Directional limits on persistent gravitational waves
from Advanced LIGO’s first observing run
B. P. Abbott,
1
R. Abbott,
1
T. D. Abbott,
2
M. R. Abernathy,
3
F. Acernese,
4
,
5
K. Ackley,
6
C. Adams,
7
T. Adams,
8
P. Addesso,
9
R. X. Adhikari,
1
V. B. Adya,
10
C. Affeldt,
10
M. Agathos,
11
K. Agatsuma,
11
N. Aggarwal,
12
O. D. Aguiar,
13
L. Aiello,
14
,
15
A. Ain,
16
P. Ajith,
17
B. Allen,
10
,
18
,
19
A. Allocca,
20
,
21
P. A. Altin,
22
A. Ananyeva,
1
S. B. Anderson,
1
W. G. Anderson,
18
S. Appert,
1
K. Arai,
1
M. C. Araya,
1
J. S. Areeda,
23
N. Arnaud,
24
K. G. Arun,
25
S. Ascenzi,
26
,
15
G. Ashton,
10
M. Ast,
27
S. M. Aston,
7
P. Astone,
28
P. Aufmuth,
19
C. Aulbert,
10
A. Avila-Alvarez,
23
S. Babak,
29
P. Bacon,
30
M. K. M. Bader,
11
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,
4
,
5
B. Barr,
36
L. Barsotti,
12
M. Barsuglia,
30
D. Barta,
38
J. Bartlett,
37
I. Bartos,
39
R. Bassiri,
40
A. Basti,
20
,
21
J. C. Batch,
37
C. Baune,
10
V. Bavigadda,
34
M. Bazzan,
41
,
42
C. Beer,
10
M. Bejger,
43
I. Belahcene,
24
M. Belgin,
44
A. S. Bell,
36
B. K. Berger,
1
G. Bergmann,
10
C. P. L. Berry,
45
D. Bersanetti,
46
,
47
A. Bertolini,
11
J. Betzwieser,
7
S. Bhagwat,
35
R. Bhandare,
48
I. A. Bilenko,
49
G. Billingsley,
1
C. R. Billman,
6
J. Birch,
7
R. Birney,
50
O. Birnholtz,
10
S. Biscans,
12
,
1
A. S. Biscoveanu,
74
A. Bisht,
19
M. Bitossi,
34
C. Biwer,
35
M. A. Bizouard,
24
J. K. Blackburn,
1
J. Blackman,
51
C. D. Blair,
52
D. G. Blair,
52
R. M. Blair,
37
S. Bloemen,
53
O. Bock,
10
M. Boer,
54
G. Bogaert,
54
A. Bohe,
29
F. Bondu,
55
R. Bonnand,
8
B. A. Boom,
11
R. Bork,
1
V. Boschi,
20
,
21
S. Bose,
56
,
16
Y. Bouffanais,
30
A. Bozzi,
34
C. Bradaschia,
21
P. R. Brady,
18
V. B. Braginsky
∗
,
49
M. Branchesi,
57
,
58
J. E. Brau,
59
T. Briant,
60
A. Brillet,
54
M. Brinkmann,
10
V. Brisson,
24
P. Brockill,
18
J. E. Broida,
61
A. F. Brooks,
1
D. A. Brown,
35
D. D. Brown,
45
N. M. Brown,
12
S. Brunett,
1
C. C. Buchanan,
2
A. Buikema,
12
T. Bulik,
62
H. J. Bulten,
63
,
11
A. Buonanno,
29
,
64
D. Buskulic,
8
C. Buy,
30
R. L. Byer,
40
M. Cabero,
10
L. Cadonati,
44
G. Cagnoli,
65
,
66
C. Cahillane,
1
J. Calder ́on Bustillo,
44
T. A. Callister,
1
E. Calloni,
67
,
5
J. B. Camp,
68
W. Campbell,
120
M. Canepa,
46
,
47
K. C. Cannon,
69
H. Cao,
70
J. Cao,
71
C. D. Capano,
10
E. Capocasa,
30
F. Carbognani,
34
S. Caride,
72
J. Casanueva Diaz,
24
C. Casentini,
26
,
15
S. Caudill,
18
M. Cavagli`a,
73
F. Cavalier,
24
R. Cavalieri,
34
G. Cella,
21
C. B. Cepeda,
1
L. Cerboni Baiardi,
57
,
58
G. Cerretani,
20
,
21
E. Cesarini,
26
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15
S. J. Chamberlin,
74
M. Chan,
36
S. Chao,
75
P. Charlton,
76
E. Chassande-Mottin,
30
B. D. Cheeseboro,
31
H. Y. Chen,
77
Y. Chen,
51
H.-P. Cheng,
6
A. Chincarini,
47
A. Chiummo,
34
T. Chmiel,
78
H. S. Cho,
79
M. Cho,
64
J. H. Chow,
22
N. Christensen,
61
Q. Chu,
52
A. J. K. Chua,
80
S. Chua,
60
S. Chung,
52
G. Ciani,
6
F. Clara,
37
J. A. Clark,
44
F. Cleva,
54
C. Cocchieri,
73
E. Coccia,
14
,
15
P.-F. Cohadon,
60
A. Colla,
81
,
28
C. G. Collette,
82
L. Cominsky,
83
M. Constancio Jr.,
13
L. Conti,
42
S. J. Cooper,
45
T. R. Corbitt,
2
N. Cornish,
84
A. Corsi,
72
S. Cortese,
34
C. A. Costa,
13
E. Coughlin,
61
M. W. Coughlin,
61
S. B. Coughlin,
85
J.-P. Coulon,
54
S. T. Countryman,
39
P. Couvares,
1
P. B. Covas,
86
E. E. Cowan,
44
D. M. Coward,
52
M. J. Cowart,
7
D. C. Coyne,
1
R. Coyne,
72
J. D. E. Creighton,
18
T. D. Creighton,
87
J. Cripe,
2
S. G. Crowder,
88
T. J. Cullen,
23
A. Cumming,
36
L. Cunningham,
36
E. Cuoco,
34
T. Dal Canton,
68
S. L. Danilishin,
36
S. D’Antonio,
15
K. Danzmann,
19
,
10
A. Dasgupta,
89
C. F. Da Silva Costa,
6
V. Dattilo,
34
I. Dave,
48
M. Davier,
24
G. S. Davies,
36
D. Davis,
35
E. J. Daw,
90
B. Day,
44
R. Day,
34
S. De,
35
D. DeBra,
40
G. Debreczeni,
38
J. Degallaix,
65
M. De Laurentis,
67
,
5
S. Del ́eglise,
60
W. Del Pozzo,
45
T. Denker,
10
T. Dent,
10
V. Dergachev,
29
R. De Rosa,
67
,
5
R. T. DeRosa,
7
R. DeSalvo,
91
J. Devenson,
50
R. C. Devine,
31
S. Dhurandhar,
16
M. C. D ́ıaz,
87
L. Di Fiore,
5
M. Di Giovanni,
92
,
93
T. Di Girolamo,
67
,
5
A. Di Lieto,
20
,
21
S. Di Pace,
81
,
28
I. Di Palma,
29
,
81
,
28
A. Di Virgilio,
21
Z. Doctor,
77
