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Narrow-band search of continuous gravitational-wave signals from Crab and Vela

pulsars in Virgo VSR4 data

J Aasi1 B P Abbott1 R Abbott1 T Abbott2 M R Abernathy1 F Acernese34 K Ackley5 C Adams6

T Adams78 T Adams8 P Addesso9 R X Adhikari1 V Adya10 C Affeldt10 M Agathos11 K Agatsuma11

N Aggarwal12 O D Aguiar13 A Ain14 P Ajith15 A Alemic16 B Allen1718 A Allocca1920 D Amariutei5

S B Anderson1 W G Anderson18 K Arai1 M C Araya1 C Arceneaux21 J S Areeda22 G Ashton23

S Ast24 S M Aston6 P Astone25 P Aufmuth24 C Aulbert17 B E Aylott26 S Babak27 P T Baker28

F Baldaccini2930 G Ballardin31 S W Ballmer16 J C Barayoga1 M Barbet5 S Barclay32 B C Barish1

D Barker33 F Barone34 B Barr32 L Barsotti12 M Barsuglia34 J Bartlett33 M A Barton33 I Bartos35

R Bassiri36 A Basti3720 J C Batch33 Th S Bauer11 C Baune10 V Bavigadda31 B Behnke27 M Bejger38

C Belczynski39 A S Bell32 C Bell32 M Benacquista40 J Bergman33 G Bergmann10 C P L Berry26

D Bersanetti4142 A Bertolini11 J Betzwieser6 S Bhagwat16 R Bhandare43 I A Bilenko44 G Billingsley1

J Birch6 S Biscans12 M Bitossi3120 C Biwer16 M A Bizouard45 J K Blackburn1 L Blackburn46

C D Blair47 D Blair47 S Bloemen1148 O Bock17 T P Bodiya12 M Boer49 G Bogaert49 P Bojtos50

C Bond26 F Bondu51 L Bonelli3720 R Bonnand8 R Bork1 M Born10 V Boschi20 Sukanta Bose1452

C Bradaschia20 P R Brady18 V B Braginsky44 M Branchesi5354 J E Brau55 T Briant56 D O Bridges6

A Brillet49 M Brinkmann10 V Brisson45 A F Brooks1 D A Brown16 D D Brown26 N M Brown12

S Buchman36 A Buikema12 T Bulik39 H J Bulten5711 A Buonanno58 D Buskulic8 C Buy34 L Cadonati59

G Cagnoli60 J Calderon Bustillo61 E Calloni624 J B Camp46 K C Cannon63 J Cao64 C D Capano58

F Carbognani31 S Caride65 S Caudill18 M Cavaglia21 F Cavalier45 R Cavalieri31 G Cella20 C Cepeda1

E Cesarini66 R Chakraborty1 T Chalermsongsak1 S J Chamberlin18 S Chao67 P Charlton68

E Chassande-Mottin34 Y Chen69 A Chincarini42 A Chiummo31 H S Cho70 M Cho58 J H Chow71

N Christensen72 Q Chu47 S Chua56 S Chung47 G Ciani5 F Clara33 J A Clark59 F Cleva49 E Coccia7374

P-F Cohadon56 A Colla7525 C Collette76 M Colombini30 L Cominsky77 M Constancio Jr13 A Conte7525

D Cook33 T R Corbitt2 N Cornish28 A Corsi78 C A Costa13 M W Coughlin72 J-P Coulon49

S Countryman35 P Couvares16 D M Coward47 M J Cowart6 D C Coyne1 R Coyne78 K Craig32

J D E Creighton18 T D Creighton40 J Cripe2 S G Crowder79 A Cumming32 L Cunningham32 E Cuoco31

C Cutler69 K Dahl10 T Dal Canton17 M Damjanic10 S L Danilishin47 S DrsquoAntonio66 K Danzmann2410

L Dartez40 V Dattilo31 I Dave43 H Daveloza40 M Davier45 G S Davies32 E J Daw80 R Day31

D DeBra36 G Debreczeni81 J Degallaix60 M De Laurentis624 S Deleglise56 W Del Pozzo26 T Denker10

T Dent17 H Dereli49 V Dergachev1 R De Rosa624 R T DeRosa2 R DeSalvo9 S Dhurandhar14 M Dıaz40

L Di Fiore4 A Di Lieto3720 I Di Palma27 A Di Virgilio20 G Dojcinoski82 V Dolique60 E Dominguez83

F Donovan12 K L Dooley10 S Doravari6 R Douglas32 T P Downes18 M Drago8485 J C Driggers1 Z Du64

M Ducrot8 S Dwyer33 T Eberle10 T Edo80 M Edwards7 M Edwards72 A Effler2 H-B Eggenstein17

P Ehrens1 J Eichholz5 S S Eikenberry5 R Essick12 T Etzel1 M Evans12 T Evans6 M Factourovich35

V Fafone7366 S Fairhurst7 X Fan32 Q Fang47 S Farinon42 B Farr86 W M Farr26 M Favata82 M Fays7

H Fehrmann17 M M Fejer36 D Feldbaum56 I Ferrante3720 E C Ferreira13 F Ferrini31 F Fidecaro3720

I Fiori31 R P Fisher16 R Flaminio60 J-D Fournier49 S Franco45 S Frasca7525 F Frasconi20 Z Frei50

A Freise26 R Frey55 T T Fricke10 P Fritschel12 V V Frolov6 S Fuentes-Tapia40 P Fulda5 M Fyffe6

J R Gair87 L Gammaitoni2930 S Gaonkar14 F Garufi624 A Gatto34 N Gehrels46 G Gemme42 B Gendre49

E Genin31 A Gennai20 L A Gergely88 S Ghosh1148 J A Giaime62 K D Giardina6 A Giazotto20

J Gleason5 E Goetz17 R Goetz5 L Gondan50 G Gonzalez2 N Gordon32 M L Gorodetsky44 S Gossan69

S Goszligler10 R Gouaty8 C Graf32 P B Graff46 M Granata60 A Grant32 S Gras12 C Gray33 G Greco5453

R J S Greenhalgh89 A M Gretarsson90 P Groot48 H Grote10 S Grunewald27 G M Guidi5354 C J Guido6

X Guo64 K Gushwa1 E K Gustafson1 R Gustafson65 J Hacker22 E D Hall1 G Hammond32 M Hanke10

J Hanks33 C Hanna91 M D Hannam7 J Hanson6 T Hardwick552 J Harms54 G M Harry92 I W Harry27

M Hart32 M T Hartman5 C-J Haster26 K Haughian32 S Hee87 A Heidmann56 M Heintze56 G Heinzel10

H Heitmann49 P Hello45 G Hemming31 M Hendry32 I S Heng32 A W Heptonstall1 M Heurs10

M Hewitson10 S Hild32 D Hoak59 K A Hodge1 D Hofman60 S E Hollitt93 K Holt6 P Hopkins7

D J Hosken93 J Hough32 E Houston32 E J Howell47 Y M Hu32 E Huerta94 B Hughey90 S Husa61

S H Huttner32 M Huynh18 T Huynh-Dinh6 A Idrisy91 N Indik17 D R Ingram33 R Inta91 G Islas22

J C Isler16 T Isogai12 B R Iyer95 K Izumi33 M Jacobson1 H Jang96 P Jaranowski97 S Jawahar98

2

Y Ji64 F Jimenez-Forteza61 W W Johnson2 D I Jones23 R Jones32 RJG Jonker11 L Ju47 Haris K99

V Kalogera86 S Kandhasamy21 G Kang96 J B Kanner1 M Kasprzack4531 E Katsavounidis12 W Katzman6

H Kaufer24 S Kaufer24 T Kaur47 K Kawabe33 F Kawazoe10 F Kefelian49 G M Keiser36 D Keitel17

D B Kelley16 W Kells1 D G Keppel17 J S Key40 A Khalaidovski10 F Y Khalili44 E A Khazanov100

C Kim10196 K Kim102 N G Kim96 N Kim36 Y-M Kim70 E J King93 P J King33 D L Kinzel6

J S Kissel33 S Klimenko5 J Kline18 S Koehlenbeck10 K Kokeyama2 V Kondrashov1 M Korobko10

W Z Korth1 I Kowalska39 D B Kozak1 V Kringel10 B Krishnan17 A Krolak103104 C Krueger24

G Kuehn10 A Kumar105 P Kumar16 L Kuo67 A Kutynia103 M Landry33 B Lantz36 S Larson86

P D Lasky106 A Lazzarini1 C Lazzaro107 C Lazzaro59 J Le86 P Leaci27 S Leavey32 E Lebigot34

E O Lebigot64 C H Lee70 H K Lee102 H M Lee101 M Leonardi8485 J R Leong10 N Leroy45 N Letendre8

Y Levin108 B Levine33 J Lewis1 T G F Li1 K Libbrecht1 A Libson12 A C Lin36 T B Littenberg86

N A Lockerbie98 V Lockett22 J Logue32 A L Lombardi59 M Lorenzini74 V Loriette109 M Lormand6

G Losurdo54 J Lough17 M J Lubinski33 H Luck2410 A P Lundgren17 R Lynch12 Y Ma47 J Macarthur32

T MacDonald36 B Machenschalk17 M MacInnis12 D M Macleod2 F Magana-Sandoval16 R Magee52

M Mageswaran1 C Maglione83 K Mailand1 E Majorana25 I Maksimovic109 V Malvezzi7366 N Man49

I Mandel26 V Mandic79 V Mangano32 V Mangano7525 G L Mansell71 M Mantovani3120 F Marchesoni11030

F Marion8 S Marka35 Z Marka35 A Markosyan36 E Maros1 F Martelli5354 L Martellini49 I W Martin32

R M Martin5 D Martynov1 J N Marx1 K Mason12 A Masserot8 T J Massinger16 F Matichard12

L Matone35 N Mavalvala12 N Mazumder99 G Mazzolo17 R McCarthy33 D E McClelland71 S McCormick6

S C McGuire111 G McIntyre1 J McIver59 K McLin77 S McWilliams94 D Meacher49 G D Meadors65

J Meidam11 M Meinders24 A Melatos106 G Mendell33 R A Mercer18 S Meshkov1 C Messenger32

P M Meyers79 F Mezzani2575 H Miao26 C Michel60 H Middleton26 E E Mikhailov112 L Milano624

A Miller113 J Miller12 M Millhouse28 Y Minenkov66 J Ming27 S Mirshekari114 C Mishra15 S Mitra14

V P Mitrofanov44 G Mitselmakher5 R Mittleman12 B Moe18 A Moggi20 M Mohan31 S D Mohanty40

S R P Mohapatra12 B Moore82 D Moraru33 G Moreno33 S R Morriss40 K Mossavi10 B Mours8

C M Mow-Lowry10 C L Mueller5 G Mueller5 S Mukherjee40 A Mullavey6 J Munch93 D Murphy35

P G Murray32 A Mytidis5 M F Nagy81 I Nardecchia7366 T Nash1 L Naticchioni7525 R K Nayak115

V Necula5 K Nedkova59 G Nelemans1148 I Neri2930 M Neri4142 G Newton32 T Nguyen71 A B Nielsen17

S Nissanke69 A H Nitz16 F Nocera31 D Nolting6 M E N Normandin40 L K Nuttall18 E Ochsner18

J OrsquoDell89 E Oelker12 G H Ogin116 J J Oh117 S H Oh117 F Ohme7 P Oppermann10 R Oram6

B OrsquoReilly6 W Ortega83 R OrsquoShaughnessy118 C Osthelder1 D J Ottaway93 R S Ottens5 H Overmier6

B J Owen91 C Padilla22 A Pai99 S Pai43 O Palashov100 C Palomba25 A Pal-Singh10 H Pan67

C Pankow18 F Pannarale7 B C Pant43 F Paoletti3120 M A Papa1827 H Paris36 A Pasqualetti31

R Passaquieti3720 D Passuello20 Z Patrick36 M Pedraza1 L Pekowsky16 A Pele33 S Penn119 A Perreca16

M Phelps1 M Pichot49 F Piergiovanni5354 V Pierro9 G Pillant31 L Pinard60 I M Pinto9 M Pitkin32

J Poeld10 R Poggiani3720 A Post17 A Poteomkin100 J Powell32 J Prasad14 V Predoi7 S Premachandra108

T Prestegard79 L R Price1 M Prijatelj31 M Principe9 S Privitera1 G A Prodi8485 L Prokhorov44

O Puncken40 M Punturo30 P Puppo25 M Purrer7 J Qin47 V Quetschke40 E Quintero1 G Quiroga83

R Quitzow-James55 F J Raab33 D S Rabeling71 I Racz81 H Radkins33 P Raffai50 S Raja43

G Rajalakshmi120 M Rakhmanov40 K Ramirez40 P Rapagnani7525 V Raymond1 M Razzano3720 V Re7366

C M Reed33 T Regimbau49 L Rei42 S Reid121 D H Reitze15 O Reula83 F Ricci7525 K Riles65

N A Robertson132 R Robie32 F Robinet45 A Rocchi66 L Rolland8 J G Rollins1 V Roma55 R Romano34

G Romanov112 J H Romie6 D Rosinska12238 S Rowan32 A Rudiger10 P Ruggi31 K Ryan33 S Sachdev1

T Sadecki33 L Sadeghian18 M Saleem99 F Salemi17 L Sammut106 V Sandberg33 J R Sanders65

V Sannibale1 I Santiago-Prieto32 B Sassolas60 B S Sathyaprakash7 P R Saulson16 R Savage33

A Sawadsky24 J Scheuer86 R Schilling10 P Schmidt71 R Schnabel10123 R M S Schofield55 E Schreiber10

D Schuette10 B F Schutz727 J Scott32 S M Scott71 D Sellers6 A S Sengupta124 D Sentenac31

V Sequino7366 R Serafinelli7525 A Sergeev100 G Serna22 A Sevigny33 D A Shaddock71 S Shah1148

M S Shahriar86 M Shaltev17 Z Shao1 B Shapiro36 P Shawhan58 D H Shoemaker12 T L Sidery26

K Siellez49 X Siemens18 D Sigg33 A D Silva13 D Simakov10 A Singer1 L Singer1 R Singh2 A M Sintes61

B J J Slagmolen71 J R Smith22 M R Smith1 R J E Smith1 N D Smith-Lefebvre1 E J Son117

B Sorazu32 T Souradeep14 A Staley35 J Stebbins36 M Steinke10 J Steinlechner32 S Steinlechner32

D Steinmeyer10 B C Stephens18 S Steplewski52 S Stevenson26 R Stone40 K A Strain32 N Straniero60

3

S Strigin44 R Sturani114 A L Stuver6 T Z Summerscales125 P J Sutton7 B Swinkels31 M Szczepanczyk90

G Szeifert50 M Tacca34 D Talukder55 D B Tanner5 M Tapai88 S P Tarabrin10 A Taracchini58 R Taylor1

G Tellez40 T Theeg10 M P Thirugnanasambandam1 M Thomas6 P Thomas33 K A Thorne6 K S Thorne69

E Thrane1 V Tiwari5 C Tomlinson80 M Tonelli3720 C V Torres40 C I Torrie132 F Travasso2930

G Traylor6 M Tse12 D Tshilumba76 D Ugolini126 C S Unnikrishnan120 A L Urban18 S A Usman16

H Vahlbruch24 G Vajente1 G Vajente3720 G Valdes40 M Vallisneri69 N van Bakel11 M van Beuzekom11

J F J van den Brand5711 C van den Broeck11 M V van der Sluys1148 J van Heijningen11 A A van Veggel32

S Vass1 M Vasuth81 R Vaulin12 A Vecchio26 G Vedovato107 J Veitch26 J Veitch11 P J Veitch93

K Venkateswara127 D Verkindt8 F Vetrano5354 A Vicere5354 R Vincent-Finley111 J-Y Vinet49 S Vitale12

T Vo33 H Vocca2930 C Vorvick33 W D Vousden26 S P Vyatchanin44 A R Wade71 L Wade18 M Wade18

M Walker2 L Wallace1 S Walsh18 H Wang26 M Wang26 X Wang64 R L Ward71 J Warner33 M Was10

B Weaver33 L-W Wei49 M Weinert10 A J Weinstein1 R Weiss12 T Welborn6 L Wen47 P Wessels10

T Westphal10 K Wette17 J T Whelan11817 D J White80 B F Whiting5 C Wilkinson33 L Williams5

R Williams1 A R Williamson7 J L Willis113 B Willke2410 M Wimmer10 W Winkler10 C C Wipf12

H Wittel10 G Woan32 J Worden33 S Xie76 J Yablon86 I Yakushin6 W Yam12 H Yamamoto1

C C Yancey58 Q Yang64 M Yvert8 A Zadrozny103 M Zanolin90 J-P Zendri107 Fan Zhang1264 L Zhang1

M Zhang112 Y Zhang118 C Zhao47 M Zhou86 X J Zhu47 M E Zucker12 S Zuraw59 and J Zweizig1

1LIGO California Institute of Technology Pasadena CA 91125 USA2Louisiana State University Baton Rouge LA 70803 USA

3Universita di Salerno Fisciano I-84084 Salerno Italy4INFN Sezione di Napoli Complesso Universitario di Monte SAngelo I-80126 Napoli Italy

5University of Florida Gainesville FL 32611 USA6LIGO Livingston Observatory Livingston LA 70754 USA7Cardiff University Cardiff CF24 3AA United Kingdom

8Laboratoire drsquoAnnecy-le-Vieux de Physique des Particules (LAPP)Universite de Savoie CNRSIN2P3 F-74941 Annecy-le-Vieux France

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

10Experimental Group Albert-Einstein-Institut Max-Planck-Institut fur Gravitationsphysik D-30167 Hannover Germany11Nikhef Science Park 1098 XG Amsterdam The Netherlands

12LIGO Massachusetts Institute of Technology Cambridge MA 02139 USA13Instituto Nacional de Pesquisas Espaciais 12227-010 - Sao Jose dos Campos SP Brazil

14Inter-University Centre for Astronomy and Astrophysics Pune - 411007 India15International Centre for Theoretical Sciences Tata Institute of Fundamental Research Bangalore 560012 India

16Syracuse University Syracuse NY 13244 USA17Data Analysis Group Albert-Einstein-Institut Max-Planck-Institut fur Gravitationsphysik D-30167 Hannover Germany

18University of WisconsinndashMilwaukee Milwaukee WI 53201 USA19Universita di Siena I-53100 Siena Italy

20INFN Sezione di Pisa I-56127 Pisa Italy21The University of Mississippi University MS 38677 USA

22California State University Fullerton Fullerton CA 92831 USA23University of Southampton Southampton SO17 1BJ United Kingdom

24Leibniz Universitat Hannover D-30167 Hannover Germany25INFN Sezione di Roma I-00185 Roma Italy

26University of Birmingham Birmingham B15 2TT United Kingdom27Albert-Einstein-Institut Max-Planck-Institut fur Gravitationsphysik D-14476 Golm Germany

28Montana State University Bozeman MT 59717 USA29Universita di Perugia I-06123 Perugia Italy

30INFN Sezione di Perugia I-06123 Perugia Italy31European Gravitational Observatory (EGO) I-56021 Cascina Pisa Italy

32SUPA University of Glasgow Glasgow G12 8QQ United Kingdom33LIGO Hanford Observatory Richland WA 99352 USA

34APC AstroParticule et Cosmologie Universite Paris DiderotCNRSIN2P3 CEAIrfu Observatoire de Paris Sorbonne Paris Cite 10

rue Alice Domon et Leonie Duquet F-75205 Paris Cedex 13 France35Columbia University New York NY 10027 USA36Stanford University Stanford CA 94305 USA

37Universita di Pisa I-56127 Pisa Italy38CAMK-PAN 00-716 Warsaw Poland

39Astronomical Observatory Warsaw University 00-478 Warsaw Poland

4

40The University of Texas at Brownsville Brownsville TX 78520 USA41Universita degli Studi di Genova I-16146 Genova Italy

42INFN Sezione di Genova I-16146 Genova Italy43RRCAT Indore MP 452013 India

44Faculty of Physics Lomonosov Moscow State University Moscow 119991 Russia45LAL Universite Paris-Sud IN2P3CNRS F-91898 Orsay France46NASAGoddard Space Flight Center Greenbelt MD 20771 USA47University of Western Australia Crawley WA 6009 Australia

48Department of AstrophysicsIMAPP Radboud University NijmegenPO Box 9010 6500 GL Nijmegen The Netherlands

49ARTEMIS Universite Nice-Sophia-Antipolis CNRS and Observatoire de la Cote drsquoAzur F-06304 Nice France50MTA Eotvos University lsquoLenduletrsquo A R G Budapest 1117 Hungary

51Institut de Physique de Rennes CNRS Universite de Rennes 1 F-35042 Rennes France52Washington State University Pullman WA 99164 USA

53Universita degli Studi di Urbino rsquoCarlo Borsquo I-61029 Urbino Italy54INFN Sezione di Firenze I-50019 Sesto Fiorentino Firenze Italy

55University of Oregon Eugene OR 97403 USA56Laboratoire Kastler Brossel ENS CNRS UPMC

Universite Pierre et Marie Curie F-75005 Paris France57VU University Amsterdam 1081 HV Amsterdam The Netherlands

58University of Maryland College Park MD 20742 USA59University of Massachusetts Amherst Amherst MA 01003 USA

60Laboratoire des Materiaux Avances (LMA) IN2P3CNRSUniversite de Lyon F-69622 Villeurbanne Lyon France

61Universitat de les Illes BalearsmdashIEEC E-07122 Palma de Mallorca Spain62Universita di Napoli rsquoFederico IIrsquo Complesso Universitario di Monte SAngelo I-80126 Napoli Italy

63Canadian Institute for Theoretical AstrophysicsUniversity of Toronto Toronto Ontario M5S 3H8 Canada

64Tsinghua University Beijing 100084 China65University of Michigan Ann Arbor MI 48109 USA

66INFN Sezione di Roma Tor Vergata I-00133 Roma Italy67National Tsing Hua University Hsinchu Taiwan 300

68Charles Sturt University Wagga Wagga NSW 2678 Australia69Caltech-CaRT Pasadena CA 91125 USA

70Pusan National University Busan 609-735 Korea71Australian National University Canberra ACT 0200 Australia

72Carleton College Northfield MN 55057 USA73Universita di Roma Tor Vergata I-00133 Roma Italy

74INFN Gran Sasso Science Institute I-67100 LrsquoAquila Italy75Universita di Roma rsquoLa Sapienzarsquo I-00185 Roma Italy

76University of Brussels Brussels 1050 Belgium77Sonoma State University Rohnert Park CA 94928 USA

78Texas Tech University Lubbock TX 79409 USA79University of Minnesota Minneapolis MN 55455 USA

80The University of Sheffield Sheffield S10 2TN United Kingdom81Wigner RCP RMKI H-1121 Budapest Konkoly Thege Miklos ut 29-33 Hungary

82Montclair State University Montclair NJ 07043 USA83Argentinian Gravitational Wave Group Cordoba Cordoba 5000 Argentina

84Universita di Trento I-38123 Povo Trento Italy85INFN Trento Institute for Fundamental Physics and Applications I-38123 Povo Trento Italy

86Northwestern University Evanston IL 60208 USA87University of Cambridge Cambridge CB2 1TN United Kingdom

88University of Szeged Dom ter 9 Szeged 6720 Hungary89Rutherford Appleton Laboratory HSIC Chilton Didcot Oxon OX11 0QX United Kingdom

90Embry-Riddle Aeronautical University Prescott AZ 86301 USA91The Pennsylvania State University University Park PA 16802 USA

92American University Washington DC 20016 USA93University of Adelaide Adelaide SA 5005 Australia

94West Virginia University Morgantown WV 26506 USA95Raman Research Institute Bangalore Karnataka 560080 India

96Korea Institute of Science and Technology Information Daejeon 305-806 Korea97University of Bia lystok 15-424 Bia lystok Poland

98SUPA University of Strathclyde Glasgow G1 1XQ United Kingdom99IISER-TVM CET Campus Trivandrum Kerala 695016 India

5

100Institute of Applied Physics Nizhny Novgorod 603950 Russia101Seoul National University Seoul 151-742 Korea

102Hanyang University Seoul 133-791 Korea103NCBJ 05-400 Swierk-Otwock Poland

104IM-PAN 00-956 Warsaw Poland105Institute for Plasma Research Bhat Gandhinagar 382428 India

106The University of Melbourne Parkville VIC 3010 Australia107INFN Sezione di Padova I-35131 Padova Italy

108Monash University Victoria 3800 Australia109ESPCI CNRS F-75005 Paris France

110Universita di Camerino Dipartimento di Fisica I-62032 Camerino Italy111Southern University and AampM College Baton Rouge LA 70813 USA

112College of William and Mary Williamsburg VA 23187 USA113Abilene Christian University Abilene TX 79699 USA

114Instituto de Fısica Teorica Univ Estadual PaulistaICTP South AmericanInstitute for Fundamental Research Sao Paulo SP 01140-070 Brazil

115IISER-Kolkata Mohanpur West Bengal 741252 India116Whitman College 280 Boyer Ave Walla Walla WA 9936 USA

117National Institute for Mathematical Sciences Daejeon 305-390 Korea118Rochester Institute of Technology Rochester NY 14623 USA119Hobart and William Smith Colleges Geneva NY 14456 USA

120Tata Institute for Fundamental Research Mumbai 400005 India121SUPA University of the West of Scotland Paisley PA1 2BE United Kingdom

122Institute of Astronomy 65-265 Zielona Gora Poland123Universitat Hamburg D-22761 Hamburg Germany

124Indian Institute of Technology Gandhinagar Ahmedabad Gujarat 382424 India125Andrews University Berrien Springs MI 49104 USA

126Trinity University San Antonio TX 78212 USA127University of Washington Seattle WA 98195 USA

In this paper we present the results of a coherent narrow-band search for continuous gravitational-wave signals from the Crab and Vela pulsars conducted on Virgo VSR4 data In order to take intoaccount a possible small mismatch between the gravitational wave frequency and two times thestar rotation frequency inferred from measurement of the electromagnetic pulse rate a range of002 Hz around two times the star rotational frequency has been searched for both the pulsars Noevidence for a signal has been found and 95 confidence level upper limits have been computedboth assuming polarization parameters are completely unknown and that they are known with someuncertainty as derived from X-ray observations of the pulsar wind torii For Vela the upper limitsare comparable to the spin-down limit computed assuming that all the observed spin-down is dueto the emission of gravitational waves For Crab the upper limits are about a factor of two belowthe spin-down limit and represent a significant improvement with respect to past analysis This isthe first time the spin-down limit is significantly overcome in a narrow-band search

I INTRODUCTION

Continuous gravitational-wave signals (CW) are emit-ted by spinning neutron stars if asymmetric with respectto the rotation axis The asymmetry can be due to var-ious mechanisms like a non-axismmetric residual strainfrom the starrsquos birth or a strong internal magnetic fieldnot aligned to the rotation axis see eg [1] When thesource parameters position frequency spin-down areknown with high accuracy targeted searches can be doneusing optimal analysis methods based on matched filter-ing This happens for instance with known pulsars forwhich accurate ephemerides are often obtained from elec-tromagnetic (EM) observations especially in the radiogamma-ray and X-ray band This means that a strict cor-relation between the gravitational wave frequency f0 andthe starrsquos measured rotational frequency frot is assumedIn the classical case of a non-axisymmetric neutron star

rotating around one of its principal axes of inertia thegravitational frequency would be exactly twice the rota-tion frequency of the star Several targeted searches forCW have been conducted on data from first generationinterferometric detectors No evidence for a signal hasbeen found but interesting upper limits have been placedin a few cases [2ndash4] Given the uncertainties on gravita-tional wave emission mechanisms and also the lack of afull detailed picture of the electro-magnetic emission ge-ometry it is not obvious at all that the gravitational-waveemission takes place at exactly twice the star measuredpulse rate or that such relation holds for observationtimes of months to years For instance if a neutron staris made of a crust and a core rotating at slightly differentrates and if the gravitational-wave emission is dominatedby an asymmetry in the core then a search targeted at2frot would assume a wrong signal frequency We thenconsider here that the signal frequency can be slightly

6

different with respect to f0 = 2frot and can vary in theinterval

f(t) isin [f0(t)(1minus δ) f0(t)(1 + δ)] (1)

where δ is a small positive real number Following thediscussion in [5] if the star crust and core are linkedby some torque that tends to enforce corotation on atimescale τc then δ sim τcτsd where τsd sim f0f0 is thecharacteristic spin-down time A relation of the form ofEq (1) also holds in the case the gravitational radiationis produced by free precession of a nearly bi-axial star [6]in which case δ is of the order of (Izz minus Ixx)Ixx whereIxx and Izz are the star moments of inertia with respectto a principal axis on the equatorial plane and alignedwith the rotation axis respectively In general a value ofδ of the order of say 10minus4 corresponds to τc sim 10minus4τsdwhich depending on the specific targeted pulsar can beseveral months or years This would be comparable orlarger than the longest timescale observed in pulsar glitchrecovery where a recoupling between the two componentsmight occur In terms of free precession δ sim 10minus4 is onthe high end of the range of deformations that neutronstars could sustain [7ndash9]Narrow-band searches have not received much atten-

tion until now one notable exception being the Crabnarrow-band search over the first 9 months of LIGO S5data [5] based on the F -statistic [10] In previous work[11] an optimal method based on matched filtering in thespace of signal Fourier components has been proposedto carry out a search for CW over a small frequency andspin-down range around the values inferred from EM ob-servations In this paper we describe the application ofsuch a method to a narrow-band search of CW from Craband Vela pulsars in the data of Virgo VSR4 run As noevidence for a signal is found we place upper limits onsignal amplitudeThe outline of the paper is the following In Sec II the

expected gravitational wave signal from spinning neutronstars is introduced In Sec III the main characteristics ofVirgo VSR4 data are presented In Sec IV the analysismethod is described while in Sec V the analysis resultsare discussed Sec VI is dedicated to the validation testsof the analysis procedure Finally Sec VII contains theconclusions

II THE GRAVITATIONAL WAVE SIGNAL

A non-axisymmetric neutron star steadily spinningabout one of its principal axis emits a quadrupolargravitational-wave signal at twice the star rotational fre-quency frot which at the detector can be described [12]as

h(t) = H0(H+A+(t) +HtimesA

times(t))eı(ω0(t)t+Φ0) (2)

where taking the real part is understood The constantΦ0 is the initial signal phase The angular signal fre-

quency ω0(t) = 4πfrot(t) is a function of time Conse-quently the signal phase is not that of a simple monochro-matic signal and depends on both the intrinsic rotationalfrequency and frequency derivatives of the pulsar and onDoppler and propagation effects which include the Ein-stein delay and possibly the Shapiro delay These varia-tions are corrected in the time-domain in a way describedin Sec IV The two complex amplitudes H+ and Htimes aregiven respectively by

H+ =cos 2ψ minus ıη sin 2ψ

radic

1 + η2(3)

Htimes =sin 2ψ + ıη cos 2ψ

radic

1 + η2(4)

in which η is the ratio of the polarization ellipse semi-minor to semi-major axis and the polarization angle ψdefines the direction of the major axis with respect to thecelestial parallel of the source (measured counterclock-wise) The functions A+ and Atimes describe the detectorresponse as a function of time and are a linear combi-nation of terms depending on the Earth sidereal angularfrequency Ωoplus

A+ = a0 + a1c cosΩoplust+ a1s sinΩoplust+

a2c cos 2Ωoplust+ a2s sin 2Ωoplust (5)

Atimes = b1c cosΩoplust+ b1s sinΩoplust+

b2c cos 2Ωoplust+ b2s sin 2Ωoplust (6)

where the coefficients depend on the source position anddetector position and orientation on the Earth [12]As discussed eg in [3] the overall wave amplitude H0

and η are related to the ldquostandardrdquo signal amplitude

h0 =4π2G

c4Izzεf

20

d(7)

and to the angle ι between the star rotation axis and theline of sight to the source by

H0 = h0

radic

1 + 6 cos2 ι+ cos4 ι

4(8)

η = minus2 cos ι

1 + cos2 ι (9)

In Eq (7) G is the gravitational constant c is the lightvelocity Izz is the star moment of inertia with respectto the principal axis aligned with the rotation axis ε =IxxminusIyyIzz

is the equatorial ellipticity expressed in terms ofprincipal moments of inertia and d is the source distance

7

Equating the gravitational-wave luminosity to the ki-netic energy lost as the pulsar spins down gives us theso-called spin-down limit on gravitational-wave strain

hsd0 =

(

5

2

GIzz frotc3d2frot

)12

=

806times 10minus19 I1238

dkpc

(

|frot|

Hzs

)12(

frotHz

)minus12

(10)

where I38 is the starrsquos moment of inertia in units of1038 kgm2 frot is the time derivative of the rotationalfrequency and dkpc is the distance to the pulsar in kilo-parsecs The spin-down limit on the signal amplitudecorresponds to an upper limit on the starrsquos fiducial ellip-ticity

