The everincreasing number of detections of gravitational waves GWs from compact binaries by the Advanced LIGO and Advanced Virgo detectors allows us to perform evermore sensitive tests of general relativity GR in the dynamical and strongfield regime of gravity. We perform a suite of tests of GR using the compact binary signals observed during the second half of the third observing run of those detectors. We restrict our analysis to the 15 confident signals that have false alarm rates leq 103 rm yr1. In addition to signals consistent with binary black hole BH mergers the new events include GW200115042309 a signal consistent with a neutron starBH merger. We find the residual power after subtracting the best fit waveform from the data for each event to be consistent with the detector noise. Additionally we find all the postNewtonian deformation coefficients to be consistent with the predictions from GR with an improvement by a factor of 2 in the 1PN parameter. We also find that the spininduced quadrupole moments of the binary BH constituents are consistent with those of Kerr BHs in GR. We find no evidence for dispersion of GWs nonGR modes of polarization or postmerger echoes in the events that were analyzed. We update the bound on the mass of the graviton at 90 credibility to mg leq 1.27 times 1023 mathrmeVc2. The final mass and final spin as inferred from the premerger and postmerger parts of the waveform are consistent with each other. The studies of the properties of the remnant BHs including deviations of the quasinormal mode frequencies and damping times show consistency with the predictions of GR. In addition to considering signals individually we also combine results from the catalog of GW signals to calculate more precise population constraints. We find no evidence in support of physics beyond GR.
['General Relativity and Quantum Cosmology', 'Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Theory']
Tests of General Relativity with GWTC3
{'Tests of General Relativity with GWTC-3': "R. Abbott, 1 H. Abe, 2 F. Acernese, 3, 4 K. Ackley , 5 N. Adhikari , 6 R. X. Adhikari , 1 V. K. Adkins, 7 V. B. Adya, 8 C. A GLYPH<11> eldt, 9, 10 D. Agarwal, 11 M. Agathos , 12, 13 K. Agatsuma , 14 N. Aggarwal, 15 O. D. Aguiar , 16 L. Aiello , 17 A. Ain, 18 P. Ajith , 19 T. Akutsu , 20, 21 P. F. de Alarc'on, 22 S. Albanesi, 23, 24 R. A. Alfaidi, 25 A. Allocca , 26, 4 P. A. Altin , 8 A. Amato , 27 C. Anand, 5 S. Anand, 1 A. Ananyeva, 1 S. B. Anderson , 1 W. G. Anderson , 6 M. Ando, 28, 29 T. Andrade, 30 N. Andres , 31 M. Andr'es-Carcasona , 32 T. Andri'c , 33 S. V. Angelova, 34 S. Ansoldi, 35, 36 J. M. Antelis , 37 S. Antier , 38, 39 T. Apostolatos, 40 E. Z. Appavuravther, 41, 42 S. Appert, 1 S. K. Apple, 43 K. Arai , 1 A. Araya , 44 M. C. Araya , 1 J. S. Areeda , 45 M. Ar'ene, 46 N. Aritomi , 20 N. Arnaud , 47, 48 M. Arogeti, 49 S. M. Aronson, 7 K. G. Arun , 50 H. Asada , 51 Y. Asali, 52 G. Ashton , 53 Y. Aso , 54, 55 M. Assiduo, 56, 57 S. Assis de Souza Melo, 48 S. M. Aston, 58 P. Astone , 59 F. Aubin , 57 K. AultONeal , 37 C. Austin, 7 S. Babak , 46 F. Badaracco , 60 M. K. M. Bader, 61 C. Badger, 62 S. Bae , 63 Y. Bae, 64 A. M. Baer, 65 S. Bagnasco , 24 Y. Bai, 1 J. Baird, 46 R. Bajpai , 66 T. Baka, 67 M. Ball, 68 G. Ballardin, 48 S. W. Ballmer, 69 A. Balsamo, 65 G. Baltus , 70 S. Banagiri , 15 B. Banerjee , 33 D. Bankar , 11 J. C. Barayoga, 1 C. Barbieri, 71, 72, 73 B. C. Barish, 1 D. Barker, 74 P. Barneo , 30 F. Barone , 75, 4 B. Barr , 25 L. Barsotti , 76 M. Barsuglia , 46 D. Barta , 77 J. Bartlett, 74 M. A. Barton , 25 I. Bartos, 78 S. Basak, 19 R. Bassiri , 79 A. Basti, 80, 18 M. Bawaj , 41, 81 J. C. Bayley , 25 M. Bazzan, 82, 83 B. R. Becher, 84 B. B'ecsy , 85 V. M. Bedakihale, 86 F. Beirnaert , 87 M. Bejger , 88 I. Belahcene, 47 V. Benedetto, 89 D. Beniwal, 90 M. G. Benjamin, 91 T. F. Bennett, 92 J. D. Bentley , 14 M. BenYaala, 34 S. Bera, 11 M. Berbel , 93 F. Bergamin, 9, 10 B. K. Berger , 79 S. Bernuzzi , 13 C. P. L. Berry , 25 D. Bersanetti , 94 A. Bertolini, 61 J. Betzwieser , 58 D. Beveridge , 95 R. Bhandare, 96 A. V. Bhandari, 11 U. Bhardwaj , 39, 61 R. Bhatt, 1 D. Bhattacharjee , 97 S. Bhaumik , 78 A. Bianchi, 61, 98 I. A. Bilenko, 99 G. Billingsley , 1 S. Bini, 100, 101 R. Birney, 102 O. Birnholtz , 103 S. Biscans, 1, 76 M. Bischi, 56, 57 S. Biscoveanu , 76 A. Bisht, 9, 10 B. Biswas , 11 M. Bitossi, 48, 18 M.-A. Bizouard , 38 J. K. Blackburn , 1 C. D. Blair, 95 D. G. Blair, 95 R. M. Blair, 74 F. Bobba, 104, 105 N. Bode, 9, 10 M. Boer, 38 G. Bogaert, 38 M. Boldrini, 106, 59 G. N. Bolingbroke , 90 L. D. Bonavena, 82 F. Bondu, 107 E. Bonilla , 79 R. Bonnand , 31 P. Booker, 9, 10 B. A. Boom, 61 R. Bork, 1 V. Boschi , 18 N. Bose, 108 S. Bose, 11 V. Bossilkov, 95 V. Boudart , 70 Y. Bou GLYPH<11> anais, 82, 83 A. Bozzi, 48 C. Bradaschia, 18 P. R. Brady , 6 A. Bramley, 58 A. Branch, 58 M. Branchesi , 33, 109 J. E. Brau , 68 M. Breschi , 13 T. Briant , 110 J. H. Briggs, 25 A. Brillet, 38 M. Brinkmann, 9, 10 P. Brockill, 6 A. F. Brooks , 1 J. Brooks, 48 D. D. Brown, 90 S. Brunett, 1 G. Bruno, 60 R. Bruntz , 65 J. Bryant, 14 F. Bucci, 57 T. Bulik, 111 H. J. Bulten, 61 A. Buonanno , 112, 113 K. Burtnyk, 74 R. Buscicchio , 14 D. Buskulic, 31 C. Buy , 114 R. L. Byer, 79 G. S. Cabourn Davies , 53 G. Cabras , 35, 36 R. Cabrita , 60 L. Cadonati , 49 M. Caesar, 115 G. Cagnoli , 27 C. Cahillane, 74 J. Calder'on Bustillo, 116 J. D. Callaghan, 25 T. A. Callister, 117, 118 E. Calloni, 26, 4 J. Cameron, 95 J. B. Camp, 119 M. Canepa, 120, 94 S. Canevarolo, 67 M. Cannavacciuolo, 104 K. C. Cannon , 29 H. Cao, 90 Z. Cao , 121 E. Capocasa , 46, 20 E. Capote, 69 G. Carapella, 104, 105 F. Carbognani, 48 M. Carlassara, 9, 10 J. B. Carlin , 122 M. F. Carney, 15 M. Carpinelli, 123, 124, 48 G. Carrillo, 68 G. Carullo , 80, 18 T. L. Carver, 17 J. Casanueva Diaz, 48 C. Casentini, 125, 126 G. Castaldi, 127 S. Caudill, 61, 67 M. Cavagli'a , 97 F. Cavalier , 47 R. Cavalieri , 48 G. Cella , 18 P. Cerd'a-Dur'an, 128 E. Cesarini , 126 W. Chaibi, 38 S. Chalathadka Subrahmanya , 129 E. Champion , 130 C.-H. Chan, 131 C. Chan, 29 C. L. Chan , 132 K. Chan, 132 M. Chan, 133 K. Chandra, 108 I. P. Chang, 131 P. Chanial , 48 S. Chao, 131 C. Chapman-Bird , 25 P. Charlton , 134 E. A. Chase , 15 E. Chassande-Mottin , 46 C. Chatterjee , 95 Debarati Chatterjee , 11 Deep Chatterjee, 6 M. Chaturvedi, 96 S. Chaty , 46 K. Chatziioannou , 1 C. Chen , 135, 136 D. Chen , 54 H. Y. Chen , 76 J. Chen, 131 K. Chen, 137 X. Chen, 95 Y.-B. Chen, 138 Y.