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Implementation of an -statistic all-sky search for continuous gravitational waves in Virgo

VSR1 data

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2014 Class. Quantum Grav. 31 165014

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Implementation of an F -statistic all-skysearch for continuous gravitational wavesin Virgo VSR1 data

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 Brückner24,S Buchman20, T Bulik34, H J Bulten10,53, A Buonanno54,R Burman41, D Buskulic3, C Buy30, L Cadonati55, G Cagnoli45,J Calderón 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, F Cavalier39, R Cavalieri27, C Celerier20, G Cella18,C Cepeda1, E Cesarini62, R Chakraborty1,T Chalermsongsak1, S J Chamberlin16, S Chao63,P Charlton64, E Chassande Mottin30, X Chen41, Y Chen65,

Classical and Quantum Gravity

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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, 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 Danzmann9,17, V Dattilo27, H Daveloza79,M Davier39, G S Davies29, E J Daw80, R Day27, T Dayanga47,G Debreczeni81, J Degallaix45, S Deléglise52, W Del Pozzo10,T Denker9, T Dent9, H Dereli43, V Dergachev1, R De Rosa4,57,R T DeRosa2, R DeSalvo82, S Dhurandhar13, M Díaz79,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 Endrőczi81, 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 Ghosh10,42,47, J A Giaime7,2, K D Giardina7, A Giazotto18,C Gill29, J Gleason6, E Goetz9, R Goetz6, L Gondan88,G González2, N Gordon29, M L Gorodetsky38, S Gossan65,S Goßler9, R Gouaty3, C Gräf29, 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,

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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 Jiménez 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 Kéfélian43,G M Keiser20, D Keitel9, D B Kelley15, W Kells1,A Khalaidovski9, F Y Khalili38, E A Khazanov99, C Kim95,100,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 Królak83,102,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 Li1,10, 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 Lück9,17, 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 Márka31,Z Márka31, 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 Mazzolo9,17, 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,

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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 Pierro35,82, M Pietka117, L Pinard45, I M Pinto35,82, 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 Rácz81,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 Rosińska33,120, S Rowan29, A Rüdiger9,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 Shah10,42, 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,

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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 Sturani49,50,122, 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 Tse11,31,D Ugolini124, C S Unnikrishnan14, A L Urban16, K Urbanek20,H Vahlbruch17, G Vajente18,32, G Valdes79, M Vallisneri65,J F J vanden Brand10,53, C VanDen Broeck10,S vander Putten10, M V vander Sluys10,42, J van Heijningen10,A A van Veggel29, S Vass1, M Vasúth81, R Vaulin11,A Vecchio24, G Vedovato105, J Veitch10, P J Veitch93,K Venkateswara125, D Verkindt3, S S Verma41, F Vetrano49,50,A Viceré49,50, 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 Zadrożny83, M Zanolin74,J P Zendri105, Fan Zhang11,59, L Zhang1, C Zhao41, X J Zhu41,M E Zucker11, S Zuraw55 and J Zweizig1

1 LIGO—California Institute of Technology, Pasadena, CA 91125, USA2 Louisiana State University, Baton Rouge, LA 70803, USA3 Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Université deSavoie, CNRS/IN2P3, F-74941 Annecy-le-Vieux, France4 INFN, Sezione di Napoli, Complesso Universitario di Monte S.Angelo, I-80126Napoli, Italy5 Università di Salerno, Fisciano, I-84084 Salerno, Italy6 University of Florida, Gainesville, FL 32611, USA7 LIGO—Livingston Observatory, Livingston, LA 70754, USA8Cardiff University, Cardiff, CF24 3AA, UK

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9 Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-30167Hannover, Germany10Nikhef, Science Park, 1098 XG Amsterdam, The Netherlands11 LIGO—Massachusetts Institute of Technology, Cambridge, MA 02139, USA12 Instituto Nacional de Pesquisas Espaciais, 12227-010—São José dos Campos, SP,Brazil13 Inter-University Centre for Astronomy and Astrophysics, Pune—411007, India14 Tata Institute for Fundamental Research, Mumbai 400005, India15 Syracuse University, Syracuse, NY 13244, USA16University of Wisconsin–Milwaukee, Milwaukee, WI 53201, USA17 Leibniz Universität Hannover, D-30167 Hannover, Germany18 INFN, Sezione di Pisa, I-56127 Pisa, Italy19 Università di Siena, I-53100 Siena, Italy20 Stanford University, Stanford, CA 94305, USA21 The University of Mississippi, University, MS 38677, USA22 California State University Fullerton, Fullerton, CA 92831, USA23 INFN, Sezione di Roma, I-00185 Roma, Italy24 University of Birmingham, Birmingham B15 2TT, UK25Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, D-14476 Golm,Germany26Montana State University, Bozeman, MT 59717, USA27 European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy28 LIGO—Hanford Observatory, Richland, WA 99352, USA29 SUPA, University of Glasgow, Glasgow G12 8QQ, UK30APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cité, 10, rue Alice Domon et LéonieDuquet, F-75205 Paris Cedex 13, France31 Columbia University, New York, NY 10027, USA32Università di Pisa, I-56127 Pisa, Italy33 CAMK-PAN, 00-716 Warsaw, Poland34Astronomical Observatory Warsaw University, 00-478 Warsaw, Poland35 INFN, Sezione di Genova, I-16146 Genova, Italy36 Università degli Studi di Genova, I-16146 Genova, Italy37 San Jose State University, San Jose, CA 95192, USA38 Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia39 LAL, Université Paris-Sud, IN2P3/CNRS, F-91898 Orsay, France40 NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA41University of Western Australia, Crawley, WA 6009, Australia42 Department of Astrophysics/IMAPP, Radboud University Nijmegen, PO Box 9010,6500 GL Nijmegen, The Netherlands43 Université Nice-Sophia-Antipolis, CNRS, Observatoire de la Côte d’Azur, F-06304Nice, France44 Institut de Physique de Rennes, CNRS, Université de Rennes 1, F-35042 Rennes,France45 Laboratoire des Matériaux Avancés (LMA), IN2P3/CNRS, Université de Lyon,F-69622 Villeurbanne, Lyon, France46 Centre for Astronomy, Nicolaus Copernicus University, 87-100 Toruń, Poland47Washington State University, Pullman, WA 99164, USA48 INFN, Sezione di Perugia, I-06123 Perugia, Italy49 INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy50 Università degli Studi di Urbino ’Carlo Bo’, I-61029 Urbino, Italy51 University of Oregon, Eugene, OR 97403, USA