V. Dolique,
65
F. Donovan,
12
K. L. Dooley,
73
S. Doravari,
10
I. Dorrington,
94
R. Douglas,
36
M. Dovale
́
Alvarez,
45
T. P. Downes,
18
M. Drago,
10
R. W. P. Drever,
1
J. C. Driggers,
37
Z. Du,
71
M. Ducrot,
8
S. E. Dwyer,
37
T. B. Edo,
90
M. C. Edwards,
61
A. Effler,
7
H.-B. Eggenstein,
10
P. Ehrens,
1
J. Eichholz,
1
S. S. Eikenberry,
6
R. C. Essick,
12
Z. Etienne,
31
T. Etzel,
1
M. Evans,
12
T. M. Evans,
7
R. Everett,
74
M. Factourovich,
39
V. Fafone,
26
,
15
,
14
H. Fair,
35
S. Fairhurst,
94
X. Fan,
71
S. Farinon,
47
B. Farr,
77
W. M. Farr,
45
E. J. Fauchon-Jones,
94
M. Favata,
95
M. Fays,
94
H. Fehrmann,
10
M. M. Fejer,
40
A. Fern ́andez Galiana,
12
I. Ferrante,
20
,
21
E. C. Ferreira,
13
F. Ferrini,
34
F. Fidecaro,
20
,
21
I. Fiori,
34
D. Fiorucci,
30
R. P. Fisher,
35
R. Flaminio,
65
,
96
M. Fletcher,
36
H. Fong,
97
S. S. Forsyth,
44
J.-D. Fournier,
54
S. Frasca,
81
,
28
F. Frasconi,
21
Z. Frei,
98
A. Freise,
45
R. Frey,
59
V. Frey,
24
E. M. Fries,
1
P. Fritschel,
12
V. V. Frolov,
7
P. Fulda,
6
,
68
M. Fyffe,
7
H. Gabbard,
10
B. U. Gadre,
16
S. M. Gaebel,
45
J. R. Gair,
99
L. Gammaitoni,
32
S. G. Gaonkar,
16
F. Garufi,
67
,
5
G. Gaur,
100
V. Gayathri,
101
N. Gehrels,
68
G. Gemme,
47
E. Genin,
34
A. Gennai,
21
J. George,
48
L. Gergely,
102
V. Germain,
8
S. Ghonge,
17
Abhirup Ghosh,
17
Archisman Ghosh,
11
,
17
S. Ghosh,
53
,
11
J. A. Giaime,
2
,
7
K. D. Giardina,
7
A. Giazotto,
21
K. Gill,
103
A. Glaefke,
36
E. Goetz,
10
R. Goetz,
6
L. Gondan,
98
G. Gonz ́alez,
2
J. M. Gonzalez Castro,
20
,
21
A. Gopakumar,
104
M. L. Gorodetsky,
49
S. E. Gossan,
1
M. Gosselin,
34
arXiv:1612.02030v4 [gr-qc] 30 Jan 2017
2
R. Gouaty,
8
A. Grado,
105
,
5
C. Graef,
36
M. Granata,
65
A. Grant,
36
S. Gras,
12
C. Gray,
37
G. Greco,
57
,
58
A. C. Green,
45
P. Groot,
53
H. Grote,
10
S. Grunewald,
29
G. M. Guidi,
57
,
58
X. Guo,
71
A. Gupta,
16
M. K. Gupta,
89
K. E. Gushwa,
1
E. K. Gustafson,
1
R. Gustafson,
106
J. J. Hacker,
23
B. R. Hall,
56
E. D. Hall,
1
G. Hammond,
36
M. Haney,
104
M. M. Hanke,
10
J. Hanks,
37
C. Hanna,
74
M. D. Hannam,
94
J. Hanson,
7
T. Hardwick,
2
J. Harms,
57
,
58
G. M. Harry,
3
I. W. Harry,
29
M. J. Hart,
36
M. T. Hartman,
6
C.-J. Haster,
45
,
97
K. Haughian,
36
J. Healy,
107
A. Heidmann,
60
M. C. Heintze,
7
H. Heitmann,
54
P. Hello,
24
G. Hemming,
34
M. Hendry,
36
I. S. Heng,
36
J. Hennig,
36
J. Henry,
107
A. W. Heptonstall,
1
M. Heurs,
10
,
19
S. Hild,
36
D. Hoak,
34
D. Hofman,
65
K. Holt,
7
D. E. Holz,
77
P. Hopkins,
94
J. Hough,
36
E. A. Houston,
36
E. J. Howell,
52
Y. M. Hu,
10
E. A. Huerta,
108
D. Huet,
24
B. Hughey,
103
S. Husa,
86
S. H. Huttner,
36
T. Huynh-Dinh,
7
N. Indik,
10
D. R. Ingram,
37
R. Inta,
72
H. N. Isa,
36
J.-M. Isac,
60
M. Isi,
1
T. Isogai,
12
B. R. Iyer,
17
K. Izumi,
37
T. Jacqmin,
60
K. Jani,
44
P. Jaranowski,
109
S. Jawahar,
110
F. Jim ́enez-Forteza,
86
W. W. Johnson,
2
D. I. Jones,
111
R. Jones,
36
R. J. G. Jonker,
11
L. Ju,
52
J. Junker,
10
C. V. Kalaghatgi,
94
V. Kalogera,
85
S. Kandhasamy,
73
G. Kang,
79
J. B. Kanner,
1
S. Karki,
59
K. S. Karvinen,
10
M. Kasprzack,
2
E. Katsavounidis,
12
W. Katzman,
7
S. Kaufer,
19
T. Kaur,
52
K. Kawabe,
37
F. K ́ef ́elian,
54
D. Keitel,
86
D. B. Kelley,
35
R. Kennedy,
90
J. S. Key,
112
F. Y. Khalili,
49
I. Khan,
14
S. Khan,
94
Z. Khan,
89
E. A. Khazanov,
113
N. Kijbunchoo,
37
Chunglee Kim,
114
J. C. Kim,
115
Whansun Kim,
116
W. Kim,
70
Y.-M. Kim,
117
,
114
S. J. Kimbrell,
44
E. J. King,
70
P. J. King,
37
R. Kirchhoff,
10
J. S. Kissel,
37
B. Klein,
85
L. Kleybolte,
27
S. Klimenko,
6
P. Koch,
10
S. M. Koehlenbeck,
10
S. Koley,
11
V. Kondrashov,
1
A. Kontos,
12
M. Korobko,
27
W. Z. Korth,
1
I. Kowalska,
62
D. B. Kozak,
1
C. Kr ̈amer,
10
V. Kringel,
10
A. Kr ́olak,
118
,
119
G. Kuehn,
10
P. Kumar,
97
R. Kumar,
89
L. Kuo,
75
A. Kutynia,
118
B. D. Lackey,
29
,
35
M. Landry,
37
R. N. Lang,
18
J. Lange,
107
B. Lantz,
40
R. K. Lanza,
12
A. Lartaux-Vollard,
24
P. D. Lasky,
120
M. Laxen,
7
A. Lazzarini,
1
C. Lazzaro,
42
P. Leaci,
81
,
28
S. Leavey,
36
E. O. Lebigot,
30
C. H. Lee,
117
H. K. Lee,
121
H. M. Lee,
114
K. Lee,
36
J. Lehmann,
10
A. Lenon,
31
M. Leonardi,
92
,
93
J. R. Leong,
10
N. Leroy,
24
N. Letendre,
8
Y. Levin,
120
T. G. F. Li,
122
A. Libson,
12
T. B. Littenberg,
123
