Constraints on cosmic strings using data from the first Advanced LIGO observing run

published as:

Constraints on cosmic strings using data from the first

Advanced LIGO observing run

B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration)

Phys. Rev. D 97, 102002 — Published 8 May 2018

B. P. Abbott,1 _{R. Abbott,}1 _{T. D. Abbott,}2 _{F. Acernese,}3,4 _{K. Ackley,}5 _{C. Adams,}6 _{T. Adams,}7 _{P. Addesso,}8 R. X. Adhikari,1 _{V. B. Adya,}9 _{C. Affeldt,}9 _{M. Afrough,}10 _{B. Agarwal,}11 _{M. Agathos,}12 _{K. Agatsuma,}13 N. Aggarwal,14_{O. D. Aguiar,}15 _{L. Aiello,}16,17 _{A. Ain,}18 _{P. Ajith,}19 _{B. Allen,}9,20,21 _{G. Allen,}11 _{A. Allocca,}22,23

P. A. Altin,24 _{A. Amato,}25 _{A. Ananyeva,}1 _{S. B. Anderson,}1 _{W. G. Anderson,}20_{S. Antier,}26 _{S. Appert,}1 K. Arai,1 _{M. C. Araya,}1 _{J. S. Areeda,}27_{N. Arnaud,}26,28 _{K. G. Arun,}29 _{S. Ascenzi,}30,17 _{G. Ashton,}9 _{M. Ast,}31

S. M. Aston,6 _{P. Astone,}32 _{P. Aufmuth,}21 _{C. Aulbert,}9 _{K. AultONeal,}33 _{A. Avila-Alvarez,}27 _{S. Babak,}34 P. Bacon,35_{M. K. M. Bader,}13_{S. Bae,}36_{P. T. Baker,}37,38 _{F. Baldaccini,}39,40 _{G. Ballardin,}28_{S. W. Ballmer,}41 S. Banagiri,42 J. C. Barayoga,1S. E. Barclay,43B. C. Barish,1D. Barker,44F. Barone,3,4 B. Barr,43 L. Barsotti,14

M. Barsuglia,35_{D. Barta,}45_{J. Bartlett,}44 _{I. Bartos,}46 _{R. Bassiri,}47 _{A. Basti,}22,23 _{J. C. Batch,}44 _{C. Baune,}9 M. Bawaj,48,40 _{M. Bazzan,}49,50 _{B. B´ecsy,}51 _{C. Beer,}9 _{M. Bejger,}52 _{I. Belahcene,}26 _{A. S. Bell,}43 _{B. K. Berger,}1 G. Bergmann,9 C. P. L. Berry,53D. Bersanetti,54,55 A. Bertolini,13 J. Betzwieser,6S. Bhagwat,41R. Bhandare,56

I. A. Bilenko,57 _{G. Billingsley,}1_{C. R. Billman,}5 _{J. Birch,}6 _{R. Birney,}58 _{O. Birnholtz,}9 _{S. Biscans,}14_{A. Bisht,}21 M. Bitossi,28,23 _{C. Biwer,}41 _{M. A. Bizouard,}26 _{J. K. Blackburn,}1_{J. Blackman,}59 _{C. D. Blair,}60 _{D. G. Blair,}60 R. M. Blair,44 _{S. Bloemen,}61 _{O. Bock,}9 _{N. Bode,}9 _{M. Boer,}62 _{G. Bogaert,}62 _{A. Bohe,}34 _{F. Bondu,}63 _{R. Bonnand,}7 B. A. Boom,13 _{R. Bork,}1_{V. Boschi,}22,23 _{S. Bose,}64,18 _{Y. Bouffanais,}35 _{A. Bozzi,}28_{C. Bradaschia,}23_{P. R. Brady,}20

V. B. Braginsky∗_,57 _{M. Branchesi,}65,66 _{J. E. Brau,}67 _{T. Briant,}68 _{A. Brillet,}62 _{M. Brinkmann,}9_{V. Brisson,}26 P. Brockill,20 _{J. E. Broida,}69 _{A. F. Brooks,}1_{D. A. Brown,}41 _{D. D. Brown,}53 _{N. M. Brown,}14 _{S. Brunett,}1 C. C. Buchanan,2 _{A. Buikema,}14_{T. Bulik,}70 _{H. J. Bulten,}71,13 _{A. Buonanno,}34,72 _{D. Buskulic,}7 _{C. Buy,}35 R. L. Byer,47_{M. Cabero,}9 _{L. Cadonati,}73 _{G. Cagnoli,}25,74 _{C. Cahillane,}1 _{J. Calder´on Bustillo,}73 _{T. A. Callister,}1 E. Calloni,75,4 _{J. B. Camp,}76 _{M. Canepa,}54,55 _{P. Canizares,}61_{K. C. Cannon,}77_{H. Cao,}78_{J. Cao,}79 _{C. D. Capano,}9 E. Capocasa,35_{F. Carbognani,}28_{S. Caride,}80 _{M. F. Carney,}81_{J. Casanueva Diaz,}26_{C. Casentini,}30,17 _{S. Caudill,}20 M. Cavagli`a,10 _{F. Cavalier,}26 _{R. Cavalieri,}28_{G. Cella,}23 _{C. B. Cepeda,}1_{L. Cerboni Baiardi,}65,66 _{G. Cerretani,}22,23 E. Cesarini,30,17 _{S. J. Chamberlin,}82 _{M. Chan,}43_{S. Chao,}83 _{P. Charlton,}84_{E. Chassande-Mottin,}35_{D. Chatterjee,}20

B. D. Cheeseboro,37,38 _{H. Y. Chen,}85 _{Y. Chen,}59 _{H.-P. Cheng,}5 _{A. Chincarini,}55 _{A. Chiummo,}28 _{T. Chmiel,}81 H. S. Cho,86 _{M. Cho,}72 _{J. H. Chow,}24 _{N. Christensen,}69,62 _{Q. Chu,}60 _{A. J. K. Chua,}12 _{S. Chua,}68 A. K. W. Chung,87 _{S. Chung,}60 _{G. Ciani,}5_{R. Ciolfi,}88,89 _{C. E. Cirelli,}47_{A. Cirone,}54,55 _{F. Clara,}44 _{J. A. Clark,}73

F. Cleva,62 _{C. Cocchieri,}10 _{E. Coccia,}16,17 _{P.-F. Cohadon,}68 _{A. Colla,}90,32 _{C. G. Collette,}91 _{L. R. Cominsky,}92 M. Constancio Jr.,15 _{L. Conti,}50 _{S. J. Cooper,}53_{P. Corban,}6 _{T. R. Corbitt,}2 _{K. R. Corley,}46_{N. Cornish,}93 A. Corsi,80_{S. Cortese,}28_{C. A. Costa,}15_{M. W. Coughlin,}69_{S. B. Coughlin,}94,95 _{J.-P. Coulon,}62_{S. T. Countryman,}46

P. Couvares,1 _{P. B. Covas,}96 _{E. E. Cowan,}73 _{D. M. Coward,}60_{M. J. Cowart,}6_{D. C. Coyne,}1 _{R. Coyne,}80 J. D. E. Creighton,20 _{T. D. Creighton,}97 _{J. Cripe,}2 _{S. G. Crowder,}98 _{T. J. Cullen,}27 _{A. Cumming,}43 L. Cunningham,43 _{E. Cuoco,}28 _{T. Dal Canton,}76 _{S. L. Danilishin,}21,9 _{S. D’Antonio,}17 _{K. Danzmann,}21,9 A. Dasgupta,99 _{C. F. Da Silva Costa,}5 _{V. Dattilo,}28_{I. Dave,}56_{M. Davier,}26 _{D. Davis,}41 _{E. J. Daw,}100 _{B. Day,}73

S. De,41 _{D. DeBra,}47 _{J. Degallaix,}25 _{M. De Laurentis,}75,4 _{S. Del´eglise,}68_{W. Del Pozzo,}53,22,23 _{T. Denker,}9 T. Dent,9 _{V. Dergachev,}34_{R. De Rosa,}75,4 _{R. T. DeRosa,}6 _{R. DeSalvo,}101_{J. Devenson,}58 _{R. C. Devine,}37,38

S. Dhurandhar,18 _{M. C. D´ıaz,}97_{L. Di Fiore,}4 _{M. Di Giovanni,}102,89 _{T. Di Girolamo,}75,4,46 _{A. Di Lieto,}22,23 S. Di Pace,90,32 _{I. Di Palma,}90,32 _{F. Di Renzo,}22,23 _{Z. Doctor,}85 _{V. Dolique,}25 _{F. Donovan,}14 _{K. L. Dooley,}10 S. Doravari,9_{I. Dorrington,}95 _{R. Douglas,}43_{M. Dovale ´Alvarez,}53 _{T. P. Downes,}20 _{M. Drago,}9_{R. W. P. Drever}♯_,1

J. C. Driggers,44_{Z. Du,}79 _{M. Ducrot,}7 _{J. Duncan,}94_{S. E. Dwyer,}44 _{T. B. Edo,}100 _{M. C. Edwards,}69 _{A. Effler,}6 H.-B. Eggenstein,9_{P. Ehrens,}1_{J. Eichholz,}1_{S. S. Eikenberry,}5_{R. A. Eisenstein,}14 _{R. C. Essick,}14_{Z. B. Etienne,}37,38