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52 Laboratoire Kastler Brossel, ENS, CNRS, UPMC, Université Pierre et Marie Curie,F-75005 Paris, France53 VU University Amsterdam, 1081 HV Amsterdam, The Netherlands54 University of Maryland, College Park, MD 20742, USA55University of Massachusetts—Amherst, Amherst, MA 01003, USA56Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain57 Università di Napoli ’Federico II’, Complesso Universitario di Monte S.Angelo,I-80126 Napoli, Italy58 Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto,Ontario M5S 3H8, Canada59 Tsinghua University, Beijing 100084, Peopleʼs Republic of China60 University of Michigan, Ann Arbor, MI 48109, USA61 Rochester Institute of Technology, Rochester, NY 14623, USA62 INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy63 National Tsing Hua University, Hsinchu, Taiwan 30064 Charles Sturt University, Wagga Wagga, NSW 2678, Australia65 Caltech-CaRT, Pasadena, CA 91125, USA66 Pusan National University, Busan 609-735, Korea67 Australian National University, Canberra, ACT 0200, Australia68 Carleton College, Northfield, MN 55057, USA69 INFN, Gran Sasso Science Institute, I-67100 L’Aquila, Italy70 Università di Roma Tor Vergata, I-00133 Roma, Italy71 Università di Roma ’La Sapienza’, I-00185 Roma, Italy72 University of Brussels, Brussels 1050 Belgium73 Sonoma State University, Rohnert Park, CA 94928, USA74 Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA75 The George Washington University, Washington DC 20052, USA76University of Cambridge, Cambridge CB2 1TN, UK77Northwestern University, Evanston, IL 60208, USA78University of Minnesota, Minneapolis, MN 55455, USA79 The University of Texas at Brownsville, Brownsville, TX 78520, USA80 The University of Sheffield, Sheffield S10 2TN, UK81Wigner RCP, RMKI, H-1121 Budapest, Konkoly Thege Miklósút 29-33, Hungary82 University of Sannio at Benevento, I-82100 Benevento, Italy83 NCBJ, 05-400Świerk-Otwock, Poland84 INFN, Gruppo Collegato di Trento, I-38050 Povo, Trento, Italy85 Università di Trento, I-38050 Povo, Trento, Italy86Montclair State University, Montclair, NJ 07043, USA87 The Pennsylvania State University, University Park, PA 16802, USA88MTA Eötvös University, ‘Lendulet’ ARG, Budapest 1117, Hungary89 Università di Perugia, I-06123 Perugia, Italy90 Rutherford Appleton Laboratory, HSIC, Chilton, Didcot, Oxon OX11 0QX, UK91 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada92American University, Washington, DC 20016, USA93University of Adelaide, Adelaide, SA 5005, Australia94 Raman Research Institute, Bangalore, Karnataka 560080, India95 Korea Institute of Science and Technology Information, Daejeon 305-806, Korea96 Białystok University, 15-424 Białystok, Poland97University of Southampton, Southampton SO17 1BJ, UK98 IISER-TVM, CET Campus, Trivandrum Kerala 695016, India99 Institute of Applied Physics, Nizhny Novgorod, 603950, Russia100 Seoul National University, Seoul 151-742, Korea101 Hanyang University, Seoul 133-791, Korea

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102 IM-PAN, 00-956 Warsaw, Poland103 Institute for Plasma Research, Bhat, Gandhinagar 382428, India104 The University of Melbourne, Parkville, VIC 3010, Australia105 INFN, Sezione di Padova, I-35131 Padova, Italy106Monash University, Victoria 3800, Australia107 SUPA, University of Strathclyde, Glasgow, G1 1XQ, UK108 ESPCI, CNRS, F-75005 Paris, France109 Argentinian Gravitational Wave Group, Cordoba Cordoba 5000, Argentina110 Università di Camerino, Dipartimento di Fisica, I-62032 Camerino, Italy111 The University of Texas at Austin, Austin, TX 78712, USA112 Southern University and A&M College, Baton Rouge, LA 70813, USA113 College of William and Mary, Williamsburg, VA 23187, USA114 IISER-Kolkata, Mohanpur, West Bengal 741252, India115 National Institute for Mathematical Sciences, Daejeon 305-390, Korea116 Hobart and William Smith Colleges, Geneva, NY 14456, USA117 Gjøvik videregående skole, Pb 534, NO-2803 Gjøvik, Norway118 RRCAT, Indore MP 452013, India119 SUPA, University of the West of Scotland, Paisley, PA1 2BE, UK120 Institute of Astronomy, 65-265 Zielona Góra, Poland121 Indian Institute of Technology, Gandhinagar Ahmedabad Gujarat 382424, India122 Instituto de Física Teórica, Univ. Estadual Paulista/International Center forTheoretical Physics—South American Institue for Research, São Paulo SP 01140-070,Brazil123 Andrews University, Berrien Springs, MI 49104, USA124 Trinity University, San Antonio, TX 78212, USA125 University of Washington, Seattle, WA 98195, USA126 Southeastern Louisiana University, Hammond, LA 70402, USA127 Abilene Christian University, Abilene, TX 79699, USA

Received 7 March 2014, revised 16 June 2014Accepted for publication 2 July 2014Published 5 August 2014

AbstractWe present an implementation of the -statistic to carry out the first search indata from the Virgo laser interferometric gravitational wave detector for per-iodic gravitational waves from a priori unknown, isolated rotating neutronstars. We searched a frequency f0 range from 100 Hz to 1 kHz and the fre-quency dependent spindown f1 range from − × −f1.6( 100 Hz) 100

9 Hz s−1 tozero. A large part of this frequency–spindown space was unexplored by any ofthe all-sky searches published so far. Our method consisted of a coherentsearch over two-day periods using the -statistic, followed by a search forcoincidences among the candidates from the two-day segments. We haveintroduced a number of novel techniques and algorithms that allow the use ofthe fast Fourier transform (FFT) algorithm in the coherent part of the searchresulting in a fifty-fold speed-up in computation of the -statistic with respectto the algorithm used in the other pipelines. No significant gravitational wavesignal was found. The sensitivity of the search was estimated by injectingsignals into the data. In the most sensitive parts of the detector band more than90% of signals would have been detected with dimensionless gravitational-wave amplitude greater than × −5 10 24.