J. Liu,
52
N. A. Lockerbie,
110
A. L. Lombardi,
44
L. T. London,
94
J. E. Lord,
35
M. Lorenzini,
14
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15
V. Loriette,
124
M. Lormand,
7
G. Losurdo,
21
J. D. Lough,
10
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19
G. Lovelace,
23
H. L ̈uck,
19
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10
A. P. Lundgren,
10
R. Lynch,
12
Y. Ma,
51
S. Macfoy,
50
B. Machenschalk,
10
M. MacInnis,
12
D. M. Macleod,
2
F. Maga ̃na-Sandoval,
35
E. Majorana,
28
I. Maksimovic,
124
V. Malvezzi,
26
,
15
N. Man,
54
V. Mandic,
125
V. Mangano,
36
G. L. Mansell,
22
M. Manske,
18
M. Mantovani,
34
F. Marchesoni,
126
,
33
F. Marion,
8
S. M ́arka,
39
Z. M ́arka,
39
A. S. Markosyan,
40
E. Maros,
1
F. Martelli,
57
,
58
L. Martellini,
54
I. W. Martin,
36
D. V. Martynov,
12
K. Mason,
12
A. Masserot,
8
T. J. Massinger,
1
M. Masso-Reid,
36
S. Mastrogiovanni,
81
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28
A. Matas,
125
F. Matichard,
12
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1
L. Matone,
39
N. Mavalvala,
12
N. Mazumder,
56
R. McCarthy,
37
D. E. McClelland,
22
S. McCormick,
7
C. McGrath,
18
S. C. McGuire,
127
G. McIntyre,
1
J. McIver,
1
D. J. McManus,
22
T. McRae,
22
S. T. McWilliams,
31
D. Meacher,
54
,
74
G. D. Meadors,
29
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10
J. Meidam,
11
A. Melatos,
128
G. Mendell,
37
D. Mendoza-Gandara,
10
R. A. Mercer,
18
E. L. Merilh,
37
M. Merzougui,
54
S. Meshkov,
1
C. Messenger,
36
C. Messick,
74
R. Metzdorff,
60
P. M. Meyers,
125
F. Mezzani,
28
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81
H. Miao,
45
C. Michel,
65
H. Middleton,
45
E. E. Mikhailov,
129
L. Milano,
67
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5
A. L. Miller,
6
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81
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28
A. Miller,
85
B. B. Miller,
85
J. Miller,
12
M. Millhouse,
84
Y. Minenkov,
15
J. Ming,
29
S. Mirshekari,
130
C. Mishra,
17
S. Mitra,
16
V. P. Mitrofanov,
49
G. Mitselmakher,
6
R. Mittleman,
12
A. Moggi,
21
M. Mohan,
34
S. R. P. Mohapatra,
12
M. Montani,
57
,
58
B. C. Moore,
95
C. J. Moore,
80
D. Moraru,
37
G. Moreno,
37
S. R. Morriss,
87
B. Mours,
8
C. M. Mow-Lowry,
45
G. Mueller,
6
A. W. Muir,
94
Arunava Mukherjee,
17
D. Mukherjee,
18
S. Mukherjee,
87
N. Mukund,
16
A. Mullavey,
7
J. Munch,
70
E. A. M. Muniz,
23
P. G. Murray,
36
A. Mytidis,
6
K. Napier,
44
I. Nardecchia,
26
,
15
L. Naticchioni,
81
,
28
G. Nelemans,
53
,
11
T. J. N. Nelson,
7
M. Neri,
46
,
47
M. Nery,
10
A. Neunzert,
106
J. M. Newport,
3
G. Newton,
36
T. T. Nguyen,
22
A. B. Nielsen,
10
S. Nissanke,
53
,
11
A. Nitz,
10
A. Noack,
10
F. Nocera,
34
D. Nolting,
7
M. E. N. Normandin,
87
L. K. Nuttall,
35
J. Oberling,
37
E. Ochsner,
18
E. Oelker,
12
G. H. Ogin,
131
J. J. Oh,
116
S. H. Oh,
116
F. Ohme,
94
,
10
M. Oliver,
86
P. Oppermann,
10
Richard J. Oram,
7
B. O’Reilly,
7
R. O’Shaughnessy,
107
D. J. Ottaway,
70
H. Overmier,
7
B. J. Owen,
72
A. E. Pace,
74
J. Page,
123
A. Pai,
101
S. A. Pai,
48
J. R. Palamos,
59
O. Palashov,
113
C. Palomba,
28
A. Pal-Singh,
27
H. Pan,
75
C. Pankow,
85
F. Pannarale,
94
B. C. Pant,
48
F. Paoletti,
34
,
21
A. Paoli,
34
M. A. Papa,
29
,
18
,
10
H. R. Paris,
40
W. Parker,
7
D. Pascucci,
36
A. Pasqualetti,
34
R. Passaquieti,
20
,
21
D. Passuello,
21
B. Patricelli,
20
,
21
B. L. Pearlstone,
36
M. Pedraza,
1
R. Pedurand,
65
,
132
L. Pekowsky,
35
A. Pele,
7
S. Penn,
133
C. J. Perez,
37
A. Perreca,
1
L. M. Perri,
85
H. P. Pfeiffer,
97
M. Phelps,
36
O. J. Piccinni,
81
,
28
M. Pichot,
54
F. Piergiovanni,
57
,
58
V. Pierro,
9
G. Pillant,
34
L. Pinard,
65
I. M. Pinto,
9
M. Pitkin,
36
M. Poe,
18
R. Poggiani,
20
,
21
P. Popolizio,
34
A. Post,
10
J. Powell,
36
J. Prasad,
16
J. W. W. Pratt,
103
V. Predoi,
94
T. Prestegard,
125
,
18
M. Prijatelj,
10
,
34
M. Principe,
9
S. Privitera,
29
3
G. A. Prodi,
92
,
93
L. G. Prokhorov,
49
O. Puncken,
10
M. Punturo,
33
P. Puppo,
28
M. P ̈urrer,
29
H. Qi,
18
J. Qin,
52
S. Qiu,
120
V. Quetschke,
87
E. A. Quintero,
1
R. Quitzow-James,
59
F. J. Raab,
37
D. S. Rabeling,
22
H. Radkins,
37
P. Raffai,
98
S. Raja,
48
C. Rajan,
48
M. Rakhmanov,
87
P. Rapagnani,
81
,
28
V. Raymond,
29