T. Etzel,1_{M. Evans,}14 _{T. M. Evans,}6_{M. Factourovich,}46_{V. Fafone,}30,17,16 _{H. Fair,}41 _{S. Fairhurst,}95 _{X. Fan,}79 S. Farinon,55 _{B. Farr,}85_{W. M. Farr,}53 _{E. J. Fauchon-Jones,}95_{M. Favata,}103_{M. Fays,}95_{H. Fehrmann,}9 _{J. Feicht,}1 M. M. Fejer,47_{A. Fernandez-Galiana,}14_{I. Ferrante,}22,23 _{E. C. Ferreira,}15_{F. Ferrini,}28_{F. Fidecaro,}22,23 _{I. Fiori,}28 D. Fiorucci,35 _{R. P. Fisher,}41 _{M. Fitz-Axen,}42 _{R. Flaminio,}25,104 _{M. Fletcher,}43_{H. Fong,}105_{P. W. F. Forsyth,}24

S. S. Forsyth,73 _{J.-D. Fournier,}62 _{S. Frasca,}90,32 _{F. Frasconi,}23 _{Z. Frei,}51 _{A. Freise,}53 _{R. Frey,}67 _{V. Frey,}26 E. M. Fries,1 _{P. Fritschel,}14 _{V. V. Frolov,}6 _{P. Fulda,}5,76 _{M. Fyffe,}6 _{H. Gabbard,}9 _{M. Gabel,}106_{B. U. Gadre,}18 S. M. Gaebel,53 _{J. R. Gair,}107 _{L. Gammaitoni,}39 _{M. R. Ganija,}78 _{S. G. Gaonkar,}18 _{F. Garufi,}75,4 _{S. Gaudio,}33 G. Gaur,108_{V. Gayathri,}109 _{N. Gehrels}†_,76 _{G. Gemme,}55 _{E. Genin,}28 _{A. Gennai,}23 _{D. George,}11_{J. George,}56 L. Gergely,110_{V. Germain,}7 _{S. Ghonge,}73_{Abhirup Ghosh,}19 _{Archisman Ghosh,}19,13 _{S. Ghosh,}61,13 _{J. A. Giaime,}2,6

K. D. Giardina,6 _{A. Giazotto}§_,23 _{K. Gill,}33 _{L. Glover,}101 _{E. Goetz,}9 _{R. Goetz,}5_{S. Gomes,}95 _{G. Gonz´alez,}2 J. M. Gonzalez Castro,22,23 _{A. Gopakumar,}111 _{M. L. Gorodetsky,}57 _{S. E. Gossan,}1_{M. Gosselin,}28 _{R. Gouaty,}7

P. Groot,61 _{H. Grote,}9 _{S. Grunewald,}34_{P. Gruning,}26 _{G. M. Guidi,}65,66 _{X. Guo,}79 _{A. Gupta,}82 _{M. K. Gupta,}99 K. E. Gushwa,1_{E. K. Gustafson,}1 _{R. Gustafson,}113_{B. R. Hall,}64 _{E. D. Hall,}1 _{G. Hammond,}43 _{M. Haney,}111

M. M. Hanke,9 _{J. Hanks,}44 _{C. Hanna,}82 _{M. D. Hannam,}95 _{O. A. Hannuksela,}87 _{J. Hanson,}6 _{T. Hardwick,}2 J. Harms,65,66 _{G. M. Harry,}114 _{I. W. Harry,}34 _{M. J. Hart,}43 _{C.-J. Haster,}105 _{K. Haughian,}43 _{J. Healy,}115 A. Heidmann,68 _{M. C. Heintze,}6_{H. Heitmann,}62 _{P. Hello,}26_{G. Hemming,}28 _{M. Hendry,}43_{I. S. Heng,}43 _{J. Hennig,}43

J. Henry,115_{A. W. Heptonstall,}1 _{M. Heurs,}9,21 _{S. Hild,}43 _{D. Hoak,}28 _{D. Hofman,}25 _{K. Holt,}6 _{D. E. Holz,}85 P. Hopkins,95 _{C. Horst,}20 _{J. Hough,}43 _{E. A. Houston,}43 _{E. J. Howell,}60 _{Y. M. Hu,}9 _{E. A. Huerta,}11 _{D. Huet,}26 B. Hughey,33 S. Husa,96 S. H. Huttner,43 T. Huynh-Dinh,6 N. Indik,9 D. R. Ingram,44R. Inta,80G. Intini,90,32 H. N. Isa,43_{J.-M. Isac,}68 _{M. Isi,}1_{B. R. Iyer,}19_{K. Izumi,}44_{T. Jacqmin,}68_{K. Jani,}73_{P. Jaranowski,}116_{S. Jawahar,}117

F. Jim´enez-Forteza,96 _{W. W. Johnson,}2_{D. I. Jones,}118 _{R. Jones,}43 _{R. J. G. Jonker,}13_{L. Ju,}60 _{J. Junker,}9 C. V. Kalaghatgi,95_{V. Kalogera,}94_{S. Kandhasamy,}6_{G. Kang,}36_{J. B. Kanner,}1 _{S. Karki,}67_{K. S. Karvinen,}9

M. Kasprzack,2 _{M. Katolik,}11 _{E. Katsavounidis,}14_{W. Katzman,}6 _{S. Kaufer,}21_{K. Kawabe,}44 _{F. K´ef´elian,}62 D. Keitel,43 _{A. J. Kemball,}11 _{R. Kennedy,}100 _{C. Kent,}95 _{J. S. Key,}119_{F. Y. Khalili,}57 _{I. Khan,}16,17 _{S. Khan,}9

Z. Khan,99 _{E. A. Khazanov,}120_{N. Kijbunchoo,}44 _{Chunglee Kim,}121_{J. C. Kim,}122 _{W. Kim,}78 _{W. S. Kim,}123 Y.-M. Kim,86,121 _{S. J. Kimbrell,}73 _{E. J. King,}78 _{P. J. King,}44 _{R. Kirchhoff,}9 _{J. S. Kissel,}44 _{L. Kleybolte,}31

S. Klimenko,5_{P. Koch,}9 _{S. M. Koehlenbeck,}9 _{S. Koley,}13 _{V. Kondrashov,}1_{A. Kontos,}14_{M. Korobko,}31 W. Z. Korth,1 _{I. Kowalska,}70 _{D. B. Kozak,}1_{C. Kr¨amer,}9 _{V. Kringel,}9_{B. Krishnan,}9 _{A. Kr´olak,}124,125 _{G. Kuehn,}9

P. Kumar,105 _{R. Kumar,}99 _{S. Kumar,}19_{L. Kuo,}83 _{A. Kutynia,}124_{S. Kwang,}20_{B. D. Lackey,}34_{K. H. Lai,}87 M. Landry,44 _{R. N. Lang,}20 _{J. Lange,}115_{B. Lantz,}47 _{R. K. Lanza,}14 _{A. Lartaux-Vollard,}26_{P. D. Lasky,}126 M. Laxen,6 _{A. Lazzarini,}1 _{C. Lazzaro,}50_{P. Leaci,}90,32 _{S. Leavey,}43 _{C. H. Lee,}86_{H. K. Lee,}127 _{H. M. Lee,}121 H. W. Lee,122 K. Lee,43 J. Lehmann,9 A. Lenon,37,38 M. Leonardi,102,89 N. Leroy,26 N. Letendre,7 Y. Levin,126

T. G. F. Li,87 _{A. Libson,}14 _{T. B. Littenberg,}128 _{J. Liu,}60 _{R. K. L. Lo,}87 _{N. A. Lockerbie,}117 _{L. T. London,}95 J. E. Lord,41 _{M. Lorenzini,}16,17 _{V. Loriette,}129_{M. Lormand,}6 _{G. Losurdo,}23 _{J. D. Lough,}9,21 _{C. O. Lousto,}115 G. Lovelace,27H. L¨uck,21,9 D. Lumaca,30,17 A. P. Lundgren,9 R. Lynch,14 Y. Ma,59S. Macfoy,58 B. Machenschalk,9

M. MacInnis,14 _{D. M. Macleod,}2 _{I. Maga˜na Hernandez,}87 _{F. Maga˜na-Sandoval,}41 _{L. Maga˜na Zertuche,}41 R. M. Magee,82 _{E. Majorana,}32_{I. Maksimovic,}129 _{N. Man,}62 _{V. Mandic,}42 _{V. Mangano,}43 _{G. L. Mansell,}24

M. Manske,20 _{M. Mantovani,}28 _{F. Marchesoni,}48,40 _{F. Marion,}7 _{S. M´arka,}46 _{Z. M´arka,}46 _{C. Markakis,}11 A. S. Markosyan,47_{E. Maros,}1 _{F. Martelli,}65,66 _{L. Martellini,}62_{I. W. Martin,}43 _{D. V. Martynov,}14 _{K. Mason,}14 A. Masserot,7_{T. J. Massinger,}1 _{M. Masso-Reid,}43 _{S. Mastrogiovanni,}90,32_{A. Matas,}42 _{F. Matichard,}14 _{L. Matone,}46

N. Mavalvala,14 N. Mazumder,64 R. McCarthy,44 D. E. McClelland,24 S. McCormick,6 L. McCuller,14 S. C. McGuire,130_{G. McIntyre,}1_{J. McIver,}1 _{D. J. McManus,}24 _{T. McRae,}24 _{S. T. McWilliams,}37,38 _{D. Meacher,}82

G. D. Meadors,34,9 _{J. Meidam,}13 _{E. Mejuto-Villa,}8 _{A. Melatos,}131 _{G. Mendell,}44 _{R. A. Mercer,}20_{E. L. Merilh,}44 M. Merzougui,62 _{S. Meshkov,}1_{C. Messenger,}43 _{C. Messick,}82 _{R. Metzdorff,}68 _{P. M. Meyers,}42 _{F. Mezzani,}32,90