εsd = 0237

(

hsd010minus24

)

fminus2rot I

minus138 dkpc (11)

This quantity for a given neutron star equation of statecan be related to the physical ellipticity of the star sur-face [9] Setting a gravitational-wave upper limit belowthe spin-down limit is an important milestone in CWsearches as it allows us to constrain the fraction of spin-down energy due to the emission of gravitational-waves which gives insight into the spin-down energy budgetof the pulsar On the other hand the l = m = 2 massquadrupole moment Q22 is related to the gravitational-wave amplitude by

Q22 =

radic

15

8πIzzε = h0

(

c4d

16π2Gf2rot

)

radic

15

8π (12)

see eg [7] and is independent of any assumptions aboutthe starrsquos equation of state and its moment of inertia

III INSTRUMENTAL PERFORMANCE IN

VIRGO VSR4 RUN

Interferometric gravitational wave detectors LIGO[13] Virgo [14] GEO [15] have collected a large amountof data in recent years For the analysis described in thispaper we have used calibrated data sampled at 4096 Hzfrom Virgo VSR4 science run The run extended fromJune 3rd 2011 (1027 UTC) to September 5th 2011(1326 UTC) with a duty factor of about 81 corre-sponding to an effective duration of 76 days Calibra-tion uncertainties amounted to 75 in amplitude and(40+50fkHz) mrad in phase up to 500 Hz where fkHz isthe frequency in kilohertz The uncertainty on the ampli-tude will contribute to the uncertainty on the upper limiton signal amplitude together with that coming from thefinite size of the Monte Carlo simulation used to computeit see Sec V The calibration error on the phase can beshown to have a negligible impact on the analysis [3]

The low-frequency sensitivity of VSR4 was significantlybetter than that of previous Virgo runs especially due tothe use of monolithic mirror suspensions and basicallyin agreement with the planned sensitivity of the initialVirgo interferometer In Fig 1 a typical VSR4 strainsensitivity curve is shown In Fig 2 the average powerspectrum around Crab and Vela reference frequency isplotted Note the large sensitivity improvement aroundthe Vela frequency after removal of an instrumental dis-turbance about one month after the beginning of therun see the figure caption for more details

IV SEARCH DESCRIPTION

The analysis pipeline consists of several steps schemat-ically depicted in Fig 3 which we summarize here Moredetails are given in [11] The starting point is a collectionof windowed and interlaced (by half) ldquoshortrdquo (1024 sec-onds) FFTs (the Short FFT Database - SFDB) built fromcalibrated detector data [16] At this stage a first clean-ing procedure is applied to the data in order to removebig and short time duration disturbances that could re-duce the search sensitivity A small frequency band isthen extracted from the SFDB but large enough to con-tain the Doppler and spin-down variations of the signalpossibly emitted by the target pulsar In the analysesdescribed in this paper for instance it was of 015 HzAt this point we go back to the time domain (the sam-pling frequency is still the original one 4096 Hz) andmake barycentric and spin-down corrections Due to theDoppler effect the received frequency f(t) is related tothe emitted frequency f0(t) by the well-known relation(valid in the non-relativistic approximation)

f(t) =1

2π

dΦ(t)

dt= f0(t)

(

1 +~v middot n

c

)

(13)

where Φ(t) is the signal phase ~v = ~vrev+~vrot is the detec-tor velocity with respect to the Solar system barycenter(SSB) given by the sum of the Earthrsquos orbital velocity~vrev and rotational velocity ~vrot while n is the versoridentifying the source position In practice the Dopplereffect is efficiently corrected in the time domain by chang-ing the time stamp t of data samples according to

τ1 = t+~r(t) middot n

c= t+∆R (14)

where ~r is the detector position in the SSB The correc-tion term ∆R called Roemer delay amounts up to about500 seconds and corresponds to the time taken by a signaltraveling at the speed of light to cover the distance be-tween the detector and the SSB Unlike Eq (13) Eq (14)does not depend on the frequency which means that onesingle correction holds for every frequency In fact thereare other smaller relativistic effects that must be takeninto account when making barycentric corrections Oneis the Einstein delay ∆E due to the Earth motion and

8

102

103

10minus23

10minus22

10minus21

10minus20

10minus19

Frequency [Hz]

h n [Hzminus

12 ]

FIG 1 A typical sensitivity curve of Virgo VSR4 run expressed in terms of noise spectral density

032 034 036 038 04 042 04410

minus3

10minus2

Frequency minus220 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

036 038 04 042 044 046 048 05

10minus4

10minus3

Frequency minus 590 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

FIG 2 Average spectrum of VSR4 data around Vela (top plot) and Crab (bottom plot) reference frequency (identified by thevertical black lines) For Vela two spectra are in fact shown in the same plot The upper one (black curve) has been computedover the first 29 days of the run when the frequency region around Vela was affected by an instrumental disturbance Thiswas due to a non-linear coupling between the DARM control line at 379 Hz and calibration lines at 356 Hz and 3565 Hz andwas moved away from the Vela region by shifting of 5 Hertz the frequency of calibration lines The lower plot (blue curve) isthe average spectrum computed after the removal of the disturbance The large spectral disturbance appearing near the Crabreference frequency is due to a poorly understood coupling between a line at 60 Hz part of a 10-Hertz comb of lines of likelymagnetic origin and the fundamental pendulum frequency of the Virgo mirror system at 0594 Hz

9

$amp())+-amp0

12$amp(3-45

amp(67(898)

(amp-8

ltamp9(8amp$

-ampamp(-8

=00gt$8)-8

-ampamp(-8A((-800

1B-amp(Bamp(67(895

C(3-4-B

-7$(amp0

D-7)(08)$)(

Egtgt(amp$3$-amp

gtamp3((amp(03-8

95

(

F8)0gt$8)-8

-ampamp(-8

0gt$8)-8

-ampamp(-8

8

C(3-4-B

-7$(amp0

C(3-4-B

-7$(amp0

A((-800

1B-amp(Bamp(67(895

A((-800

1B-amp(Bamp(67(895

1gt47(nɺf

FIG 3 Scheme of the narrow-band analysis pipeline The starting point is constituted by detector calibrated data sampled at4096 Hz After barycentric and spin-down corrections the data are down-sampled at 1 Hz The number of different spin-downcorrections applied to the data nf and then the number of produced corrected time series is given by Eq (21) See text and[11] for more details

the gravitational redshift at the Earth geocenter due tothe solar system bodies and that amounts to about 2milliseconds at most Another effect is the Shapiro delay

∆S which takes into account the deflection of a signalpassing near a massive body and which can be shownto be negligible for CW searches unless the source lineof sight passes very near the Sunrsquos limb See eg [11]for explicit expressions of ∆E and ∆S Overall we canmake the full barycentric corrections by introducing thenew time variable

τ1 = t+∆R +∆E minus∆S (15)

This transformation corresponds to referring the datacollected at the detector site to the SSB which can beconsidered an inertial reference frame to a very good ap-proximation In practice for the Crab and Vela anal-yses described in this paper the Shapiro delay can beneglectedFor given values of the signal frequency f and fre-

quency derivatives f f we could take into account thespin-down in a similar way by re-scaling time accordingto

τ2 = t+f

2f(tminus t0)

2 +f

6f(tminus t0)

3 (16)

where t0 is the initial time of the data set and higherorder terms can be included if neededNote that re-scaling the time in this way to make the

signal monochromatic assumes that the intrinsic phaseevolution of the pulsarrsquos GW signal is described by a Tay-lor expansion over the entire observation period Elec-tromagnetic observations of pulsars show that young

rapidly spinning pulsars demonstrate deviations from aTaylor expansion when spinning down a phenomenonknown as timing noise Given our lack of knowledge ofthe exact mechanism that might cause the gravitationalwave frequency to deviate from (twice) the observed elec-tromagnetic frequency we cannot be sure if we shouldexpect the timing noise to be present in the signal weare searching for (see [17] for discussion) However astudy based on the monthly Crab ephemeris data [18]has shown that in the Crab one of the noisiest of pul-sars the effect of timing noise over the duration of theobservation period is negligible producing a mismatchbetween a Taylor expansion and a signal based on theactual lsquonoisyrsquo time series of less than 1 This is smallconfirming that timing noise is likely to have a negligibleeffect on our analysis

In practice the spin-down correction is applied afterbarycentric corrections then the time t that appears inEq (16) is in fact the re-scaled time τ1 of Eq (15) Inthis case the correction depends explicitly on the searchfrequency f The number of frequency bins which coverthe range ∆f = 2f0δ corresponding to Eq (1) is

nf =

[

∆f

δf

]

= [∆f middot T ] asymp 63times 105(

∆f

002 Hz

)(

T

1 yr

)

(17)with δf = 1

T being the frequency spacing with T thetotal observation time and where [ middot ] stands for thenearest integer Similarly we take the width of the firstorder spin-down range as ∆f = 2|f0|δ and ∆f = 2|f0|δfor the second order Let us consider for the momentonly the first order term For a fixed value of f we donot want to make an explicit spin-down correction for

10

each value of f So we fix the value of the frequency inthe denominator of Eq (16) for instance at the referencefrequency f0 and exploit the fact that the same correc-tion that holds for the pair (f0 f0) is also valid for all the

pairs (f f) such that

f0f0

=f

f(18)

In practice this means that at each frequency f we ex-

plore a range of first order spin-down values [fminus ∆f2 f+

∆f2 ] with

f = f0f

f0(19)

The number of bins in the spin-down term of the firstorder can be computed by considering the ldquonaturalrdquo stepδf = 1

2T 2 and is given by

n(1) equiv nf =[

2∆f middot T 2]

(20)

asymp 400

(

f010minus10 Hzs

)

(

δ

10minus3

)(

T

1 yr

)2

Following the same reasoning for the second order spin-down we arrive at the following expression for the numberof steps that should be taken into account

n(2) equiv nf =[

∆f middot T 3]

(21)

asymp 06

(

f010minus20 Hzs2

)

(

δ

10minus3

)(

T

1 yr

)3

For observation times of the order of a year reasonablevalues of second order spin-down and values of δ typicalof narrow-band searches nf lt 1 which means that we donot need to consider more values other than the ldquocentralrdquoone In Tab I the Crab and Vela EM-inferred positionaland rotational parameters are shown Also estimationsof the polarization parameters are given which are usedin the computation of upper limits see Sec V In Fig 4the portion of the f minus f plane actually covered in theCrab and Vela narrow-band search is shownOnce barycentric and spin-down corrections have been

done the data are down-sampled to a much lower ratewith respect to the original one 1 Hz in the present caseThis strongly reduces the amount of data to be handledin the next step of the analysis At this stage for a givensource we have nf corrected time series one for eachvalue of first order spin-down For the current analysis wehave nf = 33 for Crab and nf = 3 for Vela see Tab II)where also other relevant quantities are given In partic-ular the total number of points in the frequencyspin-down plane is about 528 times 106 for Crab analysis and48 times 105 for Vela analysis For each time series we ap-ply a final cleaning step by removing the largest outliersThese are identified by histogramming the logarithm of

the absolute value of the data amplitude and choosinga threshold approximately marking the beginning of thenon-Gaussian tail of data distribution In Fig 5 the his-togram of Crab and Vela data amplitude correspond-ing to the ldquocentralrdquo time series are shown For Crab athreshold of 15 times 10minus21 corresponding to -182 in thefigure x-axis has been used to remove outliers while forVela a value of 12 times 10minus20 corresponding to -092 hasbeen used Correspondingly the fraction of removed datais of about 18 for Vela and 19 for Crab By apply-ing the Kolmogorov-Smirnov test we have verified thatfor each pulsar the data distribution of the various timeseries are fully in agreement thus justifying the use of thesame threshold for all of themAt this point the detection statistic is computed for ev-

ery frequency and spin-down value in the explored rangeThe detection statistic we use is based on the so-called 5-vectors the same used for pulsar targeted searches [3 4]and is here briefly described Once barycentric and spin-down corrections have been applied a CW signal withfrequency f0 present in the data would be monochro-matic apart from an amplitude and phase sidereal mod-ulation due to the time-varying detector beam patternfunctions and given by Eq (2) with ω0(t) constant andequal to 2πf0 From Eqs (256) it follows that the signalis completely described by its Fourier components at the5 angular frequencies ω0 ω0plusmnΩoplus ω0plusmn2Ωoplus This set of5 complex numbers constitutes the signal 5-vector Givena generic time series g(t) the corresponding 5-vector isdefined as

G =

int

T

g(t)eminusı(ω0minuskΩoplus)tdt (22)

where k = [minus2minus1 2] and T is the observation timeLet us indicate with X the data 5-vector and withA

+ Atimes the signal plus and cross 5-vectors obtained byapplying the definition of Eq (22) to Eqs (56) Thesetwo last quantities depend only on known parameters andform the signal templates Once the 5-vectors of data andof signal templates have been computed the two complexnumbers

H+times =X middotA+times

|A+times|2(23)

are built see [3 12] for more details They correspond tocomputing matched filters between the data and the sig-nal templates and it can be shown assuming the noiseis Gaussian with mean value zero that they are esti-mators of the signal plus and cross complex amplitudesH0e

ıΦ0H+ H0eıΦ0Htimes These estimators are used to

build the detection statistic

S = |A+|4|H+|2 + |Atimes|4|Htimes|

2 (24)

The maximum of the detection statistic over thesearched parameter space Smax is determined Thisis the loudest candidate and is identified by a triple

11

22374 22376 22378 2238 22382 22384 22386 22388 2239 22392 22394minus3149

minus3148

minus3147

minus3146

minus3145

minus3144

minus3143x 10

minus11

Frequency [Hz]S

pinminus

dow

n [H

zs]

Covered parameter space for Vela pulsar

59434 59436 59438 5944 59442 59444 59446 59448 5945 59452 59454minus7429

minus7428

minus7427

minus7426

minus7425

minus7424

minus7423x 10

minus10

Frequency [Hz]

Spi

nminusdo

wn

[Hz

s]

Covered parameter space for Crab pulsar

FIG 4 The parallelograms delimit the portion of the f minus f plane covered in the narrow-band search for the Vela (upper plot)

and Crab (bottom plot) pulsars Only one value of f has been considered in the analysis The total number of points is givenin the last column of Tab II

minus3 minus25 minus2 minus15 minus1 minus05 0 05 1

102

103

104

105

log10(abs(real(data1020)))

minus3 minus28 minus26 minus24 minus22 minus2 minus18 minus16 minus14 minus12 minus110

1

102

103

104

105

log10(abs(real(data1020)))

FIG 5 Histogram of the logarithm of data amplitude for Vela (top plot) and the Crab (bottom plot) used to select the thresholdfor the removal of outliers For Crab the threshold has been put at -182 corresponding to an amplitude of sim 15 times 10minus21while for Vela a value of -092 corresponding to an amplitude of sim 12times 10minus20 has been chosen

12

TABLE I The Crab and Vela reference parameters α and δ are the equatorial coordinates f0 f0 f0 are the gravitational-wavefrequency parameters The reference epoch for the Crab position is MJD 54632 while for the rotational parameters is MJD55696 The reference epoch for Vela is MJD 53576 both for position and GW frequency parameters The Crab ephemeris hasbeen obtained from a fit of the Jodrell Bank monthly ephemeris [18] while for Vela they have been derived from Hartebeesthoekradio telescope observations [19] Also estimations of the polarization parameters and their associated uncertainty obtainedfrom the analysis of Chandra X-ray observations of the pulsar wind nebula torus [20 21] are given in the last two columnsThe analysis presented in [20 21] does not allow to determine the sense of the starrsquos spin so values of ι and ψ correspondingto ι rarr 180 minus ι ψ rarr ψ + 180 are also possible However the upper limits on strain amplitude reported in this paper are notsensitive to the sense of rotation

Source α [hhmmss] δ[deg] f0[Hz] f0[Hzs] f0[Hzs2] ι[deg] ψ[deg]

Crab 05 34 3197 220145 594448 minus74183 times 10minus10 26307 times 10minus20 62o2plusmn 1o9 35o2plusmn 1o5

Vela 08 35 2061 minus451764 223840 minus31460 times 10minus11 12848 times 10minus21 63o6plusmn 0o6 40o6plusmn 0o1

TABLE II Main quantities related to the parameter space of VSR4 narrow-band search ∆f is the frequency range δ is thewidth parameter defined in Eq (1) ∆f is the first order spin-down range nf is the number of frequency bins nf and nf arerespectively the number of bins for the first and second order spin-down ntot = nf middot nf is the total number of points in theparameter space

Source ∆f [Hz] δ ∆f [Hzs] nf nf nf ntot

Crab 002 168times 10minus4 249times 10minus13 16times 105 33 1 528times 106

Vela 002 447times 10minus4 281times 10minus14 16times 105 3 1 480times 105

(Smax fSmax fSmax

) This means we need to computentot = nf middot nf values of the detection statisticThe maximum value Smax is used to assess detection

significance by computing the p-value that is the prob-ability that a value of the detection statistic equal orlarger than Smax can be obtained in the absence ofany signal It implies the need to compute the noise-only distribution of the detection statistic This is amulti-dimensional probability distribution (with dimen-sion ntot) which would be difficult to compute and to han-dle In practice as discussed in [11] the p-value is com-puted considering the single-trial noise probability distri-bution that is the same that would be used for a targetedsearch and choosing a suitable threshold on it to dis-criminate between interesting (that is deserving a deeperstudy) and not interesting candidates As shown in [11]by setting an overall p-value p0 (over the full multi-dimensional distribution) the corresponding threshold on

the single-trial distribution is pthr = 1minus (1minus p0)1

ntot Inour case by setting eg p0 = 001 the resulting pthrwould be of the order of 10minus8 In principle we wouldlike to generate the noise-only probability distribution bycomputing the detection statistic at several ldquooff-sourcerdquofrequency bins that is frequencies near but outside theexplored range where we are assuming the signal couldbe This is what is typically done in targeted searches [4]In the present analysis however in order to appreciate p-values of the order of pthr we should consider of the orderof 108 off-source frequency bins This is computationallyimpractical For this reason we use the theoretical dis-tribution which assumes the noise is Gaussian given in[11]

f(S) =eminus S

σ2X

|Atimes|2 minus eminus S

σ2X

|A+|2

σ2X (|Atimes|2 minus |A+|2)

(25)

where σ2X = σ2 middot T with σ2 being the noise variance

If Smax is compatible with noise at a given confidencelevel eg 1 we compute an upper limit on signalstrength using the same method described in [4] other-wise in case of detection signal parameters are estimatedthrough suitable combinations of the real and imaginaryparts of H+ and Htimes as explained in [12]

V RESULTS

Analysis results are summarized in Tab III The databoth for the Vela and Crab searches are compatible withnoise In particular we find p-values equal to 033 and0013 for Vela and the Crab respectively larger thanp0 = 001 chosen to identify interesting candidates ForCrab however the obtained p-value is very near to thechosen threshold Altough a reasonable choice of p0 israther arbitrary (we could have for instance chosen 10minus3)we have decided to study in some detail the candidatersquosproperties In particular we have considered the distri-bution of the top ten candidates in the frequencyspin-down plane They appear to be randomly distributedwithout the clustering that we would expect in presenceof a signal To verify this hypothesis we have addeda simulated signal to the data with the same parame-ters as the Crab but with a slightly different frequencyand spin-down and with an amplitude h0 ≃ 41times 10minus25

such that the resulting loudest candidate has a value ofthe detection statistic approximately equal to the loud-est candidate of the actual analysis We have then runthe full analysis on this new data set and looked again atthe top ten candidate distribution in the frequencyspin-down plane In this case we indeed observe a clusteringwith 4 out of 10 candidates having a frequency within

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

3

S Strigin44 R Sturani114 A L Stuver6 T Z Summerscales125 P J Sutton7 B Swinkels31 M Szczepanczyk90

G Szeifert50 M Tacca34 D Talukder55 D B Tanner5 M Tapai88 S P Tarabrin10 A Taracchini58 R Taylor1

G Tellez40 T Theeg10 M P Thirugnanasambandam1 M Thomas6 P Thomas33 K A Thorne6 K S Thorne69

E Thrane1 V Tiwari5 C Tomlinson80 M Tonelli3720 C V Torres40 C I Torrie132 F Travasso2930

G Traylor6 M Tse12 D Tshilumba76 D Ugolini126 C S Unnikrishnan120 A L Urban18 S A Usman16

H Vahlbruch24 G Vajente1 G Vajente3720 G Valdes40 M Vallisneri69 N van Bakel11 M van Beuzekom11

J F J van den Brand5711 C van den Broeck11 M V van der Sluys1148 J van Heijningen11 A A van Veggel32

S Vass1 M Vasuth81 R Vaulin12 A Vecchio26 G Vedovato107 J Veitch26 J Veitch11 P J Veitch93

K Venkateswara127 D Verkindt8 F Vetrano5354 A Vicere5354 R Vincent-Finley111 J-Y Vinet49 S Vitale12

T Vo33 H Vocca2930 C Vorvick33 W D Vousden26 S P Vyatchanin44 A R Wade71 L Wade18 M Wade18

M Walker2 L Wallace1 S Walsh18 H Wang26 M Wang26 X Wang64 R L Ward71 J Warner33 M Was10

B Weaver33 L-W Wei49 M Weinert10 A J Weinstein1 R Weiss12 T Welborn6 L Wen47 P Wessels10

T Westphal10 K Wette17 J T Whelan11817 D J White80 B F Whiting5 C Wilkinson33 L Williams5

R Williams1 A R Williamson7 J L Willis113 B Willke2410 M Wimmer10 W Winkler10 C C Wipf12

H Wittel10 G Woan32 J Worden33 S Xie76 J Yablon86 I Yakushin6 W Yam12 H Yamamoto1

C C Yancey58 Q Yang64 M Yvert8 A Zadrozny103 M Zanolin90 J-P Zendri107 Fan Zhang1264 L Zhang1

M Zhang112 Y Zhang118 C Zhao47 M Zhou86 X J Zhu47 M E Zucker12 S Zuraw59 and J Zweizig1

1LIGO California Institute of Technology Pasadena CA 91125 USA2Louisiana State University Baton Rouge LA 70803 USA

3Universita di Salerno Fisciano I-84084 Salerno Italy4INFN Sezione di Napoli Complesso Universitario di Monte SAngelo I-80126 Napoli Italy

5University of Florida Gainesville FL 32611 USA6LIGO Livingston Observatory Livingston LA 70754 USA7Cardiff University Cardiff CF24 3AA United Kingdom

8Laboratoire drsquoAnnecy-le-Vieux de Physique des Particules (LAPP)Universite de Savoie CNRSIN2P3 F-74941 Annecy-le-Vieux France

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

10Experimental Group Albert-Einstein-Institut Max-Planck-Institut fur Gravitationsphysik D-30167 Hannover Germany11Nikhef Science Park 1098 XG Amsterdam The Netherlands

12LIGO Massachusetts Institute of Technology Cambridge MA 02139 USA13Instituto Nacional de Pesquisas Espaciais 12227-010 - Sao Jose dos Campos SP Brazil

14Inter-University Centre for Astronomy and Astrophysics Pune - 411007 India15International Centre for Theoretical Sciences Tata Institute of Fundamental Research Bangalore 560012 India

16Syracuse University Syracuse NY 13244 USA17Data Analysis Group Albert-Einstein-Institut Max-Planck-Institut fur Gravitationsphysik D-30167 Hannover Germany

18University of WisconsinndashMilwaukee Milwaukee WI 53201 USA19Universita di Siena I-53100 Siena Italy

20INFN Sezione di Pisa I-56127 Pisa Italy21The University of Mississippi University MS 38677 USA

22California State University Fullerton Fullerton CA 92831 USA23University of Southampton Southampton SO17 1BJ United Kingdom

24Leibniz Universitat Hannover D-30167 Hannover Germany25INFN Sezione di Roma I-00185 Roma Italy

26University of Birmingham Birmingham B15 2TT United Kingdom27Albert-Einstein-Institut Max-Planck-Institut fur Gravitationsphysik D-14476 Golm Germany

28Montana State University Bozeman MT 59717 USA29Universita di Perugia I-06123 Perugia Italy

30INFN Sezione di Perugia I-06123 Perugia Italy31European Gravitational Observatory (EGO) I-56021 Cascina Pisa Italy

32SUPA University of Glasgow Glasgow G12 8QQ United Kingdom33LIGO Hanford Observatory Richland WA 99352 USA

34APC AstroParticule et Cosmologie Universite Paris DiderotCNRSIN2P3 CEAIrfu Observatoire de Paris Sorbonne Paris Cite 10

rue Alice Domon et Leonie Duquet F-75205 Paris Cedex 13 France35Columbia University New York NY 10027 USA36Stanford University Stanford CA 94305 USA

37Universita di Pisa I-56127 Pisa Italy38CAMK-PAN 00-716 Warsaw Poland

39Astronomical Observatory Warsaw University 00-478 Warsaw Poland

4

40The University of Texas at Brownsville Brownsville TX 78520 USA41Universita degli Studi di Genova I-16146 Genova Italy

42INFN Sezione di Genova I-16146 Genova Italy43RRCAT Indore MP 452013 India

44Faculty of Physics Lomonosov Moscow State University Moscow 119991 Russia45LAL Universite Paris-Sud IN2P3CNRS F-91898 Orsay France46NASAGoddard Space Flight Center Greenbelt MD 20771 USA47University of Western Australia Crawley WA 6009 Australia

48Department of AstrophysicsIMAPP Radboud University NijmegenPO Box 9010 6500 GL Nijmegen The Netherlands

49ARTEMIS Universite Nice-Sophia-Antipolis CNRS and Observatoire de la Cote drsquoAzur F-06304 Nice France50MTA Eotvos University lsquoLenduletrsquo A R G Budapest 1117 Hungary

51Institut de Physique de Rennes CNRS Universite de Rennes 1 F-35042 Rennes France52Washington State University Pullman WA 99164 USA

53Universita degli Studi di Urbino rsquoCarlo Borsquo I-61029 Urbino Italy54INFN Sezione di Firenze I-50019 Sesto Fiorentino Firenze Italy

55University of Oregon Eugene OR 97403 USA56Laboratoire Kastler Brossel ENS CNRS UPMC

Universite Pierre et Marie Curie F-75005 Paris France57VU University Amsterdam 1081 HV Amsterdam The Netherlands

58University of Maryland College Park MD 20742 USA59University of Massachusetts Amherst Amherst MA 01003 USA

60Laboratoire des Materiaux Avances (LMA) IN2P3CNRSUniversite de Lyon F-69622 Villeurbanne Lyon France

61Universitat de les Illes BalearsmdashIEEC E-07122 Palma de Mallorca Spain62Universita di Napoli rsquoFederico IIrsquo Complesso Universitario di Monte SAngelo I-80126 Napoli Italy

63Canadian Institute for Theoretical AstrophysicsUniversity of Toronto Toronto Ontario M5S 3H8 Canada

64Tsinghua University Beijing 100084 China65University of Michigan Ann Arbor MI 48109 USA

66INFN Sezione di Roma Tor Vergata I-00133 Roma Italy67National Tsing Hua University Hsinchu Taiwan 300

68Charles Sturt University Wagga Wagga NSW 2678 Australia69Caltech-CaRT Pasadena CA 91125 USA

70Pusan National University Busan 609-735 Korea71Australian National University Canberra ACT 0200 Australia

72Carleton College Northfield MN 55057 USA73Universita di Roma Tor Vergata I-00133 Roma Italy

74INFN Gran Sasso Science Institute I-67100 LrsquoAquila Italy75Universita di Roma rsquoLa Sapienzarsquo I-00185 Roma Italy

76University of Brussels Brussels 1050 Belgium77Sonoma State University Rohnert Park CA 94928 USA

78Texas Tech University Lubbock TX 79409 USA79University of Minnesota Minneapolis MN 55455 USA

80The University of Sheffield Sheffield S10 2TN United Kingdom81Wigner RCP RMKI H-1121 Budapest Konkoly Thege Miklos ut 29-33 Hungary

82Montclair State University Montclair NJ 07043 USA83Argentinian Gravitational Wave Group Cordoba Cordoba 5000 Argentina

84Universita di Trento I-38123 Povo Trento Italy85INFN Trento Institute for Fundamental Physics and Applications I-38123 Povo Trento Italy

86Northwestern University Evanston IL 60208 USA87University of Cambridge Cambridge CB2 1TN United Kingdom

88University of Szeged Dom ter 9 Szeged 6720 Hungary89Rutherford Appleton Laboratory HSIC Chilton Didcot Oxon OX11 0QX United Kingdom

90Embry-Riddle Aeronautical University Prescott AZ 86301 USA91The Pennsylvania State University University Park PA 16802 USA

92American University Washington DC 20016 USA93University of Adelaide Adelaide SA 5005 Australia

94West Virginia University Morgantown WV 26506 USA95Raman Research Institute Bangalore Karnataka 560080 India

96Korea Institute of Science and Technology Information Daejeon 305-806 Korea97University of Bia lystok 15-424 Bia lystok Poland

98SUPA University of Strathclyde Glasgow G1 1XQ United Kingdom99IISER-TVM CET Campus Trivandrum Kerala 695016 India

5

100Institute of Applied Physics Nizhny Novgorod 603950 Russia101Seoul National University Seoul 151-742 Korea

102Hanyang University Seoul 133-791 Korea103NCBJ 05-400 Swierk-Otwock Poland

104IM-PAN 00-956 Warsaw Poland105Institute for Plasma Research Bhat Gandhinagar 382428 India

106The University of Melbourne Parkville VIC 3010 Australia107INFN Sezione di Padova I-35131 Padova Italy

108Monash University Victoria 3800 Australia109ESPCI CNRS F-75005 Paris France

110Universita di Camerino Dipartimento di Fisica I-62032 Camerino Italy111Southern University and AampM College Baton Rouge LA 70813 USA

112College of William and Mary Williamsburg VA 23187 USA113Abilene Christian University Abilene TX 79699 USA

114Instituto de Fısica Teorica Univ Estadual PaulistaICTP South AmericanInstitute for Fundamental Research Sao Paulo SP 01140-070 Brazil

115IISER-Kolkata Mohanpur West Bengal 741252 India116Whitman College 280 Boyer Ave Walla Walla WA 9936 USA

117National Institute for Mathematical Sciences Daejeon 305-390 Korea118Rochester Institute of Technology Rochester NY 14623 USA119Hobart and William Smith Colleges Geneva NY 14456 USA

120Tata Institute for Fundamental Research Mumbai 400005 India121SUPA University of the West of Scotland Paisley PA1 2BE United Kingdom

122Institute of Astronomy 65-265 Zielona Gora Poland123Universitat Hamburg D-22761 Hamburg Germany

124Indian Institute of Technology Gandhinagar Ahmedabad Gujarat 382424 India125Andrews University Berrien Springs MI 49104 USA

126Trinity University San Antonio TX 78212 USA127University of Washington Seattle WA 98195 USA

In this paper we present the results of a coherent narrow-band search for continuous gravitational-wave signals from the Crab and Vela pulsars conducted on Virgo VSR4 data In order to take intoaccount a possible small mismatch between the gravitational wave frequency and two times thestar rotation frequency inferred from measurement of the electromagnetic pulse rate a range of002 Hz around two times the star rotational frequency has been searched for both the pulsars Noevidence for a signal has been found and 95 confidence level upper limits have been computedboth assuming polarization parameters are completely unknown and that they are known with someuncertainty as derived from X-ray observations of the pulsar wind torii For Vela the upper limitsare comparable to the spin-down limit computed assuming that all the observed spin-down is dueto the emission of gravitational waves For Crab the upper limits are about a factor of two belowthe spin-down limit and represent a significant improvement with respect to past analysis This isthe first time the spin-down limit is significantly overcome in a narrow-band search

I INTRODUCTION

Continuous gravitational-wave signals (CW) are emit-ted by spinning neutron stars if asymmetric with respectto the rotation axis The asymmetry can be due to var-ious mechanisms like a non-axismmetric residual strainfrom the starrsquos birth or a strong internal magnetic fieldnot aligned to the rotation axis see eg [1] When thesource parameters position frequency spin-down areknown with high accuracy targeted searches can be doneusing optimal analysis methods based on matched filter-ing This happens for instance with known pulsars forwhich accurate ephemerides are often obtained from elec-tromagnetic (EM) observations especially in the radiogamma-ray and X-ray band This means that a strict cor-relation between the gravitational wave frequency f0 andthe starrsquos measured rotational frequency frot is assumedIn the classical case of a non-axisymmetric neutron star

rotating around one of its principal axes of inertia thegravitational frequency would be exactly twice the rota-tion frequency of the star Several targeted searches forCW have been conducted on data from first generationinterferometric detectors No evidence for a signal hasbeen found but interesting upper limits have been placedin a few cases [2ndash4] Given the uncertainties on gravita-tional wave emission mechanisms and also the lack of afull detailed picture of the electro-magnetic emission ge-ometry it is not obvious at all that the gravitational-waveemission takes place at exactly twice the star measuredpulse rate or that such relation holds for observationtimes of months to years For instance if a neutron staris made of a crust and a core rotating at slightly differentrates and if the gravitational-wave emission is dominatedby an asymmetry in the core then a search targeted at2frot would assume a wrong signal frequency We thenconsider here that the signal frequency can be slightly

