. Magee,
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M. Mageswaran,
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I. Mandel,
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V. Mandic,
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V. Mangano,
22
G. L. Mansell,
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S. M
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24
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R. M. Martin,
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J. N. Marx,
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S. C. McGuire,
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G. McIntyre,
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J. McIver,
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D. J. McManus,
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A. Melatos,
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G. Mendell,
23
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E. Merilh,
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S. Meshkov,
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C. Messenger,
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C. Messick,
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E. E. Mikhailov,
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K. N. Mukund,
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J. Ming,
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S. Mirshekari,
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C. Mishra,
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arXiv:1602.03845v1 [gr-qc] 11 Feb 2016
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G. Mitselmakher,
3
R. Mittleman,
7
S. R. P. Mohapatra,
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B. C. Moore,
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C. J. Moore,
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D. Moraru,
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K. Mossavi,
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C. M. Mow-Lowry,
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C. L. Mueller,
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A. W. Muir,
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Arunava Mukherjee,
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D. Mukherjee,
11
S. Mukherjee,
53
A. Mullavey,
4
J. Munch,
65
D. J. Murphy,
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P. G. Murray,
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A. Mytidis,
3
R. K. Nayak,
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V. Necula,
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K. Nedkova,
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A. Neunzert,
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G. Newton,
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T. T. Nguyen,
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A. B. Nielsen,
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A. Nitz,
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M. E. N. Normandin,
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L. K. Nuttall,
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E. Oelker,
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J. J. Oh,
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S. H. Oh,
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P. Oppermann,
6
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O. Palashov,
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6
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A. Pele,
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1
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5
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6
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1
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5
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19
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87
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12
G. Vajente,
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1
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A. Vecchio,
26
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6
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3
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90
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12
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6
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6
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12
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6
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1
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6
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12
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22
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23
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4
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1
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34
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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
Deceased, May 2015.
Deceased, March 2015.
(LIGO Scientific Collaboration)
1
LIGO, California Institute of Technology, Pasadena, CA 91125, USA
2
Louisiana State University, Baton Rouge, LA 70803, USA
3
University of Florida, Gainesville, FL 32611, USA
4
LIGO Livingston Observatory, Livingston, LA 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 f ̈ur Gravitationsphysik, D-30167 Hannover, Germany
7
LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
8
Instituto Nacional de Pesquisas Espaciais, 12227-010 S ̃ao Jos ́e 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, WI 53201, USA
12
Leibniz Universit ̈at Hannover, D-30167 Hannover, Germany
13
Australian National University, Canberra, Australian Capital Territory 0200, Australia
14
The University of Mississippi, University, MS 38677, USA
15
California State University Fullerton, Fullerton, CA 92831, USA
16
Chennai Mathematical Institute, Chennai, India
17
University of Southampton, Southampton SO17 1BJ, United Kingdom
18
Universit ̈at Hamburg, D-22761 Hamburg, Germany
19
Albert-Einstein-Institut, Max-Planck-Institut f ̈ur Gravitationsphysik, D-14476 Potsdam-Golm, Germany
3
20
Montana State University, Bozeman, MT 59717, USA
21
Syracuse University, Syracuse, NY 13244, USA
22
SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom
23
LIGO Hanford Observatory, Richland, WA 99352, USA
24
Columbia University, New York, NY 10027, USA
25
Stanford University, Stanford, CA 94305, USA
26
University of Birmingham, Birmingham B15 2TT, United Kingdom
27
RRCAT, Indore MP 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 E ̈otv ̈os University, “Lendulet” Astrophysics Research Group, Budapest 1117, Hungary
32
Washington State University, Pullman, WA 99164, USA
33
University of Oregon, Eugene, OR 97403, USA
34
University of Maryland, College Park, MD 20742, USA
35
Center for Relativistic Astrophysics and School of Physics,
Georgia Institute of Technology, Atlanta, GA 30332, USA
36
Universitat de les Illes Balears, IAC3—IEEC, E-07122 Palma de Mallorca, Spain
37
NASA
/
Goddard Space Flight Center, Greenbelt, MD 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, MI 48109, USA
41
National Tsing Hua University, Hsinchu City, Taiwan 30013, R.O.C.