H. Miao,53 _{C. Michel,}25 _{H. Middleton,}53 _{E. E. Mikhailov,}132 _{L. Milano,}75,4 _{A. L. Miller,}5 _{A. Miller,}90,32 B. B. Miller,94 _{J. Miller,}14 _{M. Millhouse,}93 _{O. Minazzoli,}62 _{Y. Minenkov,}17 _{J. Ming,}34 _{C. Mishra,}133 _{S. Mitra,}18

V. P. Mitrofanov,57_{G. Mitselmakher,}5 _{R. Mittleman,}14 _{A. Moggi,}23_{M. Mohan,}28 _{S. R. P. Mohapatra,}14 M. Montani,65,66 _{B. C. Moore,}103_{C. J. Moore,}12_{D. Moraru,}44_{G. Moreno,}44 _{S. R. Morriss,}97_{B. Mours,}7 C. M. Mow-Lowry,53 _{G. Mueller,}5 _{A. W. Muir,}95 _{Arunava Mukherjee,}9 _{D. Mukherjee,}20 _{S. Mukherjee,}97 N. Mukund,18 _{A. Mullavey,}6 _{J. Munch,}78 _{E. A. M. Muniz,}41 _{P. G. Murray,}43 _{K. Napier,}73_{I. Nardecchia,}30,17 L. Naticchioni,90,32 _{R. K. Nayak,}134 _{G. Nelemans,}61,13 _{T. J. N. Nelson,}6 _{M. Neri,}54,55 _{M. Nery,}9 _{A. Neunzert,}113 J. M. Newport,114_{G. Newton}‡_,43 _{K. K. Y. Ng,}87 _{T. T. Nguyen,}24 _{D. Nichols,}61 _{A. B. Nielsen,}9 _{S. Nissanke,}61,13 A. Nitz,9 _{A. Noack,}9 _{F. Nocera,}28_{D. Nolting,}6 _{M. E. N. Normandin,}97 _{L. K. Nuttall,}41 _{J. Oberling,}44_{E. Ochsner,}20

E. Oelker,14 _{G. H. Ogin,}106 _{J. J. Oh,}123_{S. H. Oh,}123 _{F. Ohme,}9 _{M. Oliver,}96 _{P. Oppermann,}9_{Richard J. Oram,}6 B. O’Reilly,6 _{R. Ormiston,}42_{L. F. Ortega,}5 _{R. O’Shaughnessy,}115_{D. J. Ottaway,}78_{H. Overmier,}6_{B. J. Owen,}80

A. E. Pace,82 _{J. Page,}128_{M. A. Page,}60_{A. Pai,}109 _{S. A. Pai,}56_{J. R. Palamos,}67_{O. Palashov,}120_{C. Palomba,}32 A. Pal-Singh,31_{H. Pan,}83_{B. Pang,}59 _{P. T. H. Pang,}87_{C. Pankow,}94_{F. Pannarale,}95 _{B. C. Pant,}56_{F. Paoletti,}23

A. Paoli,28 _{M. A. Papa,}34,20,9 _{H. R. Paris,}47 _{W. Parker,}6 _{D. Pascucci,}43 _{A. Pasqualetti,}28 _{R. Passaquieti,}22,23 D. Passuello,23 _{B. Patricelli,}135,23 _{B. L. Pearlstone,}43_{M. Pedraza,}1_{R. Pedurand,}25,136 _{L. Pekowsky,}41 _{A. Pele,}6

S. Penn,137_{C. J. Perez,}44 _{A. Perreca,}1,102,89_{L. M. Perri,}94 _{H. P. Pfeiffer,}105 _{M. Phelps,}43 _{O. J. Piccinni,}90,32 M. Pichot,62 _{F. Piergiovanni,}65,66 _{V. Pierro,}8 _{G. Pillant,}28 _{L. Pinard,}25 _{I. M. Pinto,}8 _{M. Pitkin,}43

R. Poggiani,22,23 _{P. Popolizio,}28_{E. K. Porter,}35_{A. Post,}9_{J. Powell,}43_{J. Prasad,}18_{J. W. W. Pratt,}33_{V. Predoi,}95 T. Prestegard,20 _{M. Prijatelj,}9 _{M. Principe,}8 _{S. Privitera,}34 _{R. Prix,}9 _{G. A. Prodi,}102,89 _{L. G. Prokhorov,}57

E. A. Quintero,1 _{R. Quitzow-James,}67_{F. J. Raab,}44 _{D. S. Rabeling,}24 _{H. Radkins,}44 _{P. Raffai,}51 _{S. Raja,}56 C. Rajan,56 _{M. Rakhmanov,}97 _{K. E. Ramirez,}97 _{P. Rapagnani,}90,32 _{V. Raymond,}34 _{M. Razzano,}22,23 _{J. Read,}27

T. Regimbau,62 _{L. Rei,}55 _{S. Reid,}58 _{D. H. Reitze,}1,5 _{H. Rew,}132 _{S. D. Reyes,}41 _{F. Ricci,}90,32 _{P. M. Ricker,}11 S. Rieger,9 _{K. Riles,}113 _{M. Rizzo,}115 _{N. A. Robertson,}1,43 _{R. Robie,}43 _{F. Robinet,}26 _{A. Rocchi,}17 _{L. Rolland,}7 J. G. Rollins,1 _{V. J. Roma,}67 _{J. D. Romano,}97 _{R. Romano,}3,4 _{C. L. Romel,}44 _{J. H. Romie,}6_{D. Rosi´nska,}138,52 M. P. Ross,139 _{S. Rowan,}43_{A. R¨udiger,}9 _{P. Ruggi,}28 _{K. Ryan,}44 _{S. Sachdev,}1 _{T. Sadecki,}44_{L. Sadeghian,}20 M. Sakellariadou,140 _{L. Salconi,}28 _{M. Saleem,}109_{F. Salemi,}9 _{A. Samajdar,}134 _{L. Sammut,}126 _{L. M. Sampson,}94

E. J. Sanchez,1V. Sandberg,44B. Sandeen,94 J. R. Sanders,41 B. Sassolas,25P. R. Saulson,41 O. Sauter,113 R. L. Savage,44 _{A. Sawadsky,}21 _{P. Schale,}67 _{J. Scheuer,}94 _{E. Schmidt,}33 _{J. Schmidt,}9 _{P. Schmidt,}1,61 R. Schnabel,31 _{R. M. S. Schofield,}67 _{A. Sch¨onbeck,}31 _{E. Schreiber,}9 _{D. Schuette,}9,21 _{B. W. Schulte,}9 B. F. Schutz,95,9 _{S. G. Schwalbe,}33 _{J. Scott,}43 _{S. M. Scott,}24 _{E. Seidel,}11 _{D. Sellers,}6 _{A. S. Sengupta,}141 D. Sentenac,28_{V. Sequino,}30,17 _{A. Sergeev,}120 _{D. A. Shaddock,}24_{T. J. Shaffer,}44_{A. A. Shah,}128_{M. S. Shahriar,}94

L. Shao,34 _{B. Shapiro,}47 _{P. Shawhan,}72 _{A. Sheperd,}20 _{D. H. Shoemaker,}14 _{D. M. Shoemaker,}73 _{K. Siellez,}73 X. Siemens,20 _{M. Sieniawska,}52_{D. Sigg,}44 _{A. D. Silva,}15 _{A. Singer,}1 _{L. P. Singer,}76 _{A. Singh,}34,9,21 _{R. Singh,}2 A. Singhal,16,32 _{A. M. Sintes,}96 _{B. J. J. Slagmolen,}24_{B. Smith,}6 _{J. R. Smith,}27 _{R. J. E. Smith,}1 _{E. J. Son,}123 J. A. Sonnenberg,20_{B. Sorazu,}43_{F. Sorrentino,}55_{T. Souradeep,}18 _{A. P. Spencer,}43 _{A. K. Srivastava,}99_{A. Staley,}46

D.A. Steer,35 _{M. Steinke,}9 _{J. Steinlechner,}43,31 _{S. Steinlechner,}31 _{D. Steinmeyer,}9,21 _{B. C. Stephens,}20 _{R. Stone,}97 K. A. Strain,43 _{G. Stratta,}65,66 _{S. E. Strigin,}57 _{R. Sturani,}142_{A. L. Stuver,}6 _{T. Z. Summerscales,}143_{L. Sun,}131

S. Sunil,99 _{P. J. Sutton,}95 _{B. L. Swinkels,}28 _{M. J. Szczepa´nczyk,}33 _{M. Tacca,}35_{D. Talukder,}67_{D. B. Tanner,}5 M. T´apai,110_{A. Taracchini,}34 _{J. A. Taylor,}128 _{R. Taylor,}1 _{T. Theeg,}9 _{E. G. Thomas,}53_{M. Thomas,}6_{P. Thomas,}44

K. A. Thorne,6 K. S. Thorne,59 E. Thrane,126S. Tiwari,16,89 V. Tiwari,95 K. V. Tokmakov,117 K. Toland,43 M. Tonelli,22,23 _{Z. Tornasi,}43_{C. I. Torrie,}1 _{D. T¨oyr¨a,}53 _{F. Travasso,}28,40_{G. Traylor,}6_{D. Trifir`o,}10 _{J. Trinastic,}5 M. C. Tringali,102,89 _{L. Trozzo,}144,23 _{K. W. Tsang,}13 _{M. Tse,}14 _{R. Tso,}1 _{D. Tuyenbayev,}97_{K. Ueno,}20 _{D. Ugolini,}145

C. S. Unnikrishnan,111A. L. Urban,1S. A. Usman,95 H. Vahlbruch,21 G. Vajente,1 G. Valdes,97 M. Vallisneri,59 N. van Bakel,13 _{M. van Beuzekom,}13 _{J. F. J. van den Brand,}71,13 _{C. Van Den Broeck,}13_{D. C. Vander-Hyde,}41