6

different with respect to f0 = 2frot and can vary in theinterval

f(t) isin [f0(t)(1minus δ) f0(t)(1 + δ)] (1)

where δ is a small positive real number Following thediscussion in [5] if the star crust and core are linkedby some torque that tends to enforce corotation on atimescale τc then δ sim τcτsd where τsd sim f0f0 is thecharacteristic spin-down time A relation of the form ofEq (1) also holds in the case the gravitational radiationis produced by free precession of a nearly bi-axial star [6]in which case δ is of the order of (Izz minus Ixx)Ixx whereIxx and Izz are the star moments of inertia with respectto a principal axis on the equatorial plane and alignedwith the rotation axis respectively In general a value ofδ of the order of say 10minus4 corresponds to τc sim 10minus4τsdwhich depending on the specific targeted pulsar can beseveral months or years This would be comparable orlarger than the longest timescale observed in pulsar glitchrecovery where a recoupling between the two componentsmight occur In terms of free precession δ sim 10minus4 is onthe high end of the range of deformations that neutronstars could sustain [7ndash9]Narrow-band searches have not received much atten-

tion until now one notable exception being the Crabnarrow-band search over the first 9 months of LIGO S5data [5] based on the F -statistic [10] In previous work[11] an optimal method based on matched filtering in thespace of signal Fourier components has been proposedto carry out a search for CW over a small frequency andspin-down range around the values inferred from EM ob-servations In this paper we describe the application ofsuch a method to a narrow-band search of CW from Craband Vela pulsars in the data of Virgo VSR4 run As noevidence for a signal is found we place upper limits onsignal amplitudeThe outline of the paper is the following In Sec II the

expected gravitational wave signal from spinning neutronstars is introduced In Sec III the main characteristics ofVirgo VSR4 data are presented In Sec IV the analysismethod is described while in Sec V the analysis resultsare discussed Sec VI is dedicated to the validation testsof the analysis procedure Finally Sec VII contains theconclusions

II THE GRAVITATIONAL WAVE SIGNAL

A non-axisymmetric neutron star steadily spinningabout one of its principal axis emits a quadrupolargravitational-wave signal at twice the star rotational fre-quency frot which at the detector can be described [12]as

h(t) = H0(H+A+(t) +HtimesA

times(t))eı(ω0(t)t+Φ0) (2)

where taking the real part is understood The constantΦ0 is the initial signal phase The angular signal fre-

quency ω0(t) = 4πfrot(t) is a function of time Conse-quently the signal phase is not that of a simple monochro-matic signal and depends on both the intrinsic rotationalfrequency and frequency derivatives of the pulsar and onDoppler and propagation effects which include the Ein-stein delay and possibly the Shapiro delay These varia-tions are corrected in the time-domain in a way describedin Sec IV The two complex amplitudes H+ and Htimes aregiven respectively by

H+ =cos 2ψ minus ıη sin 2ψ

radic

1 + η2(3)

Htimes =sin 2ψ + ıη cos 2ψ

radic

1 + η2(4)

in which η is the ratio of the polarization ellipse semi-minor to semi-major axis and the polarization angle ψdefines the direction of the major axis with respect to thecelestial parallel of the source (measured counterclock-wise) The functions A+ and Atimes describe the detectorresponse as a function of time and are a linear combi-nation of terms depending on the Earth sidereal angularfrequency Ωoplus

A+ = a0 + a1c cosΩoplust+ a1s sinΩoplust+

a2c cos 2Ωoplust+ a2s sin 2Ωoplust (5)

Atimes = b1c cosΩoplust+ b1s sinΩoplust+

b2c cos 2Ωoplust+ b2s sin 2Ωoplust (6)

where the coefficients depend on the source position anddetector position and orientation on the Earth [12]As discussed eg in [3] the overall wave amplitude H0

and η are related to the ldquostandardrdquo signal amplitude

h0 =4π2G

c4Izzεf

20

d(7)

and to the angle ι between the star rotation axis and theline of sight to the source by

H0 = h0

radic

1 + 6 cos2 ι+ cos4 ι

4(8)

η = minus2 cos ι

1 + cos2 ι (9)

In Eq (7) G is the gravitational constant c is the lightvelocity Izz is the star moment of inertia with respectto the principal axis aligned with the rotation axis ε =IxxminusIyyIzz

is the equatorial ellipticity expressed in terms ofprincipal moments of inertia and d is the source distance

7

Equating the gravitational-wave luminosity to the ki-netic energy lost as the pulsar spins down gives us theso-called spin-down limit on gravitational-wave strain

hsd0 =

(

5

2

GIzz frotc3d2frot

)12

=

806times 10minus19 I1238

dkpc

(

|frot|

Hzs

)12(

frotHz

)minus12

(10)

where I38 is the starrsquos moment of inertia in units of1038 kgm2 frot is the time derivative of the rotationalfrequency and dkpc is the distance to the pulsar in kilo-parsecs The spin-down limit on the signal amplitudecorresponds to an upper limit on the starrsquos fiducial ellip-ticity

εsd = 0237

(

hsd010minus24

)

fminus2rot I

minus138 dkpc (11)

This quantity for a given neutron star equation of statecan be related to the physical ellipticity of the star sur-face [9] Setting a gravitational-wave upper limit belowthe spin-down limit is an important milestone in CWsearches as it allows us to constrain the fraction of spin-down energy due to the emission of gravitational-waves which gives insight into the spin-down energy budgetof the pulsar On the other hand the l = m = 2 massquadrupole moment Q22 is related to the gravitational-wave amplitude by

Q22 =

radic

15

8πIzzε = h0

(

c4d

16π2Gf2rot

)

radic

15

8π (12)

see eg [7] and is independent of any assumptions aboutthe starrsquos equation of state and its moment of inertia

III INSTRUMENTAL PERFORMANCE IN

VIRGO VSR4 RUN

Interferometric gravitational wave detectors LIGO[13] Virgo [14] GEO [15] have collected a large amountof data in recent years For the analysis described in thispaper we have used calibrated data sampled at 4096 Hzfrom Virgo VSR4 science run The run extended fromJune 3rd 2011 (1027 UTC) to September 5th 2011(1326 UTC) with a duty factor of about 81 corre-sponding to an effective duration of 76 days Calibra-tion uncertainties amounted to 75 in amplitude and(40+50fkHz) mrad in phase up to 500 Hz where fkHz isthe frequency in kilohertz The uncertainty on the ampli-tude will contribute to the uncertainty on the upper limiton signal amplitude together with that coming from thefinite size of the Monte Carlo simulation used to computeit see Sec V The calibration error on the phase can beshown to have a negligible impact on the analysis [3]

The low-frequency sensitivity of VSR4 was significantlybetter than that of previous Virgo runs especially due tothe use of monolithic mirror suspensions and basicallyin agreement with the planned sensitivity of the initialVirgo interferometer In Fig 1 a typical VSR4 strainsensitivity curve is shown In Fig 2 the average powerspectrum around Crab and Vela reference frequency isplotted Note the large sensitivity improvement aroundthe Vela frequency after removal of an instrumental dis-turbance about one month after the beginning of therun see the figure caption for more details

IV SEARCH DESCRIPTION

The analysis pipeline consists of several steps schemat-ically depicted in Fig 3 which we summarize here Moredetails are given in [11] The starting point is a collectionof windowed and interlaced (by half) ldquoshortrdquo (1024 sec-onds) FFTs (the Short FFT Database - SFDB) built fromcalibrated detector data [16] At this stage a first clean-ing procedure is applied to the data in order to removebig and short time duration disturbances that could re-duce the search sensitivity A small frequency band isthen extracted from the SFDB but large enough to con-tain the Doppler and spin-down variations of the signalpossibly emitted by the target pulsar In the analysesdescribed in this paper for instance it was of 015 HzAt this point we go back to the time domain (the sam-pling frequency is still the original one 4096 Hz) andmake barycentric and spin-down corrections Due to theDoppler effect the received frequency f(t) is related tothe emitted frequency f0(t) by the well-known relation(valid in the non-relativistic approximation)

f(t) =1

2π

dΦ(t)

dt= f0(t)

(

1 +~v middot n

c

)

(13)

where Φ(t) is the signal phase ~v = ~vrev+~vrot is the detec-tor velocity with respect to the Solar system barycenter(SSB) given by the sum of the Earthrsquos orbital velocity~vrev and rotational velocity ~vrot while n is the versoridentifying the source position In practice the Dopplereffect is efficiently corrected in the time domain by chang-ing the time stamp t of data samples according to

τ1 = t+~r(t) middot n

c= t+∆R (14)

where ~r is the detector position in the SSB The correc-tion term ∆R called Roemer delay amounts up to about500 seconds and corresponds to the time taken by a signaltraveling at the speed of light to cover the distance be-tween the detector and the SSB Unlike Eq (13) Eq (14)does not depend on the frequency which means that onesingle correction holds for every frequency In fact thereare other smaller relativistic effects that must be takeninto account when making barycentric corrections Oneis the Einstein delay ∆E due to the Earth motion and

8

102

103

10minus23

10minus22

10minus21

10minus20

10minus19

Frequency [Hz]

h n [Hzminus

12 ]

FIG 1 A typical sensitivity curve of Virgo VSR4 run expressed in terms of noise spectral density

032 034 036 038 04 042 04410

minus3

10minus2

Frequency minus220 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

036 038 04 042 044 046 048 05

10minus4

10minus3

Frequency minus 590 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

FIG 2 Average spectrum of VSR4 data around Vela (top plot) and Crab (bottom plot) reference frequency (identified by thevertical black lines) For Vela two spectra are in fact shown in the same plot The upper one (black curve) has been computedover the first 29 days of the run when the frequency region around Vela was affected by an instrumental disturbance Thiswas due to a non-linear coupling between the DARM control line at 379 Hz and calibration lines at 356 Hz and 3565 Hz andwas moved away from the Vela region by shifting of 5 Hertz the frequency of calibration lines The lower plot (blue curve) isthe average spectrum computed after the removal of the disturbance The large spectral disturbance appearing near the Crabreference frequency is due to a poorly understood coupling between a line at 60 Hz part of a 10-Hertz comb of lines of likelymagnetic origin and the fundamental pendulum frequency of the Virgo mirror system at 0594 Hz

9

$amp())+-amp0

12$amp(3-45

amp(67(898)

(amp-8

ltamp9(8amp$

-ampamp(-8

=00gt$8)-8

-ampamp(-8A((-800

1B-amp(Bamp(67(895

C(3-4-B

-7$(amp0

D-7)(08)$)(

Egtgt(amp$3$-amp

gtamp3((amp(03-8

95

(

F8)0gt$8)-8

-ampamp(-8

0gt$8)-8

-ampamp(-8

8

C(3-4-B

-7$(amp0

C(3-4-B

-7$(amp0

A((-800

1B-amp(Bamp(67(895

A((-800

1B-amp(Bamp(67(895

1gt47(nɺf

FIG 3 Scheme of the narrow-band analysis pipeline The starting point is constituted by detector calibrated data sampled at4096 Hz After barycentric and spin-down corrections the data are down-sampled at 1 Hz The number of different spin-downcorrections applied to the data nf and then the number of produced corrected time series is given by Eq (21) See text and[11] for more details

the gravitational redshift at the Earth geocenter due tothe solar system bodies and that amounts to about 2milliseconds at most Another effect is the Shapiro delay

∆S which takes into account the deflection of a signalpassing near a massive body and which can be shownto be negligible for CW searches unless the source lineof sight passes very near the Sunrsquos limb See eg [11]for explicit expressions of ∆E and ∆S Overall we canmake the full barycentric corrections by introducing thenew time variable

τ1 = t+∆R +∆E minus∆S (15)

This transformation corresponds to referring the datacollected at the detector site to the SSB which can beconsidered an inertial reference frame to a very good ap-proximation In practice for the Crab and Vela anal-yses described in this paper the Shapiro delay can beneglectedFor given values of the signal frequency f and fre-

quency derivatives f f we could take into account thespin-down in a similar way by re-scaling time accordingto

τ2 = t+f

2f(tminus t0)

2 +f

6f(tminus t0)

3 (16)

where t0 is the initial time of the data set and higherorder terms can be included if neededNote that re-scaling the time in this way to make the

signal monochromatic assumes that the intrinsic phaseevolution of the pulsarrsquos GW signal is described by a Tay-lor expansion over the entire observation period Elec-tromagnetic observations of pulsars show that young

rapidly spinning pulsars demonstrate deviations from aTaylor expansion when spinning down a phenomenonknown as timing noise Given our lack of knowledge ofthe exact mechanism that might cause the gravitationalwave frequency to deviate from (twice) the observed elec-tromagnetic frequency we cannot be sure if we shouldexpect the timing noise to be present in the signal weare searching for (see [17] for discussion) However astudy based on the monthly Crab ephemeris data [18]has shown that in the Crab one of the noisiest of pul-sars the effect of timing noise over the duration of theobservation period is negligible producing a mismatchbetween a Taylor expansion and a signal based on theactual lsquonoisyrsquo time series of less than 1 This is smallconfirming that timing noise is likely to have a negligibleeffect on our analysis

In practice the spin-down correction is applied afterbarycentric corrections then the time t that appears inEq (16) is in fact the re-scaled time τ1 of Eq (15) Inthis case the correction depends explicitly on the searchfrequency f The number of frequency bins which coverthe range ∆f = 2f0δ corresponding to Eq (1) is

nf =

[

∆f

δf

]

= [∆f middot T ] asymp 63times 105(

∆f

002 Hz

)(

T

1 yr

)

(17)with δf = 1

T being the frequency spacing with T thetotal observation time and where [ middot ] stands for thenearest integer Similarly we take the width of the firstorder spin-down range as ∆f = 2|f0|δ and ∆f = 2|f0|δfor the second order Let us consider for the momentonly the first order term For a fixed value of f we donot want to make an explicit spin-down correction for

10

each value of f So we fix the value of the frequency inthe denominator of Eq (16) for instance at the referencefrequency f0 and exploit the fact that the same correc-tion that holds for the pair (f0 f0) is also valid for all the

pairs (f f) such that

f0f0

=f

f(18)

In practice this means that at each frequency f we ex-

plore a range of first order spin-down values [fminus ∆f2 f+

∆f2 ] with

f = f0f

f0(19)

The number of bins in the spin-down term of the firstorder can be computed by considering the ldquonaturalrdquo stepδf = 1

2T 2 and is given by

n(1) equiv nf =[

2∆f middot T 2]

(20)

asymp 400

(

f010minus10 Hzs

)

(

δ

10minus3

)(

T

1 yr

)2

Following the same reasoning for the second order spin-down we arrive at the following expression for the numberof steps that should be taken into account

n(2) equiv nf =[

∆f middot T 3]

(21)

asymp 06

(

f010minus20 Hzs2

)

(

δ

10minus3

)(

T

1 yr

)3

For observation times of the order of a year reasonablevalues of second order spin-down and values of δ typicalof narrow-band searches nf lt 1 which means that we donot need to consider more values other than the ldquocentralrdquoone In Tab I the Crab and Vela EM-inferred positionaland rotational parameters are shown Also estimationsof the polarization parameters are given which are usedin the computation of upper limits see Sec V In Fig 4the portion of the f minus f plane actually covered in theCrab and Vela narrow-band search is shownOnce barycentric and spin-down corrections have been

done the data are down-sampled to a much lower ratewith respect to the original one 1 Hz in the present caseThis strongly reduces the amount of data to be handledin the next step of the analysis At this stage for a givensource we have nf corrected time series one for eachvalue of first order spin-down For the current analysis wehave nf = 33 for Crab and nf = 3 for Vela see Tab II)where also other relevant quantities are given In partic-ular the total number of points in the frequencyspin-down plane is about 528 times 106 for Crab analysis and48 times 105 for Vela analysis For each time series we ap-ply a final cleaning step by removing the largest outliersThese are identified by histogramming the logarithm of

the absolute value of the data amplitude and choosinga threshold approximately marking the beginning of thenon-Gaussian tail of data distribution In Fig 5 the his-togram of Crab and Vela data amplitude correspond-ing to the ldquocentralrdquo time series are shown For Crab athreshold of 15 times 10minus21 corresponding to -182 in thefigure x-axis has been used to remove outliers while forVela a value of 12 times 10minus20 corresponding to -092 hasbeen used Correspondingly the fraction of removed datais of about 18 for Vela and 19 for Crab By apply-ing the Kolmogorov-Smirnov test we have verified thatfor each pulsar the data distribution of the various timeseries are fully in agreement thus justifying the use of thesame threshold for all of themAt this point the detection statistic is computed for ev-

ery frequency and spin-down value in the explored rangeThe detection statistic we use is based on the so-called 5-vectors the same used for pulsar targeted searches [3 4]and is here briefly described Once barycentric and spin-down corrections have been applied a CW signal withfrequency f0 present in the data would be monochro-matic apart from an amplitude and phase sidereal mod-ulation due to the time-varying detector beam patternfunctions and given by Eq (2) with ω0(t) constant andequal to 2πf0 From Eqs (256) it follows that the signalis completely described by its Fourier components at the5 angular frequencies ω0 ω0plusmnΩoplus ω0plusmn2Ωoplus This set of5 complex numbers constitutes the signal 5-vector Givena generic time series g(t) the corresponding 5-vector isdefined as

G =

int

T

g(t)eminusı(ω0minuskΩoplus)tdt (22)

where k = [minus2minus1 2] and T is the observation timeLet us indicate with X the data 5-vector and withA

+ Atimes the signal plus and cross 5-vectors obtained byapplying the definition of Eq (22) to Eqs (56) Thesetwo last quantities depend only on known parameters andform the signal templates Once the 5-vectors of data andof signal templates have been computed the two complexnumbers

H+times =X middotA+times

|A+times|2(23)

are built see [3 12] for more details They correspond tocomputing matched filters between the data and the sig-nal templates and it can be shown assuming the noiseis Gaussian with mean value zero that they are esti-mators of the signal plus and cross complex amplitudesH0e

ıΦ0H+ H0eıΦ0Htimes These estimators are used to

build the detection statistic

S = |A+|4|H+|2 + |Atimes|4|Htimes|

2 (24)

The maximum of the detection statistic over thesearched parameter space Smax is determined Thisis the loudest candidate and is identified by a triple

11

22374 22376 22378 2238 22382 22384 22386 22388 2239 22392 22394minus3149

minus3148

minus3147

minus3146

minus3145

minus3144

minus3143x 10

minus11

Frequency [Hz]S

pinminus

dow

n [H

zs]

Covered parameter space for Vela pulsar

59434 59436 59438 5944 59442 59444 59446 59448 5945 59452 59454minus7429

minus7428

minus7427

minus7426

minus7425

minus7424

minus7423x 10

minus10

Frequency [Hz]

Spi

nminusdo

wn

[Hz

s]

Covered parameter space for Crab pulsar

FIG 4 The parallelograms delimit the portion of the f minus f plane covered in the narrow-band search for the Vela (upper plot)

and Crab (bottom plot) pulsars Only one value of f has been considered in the analysis The total number of points is givenin the last column of Tab II

minus3 minus25 minus2 minus15 minus1 minus05 0 05 1

102

103

104

105

log10(abs(real(data1020)))

minus3 minus28 minus26 minus24 minus22 minus2 minus18 minus16 minus14 minus12 minus110

1

102

103

104

105

log10(abs(real(data1020)))

FIG 5 Histogram of the logarithm of data amplitude for Vela (top plot) and the Crab (bottom plot) used to select the thresholdfor the removal of outliers For Crab the threshold has been put at -182 corresponding to an amplitude of sim 15 times 10minus21while for Vela a value of -092 corresponding to an amplitude of sim 12times 10minus20 has been chosen

12

TABLE I The Crab and Vela reference parameters α and δ are the equatorial coordinates f0 f0 f0 are the gravitational-wavefrequency parameters The reference epoch for the Crab position is MJD 54632 while for the rotational parameters is MJD55696 The reference epoch for Vela is MJD 53576 both for position and GW frequency parameters The Crab ephemeris hasbeen obtained from a fit of the Jodrell Bank monthly ephemeris [18] while for Vela they have been derived from Hartebeesthoekradio telescope observations [19] Also estimations of the polarization parameters and their associated uncertainty obtainedfrom the analysis of Chandra X-ray observations of the pulsar wind nebula torus [20 21] are given in the last two columnsThe analysis presented in [20 21] does not allow to determine the sense of the starrsquos spin so values of ι and ψ correspondingto ι rarr 180 minus ι ψ rarr ψ + 180 are also possible However the upper limits on strain amplitude reported in this paper are notsensitive to the sense of rotation

Source α [hhmmss] δ[deg] f0[Hz] f0[Hzs] f0[Hzs2] ι[deg] ψ[deg]

Crab 05 34 3197 220145 594448 minus74183 times 10minus10 26307 times 10minus20 62o2plusmn 1o9 35o2plusmn 1o5

Vela 08 35 2061 minus451764 223840 minus31460 times 10minus11 12848 times 10minus21 63o6plusmn 0o6 40o6plusmn 0o1

TABLE II Main quantities related to the parameter space of VSR4 narrow-band search ∆f is the frequency range δ is thewidth parameter defined in Eq (1) ∆f is the first order spin-down range nf is the number of frequency bins nf and nf arerespectively the number of bins for the first and second order spin-down ntot = nf middot nf is the total number of points in theparameter space

Source ∆f [Hz] δ ∆f [Hzs] nf nf nf ntot

Crab 002 168times 10minus4 249times 10minus13 16times 105 33 1 528times 106

Vela 002 447times 10minus4 281times 10minus14 16times 105 3 1 480times 105

(Smax fSmax fSmax

) This means we need to computentot = nf middot nf values of the detection statisticThe maximum value Smax is used to assess detection

significance by computing the p-value that is the prob-ability that a value of the detection statistic equal orlarger than Smax can be obtained in the absence ofany signal It implies the need to compute the noise-only distribution of the detection statistic This is amulti-dimensional probability distribution (with dimen-sion ntot) which would be difficult to compute and to han-dle In practice as discussed in [11] the p-value is com-puted considering the single-trial noise probability distri-bution that is the same that would be used for a targetedsearch and choosing a suitable threshold on it to dis-criminate between interesting (that is deserving a deeperstudy) and not interesting candidates As shown in [11]by setting an overall p-value p0 (over the full multi-dimensional distribution) the corresponding threshold on

the single-trial distribution is pthr = 1minus (1minus p0)1

ntot Inour case by setting eg p0 = 001 the resulting pthrwould be of the order of 10minus8 In principle we wouldlike to generate the noise-only probability distribution bycomputing the detection statistic at several ldquooff-sourcerdquofrequency bins that is frequencies near but outside theexplored range where we are assuming the signal couldbe This is what is typically done in targeted searches [4]In the present analysis however in order to appreciate p-values of the order of pthr we should consider of the orderof 108 off-source frequency bins This is computationallyimpractical For this reason we use the theoretical dis-tribution which assumes the noise is Gaussian given in[11]

f(S) =eminus S

σ2X

|Atimes|2 minus eminus S

σ2X

|A+|2

σ2X (|Atimes|2 minus |A+|2)

(25)

where σ2X = σ2 middot T with σ2 being the noise variance

If Smax is compatible with noise at a given confidencelevel eg 1 we compute an upper limit on signalstrength using the same method described in [4] other-wise in case of detection signal parameters are estimatedthrough suitable combinations of the real and imaginaryparts of H+ and Htimes as explained in [12]

V RESULTS

Analysis results are summarized in Tab III The databoth for the Vela and Crab searches are compatible withnoise In particular we find p-values equal to 033 and0013 for Vela and the Crab respectively larger thanp0 = 001 chosen to identify interesting candidates ForCrab however the obtained p-value is very near to thechosen threshold Altough a reasonable choice of p0 israther arbitrary (we could have for instance chosen 10minus3)we have decided to study in some detail the candidatersquosproperties In particular we have considered the distri-bution of the top ten candidates in the frequencyspin-down plane They appear to be randomly distributedwithout the clustering that we would expect in presenceof a signal To verify this hypothesis we have addeda simulated signal to the data with the same parame-ters as the Crab but with a slightly different frequencyand spin-down and with an amplitude h0 ≃ 41times 10minus25

such that the resulting loudest candidate has a value ofthe detection statistic approximately equal to the loud-est candidate of the actual analysis We have then runthe full analysis on this new data set and looked again atthe top ten candidate distribution in the frequencyspin-down plane In this case we indeed observe a clusteringwith 4 out of 10 candidates having a frequency within

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

4

40The University of Texas at Brownsville Brownsville TX 78520 USA41Universita degli Studi di Genova I-16146 Genova Italy

42INFN Sezione di Genova I-16146 Genova Italy43RRCAT Indore MP 452013 India

44Faculty of Physics Lomonosov Moscow State University Moscow 119991 Russia45LAL Universite Paris-Sud IN2P3CNRS F-91898 Orsay France46NASAGoddard Space Flight Center Greenbelt MD 20771 USA47University of Western Australia Crawley WA 6009 Australia

48Department of AstrophysicsIMAPP Radboud University NijmegenPO Box 9010 6500 GL Nijmegen The Netherlands

49ARTEMIS Universite Nice-Sophia-Antipolis CNRS and Observatoire de la Cote drsquoAzur F-06304 Nice France50MTA Eotvos University lsquoLenduletrsquo A R G Budapest 1117 Hungary

51Institut de Physique de Rennes CNRS Universite de Rennes 1 F-35042 Rennes France52Washington State University Pullman WA 99164 USA

53Universita degli Studi di Urbino rsquoCarlo Borsquo I-61029 Urbino Italy54INFN Sezione di Firenze I-50019 Sesto Fiorentino Firenze Italy

55University of Oregon Eugene OR 97403 USA56Laboratoire Kastler Brossel ENS CNRS UPMC

Universite Pierre et Marie Curie F-75005 Paris France57VU University Amsterdam 1081 HV Amsterdam The Netherlands

58University of Maryland College Park MD 20742 USA59University of Massachusetts Amherst Amherst MA 01003 USA

60Laboratoire des Materiaux Avances (LMA) IN2P3CNRSUniversite de Lyon F-69622 Villeurbanne Lyon France

61Universitat de les Illes BalearsmdashIEEC E-07122 Palma de Mallorca Spain62Universita di Napoli rsquoFederico IIrsquo Complesso Universitario di Monte SAngelo I-80126 Napoli Italy

63Canadian Institute for Theoretical AstrophysicsUniversity of Toronto Toronto Ontario M5S 3H8 Canada

64Tsinghua University Beijing 100084 China65University of Michigan Ann Arbor MI 48109 USA

66INFN Sezione di Roma Tor Vergata I-00133 Roma Italy67National Tsing Hua University Hsinchu Taiwan 300

68Charles Sturt University Wagga Wagga NSW 2678 Australia69Caltech-CaRT Pasadena CA 91125 USA

70Pusan National University Busan 609-735 Korea71Australian National University Canberra ACT 0200 Australia

72Carleton College Northfield MN 55057 USA73Universita di Roma Tor Vergata I-00133 Roma Italy

74INFN Gran Sasso Science Institute I-67100 LrsquoAquila Italy75Universita di Roma rsquoLa Sapienzarsquo I-00185 Roma Italy

76University of Brussels Brussels 1050 Belgium77Sonoma State University Rohnert Park CA 94928 USA

78Texas Tech University Lubbock TX 79409 USA79University of Minnesota Minneapolis MN 55455 USA

80The University of Sheffield Sheffield S10 2TN United Kingdom81Wigner RCP RMKI H-1121 Budapest Konkoly Thege Miklos ut 29-33 Hungary

82Montclair State University Montclair NJ 07043 USA83Argentinian Gravitational Wave Group Cordoba Cordoba 5000 Argentina

84Universita di Trento I-38123 Povo Trento Italy85INFN Trento Institute for Fundamental Physics and Applications I-38123 Povo Trento Italy

86Northwestern University Evanston IL 60208 USA87University of Cambridge Cambridge CB2 1TN United Kingdom

88University of Szeged Dom ter 9 Szeged 6720 Hungary89Rutherford Appleton Laboratory HSIC Chilton Didcot Oxon OX11 0QX United Kingdom

90Embry-Riddle Aeronautical University Prescott AZ 86301 USA91The Pennsylvania State University University Park PA 16802 USA

92American University Washington DC 20016 USA93University of Adelaide Adelaide SA 5005 Australia

94West Virginia University Morgantown WV 26506 USA95Raman Research Institute Bangalore Karnataka 560080 India

96Korea Institute of Science and Technology Information Daejeon 305-806 Korea97University of Bia lystok 15-424 Bia lystok Poland

98SUPA University of Strathclyde Glasgow G1 1XQ United Kingdom99IISER-TVM CET Campus Trivandrum Kerala 695016 India

5

100Institute of Applied Physics Nizhny Novgorod 603950 Russia101Seoul National University Seoul 151-742 Korea

102Hanyang University Seoul 133-791 Korea103NCBJ 05-400 Swierk-Otwock Poland

104IM-PAN 00-956 Warsaw Poland105Institute for Plasma Research Bhat Gandhinagar 382428 India

106The University of Melbourne Parkville VIC 3010 Australia107INFN Sezione di Padova I-35131 Padova Italy

108Monash University Victoria 3800 Australia109ESPCI CNRS F-75005 Paris France

110Universita di Camerino Dipartimento di Fisica I-62032 Camerino Italy111Southern University and AampM College Baton Rouge LA 70813 USA

112College of William and Mary Williamsburg VA 23187 USA113Abilene Christian University Abilene TX 79699 USA

114Instituto de Fısica Teorica Univ Estadual PaulistaICTP South AmericanInstitute for Fundamental Research Sao Paulo SP 01140-070 Brazil

115IISER-Kolkata Mohanpur West Bengal 741252 India116Whitman College 280 Boyer Ave Walla Walla WA 9936 USA

117National Institute for Mathematical Sciences Daejeon 305-390 Korea118Rochester Institute of Technology Rochester NY 14623 USA119Hobart and William Smith Colleges Geneva NY 14456 USA

120Tata Institute for Fundamental Research Mumbai 400005 India121SUPA University of the West of Scotland Paisley PA1 2BE United Kingdom

122Institute of Astronomy 65-265 Zielona Gora Poland123Universitat Hamburg D-22761 Hamburg Germany

124Indian Institute of Technology Gandhinagar Ahmedabad Gujarat 382424 India125Andrews University Berrien Springs MI 49104 USA

126Trinity University San Antonio TX 78212 USA127University of Washington Seattle WA 98195 USA

In this paper we present the results of a coherent narrow-band search for continuous gravitational-wave signals from the Crab and Vela pulsars conducted on Virgo VSR4 data In order to take intoaccount a possible small mismatch between the gravitational wave frequency and two times thestar rotation frequency inferred from measurement of the electromagnetic pulse rate a range of002 Hz around two times the star rotational frequency has been searched for both the pulsars Noevidence for a signal has been found and 95 confidence level upper limits have been computedboth assuming polarization parameters are completely unknown and that they are known with someuncertainty as derived from X-ray observations of the pulsar wind torii For Vela the upper limitsare comparable to the spin-down limit computed assuming that all the observed spin-down is dueto the emission of gravitational waves For Crab the upper limits are about a factor of two belowthe spin-down limit and represent a significant improvement with respect to past analysis This isthe first time the spin-down limit is significantly overcome in a narrow-band search

I INTRODUCTION

Continuous gravitational-wave signals (CW) are emit-ted by spinning neutron stars if asymmetric with respectto the rotation axis The asymmetry can be due to var-ious mechanisms like a non-axismmetric residual strainfrom the starrsquos birth or a strong internal magnetic fieldnot aligned to the rotation axis see eg [1] When thesource parameters position frequency spin-down areknown with high accuracy targeted searches can be doneusing optimal analysis methods based on matched filter-ing This happens for instance with known pulsars forwhich accurate ephemerides are often obtained from elec-tromagnetic (EM) observations especially in the radiogamma-ray and X-ray band This means that a strict cor-relation between the gravitational wave frequency f0 andthe starrsquos measured rotational frequency frot is assumedIn the classical case of a non-axisymmetric neutron star

rotating around one of its principal axes of inertia thegravitational frequency would be exactly twice the rota-tion frequency of the star Several targeted searches forCW have been conducted on data from first generationinterferometric detectors No evidence for a signal hasbeen found but interesting upper limits have been placedin a few cases [2ndash4] Given the uncertainties on gravita-tional wave emission mechanisms and also the lack of afull detailed picture of the electro-magnetic emission ge-ometry it is not obvious at all that the gravitational-waveemission takes place at exactly twice the star measuredpulse rate or that such relation holds for observationtimes of months to years For instance if a neutron staris made of a crust and a core rotating at slightly differentrates and if the gravitational-wave emission is dominatedby an asymmetry in the core then a search targeted at2frot would assume a wrong signal frequency We thenconsider here that the signal frequency can be slightly

