arxiv:1402.4974v3 [gr-qc] 10 apr 2014

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Implementation of an F-statistic all-sky search for

continuous gravitational waves in Virgo VSR1 data

LIGO Scientific Collaboration and Virgo Collaboration

J. Aasi1, B. P. Abbott1, R. Abbott1, T. Abbott2,

M. R. Abernathy1, T. Accadia3, F. Acernese4,5, K. Ackley6,

C. Adams7, T. Adams8, P. Addesso5, R. X. Adhikari1,

C. Affeldt9, M. Agathos10, N. Aggarwal11, O. D. Aguiar12,

A. Ain13, P. Ajith14, A. Alemic15, B. Allen9,16,17,

A. Allocca18,19, D. Amariutei6, M. Andersen20,

R. Anderson1, S. B. Anderson1, W. G. Anderson16,

K. Arai1, M. C. Araya1, C. Arceneaux21, J. Areeda22,

S. M. Aston7, P. Astone23, P. Aufmuth17, C. Aulbert9,

L. Austin1, B. E. Aylott24, S. Babak25, P. T. Baker26,

G. Ballardin27, S. W. Ballmer15, J. C. Barayoga1,

M. Barbet6, B. C. Barish1, D. Barker28, F. Barone4,5,

B. Barr29, L. Barsotti11, M. Barsuglia30, M. A. Barton28,

I. Bartos31, R. Bassiri20, A. Basti18,32, J. C. Batch28,

J. Bauchrowitz9, Th. S. Bauer10, B. Behnke25, M. Bejger33,

M. G. Beker10, C. Belczynski34, A. S. Bell29, C. Bell29,

G. Bergmann9, D. Bersanetti35,36, A. Bertolini10,

J. Betzwieser7, P. T. Beyersdorf37, I. A. Bilenko38,

G. Billingsley1, J. Birch7, S. Biscans11, M. Bitossi18,

M. A. Bizouard39, E. Black1, J. K. Blackburn1,

L. Blackburn40, D. Blair41, S. Bloemen42,10, M. Blom10,

O. Bock9, T. P. Bodiya11, M. Boer43, G. Bogaert43,

C. Bogan9, C. Bond24, F. Bondu44, L. Bonelli18,32,

R. Bonnand45, R. Bork1, M. Born9, K. Borkowski46,

V. Boschi18, Sukanta Bose47,13, L. Bosi48, C. Bradaschia18,

P. R. Brady16, V. B. Braginsky38, M. Branchesi49,50,

J. E. Brau51, T. Briant52, D. O. Bridges7, A. Brillet43,

M. Brinkmann9, V. Brisson39, A. F. Brooks1,

D. A. Brown15, D. D. Brown24, F. Bruckner24,

S. Buchman20, T. Bulik34, H. J. Bulten10,53, A. Buonanno54,

R. Burman41, D. Buskulic3, C. Buy30, L. Cadonati55,

G. Cagnoli45, J. Calderon Bustillo56, E. Calloni4,57,

J. B. Camp40, P. Campsie29, K. C. Cannon58, B. Canuel27,

J. Cao59, C. D. Capano54, F. Carbognani27, L. Carbone24,

S. Caride60, A. Castiglia61, S. Caudill16, M. Cavaglia21,

F. Cavalier39, R. Cavalieri27, C. Celerier20, G. Cella18,

C. Cepeda1, E. Cesarini62, R. Chakraborty1,

T. Chalermsongsak1, S. J. Chamberlin16, S. Chao63,

http://arxiv.org/abs/1402.4974v3

F-statistic all-sky search for continuous gravitational waves in Virgo data 2

P. Charlton64, E. Chassande-Mottin30, X. Chen41,

Y. Chen65, A. Chincarini35, A. Chiummo27, H. S. Cho66,

J. Chow67, N. Christensen68, Q. Chu41, S. S. Y. Chua67,

S. Chung41, G. Ciani6, F. Clara28, J. A. Clark55, F. Cleva43,

E. Coccia69,70, P.-F. Cohadon52, A. Colla23,71, C. Collette72,

M. Colombini48, L. Cominsky73, M. Constancio Jr.12,

A. Conte23,71, D. Cook28, T. R. Corbitt2, M. Cordier37,

N. Cornish26, A. Corpuz74, A. Corsi75, C. A. Costa12,

M. W. Coughlin76, S. Coughlin77, J.-P. Coulon43,

S. Countryman31, P. Couvares15, D. M. Coward41,

M. Cowart7, D. C. Coyne1, R. Coyne75, K. Craig29,

J. D. E. Creighton16, S. G. Crowder78, A. Cumming29,

L. Cunningham29, E. Cuoco27, K. Dahl9, T. Dal Canton9,

M. Damjanic9, S. L. Danilishin41, S. D’Antonio62,

K. Danzmann17,9, V. Dattilo27, H. Daveloza79, M. Davier39,

G. S. Davies29, E. J. Daw80, R. Day27, T. Dayanga47,

G. Debreczeni81, J. Degallaix45, S. Deleglise52,

W. Del Pozzo10, T. Denker9, T. Dent9, H. Dereli43,

V. Dergachev1, R. De Rosa4,57, R. T. DeRosa2,

R. DeSalvo82, S. Dhurandhar13, M. Dıaz79, L. Di Fiore4,

A. Di Lieto18,32, I. Di Palma9, A. Di Virgilio18,

A. Donath25, F. Donovan11, K. L. Dooley9, S. Doravari7,

O. Dorosh83, S. Dossa68, R. Douglas29, T. P. Downes16,

M. Drago84,85, R. W. P. Drever1, J. C. Driggers1, Z. Du59,

S. Dwyer28, T. Eberle9, T. Edo80, M. Edwards8, A. Effler2,

H. Eggenstein9, P. Ehrens1, J. Eichholz6, S. S. Eikenberry6,

G. Endroczi81, R. Essick11, T. Etzel1, M. Evans11,

T. Evans7, M. Factourovich31, V. Fafone62,70, S. Fairhurst8,

Q. Fang41, S. Farinon35, B. Farr77, W. M. Farr24,

M. Favata86, H. Fehrmann9, M. M. Fejer20, D. Feldbaum6,7,

F. Feroz76, I. Ferrante18,32, F. Ferrini27, F. Fidecaro18,32,

L. S. Finn87, I. Fiori27, R. P. Fisher15, R. Flaminio45,

J.-D. Fournier43, S. Franco39, S. Frasca23,71, F. Frasconi18,

M. Frede9, Z. Frei88, A. Freise24, R. Frey51, T. T. Fricke9,

P. Fritschel11, V. V. Frolov7, P. Fulda6, M. Fyffe7, J. Gair76,

L. Gammaitoni48,89, S. Gaonkar13, F. Garufi4,57,

N. Gehrels40, G. Gemme35, E. Genin27, A. Gennai18,

S. Ghosh42,10,47, J. A. Giaime7,2, K. D. Giardina7,

A. Giazotto18, C. Gill29, J. Gleason6, E. Goetz9, R. Goetz6,

L. Gondan88, G. Gonzalez2, N. Gordon29,

M. L. Gorodetsky38, S. Gossan65, S. Goßler9, R. Gouaty3,

C. Graf29, P. B. Graff40, M. Granata45, A. Grant29,

S. Gras11, C. Gray28, R. J. S. Greenhalgh90,

A. M. Gretarsson74, P. Groot42, H. Grote9, K. Grover24,

S. Grunewald25, G. M. Guidi49,50, C. Guido7, K. Gushwa1,

E. K. Gustafson1, R. Gustafson60, D. Hammer16,

G. Hammond29, M. Hanke9, J. Hanks28, C. Hanna91,

J. Hanson7, J. Harms1, G. M. Harry92, I. W. Harry15,

E. D. Harstad51, M. Hart29, M. T. Hartman6,


C.-J. Haster24, K. Haughian29, A. Heidmann52,

M. Heintze6,7, H. Heitmann43, P. Hello39, G. Hemming27,

M. Hendry29, I. S. Heng29, A. W. Heptonstall1, M. Heurs9,

M. Hewitson9, S. Hild29, D. Hoak55, K. A. Hodge1,

K. Holt7, S. Hooper41, P. Hopkins8, D. J. Hosken93,

J. Hough29, E. J. Howell41, Y. Hu29, E. Huerta15,

B. Hughey74, S. Husa56, S. H. Huttner29, M. Huynh16,

T. Huynh-Dinh7, D. R. Ingram28, R. Inta87, T. Isogai11,

A. Ivanov1, B. R. Iyer94, K. Izumi28, M. Jacobson1,

E. James1, H. Jang95, P. Jaranowski96, Y. Ji59,

F. Jimenez-Forteza56, W. W. Johnson2, D. I. Jones97,

R. Jones29, R.J.G. Jonker10, L. Ju41, Haris K98,

P. Kalmus1, V. Kalogera77, S. Kandhasamy21, G. Kang95,

J. B. Kanner1, J. Karlen55, M. Kasprzack27,39,

E. Katsavounidis11, W. Katzman7, H. Kaufer17,

K. Kawabe28, F. Kawazoe9, F. Kefelian43, G. M. Keiser20,

D. Keitel9, D. B. Kelley15, W. Kells1, A. Khalaidovski9,

F. Y. Khalili38, E. A. Khazanov99, C. Kim100,95, K. Kim101,

N. Kim20, N. G. Kim95, Y.-M. Kim66, E. J. King93,

P. J. King1, D. L. Kinzel7, J. S. Kissel28, S. Klimenko6,

J. Kline16, S. Koehlenbeck9, K. Kokeyama2,

V. Kondrashov1, S. Koranda16, W. Z. Korth1,

I. Kowalska34, D. B. Kozak1, A. Kremin78, V. Kringel9,

B. Krishnan9, A. Krolak102,83, G. Kuehn9, A. Kumar103,

P. Kumar15, R. Kumar29, L. Kuo63, A. Kutynia83,

P. Kwee11, M. Landry28, B. Lantz20, S. Larson77,

P. D. Lasky104, C. Lawrie29, A. Lazzarini1, C. Lazzaro105,

P. Leaci25, S. Leavey29, E. O. Lebigot59, C.-H. Lee66,

H. K. Lee101, H. M. Lee100, J. Lee11, M. Leonardi84,85,

J. R. Leong9, A. Le Roux7, N. Leroy39, N. Letendre3,

Y. Levin106, B. Levine28, J. Lewis1, T. G. F. Li10,1,

K. Libbrecht1, A. Libson11, A. C. Lin20, T. B. Littenberg77,

V. Litvine1, N. A. Lockerbie107, V. Lockett22, D. Lodhia24,

K. Loew74, J. Logue29, A. L. Lombardi55, M. Lorenzini62,70,

V. Loriette108, M. Lormand7, G. Losurdo49, J. Lough15,