L. van der Schaaf,13 _{J. V. van Heijningen,}13 _{A. A. van Veggel,}43_{M. Vardaro,}49,50 _{V. Varma,}59 _{S. Vass,}1 M. Vas´uth,45 _{A. Vecchio,}53 _{G. Vedovato,}50 _{J. Veitch,}53 _{P. J. Veitch,}78 _{K. Venkateswara,}139 _{G. Venugopalan,}1

D. Verkindt,7 _{F. Vetrano,}65,66 _{A. Vicer´e,}65,66 _{A. D. Viets,}20 _{S. Vinciguerra,}53 _{D. J. Vine,}58 _{J.-Y. Vinet,}62 S. Vitale,14 _{T. Vo,}41_{H. Vocca,}39,40 _{C. Vorvick,}44 _{D. V. Voss,}5 _{W. D. Vousden,}53 _{S. P. Vyatchanin,}57 _{A. R. Wade,}1

L. E. Wade,81 M. Wade,81 R. Walet,13 M. Walker,2 L. Wallace,1 S. Walsh,20 G. Wang,16,66 H. Wang,53 J. Z. Wang,82 _{M. Wang,}53_{Y.-F. Wang,}87 _{Y. Wang,}60 _{R. L. Ward,}24 _{J. Warner,}44 _{M. Was,}7 _{J. Watchi,}91 B. Weaver,44 _{L.-W. Wei,}9,21 _{M. Weinert,}9 _{A. J. Weinstein,}1_{R. Weiss,}14_{L. Wen,}60 _{E. K. Wessel,}11 _{P. Weßels,}9

T. Westphal,9 _{K. Wette,}9 _{J. T. Whelan,}115_{B. F. Whiting,}5 _{C. Whittle,}126 _{D. Williams,}43 _{R. D. Williams,}1 A. R. Williamson,115 _{J. L. Willis,}146 _{B. Willke,}21,9 _{M. H. Wimmer,}9,21 _{W. Winkler,}9 _{C. C. Wipf,}1 _{H. Wittel,}9,21

G. Woan,43 _{J. Woehler,}9 _{J. Wofford,}115 _{K. W. K. Wong,}87 _{J. Worden,}44 _{J. L. Wright,}43 _{D. S. Wu,}9 G. Wu,6 _{W. Yam,}14 _{H. Yamamoto,}1 _{C. C. Yancey,}72 _{M. J. Yap,}24 _{Hang Yu,}14 _{Haocun Yu,}14 _{M. Yvert,}7 A. Zadro˙zny,124_{M. Zanolin,}33_{T. Zelenova,}28 _{J.-P. Zendri,}50 _{M. Zevin,}94_{L. Zhang,}1 _{M. Zhang,}132_{T. Zhang,}43 Y.-H. Zhang,115_{C. Zhao,}60_{M. Zhou,}94 _{Z. Zhou,}94_{S. J. Zhu,}34,9 _{X. J. Zhu,}60_{M. E. Zucker,}1,14 _{and J. Zweizig}1

(LIGO Scientific Collaboration and Virgo Collaboration) ∗_{Deceased, March 2016.} ‡_{Deceased, December 2016.} †_Deceased, February 2017. ♯_{Deceased, March 2017.} §_{Deceased, November 2017.}

1

LIGO, California Institute of Technology, Pasadena, CA 91125, USA 2

Louisiana State University, Baton Rouge, LA 70803, USA 3

Universit`a di Salerno, Fisciano, I-84084 Salerno, Italy 4

INFN, Sezione di Napoli, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy 5

University of Florida, Gainesville, FL 32611, USA 6

LIGO Livingston Observatory, Livingston, LA 70754, USA 7

Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Universit´e Savoie Mont Blanc, CNRS/IN2P3, F-74941 Annecy, France

8

University of Sannio at Benevento, I-82100 Benevento, Italy and INFN, Sezione di Napoli, I-80100 Napoli, Italy 9

Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik, D-30167 Hannover, Germany 10

The University of Mississippi, University, MS 38677, USA 11

NCSA, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 12

13

Nikhef, Science Park, 1098 XG Amsterdam, The Netherlands 14

LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 15

Instituto Nacional de Pesquisas Espaciais, 12227-010 S˜ao Jos´e dos Campos, S˜ao Paulo, Brazil 16

Gran Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy 17

INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy 18

Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India 19

International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India 20

University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA 21

Leibniz Universit¨at Hannover, D-30167 Hannover, Germany 22

Universit`a di Pisa, I-56127 Pisa, Italy 23

INFN, Sezione di Pisa, I-56127 Pisa, Italy 24

OzGrav, Australian National University, Canberra, Australian Capital Territory 0200, Australia 25

Laboratoire des Mat´eriaux Avanc´es (LMA), CNRS/IN2P3, F-69622 Villeurbanne, France 26

LAL, Univ. Paris-Sud, CNRS/IN2P3, Universit´e Paris-Saclay, F-91898 Orsay, France 27

California State University Fullerton, Fullerton, CA 92831, USA 28

European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy 29

Chennai Mathematical Institute, Chennai 603103, India 30

Universit`a di Roma Tor Vergata, I-00133 Roma, Italy 31

Universit¨at Hamburg, D-22761 Hamburg, Germany 32

INFN, Sezione di Roma, I-00185 Roma, Italy 33

Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA 34

Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik, D-14476 Potsdam-Golm, Germany 35

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 36

Korea Institute of Science and Technology Information, Daejeon 34141, Korea 37

West Virginia University, Morgantown, WV 26506, USA 38

Center for Gravitational Waves and Cosmology, West Virginia University, Morgantown, WV 26505, USA

39

Universit`a di Perugia, I-06123 Perugia, Italy 40

INFN, Sezione di Perugia, I-06123 Perugia, Italy 41

Syracuse University, Syracuse, NY 13244, USA 42

University of Minnesota, Minneapolis, MN 55455, USA 43

SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom 44

LIGO Hanford Observatory, Richland, WA 99352, USA 45

Wigner RCP, RMKI, H-1121 Budapest, Konkoly Thege Mikl´os ´ut 29-33, Hungary 46

Columbia University, New York, NY 10027, USA 47

Stanford University, Stanford, CA 94305, USA 48

Universit`a di Camerino, Dipartimento di Fisica, I-62032 Camerino, Italy 49

Universit`a di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova, Italy 50

INFN, Sezione di Padova, I-35131 Padova, Italy 51

MTA E¨otv¨os University, “Lendulet” Astrophysics Research Group, Budapest 1117, Hungary 52

Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, 00-716, Warsaw, Poland 53

University of Birmingham, Birmingham B15 2TT, United Kingdom 54

Universit`a degli Studi di Genova, I-16146 Genova, Italy 55

INFN, Sezione di Genova, I-16146 Genova, Italy 56

RRCAT, Indore MP 452013, India 57

Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia 58

SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom 59

Caltech CaRT, Pasadena, CA 91125, USA 60

OzGrav, University of Western Australia, Crawley, Western Australia 6009, Australia 61

Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands 62

Artemis, Universit´e Cˆote d’Azur, Observatoire Cˆote d’Azur, CNRS, CS 34229, F-06304 Nice Cedex 4, France 63

Institut de Physique de Rennes, CNRS, Universit´e de Rennes 1, F-35042 Rennes, France 64

Washington State University, Pullman, WA 99164, USA 65

Universit`a degli Studi di Urbino ’Carlo Bo’, I-61029 Urbino, Italy 66

INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy 67

University of Oregon, Eugene, OR 97403, USA 68

Laboratoire Kastler Brossel, UPMC-Sorbonne Universit´es, CNRS, ENS-PSL Research University, Coll`ege de France, F-75005 Paris, France

69

Carleton College, Northfield, MN 55057, USA 70

71

VU University Amsterdam, 1081 HV Amsterdam, The Netherlands 72

University of Maryland, College Park, MD 20742, USA 73

Center for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA 74

Universit´e Claude Bernard Lyon 1, F-69622 Villeurbanne, France 75

Universit`a di Napoli ’Federico II’, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy 76

NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 77

RESCEU, University of Tokyo, Tokyo, 113-0033, Japan. 78

OzGrav, University of Adelaide, Adelaide, South Australia 5005, Australia 79

Tsinghua University, Beijing 100084, China 80

Texas Tech University, Lubbock, TX 79409, USA 81

Kenyon College, Gambier, OH 43022, USA 82

The Pennsylvania State University, University Park, PA 16802, USA 83

National Tsing Hua University, Hsinchu City, 30013 Taiwan, Republic of China 84

Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia 85

University of Chicago, Chicago, IL 60637, USA 86

Pusan National University, Busan 46241, Korea 87

The Chinese University of Hong Kong, Shatin, NT, Hong Kong 88

INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy 89

INFN, Trento Institute for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy 90

Universit`a di Roma ’La Sapienza’, I-00185 Roma, Italy 91

Universit´e Libre de Bruxelles, Brussels 1050, Belgium 92

Sonoma State University, Rohnert Park, CA 94928, USA 93

Montana State University, Bozeman, MT 59717, USA 94

Center for Interdisciplinary Exploration & Research in Astrophysics (CIERA), Northwestern University, Evanston, IL 60208, USA

95

Cardiff University, Cardiff CF24 3AA, United Kingdom 96

Universitat de les Illes Balears, IAC3—IEEC, E-07122 Palma de Mallorca, Spain 97

The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA 98

Bellevue College, Bellevue, WA 98007, USA 99

Institute for Plasma Research, Bhat, Gandhinagar 382428, India 100

The University of Sheffield, Sheffield S10 2TN, United Kingdom 101

California State University, Los Angeles, 5151 State University Dr, Los Angeles, CA 90032, USA 102