6

different with respect to f0 = 2frot and can vary in theinterval

f(t) isin [f0(t)(1minus δ) f0(t)(1 + δ)] (1)

where δ is a small positive real number Following thediscussion in [5] if the star crust and core are linkedby some torque that tends to enforce corotation on atimescale τc then δ sim τcτsd where τsd sim f0f0 is thecharacteristic spin-down time A relation of the form ofEq (1) also holds in the case the gravitational radiationis produced by free precession of a nearly bi-axial star [6]in which case δ is of the order of (Izz minus Ixx)Ixx whereIxx and Izz are the star moments of inertia with respectto a principal axis on the equatorial plane and alignedwith the rotation axis respectively In general a value ofδ of the order of say 10minus4 corresponds to τc sim 10minus4τsdwhich depending on the specific targeted pulsar can beseveral months or years This would be comparable orlarger than the longest timescale observed in pulsar glitchrecovery where a recoupling between the two componentsmight occur In terms of free precession δ sim 10minus4 is onthe high end of the range of deformations that neutronstars could sustain [7ndash9]Narrow-band searches have not received much atten-

tion until now one notable exception being the Crabnarrow-band search over the first 9 months of LIGO S5data [5] based on the F -statistic [10] In previous work[11] an optimal method based on matched filtering in thespace of signal Fourier components has been proposedto carry out a search for CW over a small frequency andspin-down range around the values inferred from EM ob-servations In this paper we describe the application ofsuch a method to a narrow-band search of CW from Craband Vela pulsars in the data of Virgo VSR4 run As noevidence for a signal is found we place upper limits onsignal amplitudeThe outline of the paper is the following In Sec II the

expected gravitational wave signal from spinning neutronstars is introduced In Sec III the main characteristics ofVirgo VSR4 data are presented In Sec IV the analysismethod is described while in Sec V the analysis resultsare discussed Sec VI is dedicated to the validation testsof the analysis procedure Finally Sec VII contains theconclusions

II THE GRAVITATIONAL WAVE SIGNAL

A non-axisymmetric neutron star steadily spinningabout one of its principal axis emits a quadrupolargravitational-wave signal at twice the star rotational fre-quency frot which at the detector can be described [12]as

h(t) = H0(H+A+(t) +HtimesA

times(t))eı(ω0(t)t+Φ0) (2)

where taking the real part is understood The constantΦ0 is the initial signal phase The angular signal fre-

quency ω0(t) = 4πfrot(t) is a function of time Conse-quently the signal phase is not that of a simple monochro-matic signal and depends on both the intrinsic rotationalfrequency and frequency derivatives of the pulsar and onDoppler and propagation effects which include the Ein-stein delay and possibly the Shapiro delay These varia-tions are corrected in the time-domain in a way describedin Sec IV The two complex amplitudes H+ and Htimes aregiven respectively by

H+ =cos 2ψ minus ıη sin 2ψ

radic

1 + η2(3)

Htimes =sin 2ψ + ıη cos 2ψ

radic

1 + η2(4)

in which η is the ratio of the polarization ellipse semi-minor to semi-major axis and the polarization angle ψdefines the direction of the major axis with respect to thecelestial parallel of the source (measured counterclock-wise) The functions A+ and Atimes describe the detectorresponse as a function of time and are a linear combi-nation of terms depending on the Earth sidereal angularfrequency Ωoplus

A+ = a0 + a1c cosΩoplust+ a1s sinΩoplust+

a2c cos 2Ωoplust+ a2s sin 2Ωoplust (5)

Atimes = b1c cosΩoplust+ b1s sinΩoplust+

b2c cos 2Ωoplust+ b2s sin 2Ωoplust (6)

where the coefficients depend on the source position anddetector position and orientation on the Earth [12]As discussed eg in [3] the overall wave amplitude H0

and η are related to the ldquostandardrdquo signal amplitude

h0 =4π2G

c4Izzεf

20

d(7)

and to the angle ι between the star rotation axis and theline of sight to the source by

H0 = h0

radic

1 + 6 cos2 ι+ cos4 ι

4(8)

η = minus2 cos ι

1 + cos2 ι (9)

In Eq (7) G is the gravitational constant c is the lightvelocity Izz is the star moment of inertia with respectto the principal axis aligned with the rotation axis ε =IxxminusIyyIzz

is the equatorial ellipticity expressed in terms ofprincipal moments of inertia and d is the source distance

7

Equating the gravitational-wave luminosity to the ki-netic energy lost as the pulsar spins down gives us theso-called spin-down limit on gravitational-wave strain

hsd0 =

(

5

2

GIzz frotc3d2frot

)12

=

806times 10minus19 I1238

dkpc

(

|frot|

Hzs

)12(

frotHz

)minus12

(10)

where I38 is the starrsquos moment of inertia in units of1038 kgm2 frot is the time derivative of the rotationalfrequency and dkpc is the distance to the pulsar in kilo-parsecs The spin-down limit on the signal amplitudecorresponds to an upper limit on the starrsquos fiducial ellip-ticity

εsd = 0237

(

hsd010minus24

)

fminus2rot I

minus138 dkpc (11)

This quantity for a given neutron star equation of statecan be related to the physical ellipticity of the star sur-face [9] Setting a gravitational-wave upper limit belowthe spin-down limit is an important milestone in CWsearches as it allows us to constrain the fraction of spin-down energy due to the emission of gravitational-waves which gives insight into the spin-down energy budgetof the pulsar On the other hand the l = m = 2 massquadrupole moment Q22 is related to the gravitational-wave amplitude by

Q22 =

radic

15

8πIzzε = h0

(

c4d

16π2Gf2rot

)

radic

15

8π (12)

see eg [7] and is independent of any assumptions aboutthe starrsquos equation of state and its moment of inertia

III INSTRUMENTAL PERFORMANCE IN

VIRGO VSR4 RUN

Interferometric gravitational wave detectors LIGO[13] Virgo [14] GEO [15] have collected a large amountof data in recent years For the analysis described in thispaper we have used calibrated data sampled at 4096 Hzfrom Virgo VSR4 science run The run extended fromJune 3rd 2011 (1027 UTC) to September 5th 2011(1326 UTC) with a duty factor of about 81 corre-sponding to an effective duration of 76 days Calibra-tion uncertainties amounted to 75 in amplitude and(40+50fkHz) mrad in phase up to 500 Hz where fkHz isthe frequency in kilohertz The uncertainty on the ampli-tude will contribute to the uncertainty on the upper limiton signal amplitude together with that coming from thefinite size of the Monte Carlo simulation used to computeit see Sec V The calibration error on the phase can beshown to have a negligible impact on the analysis [3]

The low-frequency sensitivity of VSR4 was significantlybetter than that of previous Virgo runs especially due tothe use of monolithic mirror suspensions and basicallyin agreement with the planned sensitivity of the initialVirgo interferometer In Fig 1 a typical VSR4 strainsensitivity curve is shown In Fig 2 the average powerspectrum around Crab and Vela reference frequency isplotted Note the large sensitivity improvement aroundthe Vela frequency after removal of an instrumental dis-turbance about one month after the beginning of therun see the figure caption for more details

IV SEARCH DESCRIPTION

The analysis pipeline consists of several steps schemat-ically depicted in Fig 3 which we summarize here Moredetails are given in [11] The starting point is a collectionof windowed and interlaced (by half) ldquoshortrdquo (1024 sec-onds) FFTs (the Short FFT Database - SFDB) built fromcalibrated detector data [16] At this stage a first clean-ing procedure is applied to the data in order to removebig and short time duration disturbances that could re-duce the search sensitivity A small frequency band isthen extracted from the SFDB but large enough to con-tain the Doppler and spin-down variations of the signalpossibly emitted by the target pulsar In the analysesdescribed in this paper for instance it was of 015 HzAt this point we go back to the time domain (the sam-pling frequency is still the original one 4096 Hz) andmake barycentric and spin-down corrections Due to theDoppler effect the received frequency f(t) is related tothe emitted frequency f0(t) by the well-known relation(valid in the non-relativistic approximation)

f(t) =1

2π

dΦ(t)

dt= f0(t)

(

1 +~v middot n

c

)

(13)

where Φ(t) is the signal phase ~v = ~vrev+~vrot is the detec-tor velocity with respect to the Solar system barycenter(SSB) given by the sum of the Earthrsquos orbital velocity~vrev and rotational velocity ~vrot while n is the versoridentifying the source position In practice the Dopplereffect is efficiently corrected in the time domain by chang-ing the time stamp t of data samples according to

τ1 = t+~r(t) middot n

c= t+∆R (14)

where ~r is the detector position in the SSB The correc-tion term ∆R called Roemer delay amounts up to about500 seconds and corresponds to the time taken by a signaltraveling at the speed of light to cover the distance be-tween the detector and the SSB Unlike Eq (13) Eq (14)does not depend on the frequency which means that onesingle correction holds for every frequency In fact thereare other smaller relativistic effects that must be takeninto account when making barycentric corrections Oneis the Einstein delay ∆E due to the Earth motion and

8

102

103

10minus23

10minus22

10minus21

10minus20

10minus19

Frequency [Hz]

h n [Hzminus

12 ]

FIG 1 A typical sensitivity curve of Virgo VSR4 run expressed in terms of noise spectral density

032 034 036 038 04 042 04410

minus3

10minus2

Frequency minus220 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

036 038 04 042 044 046 048 05

10minus4

10minus3

Frequency minus 590 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

FIG 2 Average spectrum of VSR4 data around Vela (top plot) and Crab (bottom plot) reference frequency (identified by thevertical black lines) For Vela two spectra are in fact shown in the same plot The upper one (black curve) has been computedover the first 29 days of the run when the frequency region around Vela was affected by an instrumental disturbance Thiswas due to a non-linear coupling between the DARM control line at 379 Hz and calibration lines at 356 Hz and 3565 Hz andwas moved away from the Vela region by shifting of 5 Hertz the frequency of calibration lines The lower plot (blue curve) isthe average spectrum computed after the removal of the disturbance The large spectral disturbance appearing near the Crabreference frequency is due to a poorly understood coupling between a line at 60 Hz part of a 10-Hertz comb of lines of likelymagnetic origin and the fundamental pendulum frequency of the Virgo mirror system at 0594 Hz

9

$amp())+-amp0

12$amp(3-45

amp(67(898)

(amp-8

ltamp9(8amp$

-ampamp(-8

=00gt$8)-8

-ampamp(-8A((-800

1B-amp(Bamp(67(895

C(3-4-B

-7$(amp0

D-7)(08)$)(

Egtgt(amp$3$-amp

gtamp3((amp(03-8

95

(

F8)0gt$8)-8

-ampamp(-8

0gt$8)-8

-ampamp(-8

8

C(3-4-B

-7$(amp0

C(3-4-B

-7$(amp0

A((-800

1B-amp(Bamp(67(895

A((-800

1B-amp(Bamp(67(895

1gt47(nɺf

FIG 3 Scheme of the narrow-band analysis pipeline The starting point is constituted by detector calibrated data sampled at4096 Hz After barycentric and spin-down corrections the data are down-sampled at 1 Hz The number of different spin-downcorrections applied to the data nf and then the number of produced corrected time series is given by Eq (21) See text and[11] for more details

the gravitational redshift at the Earth geocenter due tothe solar system bodies and that amounts to about 2milliseconds at most Another effect is the Shapiro delay

∆S which takes into account the deflection of a signalpassing near a massive body and which can be shownto be negligible for CW searches unless the source lineof sight passes very near the Sunrsquos limb See eg [11]for explicit expressions of ∆E and ∆S Overall we canmake the full barycentric corrections by introducing thenew time variable

τ1 = t+∆R +∆E minus∆S (15)

This transformation corresponds to referring the datacollected at the detector site to the SSB which can beconsidered an inertial reference frame to a very good ap-proximation In practice for the Crab and Vela anal-yses described in this paper the Shapiro delay can beneglectedFor given values of the signal frequency f and fre-

quency derivatives f f we could take into account thespin-down in a similar way by re-scaling time accordingto

τ2 = t+f

2f(tminus t0)

2 +f

6f(tminus t0)

3 (16)

where t0 is the initial time of the data set and higherorder terms can be included if neededNote that re-scaling the time in this way to make the

signal monochromatic assumes that the intrinsic phaseevolution of the pulsarrsquos GW signal is described by a Tay-lor expansion over the entire observation period Elec-tromagnetic observations of pulsars show that young

rapidly spinning pulsars demonstrate deviations from aTaylor expansion when spinning down a phenomenonknown as timing noise Given our lack of knowledge ofthe exact mechanism that might cause the gravitationalwave frequency to deviate from (twice) the observed elec-tromagnetic frequency we cannot be sure if we shouldexpect the timing noise to be present in the signal weare searching for (see [17] for discussion) However astudy based on the monthly Crab ephemeris data [18]has shown that in the Crab one of the noisiest of pul-sars the effect of timing noise over the duration of theobservation period is negligible producing a mismatchbetween a Taylor expansion and a signal based on theactual lsquonoisyrsquo time series of less than 1 This is smallconfirming that timing noise is likely to have a negligibleeffect on our analysis

In practice the spin-down correction is applied afterbarycentric corrections then the time t that appears inEq (16) is in fact the re-scaled time τ1 of Eq (15) Inthis case the correction depends explicitly on the searchfrequency f The number of frequency bins which coverthe range ∆f = 2f0δ corresponding to Eq (1) is

nf =

[

∆f

δf

]

= [∆f middot T ] asymp 63times 105(

∆f

002 Hz

)(

T

1 yr

)

(17)with δf = 1

T being the frequency spacing with T thetotal observation time and where [ middot ] stands for thenearest integer Similarly we take the width of the firstorder spin-down range as ∆f = 2|f0|δ and ∆f = 2|f0|δfor the second order Let us consider for the momentonly the first order term For a fixed value of f we donot want to make an explicit spin-down correction for

10

each value of f So we fix the value of the frequency inthe denominator of Eq (16) for instance at the referencefrequency f0 and exploit the fact that the same correc-tion that holds for the pair (f0 f0) is also valid for all the

pairs (f f) such that

f0f0

=f

f(18)

In practice this means that at each frequency f we ex-

plore a range of first order spin-down values [fminus ∆f2 f+

∆f2 ] with

f = f0f

f0(19)

The number of bins in the spin-down term of the firstorder can be computed by considering the ldquonaturalrdquo stepδf = 1

2T 2 and is given by

n(1) equiv nf =[

2∆f middot T 2]

(20)

asymp 400

(

f010minus10 Hzs

)

(

δ

10minus3

)(

T

1 yr

)2

Following the same reasoning for the second order spin-down we arrive at the following expression for the numberof steps that should be taken into account

n(2) equiv nf =[

∆f middot T 3]

(21)

asymp 06

(

f010minus20 Hzs2

)

(

δ

10minus3

)(

T

1 yr

)3

For observation times of the order of a year reasonablevalues of second order spin-down and values of δ typicalof narrow-band searches nf lt 1 which means that we donot need to consider more values other than the ldquocentralrdquoone In Tab I the Crab and Vela EM-inferred positionaland rotational parameters are shown Also estimationsof the polarization parameters are given which are usedin the computation of upper limits see Sec V In Fig 4the portion of the f minus f plane actually covered in theCrab and Vela narrow-band search is shownOnce barycentric and spin-down corrections have been

done the data are down-sampled to a much lower ratewith respect to the original one 1 Hz in the present caseThis strongly reduces the amount of data to be handledin the next step of the analysis At this stage for a givensource we have nf corrected time series one for eachvalue of first order spin-down For the current analysis wehave nf = 33 for Crab and nf = 3 for Vela see Tab II)where also other relevant quantities are given In partic-ular the total number of points in the frequencyspin-down plane is about 528 times 106 for Crab analysis and48 times 105 for Vela analysis For each time series we ap-ply a final cleaning step by removing the largest outliersThese are identified by histogramming the logarithm of

the absolute value of the data amplitude and choosinga threshold approximately marking the beginning of thenon-Gaussian tail of data distribution In Fig 5 the his-togram of Crab and Vela data amplitude correspond-ing to the ldquocentralrdquo time series are shown For Crab athreshold of 15 times 10minus21 corresponding to -182 in thefigure x-axis has been used to remove outliers while forVela a value of 12 times 10minus20 corresponding to -092 hasbeen used Correspondingly the fraction of removed datais of about 18 for Vela and 19 for Crab By apply-ing the Kolmogorov-Smirnov test we have verified thatfor each pulsar the data distribution of the various timeseries are fully in agreement thus justifying the use of thesame threshold for all of themAt this point the detection statistic is computed for ev-

ery frequency and spin-down value in the explored rangeThe detection statistic we use is based on the so-called 5-vectors the same used for pulsar targeted searches [3 4]and is here briefly described Once barycentric and spin-down corrections have been applied a CW signal withfrequency f0 present in the data would be monochro-matic apart from an amplitude and phase sidereal mod-ulation due to the time-varying detector beam patternfunctions and given by Eq (2) with ω0(t) constant andequal to 2πf0 From Eqs (256) it follows that the signalis completely described by its Fourier components at the5 angular frequencies ω0 ω0plusmnΩoplus ω0plusmn2Ωoplus This set of5 complex numbers constitutes the signal 5-vector Givena generic time series g(t) the corresponding 5-vector isdefined as

G =

int

T

g(t)eminusı(ω0minuskΩoplus)tdt (22)

where k = [minus2minus1 2] and T is the observation timeLet us indicate with X the data 5-vector and withA

+ Atimes the signal plus and cross 5-vectors obtained byapplying the definition of Eq (22) to Eqs (56) Thesetwo last quantities depend only on known parameters andform the signal templates Once the 5-vectors of data andof signal templates have been computed the two complexnumbers

H+times =X middotA+times

|A+times|2(23)

are built see [3 12] for more details They correspond tocomputing matched filters between the data and the sig-nal templates and it can be shown assuming the noiseis Gaussian with mean value zero that they are esti-mators of the signal plus and cross complex amplitudesH0e

ıΦ0H+ H0eıΦ0Htimes These estimators are used to

build the detection statistic

S = |A+|4|H+|2 + |Atimes|4|Htimes|

2 (24)

The maximum of the detection statistic over thesearched parameter space Smax is determined Thisis the loudest candidate and is identified by a triple

11

22374 22376 22378 2238 22382 22384 22386 22388 2239 22392 22394minus3149

minus3148

minus3147

minus3146

minus3145

minus3144

minus3143x 10

minus11

Frequency [Hz]S

pinminus

dow

n [H

zs]

Covered parameter space for Vela pulsar

59434 59436 59438 5944 59442 59444 59446 59448 5945 59452 59454minus7429

minus7428

minus7427

minus7426

minus7425

minus7424

minus7423x 10

minus10

Frequency [Hz]

Spi

nminusdo

wn

[Hz

s]

Covered parameter space for Crab pulsar

FIG 4 The parallelograms delimit the portion of the f minus f plane covered in the narrow-band search for the Vela (upper plot)

and Crab (bottom plot) pulsars Only one value of f has been considered in the analysis The total number of points is givenin the last column of Tab II

minus3 minus25 minus2 minus15 minus1 minus05 0 05 1

102

103

104

105

log10(abs(real(data1020)))

minus3 minus28 minus26 minus24 minus22 minus2 minus18 minus16 minus14 minus12 minus110

1

102

103

104

105

log10(abs(real(data1020)))

FIG 5 Histogram of the logarithm of data amplitude for Vela (top plot) and the Crab (bottom plot) used to select the thresholdfor the removal of outliers For Crab the threshold has been put at -182 corresponding to an amplitude of sim 15 times 10minus21while for Vela a value of -092 corresponding to an amplitude of sim 12times 10minus20 has been chosen

12

TABLE I The Crab and Vela reference parameters α and δ are the equatorial coordinates f0 f0 f0 are the gravitational-wavefrequency parameters The reference epoch for the Crab position is MJD 54632 while for the rotational parameters is MJD55696 The reference epoch for Vela is MJD 53576 both for position and GW frequency parameters The Crab ephemeris hasbeen obtained from a fit of the Jodrell Bank monthly ephemeris [18] while for Vela they have been derived from Hartebeesthoekradio telescope observations [19] Also estimations of the polarization parameters and their associated uncertainty obtainedfrom the analysis of Chandra X-ray observations of the pulsar wind nebula torus [20 21] are given in the last two columnsThe analysis presented in [20 21] does not allow to determine the sense of the starrsquos spin so values of ι and ψ correspondingto ι rarr 180 minus ι ψ rarr ψ + 180 are also possible However the upper limits on strain amplitude reported in this paper are notsensitive to the sense of rotation

Source α [hhmmss] δ[deg] f0[Hz] f0[Hzs] f0[Hzs2] ι[deg] ψ[deg]

Crab 05 34 3197 220145 594448 minus74183 times 10minus10 26307 times 10minus20 62o2plusmn 1o9 35o2plusmn 1o5

Vela 08 35 2061 minus451764 223840 minus31460 times 10minus11 12848 times 10minus21 63o6plusmn 0o6 40o6plusmn 0o1

TABLE II Main quantities related to the parameter space of VSR4 narrow-band search ∆f is the frequency range δ is thewidth parameter defined in Eq (1) ∆f is the first order spin-down range nf is the number of frequency bins nf and nf arerespectively the number of bins for the first and second order spin-down ntot = nf middot nf is the total number of points in theparameter space

Source ∆f [Hz] δ ∆f [Hzs] nf nf nf ntot

Crab 002 168times 10minus4 249times 10minus13 16times 105 33 1 528times 106

Vela 002 447times 10minus4 281times 10minus14 16times 105 3 1 480times 105

(Smax fSmax fSmax

) This means we need to computentot = nf middot nf values of the detection statisticThe maximum value Smax is used to assess detection

significance by computing the p-value that is the prob-ability that a value of the detection statistic equal orlarger than Smax can be obtained in the absence ofany signal It implies the need to compute the noise-only distribution of the detection statistic This is amulti-dimensional probability distribution (with dimen-sion ntot) which would be difficult to compute and to han-dle In practice as discussed in [11] the p-value is com-puted considering the single-trial noise probability distri-bution that is the same that would be used for a targetedsearch and choosing a suitable threshold on it to dis-criminate between interesting (that is deserving a deeperstudy) and not interesting candidates As shown in [11]by setting an overall p-value p0 (over the full multi-dimensional distribution) the corresponding threshold on

the single-trial distribution is pthr = 1minus (1minus p0)1

ntot Inour case by setting eg p0 = 001 the resulting pthrwould be of the order of 10minus8 In principle we wouldlike to generate the noise-only probability distribution bycomputing the detection statistic at several ldquooff-sourcerdquofrequency bins that is frequencies near but outside theexplored range where we are assuming the signal couldbe This is what is typically done in targeted searches [4]In the present analysis however in order to appreciate p-values of the order of pthr we should consider of the orderof 108 off-source frequency bins This is computationallyimpractical For this reason we use the theoretical dis-tribution which assumes the noise is Gaussian given in[11]

f(S) =eminus S

σ2X

|Atimes|2 minus eminus S

σ2X

|A+|2

σ2X (|Atimes|2 minus |A+|2)

(25)

where σ2X = σ2 middot T with σ2 being the noise variance

If Smax is compatible with noise at a given confidencelevel eg 1 we compute an upper limit on signalstrength using the same method described in [4] other-wise in case of detection signal parameters are estimatedthrough suitable combinations of the real and imaginaryparts of H+ and Htimes as explained in [12]

V RESULTS

Analysis results are summarized in Tab III The databoth for the Vela and Crab searches are compatible withnoise In particular we find p-values equal to 033 and0013 for Vela and the Crab respectively larger thanp0 = 001 chosen to identify interesting candidates ForCrab however the obtained p-value is very near to thechosen threshold Altough a reasonable choice of p0 israther arbitrary (we could have for instance chosen 10minus3)we have decided to study in some detail the candidatersquosproperties In particular we have considered the distri-bution of the top ten candidates in the frequencyspin-down plane They appear to be randomly distributedwithout the clustering that we would expect in presenceof a signal To verify this hypothesis we have addeda simulated signal to the data with the same parame-ters as the Crab but with a slightly different frequencyand spin-down and with an amplitude h0 ≃ 41times 10minus25

such that the resulting loudest candidate has a value ofthe detection statistic approximately equal to the loud-est candidate of the actual analysis We have then runthe full analysis on this new data set and looked again atthe top ten candidate distribution in the frequencyspin-down plane In this case we indeed observe a clusteringwith 4 out of 10 candidates having a frequency within

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

5

100Institute of Applied Physics Nizhny Novgorod 603950 Russia101Seoul National University Seoul 151-742 Korea

102Hanyang University Seoul 133-791 Korea103NCBJ 05-400 Swierk-Otwock Poland

104IM-PAN 00-956 Warsaw Poland105Institute for Plasma Research Bhat Gandhinagar 382428 India

106The University of Melbourne Parkville VIC 3010 Australia107INFN Sezione di Padova I-35131 Padova Italy

108Monash University Victoria 3800 Australia109ESPCI CNRS F-75005 Paris France

110Universita di Camerino Dipartimento di Fisica I-62032 Camerino Italy111Southern University and AampM College Baton Rouge LA 70813 USA

112College of William and Mary Williamsburg VA 23187 USA113Abilene Christian University Abilene TX 79699 USA

114Instituto de Fısica Teorica Univ Estadual PaulistaICTP South AmericanInstitute for Fundamental Research Sao Paulo SP 01140-070 Brazil

115IISER-Kolkata Mohanpur West Bengal 741252 India116Whitman College 280 Boyer Ave Walla Walla WA 9936 USA

117National Institute for Mathematical Sciences Daejeon 305-390 Korea118Rochester Institute of Technology Rochester NY 14623 USA119Hobart and William Smith Colleges Geneva NY 14456 USA

120Tata Institute for Fundamental Research Mumbai 400005 India121SUPA University of the West of Scotland Paisley PA1 2BE United Kingdom

122Institute of Astronomy 65-265 Zielona Gora Poland123Universitat Hamburg D-22761 Hamburg Germany

124Indian Institute of Technology Gandhinagar Ahmedabad Gujarat 382424 India125Andrews University Berrien Springs MI 49104 USA

126Trinity University San Antonio TX 78212 USA127University of Washington Seattle WA 98195 USA

In this paper we present the results of a coherent narrow-band search for continuous gravitational-wave signals from the Crab and Vela pulsars conducted on Virgo VSR4 data In order to take intoaccount a possible small mismatch between the gravitational wave frequency and two times thestar rotation frequency inferred from measurement of the electromagnetic pulse rate a range of002 Hz around two times the star rotational frequency has been searched for both the pulsars Noevidence for a signal has been found and 95 confidence level upper limits have been computedboth assuming polarization parameters are completely unknown and that they are known with someuncertainty as derived from X-ray observations of the pulsar wind torii For Vela the upper limitsare comparable to the spin-down limit computed assuming that all the observed spin-down is dueto the emission of gravitational waves For Crab the upper limits are about a factor of two belowthe spin-down limit and represent a significant improvement with respect to past analysis This isthe first time the spin-down limit is significantly overcome in a narrow-band search

I INTRODUCTION

Continuous gravitational-wave signals (CW) are emit-ted by spinning neutron stars if asymmetric with respectto the rotation axis The asymmetry can be due to var-ious mechanisms like a non-axismmetric residual strainfrom the starrsquos birth or a strong internal magnetic fieldnot aligned to the rotation axis see eg [1] When thesource parameters position frequency spin-down areknown with high accuracy targeted searches can be doneusing optimal analysis methods based on matched filter-ing This happens for instance with known pulsars forwhich accurate ephemerides are often obtained from elec-tromagnetic (EM) observations especially in the radiogamma-ray and X-ray band This means that a strict cor-relation between the gravitational wave frequency f0 andthe starrsquos measured rotational frequency frot is assumedIn the classical case of a non-axisymmetric neutron star

rotating around one of its principal axes of inertia thegravitational frequency would be exactly twice the rota-tion frequency of the star Several targeted searches forCW have been conducted on data from first generationinterferometric detectors No evidence for a signal hasbeen found but interesting upper limits have been placedin a few cases [2ndash4] Given the uncertainties on gravita-tional wave emission mechanisms and also the lack of afull detailed picture of the electro-magnetic emission ge-ometry it is not obvious at all that the gravitational-waveemission takes place at exactly twice the star measuredpulse rate or that such relation holds for observationtimes of months to years For instance if a neutron staris made of a crust and a core rotating at slightly differentrates and if the gravitational-wave emission is dominatedby an asymmetry in the core then a search targeted at2frot would assume a wrong signal frequency We thenconsider here that the signal frequency can be slightly

6

different with respect to f0 = 2frot and can vary in theinterval

f(t) isin [f0(t)(1minus δ) f0(t)(1 + δ)] (1)

where δ is a small positive real number Following thediscussion in [5] if the star crust and core are linkedby some torque that tends to enforce corotation on atimescale τc then δ sim τcτsd where τsd sim f0f0 is thecharacteristic spin-down time A relation of the form ofEq (1) also holds in the case the gravitational radiationis produced by free precession of a nearly bi-axial star [6]in which case δ is of the order of (Izz minus Ixx)Ixx whereIxx and Izz are the star moments of inertia with respectto a principal axis on the equatorial plane and alignedwith the rotation axis respectively In general a value ofδ of the order of say 10minus4 corresponds to τc sim 10minus4τsdwhich depending on the specific targeted pulsar can beseveral months or years This would be comparable orlarger than the longest timescale observed in pulsar glitchrecovery where a recoupling between the two componentsmight occur In terms of free precession δ sim 10minus4 is onthe high end of the range of deformations that neutronstars could sustain [7ndash9]Narrow-band searches have not received much atten-

tion until now one notable exception being the Crabnarrow-band search over the first 9 months of LIGO S5data [5] based on the F -statistic [10] In previous work[11] an optimal method based on matched filtering in thespace of signal Fourier components has been proposedto carry out a search for CW over a small frequency andspin-down range around the values inferred from EM ob-servations In this paper we describe the application ofsuch a method to a narrow-band search of CW from Craband Vela pulsars in the data of Virgo VSR4 run As noevidence for a signal is found we place upper limits onsignal amplitudeThe outline of the paper is the following In Sec II the

expected gravitational wave signal from spinning neutronstars is introduced In Sec III the main characteristics ofVirgo VSR4 data are presented In Sec IV the analysismethod is described while in Sec V the analysis resultsare discussed Sec VI is dedicated to the validation testsof the analysis procedure Finally Sec VII contains theconclusions

II THE GRAVITATIONAL WAVE SIGNAL

A non-axisymmetric neutron star steadily spinningabout one of its principal axis emits a quadrupolargravitational-wave signal at twice the star rotational fre-quency frot which at the detector can be described [12]as

h(t) = H0(H+A+(t) +HtimesA

times(t))eı(ω0(t)t+Φ0) (2)

where taking the real part is understood The constantΦ0 is the initial signal phase The angular signal fre-

quency ω0(t) = 4πfrot(t) is a function of time Conse-quently the signal phase is not that of a simple monochro-matic signal and depends on both the intrinsic rotationalfrequency and frequency derivatives of the pulsar and onDoppler and propagation effects which include the Ein-stein delay and possibly the Shapiro delay These varia-tions are corrected in the time-domain in a way describedin Sec IV The two complex amplitudes H+ and Htimes aregiven respectively by

H+ =cos 2ψ minus ıη sin 2ψ

radic

1 + η2(3)

Htimes =sin 2ψ + ıη cos 2ψ

radic

1 + η2(4)

in which η is the ratio of the polarization ellipse semi-minor to semi-major axis and the polarization angle ψdefines the direction of the major axis with respect to thecelestial parallel of the source (measured counterclock-wise) The functions A+ and Atimes describe the detectorresponse as a function of time and are a linear combi-nation of terms depending on the Earth sidereal angularfrequency Ωoplus

A+ = a0 + a1c cosΩoplust+ a1s sinΩoplust+

a2c cos 2Ωoplust+ a2s sin 2Ωoplust (5)

Atimes = b1c cosΩoplust+ b1s sinΩoplust+

b2c cos 2Ωoplust+ b2s sin 2Ωoplust (6)

where the coefficients depend on the source position anddetector position and orientation on the Earth [12]As discussed eg in [3] the overall wave amplitude H0

and η are related to the ldquostandardrdquo signal amplitude

h0 =4π2G

c4Izzεf

20

d(7)

and to the angle ι between the star rotation axis and theline of sight to the source by