-R. Chen, 139 Z. Chen, 17 H. Cheng, 78 C. K. Cheong, 132 H. Y. Cheung, 132 H. Y. Chia, 78 F. Chiadini , 140, 105 C-Y. Chiang, 141 G. Chiarini, 83 R. Chierici, 142 A. Chincarini , 94 M. L. Chiofalo, 80, 18 A. Chiummo , 48 R. K. Choudhary, 95 S. Choudhary , 11 N. Christensen , 38 Q. Chu, 95 Y-K. Chu, 141 S. S. Y. Chua , 8 K. W. Chung, 62 G. Ciani , 82, 83 P. Ciecielag, 88 M. Cie'slar , 88 M. Cifaldi, 125, 126 A. A. Ciobanu, 90 R. Ciolfi , 143, 83 F. Cipriano, 38 F. Clara, 74 J. A. Clark , 1, 49 P. Clearwater, 144 S. Clesse, 145 F. Cleva, 38 E. Coccia, 33, 109 E. Codazzo , 33 P.-F. Cohadon , 110 D. E. Cohen , 47 M. Colleoni , 22 C. G. Collette, 146 A. Colombo , 71, 72 M. Colpi, 71, 72 C. M. Compton, 74 M. Constancio Jr., 16 L. Conti , 83 S. J. Cooper, 14 P. Corban, 58 T. R. Corbitt , 7 I. Cordero-Carri'on , 147 S. Corezzi, 81, 41 K. R. Corley, 52 N. J. Cornish , 85 D. Corre, 47 A. Corsi, 148 S. Cortese , 48 C. A. Costa, 16 R. Cotesta, 113 R. Cottingham, 58 M. W. Coughlin , 149 J.-P. Coulon, 38 S. T. Countryman, 52 B. Cousins , 150 P. Couvares , 1 D. M. Coward, 95 M. J. Cowart, 58 D. C. Coyne , 1 R. Coyne , 151 J. D. E. Creighton , 6 T. D. Creighton, 91 A. W. Criswell , 149 M. Croquette , 110 S. G. Crowder, 152 J. R. Cudell , 70 T. J. Cullen, 7 A. Cumming, 25 R. Cummings , 25 L. Cunningham, 25 E. Cuoco, 48, 153, 18 M. Curyło, 111 P. Dabadie, 27 T. Dal Canton , 47 S. Dall'Osso , 33 G. D'alya , 87, 154 A. Dana, 79 B. D'Angelo , 120, 94 S. Danilishin , 155, 61 S. D'Antonio, 126 K. Danzmann, 9, 10 C. Darsow-Fromm , 129 A. Dasgupta, 86 L. E. H. Datrier, 25 Sayak Datta, 11 Sayantani Datta , 50 V. Dattilo, 48 I. Dave, 96 M. Davier, 47 D. Davis , 1 M. C. Davis , 115 E. J. Daw , 156 R. Dean, 115 D. DeBra, 79 M. Deenadayalan, 11 J. Degallaix , 157 M. De Laurentis, 26, 4 S. Del'eglise , 110 V. Del Favero, 130 F. De Lillo , 60 N. De Lillo, 25 D. Dell'Aquila , 123 W. Del Pozzo, 80, 18 L. M. DeMarchi, 15 F. De Matteis, 125, 126 V. D'Emilio, 17 N. Demos, 76 T. Dent , 116 A. Depasse , 60 R. De Pietri , 158, 159 R. De Rosa , 26, 4 C. De Rossi, 48 R. DeSalvo , 127, 160 R. De Simone, 140 S. Dhurandhar, 11 M. C. D'ıaz , 91 N. A. Didio, 69 T. Dietrich , 113 L. Di Fiore, 4 C. Di Fronzo, 14  \nC. Di Giorgio , 104, 105 F. Di Giovanni , 128 M. Di Giovanni, 33 T. Di Girolamo , 26, 4 A. Di Lieto , 80, 18 A. Di Michele , 81 B. Ding, 146 S. Di Pace , 106, 59 I. Di Palma , 106, 59 F. Di Renzo , 80, 18 A. K. Divakarla, 78 Divyajyoti , 161 A. Dmitriev , 14 Z. Doctor, 15 L. Donahue, 162 L. D'Onofrio , 26, 4 F. Donovan, 76 K. L. Dooley, 17 S. Doravari , 11 M. Drago , 106, 59 J. C. Driggers , 74 Y. Drori, 1 J.-G. Ducoin, 47 P. Dupej, 25 U. Dupletsa, 33 O. Durante, 104, 105 D. D'Urso , 123, 124 P.-A. Duverne, 47 S. E. Dwyer, 74 C. Eassa, 74 P. J. Easter, 5 M. Ebersold, 163 T. Eckhardt , 129 G. Eddolls , 25 B. Edelman , 68 T. B. Edo, 1 O. Edy , 53 A. E GLYPH<15> er , 58 S. Eguchi , 133 J. Eichholz , 8 S. S. Eikenberry, 78 M. Eisenmann, 31, 20 R. A. Eisenstein, 76 A. Ejlli , 17 E. Engelby, 45 Y. Enomoto , 28 L. Errico, 26, 4 R. C. Essick , 164 H. Estell'es, 22 D. Estevez , 165 Z. Etienne, 166 T. Etzel, 1 M. Evans , 76 T. M. Evans, 58 T. Evstafyeva, 12 B. E. Ewing, 150 F. Fabrizi , 56, 57 F. Faedi, 57 V. Fafone , 125, 126, 33 H. Fair, 69 S. Fairhurst, 17 P. C. Fan , 162 A. M. Farah , 167 S. Farinon, 94 B. Farr , 68 W. M. Farr , 117, 118 E. J. Fauchon-Jones, 17 G. Favaro , 82 M. Favata , 168 M. Fays , 70 M. Fazio, 169 J. Feicht, 1 M. M. Fejer, 79 E. Fenyvesi , 77, 170 D. L. Ferguson , 171 A. Fernandez-Galiana , 76 I. Ferrante , 80, 18 T. A. Ferreira, 16 F. Fidecaro , 80, 18 P. Figura , 111 A. Fiori , 18, 80 I. Fiori , 48 M. Fishbach , 15 R. P. Fisher, 65 R. Fittipaldi, 172, 105 V. Fiumara, 173, 105 R. Flaminio, 31, 20 E. Floden, 149 H. K. Fong, 29 J. A. Font , 128, 174 B. Fornal , 160 P. W. F. Forsyth, 8 A. Franke, 129 S. Frasca, 106, 59 F. Frasconi , 18 J. P. Freed, 37 Z. Frei , 154 A. Freise , 61, 98 O. Freitas, 175 R. Frey , 68 P. Fritschel, 76 V. V. Frolov, 58 G. G. Fronz'e , 24 Y. Fujii, 176 Y. Fujikawa, 177 Y. Fujimoto, 178 P. Fulda, 78 M. Fy GLYPH<11> e, 58 H. A. Gabbard, 25 B. U. Gadre , 113 J. R. Gair , 113 J. Gais, 132 S. Galaudage, 5 R. Gamba, 13 D. Ganapathy , 76 A. Ganguly , 11 D. Gao , 179 S. G. Gaonkar, 11 B. Garaventa , 94, 120 C. Garc'ıa N'u˜nez, 102 C. Garc'ıa-Quir'os, 22 F. Garufi , 26, 4 B. Gateley, 74 V. Gayathri, 78 G.-G. Ge , 179 G. Gemme , 94 A. Gennai , 18 J. George, 96 O. Gerberding , 129 L. Gergely , 180 P. Gewecke, 129 S. Ghonge , 49 Abhirup Ghosh , 113 Archisman Ghosh , 87 Shaon Ghosh , 168 Shrobana Ghosh, 17 Tathagata Ghosh , 11 B. Giacomazzo , 71, 72, 73 L. Giacoppo, 106, 59 J. A. Giaime , 7, 58 K. D. Giardina, 58 D. R. Gibson, 102 C. Gier, 34 M. Giesler , 181 P. Giri , 18, 80 F. Gissi, 89 S. Gkaitatzis , 18, 80 J. Glanzer, 7 A. E. Gleckl, 45 P. Godwin, 150 E. Goetz , 182 R. Goetz , 78 N. Gohlke, 9, 10 J. Golomb, 1 B. Goncharov , 33 G. Gonz'alez , 7 M. Gosselin, 48 R. Gouaty, 31 D. W. Gould, 8 S. Goyal, 19 B. Grace, 8 A. Grado , 183, 4 V. Graham, 25 M. Granata , 157 V. Granata, 104 A. Grant, 25 S. Gras, 76 P. Grassia, 1 C. Gray, 74 R. Gray , 25 G. Greco, 41 A. C. Green , 78 R. Green, 17 A. M. Gretarsson, 37 E. M. Gretarsson, 37 D. Gri GLYPH<14> th, 1 W. L. Gri GLYPH<14> ths , 17 H. L. Griggs , 49 G. Grignani, 81, 41 A. Grimaldi , 100, 101 E. Grimes, 37 S. J. Grimm, 33, 109 H. Grote , 17 S. Grunewald, 113 P. Gruning, 47 A. S. Gruson, 45 D. Guerra , 128 G. M. Guidi , 56, 57 A. R. Guimaraes, 7 G. Guix'e, 30 H. K. Gulati, 86 A. M. Gunny, 76 H.-K. Guo , 160 Y. Guo, 61 Anchal Gupta, 1 Anuradha Gupta , 184 I. M. Gupta, 150 P. Gupta, 61, 67 S. K. Gupta, 108 R. Gustafson, 185 F. Guzman , 186 S. Ha, 187 I. P. W. Hadiputrawan, 137 L. Haegel , 46 S. Haino, 141 O. Halim , 36 E. D. Hall , 76 E. Z. Hamilton, 163 G. Hammond, 25 W.-B. Han , 188 M. Haney , 163 J. Hanks, 74 C. Hanna, 150 M. D. Hannam, 17 O. Hannuksela, 67, 61 H. Hansen, 74 T. J. Hansen, 37 J. Hanson, 58 T. Harder, 38 K. Haris, 61, 67 J. Harms , 33, 109 G. M. Harry , 43 I. W. Harry , 53 D. Hartwig , 129 K. Hasegawa, 189 B. Haskell, 88 C.-J. Haster , 76 J. S. Hathaway, 130 K. Hattori, 190 K. Haughian, 25 H. Hayakawa, 191 K. Hayama, 133 F. J. Hayes, 25 J. Healy , 130 A. Heidmann , 110 A. Heidt, 9, 10 M. C. Heintze, 58 J. Heinze , 9, 10 J. Heinzel, 76 H. Heitmann , 38 F. Hellman , 192 P. Hello, 47 A. F. Helmling-Cornell , 68 G. Hemming , 48 M. Hendry , 25 I. S. Heng, 25 E. Hennes , 61 J. Hennig, 193 M. H. Hennig , 193 C. Henshaw, 49 A. G. Hernandez, 92 F. Hernandez Vivanco, 5 M. Heurs , 9, 10 A. L. Hewitt , 194 S. Higginbotham, 17 S. Hild, 155, 61 P. Hill, 34 Y. Himemoto, 195 A. S. Hines, 186 N. Hirata, 20 C. Hirose, 177 T-C. Ho, 137 S. Hochheim, 9, 10 D. Hofman, 157 J. N. Hohmann, 129 D. G. Holcomb , 115 N. A. Holland, 8 I. J. Hollows , 156 Z. J. Holmes , 90 K. Holt, 58 D. E. Holz , 167 Q. Hong, 131 J. Hough, 25 S. Hourihane, 1 E. J. Howell , 95 C. G. Hoy , 17 D. Hoyland, 14 A. Hreibi, 9, 10 B-H. Hsieh, 189 H-F. Hsieh , 196 C. Hsiung, 135 Y. Hsu, 131 H-Y. Huang , 141 P. Huang , 179 Y-C. Huang , 139 Y.