Universit`a di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy 103

Montclair State University, Montclair, NJ 07043, USA 104

National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 105

Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada 106

Whitman College, 345 Boyer Avenue, Walla Walla, WA 99362 USA 107

School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom 108

University and Institute of Advanced Research, Gandhinagar Gujarat 382007, India 109

IISER-TVM, CET Campus, Trivandrum Kerala 695016, India 110

University of Szeged, D´om t´er 9, Szeged 6720, Hungary 111

Tata Institute of Fundamental Research, Mumbai 400005, India 112

INAF, Osservatorio Astronomico di Capodimonte, I-80131, Napoli, Italy 113

University of Michigan, Ann Arbor, MI 48109, USA 114

American University, Washington, D.C. 20016, USA 115

Rochester Institute of Technology, Rochester, NY 14623, USA 116

University of Bia lystok, 15-424 Bia lystok, Poland 117

SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom 118

University of Southampton, Southampton SO17 1BJ, United Kingdom 119

University of Washington Bothell, 18115 Campus Way NE, Bothell, WA 98011, USA 120

Institute of Applied Physics, Nizhny Novgorod, 603950, Russia 121

Seoul National University, Seoul 08826, Korea 122

Inje University Gimhae, South Gyeongsang 50834, Korea 123

National Institute for Mathematical Sciences, Daejeon 34047, Korea 124

NCBJ, 05-400 ´Swierk-Otwock, Poland 125

Institute of Mathematics, Polish Academy of Sciences, 00656 Warsaw, Poland 126

OzGrav, School of Physics & Astronomy, Monash University, Clayton 3800, Victoria, Australia 127

Hanyang University, Seoul 04763, Korea 128

NASA Marshall Space Flight Center, Huntsville, AL 35811, USA 129

ESPCI, CNRS, F-75005 Paris, France 130

Southern University and A&M College, Baton Rouge, LA 70813, USA 131

132

College of William and Mary, Williamsburg, VA 23187, USA 133

Indian Institute of Technology Madras, Chennai 600036, India 134

IISER-Kolkata, Mohanpur, West Bengal 741252, India 135

Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy 136

Universit´e de Lyon, F-69361 Lyon, France 137

Hobart and William Smith Colleges, Geneva, NY 14456, USA 138

Janusz Gil Institute of Astronomy, University of Zielona G´ora, 65-265 Zielona G´ora, Poland 139

University of Washington, Seattle, WA 98195, USA 140

King’s College London, University of London, London WC2R 2LS, United Kingdom 141

Indian Institute of Technology, Gandhinagar Ahmedabad Gujarat 382424, India 142

International Institute of Physics, Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil 143

Andrews University, Berrien Springs, MI 49104, USA 144

Universit`a di Siena, I-53100 Siena, Italy 145

Trinity University, San Antonio, TX 78212, USA 146

Abilene Christian University, Abilene, TX 79699, USA

Cosmic strings are topological defects which can be formed in GUT-scale phase transitions in the early universe. They are also predicted to form in the context of string theory. The main mechanism for a network of Nambu-Goto cosmic strings to lose energy is through the production of loops and the subsequent emission of gravitational waves, thus offering an experimental signature for the existence of cosmic strings. Here we report on the analysis conducted to specifically search for gravitational-wave bursts from cosmic string loops in the data of Advanced LIGO 2015-2016 observing run (O1). No evidence of such signals was found in the data, and as a result we set upper limits on the cosmic string parameters for three recent loop distribution models. In this paper, we initially derive constraints on the string tension Gµ and the intercommutation probability, using not only the burst analysis performed on the O1 data set, but also results from the previously published LIGO stochastic O1 analysis, pulsar timing arrays, cosmic microwave background and Big-Bang nucleosynthesis experiments. We show that these data sets are complementary in that they probe gravitational waves produced by cosmic string loops during very different epochs. Finally, we show that the data sets exclude large parts of the parameter space of the three loop distribution models we consider.

PACS numbers: 11.27.+d, 98.80.Cq, 11.25.-w

I. INTRODUCTION

The recent observation of gravitational waves [1] (GWs) has started a new era in astronomy [2, 3]. In the coming years Advanced LIGO [4] and Advanced Virgo [5] will be targeting a wide variety of GW sources [6]. Some of these potential sources could yield new physics and information about the universe at its earliest moments. This would be the case for the observation of GWs from cosmic strings, which are one-dimensional topological de-fects, formed after a spontaneous symmetry phase tran-sition characterized by a vacuum manifold with non-contractible loops. Cosmic strings were first introduced by Kibble [7], (for a review see for instance [8–10]). They can be generically produced in the context of Grand Uni-fied Theories [11]. Linear-type topological defects of dif-ferent forms should leave a variety of observational signa-tures, opening up a fascinating window to fundamental physics at very high energy scales. In particular, they should lens distant galaxies [12–14], produce high energy cosmic rays [15], lead to anisotropies in the cosmic mi-crowave background [16, 17], and produce GWs [18, 19]. A network of cosmic strings is primarily characterized by the string tension Gµ (c = 1), where G is Newton’s constant and µ the mass per unit length. The existence of cosmic strings can be tested using the cosmic microwave background (CMB) measurements. Confronting

experi-mental CMB data with numerical simulations of cosmic string networks [20–23], the string tension is constrained to be smaller than a few 10−7_.

Cosmic superstrings are coherent macroscopic states of fundamental superstrings (F-strings) and also D-branes extended in one macroscopic direction (D-strings). They are predicted in superstring inspired inflationary models with spacetime-wrapping D-branes [24, 25]. For cosmic superstrings, one must introduce another parameter to account for the fact that they interact probabilistically. In [26], it is suggested that this intercommutation proba-bility p must take values between 10−1_{and 1 for D-strings} and between 10−3_{and 1 for F-strings. In this paper, we} will refer to both topological strings and superstrings as “strings”, and parameterize them by p and Gµ.

Cosmic string parameters can also be accessed through GWs. Indeed, the dynamics of the network is driven by the formation of loops and the emission of GWs. In particular, cusps and kinks propagating on string loops are expected to produce powerful bursts of GWs. The superposition of these bursts gives rise to a stochastic background which can be probed over a large range of frequencies by different observations. Historically, the Big-Bang nucleosynthesis (BBN) data provided the first constraints on cosmic strings [27]. It was then surpassed by CMB bounds [28] to then be surpassed more recently by pulsar timing bounds [29]. In this paper, we report

on the search for GW burst signals produced by cos-mic string cusps and kinks using Advanced LIGO data collected between September 12, 2015 06:00 UTC and January 19, 2016 17:00 UTC [30], offering a total of Tobs = 4 163 421 s (∼ 48.2 days) of coincident data be-tween the two LIGO detectors. Moreover, combining the result from the stochastic GW background search previously published in [31], we test and constrain cos-mic string models. While the LIGO O1 burst limit re-mains weak, the stochastic bound now surpasses the BBN bound for the first time and is competitive with the CMB bound across much of the parameter space.

We will place constraints on the most up-to-date string loop distributions. In particular, we select three analytic cosmic string models (M = {1, 2, 3}) [8, 32–35] for the number density of string loops, developed in part from numerical simulations of Nambu-Goto string networks (zero thickness strings with intercommutation probabil-ity equal to unprobabil-ity), in a Friedman-Lemaˆıtre-Robertson-Walker geometry. These models are more fully described in Sec. II where their fundamental differences are also dis-cussed. Sec. III presents an overview of the experimental data sets which are used to constrain the cosmic string parameters. Finally, the resulting limits are discussed in Sec. IV.

II. COSMIC STRING MODELS

We constrain three different models of cosmic strings indexed by M . Common to all these models is the as-sumption that the width of the strings is negligible com-pared to the size of the horizon, so that the string dy-namics is given by the Nambu-Goto action. A further input is the strings intercommutation probability p. For field theory strings, and in particular U (1) Abelian-Higgs strings in the Bogomol’nyi–Prasad–Sommerfield limit [8], intercommutation occurs with effectively unit probability [36, 37], p = 1. That is, when two super-horizon (infi-nite) strings intersect, they always swap partners; and if a string intersects itself, it therefore chops off a (sub-horizon) loop. The latter can also result from string-string intersections at two points, leading to the forma-tion of two new infinite strings and a loop.

Cosmic string loops oscillate periodically in time, emit-ting GWs 1_{. A loop of invariant length ℓ, has} pe-riod T = ℓ/2 and corresponding fundamental frequency ω = 4π/ℓ. As a result it radiates GWs with frequen-cies which are multiples of ω, and decays in a lifetime τ = ℓ/γdwhere [18, 40, 41]

γd≡ ΓGµ with Γ ≃ 50 . (1)

If a loop contains kinks [41–43] (discontinuities on the tangent vector of a string) and cusps (points where the

1

Super-horizon cosmic strings also emit GWs, due to their small-scale structure [19, 38, 39].

string instantaneously reaches the speed of light), these source bursts of beamed GWs [44–46]. The incoherent superposition of these bursts give rise to a stationary and nearly Gaussian stochastic GW background. Occa-sionally, sharp and high-amplitude bursts of GWs stand above this stochastic GW background.

The three models considered here differ in the loop dis-tribution n(ℓ, t)dℓ, namely the number density of cosmic string loops of invariant length between ℓ and ℓ + dℓ at cosmic time t. To determine the consequences of these differences on their GW signal, we work in units of cosmic time t and introduce the dimensionless variables

γ ≡ ℓ/t and F (γ, t) ≡ n(ℓ, t) × t4. (2) We will often refer to γ as the relative size of loops and F as simply the loop distribution. All GWs observed to-day are formed when the string network is in its scaling regime, namely a self-similar, attractor solution in which all the typical length scales in the problem are propor-tional to cosmic time2_.