M. J. Lubinski28, H. Luck17,9, E. Luijten77,

A. P. Lundgren9, R. Lynch11, Y. Ma41, J. Macarthur29,

E. P. Macdonald8, T. MacDonald20, B. Machenschalk9,

M. MacInnis11, D. M. Macleod2, F. Magana-Sandoval15,

M. Mageswaran1, C. Maglione109, K. Mailand1,

E. Majorana23, I. Maksimovic108, V. Malvezzi62,70,

N. Man43, G. M. Manca9, I. Mandel24, V. Mandic78,

V. Mangano23,71, N. Mangini55, M. Mantovani18,

F. Marchesoni48,110, F. Marion3, S. Marka31, Z. Marka31,

A. Markosyan20, E. Maros1, J. Marque27, F. Martelli49,50,

I. W. Martin29, R. M. Martin6, L. Martinelli43,

D. Martynov1, J. N. Marx1, K. Mason11, A. Masserot3,

T. J. Massinger15, F. Matichard11, L. Matone31,

R. A. Matzner111, N. Mavalvala11, N. Mazumder98,


G. Mazzolo17,9, R. McCarthy28, D. E. McClelland67,

S. C. McGuire112, G. McIntyre1, J. McIver55, K. McLin73,

D. Meacher43, G. D. Meadors60, M. Mehmet9,

J. Meidam10, M. Meinders17, A. Melatos104, G. Mendell28,

R. A. Mercer16, S. Meshkov1, C. Messenger29, P. Meyers78,

H. Miao65, C. Michel45, E. E. Mikhailov113, L. Milano4,57,

S. Milde25, J. Miller11, Y. Minenkov62,

C. M. F. Mingarelli24, C. Mishra98, S. Mitra13,

V. P. Mitrofanov38, G. Mitselmakher6, R. Mittleman11,

B. Moe16, P. Moesta65, M. Mohan27,

S. R. P. Mohapatra15,61, D. Moraru28, G. Moreno28,

N. Morgado45, S. R. Morriss79, K. Mossavi9, B. Mours3,

C. M. Mow-Lowry9, C. L. Mueller6, G. Mueller6,

S. Mukherjee79, A. Mullavey2, J. Munch93, D. Murphy31,

P. G. Murray29, A. Mytidis6, M. F. Nagy81,

D. Nanda Kumar6, I. Nardecchia62,70, L. Naticchioni23,71,

R. K. Nayak114, V. Necula6, G. Nelemans42,10, I. Neri48,89,

M. Neri35,36, G. Newton29, T. Nguyen67, A. Nitz15,

F. Nocera27, D. Nolting7, M. E. N. Normandin79,

L. K. Nuttall16, E. Ochsner16, J. O’Dell90, E. Oelker11,

J. J. Oh115, S. H. Oh115, F. Ohme8, P. Oppermann9,

B. O’Reilly7, R. O’Shaughnessy16, C. Osthelder1,

D. J. Ottaway93, R. S. Ottens6, H. Overmier7,

B. J. Owen87, C. Padilla22, A. Pai98, O. Palashov99,

C. Palomba23, H. Pan63, Y. Pan54, C. Pankow16,

F. Paoletti18,27, R. Paoletti18,19, M. A. Papa16,25, H. Paris28,

A. Pasqualetti27, R. Passaquieti18,32, D. Passuello18,

M. Pedraza1, S. Penn116, A. Perreca15, M. Phelps1,

M. Pichot43, M. Pickenpack9, F. Piergiovanni49,50,

V. Pierro82,35, M. Pietka117, L. Pinard45, I. M. Pinto82,35,

M. Pitkin29, J. Poeld9, R. Poggiani18,32, A. Poteomkin99,

J. Powell29, J. Prasad13, S. Premachandra106,

T. Prestegard78, L. R. Price1, M. Prijatelj27, S. Privitera1,

G. A. Prodi84,85, L. Prokhorov38, O. Puncken79,

M. Punturo48, P. Puppo23, J. Qin41, V. Quetschke79,

E. Quintero1, G. Quiroga109, R. Quitzow-James51,

F. J. Raab28, D. S. Rabeling10,53, I. Racz81, H. Radkins28,

P. Raffai88, S. Raja118, G. Rajalakshmi14, M. Rakhmanov79,

C. Ramet7, K. Ramirez79, P. Rapagnani23,71, V. Raymond1,

V. Re62,70, J. Read22, C. M. Reed28, T. Regimbau43,

S. Reid119, D. H. Reitze1,6, E. Rhoades74, F. Ricci23,71,

K. Riles60, N. A. Robertson1,29, F. Robinet39, A. Rocchi62,

M. Rodruck28, L. Rolland3, J. G. Rollins1, R. Romano4,5,

G. Romanov113, J. H. Romie7, D. Rosinska33,120,

S. Rowan29, A. Rudiger9, P. Ruggi27, K. Ryan28,

F. Salemi9, L. Sammut104, V. Sandberg28, J. R. Sanders60,

V. Sannibale1, I. Santiago-Prieto29, E. Saracco45,

B. Sassolas45, B. S. Sathyaprakash8, P. R. Saulson15,

R. Savage28, J. Scheuer77, R. Schilling9, R. Schnabel9,17,


R. M. S. Schofield51, E. Schreiber9, D. Schuette9,

B. F. Schutz8,25, J. Scott29, S. M. Scott67, D. Sellers7,

A. S. Sengupta121, D. Sentenac27, V. Sequino62,70,

A. Sergeev99, D. Shaddock67, S. Shah42,10, M. S. Shahriar77,

M. Shaltev9, B. Shapiro20, P. Shawhan54,

D. H. Shoemaker11, T. L. Sidery24, K. Siellez43,

X. Siemens16, D. Sigg28, D. Simakov9, A. Singer1,

L. Singer1, R. Singh2, A. M. Sintes56, B. J. J. Slagmolen67,

J. Slutsky9, J. R. Smith22, M. Smith1, R. J. E. Smith1,

N. D. Smith-Lefebvre1, E. J. Son115, B. Sorazu29,

T. Souradeep13, L. Sperandio62,70, A. Staley31,

J. Stebbins20, J. Steinlechner9, S. Steinlechner9,

B. C. Stephens16, S. Steplewski47, S. Stevenson24,

R. Stone79, D. Stops24, K. A. Strain29, N. Straniero45,

S. Strigin38, R. Sturani122,49,50, A. L. Stuver7,

T. Z. Summerscales123, S. Susmithan41, P. J. Sutton8,

B. Swinkels27, M. Tacca30, D. Talukder51, D. B. Tanner6,

S. P. Tarabrin9, R. Taylor1, A. P. M. ter Braack10,

M. P. Thirugnanasambandam1, M. Thomas7, P. Thomas28,

K. A. Thorne7, K. S. Thorne65, E. Thrane1, V. Tiwari6,

K. V. Tokmakov107, C. Tomlinson80, A. Toncelli18,32,

M. Tonelli18,32, O. Torre18,19, C. V. Torres79, C. I. Torrie1,29,

F. Travasso48,89, G. Traylor7, M. Tse31,11, D. Ugolini124,

C. S. Unnikrishnan14, A. L. Urban16, K. Urbanek20,

H. Vahlbruch17, G. Vajente18,32, G. Valdes79,

M. Vallisneri65, J. F. J. van den Brand10,53,

C. Van Den Broeck10, S. van der Putten10,

M. V. van der Sluys42,10, J. van Heijningen10,

A. A. van Veggel29, S. Vass1, M. Vasuth81, R. Vaulin11,

A. Vecchio24, G. Vedovato105, J. Veitch10, P. J. Veitch93,

K. Venkateswara125, D. Verkindt3, S. S. Verma41,

F. Vetrano49,50, A. Vicere49,50, R. Vincent-Finley112,

J.-Y. Vinet43, S. Vitale11, T. Vo28, H. Vocca48,89,

C. Vorvick28, W. D. Vousden24, S. P. Vyachanin38,

A. Wade67, L. Wade16, M. Wade16, M. Walker2,

L. Wallace1, M. Wang24, X. Wang59, R. L. Ward67,

M. Was9, B. Weaver28, L.-W. Wei43, M. Weinert9,

A. J. Weinstein1, R. Weiss11, T. Welborn7, L. Wen41,

P. Wessels9, M. West15, T. Westphal9, K. Wette9,

J. T. Whelan61, D. J. White80, B. F. Whiting6,

K. Wiesner9, C. Wilkinson28, K. Williams112, L. Williams6,

R. Williams1, T. Williams126, A. R. Williamson8,

J. L. Willis127, B. Willke17,9, M. Wimmer9, W. Winkler9,

C. C. Wipf11, A. G. Wiseman16, H. Wittel9, G. Woan29,

J. Worden28, J. Yablon77, I. Yakushin7, H. Yamamoto1,

C. C. Yancey54, H. Yang65, Z. Yang59, S. Yoshida126,

M. Yvert3, A. Zadrozny83, M. Zanolin74, J.-P. Zendri105,

Fan Zhang11,59, L. Zhang1, C. Zhao41, X. J. Zhu41,

M. E. Zucker11, S. Zuraw55, and J. Zweizig1