M. Razzano,
20
,
21
V. Re,
26
J. Read,
23
T. Regimbau,
54
L. Rei,
47
S. Reid,
50
D. H. Reitze,
1
,
6
H. Rew,
129
S. D. Reyes,
35
E. Rhoades,
103
F. Ricci,
81
,
28
K. Riles,
106
M. Rizzo,
107
N. A. Robertson,
1
,
36
R. Robie,
36
F. Robinet,
24
A. Rocchi,
15
L. Rolland,
8
J. G. Rollins,
1
V. J. Roma,
59
J. D. Romano,
87
R. Romano,
4
,
5
J. H. Romie,
7
D. Rosi ́nska,
134
,
43
S. Rowan,
36
A. R ̈udiger,
10
P. Ruggi,
34
K. Ryan,
37
S. Sachdev,
1
T. Sadecki,
37
L. Sadeghian,
18
M. Sakellariadou,
135
L. Salconi,
34
M. Saleem,
101
F. Salemi,
10
A. Samajdar,
136
L. Sammut,
120
L. M. Sampson,
85
E. J. Sanchez,
1
V. Sandberg,
37
J. R. Sanders,
35
B. Sassolas,
65
B. S. Sathyaprakash,
74
,
94
P. R. Saulson,
35
O. Sauter,
106
R. L. Savage,
37
A. Sawadsky,
19
P. Schale,
59
J. Scheuer,
85
S. Schlassa,
61
E. Schmidt,
103
J. Schmidt,
10
P. Schmidt,
1
,
51
R. Schnabel,
27
R. M. S. Schofield,
59
A. Sch ̈onbeck,
27
E. Schreiber,
10
D. Schuette,
10
,
19
B. F. Schutz,
94
,
29
S. G. Schwalbe,
103
J. Scott,
36
S. M. Scott,
22
D. Sellers,
7
A. S. Sengupta,
137
D. Sentenac,
34
V. Sequino,
26
,
15
A. Sergeev,
113
Y. Setyawati,
53
,
11
D. A. Shaddock,
22
T. J. Shaffer,
37
M. S. Shahriar,
85
B. Shapiro,
40
P. Shawhan,
64
A. Sheperd,
18
D. H. Shoemaker,
12
D. M. Shoemaker,
44
K. Siellez,
44
X. Siemens,
18
M. Sieniawska,
43
D. Sigg,
37
A. D. Silva,
13
A. Singer,
1
L. P. Singer,
68
A. Singh,
29
,
10
,
19
R. Singh,
2
A. Singhal,
14
A. M. Sintes,
86
B. J. J. Slagmolen,
22
B. Smith,
7
J. R. Smith,
23
R. J. E. Smith,
1
E. J. Son,
116
B. Sorazu,
36
F. Sorrentino,
47
T. Souradeep,
16
A. P. Spencer,
36
A. K. Srivastava,
89
A. Staley,
39
M. Steinke,
10
J. Steinlechner,
36
S. Steinlechner,
27
,
36
D. Steinmeyer,
10
,
19
B. C. Stephens,
18
S. P. Stevenson,
45
R. Stone,
87
K. A. Strain,
36
N. Straniero,
65
G. Stratta,
57
,
58
S. E. Strigin,
49
R. Sturani,
130
A. L. Stuver,
7
T. Z. Summerscales,
138
L. Sun,
128
S. Sunil,
89
P. J. Sutton,
94
B. L. Swinkels,
34
M. J. Szczepa ́nczyk,
103
M. Tacca,
30
D. Talukder,
59
D. B. Tanner,
6
D. Tao,
61
M. T ́apai,
102
A. Taracchini,
29
R. Taylor,
1
T. Theeg,
10
E. G. Thomas,
45
M. Thomas,
7
P. Thomas,
37
K. A. Thorne,
7
E. Thrane,
120
T. Tippens,
44
S. Tiwari,
14
,
93
V. Tiwari,
94
K. V. Tokmakov,
110
K. Toland,
36
C. Tomlinson,
90
M. Tonelli,
20
,
21
Z. Tornasi,
36
C. I. Torrie,
1
D. T ̈oyr ̈a,
45
F. Travasso,
32
,
33
G. Traylor,
7
D. Trifir`o,
73
J. Trinastic,
6
M. C. Tringali,
92
,
93
L. Trozzo,
139
,
21
M. Tse,
12
R. Tso,
1
M. Turconi,
54
D. Tuyenbayev,
87
D. Ugolini,
140
C. S. Unnikrishnan,
104
A. L. Urban,
1
S. A. Usman,
94
H. Vahlbruch,
19
G. Vajente,
1
G. Valdes,
87
N. van Bakel,
11
M. van Beuzekom,
11
J. F. J. van den Brand,
63
,
11
C. Van Den Broeck,
11
D. C. Vander-Hyde,
35
L. van der Schaaf,
11
J. V. van Heijningen,
11
A. A. van Veggel,
36
M. Vardaro,
41
,
42
V. Varma,
51
S. Vass,
1
M. Vas ́uth,
38
A. Vecchio,
45
G. Vedovato,
42
J. Veitch,
45
P. J. Veitch,
70
K. Venkateswara,
141
G. Venugopalan,
1
D. Verkindt,
8
F. Vetrano,
57
,
58
A. Vicer ́e,
57
,
58
A. D. Viets,
18
S. Vinciguerra,
45
D. J. Vine,
50
J.-Y. Vinet,
54
S. Vitale,
12
T. Vo,
35
H. Vocca,
32
,
33
C. Vorvick,
37
D. V. Voss,
6
W. D. Vousden,
45
S. P. Vyatchanin,
49
A. R. Wade,
1
L. E. Wade,
78
M. Wade,
78
M. Walker,
2
L. Wallace,
1
S. Walsh,
29
,
10
G. Wang,
14
,
58
H. Wang,
45
M. Wang,
45
Y. Wang,
52
R. L. Ward,
22
J. Warner,
37
M. Was,
8
J. Watchi,
82
B. Weaver,
37
L.-W. Wei,
54
M. Weinert,
10
A. J. Weinstein,
1
R. Weiss,
12
L. Wen,
52
P. Weßels,
10
T. Westphal,
10
K. Wette,
10
J. T. Whelan,
107
B. F. Whiting,
6
C. Whittle,
120
D. Williams,
36
R. D. Williams,
1
A. R. Williamson,
94
J. L. Willis,
142
B. Willke,
19
,
10
M. H. Wimmer,
10
,
19
W. Winkler,
10
C. C. Wipf,
1
H. Wittel,
10
,
19
G. Woan,
36
J. Woehler,
10
J. Worden,
37
J. L. Wright,
36
D. S. Wu,
10
G. Wu,
7
W. Yam,
12
H. Yamamoto,
1
C. C. Yancey,
64
M. J. Yap,
22
Hang Yu,
12
Haocun Yu,
12
M. Yvert,
8
A. Zadro ̇zny,
118
L. Zangrando,
42
M. Zanolin,
103
J.-P. Zendri,
42
M. Zevin,
85
L. Zhang,
1
M. Zhang,
129
T. Zhang,
36
Y. Zhang,
107
C. Zhao,
52
M. Zhou,
85
Z. Zhou,
85
S. J. Zhu,
29
,
10
X. J. Zhu,
52
M. E. Zucker,
1
,
12
and J. Zweizig
1
(LIGO Scientific Collaboration and Virgo Collaboration)
∗
Deceased, March 2016.