-J. Huang , 141 Yiting Huang, 152 Yiwen Huang, 76 M. T. Hubner , 5 A. D. Huddart, 197 B. Hughey, 37 D. C. Y. Hui , 198 V. Hui , 31 S. Husa, 22 S. H. Huttner, 25 R. Huxford, 150 T. Huynh-Dinh, 58 S. Ide, 199 B. Idzkowski , 111 A. Iess, 125, 126 K. Inayoshi , 200 Y. Inoue, 137 P. Iosif , 201 M. Isi , 76 K. Isleif, 129 K. Ito, 202 Y. Itoh , 178, 203 B. R. Iyer , 19 V. JaberianHamedan , 95 T. Jacqmin , 110 P.-E. Jacquet , 110 S. J. Jadhav, 204 S. P. Jadhav , 11 T. Jain, 12 A. L. James , 17 A. Z. Jan , 171 K. Jani, 205 J. Janquart, 67, 61 K. Janssens , 206, 38 N. N. Janthalur, 204 P. Jaranowski , 207 D. Jariwala, 78 R. Jaume , 22 A. C. Jenkins , 62 K. Jenner, 90 C. Jeon, 208 W. Jia, 76 J. Jiang , 78 H.-B. Jin , 209, 210 G. R. Johns, 65 N. K. Johnson-McDaniel , 184 R. Johnston, 25 A. W. Jones , 95 D. I. Jones, 211 P. Jones, 14 R. Jones, 25 P. Joshi, 150 L. Ju , 95 A. Jue, 160 P. Jung , 64 K. Jung, 187 J. Junker , 9, 10 V. Juste, 165 K. Kaihotsu, 202 T. Kajita , 212 M. Kakizaki , 190 C. V. Kalaghatgi, 17, 67, 61, 213 V. Kalogera , 15 B. Kamai, 1 M. Kamiizumi , 191 N. Kanda , 178, 203 S. Kandhasamy , 11 G. Kang , 214 J. B. Kanner, 1 Y. Kao, 131 S. J. Kapadia, 19 D. P. Kapasi , 8 C. Karathanasis , 32 S. Karki, 97 R. Kashyap, 150 M. Kasprzack , 1 W. Kastaun, 9, 10 T. Kato, 189 S. Katsanevas , 48 E. Katsavounidis, 76 W. Katzman, 58 T. Kaur, 95 K. Kawabe, 74 K. Kawaguchi , 189 F. K'ef'elian, 38 D. Keitel , 22 J. S. Key , 215 S. Khadka, 79 F. Y. Khalili , 99 S. Khan , 17 T. Khanam, 148 E. A. Khazanov, 216 N. Khetan, 33, 109 M. Khursheed, 96 N. Kijbunchoo , 8 A. Kim, 15 C. Kim , 208 J. C. Kim, 217 J. Kim , 218 K. Kim , 208 W. S. Kim, 64 Y.-M. Kim , 187 C. Kimball, 15 N. Kimura, 191 M. Kinley-Hanlon , 25 R. Kirchho GLYPH<11> , 9, 10 J. S. Kissel , 74 S. Klimenko, 78 T. Klinger, 12 A. M. Knee , 182 T. D. Knowles, 166 N. Knust, 9, 10 E. Knyazev, 76 Y. Kobayashi, 178 P. Koch, 9, 10 G. Koekoek, 61, 155 K. Kohri, 219 K. Kokeyama , 220 S. Koley , 33 P. Kolitsidou , 17 M. Kolstein , 32 K. Komori, 76  \nV. Kondrashov, 1 A. K. H. Kong , 196 A. Kontos , 84 N. Koper, 9, 10 M. Korobko , 129 M. Kovalam, 95 N. Koyama, 177 D. B. Kozak, 1 C. Kozakai , 54 V. Kringel, 9, 10 N. V. Krishnendu , 9, 10 A. Kr'olak , 221, 222 G. Kuehn, 9, 10 F. Kuei, 131 P. Kuijer , 61 S. Kulkarni, 184 A. Kumar, 204 Prayush Kumar , 19 Rahul Kumar, 74 Rakesh Kumar, 86 J. Kume, 29 K. Kuns , 76 Y. Kuromiya, 202 S. Kuroyanagi , 223, 224 K. Kwak , 187 G. Lacaille, 25 P. Lagabbe, 31 D. Laghi , 114 E. Lalande, 225 M. Lalleman, 206 T. L. Lam, 132 A. Lamberts, 38, 226 M. Landry, 74 B. B. Lane, 76 R. N. Lang , 76 J. Lange, 171 B. Lantz , 79 I. La Rosa, 31 A. Lartaux-Vollard, 47 P. D. Lasky , 5 M. Laxen , 58 A. Lazzarini , 1 C. Lazzaro, 82, 83 P. Leaci , 106, 59 S. Leavey , 9, 10 S. LeBohec, 160 Y. K. Lecoeuche , 182 E. Lee, 189 H. M. Lee , 227 H. W. Lee , 217 K. Lee , 228 R. Lee , 139 I. N. Legred, 1 J. Lehmann, 9, 10 A. Lemaˆıtre, 229 M. Lenti , 57, 230 M. Leonardi , 20 E. Leonova, 39 N. Leroy , 47 N. Letendre, 31 C. Levesque, 225 Y. Levin, 5 J. N. Leviton, 185 K. Leyde, 46 A. K. Y. Li, 1 B. Li, 131 J. Li, 15 K. L. Li , 231 P. Li, 232 T. G. F. Li, 132 X. Li , 138 C-Y. Lin , 233 E. T. Lin , 196 F-K. Lin, 141 F-L. Lin , 234 H. L. Lin , 137 L. C.-C. Lin , 231 F. Linde, 213, 61 S. D. Linker, 127, 92 J. N. Linley, 25 T. B. Littenberg, 235 G. C. Liu , 135 J. Liu , 95 K. Liu, 131 X. Liu, 6 F. Llamas, 91 R. K. L. Lo , 1 T. Lo, 131 L. T. London, 39, 76 A. Longo , 236 D. Lopez, 163 M. Lopez Portilla, 67 M. Lorenzini , 125, 126 V. Loriette, 237 M. Lormand, 58 G. Losurdo , 18 T. P. Lott, 49 J. D. Lough , 9, 10 C. O. Lousto , 130 G. Lovelace, 45 J. F. Lucaccioni, 238 H. Luck, 9, 10 D. Lumaca , 125, 126 A. P. Lundgren, 53 L.-W. Luo , 141 J. E. Lynam, 65 M. Ma'arif, 137 R. Macas , 53 J. B. Machtinger, 15 M. MacInnis, 76 D. M. Macleod , 17 I. A. O. MacMillan , 1 A. Macquet, 38 I. Maga˜na Hernandez, 6 C. Magazz'u , 18 R. M. Magee , 1 R. Maggiore , 14 M. Magnozzi , 94, 120 S. Mahesh, 166 E. Majorana , 106, 59 I. Maksimovic, 237 S. Maliakal, 1 A. Malik, 96 N. Man, 38 V. Mandic , 149 V. Mangano , 106, 59 G. L. Mansell, 74, 76 M. Manske , 6 M. Mantovani , 48 M. Mapelli , 82, 83 F. Marchesoni, 42, 41, 239 D. Mar'ın Pina , 30 F. Marion, 31 Z. Mark, 138 S. M'arka , 52 Z. M'arka , 52 C. Markakis, 12 A. S. Markosyan, 79 A. Markowitz, 1 E. Maros, 1 A. Marquina, 147 S. Marsat , 46 F. Martelli, 56, 57 I. W. Martin , 25 R. M. Martin, 168 M. Martinez, 32 V. A. Martinez, 78 V. Martinez, 27 K. Martinovic, 62 D. V. Martynov, 14 E. J. Marx, 76 H. Masalehdan , 129 K. Mason, 76 E. Massera, 156 A. Masserot, 31 M. Masso-Reid , 25 S. Mastrogiovanni , 46 A. Matas, 113 M. Mateu-Lucena , 22 F. Matichard, 1, 76 M. Matiushechkina , 9, 10 N. Mavalvala , 76 J. J. McCann, 95 R. McCarthy, 74 D. E. McClelland , 8 P. K. McClincy, 150 S. McCormick, 58 L. McCuller, 76 G. I. McGhee, 25 S. C. McGuire, 58 C. McIsaac, 53 J. McIver , 182 T. McRae, 8 S. T. McWilliams, 166 D. Meacher , 6 M. Mehmet , 9, 10 A. K. Mehta, 113 Q. Meijer, 67 A. Melatos, 122 D. A. Melchor, 45 G. Mendell, 74 A. Menendez-Vazquez, 32 C. S. Menoni , 169 R. A. Mercer, 6 L. Mereni, 157 K. Merfeld, 68 E. L. Merilh, 58 J. D. Merritt, 68 M. Merzougui, 38 S. Meshkov, 1, a C. Messenger , 25 C. Messick, 76 P. M. Meyers , 122 F. Meylahn , 9, 10 A. Mhaske, 11 A. Miani , 100, 101 H. Miao, 14 I. Michaloliakos , 78 C. Michel , 157 Y. Michimura , 28 H. Middleton , 122 D. P. Mihaylov , 113 L. Milano, 26, b A. L. Miller, 60 A. Miller, 92 B. Miller, 39, 61 M. Millhouse, 122 J. C. Mills, 17 E. Milotti, 240, 36 Y. Minenkov, 126 N. Mio, 241 Ll. M. Mir, 32 M. Miravet-Ten'es , 128 A. Mishkin, 78 C. Mishra, 161 T. Mishra , 78 T. Mistry, 156 S. Mitra , 11 V. P. Mitrofanov , 99 G. Mitselmakher , 78 R. Mittleman, 76 O. Miyakawa , 191 K. Miyo , 191 S. Miyoki , 191 Geo GLYPH<11> rey Mo , 76 L. M. Moda GLYPH<11> eri , 22 E. Moguel, 238 K. Mogushi, 97 S. R. P. Mohapatra, 76 S. R. Mohite , 6 I. Molina, 45 M. Molina-Ruiz , 192 M. Mondin, 92 M. Montani, 56, 57 C. J. Moore, 14 J. Moragues, 22 D. Moraru, 74 F. Morawski, 88 A. More , 11 C. Moreno , 37 G. Moreno, 74 Y. Mori, 202 S. Morisaki , 6 N. Morisue, 178 Y. Moriwaki, 190 B. Mours , 165 C. M. Mow-Lowry , 61, 98 S. Mozzon , 53 F. Muciaccia, 106, 59 Arunava Mukherjee, 242 D. Mukherjee , 150 Soma Mukherjee, 91 Subroto Mukherjee, 86 Suvodip Mukherjee , 164, 39 N. Mukund , 9, 10 A. Mullavey, 58 J. Munch, 90 E. A. Mu˜niz , 69 P. G. Murray , 25 R. Musenich , 94, 120 S. Muusse, 90 S. L. Nadji, 9, 10 K. Nagano , 243 A. Nagar, 24, 244 K. Nakamura , 20 H. Nakano , 245 M. Nakano, 189 Y. Nakayama, 202 V. Napolano, 48 I. Nardecchia , 125, 126 T. Narikawa, 189 H. Narola, 67 L. Naticchioni , 59 B. Nayak, 92 R. K. Nayak , 246 B. F. Neil, 95 J. Neilson, 89, 105 A. Nelson, 186 T. J. N. Nelson, 58 M. Nery, 9, 10 P. Neubauer, 238 A. Neunzert, 215 K. Y. Ng, 76 S. W. S. Ng , 90 C. Nguyen , 46 P. Nguyen, 68 T. Nguyen, 76 L. Nguyen Quynh , 247 J. Ni, 149 W.