1LIGO - California Institute of Technology, Pasadena, CA 91125, USA2Louisiana State University, Baton Rouge, LA 70803, USA3Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Universitede Savoie, CNRS/IN2P3, F-74941 Annecy-le-Vieux, France4INFN, Sezione di Napoli, Complesso Universitario di Monte S.Angelo, I-80126Napoli, Italy5Universita di Salerno, Fisciano, I-84084 Salerno, Italy6University of Florida, Gainesville, FL 32611, USA7LIGO - Livingston Observatory, Livingston, LA 70754, USA8Cardiff University, Cardiff, CF24 3AA, United Kingdom9Albert-Einstein-Institut, Max-Planck-Institut fur Gravitationsphysik, D-30167Hannover, Germany10Nikhef, Science Park, 1098 XG Amsterdam, The Netherlands11LIGO - Massachusetts Institute of Technology, Cambridge, MA 02139, USA12Instituto Nacional de Pesquisas Espaciais, 12227-010 - Sao Jose dos Campos,SP, Brazil13Inter-University Centre for Astronomy and Astrophysics, Pune - 411007, India14Tata Institute for Fundamental Research, Mumbai 400005, India15Syracuse University, Syracuse, NY 13244, USA16University of Wisconsin–Milwaukee, Milwaukee, WI 53201, USA17Leibniz Universitat Hannover, D-30167 Hannover, Germany18INFN, Sezione di Pisa, I-56127 Pisa, Italy19Universita di Siena, I-53100 Siena, Italy20Stanford University, Stanford, CA 94305, USA21The University of Mississippi, University, MS 38677, USA22California State University Fullerton, Fullerton, CA 92831, USA23INFN, Sezione di Roma, I-00185 Roma, Italy24University of Birmingham, Birmingham, B15 2TT, United Kingdom25Albert-Einstein-Institut, Max-Planck-Institut fur Gravitationsphysik, D-14476Golm, Germany26Montana State University, Bozeman, MT 59717, USA27European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy28LIGO - Hanford Observatory, Richland, WA 99352, USA29SUPA, University of Glasgow, Glasgow, G12 8QQ, United Kingdom30APC, AstroParticule et Cosmologie, Universite Paris Diderot, CNRS/IN2P3,CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cite, 10, rue Alice Domon etLeonie Duquet, F-75205 Paris Cedex 13, France31Columbia University, New York, NY 10027, USA32Universita di Pisa, I-56127 Pisa, Italy33CAMK-PAN, 00-716 Warsaw, Poland34Astronomical Observatory Warsaw University, 00-478 Warsaw, Poland35INFN, Sezione di Genova, I-16146 Genova, Italy36Universita degli Studi di Genova, I-16146 Genova, Italy37San Jose State University, San Jose, CA 95192, USA38Faculty of Physics, Lomonosov Moscow State University, Moscow 119991,Russia39LAL, Universite Paris-Sud, IN2P3/CNRS, F-91898 Orsay, France40NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA41University of Western Australia, Crawley, WA 6009, Australia42Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box9010, 6500 GL Nijmegen, The Netherlands43Universite Nice-Sophia-Antipolis, CNRS, Observatoire de la Cote d’Azur,F-06304 Nice, France44Institut de Physique de Rennes, CNRS, Universite de Rennes 1, F-35042Rennes, France45Laboratoire des Materiaux Avances (LMA), IN2P3/CNRS, Universite deLyon, F-69622 Villeurbanne, Lyon, France46Centre for Astronomy, Nicolaus Copernicus University, 87-100 Torun, Poland47Washington State University, Pullman, WA 99164, USA48INFN, Sezione di Perugia, I-06123 Perugia, Italy49INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy


50Universita degli Studi di Urbino ’Carlo Bo’, I-61029 Urbino, Italy51University of Oregon, Eugene, OR 97403, USA52Laboratoire Kastler Brossel, ENS, CNRS, UPMC, Universite Pierre et MarieCurie, F-75005 Paris, France53VU University Amsterdam, 1081 HV Amsterdam, The Netherlands54University of Maryland, College Park, MD 20742, USA55University of Massachusetts - Amherst, Amherst, MA 01003, USA56Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain57Universita di Napoli ’Federico II’, Complesso Universitario di Monte S.Angelo,I-80126 Napoli, Italy58Canadian Institute for Theoretical Astrophysics, University of Toronto,Toronto, Ontario, M5S 3H8, Canada59Tsinghua University, Beijing 100084, China60University of Michigan, Ann Arbor, MI 48109, USA61Rochester Institute of Technology, Rochester, NY 14623, USA62INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy63National Tsing Hua University, Hsinchu Taiwan 30064Charles Sturt University, Wagga Wagga, NSW 2678, Australia65Caltech-CaRT, Pasadena, CA 91125, USA66Pusan National University, Busan 609-735, Korea67Australian National University, Canberra, ACT 0200, Australia68Carleton College, Northfield, MN 55057, USA69INFN, Gran Sasso Science Institute, I-67100 L’Aquila, Italy70Universita di Roma Tor Vergata, I-00133 Roma, Italy71Universita di Roma ’La Sapienza’, I-00185 Roma, Italy72University of Brussels, Brussels 1050 Belgium73Sonoma State University, Rohnert Park, CA 94928, USA74Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA75The George Washington University, Washington, DC 20052, USA76University of Cambridge, Cambridge, CB2 1TN, United Kingdom77Northwestern University, Evanston, IL 60208, USA78University of Minnesota, Minneapolis, MN 55455, USA79The University of Texas at Brownsville, Brownsville, TX 78520, USA80The University of Sheffield, Sheffield S10 2TN, United Kingdom81Wigner RCP, RMKI, H-1121 Budapest, Konkoly Thege Miklosut 29-33,Hungary82University of Sannio at Benevento, I-82100 Benevento, Italy83NCBJ, 05-400Swierk-Otwock, Poland84INFN, Gruppo Collegato di Trento, I-38050 Povo, Trento, Italy85Universita di Trento, I-38050 Povo, Trento, Italy86Montclair State University, Montclair, NJ 07043, USA87The Pennsylvania State University, University Park, PA 16802, USA88MTA Eotvos University, ‘Lendulet’ A. R. G., Budapest 1117, Hungary89Universita di Perugia, I-06123 Perugia, Italy90Rutherford Appleton Laboratory, HSIC, Chilton, Didcot, Oxon, OX11 0QX,United Kingdom91Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5,Canada92American University, Washington, DC 20016, USA93University of Adelaide, Adelaide, SA 5005, Australia94Raman Research Institute, Bangalore, Karnataka 560080, India95Korea Institute of Science and Technology Information, Daejeon 305-806,Korea96Bia lystok University, 15-424 Bia lystok, Poland97University of Southampton, Southampton, SO17 1BJ, United Kingdom98IISER-TVM, CET Campus, Trivandrum Kerala 695016, India99Institute of Applied Physics, Nizhny Novgorod, 603950, Russia100Seoul National University, Seoul 151-742, Korea101Hanyang University, Seoul 133-791, Korea102IM-PAN, 00-956 Warsaw, Poland103Institute for Plasma Research, Bhat, Gandhinagar 382428, India


104The University of Melbourne, Parkville, VIC 3010, Australia105INFN, Sezione di Padova, I-35131 Padova, Italy106Monash University, Victoria 3800, Australia107SUPA, University of Strathclyde, Glasgow, G1 1XQ, United Kingdom108ESPCI, CNRS, F-75005 Paris, France109Argentinian Gravitational Wave Group, Cordoba Cordoba 5000, Argentina110Universita di Camerino, Dipartimento di Fisica, I-62032 Camerino, Italy111The University of Texas at Austin, Austin, TX 78712, USA112Southern University and A&M College, Baton Rouge, LA 70813, USA113College of William and Mary, Williamsburg, VA 23187, USA114IISER-Kolkata, Mohanpur, West Bengal 741252, India115National Institute for Mathematical Sciences, Daejeon 305-390, Korea116Hobart and William Smith Colleges, Geneva, NY 14456, USA117Gjøvik videregaende skole, P.b. 534, N-2803 Gjøvik, Norway118RRCAT, Indore MP 452013, India119SUPA, University of the West of Scotland, Paisley, PA1 2BE, UnitedKingdom120Institute of Astronomy, 65-265 Zielona Gora, Poland121Indian Institute of Technology, Gandhinagar Ahmedabad Gujarat 382424,India122Instituto de Fısica Teorica, Univ. Estadual Paulista/International Center forTheoretical Physics-South American Institue for Research, Sao Paulo SP01140-070, Brazil123Andrews University, Berrien Springs, MI 49104, USA124Trinity University, San Antonio, TX 78212, USA125University of Washington, Seattle, WA 98195, USA126Southeastern Louisiana University, Hammond, LA 70402, USA127Abilene Christian University, Abilene, TX 79699, USA