1
LIGO, California Institute of Technology, Pasadena, CA 91125, USA
2
Louisiana State University, Baton Rouge, LA 70803, USA
3
American University, Washington, D.C. 20016, USA
4
Universit`a di Salerno, Fisciano, I-84084 Salerno, Italy
5
INFN, Sezione di Napoli, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy
6
University of Florida, Gainesville, FL 32611, USA
7
LIGO Livingston Observatory, Livingston, LA 70754, USA
8
Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP),
Universit ́e Savoie Mont Blanc, CNRS/IN2P3, F-74941 Annecy-le-Vieux, France
9
University of Sannio at Benevento, I-82100 Benevento,
Italy and INFN, Sezione di Napoli, I-80100 Napoli, Italy
10
Albert-Einstein-Institut, Max-Planck-Institut f ̈ur Gravitationsphysik, D-30167 Hannover, Germany
11
Nikhef, Science Park, 1098 XG Amsterdam, The Netherlands
4
12
LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
13
Instituto Nacional de Pesquisas Espaciais, 12227-010 S ̃ao Jos ́e dos Campos, S ̃ao Paulo, Brazil
14
INFN, Gran Sasso Science Institute, I-67100 L’Aquila, Italy
15
INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy
16
Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India
17
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
18
University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
19
Leibniz Universit ̈at Hannover, D-30167 Hannover, Germany
20
Universit`a di Pisa, I-56127 Pisa, Italy
21
INFN, Sezione di Pisa, I-56127 Pisa, Italy
22
Australian National University, Canberra, Australian Capital Territory 0200, Australia
23
California State University Fullerton, Fullerton, CA 92831, USA
24
LAL, Univ. Paris-Sud, CNRS/IN2P3, Universit ́e Paris-Saclay, F-91898 Orsay, France
25
Chennai Mathematical Institute, Chennai 603103, India
26
Universit`a di Roma Tor Vergata, I-00133 Roma, Italy
27
Universit ̈at Hamburg, D-22761 Hamburg, Germany
28
INFN, Sezione di Roma, I-00185 Roma, Italy
29
Albert-Einstein-Institut, Max-Planck-Institut f ̈ur Gravitationsphysik, D-14476 Potsdam-Golm, Germany
30
APC, AstroParticule et Cosmologie, Universit ́e Paris Diderot,
CNRS/IN2P3, CEA/Irfu, Observatoire de Paris,
Sorbonne Paris Cit ́e, F-75205 Paris Cedex 13, France
31
West Virginia University, Morgantown, WV 26506, USA
32
Universit`a di Perugia, I-06123 Perugia, Italy
33
INFN, Sezione di Perugia, I-06123 Perugia, Italy
34
European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy
35
Syracuse University, Syracuse, NY 13244, USA
36
SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom
37
LIGO Hanford Observatory, Richland, WA 99352, USA
38
Wigner RCP, RMKI, H-1121 Budapest, Konkoly Thege Mikl ́os ́ut 29-33, Hungary
39
Columbia University, New York, NY 10027, USA
40
Stanford University, Stanford, CA 94305, USA
41
Universit`a di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova, Italy
42
INFN, Sezione di Padova, I-35131 Padova, Italy
43
Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, 00-716, Warsaw, Poland
44
Center for Relativistic Astrophysics and School of Physics,
Georgia Institute of Technology, Atlanta, GA 30332, USA
45
University of Birmingham, Birmingham B15 2TT, United Kingdom
46
Universit`a degli Studi di Genova, I-16146 Genova, Italy
47
INFN, Sezione di Genova, I-16146 Genova, Italy
48
RRCAT, Indore MP 452013, India
49
Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia
50
SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom
51
Caltech CaRT, Pasadena, CA 91125, USA
52
University of Western Australia, Crawley, Western Australia 6009, Australia
53
Department of Astrophysics/IMAPP, Radboud University Nijmegen,
P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
54
Artemis, Universit ́e Cˆote d’Azur, CNRS, Observatoire Cˆote d’Azur, CS 34229, F-06304 Nice Cedex 4, France
55
Institut de Physique de Rennes, CNRS, Universit ́e de Rennes 1, F-35042 Rennes, France
56
Washington State University, Pullman, WA 99164, USA
57
Universit`a degli Studi di Urbino ’Carlo Bo’, I-61029 Urbino, Italy
58
INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy
59
University of Oregon, Eugene, OR 97403, USA
60
Laboratoire Kastler Brossel, UPMC-Sorbonne Universit ́es, CNRS,
ENS-PSL Research University, Coll`ege de France, F-75005 Paris, France
61
Carleton College, Northfield, MN 55057, USA
62
Astronomical Observatory Warsaw University, 00-478 Warsaw, Poland
63
VU University Amsterdam, 1081 HV Amsterdam, The Netherlands
64
University of Maryland, College Park, MD 20742, USA
65
Laboratoire des Mat ́eriaux Avanc ́es (LMA), CNRS/IN2P3, F-69622 Villeurbanne, France
66
Universit ́e Claude Bernard Lyon 1, F-69622 Villeurbanne, France
67
Universit`a di Napoli ’Federico II’, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy
68
NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
69
RESCEU, University of Tokyo, Tokyo, 113-0033, Japan.