The models considered here were developed (in part) using numerical simulations of Nambu-Goto strings, for which p = 1. As mentioned above, cosmic superstrings intercommute with probability p < 1. The effect of a reduced intercommutation probability on the loop distri-bution has been studied in [47]. Following this reference we take Fp<1 = F /p 3, leading to an increased density of strings [48] and to an enhancement of various obser-vational signatures.

A. Model M= 1: original large loop distribution The first model we consider is the oldest, developed in [8, 32]. It assumes that, in the scaling regime, all loops chopped off the infinite string network are formed with the same relative size, which we denote by α. At time t, the distribution of loops of length ℓ to ℓ + dℓ contains loops chopped off the infinite string network at earlier times, and diluted by the expansion of the universe and by the emission of GWs. Assuming that loops do not self-intersect once formed, and taking into account that the length of a loop decays at the rate dℓ/dt = −γd, the scaling loop distribution (for γ ≤ α) in the radiation era is given by [8]

F_rad(1)(γ) = Crad (γ + γd)5/2

Θ(α − γ), (3)

2

Scaling breaks down for a short time in the transition between the radiation and matter eras, and similarly in the transition to dark energy domination.

3

In [47] the exponent of the power law behavior was found to be slightly different, namely 0.6. Since our goal here is to highlight the effect of p < 1, we used a simple dependence of 1/p as many others in the litterature have done.

where Θ is the Heaviside function, and the superscript (1) stands for model M = 1. Some of these loops formed in the radiation era can survive into the matter era, mean-ing that in the matter era the loop distribution has two components. Those loops surviving from the radiation era have distribution

Fmat(1),a(γ, t) = Crad (γ + γd)5/2  teq t 1/2 Θ(−γ + β(t)), (4) with teq the time of the radiation to matter transition, and where the lower bound, β(t), is the length in scaling units, of the last loops formed in the radiation era at time teq: β(t) = αteq t − γd  1 − teq t  . (5)

The loops formed in the matter era itself have a distri-bution

Fmat(1),b(γ, t) =

Cmat

(γ + γd)2Θ(α − γ)Θ(γ − β(t)). (6) The normalisation constants Cradand Cmatcannot be de-termined from analytical arguments, but rather are fixed by matching with numerical simulations of Nambu-Goto strings. Following [8, 32]: we set them to

Crad≃ 1.6 , Cmat≃ 0.48 . (7)

Furthermore we shall assume that α ≃ 0.1. The loop distribution in the matter era is thus given by the sum of distributions in Eqs. 4 and 6.

The loop distribution F(1) _{is plotted in Fig. 1 for} dif-ferent redshift values and fixing Gµ at 10−8_{. A} discon-tinuity, visible for low redshift values, results from the radiation-matter transition which is modeled by Heavi-side functions. For t < teq, the loop distribution is en-tirely determined by Eq. 3 and is time independent.

B. ModelM = 2: large loop Nambu-Goto distribution of Blanco-Pillado et al.

Rather than postulating that all loops are formed with a given size αt at time t as in model 1, the loop production function can be determined from numerical simulations. This approach was taken in [33], determining the rate of production of loops of size ℓ and momentum ~p at time t. Armed with this information, n(ℓ, t) is determined analytically as in model 1 with the additional assumption that the momentum dependence of the loop production function is weak so that it can be integrated out.

In the radiation era, the scaling distribution reads F_rad(2)(γ) = 0.18

(γ + γd)5/2

Θ(0.1 − γ), (8)

where the superscript (2) stands for model 2. In the matter era, analogously to above, there are two contri-butions. The loops left over from the radiation era can be

deduced from above, whereas loops formed in the matter era have distribution

Fmat(2),b(γ, t) =

0.27 − 0.45γ0.31 (γ + γd)2

Θ(0.18 − γ)Θ(γ − β(t)), (9) where β(t) is given in Eq. (5) with α = 0.1.

The loop distribution of model 2 is plotted in Fig. 1. Notice that in the radiation era, the distributions in mod-els 1 and 2 take the same functional form, though their normalisation differs by a factor of order 10. In the mat-ter era, the functional form is slightly different and the normalisation is smaller by a factor of order 2. The au-thors of [33] attribute this reduction in the number of loops to two effects: (i) only about 10% of the power is radiated into large loops – indeed, most of it is lost di-rectly into smaller loops which radiate away very quickly; (ii) most of the energy leaving the network goes into loop kinetic energy which is lost to redshifting.

C. Model M= 3: large loop Nambu-Goto distribution of Ringeval et al.

This analytical model was presented in [34], and is based in part on the numerical simulations of [35].

As opposed to model 2, here the (different) numerical simulation is not used to determine the loop production function at time t, but rather the distribution of non-self intersecting loops at time t. The analytical modeling also differs from that of model 2 in that an extra ingredient is added: not only do loops emit GWs — which decreases their length ℓ — but this GW emission back-reacts on the loops. Back-reaction smooths out the loops on the smallest scales (in particular any kinks), thus hindering the formation of smaller loops [43, 49]. Hence, the distri-butions of models 2 and 3 differ for the smallest loops.

Physically, therefore, the model of [34] contains a fur-ther length scale γc, the so-called “gravitational back-reaction scale”, with

γc< γd,

where γd is the gravitational decay scale introduced above. Following the numerical simulation of [35],

γc= Υ(Gµ)1+2χ where Υ ∼ 10 and χ = 1−P/2, (10) with P = 1.41−0.07+0.08   mat, P = 1.60 +0.21 −0.15   rad. (11) The resulting distribution of loops is given in [34]. In this paper, we work with the asymptotic expressions given in section 2.4 of [34], valid in the scaling regime (t ≫ tini). Hence the contribution of those loops formed

in the radiation era, but which persist into the matter era, are neglected. The loop distribution has three distinct regimes with different power-law behaviours, depending on whether the loops are smaller than γc (γ ≤ γc); of

intermediate length (γc ≤ γ ≤ γd); or larger than γd (that is γd ≤ γ ≤ γmax). Here γmax = 1/(1 − ν) is the largest allowed (horizon-sized) loop, in units of cosmic time, where the power-law time evolution of the scale factor of the universe, a ∼ tν_{, is}

ν = 2 3     mat , ν = 1 2     rad . (12)

Hence, γmax= 2, or γmax= 3, depending on whether we are in the radiation-dominated or matter-dominated era, respectively. More explicitly,

• For loops with length scale large compared to γd: F(3)_(γ

d≪ γ < γmax) ≃ C (γ + γd)P +1

. (13)

• For loops with length scale in the range γc< γ ≪ γd: F(3)(γc < γ ≪ γd) ≃ C(3ν − 2χ − 1) 2 − 2χ 1 γd 1 γP . (14) • For loops with length scale smaller than γc the distri-bution is γ independent: F(3)_{(γ ≪ γ} c≪ γd) ≃ C(3ν − 2χ − 1) 2 − 2χ 1 γP c 1 γd . (15) Here, C is given by C = C0(1 − ν)3−P (16) where C0 = 0.09−0.03+0.03  _mat, C0 = 0.21+0.13−0.12  _rad. (17) In the case of large loops (Eq. 13), C normalizes the distribution. In the radiation era where ν = 1/2,

C ∼ 0.08 (radiation)

(a factor of about 20 smaller than model 1), and in the matter era where ν = 2/3,

C ∼ 0.016 (matter)

(a factor of about 30 smaller than model 1).

The three loop regimes are well-visible when plotting the loop distribution: see Fig. 1. Regarding the GW signal, the most significant difference between model 3 and the two previous models is in the very small loop regime (γ ≪ γc). Comparing Eq. 15 with Eq. 4 and Eq. 8, for models 3, 1 and 2 respectively, in the radiation era, we find F(3) F(1,2)    _γ≪γc ∝ (Gµ)−0.74 , (18)

where the proportionality constant is 2.5×10−2_{for model} 1 and approximately ten times larger for model 2. For a typical value of Gµ = 10−8_{, and relative to model 1,} there are ∼ 2×104_{more very small loops in the radiation} era in model 3. As we will see in Sec. III, such a high number of small loops in model 3 will have important consequences in the rate of GW events we can detect and on the amplitude of the stochastic gravitational wave background.

III. CONSTRAINING COSMIC STRINGS

MODELS WITH GW DATA

A. Gravitational waves from cosmic strings GW bursts are emitted by both cusps and kinks on cosmic string loops, the frequency-domain waveform of which was calculated in [44, 45, 50]:

h(ℓ, z, f ) = Aq(ℓ, z)f−qΘ(fh− f )Θ(f − fℓ), (19) where q = 4/3 for cusps, q = 5/3 for kinks, and Aq(ℓ, z) is the signal amplitude produced by a cusp/kink propa-gating on a loop of size ℓ at redshift z. This waveform is linearily polarized and is only valid if the beaming angle θm(ℓ, z, f ) ≡ (g2f (1 + z)ℓ)−1/3< 1. (20) Here g2 is an ignorance factor assumed to be 1 in this work (see [32]). In order to detect the GW, the angle subtended by the line of sight and the cusp/kink on a loop of typical invariant length ℓ at redshift z, must be smaller than θm. This condition then determines the high-frequency cutoff fh in Eq. 19. The low-frequency cutoff fℓ — though in principle determined by the kink amplitude, or by the size of the feature that produces the cusp — is in practice given by the lower end of the GW detector’s sensitive band. The amplitude Aq(ℓ, z) is given by [44]

Aq(ℓ, z) = g1 Gµℓ 2−q

(1 + z)q−1_r(z) , (21)

where the proper distance to the source is given by r(z) = H0−1ϕr(z). Here, H0is the Hubble parameter to-day and ϕr(z) is determined in terms of the cosmological parameters and expressed in Appx. A. Finally g1gathers together a certain number of uncertainties which enter into the calculation of the cusp and kink waveform (in-cluding the amplitude of the cusp/kink, as well as numer-ical factors of order 1, see [32, 44]). We will set g1= 1. In the following, we will use Eq. 21 to conveniently choose 2 variables out of ℓ, z and Aq. Similarly, we will use Eq. 19 to substitute Aq for the strain amplitude h.