42
Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia
43
University of Chicago, Chicago, IL 60637, USA
44
Caltech CaRT, Pasadena, CA 91125, USA
45
Korea Institute of Science and Technology Information, Daejeon 305-806, Korea
46
Carleton College, Northfield, MN 55057, USA
47
University of Brussels, Brussels 1050, Belgium
48
Sonoma State University, Rohnert Park, CA 94928, USA
49
Texas Tech University, Lubbock, TX 79409, USA
50
Cardi
ff
University, Cardi
ff
CF24 3AA, United Kingdom
51
University of Minnesota, Minneapolis, MN 55455, USA
52
The University of Melbourne, Parkville, Victoria 3010, Australia
53
The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA
54
The University of She
ffi
eld, She
ffi
eld S10 2TN, United Kingdom
55
Montclair State University, Montclair, NJ 07043, USA
56
The Pennsylvania State University, University Park, PA 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, D ́om t ́er 9, Szeged 6720, Hungary
61
Embry-Riddle Aeronautical University, Prescott, AZ 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, MA 01003, USA
65
University of Adelaide, Adelaide, South Australia 5005, Australia
66
West Virginia University, Morgantown, WV 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, IL 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, NY 14623, USA
74
Monash University, Victoria 3800, Australia
75
Seoul National University, Seoul 151-742, Korea
76
University of Alabama in Huntsville, Huntsville, AL 35899, USA
77
Southern University and A
&
M College, Baton Rouge, LA 70813, USA
78
College of William and Mary, Williamsburg, VA 23187, USA
79
Instituto de F ́ısica Te ́orica, University Estadual Paulista
/
ICTP South
American Institute for Fundamental Research, S ̃ao Paulo SP 01140-070, Brazil
80
University of Cambridge, Cambridge CB2 1TN, United Kingdom
81
IISER-Kolkata, Mohanpur, West Bengal 741252, India
4
82
Rutherford Appleton Laboratory, HSIC, Chilton, Didcot, Oxon OX11 0QX, United Kingdom
83
Whitman College, 280 Boyer Ave, Walla Walla, WA 9936, USA
84
National Institute for Mathematical Sciences, Daejeon 305-390, Korea
85
Hobart and William Smith Colleges, Geneva, NY 14456, USA
86
Andrews University, Berrien Springs, MI 49104, USA
87
Trinity University, San Antonio, TX 78212, USA
88
University of Washington, Seattle, WA 98195, USA
89
Kenyon College, Gambier, OH 43022, USA
90
Abilene Christian University, Abilene, TX 79699, USA
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 di
ff
erential 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.
PACS numbers: 04.30.-w, 04.80.Nn, 95.55.Ym
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 Reser-
vation 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 multi-stage
pendulums to form two perpendicular optical cavities (arms) in
a modified Michelson configuration, as shown in Fig. 1. GW
strain causes apparent di
ff
erential variations of the arm lengths
which generate power fluctuations in the interferometer’s GW
readout port. These power fluctuations, measured by photodi-
odes, serve as both the GW readout signal and an error signal
for controlling the di
ff
erential arm length [3].’
Feedback control of the di
ff
erential arm length degree of
freedom (along with the interferometer’s other length and an-
gular 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
(
f
), applying a set of
digital filters to produce a control signal
d
ctrl
(
f
), then send-
ing the control signal to the test mass actuator systems which
displace the mirrors. Without this control system, di
ff
eren-
tial length variations arising from either displacement noise
or a passing GW would cause an unsuppressed (free-running)
change in di
ff
erential length,
L
free
=
L
x
L
y
=
hL
, where
L
(
L
x
+
L
y
)
/
2 is the average length of each detector’s arms,
with lengths
L
x
and
L
y
, and
h
is the sensed strain,
h
L
free
/
L
.
Corresponding Author: spokesperson@ligo.org
In the presence of feedback control, however, this free-running
displacement is suppressed to a smaller, residual length change
given by
L
res
= ∆
L
free
(
f
)
/
[1
+
G
(
f
)], where
G
(
f
) is the
open loop transfer function of the di
ff
erential arm length servo.