-T. Ni , 209, 179, 139 S. A. Nichols, 7 T. Nishimoto, 189 A. Nishizawa , 29 S. Nissanke, 39, 61 E. Nitoglia , 142 F. Nocera, 48 M. Norman, 17 C. North, 17 S. Nozaki, 190 G. Nurbek, 91 L. K. Nuttall , 53 Y. Obayashi , 189 J. Oberling, 74 B. D. O'Brien, 78 J. O'Dell, 197 E. Oelker , 25 W. Ogaki, 189 G. Oganesyan, 33, 109 J. J. Oh , 64 K. Oh , 198 S. H. Oh , 64 M. Ohashi , 191 T. Ohashi, 178 M. Ohkawa , 177 F. Ohme , 9, 10 H. Ohta, 29 M. A. Okada, 16 Y. Okutani, 199 C. Olivetto, 48 K. Oohara , 189, 248 R. Oram, 58 B. O'Reilly , 58 R. G. Ormiston, 149 N. D. Ormsby, 65 R. O'Shaughnessy , 130 E. O'Shea , 181 S. Oshino , 191 S. Ossokine , 113 C. Osthelder, 1 S. Otabe, 2 D. J. Ottaway , 90 H. Overmier, 58 A. E. Pace, 150 G. Pagano, 80, 18 R. Pagano, 7 M. A. Page, 95 G. Pagliaroli, 33, 109 A. Pai, 108 S. A. Pai, 96 S. Pal, 246 J. R. Palamos, 68 O. Palashov, 216 C. Palomba , 59 H. Pan, 131 K.-C. Pan , 139, 196 P. K. Panda, 204 P. T. H. Pang, 61, 67 C. Pankow, 15 F. Pannarale , 106, 59 B. C. Pant, 96 F. H. Panther, 95 F. Paoletti , 18 A. Paoli, 48 A. Paolone, 59, 249 G. Pappas, 201 A. Parisi , 135 H. Park, 6 J. Park , 250 W. Parker , 58 D. Pascucci , 61, 87 A. Pasqualetti, 48 R. Passaquieti , 80, 18 D. Passuello, 18 M. Patel, 65 M. Pathak, 90 B. Patricelli , 48, 18 A. S. Patron, 7 S. Paul , 68 E. Payne, 5 M. Pedraza, 1 R. Pedurand, 105 M. Pegoraro, 83 A. Pele, 58 F. E. Pe˜na Arellano , 191 S. Penano, 79 S. Penn , 251 A. Perego, 100, 101 A. Pereira, 27 T. Pereira , 252 C. J. Perez, 74 C. P'erigois, 31 C. C. Perkins, 78 A. Perreca , 100, 101 S. Perri'es, 142 D. Pesios, 201 J. Petermann , 129 D. Petterson, 1 H. P. Pfei GLYPH<11> er , 113 H. Pham, 58 K. A. Pham , 149 K. S. Phukon , 61, 213 H. Phurailatpam, 132 O. J. Piccinni , 59 M. Pichot , 38 M. Piendibene, 80, 18 F. Piergiovanni, 56, 57 L. Pierini , 106, 59  \nV. Pierro , 89, 105 G. Pillant, 48 M. Pillas, 47 F. Pilo, 18 L. Pinard, 157 C. Pineda-Bosque, 92 I. M. Pinto, 89, 105, 253 M. Pinto, 48 B. J. Piotrzkowski, 6 K. Piotrzkowski, 60 M. Pirello, 74 M. D. Pitkin , 194 A. Placidi , 41, 81 E. Placidi, 106, 59 M. L. Planas , 22 W. Plastino , 254, 236 C. Pluchar, 255 R. Poggiani , 80, 18 E. Polini , 31 D. Y. T. Pong, 132 S. Ponrathnam, 11 E. K. Porter, 46 R. Poulton , 48 A. Poverman, 84 J. Powell, 144 M. Pracchia, 31 T. Pradier, 165 A. K. Prajapati, 86 K. Prasai, 79 R. Prasanna, 204 G. Pratten , 14 M. Principe, 89, 253, 105 G. A. Prodi , 256, 101 L. Prokhorov, 14 P. Prosposito, 125, 126 L. Prudenzi, 113 A. Puecher, 61, 67 M. Punturo , 41 F. Puosi, 18, 80 P. Puppo, 59 M. Purrer , 113 H. Qi , 17 N. Quartey, 65 V. Quetschke, 91 P. J. Quinonez, 37 R. Quitzow-James, 97 N. Qutob, 257 F. J. Raab, 74 G. Raaijmakers, 39, 61 H. Radkins, 74 N. Radulesco, 38 P. Ra GLYPH<11> ai , 154 S. X. Rail, 225 S. Raja, 96 C. Rajan, 96 K. E. Ramirez , 58 T. D. Ramirez, 45 A. Ramos-Buades , 113 J. Rana, 150 P. Rapagnani, 106, 59 A. Ray, 6 V. Raymond , 17 N. Raza , 182 M. Razzano , 80, 18 J. Read, 45 L. A. Rees, 43 T. Regimbau, 31 L. Rei , 94 S. Reid, 34 S. W. Reid, 65 D. H. Reitze, 1, 78 P. Relton , 17 A. Renzini, 1 P. Rettegno , 23, 24 B. Revenu , 46 A. Reza, 61 M. Rezac, 45 F. Ricci, 106, 59 D. Richards, 197 J. W. Richardson , 258 L. Richardson, 186 G. Riemenschneider, 23, 24 K. Riles , 185 S. Rinaldi , 80, 18 K. Rink , 182 N. A. Robertson, 1 R. Robie, 1 F. Robinet, 47 A. Rocchi , 126 S. Rodriguez, 45 L. Rolland , 31 J. G. Rollins , 1 M. Romanelli, 107 R. Romano, 3, 4 C. L. Romel, 74 A. Romero , 32 I. M. Romero-Shaw, 5 J. H. Romie, 58 S. Ronchini , 33, 109 L. Rosa, 4, 26 C. A. Rose, 6 D. Rosi'nska, 111 M. P. Ross , 259 S. Rowan, 25 S. J. Rowlinson, 14 S. Roy, 67 Santosh Roy, 11 Soumen Roy, 260 D. Rozza , 123, 124 P. Ruggi, 48 K. Ruiz-Rocha, 205 K. Ryan, 74 S. Sachdev, 150 T. Sadecki, 74 J. Sadiq , 116 S. Saha , 196 Y. Saito, 191 K. Sakai, 261 M. Sakellariadou , 62 S. Sakon, 150 O. S. Salafia , 73, 72, 71 F. Salces-Carcoba , 1 L. Salconi, 48 M. Saleem , 149 F. Salemi , 100, 101 A. Samajdar , 72 E. J. Sanchez, 1 J. H. Sanchez, 45 L. E. Sanchez, 1 N. Sanchis-Gual , 262 J. R. Sanders, 263 A. Sanuy , 30 T. R. Saravanan, 11 N. Sarin, 5 B. Sassolas, 157 H. Satari, 95 B. S. Sathyaprakash , 150, 17 O. Sauter , 78 R. L. Savage , 74 V. Savant, 11 T. Sawada , 178 H. L. Sawant, 11 S. Sayah, 157 D. Schaetzl, 1 M. Scheel, 138 J. Scheuer, 15 M. G. Schiworski , 90 P. Schmidt , 14 S. Schmidt, 67 R. Schnabel , 129 M. Schneewind, 9, 10 R. M. S. Schofield, 68 A. Schonbeck, 129 B. W. Schulte, 9, 10 B. F. Schutz, 17, 9, 10 E. Schwartz , 17 J. Scott , 25 S. M. Scott , 8 M. Seglar-Arroyo , 31 Y. Sekiguchi , 264 D. Sellers, 58 A. S. Sengupta, 260 D. Sentenac, 48 E. G. Seo, 132 V. Sequino, 26, 4 A. Sergeev, 216 Y. Setyawati , 9, 10, 67 T. Sha GLYPH<11> er, 74 M. S. Shahriar , 15 M. A. Shaikh , 19 B. Shams, 160 L. Shao , 200 A. Sharma, 33, 109 P. Sharma, 96 P. Shawhan , 112 N. S. Shcheblanov , 229 A. Sheela, 161 Y. Shikano , 265, 266 M. Shikauchi, 29 H. Shimizu , 267 K. Shimode , 191 H. Shinkai , 268 T. Shishido, 55 A. Shoda , 20 D. H. Shoemaker , 76 D. M. Shoemaker , 171 S. ShyamSundar, 96 M. Sieniawska, 60 D. Sigg , 74 L. Silenzi , 41, 42 L. P. Singer , 119 D. Singh , 150 M. K. Singh , 19 N. Singh , 111 A. Singha , 155, 61 A. M. Sintes , 22 V. Sipala, 123, 124 V. Skliris, 17 B. J. J. Slagmolen , 8 T. J. Slaven-Blair, 95 J. Smetana, 14 J. R. Smith , 45 L. Smith, 25 R. J. E. Smith , 5 J. Soldateschi , 230, 269, 57 S. N. Somala , 270 K. Somiya , 2 I. Song , 196 K. Soni , 11 V. Sordini, 142 F. Sorrentino, 94 N. Sorrentino , 80, 18 R. Soulard, 38 T. Souradeep, 271, 11 E. Sowell, 148 V. Spagnuolo, 155, 61 A. P. Spencer , 25 M. Spera , 82, 83 P. Spinicelli, 48 A. K. Srivastava, 86 V. Srivastava, 69 K. Staats, 15 C. Stachie, 38 F. Stachurski, 25 D. A. Steer , 46 J. Steinho GLYPH<11> , 113 J. Steinlechner, 155, 61 S. Steinlechner , 155, 61 N. Stergioulas, 201 D. J. Stops, 14 M. Stover, 238 K. A. Strain , 25 L. C. Strang, 122 G. Stratta , 272, 59 M. D. Strong, 7 A. Strunk, 74 R. Sturani, 252 A. L. Stuver , 115 M. Suchenek, 88 S. Sudhagar , 11 V. Sudhir , 76 R. Sugimoto , 273, 243 H. G. Suh , 6 A. G. Sullivan , 52 J. M. Sullivan, 257 T. Z. Summerscales , 274 L. Sun , 8 S. Sunil, 86 A. Sur , 88 J. Suresh , 29 P. J. Sutton , 17 Takamasa Suzuki , 177 Takanori Suzuki, 2 Toshikazu Suzuki, 189 B. L. Swinkels , 61 M. J. Szczepa'nczyk , 78 P. Szewczyk, 111 M. Tacca, 61 H. Tagoshi, 189 S. C. Tait , 25 H. Takahashi , 275 R. Takahashi , 20 S. Takano, 28 H. Takeda , 28 M. Takeda, 178 C. J. Talbot, 34 C. Talbot, 1 K. Tanaka, 276 Taiki Tanaka, 189 Takahiro Tanaka , 277 A. J. Tanasijczuk, 60 S. Tanioka , 191 D. B. Tanner, 78 D. Tao, 1 L. Tao , 78 