H0 = h0

radic

1 + 6 cos2 ι+ cos4 ι

4(8)

η = minus2 cos ι

1 + cos2 ι (9)

In Eq (7) G is the gravitational constant c is the lightvelocity Izz is the star moment of inertia with respectto the principal axis aligned with the rotation axis ε =IxxminusIyyIzz

is the equatorial ellipticity expressed in terms ofprincipal moments of inertia and d is the source distance

7

Equating the gravitational-wave luminosity to the ki-netic energy lost as the pulsar spins down gives us theso-called spin-down limit on gravitational-wave strain

hsd0 =

(

5

2

GIzz frotc3d2frot

)12

=

806times 10minus19 I1238

dkpc

(

|frot|

Hzs

)12(

frotHz

)minus12

(10)

where I38 is the starrsquos moment of inertia in units of1038 kgm2 frot is the time derivative of the rotationalfrequency and dkpc is the distance to the pulsar in kilo-parsecs The spin-down limit on the signal amplitudecorresponds to an upper limit on the starrsquos fiducial ellip-ticity

εsd = 0237

(

hsd010minus24

)

fminus2rot I

minus138 dkpc (11)

This quantity for a given neutron star equation of statecan be related to the physical ellipticity of the star sur-face [9] Setting a gravitational-wave upper limit belowthe spin-down limit is an important milestone in CWsearches as it allows us to constrain the fraction of spin-down energy due to the emission of gravitational-waves which gives insight into the spin-down energy budgetof the pulsar On the other hand the l = m = 2 massquadrupole moment Q22 is related to the gravitational-wave amplitude by

Q22 =

radic

15

8πIzzε = h0

(

c4d

16π2Gf2rot

)

radic

15

8π (12)

see eg [7] and is independent of any assumptions aboutthe starrsquos equation of state and its moment of inertia

III INSTRUMENTAL PERFORMANCE IN

VIRGO VSR4 RUN

Interferometric gravitational wave detectors LIGO[13] Virgo [14] GEO [15] have collected a large amountof data in recent years For the analysis described in thispaper we have used calibrated data sampled at 4096 Hzfrom Virgo VSR4 science run The run extended fromJune 3rd 2011 (1027 UTC) to September 5th 2011(1326 UTC) with a duty factor of about 81 corre-sponding to an effective duration of 76 days Calibra-tion uncertainties amounted to 75 in amplitude and(40+50fkHz) mrad in phase up to 500 Hz where fkHz isthe frequency in kilohertz The uncertainty on the ampli-tude will contribute to the uncertainty on the upper limiton signal amplitude together with that coming from thefinite size of the Monte Carlo simulation used to computeit see Sec V The calibration error on the phase can beshown to have a negligible impact on the analysis [3]

The low-frequency sensitivity of VSR4 was significantlybetter than that of previous Virgo runs especially due tothe use of monolithic mirror suspensions and basicallyin agreement with the planned sensitivity of the initialVirgo interferometer In Fig 1 a typical VSR4 strainsensitivity curve is shown In Fig 2 the average powerspectrum around Crab and Vela reference frequency isplotted Note the large sensitivity improvement aroundthe Vela frequency after removal of an instrumental dis-turbance about one month after the beginning of therun see the figure caption for more details

IV SEARCH DESCRIPTION

The analysis pipeline consists of several steps schemat-ically depicted in Fig 3 which we summarize here Moredetails are given in [11] The starting point is a collectionof windowed and interlaced (by half) ldquoshortrdquo (1024 sec-onds) FFTs (the Short FFT Database - SFDB) built fromcalibrated detector data [16] At this stage a first clean-ing procedure is applied to the data in order to removebig and short time duration disturbances that could re-duce the search sensitivity A small frequency band isthen extracted from the SFDB but large enough to con-tain the Doppler and spin-down variations of the signalpossibly emitted by the target pulsar In the analysesdescribed in this paper for instance it was of 015 HzAt this point we go back to the time domain (the sam-pling frequency is still the original one 4096 Hz) andmake barycentric and spin-down corrections Due to theDoppler effect the received frequency f(t) is related tothe emitted frequency f0(t) by the well-known relation(valid in the non-relativistic approximation)

f(t) =1

2π

dΦ(t)

dt= f0(t)

(

1 +~v middot n

c

)

(13)

where Φ(t) is the signal phase ~v = ~vrev+~vrot is the detec-tor velocity with respect to the Solar system barycenter(SSB) given by the sum of the Earthrsquos orbital velocity~vrev and rotational velocity ~vrot while n is the versoridentifying the source position In practice the Dopplereffect is efficiently corrected in the time domain by chang-ing the time stamp t of data samples according to

τ1 = t+~r(t) middot n

c= t+∆R (14)

where ~r is the detector position in the SSB The correc-tion term ∆R called Roemer delay amounts up to about500 seconds and corresponds to the time taken by a signaltraveling at the speed of light to cover the distance be-tween the detector and the SSB Unlike Eq (13) Eq (14)does not depend on the frequency which means that onesingle correction holds for every frequency In fact thereare other smaller relativistic effects that must be takeninto account when making barycentric corrections Oneis the Einstein delay ∆E due to the Earth motion and

8

102

103

10minus23

10minus22

10minus21

10minus20

10minus19

Frequency [Hz]

h n [Hzminus

12 ]

FIG 1 A typical sensitivity curve of Virgo VSR4 run expressed in terms of noise spectral density

032 034 036 038 04 042 04410

minus3

10minus2

Frequency minus220 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

036 038 04 042 044 046 048 05

10minus4

10minus3

Frequency minus 590 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

FIG 2 Average spectrum of VSR4 data around Vela (top plot) and Crab (bottom plot) reference frequency (identified by thevertical black lines) For Vela two spectra are in fact shown in the same plot The upper one (black curve) has been computedover the first 29 days of the run when the frequency region around Vela was affected by an instrumental disturbance Thiswas due to a non-linear coupling between the DARM control line at 379 Hz and calibration lines at 356 Hz and 3565 Hz andwas moved away from the Vela region by shifting of 5 Hertz the frequency of calibration lines The lower plot (blue curve) isthe average spectrum computed after the removal of the disturbance The large spectral disturbance appearing near the Crabreference frequency is due to a poorly understood coupling between a line at 60 Hz part of a 10-Hertz comb of lines of likelymagnetic origin and the fundamental pendulum frequency of the Virgo mirror system at 0594 Hz

9

$amp())+-amp0

12$amp(3-45

amp(67(898)

(amp-8

ltamp9(8amp$

-ampamp(-8

=00gt$8)-8

-ampamp(-8A((-800

1B-amp(Bamp(67(895

C(3-4-B

-7$(amp0

D-7)(08)$)(

Egtgt(amp$3$-amp

gtamp3((amp(03-8

95

(

F8)0gt$8)-8

-ampamp(-8

0gt$8)-8

-ampamp(-8

8

C(3-4-B

-7$(amp0

C(3-4-B

-7$(amp0

A((-800

1B-amp(Bamp(67(895

A((-800

1B-amp(Bamp(67(895

1gt47(nɺf

FIG 3 Scheme of the narrow-band analysis pipeline The starting point is constituted by detector calibrated data sampled at4096 Hz After barycentric and spin-down corrections the data are down-sampled at 1 Hz The number of different spin-downcorrections applied to the data nf and then the number of produced corrected time series is given by Eq (21) See text and[11] for more details

the gravitational redshift at the Earth geocenter due tothe solar system bodies and that amounts to about 2milliseconds at most Another effect is the Shapiro delay

∆S which takes into account the deflection of a signalpassing near a massive body and which can be shownto be negligible for CW searches unless the source lineof sight passes very near the Sunrsquos limb See eg [11]for explicit expressions of ∆E and ∆S Overall we canmake the full barycentric corrections by introducing thenew time variable

τ1 = t+∆R +∆E minus∆S (15)

This transformation corresponds to referring the datacollected at the detector site to the SSB which can beconsidered an inertial reference frame to a very good ap-proximation In practice for the Crab and Vela anal-yses described in this paper the Shapiro delay can beneglectedFor given values of the signal frequency f and fre-

quency derivatives f f we could take into account thespin-down in a similar way by re-scaling time accordingto

τ2 = t+f

2f(tminus t0)

2 +f

6f(tminus t0)

3 (16)

where t0 is the initial time of the data set and higherorder terms can be included if neededNote that re-scaling the time in this way to make the

signal monochromatic assumes that the intrinsic phaseevolution of the pulsarrsquos GW signal is described by a Tay-lor expansion over the entire observation period Elec-tromagnetic observations of pulsars show that young

rapidly spinning pulsars demonstrate deviations from aTaylor expansion when spinning down a phenomenonknown as timing noise Given our lack of knowledge ofthe exact mechanism that might cause the gravitationalwave frequency to deviate from (twice) the observed elec-tromagnetic frequency we cannot be sure if we shouldexpect the timing noise to be present in the signal weare searching for (see [17] for discussion) However astudy based on the monthly Crab ephemeris data [18]has shown that in the Crab one of the noisiest of pul-sars the effect of timing noise over the duration of theobservation period is negligible producing a mismatchbetween a Taylor expansion and a signal based on theactual lsquonoisyrsquo time series of less than 1 This is smallconfirming that timing noise is likely to have a negligibleeffect on our analysis

In practice the spin-down correction is applied afterbarycentric corrections then the time t that appears inEq (16) is in fact the re-scaled time τ1 of Eq (15) Inthis case the correction depends explicitly on the searchfrequency f The number of frequency bins which coverthe range ∆f = 2f0δ corresponding to Eq (1) is

nf =

[

∆f

δf

]

= [∆f middot T ] asymp 63times 105(

∆f

002 Hz

)(

T

1 yr

)

(17)with δf = 1

T being the frequency spacing with T thetotal observation time and where [ middot ] stands for thenearest integer Similarly we take the width of the firstorder spin-down range as ∆f = 2|f0|δ and ∆f = 2|f0|δfor the second order Let us consider for the momentonly the first order term For a fixed value of f we donot want to make an explicit spin-down correction for

10

each value of f So we fix the value of the frequency inthe denominator of Eq (16) for instance at the referencefrequency f0 and exploit the fact that the same correc-tion that holds for the pair (f0 f0) is also valid for all the

pairs (f f) such that

f0f0

=f

f(18)

In practice this means that at each frequency f we ex-

plore a range of first order spin-down values [fminus ∆f2 f+

∆f2 ] with

f = f0f

f0(19)

The number of bins in the spin-down term of the firstorder can be computed by considering the ldquonaturalrdquo stepδf = 1

2T 2 and is given by

n(1) equiv nf =[

2∆f middot T 2]

(20)

asymp 400

(

f010minus10 Hzs

)

(

δ

10minus3

)(

T

1 yr

)2

Following the same reasoning for the second order spin-down we arrive at the following expression for the numberof steps that should be taken into account

n(2) equiv nf =[

∆f middot T 3]

(21)

asymp 06

(

f010minus20 Hzs2

)

(

δ

10minus3

)(

T

1 yr

)3

For observation times of the order of a year reasonablevalues of second order spin-down and values of δ typicalof narrow-band searches nf lt 1 which means that we donot need to consider more values other than the ldquocentralrdquoone In Tab I the Crab and Vela EM-inferred positionaland rotational parameters are shown Also estimationsof the polarization parameters are given which are usedin the computation of upper limits see Sec V In Fig 4the portion of the f minus f plane actually covered in theCrab and Vela narrow-band search is shownOnce barycentric and spin-down corrections have been

done the data are down-sampled to a much lower ratewith respect to the original one 1 Hz in the present caseThis strongly reduces the amount of data to be handledin the next step of the analysis At this stage for a givensource we have nf corrected time series one for eachvalue of first order spin-down For the current analysis wehave nf = 33 for Crab and nf = 3 for Vela see Tab II)where also other relevant quantities are given In partic-ular the total number of points in the frequencyspin-down plane is about 528 times 106 for Crab analysis and48 times 105 for Vela analysis For each time series we ap-ply a final cleaning step by removing the largest outliersThese are identified by histogramming the logarithm of

the absolute value of the data amplitude and choosinga threshold approximately marking the beginning of thenon-Gaussian tail of data distribution In Fig 5 the his-togram of Crab and Vela data amplitude correspond-ing to the ldquocentralrdquo time series are shown For Crab athreshold of 15 times 10minus21 corresponding to -182 in thefigure x-axis has been used to remove outliers while forVela a value of 12 times 10minus20 corresponding to -092 hasbeen used Correspondingly the fraction of removed datais of about 18 for Vela and 19 for Crab By apply-ing the Kolmogorov-Smirnov test we have verified thatfor each pulsar the data distribution of the various timeseries are fully in agreement thus justifying the use of thesame threshold for all of themAt this point the detection statistic is computed for ev-

ery frequency and spin-down value in the explored rangeThe detection statistic we use is based on the so-called 5-vectors the same used for pulsar targeted searches [3 4]and is here briefly described Once barycentric and spin-down corrections have been applied a CW signal withfrequency f0 present in the data would be monochro-matic apart from an amplitude and phase sidereal mod-ulation due to the time-varying detector beam patternfunctions and given by Eq (2) with ω0(t) constant andequal to 2πf0 From Eqs (256) it follows that the signalis completely described by its Fourier components at the5 angular frequencies ω0 ω0plusmnΩoplus ω0plusmn2Ωoplus This set of5 complex numbers constitutes the signal 5-vector Givena generic time series g(t) the corresponding 5-vector isdefined as

G =

int

T

g(t)eminusı(ω0minuskΩoplus)tdt (22)

where k = [minus2minus1 2] and T is the observation timeLet us indicate with X the data 5-vector and withA

+ Atimes the signal plus and cross 5-vectors obtained byapplying the definition of Eq (22) to Eqs (56) Thesetwo last quantities depend only on known parameters andform the signal templates Once the 5-vectors of data andof signal templates have been computed the two complexnumbers

H+times =X middotA+times

|A+times|2(23)

are built see [3 12] for more details They correspond tocomputing matched filters between the data and the sig-nal templates and it can be shown assuming the noiseis Gaussian with mean value zero that they are esti-mators of the signal plus and cross complex amplitudesH0e

ıΦ0H+ H0eıΦ0Htimes These estimators are used to

build the detection statistic

S = |A+|4|H+|2 + |Atimes|4|Htimes|

2 (24)

The maximum of the detection statistic over thesearched parameter space Smax is determined Thisis the loudest candidate and is identified by a triple

11

22374 22376 22378 2238 22382 22384 22386 22388 2239 22392 22394minus3149

minus3148

minus3147

minus3146

minus3145

minus3144

minus3143x 10

minus11

Frequency [Hz]S

pinminus

dow

n [H

zs]

Covered parameter space for Vela pulsar

59434 59436 59438 5944 59442 59444 59446 59448 5945 59452 59454minus7429

minus7428

minus7427

minus7426

minus7425

minus7424

minus7423x 10

minus10

Frequency [Hz]

Spi

nminusdo

wn

[Hz

s]

Covered parameter space for Crab pulsar

FIG 4 The parallelograms delimit the portion of the f minus f plane covered in the narrow-band search for the Vela (upper plot)

and Crab (bottom plot) pulsars Only one value of f has been considered in the analysis The total number of points is givenin the last column of Tab II

minus3 minus25 minus2 minus15 minus1 minus05 0 05 1

102

103

104

105

log10(abs(real(data1020)))

minus3 minus28 minus26 minus24 minus22 minus2 minus18 minus16 minus14 minus12 minus110

1

102

103

104

105

log10(abs(real(data1020)))

FIG 5 Histogram of the logarithm of data amplitude for Vela (top plot) and the Crab (bottom plot) used to select the thresholdfor the removal of outliers For Crab the threshold has been put at -182 corresponding to an amplitude of sim 15 times 10minus21while for Vela a value of -092 corresponding to an amplitude of sim 12times 10minus20 has been chosen

12

TABLE I The Crab and Vela reference parameters α and δ are the equatorial coordinates f0 f0 f0 are the gravitational-wavefrequency parameters The reference epoch for the Crab position is MJD 54632 while for the rotational parameters is MJD55696 The reference epoch for Vela is MJD 53576 both for position and GW frequency parameters The Crab ephemeris hasbeen obtained from a fit of the Jodrell Bank monthly ephemeris [18] while for Vela they have been derived from Hartebeesthoekradio telescope observations [19] Also estimations of the polarization parameters and their associated uncertainty obtainedfrom the analysis of Chandra X-ray observations of the pulsar wind nebula torus [20 21] are given in the last two columnsThe analysis presented in [20 21] does not allow to determine the sense of the starrsquos spin so values of ι and ψ correspondingto ι rarr 180 minus ι ψ rarr ψ + 180 are also possible However the upper limits on strain amplitude reported in this paper are notsensitive to the sense of rotation

Source α [hhmmss] δ[deg] f0[Hz] f0[Hzs] f0[Hzs2] ι[deg] ψ[deg]

Crab 05 34 3197 220145 594448 minus74183 times 10minus10 26307 times 10minus20 62o2plusmn 1o9 35o2plusmn 1o5

Vela 08 35 2061 minus451764 223840 minus31460 times 10minus11 12848 times 10minus21 63o6plusmn 0o6 40o6plusmn 0o1

TABLE II Main quantities related to the parameter space of VSR4 narrow-band search ∆f is the frequency range δ is thewidth parameter defined in Eq (1) ∆f is the first order spin-down range nf is the number of frequency bins nf and nf arerespectively the number of bins for the first and second order spin-down ntot = nf middot nf is the total number of points in theparameter space

Source ∆f [Hz] δ ∆f [Hzs] nf nf nf ntot

Crab 002 168times 10minus4 249times 10minus13 16times 105 33 1 528times 106

Vela 002 447times 10minus4 281times 10minus14 16times 105 3 1 480times 105

(Smax fSmax fSmax

) This means we need to computentot = nf middot nf values of the detection statisticThe maximum value Smax is used to assess detection

significance by computing the p-value that is the prob-ability that a value of the detection statistic equal orlarger than Smax can be obtained in the absence ofany signal It implies the need to compute the noise-only distribution of the detection statistic This is amulti-dimensional probability distribution (with dimen-sion ntot) which would be difficult to compute and to han-dle In practice as discussed in [11] the p-value is com-puted considering the single-trial noise probability distri-bution that is the same that would be used for a targetedsearch and choosing a suitable threshold on it to dis-criminate between interesting (that is deserving a deeperstudy) and not interesting candidates As shown in [11]by setting an overall p-value p0 (over the full multi-dimensional distribution) the corresponding threshold on

the single-trial distribution is pthr = 1minus (1minus p0)1

ntot Inour case by setting eg p0 = 001 the resulting pthrwould be of the order of 10minus8 In principle we wouldlike to generate the noise-only probability distribution bycomputing the detection statistic at several ldquooff-sourcerdquofrequency bins that is frequencies near but outside theexplored range where we are assuming the signal couldbe This is what is typically done in targeted searches [4]In the present analysis however in order to appreciate p-values of the order of pthr we should consider of the orderof 108 off-source frequency bins This is computationallyimpractical For this reason we use the theoretical dis-tribution which assumes the noise is Gaussian given in[11]

f(S) =eminus S

σ2X

|Atimes|2 minus eminus S

σ2X

|A+|2

σ2X (|Atimes|2 minus |A+|2)

(25)

where σ2X = σ2 middot T with σ2 being the noise variance

If Smax is compatible with noise at a given confidencelevel eg 1 we compute an upper limit on signalstrength using the same method described in [4] other-wise in case of detection signal parameters are estimatedthrough suitable combinations of the real and imaginaryparts of H+ and Htimes as explained in [12]

V RESULTS

Analysis results are summarized in Tab III The databoth for the Vela and Crab searches are compatible withnoise In particular we find p-values equal to 033 and0013 for Vela and the Crab respectively larger thanp0 = 001 chosen to identify interesting candidates ForCrab however the obtained p-value is very near to thechosen threshold Altough a reasonable choice of p0 israther arbitrary (we could have for instance chosen 10minus3)we have decided to study in some detail the candidatersquosproperties In particular we have considered the distri-bution of the top ten candidates in the frequencyspin-down plane They appear to be randomly distributedwithout the clustering that we would expect in presenceof a signal To verify this hypothesis we have addeda simulated signal to the data with the same parame-ters as the Crab but with a slightly different frequencyand spin-down and with an amplitude h0 ≃ 41times 10minus25

such that the resulting loudest candidate has a value ofthe detection statistic approximately equal to the loud-est candidate of the actual analysis We have then runthe full analysis on this new data set and looked again atthe top ten candidate distribution in the frequencyspin-down plane In this case we indeed observe a clusteringwith 4 out of 10 candidates having a frequency within

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

6

different with respect to f0 = 2frot and can vary in theinterval

f(t) isin [f0(t)(1minus δ) f0(t)(1 + δ)] (1)

where δ is a small positive real number Following thediscussion in [5] if the star crust and core are linkedby some torque that tends to enforce corotation on atimescale τc then δ sim τcτsd where τsd sim f0f0 is thecharacteristic spin-down time A relation of the form ofEq (1) also holds in the case the gravitational radiationis produced by free precession of a nearly bi-axial star [6]in which case δ is of the order of (Izz minus Ixx)Ixx whereIxx and Izz are the star moments of inertia with respectto a principal axis on the equatorial plane and alignedwith the rotation axis respectively In general a value ofδ of the order of say 10minus4 corresponds to τc sim 10minus4τsdwhich depending on the specific targeted pulsar can beseveral months or years This would be comparable orlarger than the longest timescale observed in pulsar glitchrecovery where a recoupling between the two componentsmight occur In terms of free precession δ sim 10minus4 is onthe high end of the range of deformations that neutronstars could sustain [7ndash9]Narrow-band searches have not received much atten-

tion until now one notable exception being the Crabnarrow-band search over the first 9 months of LIGO S5data [5] based on the F -statistic [10] In previous work[11] an optimal method based on matched filtering in thespace of signal Fourier components has been proposedto carry out a search for CW over a small frequency andspin-down range around the values inferred from EM ob-servations In this paper we describe the application ofsuch a method to a narrow-band search of CW from Craband Vela pulsars in the data of Virgo VSR4 run As noevidence for a signal is found we place upper limits onsignal amplitudeThe outline of the paper is the following In Sec II the

expected gravitational wave signal from spinning neutronstars is introduced In Sec III the main characteristics ofVirgo VSR4 data are presented In Sec IV the analysismethod is described while in Sec V the analysis resultsare discussed Sec VI is dedicated to the validation testsof the analysis procedure Finally Sec VII contains theconclusions

II THE GRAVITATIONAL WAVE SIGNAL

A non-axisymmetric neutron star steadily spinningabout one of its principal axis emits a quadrupolargravitational-wave signal at twice the star rotational fre-quency frot which at the detector can be described [12]as

h(t) = H0(H+A+(t) +HtimesA

times(t))eı(ω0(t)t+Φ0) (2)

where taking the real part is understood The constantΦ0 is the initial signal phase The angular signal fre-

quency ω0(t) = 4πfrot(t) is a function of time Conse-quently the signal phase is not that of a simple monochro-matic signal and depends on both the intrinsic rotationalfrequency and frequency derivatives of the pulsar and onDoppler and propagation effects which include the Ein-stein delay and possibly the Shapiro delay These varia-tions are corrected in the time-domain in a way describedin Sec IV The two complex amplitudes H+ and Htimes aregiven respectively by

H+ =cos 2ψ minus ıη sin 2ψ

radic

1 + η2(3)

Htimes =sin 2ψ + ıη cos 2ψ

radic

1 + η2(4)

in which η is the ratio of the polarization ellipse semi-minor to semi-major axis and the polarization angle ψdefines the direction of the major axis with respect to thecelestial parallel of the source (measured counterclock-wise) The functions A+ and Atimes describe the detectorresponse as a function of time and are a linear combi-nation of terms depending on the Earth sidereal angularfrequency Ωoplus

A+ = a0 + a1c cosΩoplust+ a1s sinΩoplust+

a2c cos 2Ωoplust+ a2s sin 2Ωoplust (5)

Atimes = b1c cosΩoplust+ b1s sinΩoplust+

b2c cos 2Ωoplust+ b2s sin 2Ωoplust (6)

where the coefficients depend on the source position anddetector position and orientation on the Earth [12]As discussed eg in [3] the overall wave amplitude H0

and η are related to the ldquostandardrdquo signal amplitude

h0 =4π2G

c4Izzεf

20

d(7)

and to the angle ι between the star rotation axis and theline of sight to the source by

H0 = h0

radic

1 + 6 cos2 ι+ cos4 ι

4(8)

η = minus2 cos ι

1 + cos2 ι (9)

In Eq (7) G is the gravitational constant c is the lightvelocity Izz is the star moment of inertia with respectto the principal axis aligned with the rotation axis ε =IxxminusIyyIzz

is the equatorial ellipticity expressed in terms ofprincipal moments of inertia and d is the source distance

7

Equating the gravitational-wave luminosity to the ki-netic energy lost as the pulsar spins down gives us theso-called spin-down limit on gravitational-wave strain

hsd0 =

(

5

2

GIzz frotc3d2frot

)12

=

806times 10minus19 I1238

dkpc

(

|frot|

Hzs

)12(

frotHz

)minus12

(10)

where I38 is the starrsquos moment of inertia in units of1038 kgm2 frot is the time derivative of the rotationalfrequency and dkpc is the distance to the pulsar in kilo-parsecs The spin-down limit on the signal amplitudecorresponds to an upper limit on the starrsquos fiducial ellip-ticity

εsd = 0237

(

hsd010minus24

)

fminus2rot I

minus138 dkpc (11)

This quantity for a given neutron star equation of statecan be related to the physical ellipticity of the star sur-face [9] Setting a gravitational-wave upper limit belowthe spin-down limit is an important milestone in CWsearches as it allows us to constrain the fraction of spin-down energy due to the emission of gravitational-waves which gives insight into the spin-down energy budgetof the pulsar On the other hand the l = m = 2 massquadrupole moment Q22 is related to the gravitational-wave amplitude by

Q22 =

radic

15

8πIzzε = h0

(

c4d

16π2Gf2rot

)

radic

15

8π (12)

see eg [7] and is independent of any assumptions aboutthe starrsquos equation of state and its moment of inertia

III INSTRUMENTAL PERFORMANCE IN

VIRGO VSR4 RUN

Interferometric gravitational wave detectors LIGO[13] Virgo [14] GEO [15] have collected a large amountof data in recent years For the analysis described in thispaper we have used calibrated data sampled at 4096 Hzfrom Virgo VSR4 science run The run extended fromJune 3rd 2011 (1027 UTC) to September 5th 2011(1326 UTC) with a duty factor of about 81 corre-sponding to an effective duration of 76 days Calibra-tion uncertainties amounted to 75 in amplitude and(40+50fkHz) mrad in phase up to 500 Hz where fkHz isthe frequency in kilohertz The uncertainty on the ampli-tude will contribute to the uncertainty on the upper limiton signal amplitude together with that coming from thefinite size of the Monte Carlo simulation used to computeit see Sec V The calibration error on the phase can beshown to have a negligible impact on the analysis [3]

The low-frequency sensitivity of VSR4 was significantlybetter than that of previous Virgo runs especially due tothe use of monolithic mirror suspensions and basicallyin agreement with the planned sensitivity of the initialVirgo interferometer In Fig 1 a typical VSR4 strainsensitivity curve is shown In Fig 2 the average powerspectrum around Crab and Vela reference frequency isplotted Note the large sensitivity improvement aroundthe Vela frequency after removal of an instrumental dis-turbance about one month after the beginning of therun see the figure caption for more details

IV SEARCH DESCRIPTION

The analysis pipeline consists of several steps schemat-ically depicted in Fig 3 which we summarize here Moredetails are given in [11] The starting point is a collectionof windowed and interlaced (by half) ldquoshortrdquo (1024 sec-onds) FFTs (the Short FFT Database - SFDB) built fromcalibrated detector data [16] At this stage a first clean-ing procedure is applied to the data in order to removebig and short time duration disturbances that could re-duce the search sensitivity A small frequency band isthen extracted from the SFDB but large enough to con-tain the Doppler and spin-down variations of the signalpossibly emitted by the target pulsar In the analysesdescribed in this paper for instance it was of 015 HzAt this point we go back to the time domain (the sam-pling frequency is still the original one 4096 Hz) andmake barycentric and spin-down corrections Due to theDoppler effect the received frequency f(t) is related tothe emitted frequency f0(t) by the well-known relation(valid in the non-relativistic approximation)

f(t) =1

2π

dΦ(t)

dt= f0(t)

(

1 +~v middot n

c

)

(13)

where Φ(t) is the signal phase ~v = ~vrev+~vrot is the detec-tor velocity with respect to the Solar system barycenter(SSB) given by the sum of the Earthrsquos orbital velocity~vrev and rotational velocity ~vrot while n is the versoridentifying the source position In practice the Dopplereffect is efficiently corrected in the time domain by chang-ing the time stamp t of data samples according to

τ1 = t+~r(t) middot n

c= t+∆R (14)

where ~r is the detector position in the SSB The correc-tion term ∆R called Roemer delay amounts up to about500 seconds and corresponds to the time taken by a signaltraveling at the speed of light to cover the distance be-tween the detector and the SSB Unlike Eq (13) Eq (14)does not depend on the frequency which means that onesingle correction holds for every frequency In fact thereare other smaller relativistic effects that must be takeninto account when making barycentric corrections Oneis the Einstein delay ∆E due to the Earth motion and

8

102

103

10minus23

10minus22

10minus21

10minus20

10minus19

Frequency [Hz]

h n [Hzminus

12 ]

FIG 1 A typical sensitivity curve of Virgo VSR4 run expressed in terms of noise spectral density

032 034 036 038 04 042 04410

minus3

10minus2

Frequency minus220 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

036 038 04 042 044 046 048 05

10minus4

10minus3

Frequency minus 590 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

FIG 2 Average spectrum of VSR4 data around Vela (top plot) and Crab (bottom plot) reference frequency (identified by thevertical black lines) For Vela two spectra are in fact shown in the same plot The upper one (black curve) has been computedover the first 29 days of the run when the frequency region around Vela was affected by an instrumental disturbance Thiswas due to a non-linear coupling between the DARM control line at 379 Hz and calibration lines at 356 Hz and 3565 Hz andwas moved away from the Vela region by shifting of 5 Hertz the frequency of calibration lines The lower plot (blue curve) isthe average spectrum computed after the removal of the disturbance The large spectral disturbance appearing near the Crabreference frequency is due to a poorly understood coupling between a line at 60 Hz part of a 10-Hertz comb of lines of likelymagnetic origin and the fundamental pendulum frequency of the Virgo mirror system at 0594 Hz

9

$amp())+-amp0

12$amp(3-45

amp(67(898)

(amp-8

ltamp9(8amp$

-ampamp(-8

=00gt$8)-8

-ampamp(-8A((-800

1B-amp(Bamp(67(895

C(3-4-B

-7$(amp0

D-7)(08)$)(

Egtgt(amp$3$-amp

gtamp3((amp(03-8

95

(

F8)0gt$8)-8

-ampamp(-8

0gt$8)-8

-ampamp(-8

8

C(3-4-B

-7$(amp0

C(3-4-B

-7$(amp0

A((-800

1B-amp(Bamp(67(895

A((-800

1B-amp(Bamp(67(895

1gt47(nɺf

FIG 3 Scheme of the narrow-band analysis pipeline The starting point is constituted by detector calibrated data sampled at4096 Hz After barycentric and spin-down corrections the data are down-sampled at 1 Hz The number of different spin-downcorrections applied to the data nf and then the number of produced corrected time series is given by Eq (21) See text and[11] for more details

the gravitational redshift at the Earth geocenter due tothe solar system bodies and that amounts to about 2milliseconds at most Another effect is the Shapiro delay

∆S which takes into account the deflection of a signalpassing near a massive body and which can be shownto be negligible for CW searches unless the source lineof sight passes very near the Sunrsquos limb See eg [11]for explicit expressions of ∆E and ∆S Overall we canmake the full barycentric corrections by introducing thenew time variable

τ1 = t+∆R +∆E minus∆S (15)

This transformation corresponds to referring the datacollected at the detector site to the SSB which can beconsidered an inertial reference frame to a very good ap-proximation In practice for the Crab and Vela anal-yses described in this paper the Shapiro delay can beneglectedFor given values of the signal frequency f and fre-

quency derivatives f f we could take into account thespin-down in a similar way by re-scaling time accordingto

τ2 = t+f

2f(tminus t0)

2 +f

6f(tminus t0)