Therefore, estimating the equivalent GW strain sensed by the
interferometer requires detailed characterization of, and correc-
tion for, the e
ff
ect of this loop. The e
ff
ects of other feedback
loops associated with other degrees of freedom are negligible
across the relevant frequency band, from 20 Hz to 1 kHz.
The di
ff
erential arm length feedback loop is characterized
by a sensing function
C
(
f
), a digital filter function
D
(
f
), and
an actuation function
A
(
f
), which together give the open loop
transfer function
G
(
f
)
=
A
(
f
)
D
(
f
)
C
(
f
)
.
(1)
The sensing function describes how residual arm length dis-
placements propagate to the digitized error signal,
d
err
(
f
)
C
(
f
)
L
res
(
f
); the digital filter function describes how the dig-
ital control signal is generated from the digital error signal,
d
ctrl
(
f
)
D
(
f
)
d
err
(
f
); and the actuation function describes
how the digital control signal produces a di
ff
erential displace-
ment of the arm lengths,
L
ctrl
A
(
f
)
d
ctrl
(
f
). These relation-
ships are shown schematically in Fig. 2.
Either the error signal, the control signal, or a combination
of the two can be used estimate the strain sensed by the detec-
tor [
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 applica-
tion of slow corrections to the sensing and actuation functions
convenient:
h
(
t
)
=
1
L
[
C
1
d
err
(
t
)
+
A∗
d
ctrl
(
t
)
]
,
(2)
where
A
and
C
1
are time domain filters generated from fre-
quency domain models of
A
and
C
, and
denotes convolution.
5
Input Test
Mass X
End Test
Mass X
4 km
High Range
Electrostatic
Actuator
From
Laser
To
GW Readout Port
Low Noise
Electrostatic
Actuator
Y Arm
X Arm
End Test
Mass Y
Input Test
Mass Y
4 km
Top Mass
Upper
Intermediate
Mass (U)
Test Mass (T)
Penultimate
Mass (P)
Electromagnetic
Actuators
FIG. 1. Simplified diagram of an Advanced LIGO interferometer.
Four highly reflective test masses form two Fabry–P
́
erot 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 interferom-
eter to di
ff
erential 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.
d
err
Δ
L
ctrl
Δ
L
res
d
ctrl
x
T
(PC)
Sensing
Actuation
Digital
Filter
Δ
L
free
h
C
D
A
Realtime interferometer control
Calibration pipeline
1/
L
-1
1
/C
(model)
A
(model)
FIG. 2. Block diagram of the di
ff
erential arm length feedback control
servo. The sensing function, digital filter function, and actuation
function combine to form the open loop transfer function
G
(
f
)
=
A
(
f
)
D
(
f
)
C
(
f
). The signal
x
(PC)
T
is described in section IV.
The accuracy and precision of this estimated strain rely on
characterizing the sensing and actuation functions of each de-
tector,
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. Un-
certainty in the calibration model parameters directly impacts
the uncertainty in the reconstructed detector strain signal. This
uncertainty could limit the signal-to-noise ratios of GW detec-
tion statistics, and could dominate uncertainties in estimated
astrophysical parameters, e.g., luminosity distance, sky loca-
tion, component 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 observation during which
GW150914 was detected. Sec. II describes the actuation and
sensing function models in terms of their measured parameters.
Sec. 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. Secs. V and VI discuss the measure-
ments used to determine the static statistical uncertainties and
systematic errors in the actuation and sensing function models,
respectively, and their results. Sec. VII details the systematic er-
rors in model parameters near the time of the GW150914 event
resulting from uncorrected, slow time variations. Sec. 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
(
t
). Sec. IX
discusses the inter-site 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 es-
timation 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 di
ff
erential 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
describes the interferometric re-
sponse of the detector. It converts residual test mass di
ff
erential
displacement
L
res
to the signal at the GW readout port. The
sensing function includes the response of the photodiodes, their
analog readout electronics, and e
ff
ects from the digitization
process, all of which convert GW port laser power variations to
counts (ct) of digital signal
d
err
, sampled at a rate of 16 384 Hz.
The interferometric response is determined by the arm cavity
mirror (test mass) reflectivities, the reflectivity 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
well approximated by a single-pole low-pass filter with a gain
and an additional time delay.