R. D. Tapia, 150 E. N. Tapia San Mart'ın , 61 C. Taranto, 125 A. Taruya , 278 J. D. Tasson , 162 R. Tenorio , 22 J. E. S. Terhune , 115 L. Terkowski , 129 M. P. Thirugnanasambandam, 11 M. Thomas, 58 P. Thomas, 74 E. E. Thompson, 49 J. E. Thompson , 17 S. R. Thondapu, 96 K. A. Thorne, 58 E. Thrane, 5 Shubhanshu Tiwari , 163 Srishti Tiwari, 11 V. Tiwari , 17 A. M. Toivonen, 149 A. E. Tolley , 53 T. Tomaru , 20 T. Tomura , 191 M. Tonelli, 80, 18 Z. Tornasi, 25 A. Torres-Forn'e , 128 C. I. Torrie, 1 I. Tosta e Melo , 124 D. Toyra, 8 A. Trapananti , 42, 41 F. Travasso , 41, 42 G. Traylor, 58 M. Trevor, 112 M. C. Tringali , 48 A. Tripathee , 185 L. Troiano, 279, 105 A. Trovato , 46 L. Trozzo , 4, 191 R. J. Trudeau, 1 D. Tsai, 131 K. W. Tsang, 61, 280, 67 T. Tsang , 281 J-S. Tsao, 234 M. Tse, 76 R. Tso, 138 S. Tsuchida, 178 L. Tsukada, 150 D. Tsuna, 29 T. Tsutsui , 29 K. Turbang , 282, 206 M. Turconi, 38 D. Tuyenbayev , 178 A. S. Ubhi , 14 N. Uchikata , 189 T. Uchiyama , 191 R. P. Udall, 1 A. Ueda, 283 T. Uehara , 284, 285 K. Ueno , 29 G. Ueshima, 286 C. S. Unnikrishnan, 287 A. L. Urban, 7 T. Ushiba , 191 A. Utina , 155, 61 G. Vajente , 1 A. Vajpeyi, 5 G. Valdes , 186 M. Valentini , 184, 100, 101 V. Valsan, 6 N. van Bakel, 61 M. van Beuzekom , 61 M. van Dael, 61, 288 J. F. J. van den Brand , 155, 98, 61 C. Van Den Broeck, 67, 61 D. C. Vander-Hyde, 69 H. van Haevermaet , 206 J. V. van Heijningen , 60 M. H. P. M. van Putten, 289 N. van Remortel , 206 M. Vardaro, 213, 61 A. F. Vargas, 122 V. Varma , 113 M. Vas'uth , 77 A. Vecchio , 14 G. Vedovato, 83 J. Veitch , 25 P. J. Veitch , 90 J. Venneberg , 9, 10 G. Venugopalan , 1 D. Verkindt , 31 P. Verma, 222 Y. Verma , 96 S. M. Vermeulen , 17 D. Veske , 52 F. Vetrano, 56 A. Vicer'e , 56, 57 S. Vidyant, 69 A. D. Viets , 290 A. Vijaykumar , 19 V. Villa-Ortega , 116 J.-Y. Vinet, 38 A. Virtuoso, 240, 36 S. Vitale , 76 H. Vocca, 81, 41 E. R. G. von Reis, 74 J. S. A. von Wrangel, 9, 10 C. Vorvick , 74 S. P. Vyatchanin , 99 L. E. Wade, 238 M. Wade , 238 K. J. Wagner , 130 R. Wald, 167 R. C. Walet, 61 M. Walker, 65  \n32  \nG. S. Wallace, 34 L. Wallace, 1 J. Wang , 179 J. Z. Wang, 185 W. H. Wang, 91 R. L. Ward, 8 J. Warner, 74 M. Was , 31 T. Washimi , 20 N. Y. Washington, 1 J. Watchi , 146 B. Weaver, 74 C. R. Weaving, 53 S. A. Webster, 25 M. Weinert, 9, 10 A. J. Weinstein , 1 R. Weiss, 76 C. M. Weller, 259 R. A. Weller , 205 F. Wellmann, 9, 10 L. Wen, 95 P. Weßels, 9, 10 K. Wette , 8 J. T. Whelan , 130 D. D. White, 45 B. F. Whiting , 78 C. Whittle , 76 D. Wilken, 9, 10 D. Williams , 25 M. J. Williams , 25 A. R. Williamson , 53 J. L. Willis , 1 B. Willke , 9, 10 D. J. Wilson, 255 C. C. Wipf, 1 T. Wlodarczyk, 113 G. Woan , 25 J. Woehler, 9, 10 J. K. Wo GLYPH<11> ord , 130 D. Wong, 182 I. C. F. Wong , 132 M. Wright, 25 C. Wu , 139 D. S. Wu , 9, 10 H. Wu, 139 D. M. Wysocki, 6 L. Xiao , 1 T. Yamada, 267 H. Yamamoto , 1 K. Yamamoto , 190 T. Yamamoto , 191 K. Yamashita, 202 R. Yamazaki, 199 F. W. Yang , 160 K. Z. Yang , 149 L. Yang , 169 Y.-C. Yang, 131 Y. Yang , 291 Yang Yang, 78 M. J. Yap, 8 D. W. Yeeles, 17 S.-W. Yeh, 139 A. B. Yelikar , 130 M. Ying, 131 J. Yokoyama , 29, 28 T. Yokozawa, 191 J. Yoo, 181 T. Yoshioka, 202 Hang Yu , 138 Haocun Yu , 76 H. Yuzurihara, 189 A. Zadro˙zny, 222 M. Zanolin, 37 S. Zeidler , 292 T. Zelenova, 48 J.-P. Zendri, 83 M. Zevin , 167 M. Zhan, 179 H. Zhang, 234 J. Zhang , 95 L. Zhang, 1 R. Zhang , 78 T. Zhang, 14 Y. Zhang, 186 C. Zhao , 95 G. Zhao, 146 Y. Zhao , 189, 20 Yue Zhao, 160 R. Zhou, 192 Z. Zhou, 15 X. J. Zhu , 5 Z.-H. Zhu , 121, 232 A. B. Zimmerman , 171 M. E. Zucker, 1, 76 and J. Zweizig 1  \n1 LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA  \n2  \nGraduate School of Science, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan  \n3 Dipartimento di Farmacia, Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy  \n4 INFN, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy  \n5 OzGrav, School of Physics & Astronomy, Monash University, Clayton 3800, Victoria, Australia  \n6 University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA  \n7 Louisiana State University, Baton Rouge, LA 70803, USA  \n8 OzGrav, Australian National University, Canberra, Australian Capital Territory 0200, Australia  \n9 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany  \n10 Leibniz Universitat Hannover, D-30167 Hannover, Germany  \n11 Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India  \n12 University of Cambridge, Cambridge CB2 1TN, United Kingdom  \n13 Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat Jena, D-07743 Jena, Germany  \n14 University of Birmingham, Birmingham B15 2TT, United Kingdom  \n15 Northwestern University, Evanston, IL 60208, USA  \n16 Instituto Nacional de Pesquisas Espaciais, 12227-010 S˜ao Jos'e dos Campos, S˜ao Paulo, Brazil  \n17 Cardi GLYPH<11> University, Cardi GLYPH<11> CF24 3AA, United Kingdom  \n18 INFN, Sezione di Pisa, I-56127 Pisa, Italy  \n19 International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India  \n- 20 Gravitational Wave Science Project, National Astronomical Observatory of Japan (NAOJ), Mitaka City, Tokyo 181-8588, Japan  \n21 Advanced Technology Center, National Astronomical Observatory of Japan (NAOJ), Mitaka City, Tokyo 181-8588, Japan  \n22 IAC3-IEEC, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain  \n23 Dipartimento di Fisica, Universit'a degli Studi di Torino, I-10125 Torino, Italy  \n24 INFN Sezione di Torino, I-10125 Torino, Italy  \n25 SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom  \n26 Universit'a di Napoli 'Federico II', Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy  \n27 Universit'e de Lyon, Universit'e Claude Bernard Lyon 1,  \nCNRS, Institut Lumi'ere Mati'ere, F-69622 Villeurbanne, France  \n28 Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan  \n29 Research Center for the Early Universe (RESCEU),  \nThe University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan  \n30 Institut de Ci'encies del Cosmos (ICCUB), Universitat de Barcelona, C / Mart'ı i Franqu'es 1, Barcelona, 08028, Spain  \n31 Univ. Savoie Mont Blanc, CNRS, Laboratoire d'Annecy de Physique des Particules - IN2P3, F-74000 Annecy, France  \nInstitut de F'ısica d'Altes Energies (IFAE), Barcelona Institute of Science and Technology, and ICREA, E-08193 Barcelona, Spain  \n33 Gran Sasso Science Institute (GSSI), I-67100 L'Aquila, Italy  \n34 SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom  \n35 Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Universit'a di Udine, I-33100 Udine, Italy  \n36 INFN, Sezione di Trieste, I-34127 Trieste, Italy  \n37 Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA  \n38 Artemis, Universit'e Cˆote d'Azur, Observatoire de la Cˆote d'Azur, CNRS, F-06304 Nice, France  \n39 