Abstract. We present an implementation of the F-statistic to carry out the firstsearch in data from the Virgo laser interferometric gravitational wave detector forperiodic gravitational waves from a priori unknown, isolated rotating neutronstars. We searched a frequency f0 range from 100 Hz to 1 kHz and the frequencydependent spindown f1 range from −1.6 (f0/100 Hz) × 10−9 Hz/s to zero. Alarge part of this frequency - spindown space was unexplored by any of the all-skysearches published so far. Our method consisted of a coherent search over two-day periods using the F-statistic, followed by a search for coincidences among thecandidates from the two-day segments. We have introduced a number of noveltechniques and algorithms that allow the use of the Fast Fourier Transform (FFT)algorithm in the coherent part of the search resulting in a fifty-fold speed-up incomputation of the F-statistic with respect to the algorithm used in the otherpipelines. No significant gravitational wave signal was found. The sensitivity ofthe search was estimated by injecting signals into the data. In the most sensitiveparts of the detector band more than 90% of signals would have been detectedwith dimensionless gravitational-wave amplitude greater than 5 × 10−24.

PACS numbers: 04.80.Nn, 95.55.Ym, 97.60.Gb, 07.05.Kf

1. Introduction

This paper presents results from a wide parameter search for periodic gravitationalwaves from spinning neutron stars using data from the Virgo detector [1]. The dataused in this paper were produced during Virgo’s first science run (VSR1) which startedon May 18, 2007 and ended on October 1, 2007. The VSR1 data has never beensearched for periodic gravitational waves from isolated neutron stars before. The


innovation of the search is the combination of the efficiency of the FFT algorithmtogether with a nearly optimal grid of templates.

Rotating neutron stars are promising sources of gravitational radiation in theband of ground-based laser interferometric detectors (see [2] for a review). In orderfor these objects to emit gravitational waves, they must exhibit some asymmetry,which could be due to imperfections of the neutron star crust or the influence ofstrong internal magnetic fields (see [3] for recent results). Gravitational radiation alsomay arise due to the r-modes, i.e., rotation-dominated oscillations driven unstable bythe gravitational emission (see [4] for discussion of implications of r-modes for GWsearches). Neutron star precession is another gravitational wave emission mechanism(see [5] for a recent study). Details of the above mechanisms of gravitational waveemission can be found in [6, 7, 8, 9, 10, 11].

A signal from a rotating neutron star can be modeled independently of the specificmechanism of gravitational wave emission as a nearly periodic wave the frequencyof which decreases slowly during the observation time due to the rotational energyloss. The signal registered by an Earth-based detector is both amplitude and phasemodulated because of the motion of the Earth with respect to the source.

The gravitational wave signal from such a star is expected to be very weak, andtherefore months-long segments of data must be analyzed. The maximum deformationthat a neutron star can sustain, measured by the ellipticity parameter ǫ (Eq. (7)),ranges from 5 × 10−6 for ordinary matter [7, 12] to 10−4 for strange quark matter[10, 11]. For unknown neutron stars one needs to search a very large parameter space.As a result, fully coherent, blind searches are computationally prohibitive. To performa fully coherent search of VSR1 data in real time (i.e., in time of 136 days of durationof the VSR1 run) over the parameter space proposed in this paper would require a1.4× 104 petaflop computer [13].

A natural way to reduce the computational burden is a hierarchical scheme, wherefirst short segments of data are analyzed coherently, and then results are combinedincoherently. This leads to computationally manageable searches at the expense ofthe signal-to-ratio loss. To perform the hierarchical search presented in this paper inreal time a 2.8 teraflop computer is required. One approach is to use short segmentsof the order of half an hour long so that the the signal remains within a single Fourierfrequency bin in each segment and thus a single Fourier transform suffices to extractthe whole power of the gravitational wave signal in each segment. Three schemes weredeveloped for the analysis of Fourier transforms of the short data segments and theywere used in the all-sky searches of ground interferometer data: the “stack-slide” [14],the “Hough transform” [15, 14], and the “PowerFlux” methods [14, 16, 17].

Another hierarchical scheme involves using longer segment duration for whichsignal modulations need to be taken into account. For segments of the order ofdays long, coherent analysis using the popular F -statistic [18] is computationallydemanding, but feasible. For example the hierarchical search of VSR1 data presentedin this paper requires around 1900 processor cores for the time of 136 days which wasthe duration of the VSR1 run. In the hierarchical scheme the first coherent analysisstep is followed by a post-processing step where data obtained in the first step arecombined incoherently. This scheme was implemented by the distributed volunteercomputing project Einstein@Home [19]. The E@H project performed analysis of LIGOS4 and S5 data, leading to results published in three papers. In the first two, [20, 21],the candidate signals from the coherent analysis were searched for coincidences. Inthe third paper [22] the results from the coherent search were analyzed using a Hough-


transform scheme.The search method used here is similar to the one used in the first two E@H

searches: it consists of coherent analysis by means of the F -statistic, followed by asearch for coincidences among the candidates obtained in the coherent analysis. Thereare, however, important differences: in this search we have a fixed threshold for thecoherent analysis, resulting in a variable number of candidates from analysis of eachof the data segments. Moreover, for different bands we have a variable number oftwo-day data segments (see Section 3 for details). In E@H searches the number ofcandidates used in coincidences from each data segment was the same, as was thenumber of data segments for each band. In addition, the duration of data segmentsfor coherent analysis in the two first E@H searches was 30h (48h in this case).

In this analysis we have implemented algorithms and techniques that considerablyimprove the efficiency of this search. Most importantly we have used the FFTalgorithm to evaluate the F -statistic for two-day data segments. Also we were ableto use the FFT algorithm together with a grid of templates that was only 20% denserthan the best known grid (i.e., the one with the least number of points).

Given that the data we analyzed (Virgo VSR1) had a higher noise and theduration was shorter than the LIGO S5 data we have achieved a lower sensitivitythan in the most recent E@H search [22]. However due to very good efficiency of ourcode we were able to analyze a much larger parameter space. We have analyzed a largepart of the f − f plane that was previously unexplored by any of the all-sky searches.For example in this analysis the f − f plane searched was 6.5 larger than that in thefull S5 E@H search [22] (see Figure 5). In addition, the data from the Virgo detectoris characterized by different spectral artifacts (instrumental and environmental) fromthese seen in LIGO data. As a results, certain narrow bands excluded from LIGOsearches because of highly non-Gaussian noise can be explored in this analysis.

The paper is organized as follows. In Section 2 we describe the data from theVirgo detector VSR1 run, in Section 3 we explain how the data were selected. InSection 4 the response of the detector to a gravitational wave signal from a rotatingneutron star is briefly recalled. In Section 5 we introduce the F -statistic. In Section 6we describe the search method and present the algorithms for an efficient calculationof the F -statistic. In Section 7 we describe the vetoing procedure of the candidates. InSection 8 we present the coincidence algorithm that is used for post-processing of thecandidates. Section 9 contains results of the analysis. In Section 10 we determine thesensitivity of this search, and we conclude the paper by Section 11. In Appendix A wepresent a general formula for probability that is used in the estimation of significanceof the coincidences.

2. Data from the Virgo’s first science run

The VSR1 science run spanned more than four months of data acquisition. This runstarted on May 18, 2007 at 21:00 UTC and ended on October 1, 2007 at 05:00 UTC.The detector was running close to its design sensitivity with a duty cycle of 81.0%[23]. The data were calibrated, and the time series of the gravitational wave strainh(t) was reconstructed. In the range of frequencies from 10Hz to 10kHz the systematicerror in amplitude was 6% and the phase error was 70 mrad below the frequency of1.9 kHz [24]. A snapshot of the amplitude spectral density of VSR1 data in the Virgodetector band is presented in Figure 1.


101

102

103

104

10−23

10−22

10−21

10−20

10−19

10−18

Frequency [Hz]

Sh1/

2 [Hz−

1/2 ]

25th May 2007

Figure 1. Strain amplitude spectral density√Sh of Virgo data taken on May

25, 2007 during the VSR1 run.