The sensing function is thus given by
C
(model)
(
f
)
=
K
C
1
+
i
f
/
f
C
C
R
(
f
) exp(
2
π
i
f
τ
C
)
,
(3)
where
K
C
is combined gain of the opto-mechanical response
and analog-to-digital converter (see Fig. 3). It describes, at
6
10
0
10
1
10
2
10
3
10
4
10
10
10
11
10
12
Magnitude (ct/m)
10
0
10
1
10
2
10
3
10
4
−180
−135
−90
−45
0
45
90
135
180
Phase (deg.)
Frequency (Hz)
FIG. 3. The magnitude and phase of the sensing function model
C
(
f
) 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.
a reference time, how many digital counts are produced in
d
err
in response to di
ff
erential arm length displacement. The
pole frequency,
f
C
(341 Hz and 388 Hz for H1 and L1, re-
spectively), is the characteristic frequency that describes the
attenuation of the interferometer response to high-frequency
length perturbations [
5
]. The time delay
τ
C
includes the light
travel time
L
/
c
along the length of the arms (
L
=
3994
.
5 m),
computational delay in the digital acquisition system, and the
delay introduced to approximate the complete interferometric
response as a single pole. Finally, the dimensionless quantity
C
R
(
f
) accounts for any additional frequency dependence of the
sensing function. It has features, mostly 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 di
ff
erential 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 [
7
,
8
], which
isolate the test mass from residual motion of the supporting
active isolation system [
9
]. Each suspension system also in-
cludes 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—
10
0
10
1
10
2
10
3
10
4
10
−20
10
−19
10
−18
10
−17
10
−16
10
−15
10
−14
10
−13
10
−12
10
−11
10
−10
10
−9
10
−8
Magnitude (m/ct)
Upper Intermediate Mass
Penultimate Mass
Test Mass
Total
10
0
10
1
10
2
10
3
10
4
−180
−135
−90
−45
0
45
90
135
180
Phase (deg.)
Frequency (Hz)
FIG. 4. Overall actuation transfer function
A
(
f
) and actuation func-
tions for each suspension stage
F
i
(
f
)
K
i
A
i
(
f
) for the L1 detector.
The mechanical response of the pendulums and
F
i
dictate the char-
acteristics 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 insta-
bilities. The H1 actuation function di
ff
ers slightly in scale, frequency
dependence, and digital filter choice.
the upper intermediate mass (U), the penultimate mass (P), and
the test 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
(
f
), where
i
=
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 di
ff
erential arm length can be controlled using any
combination of the four test mass suspension systems, only
one, the Y-arm end test mass, is used to create
L
ctrl
. Actuating
a single test mass a
ff
ects both the common and the di
ff
erential
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
(model)
(
f
)
=
[
F
T
(
f
)
K
T
A
T
(
f
)
+
F
P
(
f
)
K
P
A
P
(
f
)
+
F
U
(
f
)
K
U
A
U
(
f
)
]
exp(
2
π
i
f
τ
A
)
.
(4)
Here
K
i
and
A
i
(
f
) are the gain and the normalized frequency
dependence of the
i
th suspension stage actuator, measured at a
7
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
determined 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 a
ff
ect the actuation function [
10
], the
laser power resonating in the arm cavities during the observa-
tion period was low enough that radiation pressure e
ff
ects can
be ignored. The H1 and L1 suspensions and electronics are
identical by design, but there are slight di
ff
erences, 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
(
f
) that relates the di
ff
erential arm
length servo error signal to strain sensed by the interferometer:
h
(
f
)
=
(1
/
L
)
R
(
f
)
d
err
(
f
). As shown schematically in Fig. 2,
the response function is given by
R
(
f
)
=
1
+
A
(
f
)
D
(
f
)
C
(
f
)
C
(
f
)
=
1
+
G
(
f
)
C
(
f
)
.
(5)
We will use this response function to evaluate the overall accu-
racy and precision of the calibrated detector strain output. The
actuation function dominates at frequencies below the di
ff
eren-
tial arm length servo unity gain frequency, 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
Using Eqs.
(3)
and
(4)
we identify the set
Q
of parameters
that must be characterized in order to define the sensing and
actuation functions (see Table I). The remaining parameters,
F
i
(
f
) and
τ
A
, are part of the digital control system and thus
known with negligible uncertainty.