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Keywords: gravitational waves, gravitational radiation detectors, pulsars, dataanalysis: algorithms and implementationPACS numbers: 04.80.Nn, 95.55.Ym, 97.60.Gb, 07.05.Kf

(Some figures may appear in colour only in the online journal)

1. Introduction

This paper presents results from a wide parameter search for periodic gravitational wavesfrom spinning neutron stars using data from the Virgo detector [1]. The data used in this paperwere produced during Virgoʼs first science run (VSR1), which started on 18 May 2007 andended on 1 October 2007. The VSR1 data have never been searched for periodic gravitationalwaves from isolated neutron stars before. The innovation of the search is the combination ofthe efficiency of the FFT algorithm together with a nearly optimal grid of templates.

Rotating neutron stars are promising sources of gravitational radiation in the band ofground-based laser interferometric detectors (see [2] for a review). In order for these objectsto emit gravitational waves, they must exhibit some asymmetry, which could be due toimperfections of the neutron star crust or the influence of strong internal magnetic fields (see[3] for recent results). Gravitational radiation also may arise due to the r-modes, i.e., rotation-dominated oscillations driven unstable by the gravitational emission (see [4] for discussion ofimplications of r-modes for GW searches). Neutron star precession is another gravitationalwave emission mechanism (see [5] for a recent study). Details of the above mechanisms ofgravitational wave emission can be found in [6–11].

A signal from a rotating neutron star can be modeled independently of the specificmechanism of gravitational wave emission as a nearly periodic wave the frequency of whichdecreases slowly during the observation time due to the rotational energy loss. The signalregistered by an Earth-based detector is both amplitude and phase modulated because of themotion of the Earth with respect to the source.

The gravitational wave signal from such a star is expected to be very weak, and thereforemonths-long segments of data must be analyzed. Assuming the moment of inertia

=I 10 g cm45 2, the maximum deformation that a neutron star can sustain, measured by theellipticity parameter ϵ (equation (7)), ranges from 10−5 for ordinary matter, 10−3 for hybridstars to 10−1 for color-superconducting quark stars (see [11] for details). For unknown neutronstars one needs to search a very large parameter space. As a result, fully coherent, blindsearches are computationally prohibitive. To perform a fully coherent search of VSR1 data inreal time (i.e., in the time of 136 days of duration of the VSR1 run) over the parameter spaceproposed in this paper would require a ×1.4 104 petaflop computer [12].

A natural way to reduce the computational burden is a hierarchical scheme, where firstshort segments of data are analyzed coherently, and then results are combined incoherently.This leads to computationally manageable searches at the expense of the signal-to-ratio loss.To perform the hierarchical search presented in this paper in real time a 2.8 teraflop computeris required. One approach is to use short segments of the order of half an hour long so that thesignal remains within a single Fourier frequency bin in each segment and thus a single Fouriertransform suffices to extract the whole power of the gravitational wave signal in each seg-ment. Three schemes were developed for the analysis of Fourier transforms of the short datasegments and they were used in the all-sky searches of ground interferometer data: the ‘stack-slide’ [13], the ‘Hough transform’ [13, 14] and the ‘PowerFlux’ methods [13, 15, 16].

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Another hierarchical scheme involves using longer segment duration for which signalmodulations need to be taken into account. For segments of the order of days long, coherentanalysis using the popular -statistic [17] is computationally demanding, but feasible. Forexample, the hierarchical search of VSR1 data presented in this paper requires around 1900processor cores for the time of 136 days, which was the duration of the VSR1 run. In thehierarchical scheme the first coherent analysis step is followed by a post-processing stepwhere data obtained in the first step are combined incoherently. This scheme was imple-mented by the distributed volunteer computing project Einstein@Home [18]. The E@Hproject performed analysis of LIGO S4 and S5 data, leading to results published in threepapers. In the first two, [19, 20], the candidate signals from the coherent analysis weresearched for coincidences. In the third paper [21] the results from the coherent search wereanalyzed using a Hough-transform scheme.

The search method used here is similar to the one used in the first two E@H searches: itconsists of coherent analysis by means of the -statistic, followed by a search for coin-cidences among the candidates obtained in the coherent analysis. There are, however,important differences: in this search we have a fixed threshold for the coherent analysis,resulting in a variable number of candidates from analysis of each of the data segments.Moreover, for different bands we have a variable number of two-day data segments (seesection 3 for details). In E@H searches the number of candidates used in coincidences fromeach data segment was the same, as was the number of data segments for each band. Inaddition, the duration of data segments for coherent analysis in the two first E@H searcheswas 30 h (48 h 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 FFT algorithm toevaluate the -statistic for two-day data segments. Also we were able to use the FFTalgorithm together with a grid of templates that was only 20% denser than the best knowngrid (i.e., the one with the least number of points).

Given that the data we analyzed (Virgo VSR1) had higher noise and the duration wasshorter than the LIGO S5 data we have achieved a lower sensitivity than in the most recentE@H search [21]. However, due to very good efficiency of our code we were able to analyzea much larger parameter space. We have analyzed a large part of the − ˙f f plane that waspreviously unexplored by any of the all-sky searches. For example in this analysis the − ˙f fplane searched was 6.5 larger than that in the full S5 E@H search [21] (see figure 5). Inaddition, the data from the Virgo detector are characterized by different spectral artifacts(instrumental and environmental) from these seen in LIGO data. As a result, certain narrowbands excluded from LIGO searches because of highly non-Gaussian noise can be explored inthis analysis.

The paper is organized as follows. In section 2 we describe the data from the Virgodetector VSR1 run, in section 3 we explain how the data were selected. In section 4 theresponse of the detector to a gravitational wave signal from a rotating neutron star is brieflyrecalled. In section 5 we introduce the -statistic. In section 6 we describe the search methodand present the algorithms for an efficient calculation of the -statistic. In section 7 wedescribe the vetoing procedure of the candidates. In section 8 we present the coincidencealgorithm that is used for post-processing of the candidates. Section 9 contains results of theanalysis. In section 10 we determine the sensitivity of this search, and we conclude the paperby section 11. In the appendix we present a general formula for probability that is used in theestimation of significance of the coincidences.

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2. Data from the Virgoʼs first science run

The VSR1 science run spanned more than four months of data acquisition. This run started on18 May 2007 at 21:00 UTC and ended on 1 October 2007 at 05:00 UTC. The detector wasrunning close to its design sensitivity with a duty cycle of 81.0% [22]. The data werecalibrated, and the time series of the gravitational wave strain h(t) was reconstructed. In therange of frequencies from 10 Hz to 10 kHz the systematic error in amplitude was 6% and thephase error was 70 mrad below the frequency of 1.9 kHz [23]. A snapshot of the amplitudespectral density of VSR1 data in the Virgo detector band is presented in figure 1.