70
University of Adelaide, Adelaide, South Australia 5005, Australia
5
71
Tsinghua University, Beijing 100084, China
72
Texas Tech University, Lubbock, TX 79409, USA
73
The University of Mississippi, University, MS 38677, USA
74
The Pennsylvania State University, University Park, PA 16802, USA
75
National Tsing Hua University, Hsinchu City, 30013 Taiwan, Republic of China
76
Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia
77
University of Chicago, Chicago, IL 60637, USA
78
Kenyon College, Gambier, OH 43022, USA
79
Korea Institute of Science and Technology Information, Daejeon 305-806, Korea
80
University of Cambridge, Cambridge CB2 1TN, United Kingdom
81
Universit`a di Roma ’La Sapienza’, I-00185 Roma, Italy
82
University of Brussels, Brussels 1050, Belgium
83
Sonoma State University, Rohnert Park, CA 94928, USA
84
Montana State University, Bozeman, MT 59717, USA
85
Center for Interdisciplinary Exploration & Research in Astrophysics (CIERA),
Northwestern University, Evanston, IL 60208, USA
86
Universitat de les Illes Balears, IAC3—IEEC, E-07122 Palma de Mallorca, Spain
87
The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA
88
Bellevue College, Bellevue, WA 98007, USA
89
Institute for Plasma Research, Bhat, Gandhinagar 382428, India
90
The University of Sheffield, Sheffield S10 2TN, United Kingdom
91
California State University, Los Angeles, 5154 State University Dr, Los Angeles, CA 90032, USA
92
Universit`a di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy
93
INFN, Trento Institute for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy
94
Cardiff University, Cardiff CF24 3AA, United Kingdom
95
Montclair State University, Montclair, NJ 07043, USA
96
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
97
Canadian Institute for Theoretical Astrophysics,
University of Toronto, Toronto, Ontario M5S 3H8, Canada
98
MTA E ̈otv ̈os University, “Lendulet” Astrophysics Research Group, Budapest 1117, Hungary
99
School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
100
University and Institute of Advanced Research, Gandhinagar, Gujarat 382007, India
101
IISER-TVM, CET Campus, Trivandrum Kerala 695016, India
102
University of Szeged, D ́om t ́er 9, Szeged 6720, Hungary
103
Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA
104
Tata Institute of Fundamental Research, Mumbai 400005, India
105
INAF, Osservatorio Astronomico di Capodimonte, I-80131, Napoli, Italy
106
University of Michigan, Ann Arbor, MI 48109, USA
107
Rochester Institute of Technology, Rochester, NY 14623, USA
108
NCSA, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
109
University of Bia lystok, 15-424 Bia lystok, Poland
110
SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom
111
University of Southampton, Southampton SO17 1BJ, United Kingdom
112
University of Washington Bothell, 18115 Campus Way NE, Bothell, WA 98011, USA
113
Institute of Applied Physics, Nizhny Novgorod, 603950, Russia
114
Seoul National University, Seoul 151-742, Korea
115
Inje University Gimhae, 621-749 South Gyeongsang, Korea
116
National Institute for Mathematical Sciences, Daejeon 305-390, Korea
117
Pusan National University, Busan 609-735, Korea
118
NCBJ, 05-400
́
Swierk-Otwock, Poland
119
Institute of Mathematics, Polish Academy of Sciences, 00656 Warsaw, Poland
120
The School of Physics & Astronomy, Monash University, Clayton 3800, Victoria, Australia
121
Hanyang University, Seoul 133-791, Korea
122
The Chinese University of Hong Kong, Shatin, NT, Hong Kong
123
University of Alabama in Huntsville, Huntsville, AL 35899, USA
124
ESPCI, CNRS, F-75005 Paris, France
125
University of Minnesota, Minneapolis, MN 55455, USA
126
Universit`a di Camerino, Dipartimento di Fisica, I-62032 Camerino, Italy
127
Southern University and A&M College, Baton Rouge, LA 70813, USA
128
The University of Melbourne, Parkville, Victoria 3010, Australia
129
College of William and Mary, Williamsburg, VA 23187, USA
130
Instituto de F ́ısica Te ́orica, University Estadual Paulista/ICTP South
American Institute for Fundamental Research, S ̃ao Paulo SP 01140-070, Brazil
131
Whitman College, 345 Boyer Avenue, Walla Walla, WA 99362 USA
6
132
Universit ́e de Lyon, F-69361 Lyon, France
133
Hobart and William Smith Colleges, Geneva, NY 14456, USA
134
Janusz Gil Institute of Astronomy, University of Zielona G ́ora, 65-265 Zielona G ́ora, Poland
135
King’s College London, University of London, London WC2R 2LS, United Kingdom
136
IISER-Kolkata, Mohanpur, West Bengal 741252, India
137
Indian Institute of Technology, Gandhinagar Ahmedabad Gujarat 382424, India
138
Andrews University, Berrien Springs, MI 49104, USA
139
Universit`a di Siena, I-53100 Siena, Italy
140
Trinity University, San Antonio, TX 78212, USA
141
University of Washington, Seattle, WA 98195, USA
142
Abilene Christian University, Abilene, TX 79699, USA
We employ gravitational-wave radiometry to map the gravitational waves stochastic background
expected from a variety of contributing mechanisms and test the assumption of isotropy us-
ing data from Advanced LIGO’s first observing run.
We also search for persistent gravita-
tional waves from point sources with only minimal assumptions over the 20 - 1726 Hz frequency
band. Finding no evidence of gravitational waves from either point sources or a stochastic back-
ground, we set limits at 90% confidence. For broadband point sources, we report upper limits
on the gravitational wave energy flux per unit frequency in the range
F
α,
Θ
(
f
)
<
(0
.
1
−
56)
×
10
−
8
erg cm
−
2
s
−
1
Hz
−
1
(
f/
25 Hz)
α
−
1
depending on the sky location Θ and the spectral power in-
dex
α
. For extended sources, we report upper limits on the fractional gravitational wave energy
density required to close the Universe of Ω(
f,
Θ)
<
(0
.
39
−
7
.
6)
×
10
−
8
sr
−
1
(
f/
25 Hz)
α
depending on
Θ and
α
. Directed searches for narrowband gravitational waves from astrophysically interesting ob-
jects (Scorpius X-1, Supernova 1987 A, and the Galactic Center) yield median frequency-dependent
limits on strain amplitude of
h
0
<
(6
.
7
,
5
.
5
,
and 7
.
0)
×
10
−
25
respectively, at the most sensitive
detector frequencies between 130 – 175 Hz. This represents a mean improvement of a factor of 2
across the band compared to previous searches of this kind for these sky locations, considering the
different quantities of strain constrained in each case.
Introduction.