3 (16)

where t0 is the initial time of the data set and higherorder terms can be included if neededNote that re-scaling the time in this way to make the

signal monochromatic assumes that the intrinsic phaseevolution of the pulsarrsquos GW signal is described by a Tay-lor expansion over the entire observation period Elec-tromagnetic observations of pulsars show that young

rapidly spinning pulsars demonstrate deviations from aTaylor expansion when spinning down a phenomenonknown as timing noise Given our lack of knowledge ofthe exact mechanism that might cause the gravitationalwave frequency to deviate from (twice) the observed elec-tromagnetic frequency we cannot be sure if we shouldexpect the timing noise to be present in the signal weare searching for (see [17] for discussion) However astudy based on the monthly Crab ephemeris data [18]has shown that in the Crab one of the noisiest of pul-sars the effect of timing noise over the duration of theobservation period is negligible producing a mismatchbetween a Taylor expansion and a signal based on theactual lsquonoisyrsquo time series of less than 1 This is smallconfirming that timing noise is likely to have a negligibleeffect on our analysis

In practice the spin-down correction is applied afterbarycentric corrections then the time t that appears inEq (16) is in fact the re-scaled time τ1 of Eq (15) Inthis case the correction depends explicitly on the searchfrequency f The number of frequency bins which coverthe range ∆f = 2f0δ corresponding to Eq (1) is

nf =

[

∆f

δf

]

= [∆f middot T ] asymp 63times 105(

∆f

002 Hz

)(

T

1 yr

)

(17)with δf = 1

T being the frequency spacing with T thetotal observation time and where [ middot ] stands for thenearest integer Similarly we take the width of the firstorder spin-down range as ∆f = 2|f0|δ and ∆f = 2|f0|δfor the second order Let us consider for the momentonly the first order term For a fixed value of f we donot want to make an explicit spin-down correction for

10

each value of f So we fix the value of the frequency inthe denominator of Eq (16) for instance at the referencefrequency f0 and exploit the fact that the same correc-tion that holds for the pair (f0 f0) is also valid for all the

pairs (f f) such that

f0f0

=f

f(18)

In practice this means that at each frequency f we ex-

plore a range of first order spin-down values [fminus ∆f2 f+

∆f2 ] with

f = f0f

f0(19)

The number of bins in the spin-down term of the firstorder can be computed by considering the ldquonaturalrdquo stepδf = 1

2T 2 and is given by

n(1) equiv nf =[

2∆f middot T 2]

(20)

asymp 400

(

f010minus10 Hzs

)

(

δ

10minus3

)(

T

1 yr

)2

Following the same reasoning for the second order spin-down we arrive at the following expression for the numberof steps that should be taken into account

n(2) equiv nf =[

∆f middot T 3]

(21)

asymp 06

(

f010minus20 Hzs2

)

(

δ

10minus3

)(

T

1 yr

)3

For observation times of the order of a year reasonablevalues of second order spin-down and values of δ typicalof narrow-band searches nf lt 1 which means that we donot need to consider more values other than the ldquocentralrdquoone In Tab I the Crab and Vela EM-inferred positionaland rotational parameters are shown Also estimationsof the polarization parameters are given which are usedin the computation of upper limits see Sec V In Fig 4the portion of the f minus f plane actually covered in theCrab and Vela narrow-band search is shownOnce barycentric and spin-down corrections have been

done the data are down-sampled to a much lower ratewith respect to the original one 1 Hz in the present caseThis strongly reduces the amount of data to be handledin the next step of the analysis At this stage for a givensource we have nf corrected time series one for eachvalue of first order spin-down For the current analysis wehave nf = 33 for Crab and nf = 3 for Vela see Tab II)where also other relevant quantities are given In partic-ular the total number of points in the frequencyspin-down plane is about 528 times 106 for Crab analysis and48 times 105 for Vela analysis For each time series we ap-ply a final cleaning step by removing the largest outliersThese are identified by histogramming the logarithm of

the absolute value of the data amplitude and choosinga threshold approximately marking the beginning of thenon-Gaussian tail of data distribution In Fig 5 the his-togram of Crab and Vela data amplitude correspond-ing to the ldquocentralrdquo time series are shown For Crab athreshold of 15 times 10minus21 corresponding to -182 in thefigure x-axis has been used to remove outliers while forVela a value of 12 times 10minus20 corresponding to -092 hasbeen used Correspondingly the fraction of removed datais of about 18 for Vela and 19 for Crab By apply-ing the Kolmogorov-Smirnov test we have verified thatfor each pulsar the data distribution of the various timeseries are fully in agreement thus justifying the use of thesame threshold for all of themAt this point the detection statistic is computed for ev-

ery frequency and spin-down value in the explored rangeThe detection statistic we use is based on the so-called 5-vectors the same used for pulsar targeted searches [3 4]and is here briefly described Once barycentric and spin-down corrections have been applied a CW signal withfrequency f0 present in the data would be monochro-matic apart from an amplitude and phase sidereal mod-ulation due to the time-varying detector beam patternfunctions and given by Eq (2) with ω0(t) constant andequal to 2πf0 From Eqs (256) it follows that the signalis completely described by its Fourier components at the5 angular frequencies ω0 ω0plusmnΩoplus ω0plusmn2Ωoplus This set of5 complex numbers constitutes the signal 5-vector Givena generic time series g(t) the corresponding 5-vector isdefined as

G =

int

T

g(t)eminusı(ω0minuskΩoplus)tdt (22)

where k = [minus2minus1 2] and T is the observation timeLet us indicate with X the data 5-vector and withA

+ Atimes the signal plus and cross 5-vectors obtained byapplying the definition of Eq (22) to Eqs (56) Thesetwo last quantities depend only on known parameters andform the signal templates Once the 5-vectors of data andof signal templates have been computed the two complexnumbers

H+times =X middotA+times

|A+times|2(23)

are built see [3 12] for more details They correspond tocomputing matched filters between the data and the sig-nal templates and it can be shown assuming the noiseis Gaussian with mean value zero that they are esti-mators of the signal plus and cross complex amplitudesH0e

ıΦ0H+ H0eıΦ0Htimes These estimators are used to

build the detection statistic

S = |A+|4|H+|2 + |Atimes|4|Htimes|

2 (24)

The maximum of the detection statistic over thesearched parameter space Smax is determined Thisis the loudest candidate and is identified by a triple

11

22374 22376 22378 2238 22382 22384 22386 22388 2239 22392 22394minus3149

minus3148

minus3147

minus3146

minus3145

minus3144

minus3143x 10

minus11

Frequency [Hz]S

pinminus

dow

n [H

zs]

Covered parameter space for Vela pulsar

59434 59436 59438 5944 59442 59444 59446 59448 5945 59452 59454minus7429

minus7428

minus7427

minus7426

minus7425

minus7424

minus7423x 10

minus10

Frequency [Hz]

Spi

nminusdo

wn

[Hz

s]

Covered parameter space for Crab pulsar

FIG 4 The parallelograms delimit the portion of the f minus f plane covered in the narrow-band search for the Vela (upper plot)

and Crab (bottom plot) pulsars Only one value of f has been considered in the analysis The total number of points is givenin the last column of Tab II

minus3 minus25 minus2 minus15 minus1 minus05 0 05 1

102

103

104

105

log10(abs(real(data1020)))

minus3 minus28 minus26 minus24 minus22 minus2 minus18 minus16 minus14 minus12 minus110

1

102

103

104

105

log10(abs(real(data1020)))

FIG 5 Histogram of the logarithm of data amplitude for Vela (top plot) and the Crab (bottom plot) used to select the thresholdfor the removal of outliers For Crab the threshold has been put at -182 corresponding to an amplitude of sim 15 times 10minus21while for Vela a value of -092 corresponding to an amplitude of sim 12times 10minus20 has been chosen

12

TABLE I The Crab and Vela reference parameters α and δ are the equatorial coordinates f0 f0 f0 are the gravitational-wavefrequency parameters The reference epoch for the Crab position is MJD 54632 while for the rotational parameters is MJD55696 The reference epoch for Vela is MJD 53576 both for position and GW frequency parameters The Crab ephemeris hasbeen obtained from a fit of the Jodrell Bank monthly ephemeris [18] while for Vela they have been derived from Hartebeesthoekradio telescope observations [19] Also estimations of the polarization parameters and their associated uncertainty obtainedfrom the analysis of Chandra X-ray observations of the pulsar wind nebula torus [20 21] are given in the last two columnsThe analysis presented in [20 21] does not allow to determine the sense of the starrsquos spin so values of ι and ψ correspondingto ι rarr 180 minus ι ψ rarr ψ + 180 are also possible However the upper limits on strain amplitude reported in this paper are notsensitive to the sense of rotation

Source α [hhmmss] δ[deg] f0[Hz] f0[Hzs] f0[Hzs2] ι[deg] ψ[deg]

Crab 05 34 3197 220145 594448 minus74183 times 10minus10 26307 times 10minus20 62o2plusmn 1o9 35o2plusmn 1o5

Vela 08 35 2061 minus451764 223840 minus31460 times 10minus11 12848 times 10minus21 63o6plusmn 0o6 40o6plusmn 0o1

TABLE II Main quantities related to the parameter space of VSR4 narrow-band search ∆f is the frequency range δ is thewidth parameter defined in Eq (1) ∆f is the first order spin-down range nf is the number of frequency bins nf and nf arerespectively the number of bins for the first and second order spin-down ntot = nf middot nf is the total number of points in theparameter space

Source ∆f [Hz] δ ∆f [Hzs] nf nf nf ntot

Crab 002 168times 10minus4 249times 10minus13 16times 105 33 1 528times 106

Vela 002 447times 10minus4 281times 10minus14 16times 105 3 1 480times 105

(Smax fSmax fSmax

) This means we need to computentot = nf middot nf values of the detection statisticThe maximum value Smax is used to assess detection

significance by computing the p-value that is the prob-ability that a value of the detection statistic equal orlarger than Smax can be obtained in the absence ofany signal It implies the need to compute the noise-only distribution of the detection statistic This is amulti-dimensional probability distribution (with dimen-sion ntot) which would be difficult to compute and to han-dle In practice as discussed in [11] the p-value is com-puted considering the single-trial noise probability distri-bution that is the same that would be used for a targetedsearch and choosing a suitable threshold on it to dis-criminate between interesting (that is deserving a deeperstudy) and not interesting candidates As shown in [11]by setting an overall p-value p0 (over the full multi-dimensional distribution) the corresponding threshold on

the single-trial distribution is pthr = 1minus (1minus p0)1

ntot Inour case by setting eg p0 = 001 the resulting pthrwould be of the order of 10minus8 In principle we wouldlike to generate the noise-only probability distribution bycomputing the detection statistic at several ldquooff-sourcerdquofrequency bins that is frequencies near but outside theexplored range where we are assuming the signal couldbe This is what is typically done in targeted searches [4]In the present analysis however in order to appreciate p-values of the order of pthr we should consider of the orderof 108 off-source frequency bins This is computationallyimpractical For this reason we use the theoretical dis-tribution which assumes the noise is Gaussian given in[11]

f(S) =eminus S

σ2X

|Atimes|2 minus eminus S

σ2X

|A+|2

σ2X (|Atimes|2 minus |A+|2)

(25)

where σ2X = σ2 middot T with σ2 being the noise variance

If Smax is compatible with noise at a given confidencelevel eg 1 we compute an upper limit on signalstrength using the same method described in [4] other-wise in case of detection signal parameters are estimatedthrough suitable combinations of the real and imaginaryparts of H+ and Htimes as explained in [12]

V RESULTS

Analysis results are summarized in Tab III The databoth for the Vela and Crab searches are compatible withnoise In particular we find p-values equal to 033 and0013 for Vela and the Crab respectively larger thanp0 = 001 chosen to identify interesting candidates ForCrab however the obtained p-value is very near to thechosen threshold Altough a reasonable choice of p0 israther arbitrary (we could have for instance chosen 10minus3)we have decided to study in some detail the candidatersquosproperties In particular we have considered the distri-bution of the top ten candidates in the frequencyspin-down plane They appear to be randomly distributedwithout the clustering that we would expect in presenceof a signal To verify this hypothesis we have addeda simulated signal to the data with the same parame-ters as the Crab but with a slightly different frequencyand spin-down and with an amplitude h0 ≃ 41times 10minus25

such that the resulting loudest candidate has a value ofthe detection statistic approximately equal to the loud-est candidate of the actual analysis We have then runthe full analysis on this new data set and looked again atthe top ten candidate distribution in the frequencyspin-down plane In this case we indeed observe a clusteringwith 4 out of 10 candidates having a frequency within

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

7

Equating the gravitational-wave luminosity to the ki-netic energy lost as the pulsar spins down gives us theso-called spin-down limit on gravitational-wave strain

hsd0 =

(

5

2

GIzz frotc3d2frot

)12

=

806times 10minus19 I1238

dkpc

(

|frot|

Hzs

)12(

frotHz

)minus12

(10)

where I38 is the starrsquos moment of inertia in units of1038 kgm2 frot is the time derivative of the rotationalfrequency and dkpc is the distance to the pulsar in kilo-parsecs The spin-down limit on the signal amplitudecorresponds to an upper limit on the starrsquos fiducial ellip-ticity

εsd = 0237

(

hsd010minus24

)

fminus2rot I

minus138 dkpc (11)

This quantity for a given neutron star equation of statecan be related to the physical ellipticity of the star sur-face [9] Setting a gravitational-wave upper limit belowthe spin-down limit is an important milestone in CWsearches as it allows us to constrain the fraction of spin-down energy due to the emission of gravitational-waves which gives insight into the spin-down energy budgetof the pulsar On the other hand the l = m = 2 massquadrupole moment Q22 is related to the gravitational-wave amplitude by

Q22 =

radic

15

8πIzzε = h0

(

c4d

16π2Gf2rot

)

radic

15

8π (12)

see eg [7] and is independent of any assumptions aboutthe starrsquos equation of state and its moment of inertia

III INSTRUMENTAL PERFORMANCE IN

VIRGO VSR4 RUN

Interferometric gravitational wave detectors LIGO[13] Virgo [14] GEO [15] have collected a large amountof data in recent years For the analysis described in thispaper we have used calibrated data sampled at 4096 Hzfrom Virgo VSR4 science run The run extended fromJune 3rd 2011 (1027 UTC) to September 5th 2011(1326 UTC) with a duty factor of about 81 corre-sponding to an effective duration of 76 days Calibra-tion uncertainties amounted to 75 in amplitude and(40+50fkHz) mrad in phase up to 500 Hz where fkHz isthe frequency in kilohertz The uncertainty on the ampli-tude will contribute to the uncertainty on the upper limiton signal amplitude together with that coming from thefinite size of the Monte Carlo simulation used to computeit see Sec V The calibration error on the phase can beshown to have a negligible impact on the analysis [3]

The low-frequency sensitivity of VSR4 was significantlybetter than that of previous Virgo runs especially due tothe use of monolithic mirror suspensions and basicallyin agreement with the planned sensitivity of the initialVirgo interferometer In Fig 1 a typical VSR4 strainsensitivity curve is shown In Fig 2 the average powerspectrum around Crab and Vela reference frequency isplotted Note the large sensitivity improvement aroundthe Vela frequency after removal of an instrumental dis-turbance about one month after the beginning of therun see the figure caption for more details

IV SEARCH DESCRIPTION

The analysis pipeline consists of several steps schemat-ically depicted in Fig 3 which we summarize here Moredetails are given in [11] The starting point is a collectionof windowed and interlaced (by half) ldquoshortrdquo (1024 sec-onds) FFTs (the Short FFT Database - SFDB) built fromcalibrated detector data [16] At this stage a first clean-ing procedure is applied to the data in order to removebig and short time duration disturbances that could re-duce the search sensitivity A small frequency band isthen extracted from the SFDB but large enough to con-tain the Doppler and spin-down variations of the signalpossibly emitted by the target pulsar In the analysesdescribed in this paper for instance it was of 015 HzAt this point we go back to the time domain (the sam-pling frequency is still the original one 4096 Hz) andmake barycentric and spin-down corrections Due to theDoppler effect the received frequency f(t) is related tothe emitted frequency f0(t) by the well-known relation(valid in the non-relativistic approximation)

f(t) =1

2π

dΦ(t)

dt= f0(t)

(

1 +~v middot n

c

)

(13)

where Φ(t) is the signal phase ~v = ~vrev+~vrot is the detec-tor velocity with respect to the Solar system barycenter(SSB) given by the sum of the Earthrsquos orbital velocity~vrev and rotational velocity ~vrot while n is the versoridentifying the source position In practice the Dopplereffect is efficiently corrected in the time domain by chang-ing the time stamp t of data samples according to

τ1 = t+~r(t) middot n

c= t+∆R (14)

where ~r is the detector position in the SSB The correc-tion term ∆R called Roemer delay amounts up to about500 seconds and corresponds to the time taken by a signaltraveling at the speed of light to cover the distance be-tween the detector and the SSB Unlike Eq (13) Eq (14)does not depend on the frequency which means that onesingle correction holds for every frequency In fact thereare other smaller relativistic effects that must be takeninto account when making barycentric corrections Oneis the Einstein delay ∆E due to the Earth motion and

8

102

103

10minus23

10minus22

10minus21

10minus20

10minus19

Frequency [Hz]

h n [Hzminus

12 ]

FIG 1 A typical sensitivity curve of Virgo VSR4 run expressed in terms of noise spectral density

032 034 036 038 04 042 04410

minus3

10minus2

Frequency minus220 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

036 038 04 042 044 046 048 05

10minus4

10minus3

Frequency minus 590 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

FIG 2 Average spectrum of VSR4 data around Vela (top plot) and Crab (bottom plot) reference frequency (identified by thevertical black lines) For Vela two spectra are in fact shown in the same plot The upper one (black curve) has been computedover the first 29 days of the run when the frequency region around Vela was affected by an instrumental disturbance Thiswas due to a non-linear coupling between the DARM control line at 379 Hz and calibration lines at 356 Hz and 3565 Hz andwas moved away from the Vela region by shifting of 5 Hertz the frequency of calibration lines The lower plot (blue curve) isthe average spectrum computed after the removal of the disturbance The large spectral disturbance appearing near the Crabreference frequency is due to a poorly understood coupling between a line at 60 Hz part of a 10-Hertz comb of lines of likelymagnetic origin and the fundamental pendulum frequency of the Virgo mirror system at 0594 Hz

9

$amp())+-amp0

12$amp(3-45

amp(67(898)

(amp-8

ltamp9(8amp$

-ampamp(-8

=00gt$8)-8

-ampamp(-8A((-800

1B-amp(Bamp(67(895

C(3-4-B

-7$(amp0

D-7)(08)$)(

Egtgt(amp$3$-amp

gtamp3((amp(03-8

95

(

F8)0gt$8)-8

-ampamp(-8

0gt$8)-8

-ampamp(-8

8

C(3-4-B

-7$(amp0

C(3-4-B

-7$(amp0

A((-800

1B-amp(Bamp(67(895

A((-800

1B-amp(Bamp(67(895

1gt47(nɺf

FIG 3 Scheme of the narrow-band analysis pipeline The starting point is constituted by detector calibrated data sampled at4096 Hz After barycentric and spin-down corrections the data are down-sampled at 1 Hz The number of different spin-downcorrections applied to the data nf and then the number of produced corrected time series is given by Eq (21) See text and[11] for more details

the gravitational redshift at the Earth geocenter due tothe solar system bodies and that amounts to about 2milliseconds at most Another effect is the Shapiro delay

∆S which takes into account the deflection of a signalpassing near a massive body and which can be shownto be negligible for CW searches unless the source lineof sight passes very near the Sunrsquos limb See eg [11]for explicit expressions of ∆E and ∆S Overall we canmake the full barycentric corrections by introducing thenew time variable

τ1 = t+∆R +∆E minus∆S (15)

This transformation corresponds to referring the datacollected at the detector site to the SSB which can beconsidered an inertial reference frame to a very good ap-proximation In practice for the Crab and Vela anal-yses described in this paper the Shapiro delay can beneglectedFor given values of the signal frequency f and fre-

quency derivatives f f we could take into account thespin-down in a similar way by re-scaling time accordingto

τ2 = t+f

2f(tminus t0)

2 +f

6f(tminus t0)

3 (16)

where t0 is the initial time of the data set and higherorder terms can be included if neededNote that re-scaling the time in this way to make the

signal monochromatic assumes that the intrinsic phaseevolution of the pulsarrsquos GW signal is described by a Tay-lor expansion over the entire observation period Elec-tromagnetic observations of pulsars show that young

rapidly spinning pulsars demonstrate deviations from aTaylor expansion when spinning down a phenomenonknown as timing noise Given our lack of knowledge ofthe exact mechanism that might cause the gravitationalwave frequency to deviate from (twice) the observed elec-tromagnetic frequency we cannot be sure if we shouldexpect the timing noise to be present in the signal weare searching for (see [17] for discussion) However astudy based on the monthly Crab ephemeris data [18]has shown that in the Crab one of the noisiest of pul-sars the effect of timing noise over the duration of theobservation period is negligible producing a mismatchbetween a Taylor expansion and a signal based on theactual lsquonoisyrsquo time series of less than 1 This is smallconfirming that timing noise is likely to have a negligibleeffect on our analysis

In practice the spin-down correction is applied afterbarycentric corrections then the time t that appears inEq (16) is in fact the re-scaled time τ1 of Eq (15) Inthis case the correction depends explicitly on the searchfrequency f The number of frequency bins which coverthe range ∆f = 2f0δ corresponding to Eq (1) is

nf =

[

∆f

δf

]

= [∆f middot T ] asymp 63times 105(

∆f

002 Hz

)(

T

1 yr

)

(17)with δf = 1

T being the frequency spacing with T thetotal observation time and where [ middot ] stands for thenearest integer Similarly we take the width of the firstorder spin-down range as ∆f = 2|f0|δ and ∆f = 2|f0|δfor the second order Let us consider for the momentonly the first order term For a fixed value of f we donot want to make an explicit spin-down correction for

10

each value of f So we fix the value of the frequency inthe denominator of Eq (16) for instance at the referencefrequency f0 and exploit the fact that the same correc-tion that holds for the pair (f0 f0) is also valid for all the

pairs (f f) such that

f0f0

=f

f(18)

In practice this means that at each frequency f we ex-

plore a range of first order spin-down values [fminus ∆f2 f+

∆f2 ] with

f = f0f

f0(19)

The number of bins in the spin-down term of the firstorder can be computed by considering the ldquonaturalrdquo stepδf = 1

2T 2 and is given by

n(1) equiv nf =[

2∆f middot T 2]

(20)

asymp 400

(

f010minus10 Hzs

)

(

δ

10minus3

)(

T

1 yr

)2

Following the same reasoning for the second order spin-down we arrive at the following expression for the numberof steps that should be taken into account

n(2) equiv nf =[

∆f middot T 3]

(21)

asymp 06

(

f010minus20 Hzs2

)

(

δ

10minus3

)(

T

1 yr

)3

For observation times of the order of a year reasonablevalues of second order spin-down and values of δ typicalof narrow-band searches nf lt 1 which means that we donot need to consider more values other than the ldquocentralrdquoone In Tab I the Crab and Vela EM-inferred positionaland rotational parameters are shown Also estimationsof the polarization parameters are given which are usedin the computation of upper limits see Sec V In Fig 4the portion of the f minus f plane actually covered in theCrab and Vela narrow-band search is shownOnce barycentric and spin-down corrections have been

done the data are down-sampled to a much lower ratewith respect to the original one 1 Hz in the present caseThis strongly reduces the amount of data to be handledin the next step of the analysis At this stage for a givensource we have nf corrected time series one for eachvalue of first order spin-down For the current analysis wehave nf = 33 for Crab and nf = 3 for Vela see Tab II)where also other relevant quantities are given In partic-ular the total number of points in the frequencyspin-down plane is about 528 times 106 for Crab analysis and48 times 105 for Vela analysis For each time series we ap-ply a final cleaning step by removing the largest outliersThese are identified by histogramming the logarithm of

the absolute value of the data amplitude and choosinga threshold approximately marking the beginning of thenon-Gaussian tail of data distribution In Fig 5 the his-togram of Crab and Vela data amplitude correspond-ing to the ldquocentralrdquo time series are shown For Crab athreshold of 15 times 10minus21 corresponding to -182 in thefigure x-axis has been used to remove outliers while forVela a value of 12 times 10minus20 corresponding to -092 hasbeen used Correspondingly the fraction of removed datais of about 18 for Vela and 19 for Crab By apply-ing the Kolmogorov-Smirnov test we have verified thatfor each pulsar the data distribution of the various timeseries are fully in agreement thus justifying the use of thesame threshold for all of themAt this point the detection statistic is computed for ev-

ery frequency and spin-down value in the explored rangeThe detection statistic we use is based on the so-called 5-vectors the same used for pulsar targeted searches [3 4]and is here briefly described Once barycentric and spin-down corrections have been applied a CW signal withfrequency f0 present in the data would be monochro-matic apart from an amplitude and phase sidereal mod-ulation due to the time-varying detector beam patternfunctions and given by Eq (2) with ω0(t) constant andequal to 2πf0 From Eqs (256) it follows that the signalis completely described by its Fourier components at the5 angular frequencies ω0 ω0plusmnΩoplus ω0plusmn2Ωoplus This set of5 complex numbers constitutes the signal 5-vector Givena generic time series g(t) the corresponding 5-vector isdefined as

G =

int

T

g(t)eminusı(ω0minuskΩoplus)tdt (22)

where k = [minus2minus1 2] and T is the observation timeLet us indicate with X the data 5-vector and withA

+ Atimes the signal plus and cross 5-vectors obtained byapplying the definition of Eq (22) to Eqs (56) Thesetwo last quantities depend only on known parameters andform the signal templates Once the 5-vectors of data andof signal templates have been computed the two complexnumbers

H+times =X middotA+times

|A+times|2(23)

are built see [3 12] for more details They correspond tocomputing matched filters between the data and the sig-nal templates and it can be shown assuming the noiseis Gaussian with mean value zero that they are esti-mators of the signal plus and cross complex amplitudesH0e

ıΦ0H+ H0eıΦ0Htimes These estimators are used to

build the detection statistic

S = |A+|4|H+|2 + |Atimes|4|Htimes|

2 (24)

The maximum of the detection statistic over thesearched parameter space Smax is determined Thisis the loudest candidate and is identified by a triple

11

22374 22376 22378 2238 22382 22384 22386 22388 2239 22392 22394minus3149

minus3148

minus3147

minus3146

minus3145

minus3144

minus3143x 10

minus11

Frequency [Hz]S

pinminus

dow

n [H

zs]

Covered parameter space for Vela pulsar

59434 59436 59438 5944 59442 59444 59446 59448 5945 59452 59454minus7429

minus7428

minus7427

minus7426

minus7425

minus7424

minus7423x 10

minus10

Frequency [Hz]

Spi

nminusdo

wn

[Hz

s]

Covered parameter space for Crab pulsar

FIG 4 The parallelograms delimit the portion of the f minus f plane covered in the narrow-band search for the Vela (upper plot)

and Crab (bottom plot) pulsars Only one value of f has been considered in the analysis The total number of points is givenin the last column of Tab II

minus3 minus25 minus2 minus15 minus1 minus05 0 05 1

102

103

104

105

log10(abs(real(data1020)))

minus3 minus28 minus26 minus24 minus22 minus2 minus18 minus16 minus14 minus12 minus110

1

102

103

104

105

log10(abs(real(data1020)))

FIG 5 Histogram of the logarithm of data amplitude for Vela (top plot) and the Crab (bottom plot) used to select the thresholdfor the removal of outliers For Crab the threshold has been put at -182 corresponding to an amplitude of sim 15 times 10minus21while for Vela a value of -092 corresponding to an amplitude of sim 12times 10minus20 has been chosen

12

TABLE I The Crab and Vela reference parameters α and δ are the equatorial coordinates f0 f0 f0 are the gravitational-wavefrequency parameters The reference epoch for the Crab position is MJD 54632 while for the rotational parameters is MJD55696 The reference epoch for Vela is MJD 53576 both for position and GW frequency parameters The Crab ephemeris hasbeen obtained from a fit of the Jodrell Bank monthly ephemeris [18] while for Vela they have been derived from Hartebeesthoekradio telescope observations [19] Also estimations of the polarization parameters and their associated uncertainty obtainedfrom the analysis of Chandra X-ray observations of the pulsar wind nebula torus [20 21] are given in the last two columnsThe analysis presented in [20 21] does not allow to determine the sense of the starrsquos spin so values of ι and ψ correspondingto ι rarr 180 minus ι ψ rarr ψ + 180 are also possible However the upper limits on strain amplitude reported in this paper are notsensitive to the sense of rotation

Source α [hhmmss] δ[deg] f0[Hz] f0[Hzs] f0[Hzs2] ι[deg] ψ[deg]

Crab 05 34 3197 220145 594448 minus74183 times 10minus10 26307 times 10minus20 62o2plusmn 1o9 35o2plusmn 1o5

Vela 08 35 2061 minus451764 223840 minus31460 times 10minus11 12848 times 10minus21 63o6plusmn 0o6 40o6plusmn 0o1

TABLE II Main quantities related to the parameter space of VSR4 narrow-band search ∆f is the frequency range δ is thewidth parameter defined in Eq (1) ∆f is the first order spin-down range nf is the number of frequency bins nf and nf arerespectively the number of bins for the first and second order spin-down ntot = nf middot nf is the total number of points in theparameter space

Source ∆f [Hz] δ ∆f [Hzs] nf nf nf ntot

Crab 002 168times 10minus4 249times 10minus13 16times 105 33 1 528times 106

Vela 002 447times 10minus4 281times 10minus14 16times 105 3 1 480times 105

(Smax fSmax fSmax

) This means we need to computentot = nf middot nf values of the detection statisticThe maximum value Smax is used to assess detection

significance by computing the p-value that is the prob-ability that a value of the detection statistic equal orlarger than Smax can be obtained in the absence ofany signal It implies the need to compute the noise-only distribution of the detection statistic This is amulti-dimensional probability distribution (with dimen-sion ntot) which would be difficult to compute and to han-dle In practice as discussed in [11] the p-value is com-puted considering the single-trial noise probability distri-bution that is the same that would be used for a targetedsearch and choosing a suitable threshold on it to dis-criminate between interesting (that is deserving a deeperstudy) and not interesting candidates As shown in [11]by setting an overall p-value p0 (over the full multi-dimensional distribution) the corresponding threshold on

the single-trial distribution is pthr = 1minus (1minus p0)1

ntot Inour case by setting eg p0 = 001 the resulting pthrwould be of the order of 10minus8 In principle we wouldlike to generate the noise-only probability distribution bycomputing the detection statistic at several ldquooff-sourcerdquofrequency bins that is frequencies near but outside theexplored range where we are assuming the signal couldbe This is what is typically done in targeted searches [4]In the present analysis however in order to appreciate p-values of the order of pthr we should consider of the orderof 108 off-source frequency bins This is computationallyimpractical For this reason we use the theoretical dis-tribution which assumes the noise is Gaussian given in[11]

f(S) =eminus S

σ2X

|Atimes|2 minus eminus S

σ2X

|A+|2

σ2X (|Atimes|2 minus |A+|2)

(25)

where σ2X = σ2 middot T with σ2 being the noise variance

If Smax is compatible with noise at a given confidencelevel eg 1 we compute an upper limit on signalstrength using the same method described in [4] other-wise in case of detection signal parameters are estimatedthrough suitable combinations of the real and imaginaryparts of H+ and Htimes as explained in [12]

V RESULTS

Analysis results are summarized in Tab III The databoth for the Vela and Crab searches are compatible withnoise In particular we find p-values equal to 033 and0013 for Vela and the Crab respectively larger thanp0 = 001 chosen to identify interesting candidates ForCrab however the obtained p-value is very near to thechosen threshold Altough a reasonable choice of p0 israther arbitrary (we could have for instance chosen 10minus3)we have decided to study in some detail the candidatersquosproperties In particular we have considered the distri-bution of the top ten candidates in the frequencyspin-down plane They appear to be randomly distributedwithout the clustering that we would expect in presenceof a signal To verify this hypothesis we have addeda simulated signal to the data with the same parame-ters as the Crab but with a slightly different frequencyand spin-down and with an amplitude h0 ≃ 41times 10minus25

such that the resulting loudest candidate has a value ofthe detection statistic approximately equal to the loud-est candidate of the actual analysis We have then runthe full analysis on this new data set and looked again atthe top ten candidate distribution in the frequencyspin-down plane In this case we indeed observe a clusteringwith 4 out of 10 candidates having a frequency within

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

8

102

103

10minus23

10minus22

10minus21

10minus20

10minus19

Frequency [Hz]

h n [Hzminus

12 ]

FIG 1 A typical sensitivity curve of Virgo VSR4 run expressed in terms of noise spectral density

032 034 036 038 04 042 04410

minus3

10minus2

Frequency minus220 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

036 038 04 042 044 046 048 05

10minus4

10minus3

Frequency minus 590 [Hz]

Pow

er s

pect

rum

1040

[1H

z]