3. Data selection

The analysis input consists of narrow-band time-domain data segments. In order toobtain these sequences from VSR1 data, we have used the software described in [25],and extracted the segments from the Short Fourier Transform Database (SFDB). Thetime domain data sequences were extracted with a certain sampling time ∆t, timeduration Tobs and offset frequency foff . Thus each time domain sequence has theband [foff , foff + B], where B = 1/2 × 1/∆t. We choose the segment duration to beexactly two sidereal days and the sampling time equal to 1/2 s, i.e., the bandwidthB = 1 Hz. As a result each narrow band time segment contains N = 344 656 datapoints. We have considered 67 two-day time frames for the analysis. The starting timeof the analysis (the time of the first sample of the data in the first time frame) wasMay 19, 2007 at 00:00 UTC. The data in each time frame d = 1, . . . , 67 were dividedinto narrow band sequences of 1Hz band each. The bands overlap by 2−5 Hz resultingin 929 bands numbered from 0 to 928. The relation between the band number b andthe offset frequency foff is thus given by

foff [Hz] = 100 + (1− 2−5) b. (1)

Consequently, we have 62 243 narrow band data segments. From this set we selectedgood data using the following conditions. Let N0 be the number of zeros in a givendata segment (data point set to zero means that at that time there was no sciencedata). Let [foff , foff + B] be the band of a given data segment. Let Smin and Smax

be the minimum and the maximum of the amplitude spectral density in the interval[foff +0.05B, foff +B−0.05B]. Spectral density was estimated by dividing the data ofa given segment into short stretches and averaging spectra of all the short stretches.We consider the data segment as good data and use it in this analysis, if the following


two criteria are met:

1. N0/N < 1/4,

2. Smax/Smin ≤ 1.1.

20 419 data segments met the above two criteria. Figure 2 shows a time-frequencydistribution of the good data segments. The good data appeared in 50 out of 67 two-

Virgo VSR1 Data (May 18 − Oct 1, 2007)

Fre

quen

cy [H

z]

Time frame number 5 10 15 20 25 30 35 40 45 50 55 60 65

100

200

300

400

500

600

700

800

900

Figure 2. A map of the VSR1 data. Grey - good data selected for the analysis,white - data not analyzed because of large number of missing data or a strongvariation of the spectrum, black - bad data where number of missing data isgreater than 50% or there are strong lines from mirror wires, electronics, etc.

day time frames and in 785 out of 929 bands. The distribution of good data segmentsin time frames and in bands is given in Figure 3. In Figure 4 we plot a snapshotspectral density of VSR1 data presented in Figure 1 and an estimate of the spectraldensity of the data that was used in the analysis. We estimate the spectral density ineach band by a harmonic mean of the spectral densities of each of the two-day timesegments chosen for the analysis in the band. As the data spectrum in each band isapproximately white, we estimate the spectral density in each time segment by 2σ2∆t,where σ2 is the variance of the data in a segment and ∆t is the 1/2s sampling time.We plot the band from 100Hz to 1 kHz which we have chosen for this search.

4. The response of the detector

The dimensionless noise-free response h of a gravitational-wave detector to a weakplane gravitational wave in the long wavelength approximation, i.e., when thecharacteristic size of the detector is much smaller than the reduced wavelength λ/(2π)


0

5

10

15

20

25

30

35

40

45

50T

ime

Fram

esN

umbe

r

0 100 200 300 400 500 600 700 800

Sorted band index

0

50

100

150

200

250

300

350

400

450

500

550

Ban

dsN

umbe

r

0 5 10 15 20 25 30 35 40 45 50

Sorted frame index

Figure 3. Number of good data segments. Left panel - distribution of datasegments in frequency bands. The number of data segments chosen for the analysisin a given frequency band varies from 1 to 50. Right panel - distribution of datasegments time frames. The number of data segments in a given time frame variesfrom 54 to 520. The smallest, outlying number of bands of 54 is in the 5th timeframe. In the remaining frames the number of good bands is greater than 300.

100 200 300 400 500 600 700 800 900 1000

10−22

10−21

10−20

10−19

Frequency [Hz]

Sh1/

2 [Hz−

1/2 ]

25th May 2007VSR1 good data

Figure 4. A snapshot of strain amplitude spectral density√Sh of the VSR1

data (grey curve) in comparison with the spectral density estimated from thedata used in the analysis (black dots). The spectrum in each band is obtainedfrom a harmonic mean of the spectral densities of the data in each segment chosenfor the analysis in the band.

of the wave, can be written as the linear combination of the two independent wavepolarizations h+ and h×,

h(t) = F+(t)h+(t) + F×(t)h×(t), (2)

where F+ and F× are the detector’s beam-pattern functions,

F+(t) = a(t) cos 2ψ + b(t) sin 2ψ, (3)


F×(t) = b(t) cos 2ψ − a(t) sin 2ψ. (4)

The beam-patterns F+ and F× are linear combinations of sin 2ψ and cos 2ψ, where ψis the polarization angle of the wave. The functions a(t) and b(t) are the amplitudemodulation functions, that depend on the location and orientation of the detector onthe Earth and on the position of the gravitational-wave source in the sky, described inthe equatorial coordinate system by the right ascension α and the declination δ angles.They are periodic functions of time with the period of one and two sidereal days. Theanalytic form of the functions a(t) and b(t) for the case of interferometric detectors isgiven by Eqs. (12) and (13) of [18]. For a rotating nonaxisymmetric neutron star, thewave polarization functions are of the form

h+(t) = h0+ cos(φ(t) + φ0), h×(t) = h0× sin(φ(t) + φ0), (5)

where h0+ and h0× are constant amplitudes of the two polarizations and φ(t) + φ0 isthe phase of the wave, φ0 being the initial phase of the waveform. The amplitudesh0+ and h0× depend on the physical mechanism responsible for the gravitationalradiation, e.g., if a neutron star is a triaxial ellipsoid rotating around a principal axiswith frequency f , then these amplitudes are

h0+ =1

2h0(1 + cos2 ι), h0× = h0 cos ι, (6)

where ι is the angle between the star’s angular momentum vector and the directionfrom the star to the Earth, and the amplitude h0 is given by

h0 =16π2G

c4ǫIf2

r. (7)

Here I is the star’s moment of inertia with respect to the rotation axis, r is thedistance to the star, and ǫ is the star’s ellipticity defined by ǫ = |I1 − I2|/I, whereI1 and I2 are moments of inertia with respect to the principal axes orthogonal tothe rotation axis. We assume that the gravitational waveform given by Eqs. (2)–(5) is almost monochromatic around some angular frequency ω0, which we define asinstantaneous angular frequency evaluated at the solar system barycenter (SSB) att = 0, and we assume that the frequency evolution is accurately described by onespindown parameter ω1. Then the phase φ(t) is given by

φ(t) = ω0t+ ω1t2 +

n · rd(t)c

(ω0 + 2ω1t), (8)

where, neglecting the relativistic effects, rd(t) is the vector that joins the SSB withthe detector, and n is the unit vector pointing from SSB to the source. In equatorialcoordinates (δ, α) we have n = (cos δ cosα, cos δ sinα, sin δ).

5. The F - statistic

A method to search for gravitational wave signals from a rotating neutron star in adetector data x(t), t = 1, ..., N uses the F -statistic, described in [18]. The F -statisticis obtained by maximizing the likelihood function with respect to the four unknownparameters - h0, φ0, ι, and ψ. This leaves a function of only the remaining fourparameters - ω0, ω1, δ, and α. Thus the dimension of the parameter space that weneed to search decreases from 8 to 4. In this analysis we shall use an observationtime Tobs equal to the integer multiple of sidereal days. Since the bandwidth of thesignal over our coherent observation time of two days is very small, we can assume


that over this band the spectral density of the noise is white (constant). Under theseassumptions the F -statistic is given by [26, 27]

F ≈ 2

σ2

( |Fa|2〈a2〉 +

|Fb|2〈b2〉

)

, (9)

where σ2 is the variance of the data, and

Fa :=

N∑

t=1

x(t) a(t) exp[−iφ(t)], (10)

Fb :=

N∑

t=1

x(t) b(t) exp[−iφ(t)].

⟨

a2⟩

=

N∑

t=1

a(t)2,⟨

b2⟩

=

N∑

t=1

b(t)2. (11)

6. Description of the search

The search consists of two parts; the first part is a coherent search of two-day datasegments, where we search a 4-parameter space defined by angular frequency ω0,angular frequency derivative ω1, declination δ, and right ascension α. The search isperformed on a 4-dimensional grid in the parameter space described in Section 6.2. Weset a fixed threshold of 20 for the F -statistic for each data segment. This correspondsto a threshold of 6 for the signal-to-noise ratio. All the threshold crossings are recordedtogether with corresponding 4 parameters of the grid point and the signal-to-noise ratioρ. The signal-to-noise is calculated from the value of the F -statistic at the thresholdcrossing as

ρ =√

2(F − 2). (12)

In this way for each narrow band segment we obtain a set of candidates. Thecandidates are then subject to the vetoing procedure described in Section 7. Thesecond part of the search is the post-processing stage involving search for coincidencesamong the candidates. The coincidence procedure is described in Section 8.

6.1. Choice of the parameter space

We have searched the frequency band from 100 Hz to 1 kHz over the entire sky. Wehave followed [13] to constrain the maximum value of the parameter ω1 for a givenfrequency ω0 by |ω1| ≤ ω0/(2τmin), where τmin is the minimum spindown age‡. Wehave chosen τmin = 1000yr for the whole frequency band searched. Also, in this searchwe have considered only the negative values for the parameter ω1, thus assuming thatthe rotating neutron star is spinning down. This gives the frequency-dependent rangeof the spindown parameter f1 where f1 = ω1/(2π):

|f1| ≤ 1.6× 10−9 f

100Hz

1000yr

τmin

[Hzs−1] (13)

We have considered only one frequency derivative. Estimates taking into accountparameter correlations (see [13] Figure 6 and Eqs. (6.2) - (6.6)) show that even for

‡ The factor of two in this formula appears here because the spindown parameter f used in [13] istwice the spindown parameter used in this work.