Combinations of these scalar and frequency dependent pa-
rameters form a set
Q
meas
of measurable quantities:
Q
meas
=
{
K
T
A
T
(
f
)
,
K
P
A
P
(
f
)
,
K
U
A
U
(
f
)
,
K
C
C
R
(
f
)
/
(1
+
i
f
/
f
C
)
}
,
(6)
where each element includes its associated time delay
τ
A
or
τ
C
. Each quantity
q
meas
Q
meas
is measured using sinusoidal
excitations injected at various points in the control loop while
the detector is in its lowest noise state. Each measurement
consists of excitations that are injected consecutively at discrete
frequencies. Measurements made at a reference time
t
0
are
used to construct a model
q
0
for
q
meas
. The models contain
TABLE I. The set of di
ff
erential arm length control loop parame-
ters that must be characterized to define the sensing and actuation
functions.
Parameter Description
A
T
(
f
)
Normalized test mass actuation function
A
P
(
f
)
Normalized penultimate mass actuation function
A
U
(
f
)
Normalized upper intermediate mass actuation function
C
R
(
f
)
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
frequency-dependent statistical uncertainty derived from the
coherence
γ
2
(
f
) between excitation and response during the
measurements. At each frequency, the standard deviation
σ
q
0
(i.e., the 68% confidence interval) of
q
0
is given by
σ
q
0
(
f
)
=
[
1
γ
2
(
f
)
2
N
γ
2
(
f
)
]
1
/
2
,
(7)
where
N
is the number of measurements performed at each fre-
quency point [
11
]. The measurements are repeated at di
ff
erent
times to validate and refine the estimates of
σ
q
0
(
f
).
Where a reference model
q
0
does not completely describe
the frequency dependence in a measured quantity
q
meas
, we
use the di
ff
erence between the model and the measurements to
define a known, complex systematic error
δ
q
=
q
meas
q
0
. For
most parameters, instead of dealing directly with the known
systematic error
δ
q
, it is convenient to work with the complex
quantity 1
+
δ
q
/
q
0
=
q
meas
/
q
0
. This is the systematic error
expressed as a multiplicative correction factor in the frequency
domain which must be applied to
q
0
in order to obtain the
measured value
q
meas
. The estimate of
δ
q
, which also includes
additional measurements made after the reference time, has
a statistical uncertainty
σ
δ
q
(
f
). This uncertainty is informed
by the covariance of a low-order spline fit to the frequency-
dependent correction factor. The variance of the ratio of the
measured correction factors to the modeled fit is computed and
added together with
σ
2
δ
q
(
f
) to form the statistical uncertainty
of the fit.
In total, the frequency-dependent estimate of
q
is
q
(
f
)
=
q
0
(
f
)
+
δ
q
(
f
)
±
[
σ
2
q
0
(
f
)
+
σ
2
δ
q
(
f
)
]
1
/
2
,
(8)
where we assume that statistical uncertainties in
q
0
and
δ
q
result from Gaussian distributions with zero mean and standard
deviations
σ
q
0
and
σ
δ
q
, respectively.
The frequency-independent parameters
K
i
and
f
C
are mon-
itored continuously during data taking to track small, slow
variations. Tracking is achieved using a set of sinusoidal exci-
tations at select frequencies, typically referred to as
calibration
lines
. The observed time-dependence in
K
i
and
f
C
is treated
as a systematic error
δ
q
(
t
) with an associated statistical un-
certainty
σ
δ
q
(independent of time) that is governed by the
8
signal-to-noise ratio of the continuous excitation. For these
parameters, the total uncertainty estimates are of the form
q
(
t
)
=
q
0
+
δ
q
(
t
)
±
σ
δ
q
.
(9)
In summary, the total calibration uncertainty of
h
(
t
) con-
sists of systematic errors
δ
q
with known values, but the data is
uncorrected for these errors. Future calibration of Advanced
LIGO data will correct for these e
ff
ects. The other source of
uncertainty is statistical uncertainty, arising from uncertainty
on the measured values, deviations from the modeled correc-
tion factors, and covariance of the fit to the measured data. In
Secs. V, VI, and VII, we describe the models of each parameter
in
Q
at the reference time
t
0
, and discuss estimates of statistical
uncertainty
σ
q
0
, systematic error
δ
q
, and statistical uncertainty
in the systematic error
σ
δ
q
. 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
(
t
).