GRAPPA, Anton Pannekoek Institute for Astronomy and Institute for High-Energy Physics,  \nUniversity of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands  \n40 National and Kapodistrian University of Athens, School of Science Building, 2nd floor, Panepistimiopolis, 15771 Ilissia, Greece  \n41 INFN, Sezione di Perugia, I-06123 Perugia, Italy  \n42 Universit'a di Camerino, Dipartimento di Fisica, I-62032 Camerino, Italy  \n43 American University, Washington, D.C. 20016, USA  \n44 Earthquake Research Institute, The University of Tokyo, Bunkyo-ku, Tokyo 113-0032, Japan  \n45 California State University Fullerton, Fullerton, CA 92831, USA  \n46 Universit'e de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, France  \n47 Universit'e Paris-Saclay, CNRS / IN2P3, IJCLab, 91405 Orsay, France  \n48 European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy 49 Georgia Institute of Technology, Atlanta, GA 30332, USA 50 Chennai Mathematical Institute, Chennai 603103, India 51 Department of Mathematics and Physics, 52 Columbia University, New York, NY 10027, USA 53 University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom 54 Kamioka Branch, National Astronomical Observatory of Japan (NAOJ), Kamioka-cho, Hida City, Gifu 506-1205, Japan 55 The Graduate University for Advanced Studies (SOKENDAI), Mitaka City, Tokyo 181-8588, Japan 56 Universit'a degli Studi di Urbino 'Carlo Bo', I-61029 Urbino, Italy 57 INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy 58 LIGO Livingston Observatory, Livingston, LA 70754, USA 59 INFN, Sezione di Roma, I-00185 Roma, Italy 60 Universit'e catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium 61 Nikhef, Science Park 105, 1098 XG Amsterdam, Netherlands 62 King's College London, University of London, London WC2R 2LS, United Kingdom 63 Korea Institute of Science and Technology Information, Daejeon 34141, Republic of Korea 64 National Institute for Mathematical Sciences, Daejeon 34047, Republic of Korea 65 Christopher Newport University, Newport News, VA 23606, USA 66 School of High Energy Accelerator Science, The Graduate University for Advanced Studies (SOKENDAI), Tsukuba City, Ibaraki 305-0801, Japan 67 Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands 68 University of Oregon, Eugene, OR 97403, USA 69 Syracuse University, Syracuse, NY 13244, USA 70 Universit'e de Li'ege, B-4000 Li'ege, Belgium 71 Universit'a degli Studi di Milano-Bicocca, I-20126 Milano, Italy 72 INFN, Sezione di Milano-Bicocca, I-20126 Milano, Italy 73 INAF, Osservatorio Astronomico di Brera sede di Merate, I-23807 Merate, Lecco, Italy 74 LIGO Hanford Observatory, Richland, WA 99352, USA 75 Dipartimento di Medicina, Chirurgia e Odontoiatria 'Scuola Medica Salernitana', Universit'a di Salerno, I-84081 Baronissi, Salerno, Italy 76 LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 77 Wigner RCP, RMKI, H-1121 Budapest, Konkoly Thege Mikl'os 'ut 29-33, Hungary 78 University of Florida, Gainesville, FL 32611, USA 79 Stanford University, Stanford, CA 94305, USA 80 Universit'a di Pisa, I-56127 Pisa, Italy 81 Universit'a di Perugia, I-06123 Perugia, Italy 82 Universit'a di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova, Italy 83 INFN, Sezione di Padova, I-35131 Padova, Italy 84 Bard College, Annandale-On-Hudson, NY 12504, USA 85 Montana State University, Bozeman, MT 59717, USA 86 Institute for Plasma Research, Bhat, Gandhinagar 382428, India 87 Universiteit Gent, B-9000 Gent, Belgium 88 Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, 00-716, Warsaw, Poland 89 Dipartimento di Ingegneria, Universit'a del Sannio, I-82100 Benevento, Italy 90 OzGrav, University of Adelaide, Adelaide, South Australia 5005, Australia 91 The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA 92 California State University, Los Angeles, Los Angeles, CA 90032, USA 93 Departamento de Matem'aticas, Universitat Aut'onoma de Barcelona, Edificio C Facultad de Ciencias 08193 Bellaterra (Barcelona), Spain 94 INFN, Sezione di Genova, I-16146 Genova, Italy 95 OzGrav, University of Western Australia, Crawley, Western Australia 6009, Australia 96 RRCAT, Indore, Madhya Pradesh 452013, India 97 Missouri University of Science and Technology, Rolla, MO 65409, USA 98 Vrije Universiteit Amsterdam, 1081 HV Amsterdam, Netherlands 99 Lomonosov Moscow State University, Moscow 119991, Russia 100 Universit'a di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy 101 INFN, Trento Institute for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy 102 SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom 103 Bar-Ilan University, Ramat Gan, 5290002, Israel 104 Dipartimento di Fisica 'E.R. Caianiello', Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy 105 INFN, Sezione di Napoli, Gruppo Collegato di Salerno,  \nComplesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy  \n136  \n157  \n131  \n111  \n106 Universit'a di Roma 'La Sapienza', I-00185 Roma, Italy  \n107 Univ Rennes, CNRS, Institut FOTON - UMR6082, F-3500 Rennes, France  \n108 Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India  \n109 INFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy  \n110 Laboratoire Kastler Brossel, Sorbonne Universit'e, CNRS,  \nENS-Universit'e PSL, Coll'ege de France, F-75005 Paris, France  \nAstronomical Observatory Warsaw University, 00-478 Warsaw, Poland  \n112 University of Maryland, College Park, MD 20742, USA  \n113 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-14476 Potsdam, Germany  \n114  \nL2IT, Laboratoire des 2 Infinis - Toulouse, Universit'e de Toulouse,  \nCNRS / IN2P3, UPS, F-31062 Toulouse Cedex 9, France  \n115 Villanova University, Villanova, PA 19085, USA  \n116 IGFAE, Universidade de Santiago de Compostela, 15782 Spain  \n117 Stony Brook University, Stony Brook, NY 11794, USA  \n118 Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA  \n119  \nNASA Goddard Space Flight Center, Greenbelt, MD 20771, USA  \n120 Dipartimento di Fisica, Universit'a degli Studi di Genova, I-16146 Genova, Italy  \n121 Department of Astronomy, Beijing Normal University, Beijing 100875, China  \n122 OzGrav, University of Melbourne, Parkville, Victoria 3010, Australia  \n123 Universit'a degli Studi di Sassari, I-07100 Sassari, Italy  \n124  \nINFN, Laboratori Nazionali del Sud, I-95125 Catania, Italy  \n125 Universit'a di Roma Tor Vergata, I-00133 Roma, Italy  \n126 INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy  \n127 University of Sannio at Benevento, I-82100 Benevento,  \nItaly and INFN, Sezione di Napoli, I-80100 Napoli, Italy  \n128 Departamento de Astronom'ıa y Astrof'ısica, Universitat de Val'encia, E-46100 Burjassot, Val'encia, Spain  \n129 Universitat Hamburg, D-22761 Hamburg, Germany  \n130 Rochester Institute of Technology, Rochester, NY 14623, USA  \nNational Tsing Hua University, Hsinchu City, 30013 Taiwan, Republic of China  \n132 The Chinese University of Hong Kong, Shatin, NT, Hong Kong  \n133 Department of Applied Physics, Fukuoka University, Jonan, Fukuoka City, Fukuoka 814-0180, Japan  \n134 OzGrav, Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia  \n135  \nDepartment of Physics, Tamkang University, Danshui Dist., New Taipei City 25137, Taiwan  \nDepartment of Physics and Institute of Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan  \n137 Department of Physics, Center for High Energy and High Field Physics,  \nNational Central University, Zhongli District, Taoyuan City 32001, Taiwan  \n138 CaRT, California Institute of Technology, Pasadena, CA 91125, USA  \n139 Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan  \n140 Dipartimento di Ingegneria Industriale (DIIN), Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy  \n141 Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan  \n142 Universit'e Lyon, Universit'e Claude Bernard Lyon 1, CNRS,  \nIP2I Lyon / IN2P3, UMR 5822, F-69622 Villeurbanne, France  \n143 INAF, Osservatorio Astronomico di Padova, I-35122 Padova, Italy  \n144 OzGrav, Swinburne University of Technology, Hawthorn VIC 3122, Australia  \n145 Universit'e libre de Bruxelles, Avenue Franklin Roosevelt 50 - 1050 Bruxelles, Belgium  \n146 Universit'e Libre de Bruxelles, Brussels 1050, Belgium  \nDepartamento de Matem'aticas, Universitat de Val'encia, E-46100 Burjassot, Val'encia, Spain  \n148 Texas Tech University, Lubbock, TX 79409, USA  \n149 University of Minnesota, Minneapolis, MN 55455, USA  \n150 The Pennsylvania State University, University Park, PA 16802, USA  \n151 University of Rhode Island, Kingston, RI 02881, USA  \n152 Bellevue College, Bellevue, WA 98007, USA  \n153 Scuola Normale Superiore, Piazza dei Cavalieri, 7 - 56126 Pisa, Italy  \n154 Eotvos University, Budapest 1117, Hungary  \n155  \nMaastricht University, P.O. Box 616, 6200 MD Maastricht, Netherlands  \n156 The University of She GLYPH<14> eld, She GLYPH<14> eld S10 2TN, United Kingdom  \nUniversit'e Lyon, Universit'e Claude Bernard Lyon 1, CNRS, Laboratoire des Mat'eriaux Avanc'es (LMA),  \nIP2I Lyon / IN2P3, UMR 5822, F-69622 Villeurbanne, France  \n158 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit'a di Parma, I-43124 Parma, Italy  \n159 INFN, Sezione di Milano Bicocca, Gruppo Collegato di Parma, I-43124 Parma, Italy  \n160 The University of Utah, Salt Lake City, UT 84112, USA  \n161 Indian Institute of Technology Madras, Chennai 600036, India  \n162 Carleton College, Northfield, MN 55057, USA  \n163 University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland  \n147  \n210  \n213  \n174  \n179  \n164 Perimeter Institute, Waterloo, ON N2L 2Y5, Canada  \n165 Universit'e de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France  \n166 West Virginia University, Morgantown, WV 26506, USA  \n167 University of Chicago, Chicago, IL 60637, USA  \n168 Montclair State University, Montclair, NJ 07043, USA  \n169 Colorado State University, Fort Collins, CO 80523, USA  \n170 Institute for Nuclear Research, Bem t'er 18 / c, H-4026 Debrecen, Hungary  \n171 University of Texas, Austin, TX 78712, USA  \n172 CNR-SPIN, c / o Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy  \n173 Scuola di Ingegneria, Universit'a della Basilicata, I-85100 Potenza, Italy  \nObservatori Astron'omic, Universitat de Val'encia, E-46980 Paterna, Val'encia, Spain  \n175 Centro de F'ısica das Universidades do Minho e do Porto,  \nUniversidade do Minho, Campus de Gualtar, PT-4710 - 057 Braga, Portugal  \n176 Department of Astronomy, The University of Tokyo, Mitaka City, Tokyo 181-8588, Japan  \n177  \nFaculty of Engineering, Niigata University, Nishi-ku, Niigata City, Niigata 950-2181, Japan  \n178 Department of Physics, Graduate School of Science,  \nOsaka City University, Sumiyoshi-ku, Osaka City, Osaka 558-8585, Japan  \nState Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,  \nInnovation Academy for Precision Measurement Science and Technology (APM),  \nChinese Academy of Sciences, Xiao Hong Shan, Wuhan 430071, China  \n180 University of Szeged, D'om t'er 9, Szeged 6720, Hungary  \n181 Cornell University, Ithaca, NY 14850, USA  \n182 University of British Columbia, Vancouver, BC V6T 1Z4, Canada  \nINAF, Osservatorio Astronomico di Capodimonte, I-80131 Napoli, Italy  \n184 The University of Mississippi, University, MS 38677, USA  \n185 University of Michigan, Ann Arbor, MI 48109, USA  \n186 Texas A & M University, College Station, TX 77843, USA  \n187 Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea  \n188 Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China  \n189 Institute for Cosmic Ray Research (ICRR), KAGRA Observatory,  \nThe University of Tokyo, Kashiwa City, Chiba 277-8582, Japan  \n190 Faculty of Science, University of Toyama, Toyama City, Toyama 930-8555, Japan  \n191  \nInstitute for Cosmic Ray Research (ICRR), KAGRA Observatory,  \nThe University of Tokyo, Kamioka-cho, Hida City, Gifu 506-1205, Japan  \n192 University of California, Berkeley, CA 94720, USA  \n193 Maastricht University, 6200 MD, Maastricht, Netherlands  \n194 Lancaster University, Lancaster LA1 4YW, United Kingdom  \n195 College of Industrial Technology, Nihon University, Narashino City, Chiba 275-8575, Japan  \n196 Institute of Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan  \n197 Rutherford Appleton Laboratory, Didcot OX11 0DE, United Kingdom  \n198 Department of Astronomy & Space Science, Chungnam National University, Yuseong-gu, Daejeon 34134, Republic of Korea  \n199 Department of Physical Sciences, Aoyama Gakuin University, Sagamihara City, Kanagawa 252-5258, Japan  \n200 Kavli Institute for Astronomy and Astrophysics, Peking University, Haidian District, Beijing 100871, China  \n201 Aristotle University of Thessaloniki, University Campus, 54124 Thessaloniki, Greece  \n202 Graduate School of Science and Engineering, University of Toyama, Toyama City, Toyama 930-8555, Japan  \n203  \nNambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),  \nOsaka City University, Sumiyoshi-ku, Osaka City, Osaka 558-8585, Japan  \n204 Directorate of Construction, Services & Estate Management, Mumbai 400094, India  \n205 Vanderbilt University, Nashville, TN 37235, USA  \n206  \nUniversiteit Antwerpen, Prinsstraat 13, 2000 Antwerpen, Belgium  \n207 University of Białystok, 15-424 Białystok, Poland  \n208 Ewha Womans University, Seoul 03760, Republic of Korea  \n209 National Astronomical Observatories, Chinese Academic of Sciences, Chaoyang District, Beijing, China  \nSchool of Astronomy and Space Science, University of Chinese Academy of Sciences, Chaoyang District, Beijing, China  \n211 University of