3. Data selection

The analysis input consists of narrow-band time-domain data segments. In order to obtainthese sequences from VSR1 data, we have used the software described in [24], and extractedthe segments from the Short Fourier Transform Database (SFDB). The time domain datasequences were extracted with a certain sampling time Δt, time duration Tobs and offsetfrequency foff . Thus each time domain sequence has the band +f f B[ , ]off off , where

Δ= ×B t1 2 1 . We choose the segment duration to be exactly two sidereal days and thesampling time equal to 1 2 s, i.e., the bandwidth B = 1 Hz. As a result each narrow band timesegment contains =N 344 656 data points. We have considered 67 two-day time frames forthe analysis. The starting time of the analysis (the time of the first sample of the data in thefirst time frame) was 19 May 2007 at 00:00 UTC. The data in each time frame =d 1 ,..., 67were divided into narrow band sequences of 1 Hz band each. The bands overlap by −2 5 Hzresulting in 929 bands numbered from 0 to 928. The relation between the band number b andthe offset frequency foff is thus given by

= + − −( )f b[Hz] 100 1 2 . (1)off5

Consequently, we have 62 243 narrow band data segments. From this set we selected gooddata using the following conditions. Let N0 be the number of zeros in a given data segment (adata value at a given time is set exactly to 0 whenever at that time there are no science data).Let +f f B[ , ]off off be the band of a given data segment. Let Smin and Smax be the minimumand the maximum of the amplitude spectral density in the interval

+ + −f B f B B[ 0.05 , 0.05 ]off off . Spectral density was estimated by dividing the data of agiven segment into short stretches and averaging spectra of all the short stretches. Weconsider the data segment as good data and use it in this analysis, if the following two criteria,selected as a rule of thumb, are met:

1. <N N 1 40 ,2. ⩽S S 1.1max min .

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-day timeframes and in 785 out of 929 bands. The distribution of good data segments in time framesand in bands is given in figure 3. In figure 4 we plot a snapshot spectral density of VSR1 datapresented in figure 1 and an estimate of the spectral density of the data that were used in theanalysis. We estimate the spectral density in each band by a harmonic mean of the spectraldensities of each of the two-day time segments chosen for the analysis in the band. As thedata spectrum in each band is approximately white, we estimate the spectral density in each

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time segment by σ Δt2 2 , where σ2 is the variance of the data in a segment and Δt is the 1 2 ssampling time. We plot the band from 100 Hz 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 weak planegravitational wave in the long wavelength approximation, i.e., when the characteristic size of

Figure 1. Strain amplitude spectral density Sh of Virgo data taken on 25 May 2007during the VSR1 run.

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 strong variation of thespectrum, black—bad data where number of missing data is greater than 50% or thereare strong lines from mirror wires, electronics, etc.

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the detector is much smaller than the reduced wavelength λ π(2 ) of the wave, can be writtenas the linear combination of the two independent wave polarizations +h and ×h ,

= ++ + × ×h t F t h t F t h t( ) ( ) ( ) ( ) ( ), (2)

where +F and ×F are the detectorʼs beam-pattern functions,

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

Figure 3. Number of good data segments. Left panel—distribution of data segments infrequency bands. The number of data segments chosen for the analysis in a givenfrequency band varies from 1 to 50. Right panel—distribution of data segments timeframes. The number of data segments in a given time frame varies from 54 to 520. Thesmallest, outlying number of bands of 54 is in the 5th time frame. In the remainingframes the number of good bands is greater than 300.

Figure 4. A snapshot of strain amplitude spectral density Sh of the VSR1 data (solidcurve) in comparison with the spectral density estimated from the data used in theanalysis (dots). The spectrum in each band is obtained from a harmonic mean of thespectral densities of the data in each segment chosen for the analysis in the band.

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

The beam-patterns +F and ×F are linear combinations of ψsin 2 and ψcos 2 , where ψ is thepolarization angle of the wave. The functions a(t) and b(t) are the amplitude modulationfunctions, that depend on the location and orientation of the detector on the Earth and on theposition of the gravitational-wave source in the sky, described in the equatorial coordinatesystem by the right ascension α and the declination δ angles. They are periodic functions oftime with the period of one and two sidereal days. The analytic form of the functions a(t) andb(t) for the case of interferometric detectors is given by equations (12) and (13) of [17]. For arotating non-axisymmetric neutron star, the wave polarization functions are of the form

ϕ ϕ ϕ ϕ= + = ++ + × ×( ) ( )h t h t h t h t( ) cos ( ) , ( ) sin ( ) , (5)0 0 0 0

where +h0 and ×h0 are constant amplitudes of the two polarizations and ϕ ϕ+t( ) 0 is thephase of the wave, ϕ0 being the initial phase of the waveform. The amplitudes +h0 and ×h0

depend on the physical mechanism responsible for the gravitational radiation, e.g., if aneutron star is a triaxial ellipsoid rotating around a principal axis with frequency f, then theseamplitudes are

ι ι= + =+ ×( )h h h h1

21 cos , cos , (6)0 0

20 0

where ι is the angle between the starʼs angular momentum vector and the direction from thestar to the Earth, and the amplitude h0 is given by

π ϵ=hG

c

If

r

16. (7)0

2

4

2

Here I is the starʼs moment of inertia with respect to the rotation axis, r is the distance to thestar and ϵ is the starʼs ellipticity defined by ϵ = | − |I I I1 2 , where I1 and I2 are moments ofinertia with respect to the principal axes orthogonal to the rotation axis. We assume that thegravitational waveform given by equations (2)–(5) is almost monochromatic around someangular frequency ω0, which we define as instantaneous angular frequency evaluated at thesolar system barycenter (SSB) at t = 0, and we assume that the frequency evolution isaccurately described by one spindown parameter ω1. Then the phase ϕ t( ) is given by

ϕ ω ω ω ω= + +·

+( )t t tt

ct

n r( )

( )2 , (8)d

0 12

0 1

where, neglecting the relativistic effects, tr ( )d is the vector that joins the SSB with thedetector, and n is the unit vector pointing from SSB to the source. In equatorial coordinatesδ( , α) we have δ α δ α δ=n (cos cos , cos sin , sin ).