IV. RADIATION PRESSURE ACTUATOR
The primary method for calibrating the actuation function
A
and sensing function
C
is an independent radiation pressure
actuator called the
photon calibrator
(PC). A similar system
was also used for calibration of the initial LIGO detectors [
12
].
Each detector is equipped with two photon calibrator sys-
tems, one for each end test mass, positioned outside the vac-
uum enclosure at the ends of the interferometer arms. For each
system, 1047 nm light from an auxiliary, power-modulated,
Nd
3
+
:YLF laser is directed into the vacuum envelope and re-
flects 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 InGaAs photodetector
mounted on an integrating sphere and is calibrated using a stan-
dard that is traceable to the National Institute of Standards and
Technology (NIST) [
13
]. Power modulation is accomplished
via an acousto-optic modulator that is part of an optical fol-
lower servo that ensures that the power modulation follows the
requested waveform [
14
]. After modulation, the laser beam is
divided optically and projected onto the mirror in two diamet-
rically 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 [12, 15–17].
The laser power modulation induces a modulated displace-
ment of the test mass that is given by [12]
x
(PC)
T
(
f
)
=
2
P
(
f
)
c
s
(
f
) cos
θ
(
1
+
M
T
I
T
~
a
·
~
b
)
.
(10)
This modulated displacement is shown schematically on the
left of Fig. 2. The terms entering this formula are as fol-
lows:
f
is the frequency of the power modulation,
P
(
f
) is the
power modulation amplitude,
c
is the speed of light,
s
(
f
) is
the mechanical compliance of the suspended mirror,
θ
'
8
.
8
is the angle of incidence on the mirror,
M
T
=
39
.
6 kg and
I
T
=
0
.
415
kg m
2
are the mass and rotational moment of inertia
of the mirror, and
~
a
and
~
b
are displacement vectors from the
center of the optical surface to the photon calibrator center of
force and the main interferometer beam, respectively. These
displacements determine the amount of unwanted induced ro-
tation of the mirror.
The compliance
s
(
f
) of the suspended mirror can be ap-
proximated by treating the mirror as rigid body that is free
to move along the optical axis of the arm cavity:
s
(
f
)
'
1
/
[
M
T
(2
π
f
)
2
]. Cross-couplings between other degrees of
freedom of the multi-stage suspension system, however, re-
quire that
s
(
f
) be computed with a full, rigid-body model of
the quadruple suspension. This model has been validated by
previous measurements [
18
] 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
(PC)
T
(
f
), is
'
0
.
8%.
Measurements made during and after the observation period
revealed that the estimate of
x
(PC)
T
includes systematic errors
δ
x
(PC)
T
, resulting in frequency-independent correction factors of
1.013 and 1.002 for H1 and L1, respectively. These errors are
combined with other known systematic errors in the response
function uncertainty estimates as described in Sec. VIII.
V. ACTUATION FUNCTION CALIBRATION
The actuation strength for each suspension stage can be
determined by comparing the interferometer’s response to an
excitation from the suspension stage actuator with one from
the photon calibrator. The measured transfer function of the
i
th suspension stage actuation is given by
[
K
i
A
i
(
f
)]
(meas)
=
x
(PC)
T
(
f
)
d
err
(
f
)
×
d
err
(
f
)
exc
i
(
f
)
,
(11)
where
exc
i
(
f
) is the excitation applied to the
i
th stage (
i
{
T
,
P
,
U
}
).
Figs. 5 and 6 show the measured correction factors
q
meas
/
q
0
=
1
+
δ
q
/
q
0
for the actuators of the lower three sus-
pension stages of the H1 and L1 interferometers, and their
corresponding systematic error estimates determined by low-
order spline fits. 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 di
ff
erential 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.
Recent measurements have shown that these result from un-
modeled mechanical resonances as well as the non-negligible
inductance of the electromagnetic coil actuators, which were
not included in the model at the reference time. As shown in