For a given loop distribution model M , in the following we use the GW burst rate derived in [32] and recalled in Appx. B: d2_R(M) q dzdh (h, z, f ) = 2NqH0−3ϕV(z) (2 − q)(1 + z)ht4_(z) ×F(M) ℓ(hf q_{, z)} t(z) , t(z)  ×∆q(hfq, z, f ). (22)

The first two lines on the right-hand side give the num-ber of cusp/kink features per unit space-time volume on loops of size ℓ, where Nq is the number of cusps/kinks per oscillation period T = ℓ/2 of the loop. In this pa-per, the number of cusps/kinks per loop oscillation is set

12 − 10 ₁₀−10 ₁₀−8 ₁₀−6 ₁₀−4 ₁₀−2 ₁ γ Loop size 1 4 10 8 10 12 10 16 10 20 10 22 10 Loop distribution Model 1 -2 z=10 2 z=10 z=1 4 z=10 6 z=10 12 − 10 ₁₀−10 ₁₀−8 ₁₀−6 ₁₀−4 ₁₀−2 ₁ γ Loop size 1 4 10 8 10 12 10 16 10 20 10 22 10 Loop distribution Model 2 -2 z=10 2 z=10 z=1 4 z=10 6 z=10 12 − 10 ₁₀−10 ₁₀−8 ₁₀−6 ₁₀−4 ₁₀−2 ₁ γ Loop size 1 4 10 8 10 12 10 16 10 20 10 22 10 Loop distribution Model 3 -2 z=10 2 z=10 z=1 4 z=10 6 z=10

FIG. 1: Loop size distributions predicted by three models: M = 1, 2, 3. For each model, the loop distribution, F (γ, t(z)), is plotted for different redshift values and fixing Gµ at 10−8_.

to 1 although some models [51] suggest that this num-ber can be much larger than one. Cosmic time is given by t(z) = ϕt(z)/H0 and the proper volume element is dV (z) = H0−3ϕV(z)dz where ϕt(z) and ϕV(z) are given in Appx. A. Finally ∆q, which is fully derived in Appx.B, is the fraction of GW events of amplitude Aq that are ob-servable at frequency f and redshift z.

B. Gravitational-wave bursts

We searched the Advanced LIGO O1 data (2015-2016) [30] for individual bursts of GWs from cusps and kinks. The search for cusp signals was previously con-ducted using initial LIGO and Virgo data and no signal was found [52].

For this paper, we use the same analysis pipeline to search for both cusp and kink signals. We perform a Wiener-filter analysis to identify events matching the waveform predicted by the theory [44, 45, 50] and given in Eq. 19. GW events are detected by matching the data to a bank of waveforms parameterized by the high-frequency cutoff fh, with 30 Hz < fh < 4096 Hz. Then resulting events detected at Hanford and at LIGO-Livingston are set in time coincidence to reject detector noise artifacts mimicking cosmic string signals. Finally, a multivariate likelihood ratio [53] is computed to rank co-incident events and infer probability to be signal or noise. The analysis method is described in [52]. In this paper we only report on the results obtained from the analysis of new O1 LIGO data.

The upper plots in Fig. 2 present the final event rate as a function of the likelihood ratio Λ for the cusp and kink search. The rate of accidental coincident events be-tween the two detectors (background) is estimated by performing the analysis over 6000 time-shifted LIGO-Livingston data sets. This background data set virtually offers 2.5 × 1010 _{s (∼ 790.7 years) of double-coincidence} time. For both cusps and kinks, the candidate ranking values are compatible with the expected background dis-tribution, so no signal was found. The highest-ranked event is measured with Λh≃ 232 for cusps and Λh≃ 611 for kinks. These events were scrutinized and were found to belong to a known category of noise transients called

“blips” described in [54], matching very well the wave-form of cusp and kink signals.

The sensitivity to cusp and kink GW events is esti-mated experimentally by injecting simulated signals of known amplitude Aq in the data. We measure the detec-tion efficiency eq(Aq) as the fraction of simulated signals recovered with Λ > Λh, which is associated to a false alarm rate of 1/Tobs = 2.40 × 10−7 Hz. The detection efficiencies are displayed in the bottom plots in Fig. 2. The sensitivity curve of the 2005-2010 LIGO-Virgo cusp search is also plotted, and should be compared with the O1 LIGO sensitivity measured for an equivalent false-alarm rate of 1.85 × 10−8_{Hz [52]. The sensitivity to} cos-mic string signals is improved by a factor 10. This gain is explained by the significant sensitivity improvement at low frequencies of Advanced detectors [30].

Since no signal from cosmic string was found in LIGO O1 data, it is possible to constrain cosmic string param-eters using models 1, 2 and 3. To generate statistical statements about our ability to detect true GW signals, we adopt the loudest event statistic [55]. We compute an effective detection rate for a given loop distribution model M : R(M)q (Gµ, p) = Z +∞ 0 dAq eq(Aq) (23) × Z +∞ 0 dzd 2_R(M) q dzdAq (Aq, z, f∗; Gµ, p),(24) where the predicted rate is given by Eq. 22 with the change of variables Aq = hf−q. The frequency f∗ = 30 Hz is the lowest high-frequency cutoff used in the search template bank as it provides the maximum angle between the line of sight and the cusp/kink on the loop. The parameter space of model M , (Gµ, p), is scanned and excluded at a 95%level when R(M)q exceeds 2.996/Tobs

which is the rate expected from a random Poisson process over an observation time Tobs. The resulting constraints are shown in Fig. 6 and will be discussed in Sec IV.

C. Stochastic gravitational-wave background Cosmic string networks also generate a stochastic back-ground of GWs, which is measured using the energy

den-1 10 ₁₀2 3 10 ₁₀4 5 10 Λ Likelihood ratio 11 − 10 10 − 10 9 − 10 8 − 10 7 − 10 6 − 10 5 − 10 4 − 10 Event rate [Hz]

Cusp search - LIGO O1 Background Candidates 1 10 ₁₀2 3 10 ₁₀4 5 10 Λ Likelihood ratio 11 − 10 10 − 10 9 − 10 8 − 10 7 − 10 6 − 10 5 − 10 4 − 10 Event rate [Hz]

Kink search - LIGO O1 Background Candidates 22 − 10 ₁₀−21 ₁₀−20 ₁₀−19 ₁₀−18 ] -1/3 [s cusps Signal amplitude, A 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Detection efficiency

Sensitivity to cusp signals Hz -7 10

×

LIGO O1, FAR=2.40 Hz -8 10

×

LIGO O1, FAR=1.85 Hz -8 10 × LIGO 2005-2010, FAR=1.85 21 − 10 10−20 10−19 10−18 ₁₀−17 ] -2/3 [s kinks Signal amplitude, A 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Detection efficiency

Sensitivity to kink signals

Hz -7 10

×

LIGO O1, FAR=2.40

FIG. 2: In the upper plots, the red points show the measured cumulative cusp (left-hand plot) and kink (right-hand plot) GW burst rate (using Tobs as normalization) as a function of the likelihood ratio Λ. The black line shows the expected background of the search with the ±1σ statistical error represented by the hatched area. In both cases, the

highest-ranked event (Λh≃ 232 and Λh≃ 611) is consistent with the background. The lower plots show the sensitivity of the search as a function of the cusp/kink signal amplitude. This is measured by the fraction of simulated cusp/kink events recovered with Λ > Λh. The sensitivity to cusp signals is also measured for a false-alarm

rate (FAR) of 1.85 × 10−6 _{Hz to be compared with the sensitivity of the previous LIGO-Virgo burst search [52]} (dashed lines). sity ΩGW(f ) = f ρc dρGW df , (25)

where dρGW is the energy density of GWs in the

fre-quency range f to f + df and ρc is the critical energy

density of the Universe. Following the method outlined in [56], the GW energy density is given by:

Ω(M)_GW(f ; Gµ, p) =4π 2 3H2 0 f3Z h∗ 0 dh h2 × Z +∞ 0 dzd 2_R(M) dzdh (h, z, f ; Gµ, p), (26) where the spectrum is computed for a specific choice of free parameters Gµ and p, and the maximum strain am-plitude h∗_{is defined below. This equation gives the}

con-tribution to the stochastic background from the super-position of unresolved signals from cosmic string cusps and kinks, and we shall determine the total GW energy density due to cosmic strings is by summing the two. Note that this calculation underestimates the stochastic

background since it only includes the high-frequency tribution from kinks and cusps. The low-frequency con-tribution from the smooth part of loops may be impor-tant, and has been discussed in [57–60]. Neglecting this contribution, conservative constraints will be derived.

To compute the integrals in Eq. 26 we adopt the numerical method described in Appx. B. As observed in [50], the integration over the strain amplitude is per-formed up to h∗ _{to exclude the individually resolvable} powerful and rare bursts. The maximum strain ampli-tude h∗ _{determined by solving the equation}

Z +∞ h∗ dh Z +∞ 0 dzd 2_R(M) dzdh (h, z, f ) = f. (27) This encodes the fact that when the burst rate is larger than f , individual bursts are not resolved.