5. The F -statistic

A method to search for gravitational wave signals from a rotating neutron star in detector data= …x t t N( ), 1, , uses the -statistic, described in [17]. The -statistic is obtained by

maximizing the likelihood ratio function with respect to the four unknown parameters—h0,ϕ0, ι, and ψ. This leaves a function of only the remaining four parameters—ω0, ω1, δ, and α.Thus the dimension of the parameter space that we need to search decreases from 8 to 4. Inthis analysis we shall use an observation time Tobs equal to the integer multiple of siderealdays. Since the bandwidth of the signal over our coherent observation time of two days isvery small, we can assume that over this band the spectral density of the noise is white

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(constant). Under these assumptions the -statistic is given by [25, 26]

σ≈

⟨ ⟩+

⟨ ⟩

⎛⎝⎜⎜

⎞⎠⎟⎟ F

a

F

b

2, (9)

a b

2

2

2

2

2

where σ2 is the variance of the data, and

∑

∑

ϕ

ϕ

= −

= −

=

=

F x t a t t

F x t b t t

: ( ) ( ) exp [ i ( )],

: ( ) ( ) exp [ i ( )]. (10)

a

t

N

b

t

N1

1

∑ ∑⟨ ⟩ = ⟨ ⟩ == =

a a t b b t( ) , ( ) . (11)t

N

t

N2

1

2 2

1

2

6. Description of the search

The search consists of two parts; the first part is a coherent search of two-day data segments,where we search a 4-parameter space defined by angular frequency ω0, angular frequencyderivative ω1, declination δ and right ascension α. The search is performed on a 4-dimen-sional grid in the parameter space described in section 6.2. We set a fixed threshold of 20 forthe -statistic for each data segment. This corresponds to a threshold of 6 for the signal-to-noise ratio. All the threshold crossings are recorded together with the corresponding fourparameters of the grid point and the signal-to-noise ratio ρ. The signal-to-noise is calculatedfrom the value of the -statistic at the threshold crossing as

ρ = −2( 2) . (12)

In this way, for each narrow band segment we obtain a set of candidates. The candidates arethen subject to the vetoing procedure described in section 7. The second part of the search isthe post-processing stage involving search for coincidences among the candidates. Thecoincidence 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. We havefollowed [12] to constrain the maximum value of the parameter ω1 for a given frequency ω0

by ω ω τ| | ⩽ (2 )1 0 min , where τmin is the minimum spindown age128. We have chosenτ = 1000min yr for the whole frequency band searched. Also, in this search we have con-sidered only negative values for the parameter ω1, thus assuming that the rotating neutron staris spinning down. This gives the frequency-dependent range of the spindown parameter f1where ω π=f (2 )1 1 :

τ⩽ × − −⎡⎣ ⎤⎦f

f1.6 10

100 Hz

1000 yrHz s . (13)1

9

min

1

128 The factor of two in this formula appears here because the spindown parameter f used in [12] is twice thespindown parameter used in this work.

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We have considered only one frequency derivative. Estimates taking into account parametercorrelations (see [12] figure 6 and equations (6.2)–(6.6)) show that even for the minimumspindown age of 40 yr and for two days coherent observation time that we consider here, it issufficient to include just one spindown parameter.

In figure 5 we have compared the parameter space searched in this analysis in the − ˙f fplane with that of other recently published all-sky searches: Einstein@Home early S5 search[20], Einstein@Home full S5 [21], PowerFlux early S5 [27], PowerFlux full S5 [16].

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

Calculation of the -statistic (equation (9)) involves two sums given by equation (10). Byintroducing a new time variable called the barycentric time tb ([17, 26, 28])

= +·

t tt

c

n r:

( ). (14)b

d

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

∑

∑

ϕ ω

ϕ ω

= − −

= − −

=

=

⎡⎣ ⎤⎦

⎡⎣ ⎤⎦

[ ]

[ ]

( ) ( ) ( )

( ) ( ) ( )

F x t t a t t t t t

F x t t b t t t t t

( ) ( ) exp i ( ) exp i ,

( ) ( ) exp i ( ) exp i , (15)

a

t

N

b b s b b

b

t

N

b b s b b

1

0

1

0

b

b

where

ϕ ω ω= +·

t tt

ct

n r( ) 2

( ). (16)s

d1

21

The values of ϕ−x t t a t t t t( ( )) ( ( )) exp [ i ( ( ))]b b s b in equation (15) above that are evaluated atbarycentric time tb are obtained from the values in equation (10) evaluated at time t by an

Figure 5. Comparison of the parameter space in − ˙f f plane searched in VSR1 analysispresented in this paper (area enclosed by a thick black line) and recently publishedPowerFlux and E@H searches of the LIGO S5 data.

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interpolation procedure described in section VIA of [26]. The procedure consists ofupsampling using the Fourier transform, followed by a spline interpolation.

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 transformationdescribed by equation (14) is called resampling. In addition to the use of the FFT algorithmwe apply an interpolation of the FFT using the interbinning procedure (see [26], section VB).This results in the -statistic sampled twice as fine with respect to the standard FFT. Thisprocedure is much faster than the interpolation of the FFT obtained by padding the data withzeros and calculating a FFT that is twice as long. With the approximations described abovefor each value of the parameters ω1, δ and α, we calculate the -statistic efficiently for all thefrequency bins in the data segment of bandwidth 1 Hz.

In order to search the four-dimensional parameter space, we need to construct a four-dimensional grid. To minimize the computational cost we construct a grid that has thesmallest number of points for a certain assumed minimal match MM [29]. This problem isequivalent to the covering problem [30, 31] and it has the optimal solution in four dimensionsin the case of a lattice, i.e., a uniformly spaced grid. In order that our parameter space is alattice, the signals’ reduced Fisher matrix must have components that are independent of thevalues of the parameters. This is not the case for the signal given by equations (2)–(8); it canbe realized however for an approximate signal called the linear model described in sectionIIIB of [26]. The linear model consists of neglecting amplitude modulation of the signal anddiscarding the component of the vector tr ( )d joining the detector and the solar system bar-ycenter that is perpendicular to the ecliptic. This approximation is justified because theamplitude modulation is very slow compared to the phase modulation and the discardedcomponent in the phase is small compared to the others. As a result the linear model signalhlin(t) has a constant amplitude A0, and one can find parameters such that the phases are linearfunctions of them. We explicitly have (see section IIIB of [26] for details):

ϕ ϕ= +⎡⎣ ⎤⎦h t A t( ) cos ( ) , (17)lin 0 lin 0

where

ϕ ω ω α μ α μ= + + +t t t t t( ) ( ) ( ). (18)lin 0 12

1 1 2 2

The parameters α1 and α2 are defined by

α ω α δ ε δ εα ω α δ

= +=

: (sin cos cos sin sin ),: cos cos , (19)

1 0

2 0

where ε is the obliquity of the ecliptic, and μ t( )1 and μ t( )2 are known functions of thedetector ephemeris.