FIG 2 Average spectrum of VSR4 data around Vela (top plot) and Crab (bottom plot) reference frequency (identified by thevertical black lines) For Vela two spectra are in fact shown in the same plot The upper one (black curve) has been computedover the first 29 days of the run when the frequency region around Vela was affected by an instrumental disturbance Thiswas due to a non-linear coupling between the DARM control line at 379 Hz and calibration lines at 356 Hz and 3565 Hz andwas moved away from the Vela region by shifting of 5 Hertz the frequency of calibration lines The lower plot (blue curve) isthe average spectrum computed after the removal of the disturbance The large spectral disturbance appearing near the Crabreference frequency is due to a poorly understood coupling between a line at 60 Hz part of a 10-Hertz comb of lines of likelymagnetic origin and the fundamental pendulum frequency of the Virgo mirror system at 0594 Hz

9

$amp())+-amp0

12$amp(3-45

amp(67(898)

(amp-8

ltamp9(8amp$

-ampamp(-8

=00gt$8)-8

-ampamp(-8A((-800

1B-amp(Bamp(67(895

C(3-4-B

-7$(amp0

D-7)(08)$)(

Egtgt(amp$3$-amp

gtamp3((amp(03-8

95

(

F8)0gt$8)-8

-ampamp(-8

0gt$8)-8

-ampamp(-8

8

C(3-4-B

-7$(amp0

C(3-4-B

-7$(amp0

A((-800

1B-amp(Bamp(67(895

A((-800

1B-amp(Bamp(67(895

1gt47(nɺf

FIG 3 Scheme of the narrow-band analysis pipeline The starting point is constituted by detector calibrated data sampled at4096 Hz After barycentric and spin-down corrections the data are down-sampled at 1 Hz The number of different spin-downcorrections applied to the data nf and then the number of produced corrected time series is given by Eq (21) See text and[11] for more details

the gravitational redshift at the Earth geocenter due tothe solar system bodies and that amounts to about 2milliseconds at most Another effect is the Shapiro delay

∆S which takes into account the deflection of a signalpassing near a massive body and which can be shownto be negligible for CW searches unless the source lineof sight passes very near the Sunrsquos limb See eg [11]for explicit expressions of ∆E and ∆S Overall we canmake the full barycentric corrections by introducing thenew time variable

τ1 = t+∆R +∆E minus∆S (15)

This transformation corresponds to referring the datacollected at the detector site to the SSB which can beconsidered an inertial reference frame to a very good ap-proximation In practice for the Crab and Vela anal-yses described in this paper the Shapiro delay can beneglectedFor given values of the signal frequency f and fre-

quency derivatives f f we could take into account thespin-down in a similar way by re-scaling time accordingto

τ2 = t+f

2f(tminus t0)

2 +f

6f(tminus t0)

3 (16)

where t0 is the initial time of the data set and higherorder terms can be included if neededNote that re-scaling the time in this way to make the

signal monochromatic assumes that the intrinsic phaseevolution of the pulsarrsquos GW signal is described by a Tay-lor expansion over the entire observation period Elec-tromagnetic observations of pulsars show that young

rapidly spinning pulsars demonstrate deviations from aTaylor expansion when spinning down a phenomenonknown as timing noise Given our lack of knowledge ofthe exact mechanism that might cause the gravitationalwave frequency to deviate from (twice) the observed elec-tromagnetic frequency we cannot be sure if we shouldexpect the timing noise to be present in the signal weare searching for (see [17] for discussion) However astudy based on the monthly Crab ephemeris data [18]has shown that in the Crab one of the noisiest of pul-sars the effect of timing noise over the duration of theobservation period is negligible producing a mismatchbetween a Taylor expansion and a signal based on theactual lsquonoisyrsquo time series of less than 1 This is smallconfirming that timing noise is likely to have a negligibleeffect on our analysis

In practice the spin-down correction is applied afterbarycentric corrections then the time t that appears inEq (16) is in fact the re-scaled time τ1 of Eq (15) Inthis case the correction depends explicitly on the searchfrequency f The number of frequency bins which coverthe range ∆f = 2f0δ corresponding to Eq (1) is

nf =

[

∆f

δf

]

= [∆f middot T ] asymp 63times 105(

∆f

002 Hz

)(

T

1 yr

)

(17)with δf = 1

T being the frequency spacing with T thetotal observation time and where [ middot ] stands for thenearest integer Similarly we take the width of the firstorder spin-down range as ∆f = 2|f0|δ and ∆f = 2|f0|δfor the second order Let us consider for the momentonly the first order term For a fixed value of f we donot want to make an explicit spin-down correction for

10

each value of f So we fix the value of the frequency inthe denominator of Eq (16) for instance at the referencefrequency f0 and exploit the fact that the same correc-tion that holds for the pair (f0 f0) is also valid for all the

pairs (f f) such that

f0f0

=f

f(18)

In practice this means that at each frequency f we ex-

plore a range of first order spin-down values [fminus ∆f2 f+

∆f2 ] with

f = f0f

f0(19)

The number of bins in the spin-down term of the firstorder can be computed by considering the ldquonaturalrdquo stepδf = 1

2T 2 and is given by

n(1) equiv nf =[

2∆f middot T 2]

(20)

asymp 400

(

f010minus10 Hzs

)

(

δ

10minus3

)(

T

1 yr

)2

Following the same reasoning for the second order spin-down we arrive at the following expression for the numberof steps that should be taken into account

n(2) equiv nf =[

∆f middot T 3]

(21)

asymp 06

(

f010minus20 Hzs2

)

(

δ

10minus3

)(

T

1 yr

)3

For observation times of the order of a year reasonablevalues of second order spin-down and values of δ typicalof narrow-band searches nf lt 1 which means that we donot need to consider more values other than the ldquocentralrdquoone In Tab I the Crab and Vela EM-inferred positionaland rotational parameters are shown Also estimationsof the polarization parameters are given which are usedin the computation of upper limits see Sec V In Fig 4the portion of the f minus f plane actually covered in theCrab and Vela narrow-band search is shownOnce barycentric and spin-down corrections have been

done the data are down-sampled to a much lower ratewith respect to the original one 1 Hz in the present caseThis strongly reduces the amount of data to be handledin the next step of the analysis At this stage for a givensource we have nf corrected time series one for eachvalue of first order spin-down For the current analysis wehave nf = 33 for Crab and nf = 3 for Vela see Tab II)where also other relevant quantities are given In partic-ular the total number of points in the frequencyspin-down plane is about 528 times 106 for Crab analysis and48 times 105 for Vela analysis For each time series we ap-ply a final cleaning step by removing the largest outliersThese are identified by histogramming the logarithm of

the absolute value of the data amplitude and choosinga threshold approximately marking the beginning of thenon-Gaussian tail of data distribution In Fig 5 the his-togram of Crab and Vela data amplitude correspond-ing to the ldquocentralrdquo time series are shown For Crab athreshold of 15 times 10minus21 corresponding to -182 in thefigure x-axis has been used to remove outliers while forVela a value of 12 times 10minus20 corresponding to -092 hasbeen used Correspondingly the fraction of removed datais of about 18 for Vela and 19 for Crab By apply-ing the Kolmogorov-Smirnov test we have verified thatfor each pulsar the data distribution of the various timeseries are fully in agreement thus justifying the use of thesame threshold for all of themAt this point the detection statistic is computed for ev-

ery frequency and spin-down value in the explored rangeThe detection statistic we use is based on the so-called 5-vectors the same used for pulsar targeted searches [3 4]and is here briefly described Once barycentric and spin-down corrections have been applied a CW signal withfrequency f0 present in the data would be monochro-matic apart from an amplitude and phase sidereal mod-ulation due to the time-varying detector beam patternfunctions and given by Eq (2) with ω0(t) constant andequal to 2πf0 From Eqs (256) it follows that the signalis completely described by its Fourier components at the5 angular frequencies ω0 ω0plusmnΩoplus ω0plusmn2Ωoplus This set of5 complex numbers constitutes the signal 5-vector Givena generic time series g(t) the corresponding 5-vector isdefined as

G =

int

T

g(t)eminusı(ω0minuskΩoplus)tdt (22)

where k = [minus2minus1 2] and T is the observation timeLet us indicate with X the data 5-vector and withA

+ Atimes the signal plus and cross 5-vectors obtained byapplying the definition of Eq (22) to Eqs (56) Thesetwo last quantities depend only on known parameters andform the signal templates Once the 5-vectors of data andof signal templates have been computed the two complexnumbers

H+times =X middotA+times

|A+times|2(23)

are built see [3 12] for more details They correspond tocomputing matched filters between the data and the sig-nal templates and it can be shown assuming the noiseis Gaussian with mean value zero that they are esti-mators of the signal plus and cross complex amplitudesH0e

ıΦ0H+ H0eıΦ0Htimes These estimators are used to

build the detection statistic

S = |A+|4|H+|2 + |Atimes|4|Htimes|

2 (24)

The maximum of the detection statistic over thesearched parameter space Smax is determined Thisis the loudest candidate and is identified by a triple

11

22374 22376 22378 2238 22382 22384 22386 22388 2239 22392 22394minus3149

minus3148

minus3147

minus3146

minus3145

minus3144

minus3143x 10

minus11

Frequency [Hz]S

pinminus

dow

n [H

zs]

Covered parameter space for Vela pulsar

59434 59436 59438 5944 59442 59444 59446 59448 5945 59452 59454minus7429

minus7428

minus7427

minus7426

minus7425

minus7424

minus7423x 10

minus10

Frequency [Hz]

Spi

nminusdo

wn

[Hz

s]

Covered parameter space for Crab pulsar

FIG 4 The parallelograms delimit the portion of the f minus f plane covered in the narrow-band search for the Vela (upper plot)

and Crab (bottom plot) pulsars Only one value of f has been considered in the analysis The total number of points is givenin the last column of Tab II

minus3 minus25 minus2 minus15 minus1 minus05 0 05 1

102

103

104

105

log10(abs(real(data1020)))

minus3 minus28 minus26 minus24 minus22 minus2 minus18 minus16 minus14 minus12 minus110

1

102

103

104

105

log10(abs(real(data1020)))

FIG 5 Histogram of the logarithm of data amplitude for Vela (top plot) and the Crab (bottom plot) used to select the thresholdfor the removal of outliers For Crab the threshold has been put at -182 corresponding to an amplitude of sim 15 times 10minus21while for Vela a value of -092 corresponding to an amplitude of sim 12times 10minus20 has been chosen

12

TABLE I The Crab and Vela reference parameters α and δ are the equatorial coordinates f0 f0 f0 are the gravitational-wavefrequency parameters The reference epoch for the Crab position is MJD 54632 while for the rotational parameters is MJD55696 The reference epoch for Vela is MJD 53576 both for position and GW frequency parameters The Crab ephemeris hasbeen obtained from a fit of the Jodrell Bank monthly ephemeris [18] while for Vela they have been derived from Hartebeesthoekradio telescope observations [19] Also estimations of the polarization parameters and their associated uncertainty obtainedfrom the analysis of Chandra X-ray observations of the pulsar wind nebula torus [20 21] are given in the last two columnsThe analysis presented in [20 21] does not allow to determine the sense of the starrsquos spin so values of ι and ψ correspondingto ι rarr 180 minus ι ψ rarr ψ + 180 are also possible However the upper limits on strain amplitude reported in this paper are notsensitive to the sense of rotation

Source α [hhmmss] δ[deg] f0[Hz] f0[Hzs] f0[Hzs2] ι[deg] ψ[deg]

Crab 05 34 3197 220145 594448 minus74183 times 10minus10 26307 times 10minus20 62o2plusmn 1o9 35o2plusmn 1o5

Vela 08 35 2061 minus451764 223840 minus31460 times 10minus11 12848 times 10minus21 63o6plusmn 0o6 40o6plusmn 0o1

TABLE II Main quantities related to the parameter space of VSR4 narrow-band search ∆f is the frequency range δ is thewidth parameter defined in Eq (1) ∆f is the first order spin-down range nf is the number of frequency bins nf and nf arerespectively the number of bins for the first and second order spin-down ntot = nf middot nf is the total number of points in theparameter space

Source ∆f [Hz] δ ∆f [Hzs] nf nf nf ntot

Crab 002 168times 10minus4 249times 10minus13 16times 105 33 1 528times 106

Vela 002 447times 10minus4 281times 10minus14 16times 105 3 1 480times 105

(Smax fSmax fSmax

) This means we need to computentot = nf middot nf values of the detection statisticThe maximum value Smax is used to assess detection

significance by computing the p-value that is the prob-ability that a value of the detection statistic equal orlarger than Smax can be obtained in the absence ofany signal It implies the need to compute the noise-only distribution of the detection statistic This is amulti-dimensional probability distribution (with dimen-sion ntot) which would be difficult to compute and to han-dle In practice as discussed in [11] the p-value is com-puted considering the single-trial noise probability distri-bution that is the same that would be used for a targetedsearch and choosing a suitable threshold on it to dis-criminate between interesting (that is deserving a deeperstudy) and not interesting candidates As shown in [11]by setting an overall p-value p0 (over the full multi-dimensional distribution) the corresponding threshold on

the single-trial distribution is pthr = 1minus (1minus p0)1

ntot Inour case by setting eg p0 = 001 the resulting pthrwould be of the order of 10minus8 In principle we wouldlike to generate the noise-only probability distribution bycomputing the detection statistic at several ldquooff-sourcerdquofrequency bins that is frequencies near but outside theexplored range where we are assuming the signal couldbe This is what is typically done in targeted searches [4]In the present analysis however in order to appreciate p-values of the order of pthr we should consider of the orderof 108 off-source frequency bins This is computationallyimpractical For this reason we use the theoretical dis-tribution which assumes the noise is Gaussian given in[11]

f(S) =eminus S

σ2X

|Atimes|2 minus eminus S

σ2X

|A+|2

σ2X (|Atimes|2 minus |A+|2)

(25)

where σ2X = σ2 middot T with σ2 being the noise variance

If Smax is compatible with noise at a given confidencelevel eg 1 we compute an upper limit on signalstrength using the same method described in [4] other-wise in case of detection signal parameters are estimatedthrough suitable combinations of the real and imaginaryparts of H+ and Htimes as explained in [12]

V RESULTS

Analysis results are summarized in Tab III The databoth for the Vela and Crab searches are compatible withnoise In particular we find p-values equal to 033 and0013 for Vela and the Crab respectively larger thanp0 = 001 chosen to identify interesting candidates ForCrab however the obtained p-value is very near to thechosen threshold Altough a reasonable choice of p0 israther arbitrary (we could have for instance chosen 10minus3)we have decided to study in some detail the candidatersquosproperties In particular we have considered the distri-bution of the top ten candidates in the frequencyspin-down plane They appear to be randomly distributedwithout the clustering that we would expect in presenceof a signal To verify this hypothesis we have addeda simulated signal to the data with the same parame-ters as the Crab but with a slightly different frequencyand spin-down and with an amplitude h0 ≃ 41times 10minus25

such that the resulting loudest candidate has a value ofthe detection statistic approximately equal to the loud-est candidate of the actual analysis We have then runthe full analysis on this new data set and looked again atthe top ten candidate distribution in the frequencyspin-down plane In this case we indeed observe a clusteringwith 4 out of 10 candidates having a frequency within

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

9

$amp())+-amp0

12$amp(3-45

amp(67(898)

(amp-8

ltamp9(8amp$

-ampamp(-8

=00gt$8)-8

-ampamp(-8A((-800

1B-amp(Bamp(67(895

C(3-4-B

-7$(amp0

D-7)(08)$)(

Egtgt(amp$3$-amp

gtamp3((amp(03-8

95

(

F8)0gt$8)-8

-ampamp(-8

0gt$8)-8

-ampamp(-8

8

C(3-4-B

-7$(amp0

C(3-4-B

-7$(amp0

A((-800

1B-amp(Bamp(67(895

A((-800

1B-amp(Bamp(67(895

1gt47(nɺf

FIG 3 Scheme of the narrow-band analysis pipeline The starting point is constituted by detector calibrated data sampled at4096 Hz After barycentric and spin-down corrections the data are down-sampled at 1 Hz The number of different spin-downcorrections applied to the data nf and then the number of produced corrected time series is given by Eq (21) See text and[11] for more details

the gravitational redshift at the Earth geocenter due tothe solar system bodies and that amounts to about 2milliseconds at most Another effect is the Shapiro delay

∆S which takes into account the deflection of a signalpassing near a massive body and which can be shownto be negligible for CW searches unless the source lineof sight passes very near the Sunrsquos limb See eg [11]for explicit expressions of ∆E and ∆S Overall we canmake the full barycentric corrections by introducing thenew time variable

τ1 = t+∆R +∆E minus∆S (15)

This transformation corresponds to referring the datacollected at the detector site to the SSB which can beconsidered an inertial reference frame to a very good ap-proximation In practice for the Crab and Vela anal-yses described in this paper the Shapiro delay can beneglectedFor given values of the signal frequency f and fre-

quency derivatives f f we could take into account thespin-down in a similar way by re-scaling time accordingto

τ2 = t+f

2f(tminus t0)

2 +f

6f(tminus t0)

3 (16)

where t0 is the initial time of the data set and higherorder terms can be included if neededNote that re-scaling the time in this way to make the

signal monochromatic assumes that the intrinsic phaseevolution of the pulsarrsquos GW signal is described by a Tay-lor expansion over the entire observation period Elec-tromagnetic observations of pulsars show that young

rapidly spinning pulsars demonstrate deviations from aTaylor expansion when spinning down a phenomenonknown as timing noise Given our lack of knowledge ofthe exact mechanism that might cause the gravitationalwave frequency to deviate from (twice) the observed elec-tromagnetic frequency we cannot be sure if we shouldexpect the timing noise to be present in the signal weare searching for (see [17] for discussion) However astudy based on the monthly Crab ephemeris data [18]has shown that in the Crab one of the noisiest of pul-sars the effect of timing noise over the duration of theobservation period is negligible producing a mismatchbetween a Taylor expansion and a signal based on theactual lsquonoisyrsquo time series of less than 1 This is smallconfirming that timing noise is likely to have a negligibleeffect on our analysis

In practice the spin-down correction is applied afterbarycentric corrections then the time t that appears inEq (16) is in fact the re-scaled time τ1 of Eq (15) Inthis case the correction depends explicitly on the searchfrequency f The number of frequency bins which coverthe range ∆f = 2f0δ corresponding to Eq (1) is

nf =

[

∆f

δf

]

= [∆f middot T ] asymp 63times 105(

∆f

002 Hz

)(

T

1 yr

)

(17)with δf = 1

T being the frequency spacing with T thetotal observation time and where [ middot ] stands for thenearest integer Similarly we take the width of the firstorder spin-down range as ∆f = 2|f0|δ and ∆f = 2|f0|δfor the second order Let us consider for the momentonly the first order term For a fixed value of f we donot want to make an explicit spin-down correction for

10

each value of f So we fix the value of the frequency inthe denominator of Eq (16) for instance at the referencefrequency f0 and exploit the fact that the same correc-tion that holds for the pair (f0 f0) is also valid for all the

pairs (f f) such that

f0f0

=f

f(18)

In practice this means that at each frequency f we ex-

plore a range of first order spin-down values [fminus ∆f2 f+

∆f2 ] with

f = f0f

f0(19)

The number of bins in the spin-down term of the firstorder can be computed by considering the ldquonaturalrdquo stepδf = 1

2T 2 and is given by

n(1) equiv nf =[

2∆f middot T 2]

(20)

asymp 400

(

f010minus10 Hzs

)

(

δ

10minus3

)(

T

1 yr

)2

Following the same reasoning for the second order spin-down we arrive at the following expression for the numberof steps that should be taken into account

n(2) equiv nf =[

∆f middot T 3]

(21)

asymp 06

(

f010minus20 Hzs2

)

(

δ

10minus3

)(

T

1 yr

)3

For observation times of the order of a year reasonablevalues of second order spin-down and values of δ typicalof narrow-band searches nf lt 1 which means that we donot need to consider more values other than the ldquocentralrdquoone In Tab I the Crab and Vela EM-inferred positionaland rotational parameters are shown Also estimationsof the polarization parameters are given which are usedin the computation of upper limits see Sec V In Fig 4the portion of the f minus f plane actually covered in theCrab and Vela narrow-band search is shownOnce barycentric and spin-down corrections have been

done the data are down-sampled to a much lower ratewith respect to the original one 1 Hz in the present caseThis strongly reduces the amount of data to be handledin the next step of the analysis At this stage for a givensource we have nf corrected time series one for eachvalue of first order spin-down For the current analysis wehave nf = 33 for Crab and nf = 3 for Vela see Tab II)where also other relevant quantities are given In partic-ular the total number of points in the frequencyspin-down plane is about 528 times 106 for Crab analysis and48 times 105 for Vela analysis For each time series we ap-ply a final cleaning step by removing the largest outliersThese are identified by histogramming the logarithm of

the absolute value of the data amplitude and choosinga threshold approximately marking the beginning of thenon-Gaussian tail of data distribution In Fig 5 the his-togram of Crab and Vela data amplitude correspond-ing to the ldquocentralrdquo time series are shown For Crab athreshold of 15 times 10minus21 corresponding to -182 in thefigure x-axis has been used to remove outliers while forVela a value of 12 times 10minus20 corresponding to -092 hasbeen used Correspondingly the fraction of removed datais of about 18 for Vela and 19 for Crab By apply-ing the Kolmogorov-Smirnov test we have verified thatfor each pulsar the data distribution of the various timeseries are fully in agreement thus justifying the use of thesame threshold for all of themAt this point the detection statistic is computed for ev-

ery frequency and spin-down value in the explored rangeThe detection statistic we use is based on the so-called 5-vectors the same used for pulsar targeted searches [3 4]and is here briefly described Once barycentric and spin-down corrections have been applied a CW signal withfrequency f0 present in the data would be monochro-matic apart from an amplitude and phase sidereal mod-ulation due to the time-varying detector beam patternfunctions and given by Eq (2) with ω0(t) constant andequal to 2πf0 From Eqs (256) it follows that the signalis completely described by its Fourier components at the5 angular frequencies ω0 ω0plusmnΩoplus ω0plusmn2Ωoplus This set of5 complex numbers constitutes the signal 5-vector Givena generic time series g(t) the corresponding 5-vector isdefined as

G =

int

T

g(t)eminusı(ω0minuskΩoplus)tdt (22)

where k = [minus2minus1 2] and T is the observation timeLet us indicate with X the data 5-vector and withA

+ Atimes the signal plus and cross 5-vectors obtained byapplying the definition of Eq (22) to Eqs (56) Thesetwo last quantities depend only on known parameters andform the signal templates Once the 5-vectors of data andof signal templates have been computed the two complexnumbers

H+times =X middotA+times

|A+times|2(23)

are built see [3 12] for more details They correspond tocomputing matched filters between the data and the sig-nal templates and it can be shown assuming the noiseis Gaussian with mean value zero that they are esti-mators of the signal plus and cross complex amplitudesH0e

ıΦ0H+ H0eıΦ0Htimes These estimators are used to

build the detection statistic

S = |A+|4|H+|2 + |Atimes|4|Htimes|

2 (24)

The maximum of the detection statistic over thesearched parameter space Smax is determined Thisis the loudest candidate and is identified by a triple

11

22374 22376 22378 2238 22382 22384 22386 22388 2239 22392 22394minus3149

minus3148

minus3147

minus3146

minus3145

minus3144

minus3143x 10

minus11

Frequency [Hz]S

pinminus

dow

n [H

zs]

Covered parameter space for Vela pulsar

59434 59436 59438 5944 59442 59444 59446 59448 5945 59452 59454minus7429

minus7428

minus7427

minus7426

minus7425

minus7424

minus7423x 10

minus10

Frequency [Hz]

Spi

nminusdo

wn

[Hz

s]

Covered parameter space for Crab pulsar

FIG 4 The parallelograms delimit the portion of the f minus f plane covered in the narrow-band search for the Vela (upper plot)

and Crab (bottom plot) pulsars Only one value of f has been considered in the analysis The total number of points is givenin the last column of Tab II

minus3 minus25 minus2 minus15 minus1 minus05 0 05 1

102

103

104

105

log10(abs(real(data1020)))

minus3 minus28 minus26 minus24 minus22 minus2 minus18 minus16 minus14 minus12 minus110

1

102

103

104

105

log10(abs(real(data1020)))

FIG 5 Histogram of the logarithm of data amplitude for Vela (top plot) and the Crab (bottom plot) used to select the thresholdfor the removal of outliers For Crab the threshold has been put at -182 corresponding to an amplitude of sim 15 times 10minus21while for Vela a value of -092 corresponding to an amplitude of sim 12times 10minus20 has been chosen

12

TABLE I The Crab and Vela reference parameters α and δ are the equatorial coordinates f0 f0 f0 are the gravitational-wavefrequency parameters The reference epoch for the Crab position is MJD 54632 while for the rotational parameters is MJD55696 The reference epoch for Vela is MJD 53576 both for position and GW frequency parameters The Crab ephemeris hasbeen obtained from a fit of the Jodrell Bank monthly ephemeris [18] while for Vela they have been derived from Hartebeesthoekradio telescope observations [19] Also estimations of the polarization parameters and their associated uncertainty obtainedfrom the analysis of Chandra X-ray observations of the pulsar wind nebula torus [20 21] are given in the last two columnsThe analysis presented in [20 21] does not allow to determine the sense of the starrsquos spin so values of ι and ψ correspondingto ι rarr 180 minus ι ψ rarr ψ + 180 are also possible However the upper limits on strain amplitude reported in this paper are notsensitive to the sense of rotation

Source α [hhmmss] δ[deg] f0[Hz] f0[Hzs] f0[Hzs2] ι[deg] ψ[deg]

Crab 05 34 3197 220145 594448 minus74183 times 10minus10 26307 times 10minus20 62o2plusmn 1o9 35o2plusmn 1o5

Vela 08 35 2061 minus451764 223840 minus31460 times 10minus11 12848 times 10minus21 63o6plusmn 0o6 40o6plusmn 0o1

TABLE II Main quantities related to the parameter space of VSR4 narrow-band search ∆f is the frequency range δ is thewidth parameter defined in Eq (1) ∆f is the first order spin-down range nf is the number of frequency bins nf and nf arerespectively the number of bins for the first and second order spin-down ntot = nf middot nf is the total number of points in theparameter space

Source ∆f [Hz] δ ∆f [Hzs] nf nf nf ntot

Crab 002 168times 10minus4 249times 10minus13 16times 105 33 1 528times 106

Vela 002 447times 10minus4 281times 10minus14 16times 105 3 1 480times 105

(Smax fSmax fSmax

) This means we need to computentot = nf middot nf values of the detection statisticThe maximum value Smax is used to assess detection

significance by computing the p-value that is the prob-ability that a value of the detection statistic equal orlarger than Smax can be obtained in the absence ofany signal It implies the need to compute the noise-only distribution of the detection statistic This is amulti-dimensional probability distribution (with dimen-sion ntot) which would be difficult to compute and to han-dle In practice as discussed in [11] the p-value is com-puted considering the single-trial noise probability distri-bution that is the same that would be used for a targetedsearch and choosing a suitable threshold on it to dis-criminate between interesting (that is deserving a deeperstudy) and not interesting candidates As shown in [11]by setting an overall p-value p0 (over the full multi-dimensional distribution) the corresponding threshold on

the single-trial distribution is pthr = 1minus (1minus p0)1

ntot Inour case by setting eg p0 = 001 the resulting pthrwould be of the order of 10minus8 In principle we wouldlike to generate the noise-only probability distribution bycomputing the detection statistic at several ldquooff-sourcerdquofrequency bins that is frequencies near but outside theexplored range where we are assuming the signal couldbe This is what is typically done in targeted searches [4]In the present analysis however in order to appreciate p-values of the order of pthr we should consider of the orderof 108 off-source frequency bins This is computationallyimpractical For this reason we use the theoretical dis-tribution which assumes the noise is Gaussian given in[11]

f(S) =eminus S

σ2X

|Atimes|2 minus eminus S

σ2X

|A+|2

σ2X (|Atimes|2 minus |A+|2)

(25)

where σ2X = σ2 middot T with σ2 being the noise variance

If Smax is compatible with noise at a given confidencelevel eg 1 we compute an upper limit on signalstrength using the same method described in [4] other-wise in case of detection signal parameters are estimatedthrough suitable combinations of the real and imaginaryparts of H+ and Htimes as explained in [12]

V RESULTS

Analysis results are summarized in Tab III The databoth for the Vela and Crab searches are compatible withnoise In particular we find p-values equal to 033 and0013 for Vela and the Crab respectively larger thanp0 = 001 chosen to identify interesting candidates ForCrab however the obtained p-value is very near to thechosen threshold Altough a reasonable choice of p0 israther arbitrary (we could have for instance chosen 10minus3)we have decided to study in some detail the candidatersquosproperties In particular we have considered the distri-bution of the top ten candidates in the frequencyspin-down plane They appear to be randomly distributedwithout the clustering that we would expect in presenceof a signal To verify this hypothesis we have addeda simulated signal to the data with the same parame-ters as the Crab but with a slightly different frequencyand spin-down and with an amplitude h0 ≃ 41times 10minus25

such that the resulting loudest candidate has a value ofthe detection statistic approximately equal to the loud-est candidate of the actual analysis We have then runthe full analysis on this new data set and looked again atthe top ten candidate distribution in the frequencyspin-down plane In this case we indeed observe a clusteringwith 4 out of 10 candidates having a frequency within

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

10

each value of f So we fix the value of the frequency inthe denominator of Eq (16) for instance at the referencefrequency f0 and exploit the fact that the same correc-tion that holds for the pair (f0 f0) is also valid for all the

pairs (f f) such that

f0f0

=f

f(18)

In practice this means that at each frequency f we ex-

plore a range of first order spin-down values [fminus ∆f2 f+

∆f2 ] with

f = f0f

f0(19)

The number of bins in the spin-down term of the firstorder can be computed by considering the ldquonaturalrdquo stepδf = 1

2T 2 and is given by

n(1) equiv nf =[

2∆f middot T 2]

(20)

asymp 400

(

f010minus10 Hzs

)

(

δ

10minus3

)(

T

1 yr

)2

Following the same reasoning for the second order spin-down we arrive at the following expression for the numberof steps that should be taken into account

n(2) equiv nf =[

∆f middot T 3]

(21)

asymp 06

(

f010minus20 Hzs2

)

(

δ

10minus3

)(

T

1 yr

)3

For observation times of the order of a year reasonablevalues of second order spin-down and values of δ typicalof narrow-band searches nf lt 1 which means that we donot need to consider more values other than the ldquocentralrdquoone In Tab I the Crab and Vela EM-inferred positionaland rotational parameters are shown Also estimationsof the polarization parameters are given which are usedin the computation of upper limits see Sec V In Fig 4the portion of the f minus f plane actually covered in theCrab and Vela narrow-band search is shownOnce barycentric and spin-down corrections have been

done the data are down-sampled to a much lower ratewith respect to the original one 1 Hz in the present caseThis strongly reduces the amount of data to be handledin the next step of the analysis At this stage for a givensource we have nf corrected time series one for eachvalue of first order spin-down For the current analysis wehave nf = 33 for Crab and nf = 3 for Vela see Tab II)where also other relevant quantities are given In partic-ular the total number of points in the frequencyspin-down plane is about 528 times 106 for Crab analysis and48 times 105 for Vela analysis For each time series we ap-ply a final cleaning step by removing the largest outliersThese are identified by histogramming the logarithm of

the absolute value of the data amplitude and choosinga threshold approximately marking the beginning of thenon-Gaussian tail of data distribution In Fig 5 the his-togram of Crab and Vela data amplitude correspond-ing to the ldquocentralrdquo time series are shown For Crab athreshold of 15 times 10minus21 corresponding to -182 in thefigure x-axis has been used to remove outliers while forVela a value of 12 times 10minus20 corresponding to -092 hasbeen used Correspondingly the fraction of removed datais of about 18 for Vela and 19 for Crab By apply-ing the Kolmogorov-Smirnov test we have verified thatfor each pulsar the data distribution of the various timeseries are fully in agreement thus justifying the use of thesame threshold for all of themAt this point the detection statistic is computed for ev-

ery frequency and spin-down value in the explored rangeThe detection statistic we use is based on the so-called 5-vectors the same used for pulsar targeted searches [3 4]and is here briefly described Once barycentric and spin-down corrections have been applied a CW signal withfrequency f0 present in the data would be monochro-matic apart from an amplitude and phase sidereal mod-ulation due to the time-varying detector beam patternfunctions and given by Eq (2) with ω0(t) constant andequal to 2πf0 From Eqs (256) it follows that the signalis completely described by its Fourier components at the5 angular frequencies ω0 ω0plusmnΩoplus ω0plusmn2Ωoplus This set of5 complex numbers constitutes the signal 5-vector Givena generic time series g(t) the corresponding 5-vector isdefined as