—A stochastic gravitational-wave back-
ground (SGWB) is expected from a variety of mech-
anisms [1–5]. Given the recent observations of binary
black hole mergers GW150914 and GW151226 [6, 7], we
expect the SGWB to be nearly isotropic [8] and domi-
nated [9] by compact binary coalescences [10–12]. The
LIGO and Virgo Collaborations have pursued the search
for an isotropic stochastic background from LIGO’s first
observational run [13]. Here, we adopt an eyes-wide-open
philosophy and relax the assumption of isotropy in order
to allow for the greater range of possible signals. We
search for an anisotropic background, which could indi-
cate a richer, more interesting cosmology than current
models. We present the results of a generalized search
for a stochastic signal with an arbitrary angular distri-
bution mapped over all directions in the sky.
Our search has three components. First, we utilize a
broadband radiometer analysis [14, 15], optimized for de-
tecting a small number of resolvable point sources. This
method is not applicable to extended sources. Second,
we employ a spherical harmonic decomposition [16, 17],
which can be employed for point sources but is better
suited to extended sources. Last, we carry out a nar-
rowband radiometer search directed at the sky position
of three astrophysically interesting objects: Scorpius X-1
(Sco X-1) [18, 19], Supernova 1987 A (SN 1987A) [20, 21],
and the Galactic Center (GC) [22].
These three search methods are capable of detecting a
wide range of possible signals with only minimal assump-
tions about the signal morphology. We find no evidence
of persistent gravitational waves, and set limits on broad-
band emission of gravitational waves as a function of sky
position. We also set narrowband limits as a function of
frequency for the three selected sky positions.
Data.
—We analyze data from Advanced LIGO’s 4 km
detectors in Hanford, WA (H1) and Livingston, LA (L1)
during the first observing run (O1), from 15:00 UTC,
Sep 18, 2015 – 16:00 UTC, Jan 12, 2016. During O1,
the detectors reached an instantaneous strain sensitivity
of 7
×
10
−
24
Hz
−
1
/
2
in the most sensitive region between
100 – 300 Hz , and collected 49.67 days of coincident H1L1
data. The O1 observing run saw the first direct detection
of gravitational waves and the first direct observation of
merging black holes [6, 7].
For our analysis, the time-series data are down-
sampled to 4096 Hz from 16 kHz, and divided into 192 s,
50% overlapping, Hann-windowed segments, which are
high-pass filtered with a 16th order Butterworth digital
filter with knee frequency of 11 Hz (following [13, 23]).
We apply data quality cuts in the time domain in or-
der to remove segments associated with instrumental ar-
tifacts and hardware injections used for signal valida-
tion [24, 25]. We also exclude segments containing known
gravitational-wave signals. Finally, we apply a standard
non-stationarity cut (see, e.g., [26]), to eliminate seg-
ments that do not behave as Gaussian noise. These cuts
remove 35% of the data. With all vetoes applied, the
total live time for 192 s segments is 29.85 days.
The data segments are Fourier transformed and coarse-
grained to produce power spectra with a resolution
7
of 1
/
32 Hz. This is a finer frequency resolution than
the 1
/
4 Hz used in previous LIGO/Virgo stochastic
searches [15, 17] in order to remove many finely spaced in-
strumental lines occurring at low frequencies. Frequency
bins associated with known instrumental artifacts includ-
ing suspension violin modes [27], calibration lines, elec-
tronic lines, correlated combs, and signal injections of
persistent, monochromatic, gravitational waves are not
included in the analysis. These frequency domain cuts
remove 21% of the observing band. For a detailed de-
scription of data quality studies performed for this anal-
ysis, see the supplement [28] of [13].
The broadband searches include frequencies from 20 –
500 Hz which more than cover the regions of 99% sensi-
tivity for each of the spectral bands (see Table 1 of [13]).
The narrowband analysis covers the full 20 – 1726 Hz
band.
Method.
— The main goal of a stochastic search is to
estimate the fractional contribution of the energy density
in gravitational waves Ω
gw
to the total energy density
needed to close the Universe
ρ
c
. This is defined by
Ω
gw
(
f
) =
f
ρ
c
dρ
gw
df
(1)
where
f
is frequency and
dρ
gw
represents the energy den-
sity between
f
and
f
+
df
[29]. For a stationary and
unpolarized signal,
ρ
gw
can be factored into an angular
power
P
(Θ) and a spectral shape
H
(
f
) [30], such that
Ω
gw
(
f
) =
2
π
2
3
H
2
0
f
3
H
(
f
)
∫
d
Θ
P
(Θ)
,
(2)
with Hubble constant
H
0
= 68 km s
−
1
Mpc
−
1
from [31].
The angular power
P
(Θ) represents the gravitational
wave power at each point in the sky. To express this
in terms of the fractional energy density, we define the
energy density spectrum as a function of sky position
Ω(
f,
Θ) =
2
π
2
3
H
2
0
f
3
H
(
f
)
P
(Θ)
.
(3)
We define a similar quantity for the energy flux, where
F
(
f,
Θ) =
c
3
π
4
G
f
2
H
(
f
)
P
(Θ)
(4)
has units of erg cm
−
2
s
−
1
Hz
−
1
sr
−
1
[15, 16],
c
is the
speed of light and
G
is Newton’s gravitational constant.
Point sources versus extended sources.
—We employ
two different methods to estimate
P
(Θ) based on the
cross-correlation of data streams from a pair of detectors
[17, 29]. The radiometer method [14, 15] assumes that
the cross-correlation signal is dominated by a small num-
ber of resolvable point sources. The point source power
is given by
P
Θ
0
and the angular power spectrum is then
P
(Θ)
≡P
Θ
0
δ
2
(Θ
,
Θ
0
)
.
(5)
Although the radiometer method provides the opti-
mal method for detecting resolvable point sources, it
is not well-suited for describing diffuse or extended
sources, which may have an arbitrary angular distribu-
tion. Hence, we also implement a complementary spher-
ical harmonic decomposition (SHD) algorithm, in which
the sky map is decomposed into components
Y
lm
(Θ) with
coefficients
P
lm
[16]:
P
(Θ)
≡
∑
lm
P
lm
Y
lm
(Θ)
.
(6)
Here, the sum over
l
runs from 0 to
l
max
and
−
l
≤
m
≤
l
.
We discuss the choice of
l
max
below. While the SHD algo-
rithm has comparably worse sensitivity to point sources
than the radiometer algorithm, it accounts for the detec-
tor response, producing more accurate sky maps.
Spectral models.
—In both the radiometer algorithm
and the spherical harmonic decomposition algorithm, we
must choose a spectral shape
H
(
f
). We model the spec-
tral dependence of Ω
gw
(
f
) as a power law:
H
(
f
) =
(
f
f
ref
)
α
−
3
,
(7)
where
f
ref
is an arbitrary reference frequency and
α
is
the spectral index (see also [13]). The spectral model
will also affect the angular power spectrum, so
P
(Θ) is
implicitly a function of
α
.