In order that the grid is compatible with application of the FFT, its points should beconstrained to coincide with Fourier frequencies at which the FFT is calculated. Moreover,we observed that a numerically accurate implementation of the interpolation to the barycentrictime (see equation (14)) is so computationally demanding that it may offset the advantage ofthe FFT. Therefore we introduced another constraint in the grid such that the resampling isneeded only once per sky position for all the spindown values. Construction of the con-strained grid is described in detail in section IV of [26]. In this search we have chosen thevalue of the minimal match =MM 3 2. The workflow of the coherent part of the searchprocedure is presented in figure 6.

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7. Vetoing procedure

We apply three vetoing criteria to the candidates obtained in the coherent part of the search—line width veto, stationary line veto and polar caps veto. Data from the detector alwayscontain some periodic interferences (lines) that are detector artifacts. An important part of ourvetoing procedure was to identify the lines in the data. We have therefore performed a Fouriersearch with frequency resolution of 1/(2 days) ≃ × −5.8 10 6 Hz for periodic signals of each ofthe two-day data segments. We compared the frequencies of the significant periodic signalsidentified by our analysis with the line frequencies obtained by the Virgo LineMonitor and wefound that all the lines 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 fl accordingto the following criterion

Δ− <f f f (20)l max

where the width Δfmax is estimated as

Δ = + + +( )f f v c f T T v c r c2 , (21)lmax max 1 max obs obs max max

Figure 6. Workflow of the -statistic search pipeline.

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where vmax is the maximum value of the velocity of the detector with respect to the SSBduring the observation time, rmax is the maximum distance to the SSB during the coherentobservation time Tobs and | |f1 max is the maximum of the absolute value of the frequencyderivative. Equation (21) determines the maximum smearing of the frequency on each side ofthe line due to frequency modulation induced by the filters applied in the -statistic.

7.2. Stationary line veto

Let us consider the instantaneous frequency finst of the signal, i.e., the time derivative of thephase:

πϕ= = + +

·+ +

·( )ft

tf f t

t

cf f t

t

cf

n v n r:

1

2

d ( )

d2

( )2 2

( ), (22)d d

inst 0 1 0 1 1

where =t t tv r( ) d ( ) dd d . The frequency derivative of the instantaneous frequency is given by

= +·

+ +·( )

f t

tf

t

cf f t

t

cf

n a n vd ( )

d2

( )2 4

( ), (23)d dinst

1 0 1 1

where =t t ta v( ) d ( ) dd d . Equation (23) is the rate of change of detector response frequencyfor a source whose SSB frequency and spindown are f0 and f1. An instrumental line has aconstant detector frequency and mimics a source for which the r.h.s of equation (23) vanishes.In practice, we veto candidates with

ϵ+·

+·

<( )

ft

cf

d t t t

cf

n a n r2

( )2

( ) d(24)d d

SL1 0

2 2

1

for some ϵ > 0SL . In the search we choose ϵ = T1SL obs2 , where Tobs is the observation time.

The above stationary line veto was introduced in [13] and refined in [32]; it was used in thefirst two E@H searches [19, 20].

7.3. Polar caps veto

We observe that many of the detected lines cluster around the poles where declination δ isclose to π± 2. An interference originating from a detector will correlate well with ourtemplates if the frequency modulation in equation (22) is minimized. Assuming that =f 01and that the diurnal motion of the Earth averages to 0 over two days observation time, thishappens when the quantity | · |tn v ( )d is minimized. We find that this quantity is close tominimum independently of the value of α when δ π= ± 2. Thus we veto candidates that aretoo close to the poles; we discard all candidates with the declination angle δ within three gridcells from the poles.

8. Coincidences

In order to find coincidences among the candidates, we applied a method similar to the oneused in the first two E@H searches [19, 20]. For each band we have searched for coincidencesamong candidates in different time frames. We are able to search for coincidences only inthose bands where there were two or more time frames with data selected for the analysis. Ifwe search for a real gravitational wave signal, we must take into account frequency evolutiondue to spindown of the rotating neutron star. Thus the first step in the coincidence analysiswas to transform all frequencies ω t( )l0 of the candidates to a common fiducial reference timetf. We have chosen the fiducial time to be the time of the first sample of the latest time frame

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that we analyzed, i.e., the 67th time frame. We have

ω ω ω= + −( ) [ ]t t t T t( ) 2 ( ) 67 , (25)f l l l0 1 obs

where tl is the time of the first sample of the lth time frame. The next step was to divide theparameter space into cells. This construction of the coincidence cell was different from that inthe E@H analysis. To construct the cells in the parameter space we have used the reducedFisher matrix Γ for the linear signal model defined by equations (17) and (18). The reducedFisher matrix is the projected Fisher matrix on the 4-dimensional space spanned byparameters κ ω ω α α= ( , , , )0 1 1 2 . We define the cell in the parameter space by the condition:

∑Γ κ κ˜ ⩽ 2. (26)k l

kl k l

,

Because the ephemeris of the detector is different in each of the time frames, the reducedFisher matrix is different in each time frame. To have a common coincidence grid we havechosen the grid defined by the latest frame, i.e., the frame no. 67 as the coincidence grid. Afterthe transformation of the candidate frequencies to a reference time and construction of thecoincidence grid, the coincidence algorithm for each of the bands proceeded in the followingsteps:

1. Transform angles α and δ to α1 and α2 coordinates (see equation (19))2. Transform candidate parameters to xl coordinates defined by

∑κ==

x V e , (27)l

k

k kl l

1

4

where =V k, 1 ,.., 4kl are eigenvectors of the matrix Γ , and el are its eigenvalues. In thesecoordinates 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 four-dimensional hypercubes. If there are more than onecandidates from a given data segment in a hypercube we select the candidate that has thehighest SNR. We do sorting for each time frame in the band. If there is more than onecandidate 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 shift wesearch for coincidences.

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

The most significant coincidence in each band is the one which has the highest multi-plicity. For each most significant coincidence we have calculated the false alarm probabilityi.e., the probability that such a coincidence can occur purely by chance. The false alarmprobability is calculated using the formula explained in the appendix and given byequation (A.6). This general formula applies to a variable number of candidates in varioustime slots and also takes into account the 24 shifts of the cells in the parameter space.

9. The search

In this analysis we have searched coherently 20 419 two-day time segments of data nar-rowbanded to 1 Hz. The analysis took approximately 15 months and involved several bigcomputing clusters. During the analysis 1000–3000 computing cores were used continuously.