0 200 400 600 800 1000 1200 1400 1600Frequency [Hz]

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

Freq

uenc

y de

rivat

ive

[10−8

Hz/

s]

PowerFlux (early S5)

PowerFlux (full S5)

F-stat (VSR1)

E@H (early S5)

E@H (full S5)

Figure 5. Comparison of the parameter space in f − f plane searched in VSR1analysis presented in this paper (area enclosed by a thick black line) and recentlypublished PowerFlux and E@H searches of the LIGO S5 data.

the minimum spindown age of 40yr and for two days coherent observation time thatwe consider here, it is sufficient to include just one spindown parameter.

In Figure 5 we have compared the parameter space searched in this analysis inthe f − f plane with that of other recently published all-sky searches: Einstein@Homeearly S5 search [21], Einstein@Home full S5 [22], PowerFlux early S5 [28], PowerFluxfull S5 [17].

6.2. Efficient calculation of the F-statistic on the grid in the parameter space

Calculation of the F -statistic (Eq. (9)) involves two sums given by Eqs. (10). Byintroducing a new time variable called the barycentric time tb ([29, 18, 27])

tb := t+n · rd(t)

c. (14)

we can write these sums as discrete Fourier transforms in the following way

Fa =N∑

tb=1

x(t(tb)) a(t(tb)) exp[−iφs(t(tb))] exp[−iω0tb], (15)

Fb =N∑

tb=1

x(t(tb)) b(t(tb)) exp[−iφs(t(tb))] exp[−iω0tb],

where

φs(t) = ω1t2 + 2

n · rd(t)c

ω1t. (16)

Written in this form, the two sums can be evaluated using the Fast Fourier Transform(FFT) algorithm thus speeding up their computation dramatically. The time


transformation described by equation (14) is called resampling. In addition to theuse of the FFT algorithm we apply an interpolation of the FFT using the interbinningprocedure (see [27] Section VB). This results in the F -statistic sampled twice as finewith respect to the standard FFT. This procedure is much faster than the interpolationof the FFT obtained by padding the data with zeroes and calculating a FFT thatis twice as long. With the approximations described above for each value of theparameters ω1, δ, and α, we calculate the F -statistic efficiently for all the frequencybins in the data segment of bandwidth 1Hz.

In order to search the 4-dimensional parameter space, we need to construct a4-dimensional grid. To minimize the computational cost we construct a grid that hasthe smallest number of points for a certain assumed minimal match MM [30]. Thisproblem is equivalent to the covering problem [31, 32] and it has the optimal solutionin 4-dimensions in the case of a lattice i.e., a uniformly spaced grid. In order that ourparameter space is a lattice, the signals’ reduced Fisher matrix must have componentsthat are independent of the values of the parameters. This is not the case for thesignal given by Eqs. (2) - (8); it can be realized however for an approximate signalcalled the linear model described in Section IIIB of [27]. The linear model consists ofneglecting amplitude modulation of the signal and discarding the component of thevector rd(t) joining the detector and the solar system barycenter that is perpendicularto the ecliptic. This approximation is justified because the amplitude modulation isvery slow compared to the phase modulation and the discarded component in thephase is small compared to the others. As a result the linear model signal hlin(t) hasa constant amplitude A0, and one can find parameters such that the phases are linearfunctions of them. We explicitly have (see Section IIIB of [27] for details):

hlin(t) = A0 cos[φlin(t) + φ0], (17)

where

φlin(t) = ω0t+ ω1t2 + α1µ1(t) + α2µ2(t). (18)

The parameters α1 and α2 are defined by

α1 := ω0(sinα cos δ cos ε+ sin δ sin ε), (19)

α2 := ω0 cosα cos δ, (20)

where ε is the obliquity of the ecliptic, and µ1(t) and µ2(t) are known functions of thedetector ephemeris.

In order that the grid is compatible with application of the FFT, its points shouldbe constrained to coincide with Fourier frequencies at which the FFT is calculated.Moreover, we observed that a numerically accurate implementation of the interpolationto the barycentric time (see Eq. (14)) is so computationally demanding that it mayoffset the advantage of the FFT. Therefore we introduced another constraint in the gridsuch that the resampling is needed only once per sky position for all the spindownvalues. Construction of the constrained grid is described in detail in Section IV of[27]. In this search we have chosen the value of the minimal match MM =

√3/2. The

workflow of the coherent part of the search procedure is presented in Figure 6.

7. Vetoing procedure

We apply three vetoing criteria to the candidates obtained in the coherent part ofthe search - line width veto, stationary line veto and polar caps veto. Data from the


Figure 6. Workflow of the F-statistic search pipeline.

detector always contain some periodic interferences (lines) that are detector artifacts.An important part of our vetoing procedure was to identify the lines in the data.We have therefore performed a Fourier search with frequency resolution of 1/(2 days)≃ 5.8 × 10−6 Hz for periodic signals of each of the two-day data segments. Wecompared the frequencies of the significant periodic signals identified by our analysiswith the line frequencies obtained by the Virgo LineMonitor and we found that all thelines from the LineMonitor were detected by our Fourier search.

7.1. Line width veto

We veto all the candidates with frequency f around every known line frequency flaccording to the following criterion

|f − fl| < ∆fmax, (21)

where the width ∆fmax is estimated as

∆fmax = fl|v|max/c+ 2|f1|max(Tobs + Tobs|v|max/c+ |r|max/c), (22)

where vmax is the maximum value of the velocity of the detector with respect to theSSB during the observation time, rmax is the maximum distance to the SSB during the


coherent observation time Tobs, and |f1|max is the maximum of the absolute value ofthe frequency derivative. Eq. (22) determines the maximum smearing of the frequencyon each side of the line due to frequency modulation induced by the filters applied inthe F -statistic.

7.2. Stationary line veto

Let us consider the instantaneous frequency finst of the signal, i.e., the time derivativeof the phase:

finst :=1

2π

dφ(t)

dt= f0+2f1t+

n · vd(t)

c(f0+2f1t)+2

n · rd(t)c

f1, (23)

where vd(t) = drd(t)/dt. The frequency derivative of the instantaneous frequency isgiven by

dfinst(t)

dt= 2f1 +

n · ad(t)c

(f0 + 2f1t) + 4n · vd(t)

cf1, (24)

where ad(t) = dvd(t)/dt. Eq. (24) is the rate of change of detector response frequencyfor a source whose SSB frequency and spindown are f0 and f1. An instrumental linehas a constant detector frequency and mimics a source for which the r.h.s of Eq. (24)vanishes. In practice, we veto candidates with

∣

∣

∣

∣

2f1 +n · ad(t)

cf0 + 2

n · d2(rd(t)t)/dt2c

f1

∣

∣

∣

∣

< ǫSL (25)

for some ǫSL > 0. In the search we choose ǫSL = 1/Tobs2, where Tobs is the observation

time. The above stationary line veto was introduced in reference [14] and refined in[33]; it was used in the first two E@H searches [20, 21].

7.3. Polar caps veto

We observe that many of the detected lines cluster around the poles where declinationδ is close to ±π/2. An interference originating from a detector will correlate wellwith our templates if the frequency modulation in Eq. (23) is minimized. Assumingthat f1 = 0 and that the diurnal motion of the Earth averages to 0 over two daysobservation time, this happens when the quantity |n·vd(t)| is minimized. We find thatthis quantity is close to minimum independently of the value of α when δ = ±π/2.Thus we veto candidates that are too close to the poles; we discard all candidates withthe declination angle δ within three grid cells from the poles.

8. Coincidences

In order to find coincidences among the candidates, we applied a method similar tothe one used in the first two E@H searches [20, 21]. For each band we have searchedfor coincidences among candidates in different time frames. We are able to search forcoincidences only in those bands where there were two or more time frames with dataselected for the analysis. If we search for a real gravitational wave signal, we musttake into account frequency evolution due to spindown of the rotating neutron star.Thus the first step in the coincidence analysis was to transform all frequencies ω0(tl)of the candidates to a common fiducial reference time tf . We have chosen the fiducial


time to be the time of the first sample of the latest time frame that we analyzed i.e.,the 67th time frame. We have

ω(tf ) = ω0(tl) + 2ω1(tl)[67Tobs − tl], (26)

where tl is the time of the first sample of the lth time frame. The next step wasto divide the parameter space into cells. This construction of the coincidence cellwas different from that in the E@H analysis. To construct the cells in the parameterspace we have used the reduced Fisher matrix Γ for the linear signal model defined byEqs. (17) and (18). The reduced Fisher matrix is the projected Fisher matrix on the4-dimensional space spanned by parameters κ = (ω0, ω1, α1, α2). We define the cellin the parameter space by the condition:

∑

k,l

Γklκkκl ≤ 2. (27)

Because the ephemeris of the detector is different in each of the time frames, thereduced Fisher matrix is different in each time frame. To have a common coincidencegrid we have chosen the grid defined by the latest frame i.e., the frame no. 67 as thecoincidence grid. After the transformation of the candidate frequencies to a referencetime and construction of the coincidence grid, the coincidence algorithm for each ofthe bands proceeded in the following steps:

1. Transform angles α and δ to α1 and α2 coordinates (see Eq.(19))

2. Transform candidate parameters to xl coordinates defined by

xl =4

∑

k=1

κkVkl√el, (28)

where Vkl, k = 1, .., 4 are eigenvectors of the matrix Γ, and el are its eigenvalues.In these coordinates the Fisher matrix is proportional to the unit matrix.