G =

int

T

g(t)eminusı(ω0minuskΩoplus)tdt (22)

where k = [minus2minus1 2] and T is the observation timeLet us indicate with X the data 5-vector and withA

+ Atimes the signal plus and cross 5-vectors obtained byapplying the definition of Eq (22) to Eqs (56) Thesetwo last quantities depend only on known parameters andform the signal templates Once the 5-vectors of data andof signal templates have been computed the two complexnumbers

H+times =X middotA+times

|A+times|2(23)

are built see [3 12] for more details They correspond tocomputing matched filters between the data and the sig-nal templates and it can be shown assuming the noiseis Gaussian with mean value zero that they are esti-mators of the signal plus and cross complex amplitudesH0e

ıΦ0H+ H0eıΦ0Htimes These estimators are used to

build the detection statistic

S = |A+|4|H+|2 + |Atimes|4|Htimes|

2 (24)

The maximum of the detection statistic over thesearched parameter space Smax is determined Thisis the loudest candidate and is identified by a triple

11

22374 22376 22378 2238 22382 22384 22386 22388 2239 22392 22394minus3149

minus3148

minus3147

minus3146

minus3145

minus3144

minus3143x 10

minus11

Frequency [Hz]S

pinminus

dow

n [H

zs]

Covered parameter space for Vela pulsar

59434 59436 59438 5944 59442 59444 59446 59448 5945 59452 59454minus7429

minus7428

minus7427

minus7426

minus7425

minus7424

minus7423x 10

minus10

Frequency [Hz]

Spi

nminusdo

wn

[Hz

s]

Covered parameter space for Crab pulsar

FIG 4 The parallelograms delimit the portion of the f minus f plane covered in the narrow-band search for the Vela (upper plot)

and Crab (bottom plot) pulsars Only one value of f has been considered in the analysis The total number of points is givenin the last column of Tab II

minus3 minus25 minus2 minus15 minus1 minus05 0 05 1

102

103

104

105

log10(abs(real(data1020)))

minus3 minus28 minus26 minus24 minus22 minus2 minus18 minus16 minus14 minus12 minus110

1

102

103

104

105

log10(abs(real(data1020)))

FIG 5 Histogram of the logarithm of data amplitude for Vela (top plot) and the Crab (bottom plot) used to select the thresholdfor the removal of outliers For Crab the threshold has been put at -182 corresponding to an amplitude of sim 15 times 10minus21while for Vela a value of -092 corresponding to an amplitude of sim 12times 10minus20 has been chosen

12

TABLE I The Crab and Vela reference parameters α and δ are the equatorial coordinates f0 f0 f0 are the gravitational-wavefrequency parameters The reference epoch for the Crab position is MJD 54632 while for the rotational parameters is MJD55696 The reference epoch for Vela is MJD 53576 both for position and GW frequency parameters The Crab ephemeris hasbeen obtained from a fit of the Jodrell Bank monthly ephemeris [18] while for Vela they have been derived from Hartebeesthoekradio telescope observations [19] Also estimations of the polarization parameters and their associated uncertainty obtainedfrom the analysis of Chandra X-ray observations of the pulsar wind nebula torus [20 21] are given in the last two columnsThe analysis presented in [20 21] does not allow to determine the sense of the starrsquos spin so values of ι and ψ correspondingto ι rarr 180 minus ι ψ rarr ψ + 180 are also possible However the upper limits on strain amplitude reported in this paper are notsensitive to the sense of rotation

Source α [hhmmss] δ[deg] f0[Hz] f0[Hzs] f0[Hzs2] ι[deg] ψ[deg]

Crab 05 34 3197 220145 594448 minus74183 times 10minus10 26307 times 10minus20 62o2plusmn 1o9 35o2plusmn 1o5

Vela 08 35 2061 minus451764 223840 minus31460 times 10minus11 12848 times 10minus21 63o6plusmn 0o6 40o6plusmn 0o1

TABLE II Main quantities related to the parameter space of VSR4 narrow-band search ∆f is the frequency range δ is thewidth parameter defined in Eq (1) ∆f is the first order spin-down range nf is the number of frequency bins nf and nf arerespectively the number of bins for the first and second order spin-down ntot = nf middot nf is the total number of points in theparameter space

Source ∆f [Hz] δ ∆f [Hzs] nf nf nf ntot

Crab 002 168times 10minus4 249times 10minus13 16times 105 33 1 528times 106

Vela 002 447times 10minus4 281times 10minus14 16times 105 3 1 480times 105

(Smax fSmax fSmax

) This means we need to computentot = nf middot nf values of the detection statisticThe maximum value Smax is used to assess detection

significance by computing the p-value that is the prob-ability that a value of the detection statistic equal orlarger than Smax can be obtained in the absence ofany signal It implies the need to compute the noise-only distribution of the detection statistic This is amulti-dimensional probability distribution (with dimen-sion ntot) which would be difficult to compute and to han-dle In practice as discussed in [11] the p-value is com-puted considering the single-trial noise probability distri-bution that is the same that would be used for a targetedsearch and choosing a suitable threshold on it to dis-criminate between interesting (that is deserving a deeperstudy) and not interesting candidates As shown in [11]by setting an overall p-value p0 (over the full multi-dimensional distribution) the corresponding threshold on

the single-trial distribution is pthr = 1minus (1minus p0)1

ntot Inour case by setting eg p0 = 001 the resulting pthrwould be of the order of 10minus8 In principle we wouldlike to generate the noise-only probability distribution bycomputing the detection statistic at several ldquooff-sourcerdquofrequency bins that is frequencies near but outside theexplored range where we are assuming the signal couldbe This is what is typically done in targeted searches [4]In the present analysis however in order to appreciate p-values of the order of pthr we should consider of the orderof 108 off-source frequency bins This is computationallyimpractical For this reason we use the theoretical dis-tribution which assumes the noise is Gaussian given in[11]

f(S) =eminus S

σ2X

|Atimes|2 minus eminus S

σ2X

|A+|2

σ2X (|Atimes|2 minus |A+|2)

(25)

where σ2X = σ2 middot T with σ2 being the noise variance

If Smax is compatible with noise at a given confidencelevel eg 1 we compute an upper limit on signalstrength using the same method described in [4] other-wise in case of detection signal parameters are estimatedthrough suitable combinations of the real and imaginaryparts of H+ and Htimes as explained in [12]

V RESULTS

Analysis results are summarized in Tab III The databoth for the Vela and Crab searches are compatible withnoise In particular we find p-values equal to 033 and0013 for Vela and the Crab respectively larger thanp0 = 001 chosen to identify interesting candidates ForCrab however the obtained p-value is very near to thechosen threshold Altough a reasonable choice of p0 israther arbitrary (we could have for instance chosen 10minus3)we have decided to study in some detail the candidatersquosproperties In particular we have considered the distri-bution of the top ten candidates in the frequencyspin-down plane They appear to be randomly distributedwithout the clustering that we would expect in presenceof a signal To verify this hypothesis we have addeda simulated signal to the data with the same parame-ters as the Crab but with a slightly different frequencyand spin-down and with an amplitude h0 ≃ 41times 10minus25

such that the resulting loudest candidate has a value ofthe detection statistic approximately equal to the loud-est candidate of the actual analysis We have then runthe full analysis on this new data set and looked again atthe top ten candidate distribution in the frequencyspin-down plane In this case we indeed observe a clusteringwith 4 out of 10 candidates having a frequency within

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

11

22374 22376 22378 2238 22382 22384 22386 22388 2239 22392 22394minus3149

minus3148

minus3147

minus3146

minus3145

minus3144

minus3143x 10

minus11

Frequency [Hz]S

pinminus

dow

n [H

zs]

Covered parameter space for Vela pulsar

59434 59436 59438 5944 59442 59444 59446 59448 5945 59452 59454minus7429

minus7428

minus7427

minus7426

minus7425

minus7424

minus7423x 10

minus10

Frequency [Hz]

Spi

nminusdo

wn

[Hz

s]

Covered parameter space for Crab pulsar

FIG 4 The parallelograms delimit the portion of the f minus f plane covered in the narrow-band search for the Vela (upper plot)

and Crab (bottom plot) pulsars Only one value of f has been considered in the analysis The total number of points is givenin the last column of Tab II

minus3 minus25 minus2 minus15 minus1 minus05 0 05 1

102

103

104

105

log10(abs(real(data1020)))

minus3 minus28 minus26 minus24 minus22 minus2 minus18 minus16 minus14 minus12 minus110

1

102

103

104

105

log10(abs(real(data1020)))

FIG 5 Histogram of the logarithm of data amplitude for Vela (top plot) and the Crab (bottom plot) used to select the thresholdfor the removal of outliers For Crab the threshold has been put at -182 corresponding to an amplitude of sim 15 times 10minus21while for Vela a value of -092 corresponding to an amplitude of sim 12times 10minus20 has been chosen

12

TABLE I The Crab and Vela reference parameters α and δ are the equatorial coordinates f0 f0 f0 are the gravitational-wavefrequency parameters The reference epoch for the Crab position is MJD 54632 while for the rotational parameters is MJD55696 The reference epoch for Vela is MJD 53576 both for position and GW frequency parameters The Crab ephemeris hasbeen obtained from a fit of the Jodrell Bank monthly ephemeris [18] while for Vela they have been derived from Hartebeesthoekradio telescope observations [19] Also estimations of the polarization parameters and their associated uncertainty obtainedfrom the analysis of Chandra X-ray observations of the pulsar wind nebula torus [20 21] are given in the last two columnsThe analysis presented in [20 21] does not allow to determine the sense of the starrsquos spin so values of ι and ψ correspondingto ι rarr 180 minus ι ψ rarr ψ + 180 are also possible However the upper limits on strain amplitude reported in this paper are notsensitive to the sense of rotation

Source α [hhmmss] δ[deg] f0[Hz] f0[Hzs] f0[Hzs2] ι[deg] ψ[deg]

Crab 05 34 3197 220145 594448 minus74183 times 10minus10 26307 times 10minus20 62o2plusmn 1o9 35o2plusmn 1o5

Vela 08 35 2061 minus451764 223840 minus31460 times 10minus11 12848 times 10minus21 63o6plusmn 0o6 40o6plusmn 0o1

TABLE II Main quantities related to the parameter space of VSR4 narrow-band search ∆f is the frequency range δ is thewidth parameter defined in Eq (1) ∆f is the first order spin-down range nf is the number of frequency bins nf and nf arerespectively the number of bins for the first and second order spin-down ntot = nf middot nf is the total number of points in theparameter space

Source ∆f [Hz] δ ∆f [Hzs] nf nf nf ntot

Crab 002 168times 10minus4 249times 10minus13 16times 105 33 1 528times 106

Vela 002 447times 10minus4 281times 10minus14 16times 105 3 1 480times 105

(Smax fSmax fSmax

) This means we need to computentot = nf middot nf values of the detection statisticThe maximum value Smax is used to assess detection

significance by computing the p-value that is the prob-ability that a value of the detection statistic equal orlarger than Smax can be obtained in the absence ofany signal It implies the need to compute the noise-only distribution of the detection statistic This is amulti-dimensional probability distribution (with dimen-sion ntot) which would be difficult to compute and to han-dle In practice as discussed in [11] the p-value is com-puted considering the single-trial noise probability distri-bution that is the same that would be used for a targetedsearch and choosing a suitable threshold on it to dis-criminate between interesting (that is deserving a deeperstudy) and not interesting candidates As shown in [11]by setting an overall p-value p0 (over the full multi-dimensional distribution) the corresponding threshold on

the single-trial distribution is pthr = 1minus (1minus p0)1

ntot Inour case by setting eg p0 = 001 the resulting pthrwould be of the order of 10minus8 In principle we wouldlike to generate the noise-only probability distribution bycomputing the detection statistic at several ldquooff-sourcerdquofrequency bins that is frequencies near but outside theexplored range where we are assuming the signal couldbe This is what is typically done in targeted searches [4]In the present analysis however in order to appreciate p-values of the order of pthr we should consider of the orderof 108 off-source frequency bins This is computationallyimpractical For this reason we use the theoretical dis-tribution which assumes the noise is Gaussian given in[11]

f(S) =eminus S

σ2X

|Atimes|2 minus eminus S

σ2X

|A+|2

σ2X (|Atimes|2 minus |A+|2)

(25)

where σ2X = σ2 middot T with σ2 being the noise variance

If Smax is compatible with noise at a given confidencelevel eg 1 we compute an upper limit on signalstrength using the same method described in [4] other-wise in case of detection signal parameters are estimatedthrough suitable combinations of the real and imaginaryparts of H+ and Htimes as explained in [12]

V RESULTS

Analysis results are summarized in Tab III The databoth for the Vela and Crab searches are compatible withnoise In particular we find p-values equal to 033 and0013 for Vela and the Crab respectively larger thanp0 = 001 chosen to identify interesting candidates ForCrab however the obtained p-value is very near to thechosen threshold Altough a reasonable choice of p0 israther arbitrary (we could have for instance chosen 10minus3)we have decided to study in some detail the candidatersquosproperties In particular we have considered the distri-bution of the top ten candidates in the frequencyspin-down plane They appear to be randomly distributedwithout the clustering that we would expect in presenceof a signal To verify this hypothesis we have addeda simulated signal to the data with the same parame-ters as the Crab but with a slightly different frequencyand spin-down and with an amplitude h0 ≃ 41times 10minus25

such that the resulting loudest candidate has a value ofthe detection statistic approximately equal to the loud-est candidate of the actual analysis We have then runthe full analysis on this new data set and looked again atthe top ten candidate distribution in the frequencyspin-down plane In this case we indeed observe a clusteringwith 4 out of 10 candidates having a frequency within

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

12

TABLE I The Crab and Vela reference parameters α and δ are the equatorial coordinates f0 f0 f0 are the gravitational-wavefrequency parameters The reference epoch for the Crab position is MJD 54632 while for the rotational parameters is MJD55696 The reference epoch for Vela is MJD 53576 both for position and GW frequency parameters The Crab ephemeris hasbeen obtained from a fit of the Jodrell Bank monthly ephemeris [18] while for Vela they have been derived from Hartebeesthoekradio telescope observations [19] Also estimations of the polarization parameters and their associated uncertainty obtainedfrom the analysis of Chandra X-ray observations of the pulsar wind nebula torus [20 21] are given in the last two columnsThe analysis presented in [20 21] does not allow to determine the sense of the starrsquos spin so values of ι and ψ correspondingto ι rarr 180 minus ι ψ rarr ψ + 180 are also possible However the upper limits on strain amplitude reported in this paper are notsensitive to the sense of rotation

Source α [hhmmss] δ[deg] f0[Hz] f0[Hzs] f0[Hzs2] ι[deg] ψ[deg]

Crab 05 34 3197 220145 594448 minus74183 times 10minus10 26307 times 10minus20 62o2plusmn 1o9 35o2plusmn 1o5

Vela 08 35 2061 minus451764 223840 minus31460 times 10minus11 12848 times 10minus21 63o6plusmn 0o6 40o6plusmn 0o1

TABLE II Main quantities related to the parameter space of VSR4 narrow-band search ∆f is the frequency range δ is thewidth parameter defined in Eq (1) ∆f is the first order spin-down range nf is the number of frequency bins nf and nf arerespectively the number of bins for the first and second order spin-down ntot = nf middot nf is the total number of points in theparameter space

Source ∆f [Hz] δ ∆f [Hzs] nf nf nf ntot

Crab 002 168times 10minus4 249times 10minus13 16times 105 33 1 528times 106

Vela 002 447times 10minus4 281times 10minus14 16times 105 3 1 480times 105

(Smax fSmax fSmax

) This means we need to computentot = nf middot nf values of the detection statisticThe maximum value Smax is used to assess detection

significance by computing the p-value that is the prob-ability that a value of the detection statistic equal orlarger than Smax can be obtained in the absence ofany signal It implies the need to compute the noise-only distribution of the detection statistic This is amulti-dimensional probability distribution (with dimen-sion ntot) which would be difficult to compute and to han-dle In practice as discussed in [11] the p-value is com-puted considering the single-trial noise probability distri-bution that is the same that would be used for a targetedsearch and choosing a suitable threshold on it to dis-criminate between interesting (that is deserving a deeperstudy) and not interesting candidates As shown in [11]by setting an overall p-value p0 (over the full multi-dimensional distribution) the corresponding threshold on

the single-trial distribution is pthr = 1minus (1minus p0)1

ntot Inour case by setting eg p0 = 001 the resulting pthrwould be of the order of 10minus8 In principle we wouldlike to generate the noise-only probability distribution bycomputing the detection statistic at several ldquooff-sourcerdquofrequency bins that is frequencies near but outside theexplored range where we are assuming the signal couldbe This is what is typically done in targeted searches [4]In the present analysis however in order to appreciate p-values of the order of pthr we should consider of the orderof 108 off-source frequency bins This is computationallyimpractical For this reason we use the theoretical dis-tribution which assumes the noise is Gaussian given in[11]

f(S) =eminus S

σ2X

|Atimes|2 minus eminus S

σ2X

|A+|2

σ2X (|Atimes|2 minus |A+|2)

(25)

where σ2X = σ2 middot T with σ2 being the noise variance

If Smax is compatible with noise at a given confidencelevel eg 1 we compute an upper limit on signalstrength using the same method described in [4] other-wise in case of detection signal parameters are estimatedthrough suitable combinations of the real and imaginaryparts of H+ and Htimes as explained in [12]

V RESULTS

Analysis results are summarized in Tab III The databoth for the Vela and Crab searches are compatible withnoise In particular we find p-values equal to 033 and0013 for Vela and the Crab respectively larger thanp0 = 001 chosen to identify interesting candidates ForCrab however the obtained p-value is very near to thechosen threshold Altough a reasonable choice of p0 israther arbitrary (we could have for instance chosen 10minus3)we have decided to study in some detail the candidatersquosproperties In particular we have considered the distri-bution of the top ten candidates in the frequencyspin-down plane They appear to be randomly distributedwithout the clustering that we would expect in presenceof a signal To verify this hypothesis we have addeda simulated signal to the data with the same parame-ters as the Crab but with a slightly different frequencyand spin-down and with an amplitude h0 ≃ 41times 10minus25

such that the resulting loudest candidate has a value ofthe detection statistic approximately equal to the loud-est candidate of the actual analysis We have then runthe full analysis on this new data set and looked again atthe top ten candidate distribution in the frequencyspin-down plane In this case we indeed observe a clusteringwith 4 out of 10 candidates having a frequency within

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

13

plusmn2 bins of the injected value and 4 out of 10 candidateshaving a spin-down within plusmn3 bins of the injected valueOverall 5-6 of the top ten candidates appear to be dueto the injected signal Top ten candidates distribution inthe frequencyspin-down plane is shown in Fig 6 for bothcases The results of this test make us more confident indeclaring a non-detection also in the Crab analysis

We have computed 95 confidence level upper limitsboth assuming uniform priors on the polarization param-eters ψ and cos ι and ldquorestrictedrdquo priors described bya Gaussian distribution with mean value and standarddeviation given in Tab I In fact the upper limits wecompute are obtained from the posterior distribution ofthe signal strain amplitude conditioned to the observedvalue of the detection statistic as described in [4] ForVela the upper limits are very similar to the spin-downlimit This does not allow us to significantly constrain thefraction of spin-down energy due to the emission of grav-itational waves In the case of the Crab pulsar the upperlimits on signal strain amplitude are about 2 times be-low the spin-down limit with a corresponding constraintof about 25 on the fraction of spin-down energy dueto gravitational waves The upper limit on signal strainamplitude can be converted via Eq (11) into an upperlimit on star ellipticity of about 37times10minus4 assuming theneutron star moment of inertia is equal to the canonicalvalue of 1038kg m2 The upper limits on ellipticity arecomparable to the maximum value foreseen by some ldquoex-oticrdquo equation of state for neutron star matter [7 8] butare much larger than the maximum value predicted fora ldquostandardrdquo equations of state In Tab III upper limitson the quadrupole mass moment which are independenton the uncertain value of the star moment of inertia arealso given An uncertainty of about 8 is associated withthe upper limit on strain amplitude This has been es-timated as the square root of the quadratic sum of thecalibration error which as discussed in Sec III amountsto 75 and of the error associated to the finite size ofthe simulation used to compute the upper limits whichis about 3

While this is the first time a narrow-band search hasbeen carried out for the Vela pulsar our results forthe Crab represent an improvement with respect to theLIGOS5 search described in [5] Overall this is the firsttime the spin-down limit has been significantly overcomein any narrow-band search On the other hand our up-per limits are clearly worse than those found in Vela andCrab targeted searches [4] As explained in [11] this iswhat we expect as a consequence of the lower sensitivityof narrow-band searches due to the volume of parameterspace that is explored Also the use of VSR4 data alonefor the Crab search contributes even if to a lesser extentto reduce the search sensitivity with respect to [4] wheredata from Virgo VSR2 VSR4 and LIGO S6 were used

VI VALIDATION TESTS

We have performed validation tests of the analysispipeline both injecting software simulated signals inGaussian noise and performing a narrow-band searcharound some of the hardware injected signals in VSR4data The first kind of test has been discussed in [11]Here we focus attention on the second kind of test

For the entire duration of VSR4 run 10 simulated CWsignals have been injected in the Virgo detector by send-ing the appropriate excitations to the coils used to controlmirror position These signals were characterized by var-ious amplitudes frequency spin-down and polarizationparameters and corresponded to sources with various lo-cations in the sky In particular here we have consideredinjections named Pulsar3 Pulsar5 Pulsar8 with param-eters of signal amplitude h0 frequency f0 spin-down f0position (α0 δ0) ratio between the polarization ellipsesemi-minor and semi-major axes η0 and wave polariza-tion angle ψ0 given in Tab IV For each of the hardwareinjections we have performed a narrow-band search over afrequency range of 10minus4 Hz and over a spin-down range of1585times10minus13 Hzs around the signal injected values Thegrid in frequency and spin-down has been built in sucha way that each center bin corresponds to a frequencygiven by an integer number of bins This means that thetrue values do not coincide with a bin center The ex-plored frequency range is covered by 814 bins while thespin-down range is covered by 21 bins For each hardwareinjection the frequency and spin-down corresponding tothe loudest candidate have been selected and comparedto the true signal values Moreover the amplitude andpolarization parameters of the injected signals have beenestimated and compared to the actual values and to thevalues that would have been found in a targeted searchIn Tab V and VI test results are summarized In par-ticular in Tab V we report the estimated frequency f the error with respect to the true value ǫf measured innumber of frequency bins the estimated spin-down valuef and its error ǫf in number of spin-down bins FromTab V we see that within the discretization error boththe frequency and the spin-down are correctly recoveredfor all the hardware injections This can be also seen inFig 7 where two plots referring to Pulsar3 are shownThe upper plot represents the maximum of the detectionstatistic (maximized over frequency) as a function of thespin-down index which goes from 1 to 21 The maximumof the curve is at index number 11 which is the nearestto the injected value Note that given the very smallspin-down value of Pulsar3 and Pulsar5 much smallerthan the spin-down bin width the searched range coversboth negative and positive values of f The lower plotshows the value of the detection statistic as a function ofthe frequency bin which goes from 1 to 814 computedtaking the spin-down of the loudest candidate As be-fore the maximum is found in correspondence of the binnearest to the injected value of the frequency In Tab VIthe error in the estimation of amplitude and polarization

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

14

0 5 10 15

x 104

10

15

20

25

30

Frequency bin

Spi

nminusdo

wn

bin

FIG 6 Distribution in the frequencyspin-down plane of the top ten candidates obtained in the actual analysis (blue circles)and obtained after the addition to the data of a simulated signal with amplitude h0 = 406 times 10minus25 such that the resultingloudest candidate is approximately as loud as the one obtained in the actual analysis (red squares) The black star identify theinjection Frequency and spin-down are expressed as number of bins from the beginning of the corresponding intervals Six redsquares appear to be at the same frequency but this is just a visual effect due to the large range of frequency covered by thex-axis In fact they are very near but spread around the injection values

TABLE III Summary of the analysis results The p-value is reported in the second column hunifUL is the experimental upperlimit assuming uniform priors for the polarization parameters hrestrUL is the experimental upper limit assuming ldquorestrictedrdquopriors for the polarization parameters given by a Gaussian distribution with mean value and standard deviation given inTab I hsd is the spin-down limit computed through Eq (10) ǫUL is the upper limit on the ellipticity Q22UL is the upperlimit on the mass quadrupole moment hULhsd is the ratio between the upper limit and the spin-down limit on signal strain

amplitude EULEsd is the upper limit on the fraction of spin-down energy due to the emission of gravitational waves (thevalues among parentheses refer to the restricted case)

Source p-value hunifUL hrestrUL hsd ǫUL times 104 Q22UL times 10minus34 [kg m2] hULhsd EULEsd

Vela 033 32times 10minus24 33times 10minus24 33 times 10minus24 176(191) 134(139) 097(1) 094(1)Crab 0013 70times 10minus25 69times 10minus25 14 times 10minus24 38(37) 29(28) 050(049) 025(024)

parameters of hardware injections is given In particularin column 2 the ratio between the found and the injectedamplitude is reported in column 3 the fractional erroron η that is ǫη = ηminusη0

2 and in column 4 the fractional

error on ψ ǫψ = ψminusψ0

90 Values in parentheses refer to atargeted search for the same injections Parameters aregenerally well recovered with an accuracy just slightlyworse with respect to the targeted search case This isdue to the small error in the estimation of the signal fre-quency in a narrow-band search consequence of the finitesize of the grid step In fact looking for instance at theamplitude estimation we can note that for Pulsar8 for

which the frequency error is the largest the loss with re-spect to the targeted search is the biggest as expected

VII CONCLUSIONS

Targeted searches for continuous gravitational wavesignals assume a strict correlation between the gravita-tional wave frequency and the star rotation frequencyFor instance for a neutron star non-axisymmetric withrespect to the rotation axis the gravitational wave fre-quency is exactly two times the rotation rate However

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

15

TABLE IV Main parameters of the hardware injections used to test the narrow-band search pipeline h0 is the signal amplitudef0 is the signal frequency f0 is the spin-down (α0 δ0) is the source position in equatorial coordinates η0 is the ratio betweenthe polarization ellipse semi-minor and semi-major axes ψ0 is the wave polarization angle

Name h0 f0[Hz] f0[Hzs] α0[deg] δ0[deg] η0 ψ0 [deg]

Pulsar3 8296 times 10minus24 108857159396 minus146times 10minus17 178372574 minus33436602 01600 25439Pulsar5 3703 times 10minus24 52808324359 minus403times 10minus18 302626641 minus838391399 minus07625 minus20853Pulsar8 8067 times 10minus24 192237058812 minus865 times 10minus9 351389582 minus334185168 minus01470 97673

TABLE V Tests with hardware injections frequency and spin-down recovery For each hardware injection the estimatedfrequency and spin-down f and f and the associated errors ǫf and ǫf expressed in number of bins are given

Name f [Hz] ǫf f [Hzs] ǫf

Pulsar3 108857159411 012 0 0002Pulsar5 52808324371 010 0 00005Pulsar8 192237058796 minus013 minus86500021 times 10minus9

minus0272

2 4 6 8 10 12 14 16 18 20

04

06

08

1

12

14

16

x 1019

Spinminusdown index

Max

freq

(Det

ectio

n S

tatis

tic)

385 390 395 400 405 410 415 420 4250

2

4

6

8

10

12

14

16

x 1018

Frequency index

Det

ectio

n S

tatis

tic

FIG 7 Tests with hardware injection Pulsar3 Upper plot maximum of the detection statistic (maximized over frequency) asa function of the spin-down index (which goes from 1 to 21) Bottom plot value of the detection statistic as a function of thefrequency bin (which goes from 1 to 814) computed taking the spin-down of the loudest candidate The plots show that thefrequency and spin-down of the loudest candidate are the nearest to the true values

TABLE VI Tests with hardware injections amplitude and polarization parameter recovery For each hardware injection wereport the ratio between the estimated and injected signal amplitude hh0 the relative error on η normalized to its range ofvariation (2) ǫη and the relative error on ψ normalized to its range of variation (90 degrees) ǫψ The numbers in parenthesesrefer to a targeted search for the same signals

Name hh0 ǫη ǫψ

Pulsar3 1016(0992) 00027(00021) minus00079(minus00081)Pulsar5 1023(1002) 00080(00089) 01196(00104)Pulsar8 0871(0975) 00278(00038) 00007(00064)

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

16

it is questionable that such strict a correlation is alwaysvalid or that is maintained over long times and variousmechanisms have been proposed that could produce amismatch For this reason it is important to have in placean analysis procedure robust with respect to deviationsfrom the standard assumption made in targeted searchesand then able to perform a narrow-band search for CWsignals over a range of frequency and spin-down valuesIn this paper we have presented results of a narrow-bandsearch for CW from the Crab and Vela pulsars in thedata of Virgo VSR4 run In both cases the data appearto be fully compatible with noise and we have set 95confidence level upper limits on strain amplitude assum-ing both uniform and restricted priors on polarizationparameters The upper limits are respectively compa-rable (for Vela) and a factor of 2 below (for the Crab)to the corresponding spin-down limits and while this isthe first time a narrow-band search has been done for theVela pulsar for the Crab our results significantly improvewith respect to past analyses As expected the narrow-band search upper limits are worse than those establishedin targeted searches of the same sources This analysismethod will be applied to narrow-band searches of CWsignals from several potentially interesting pulsars in dataof advanced Virgo and LIGO detectors which will startto collect science data at the end of 2015 and are ex-pect to reach their target sensitivity about one order ofmagnitude better than first generation detectors around2018 [22]

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theUnited States National Science Foundation for the con-struction and operation of the LIGO Laboratory theScience and Technology Facilities Council of the UnitedKingdom the Max-Planck-Society and the State ofNiedersachsenGermany for support of the constructionand operation of the GEO600 detector and the ItalianIstituto Nazionale di Fisica Nucleare and the French Cen-tre National de la Recherche Scientifique for the con-struction and operation of the Virgo detector Theauthors also gratefully acknowledge the support of theresearch by these agencies and by the Australian Re-search Council the International Science Linkages pro-gram of the Commonwealth of Australia the Council ofScientific and Industrial Research of India the IstitutoNazionale di Fisica Nucleare of Italy the Spanish Min-isterio de Economıa y Competitividad the ConselleriadrsquoEconomia Hisenda i Innovacio of the Govern de les IllesBalears the Foundation for Fundamental Research onMatter supported by the Netherlands Organisation forScientific Research the Polish Ministry of Science andHigher Education the FOCUS Programme of Founda-tion for Polish Science the Royal Society the ScottishFunding Council the Scottish Universities Physics Al-liance The National Aeronautics and Space Adminis-

tration OTKA of Hungary the Lyon Institute of Ori-gins (LIO) the National Research Foundation of KoreaIndustry Canada and the Province of Ontario throughthe Ministry of Economic Development and Innovationthe National Science and Engineering Research CouncilCanadathe Carnegie Trust the Leverhulme Trust theDavid and Lucile Packard Foundation the Research Cor-poration and the Alfred P Sloan Foundation

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670

17

[1] N Andersson and et al Gen Relativ Gravit 43 409(2011)

[2] B P Abbott and et al The Astrophysical Journal 713671 (2010)

[3] J Abadie and et al The Astrophysical Journal 737 93(2011)

[4] J Aasi and et al The Astrophysical Journal 785 119(2014)

[5] B P Abbott and et al The Astrophysical Journal 683L45 (2008)

[6] D I Jones and N Andersson Mon Not R Astron Soc331 203 (2002)

[7] B J Owen Physical Review Letters 95 211101 (2005)[8] B Haskell and et al Physical Review Letters 99 231101

(2007)[9] N K Johnson-MacDaniel Physical Review D 88 044016

(2013)

[10] P Jaranowski A Krolak and B F Schutz PhysicalReview D 58 063001 (1998)

[11] P Astone A Colla S DrsquoAntonio S Frasca C Palombaand R Serafinelli Phys Rev D 89 062008 (2014)

[12] P Astone S DrsquoAntonio S Frasca and C PalombaClass Q Grav 27 195016 (2010)

[13] B Abbott and et al Rep Prog Phys 72 076901 (2009)[14] T Accadia and et al J Instrum 7 P03012 (2012)[15] H Grote (The LIGO Scientific Collaboration) Class

Quantum Grav 27 084003 (2010)[16] P Astone S Frasca and C Palomba Class Q Grav

22 S1013 (2005)[17] G Ashton D I Jones and R Prix in preparation(2014)[18] httpwwwjbmanacukpulsarcrabhtml[19] rdquoS Buchner private communicationrdquo[20] C-Y Ng and R W Romani The Astrophysical Journal

601 479 (2004)[21] C-Y Ng and R W Romani The Astrophysical Journal

673 411 (2008)[22] J Aasi and et al(2013) arXiv13040670