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In the coherent part of the search described in section 6 we have used ×9.10 1016 templates,which is the number of -statistic values computed. This resulted in 20 419 candidate filescontaining ×4.21 1010 candidates. The candidates were subject to vetoing using three vetocriteria: 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. Nearlyall candidates were vetoed by the line veto, whereas 0.20% were vetoed by the stationary linecriterion and only × −3.2 10 2% by the polar caps veto. In figure 7 we present an example ofthe candidate distribution obtained from the coherent search of one narrow band data segmentand after the vetoing procedure. In the search the celestial sphere is divided by the eclipticplane into two hemispheres. Each hemisphere is mapped onto the disc defined byα α ω+ ⩽( ) 11

222

02 , where the coordinates α1 and α2 are defined by equation (19) and ω0 is

the angular frequency. The edge of the disc, determined by condition α α ω+ =( ) 112

22

02 , is

precisely the ecliptic. At this edge the grid of templates degenerates. This degeneracy

Figure 7. Distribution of candidates after vetoing obtained in the coherent -statisticsearch of two-day data segment −30 089 i.e., a segment with the time frame numberd = 30 and the band number b = 89. The left top panel shows data and their spectrum.The right top panel shows the distribution of 2 values of the candidates incomparison to the χ2 distribution with four degrees of freedom. The bottom left panelshows the distribution of signal-to-noise ratios ρ (see equation (12)) of the candidatesas a function of the frequency. The vertical black line is the periodic interferenceidentified in the data. The candidates in the band around the line are vetoed (seeequation (21)). The right bottom panel shows the distribution of the candidates on thesky in equatorial coordinates α and δ.

Class. Quantum Grav. 31 (2014) 165014 J Aasi et al

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manifests itself as a gap in the distribution of candidates on the sky (bottom right panel infigure 7), which follows exactly the ecliptic in the equatorial coordinates δ and α.

In the next step we have searched for the significant coincidences among the candidates.We have searched for coincidences in all the frequency bands where there were two or moredata segments analyzed. In figure 8 we have plotted the highest coincidence multiplicity foreach of the bands. The highest multiplicity was 6, and it occurred in 10 bands. The multi-plicity tends to grow with the frequency, because the size of the parameter space grows as f 3.For each band we have calculated the false alarm probability corresponding to the mostsignificant coincidence using equation (A.6). The most significant coincidence occurred inband no. 401, corresponding to the frequency range of ∼ [488.4, 489.4]Hz. It was a coin-cidence of multiplicity = 5, and its false alarm probability was 14.5%. Note that a coincidencewith the highest multiplicity is not the most significant. This is because the significancedepends on the number of time frames with candidates in a given band and also on thenumber of candidates in the time frames. By adopting a criterion used by E@H searches thatthe background coincidences correspond to false alarm probability of 0.1% or greater, weconclude that we have found no significant coincidence and thus no viable gravitational wavecandidate. Considering the significance of the coincidences, we could adopt even a 10% falsealarm probability as a background. Consequently, we proceed to the final stage 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 gravitational wave signalthat can be confidently detected. To estimate the sensitivity we use a procedure developed in[19]. We determine the sensitivity of the search in each of the 785 frequency bands that wehave searched. To determine the sensitivity, we perform Monte-Carlo simulations in which,for a given amplitude h0, we randomly select the other seven parameters of the signal:

Figure 8. Coincidences among the candidates. Top panel: maximum multiplicity ofcoincidence as function of frequency. Bottom panel: corresponding coincidence falsealarm probability.

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ω ω α δ ϕ ι, , , , ,0 1 0 and ψ. We choose frequency and spindown parameters uniformly overtheir range, and source positions uniformly over the sky. We choose angles ϕ0 and ψ uni-formly over the interval π[0, 2 ] and we choose ιcos uniformly over the interval −[ 1, 1]. Foreach band we add 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 -statistic search ofeach of the data segments where the signal was added, and store all the candidates above our -statistic threshold of 20. In this coherent analysis, to make the computation manageable,we search over only parameter space consisting of ±2 grid points around the nearest gridpoint where the signal was added. Then we apply our vetoing procedure to the candidatesobtained as explained in section 7. Finally, we perform coincidence analysis of the candidatesthat survive vetoing, which is described in section 8. We define a detectable signal if it iscoincident in more than 70% of the time frames in a given band. This condition is similar tothe condition used in the two E@H searches, where a coincidence method was used [19, 20].For bands with only one frame available, the coherent search over one 2-day data segmentwas performed. In this case the injected signal is declared detected if its signal-to-noise ratioobtained in the coherent search is larger than the signal-to-noise ratio of the loudest signal inthat data segment without an injection. For each band we inject signals with five differentamplitude values, and perform 100 randomized injections for each amplitude. For eachamplitude we calculate how many signals were detected, and by interpolation we determinethe amplitude corresponding to 90% of signals detected. This amplitude was defined as the90% confidence sensitivity. Sometimes even for the highest amplitude we have not reachedthe 90% detection probability. In this case we performed injections for higher amplitudes untilthe desired level of detectability was achieved. In figure 9, as an example, we present anestimation of the sensitivity in the band b = 369 corresponding to the frequency range of

Figure 9. Estimation of sensitivity in the band no. 369. In this band there are 20 timeframes. Thus the signal is detected if it is coincident in more than 14 frames. The blackdots are amplitudes of the injected signals and the corresponding detection probabilitiesestimated from the injections. The red circle is the interpolated amplitudecorresponding to 90% detection probability. This is the 90% confidence sensitivityfor this band.

Class. Quantum Grav. 31 (2014) 165014 J Aasi et al

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∼ [457.47, 458.44] Hz. The errors in the sensitivity estimates originate from calibration errorsin the amplitude and errors due to a finite number of Monte-Carlo injections. We use 100injections; hence from a binomial statistic, one σ is equivalent to 3% fluctuation. Thus theestimated amplitude sensitivity corresponds to confidence in the range from 87% to 93%. Toestimate how this uncertainty in confidence translates into uncertainty in the amplitude wehave performed an additional set of injections for a range of amplitudes close to the estimatedsensitivity and from the slope of the confidence versus amplitude we determined theuncertainty in the amplitude. To increase the accuracy of the error estimate, we have per-formed 1000 injections for each amplitude. The uncertainty in the amplitude was not morethan 5%. The calibration errors in VSR1 data are 6% (see section 2). Adding these two typesof errors in quadrature results in the total error in sensitivity estimate to be around 7%. Thesensitivity of this search obtained through Monte-Carlo simulations for the whole bandsearched is presented in figure 10. We see from figure 10 that the sensitivity essentiallyreflects the instrumental noise curve given in figure 4. We have made a fit of the sensitivityh0

90% to the one-sided spectral density Sh(f) of detector noise described by the followingrelation:

= h RS f( )

48 hours. (28)h

090%

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

Figure 10. The 90% confidence sensitivity of the all-sky search of Virgo VSR1 data inthe band from 100 Hz to 1 kHz. The dots show the source strain amplitude h0 for which90% of sources are confidently detected by this pipeline. The thin line is the rescaledinstrumental noise curve, see equation (28).