3. Coordinates xl are rounded to the nearest integer. In this way we sort candidatesefficiently into adjacent 4-dimensional hypercubes. If there are more than onecandidates from a given data segment in a hypercube we select the candidate thathas the highest SNR. We do sorting for each time frame in the band. If there ismore than one candidate in a given hypercube we register a coincidence.

4. We shift cubes by 1/2 of their size in all possible 24 directions, and for each shiftwe search for coincidences.

This last step of the algorithm takes into account cases for which the candidate eventsare located on opposite sides of cell borders, edges, and corners and consequentlycoincidences that could not be found just by packing candidates into adjacent cells.

The most significant coincidence in each band is the one which has the highestmultiplicity. For each most significant coincidence we have calculated the false alarmprobability i.e., the probability that such a coincidence can occur purely by chance.The false alarm probability is calculated using the formula explained in Appendix Aand given by Eq. (A.6). This general formula applies to a variable number of candidatesin various time slots and also takes into account the 24 shifts of the cells in theparameter space.


9. The search

In this analysis we have searched coherently 20 419 two-day time segments of datanarrowbanded to 1 Hz. In the coherent part of the search described in Section6 we have used 9.10 × 1016 templates which is the number of F -statistic valuescomputed. This resulted in 20 419 candidate files containing 4.21 × 1010 candidates.The candidates were subject to the vetoing using the three veto criteria: line veto,polar caps veto, and stationary line veto described in Section 7. As a result ofvetoing around 24% of the candidates were discarded leaving 3.19× 1010 candidates.Nearly all candidates were vetoed by the line veto, whereas 0.20% were vetoed bythe stationary line criterion and only 3.2 × 10−2% by the polar caps veto. In Figure7 we present an example of the candidate distribution obtained from the coherentsearch of one narrow band data segment and after the vetoing procedure. In the

0 5 10 15

x 104

−0.05

0

0.05

Time [s]

Am

plitu

de

Time series

186.4 186.6 186.8 187 187.2

−22.6

−22.4

−22.2

Frequency [Hz]Spe

ctra

l den

sity

[Hz−

1/2 ]

45 50 55 600

0.05

0.1

0.15

Histogram of candidates

2 × F−statistic

186.4 186.6 186.8 187 187.26

6.5

7

7.5

8

8.5

Frequency [Hz]

Sig

nal−

to−

nois

e

Frequency distribution

Candidates in data segment: 30−089

0 100 200 300

−50

0

50

α [deg]

δ [d

eg]

Distribution on the sky

6.5

7

7.5

Figure 7. Distribution of candidates after vetoing obtained in the coherent F-statistic search of two-day data segment 30 − 089 i.e., a segment with the timeframe number d = 30 and the band number b = 89. The left top panel shows dataand its spectrum. The right top panel shows the distribution of 2F values of thecandidates in comparison to the χ2 distribution with 4 degrees of freedom. Thebottom left panel shows the distribution of signal-to-noise ratios ρ (see Eq. (12))of the candidates as a function of the frequency. The vertical black line is theperiodic interference identified in the data. The candidates in the band aroundthe line are vetoed (see Eq. (22)). The right bottom panel shows the distributionof the candidates on the sky in equatorial coordinates α and δ. The distributionshows singularity at the ecliptic. This is a consequence of the grid constructionfrom the approximate linear signal model given by Eqs.(17) - (19).


next step we have searched for the significant coincidences among the candidates. Wehave searched for coincidences in all the frequency bands where there were two ormore data segments analyzed. In Figure 8 we have plotted the highest coincidencemultiplicity for each of the bands. The highest multiplicity was 6, and it occurredin 10 bands. The multiplicity tends to grow with the frequency, because the size ofthe parameter space grows as f3. For each band we have calculated the false alarm

100 200 300 400 500 600 700 800 900 10002

3

4

5

6VSR1 all sky search

Coi

ncid

ence

mul

tiplic

ity

100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Fal

se a

larm

pro

babl

ity

Figure 8. Coincidences among the candidates. Top panel: maximum multiplicityof coincidence as function of frequency. Bottom panel: corresponding coincidencefalse alarm probability.

probability corresponding to the most significant coincidence using Eq. (A.6). Themost significant coincidence occurred in band no. 401, corresponding to the frequencyrange of ∼ [488.4, 489.4] Hz. It was a coincidence of multiplicity = 5, and its falsealarm probability was 14.5%. Note that a coincidence with the highest multiplicity isnot the most significant. This is because the significance depends on the number oftime frames with candidates in a given band and also on the number of candidates inthe time frames. By adopting a criterion used by E@H searches that the backgroundcoincidences correspond to false alarm probability of 0.1% or greater, we concludethat we have found no significant coincidence and thus no viable gravitational wavecandidate. Considering the significance of the coincidences, we could adopt even a10% false alarm probability as a background. Consequently, we proceed to the finalstage of our analysis - estimation of sensitivity of the search.

10. Sensitivity of the search

The sensitivity of the search is taken to be the amplitude h0 of the gravitationalwave signal that can be confidently detected. To estimate the sensitivity we use aprocedure developed in [20]. We determine the sensitivity of the search in each of


the 785 frequency bands that we have searched. To determine the sensitivity, weperform Monte Carlo simulations in which, for a given amplitude h0, we randomlyselect the other seven parameters of the signal: ω0, ω1, α, δ, φ0, ι, and ψ. We choosefrequency and spindown parameters uniformly over their range, and source positionsuniformly over the sky. We choose angles φ0 and ψ uniformly over the interval[0, 2π] and we choose cos ι uniformly over the interval [−1, 1]. For each band weadd the signal to all the data segments chosen for the analysis in that band. Thenwe process the data through our pipeline. First, we perform a coherent F -statisticsearch of each of the data segments where the signal was added, and store all thecandidates above our F -statistic threshold of 20. In this coherent analysis, to makethe computation manageable, we search over only parameter space consisting of ±2 grid points around the nearest grid point where the signal was added. Then weapply our vetoing procedure to the candidates obtained as explained in Section 7.Finally, we perform coincidence analysis of the candidates that survive vetoing whichis described in Section 8. We define a detectable signal if it is coincident in more than70% of the time frames in a given band. This condition is similar to the condition usedin the two E@H searches, where a coincidence method was used [20, 21]. For bandswith only one frame available the coherent search over one 2-day data segment wasperformed. In this case the injected signal is declared detected if its signal-to-noiseratio obtained in the coherent search is larger than the signal-to-noise ratio of theloudest signal in that data segment without an injection. For each band we injectsignals with 5 different amplitude values, and perform 100 randomized injections foreach amplitude. For each amplitude we calculate how many signals were detected, andby interpolation we determine the amplitude corresponding to 90% of signals detected.This amplitude was defined as the 90% confidence sensitivity. Sometimes even for thehighest amplitude we have not reached the 90% detection probability. In this case weperformed injections for higher amplitudes until the desired level of detectability wasachieved. In Figure 9, as an example, we present an estimation of the sensitivity inthe band b = 369 corresponding to the frequency range of ∼ [457.47, 458.44] Hz. Theerrors in the sensitivity estimates originate from calibration errors in the amplitudeand errors due to a finite number of Monte Carlo injections. We use 100 injections;hence from a binomial statistic, one σ is equivalent to 3% fluctuation. Thus theestimated amplitude sensitivity corresponds to confidence in the range from 87% to93%. To estimate how this uncertainty in confidence translates into uncertainty in theamplitude we have performed an additional set of injections for a range of amplitudesclose to the estimated sensitivity and from the slope of the confidence vs. amplitudewe determined the uncertainty in the amplitude. To increase the accuracy of the errorestimate, we have performed 1000 injections for each amplitude. The uncertainty inthe amplitude was not more than 5%. The calibration errors in VSR1 data are 6% (seeSection 2). Adding these two types of errors in quadrature results in the total error insensitivity estimate to be around 7%. The sensitivity of this search obtained throughMonte Carlo simulations for the whole band searched is presented in Figure 10. Wesee from Figure 10 that the sensitivity essentially reflects the instrumental noise curvegiven in Figure 4. We have made a fit of the sensitivity h90%0 to the one-sided spectraldensity Sh(f) of detector noise described by the following relation:

h90%0 = RD

√

Sh(f)

48 hours. (29)


0 1 2 3 4 5 6 7 8

x 10−24

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Source strain smplitude ho

Det

ectio

n pr

obab

ility

Sensitivity for the band no. 369. No. of frames = 20

90% confidence sensiivity

Figure 9. Estimation of sensitivity in the band no. 369. In this band thereare 20 time frames. Thus the signal is detected if it is coincident in morethan 14 frames. The black dots are amplitudes of the injected signals and thecorresponding detection probabilities estimated from the injections. The red circleis the interpolated amplitude corresponding to 90% detection probability. This isthe 90% confidence sensitivity for this band.