We can rewrite the energy density map Ω(
f,
Θ) to em-
phasize the spectral properties, such that
Ω(
f,
Θ) = Ω
α
(Θ)
(
f
f
ref
)
α
,
(8)
where
Ω
α
(Θ) =
2
π
2
3
H
2
0
f
3
ref
P
(Θ)
(9)
has units of fractional energy density per steradian
Ω
gw
sr
−
1
.
The spherical harmonic analysis presents
skymaps of Ω
α
(Θ). Note that when
P
(Θ) =
P
00
(the
monopole moment), we recover a measurement for the
energy density of the isotropic gravitational wave back-
ground. Similarly, the gravitational wave energy flux can
be expressed as
F
(
f,
Θ) =
F
α
(Θ)
(
f
f
ref
)
α
−
1
,
(10)
where
F
α
(Θ) =
c
3
π
4
G
f
2
ref
P
(Θ)
.
(11)
In the radiometer case we calculate the flux in each di-
rection
F
α,
Θ
0
=
c
3
π
4
G
f
2
ref
P
Θ
0
,
(12)
8
which is obtained by integrating Equation 11 over
the sphere for the point-source signal model de-
scribed in Equation 5.
This quantity has units of
erg cm
−
2
s
−
1
Hz
−
1
.
Following [13, 32], we choose
f
ref
=
25 Hz, corresponding to the most sensitive frequency in
the spectral band for a stochastic search with the Ad-
vanced LIGO network at design sensitivity.
We consider three spectral indices:
α
= 0, correspond-
ing to a flat energy density spectrum (expected from
models of a cosmological background),
α
= 2
/
3, cor-
responding to the expected shape from a population of
compact binary coalescences, and
α
= 3, corresponding
to a flat strain power spectral density spectrum [17, 32].
The different spectral models are summarized in Table I.
Cross Correlation.
— A stochastic background would
induce low-level correlation between the two LIGO de-
tectors. Although the signal is expected to be buried in
the detector noise, the cross-correlation signal-to-noise
ratio (SNR) grows with the square root of integration
time [29]. The cross correlation between two detectors,
with (one-sided strain) power spectral density
P
i
(
f,t
) for
detector
i
, is encoded in what is known as “the dirty
map” [16]:
X
ν
=
∑
ft
γ
∗
ν
(
f,t
)
H
(
f
)
P
1
(
f,t
)
P
2
(
f,t
)
C
(
f,t
)
.
(13)
Here,
ν
is an index, which can refer to either individual
points on the sky (the pixel basis) or different
lm
indices
(the spherical harmonic basis). The variable
C
(
f,t
) is
the cross-power spectral density measured between the
two LIGO detectors at some segment time
t
. The sum
runs over all segment times and all frequency bins. The
variable
γ
ν
(
f,t
) is a generalization of the overlap reduc-
tion function, which is a function of the separation and
relative orientation between the detectors, and charac-
terizes the frequency response of the detector pair [33];
see [16] for an exact definition.
We can think of
X
ν
as a sky map representation of
the raw cross-correlation measurement before deconvolv-
ing the detector response. The associated uncertainty is
encoded in the Fisher matrix:
Γ
μν
=
∑
ft
γ
∗
μ
(
f,t
)
H
2
(
f
)
P
1
(
f,t
)
P
2
(
f,t
)
γ
ν
(
f,t
)
,
(14)
where
∗
denotes complex conjugation.
Once
X
ν
and Γ
μν
are calculated, we have the ingredi-
ents to calculate both the radiometer map and the SHD
map. However, the inversion of Γ
μν
is required to cal-
culate the maximum likelihood estimators of GW power
ˆ
P
μ
= Γ
−
1
μν
X
ν
[16]. For the radiometer, the correlations
between neighbouring pixels can be ignored. The ra-
diometer map is given by
ˆ
P
Θ
=(Γ
ΘΘ
)
−
1
X
Θ
σ
rad
Θ
=(Γ
ΘΘ
)
−
1
/
2
,
(15)
where the standard deviation
σ
rad
Θ
is the uncertainty as-
sociated with the point source amplitude estimator
ˆ
P
Θ
,
and Γ
ΘΘ
is a diagonal entry of the Fisher matrix for a
pointlike signal. For the SHD analysis, the full Fisher
matrix Γ
μν
must be taken into account, which includes
singular eigenvalues associated with modes to which the
detector pair is insensitive. The inversion of Γ
μν
is simpli-
fied by a singular value decomposition regularization. In
this decomposition, modes associated with the smallest
eigenvalues contribute the least sensitivity to the detec-
tor network. Removing a fraction of the lowest eigen-
modes “regularizes” Γ
μν
without significantly affecting
the sensitivity (see [16]). The estimator for the SHD and
corresponding standard deviation are given by
ˆ
P
lm
=
∑
l
′
m
′
(Γ
−
1
R
)
lm,l
′
m
′
X
l
′
m
′
σ
SHD
lm
=
[
(Γ
−
1
R
)
lm,lm
]
1
/
2
,
(16)
where Γ
R
is the
regularized
Fisher matrix. We remove
1
/
3 of the lowest eigenvalues following [16, 17].
Angular scale.
—In order to carry out the calculation in
Eq. 16, we must determine a suitable angular scale, which
will depend on the angular resolution of the detector net-
work and vary with spectral index
α
. The diffraction-
limited spot size on the sky
θ
(in radians) is given by
θ
=
c
2
df
≈
50 Hz
f
α
,
(17)
where
d
= 3000 km is the separation of the LIGO detec-
tors. The frequency
f
α
corresponds to the most sensitive
frequency in the detector band for a power law with spec-
tral index
α
given the detector noise power spectra [15].
In order to determine
f
α
we find the frequency at which a
power-law with index
α
is tangent to the single-detector
“power-law integrated curve” [34]. The angular resolu-
tion scale is set by the maximum spherical harmonic or-
der
l
max
, which we can express as a function of
α
since
l
max
=
π
θ
≈
πf
α
50Hz
.
(18)
The values of
f
α
,
θ
, and
l
max
for three different values of
α
are shown in Table I. As the spectral index increases,
so does
f
α
, decreasing the angular resolution limit, thus
increasing
l
max
.
Angular power spectra.
—For the SHD map, we calcu-
late the angular power spectra
C
l
, which describe the
angular scale of structure in the clean map, using an un-
biased estimator [16, 17]
ˆ
C
l
≡
1
2
l
+ 1
∑
m
[
|
ˆ
P
lm
|
2
−
(Γ
−
1
R
)
lm,lm
]
.
(19)
Narrowband radiometer.
—The radiometer algorithm
can be applied to the detection of persistent gravitational
waves from narrowband point sources associated with a