Class. Quantum Grav. 31 (2014) 165014 J Aasi et al

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11. Conclusions

The sensitivity of this search was 50% to 2 times better, depending on the bandwidth, thanthat of the LIGO S4 search [19] and comparable to the sensitivities obtained in the earlyLIGO S5 data [20], but 2 to 5 times worse than the upper limits in the E@H full LIGO S5data search [21]. This was due to the lower noise and longer observation time for the LIGOS5 data w.r.t the Virgo VSR1 data. However, for the first time in an all-sky search we haveestimated the sensitivity in the frequency band from 400 Hz to 1 kHz and the frequencyspindown range from − × −f7.2( 400 Hz) 100

9 Hz s−1 to − × −6.0 10 9 Hz s−1, which is apreviously unexplored region in the parameter space.

The next step is to test the search method described in this paper in the Mock DataChallenge (MDC) designed by the LIGO and Virgo projects to validate and compare pipe-lines that are proposed to be used in the analysis of the forthcoming data from the advanceddetectors. It is also planned to further test the pipeline presented here with other data setscollected by the LIGO and Virgo detectors.

Acknowledgments

The authors gratefully acknowledge the support of the United States National ScienceFoundation for the construction and operation of the LIGO Laboratory, the Science andTechnology Facilities Council of the United Kingdom, the Max-Planck-Society and the Stateof Niedersachsen/Germany for support of the construction and operation of the GEO600detector, and the Italian Istituto Nazionale di Fisica Nucleare and the French Centre Nationalde la Recherche Scientifique for the construction and operation of the Virgo detector. Theauthors also gratefully acknowledge the support of the research by these agencies and by theAustralian Research Council, the International Science Linkages program of the Common-wealth of Australia, the Council of Scientific and Industrial Research of India, the IstitutoNazionale di Fisica Nucleare of Italy, the Spanish Ministerio de Economía y Competitividad,the Conselleria dʼEconomia Hisenda i Innovació of the Govern de les Illes Balears, theFoundation for Fundamental Research on Matter supported by the Netherlands Organisationfor Scientific Research, the Polish Ministry of Science and Higher Education, the FOCUSProgramme of Foundation for Polish Science, the PL-Grid Infrastructure, the Royal Society,the Scottish Funding Council, the Scottish Universities Physics Alliance, The NationalAeronautics and Space Administration, the Carnegie Trust, the Leverhulme Trust, the Davidand Lucile Packard Foundation, the Research Corporation, and the Alfred P Sloan Founda-tion. This document has been assigned LIGO Laboratory document number LIGO-P1300133.

Appendix. False alarm coincidence probability

Let us assume that for a given frequency band we analyze L non-overlapping time segments.Suppose that the search of the lth segment produces Nl candidates. Let us assume that the sizeof the parameter space for each time segment is the same, and it can be divided into thenumber Ncell of independent cells. We would like to test the null hypothesis that coincidencesamong candidates from L segments are accidental. The probability for a candidate event tofall into any given coincidence cell is equal to N1 cell. Thus probability ϵl that a givencoincidence cell is populated with one or more candidate events is given by

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ϵ = − −⎛⎝⎜

⎞⎠⎟N

1 11

. (A.1)l

N

cell

l

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

ϵ =N

N. (A.2)l

l

cell

The probability p N( )F cell that any given coincidence cell out of the total of Ncell cells containscandidate events from Cmax or more distinct data segments is given by a generalized binomialdistribution

∑ ∑ ϵ ϵ ϵ ϵ=! − !

… − … −σ Π

σ σ σ σ= ∈

+( ) ( )( )p Nn L n

1

( )1 1 , (A.3)F

n C

L

L

n n Lcell

( )

(1) ( ) ( 1) ( )

max

where ∑σ Π∈ L( ) is the sum over all the permutations of the L data sequences. Finally theprobability PF that there is max or more coincidences in one or more of the Ncell cells is

= − −( )( )P p N1 1 . (A.4)F FN

cellcell

The above formula for the false alarm coincidence probability does not take into account thecase when candidate events are located on opposite sides of cell borders, edges and corners.In order to find these coincidences the entire cell coincidence grid is shifted by half a cellwidth in all possible =2 164 combinations of the four parameter-space dimensions, andcoincidences are searched in all 16 coincidence grids. This leads to a higher number ofaccidental coincidences, and consequently equation (A.4) underestimates the false alarmprobability. Let us consider the simplest one-dimensional case. In this case we have =2 21

possible shifts (the original coincidence grid and the one shifted by half). This increasesprobability p N( )F cell by a factor of 2 if the two cell coincidence grids were independent.However the cells overlap by half and some coincidences would be counted twice. Toaccount for this we divide the cells in the coincidence grid by half resulting in N2 cell cells anddefine the false alarm probability p N(2 )F cell that any given half of the coincidence cells out ofthe total of N2 cell half cells contains candidate events from Cmax or more distinct datasegments. These are coincidences that were already counted, and consequently the false alarmprobability with the cell shift is −p N p N2 ( ) (2 )F Fcell cell . This results in the false alarmprobability

= − − −( )( )( ) ( )P p N p N1 1 2 2 . (A.5)F F F

Ncell cell

cell

To generalize the above formula to higher dimensions we need to consider further shifts anddivisions of the cells. In the four dimension case this leads to the formula for the probabilityPF

shifts that there are max or more independent coincidences in one or more of the Ncell cells inall 16 grid shifts given by

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= − −

− + + +

− + +

− +

−

⎡⎣

⎤⎦⎥

( )( )( )

)

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( )

(( ) ( ) ( )

( ) ( ) ( )( ) ( )

( )

( )

( )

P p N

p N p N p N p N

p N p N p N

p N p N

p N

1 1 2

2 2 2 2

2 2 2

2 2

2 . (A.6)

F F c

F c F c F c F c

F c F c F c

F c F c

F c

N

shifts 4

41

42

2 43

3 44

4

42

2 43

3 44

4

43

3 44

4

44

4c

By choosing a certain false alarm probability PF, we can calculate the threshold numbermax of coincidences. If we obtain more than max coincidences in our search we reject thenull hypothesis that coincidences are accidental only at the significance level of PF.

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