We find that the prefactor RD is in the range from 15.6 to 22.4 and it depends on thefrequency band and the number of the data segments in the band.

11. Conclusions

The sensitivity of this search was 50% to 2 times better, depending on the bandwidth,than that of the LIGO S4 search [20] and comparable to the sensitivities obtained inthe early LIGO S5 data [21], but 2 to 5 times worse than the upper limits in the E@Hfull LIGO S5 data search [22]. This was due to the lower noise and longer observationtime for the LIGO S5 data w.r.t the Virgo VSR1 data. However, for the first timein an all-sky search we have estimated the sensitivity in the frequency band from400Hz to 1kHz and the frequency spindown range from −7.2 (f0/400Hz)× 10−9 Hz/sto −6.0× 10−9 Hz/s, which is a previously unexplored region in the parameter space.

The next step is to test the search method described in this paper in the MockData Challenge (MDC) designed by the LIGO and Virgo projects to validate andcompare pipelines that are proposed to be used in the analysis of the forthcomingdata from the advanced detectors. It is also planned to further test the pipelinepresented here with other data sets collected by the LIGO and Virgo detectors.


100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−23

Frequency [Hz]

Sen

sitiv

ity

VSR1 all−sky search: sensitivity

SensitivityRescaled noise curve

Figure 10. The 90% confidence sensitivity of the all-sky search of Virgo VSR1data in the band from 100 Hz to 1 kHz. The dots show the source strain amplitudeh0 for which 90% of sources are confidently detected by this pipeline. The thinline is the rescaled instrumental noise curve, see Eq. (29).

Acknowledgments

The authors gratefully acknowledge the support of the United States National ScienceFoundation for the construction and operation of the LIGO Laboratory, the Scienceand Technology Facilities Council of the United Kingdom, the Max-Planck-Society,and the State of Niedersachsen/Germany for support of the construction and operationof the GEO600 detector, and the Italian Istituto Nazionale di Fisica Nucleare and theFrench Centre National de la Recherche Scientifique for the construction and operationof the Virgo detector. The authors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Research Council, the InternationalScience Linkages program of the Commonwealth of Australia, the Council of Scientificand Industrial Research of India, the Istituto Nazionale di Fisica Nucleare of Italy,the Spanish Ministerio de Economıa y Competitividad, the Conselleria d’EconomiaHisenda i Innovacio of the Govern de les Illes Balears, the Foundation for FundamentalResearch on Matter supported by the Netherlands Organisation for Scientific Research,the Polish Ministry of Science and Higher Education, the FOCUS Programme ofFoundation for Polish Science, the PL-Grid Infrastructure, the Royal Society, theScottish Funding Council, the Scottish Universities Physics Alliance, The NationalAeronautics and Space Administration, the Carnegie Trust, the Leverhulme Trust,the David and Lucile Packard Foundation, the Research Corporation, and the AlfredP. Sloan Foundation. This document has been assigned LIGO Laboratory documentnumber LIGO-P1300133.


References

[1] T Accadia et al 2012 JINST 7 P03012[2] B Abbott et al 2007 Phys. Rev. D76 082001[3] A Mastrano, P D Lasky and A Melatos 2013 (Preprint 1306.4503)[4] B J Owen 2010 Phys. Rev. D 82 104002[5] D J Jones and N Andersson 2001 Mon. Not. R. Astron. Soc. 331 203[6] L Bildsten 1998 Astrophys. J. 501 L89[7] G Ushomirsky, C Cutler and L Bildsten 2000 Mon. Not. R. Astron. Soc. 319 902[8] C Cutler 2002 Phys. Rev. D 66 084025[9] A Melatos and D J B Payne 2005 Astrophys. J. 623 1044

[10] B J Owen 2005 Phys. Rev. Lett. 95 211101[11] N K Johnson-McDaniel and B J Owen 2013 Phys. Rev. D 88 044004[12] C J Horowitz and K Kadau 2009 Phys. Rev. Lett. 102 191102[13] P R Brady, T Creighton, C Cutler and B F Schutz 1998 Phys. Rev. D 57 2101[14] B Abbott et al 2008 Phys. Rev. D77 022001[15] B Abbott et al 2005 Phys. Rev. D 72 102004[16] B Abbott et al 2009 Phys. Rev. Lett. 102 111102[17] J Abadie et al 2012 Phys. Rev. D85 022001[18] P Jaranowski, A Krolak and B Schutz 1998 Phys. Rev. D 58 063001[19] http://einstein.phys.uwm.edu[20] B Abbott et al 2009 Phys. Rev. D79 022001[21] B Abbott et al 2009 Phys. Rev. D80 042003[22] J Aasi et al 2013 Phys. Rev. D87 042001[23] F Acernese et al 2008 Class. Quant. Grav. 25 184001[24] T Accadia et al 2010 Journal of Physics: Conference Series 228 012015[25] P Astone et al 2002 Phys. Rev. D 65 022001[26] P Astone, K M Borkowski, P Jaranowski and A Krolak 2002 Phys. Rev. D 65 042003[27] P Astone, K M Borkowski, P Jaranowski, M Pietka and A Krolak 2010 Phys. Rev. D 82 022005[28] B Abbott et al 2009 Phys. Rev. Lett. 102 111102[29] B F Schutz The Detection of Gravitational Waves ed Blair D (Cambridge University Press)[30] B Owen 1996 Phys. Rev. D 53 6749[31] J H Conway and N J A Slone 1999 Sphere packings, lattices and groups (New York: Springer)[32] Prix R 2007 Phys. Rev. D 75 023004[33] H J Pletsch 2008 Phys. Rev. D 78 102005

Appendix A. False alarm coincidence probability

Let us assume that for a given frequency band we analyze L non-overlapping timesegments. Suppose that the search of the lth segment produces Nl candidates. Let usassume that the size of the parameter space for each time segment is the same, and itcan be divided into the number Ncell of independent cells. We would like to test thenull hypothesis that coincidences among candidates from L segments are accidental.The probability for a candidate event to fall into any given coincidence cell is equalto 1/Ncell. Thus probability ǫl that a given coincidence cell is populated with one ormore candidate events is given by

ǫl = 1−(

1− 1

Ncell

)Nl

. (A.1)

We may also consider independent candidates only, i.e., such that there is no morethan one candidate within one cell. If we obtain more than one candidate within agiven cell we choose the one which has the highest signal-to-noise ratio. In this case

ǫl =Nl

Ncell. (A.2)

The probability pF (Ncell) that any given coincidence cell out of the total of Ncell cellscontains candidate events from Cmax or more distinct data segments is given by a

1306.4503


generalized binomial distribution

pF (Ncell) =

L∑

n=Cmax

1

n!(L− n)!×

×∑

σ∈Π(L)

ǫσ(1) . . . ǫσ(n)(1 − ǫσ(n+1)) . . . (1− ǫσ(L)), (A.3)

where∑

σ∈Π(L) is the sum over all the permutations of the L data sequences. Finallythe probability PF that there is Cmax or more coincidences in one or more of the Ncell

cells is

PF = 1− (1− pF (Ncell))Ncell . (A.4)

The above formula for the false alarm coincidence probability does not take intoaccount the case when candidate events are located on opposite sides of cell borders,edges, and corners. In order to find these coincidences the entire cell coincidencegrid is shifted by half a cell width in all possible 24 = 16 combinations of the fourparameter-space dimensions, and coincidences are searched in all the 16 coincidencegrids. This leads to a higher number of accidental coincidences, and consequentlyEq.A.4 underestimates the false alarm probability. Let us consider the simplest one-dimensional case. In this case we have 21 = 2 possible shifts (the original coincidencegrid and the one shifted by half). This increases probability pF (Ncell) by a factor of2 if the two cell coincidence grids were independent. However the cells overlap byhalf and some coincidences would be counted twice. To account for this we divide thecells in the coincidence grid by half resulting in 2Ncell cells and define the false alarmprobability pF (2Ncell) that any given half of the coincidence cell out of the total of2Ncell half cells contains candidate events from Cmax or more distinct data segments.These are coincidences that were already counted, and consequently the false alarmprobability with the cell shift is 2pF (Ncell)−pF (2Ncell). This results in the false alarmprobability

PF = 1− (1− (2pF (Ncell)− pF (2Ncell)))Ncell . (A.5)

To generalize the above formula to higher dimensions we need to consider furthershifts and divisions of the cells. In the four dimension case this leads to the formulafor the probability P shifts

F that there are Cmax or more independent coincidences in oneor more of the Ncell cells in all 16 grid shifts given by

P shiftsF = 1−

[

1−(

24pF (Nc) (A.6)

−(

(

4

1

)

pF (2Nc) +

(

4

2

)

pF (22Nc) +

(

4

3

)

pF (23Nc) +

(

4

4

)

pF (24Nc)

)

−(

(

4

2

)

pF (22Nc) +

(

4

3

)

pF (23Nc) +

(

4

4

)

pF (24Nc)

)

−(

(

4

3

)

pF (23Nc) +

(

4

4

)

pF (24Nc)

)

−(

4

4

)

pF (24Nc)

)]Nc

By choosing a certain false alarm probability PF , we can calculate the thresholdnumber Cmax of coincidences. If we obtain more than Cmax coincidences in our searchwe reject the null hypothesis that coincidences are accidental only at the significancelevel of PF .

arxiv:1402.4974v3 [gr-qc] 10 apr 2014

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