The total energy density in gravitational waves pro-duced by cosmic strings will be composed of overlapping signals (h < h∗_{) and non-overlapping signals, namely} bursts (h > h∗_{). The LIGO-Virgo stochastic search} pipeline will detect both types of signals. This has been demonstrated for a stochastic background produced by binary neutron stars, whose signals overlap, and binary

black holes, whose signals will arrive in a non-overlapping fashion [61, 62]. In this present cosmic string study this effect is negligible: the predicted GW energy density, Ω(M)_GW, does not grow significantly (and Fig. 3, top, does not change noticeably) when h∗_{→ +∞.}

Fig. 3 (top) shows the spectra for the three models un-der consiun-deration, adding both the cusp and the kink con-tributions and assuming Gµ = 10−8_{. Model 2 spectrum} is about 10 times weaker than the spectrum of model 1 over most of the frequency range. As shown in Fig. 4 (top), the spectra are dominated by the contribution of loops in the radiation era over most of the frequency range, including the frequencies accessible to LIGO and Virgo detectors (10-1000 Hz). The difference in normal-izations of the loop distributions in the radiation era in the two models, discussed in Sec. II, is therefore the cause for the difference in spectral amplitudes. Note also that at low frequencies (∼ 10−9 _{Hz), at which pulsar timing} observations are made, the matter era loops contribute the most.

Fig. 3 (top) also shows that the spectrum for model 3 has a significantly higher amplitude than those of models 1 and 2. Fig. 4 shows that this spectrum is dominated by the contribution of small loops which, as discussed in Sec. II, are much more numerous in model 3.

Fig. 3 (bottom) shows the maximum value for the strain amplitude to consider in the integration, h∗ _{as a} function of the frequency. At LIGO-Virgo frequencies (10-1000 Hz) the spectrum originates from GWs with strain amplitudes below ∼ 10−28_.

The energy density spectra predicted by the mod-els can be compared with several observational results. First, searches for the stochastic GW background using LIGO and Virgo detectors have been performed, using the initial generation detectors (science run S6, 2009-2010) [63] and the first observation run (O1, 2015-2016) of the advanced detectors [31]. Both searches reported frequency-dependent upper limits on the energy density in GWs. To translate these upper limits into constraints on cosmic string parameters, we define the following like-lihood function: ln L(Gµ, p) ∝X i −Y (fi) − Ω(M)GW(fi; Gµ, p) 2 σ2_(f i) , (28) where Y (fi) and σ(fi) are the measurement and the as-sociated uncertainty of the GW energy density in the frequency bin fi, and Ω(M)GW(fi; Gµ, p) is the energy den-sity computed by a cosmic string model at the same frequency bin fi and for some set of model parame-ters Gµ and p. We evaluate the likelihood function across the parameter space (Gµ, p) and compute the 95%confidence contours for the initial LIGO-Virgo (S6, 41.5 < f < 169 Hz) [63] and for the most recent Ad-vanced LIGO (O1, 20 < f < 86 Hz) [31] stochastic back-ground measurements (assuming Bayesian formalism and flat priors in the log parameter space). Since a stochas-tic background of GWs has not been detected yet, these

14 − 10 ₁₀−11 −8 10 10−5 ₁₀−2 _{10 10}4 5 10 Frequency [Hz] 10 − 10 9 − 10 8 − 10 7 − 10 6 − 10 5 − 10 4 − 10 3 − 10 2 − 10 GW Ω

Gravitational-wave energy density,

-8 = 10 µ cusps+kinks, G Model M=1 Model M=2 Model M=3 14 − 10 10−11 10−8 10−5 10−2 10 104105 Frequency [Hz] 30 − 10 24 − 10 18 − 10 12 − 10 6 − 10 1 5 10

Maximum strain amplitude, h*

-8 = 10 µ cusps, G Model M=1 Model M=2 Model M=3

FIG. 3: Top: GW energy density, Ω(M)GW(f ), from cusps and kinks predicted by the three loop distribution models. The string tension Gµ has been fixed to 10−8_.

Bottom: maximum strain amplitude h∗_{used for the} integration in Eq.26.

contours define the excluded regions of the parameter space. We also compute the projected design sensitivity for the Advanced LIGO and Advanced Virgo detectors, using Eq. 28 with Y (fi) = 0 and with the projected σ(fi) for the detector network [64].

Another limit can be computed based on the Pulsar Timing Array (PTA) measurements of the pulse arrival times of millisecond pulsars [29]. This measurement pro-duces a limit on the energy density at nanohertz fre-quencies — specifically, at 95% confidence ΩPTA

GW(f = 2.8 × 10−9_{Hz) < 2.3 × 10}−10_{. We directly compare the} spectra predicted by our models (at 2.8 × 10−9 _{Hz) to} this constraint.

Finally, indirect limits on the total (integrated over frequency) energy density in GWs can be placed based on the Big-Bang Nucleosynthesis (BBN) and Cosmic Mi-crowave Background (CMB) observations. The BBN model and observations of the abundances of the light-est nuclei can be used to constrain the effective num-ber of relativistic degrees of freedom at the time of the BBN, Neff. Under the assumption that only photons and standard light neutrinos contribute to the radia-tion energy density, Neff is equal to the effective num-ber of neutrinos, corrected for the residual heating of

13 − 10 10−9 10−5 ₁₀−1 3 10 105 Frequency [Hz] 12 − 10 11 − 10 10 − 10 9 − 10 8 − 10 7 − 10 6 − 10 5 − 10 4 − 10 GW Ω

Gravitational-wave energy density,

-8 = 10

µ

M=1, cusps, G Total

Loops in radiation era Loops in matter era

from radiation era

→

from matter era

→ 13 − 10 10−9 10−5 10−1 103 105 Frequency [Hz] 10 − 10 9 − 10 8 − 10 7 − 10 6 − 10 5 − 10 4 − 10 3 − 10 GW Ω

Gravitational-wave energy density,

-8 = 10

µ

M=3, cusps, G Total

Loops in radiation era

µ G Γ > γ → µ G Γ < γ < c γ → c γ < γ →

Loops in matter era

µ G Γ > γ → µ G Γ < γ < c γ → c γ < γ →

FIG. 4: Top: GW energy density, Ω(M)GW(f ), from cusps for model 1. We have separated the contributions from loops in the radiation (z > 3366) and matter (z < 3366)

eras. Additionally, for loops in the matter era, we have separated the effect of loops produced in the matter era

from the ones produced in the radiation era (Eq. 3, Eq. 4 and Eq. 6). Bottom: GW energy density, Ω(M)GW(f ), from cusps for model 3. The effect of the three

loop size regimes is shown (Eq. 13, Eq. 14 and Eq. 15) for the matter and radiation eras.

the neutrino fluid due to electron-positron annihilation: Neff ≃ 3.046 [65]. Any deviation from this value can be attributed to extra relativistic radiation, including poten-tially GWs due to cosmic string kinks and cusps gener-ated prior to BBN. We therefore use the 95% confidence upper limit Neff − 3.046 < 1.4, obtained by comparing the BBN model and the abundances of deuterium and 4_{He [27], which translates into the following limit on the} total energy density in GWs:

ΩBBNGW(Gµ, p) =

Z 1010 Hz 10−10 _Hz

df Ω(M)_GW(f ; Gµ, p) < 1.75×10−5, (29) where the lower bound on the integrated frequency region is determined by the size of the horizon at the time of BBN [60]. In this calculation we only consider kinks and cusps generated before BBN, which implies limiting the redshift integral in Eq. 26 to z > 5.5 × 109_.

Similarly, presence of GWs at the time of photon

13 − 10 10−9 10−5 10−1 103 105 Frequency [Hz] 15 − 10 13 − 10 11 − 10 9 − 10 7 − 10 5 − 10 GW Ω

Gravitational-wave energy density,

-8 = 10 µ Cusps, G M=1, Total M=1, at photon decoupling M=1, at Big-Bang nucleosynthesis M=3, Total M=3, at photon decoupling M=3, at Big-Bang nucleosynthesis

FIG. 5: GW energy density, Ω(M)_GW(f ), from cusps for models 1 and 3. The spectra have been computed at the time of photon decoupling (zCMB= 1100) and at

the time of nucleosynthesis (zBBN= 5.5 × 109).

decoupling could alter the observed CMB and Baryon Acoustic Oscillation spectra. We apply a similar proce-dure as in the BBN case, integrating over redshifts before the photon decoupling (z > 1089) and over all frequencies above 10−15_{Hz (horizon size at the time of decoupling)} to compute the total energy density of GWs at the time of decoupling. We then compare this quantity to the pos-terior distribution obtained in [28] to compute the 95% confidence contours: ΩCMBGW (Gµ, p) = Z 1010 Hz 10−15 _Hz df Ω(M)_GW(f ; Gµ, p) < 3.7 × 10−6, (30) For reference, Fig. 5 shows the energy density spectra for models 1 and 3 using Gµ = 10−8. As expected, the contribution from the matter era loops is suppressed at the time of the BBN or of photon decoupling, resulting in the suppression of the spectra at low frequencies. To have negligible systematic errors associated to the numerical integration, we compute Eq. 29 and Eq. 30 using 200 and 250 logarithmically-spaced frequency bins respectively.

Fig. 6 shows the excluded regions in the parameter spaces of the three models considered here, based on the stochastic observational constraints discussed above.

IV. DISCUSSION

The constraints on the cosmic string tension Gµ and intercommutation probability p are shown in Fig. 6 for the three loop models under consideration: M = 1 [8, 32] (top-left), M = 2 [33] (top-right) and M = 3 [34] (bottom-left). We recall that these three models were developed for p = 1 and, as explained earlier, for smaller intercommutation probability, we used a 1/p dependence for the loop distribution.

The bounds resulting from the burst search performed on O1 data are the least constraining. For model 3 and