Southampton, Southampton SO17 1BJ, United Kingdom  \n212 Institute for Cosmic Ray Research (ICRR), The University of Tokyo, Kashiwa City, Chiba 277-8582, Japan  \nInstitute for High-Energy Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands  \n214 Chung-Ang University, Seoul 06974, Republic of Korea  \n215 University of Washington Bothell, Bothell, WA 98011, USA  \n216 Institute of Applied Physics, Nizhny Novgorod, 603950, Russia  \n217 Inje University Gimhae, South Gyeongsang 50834, Republic of Korea  \n218 Department of Physics, Myongji University, Yongin 17058, Republic of Korea  \n219 Institute of Particle and Nuclear Studies (IPNS), High Energy Accelerator  \nResearch Organization (KEK), Tsukuba City, Ibaraki 305-0801, Japan  \n183  \n220 School of Physics and Astronomy, Cardi GLYPH<11> University, Cardi GLYPH<11> , CF24 3AA, UK  \n221 Institute of Mathematics, Polish Academy of Sciences, 00656 Warsaw, Poland  \n222 National Center for Nuclear Research, 05-400 ' Swierk-Otwock, Poland 223 Instituto de Fisica Teorica, 28049 Madrid, Spain 224 Department of Physics, Nagoya University, Chikusa-ku, Nagoya, Aichi 464-8602, Japan 225 Universit'e de Montr'eal / Polytechnique, Montreal, Quebec H3T 1J4, Canada 226 Laboratoire Lagrange, Universit'e Cˆote d'Azur, Observatoire Cˆote d'Azur, CNRS, F-06304 Nice, France 227 Seoul National University, Seoul 08826, Republic of Korea 228 Sungkyunkwan University, Seoul 03063, Republic of Korea 229 NAVIER, ' Ecole des Ponts, Univ Gustave Ei GLYPH<11> el, CNRS, Marne-la-Vall'ee, France 230 Universit'a di Firenze, Sesto Fiorentino I-50019, Italy 231 Department of Physics, National Cheng Kung University, Tainan City 701, Taiwan 232 School of Physics and Technology, Wuhan University, Wuhan, Hubei, 430072, China 233 National Center for High-performance computing, National Applied Research Laboratories, Hsinchu Science Park, Hsinchu City 30076, Taiwan 234 Department of Physics, National Taiwan Normal University, sec. 4, Taipei 116, Taiwan 235 NASA Marshall Space Flight Center, Huntsville, AL 35811, USA 236 INFN, Sezione di Roma Tre, I-00146 Roma, Italy 237 ESPCI, CNRS, F-75005 Paris, France 238 Kenyon College, Gambier, OH 43022, USA 239 School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 240 Dipartimento di Fisica, Universit'a di Trieste, I-34127 Trieste, Italy 241 Institute for Photon Science and Technology, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan 242 Saha Institute of Nuclear Physics, Bidhannagar, West Bengal 700064, India 243 Institute of Space and Astronautical Science (JAXA), Chuo-ku, Sagamihara City, Kanagawa 252-0222, Japan 244 Institut des Hautes Etudes Scientifiques, F-91440 Bures-sur-Yvette, France 245 Faculty of Law, Ryukoku University, Fushimi-ku, Kyoto City, Kyoto 612-8577, Japan 246 Indian Institute of Science Education and Research, Kolkata, Mohanpur, West Bengal 741252, India 247 Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA 248 Graduate School of Science and Technology, Niigata University, Nishi-ku, Niigata City, Niigata 950-2181, Japan 249 Consiglio Nazionale delle Ricerche - Istituto dei Sistemi Complessi, Piazzale Aldo Moro 5, I-00185 Roma, Italy 250 Korea Astronomy and Space Science Institute (KASI), Yuseong-gu, Daejeon 34055, Republic of Korea 251 Hobart and William Smith Colleges, Geneva, NY 14456, USA 252 International Institute of Physics, Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil 253 Museo Storico della Fisica e Centro Studi e Ricerche 'Enrico Fermi', I-00184 Roma, Italy 254 Dipartimento di Matematica e Fisica, Universit'a degli Studi Roma Tre, I-00146 Roma, Italy 255 University of Arizona, Tucson, AZ 85721, USA 256 Universit'a di Trento, Dipartimento di Matematica, I-38123 Povo, Trento, Italy 257 School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA 258 University of California, Riverside, Riverside, CA 92521, USA 259 University of Washington, Seattle, WA 98195, USA 260 Indian Institute of Technology, Palaj, Gandhinagar, Gujarat 382355, India 261 Department of Electronic Control Engineering, National Institute of Technology, Nagaoka College, Nagaoka City, Niigata 940-8532, Japan 262 Departamento de Matem'atica da Universidade de Aveiro and Centre for Research and Development in Mathematics and Applications, Campus de Santiago, 3810-183 Aveiro, Portugal 263 Marquette University, Milwaukee, WI 53233, USA 264 Faculty of Science, Toho University, Funabashi City, Chiba 274-8510, Japan 265 Graduate School of Science and Technology, Gunma University, Maebashi, Gunma 371-8510, Japan 266 Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA 267 Accelerator Laboratory, High Energy Accelerator Research Organization (KEK), Tsukuba City, Ibaraki 305-0801, Japan 268 Faculty of Information Science and Technology, Osaka Institute of Technology, Hirakata City, Osaka 573-0196, Japan 269 INAF, Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy 270 Indian Institute of Technology Hyderabad, Sangareddy, Khandi, Telangana 502285, India 271 Indian Institute of Science Education and Research, Pune, Maharashtra 411008, India 272 Istituto di Astrofisica e Planetologia Spaziali di Roma, Via del Fosso del Cavaliere, 100, 00133 Roma RM, Italy 273 Department of Space and Astronautical Science, The Graduate University for Advanced Studies (SOKENDAI), Sagamihara City, Kanagawa 252-5210, Japan 274 Andrews University, Berrien Springs, MI 49104, USA 275 Research Center for Space Science, Advanced Research Laboratories,  \nTokyo City University, Setagaya, Tokyo 158-0082, Japan  \n278  \n289  \n276 Institute for Cosmic Ray Research (ICRR), Research Center for Cosmic Neutrinos (RCCN),  \nThe University of Tokyo, Kashiwa City, Chiba 277-8582, Japan  \n277 Department of Physics, Kyoto University, Sakyou-ku, Kyoto City, Kyoto 606-8502, Japan  \nYukawa Institute for Theoretical Physics (YITP), Kyoto University, Sakyou-ku, Kyoto City, Kyoto 606-8502, Japan  \n279 Dipartimento di Scienze Aziendali - Management and Innovation Systems (DISA-MIS),  \nUniversit'a di Salerno, I-84084 Fisciano, Salerno, Italy  \n280  \nVan Swinderen Institute for Particle Physics and Gravity,  \nUniversity of Groningen, Nijenborgh 4, 9747 AG Groningen, Netherlands  \n281 Faculty of Science, Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong  \n282 Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium  \n- 283 Applied Research Laboratory, High Energy Accelerator Research Organization (KEK), Tsukuba City, Ibaraki 305-0801, Japan  \n284  \nDepartment of Communications Engineering, National Defense Academy of Japan, Yokosuka City, Kanagawa 239-8686, Japan  \n285  \nDepartment of Physics, University of Florida, Gainesville, FL 32611, USA  \n286 Department of Information and Management Systems Engineering,  \nNagaoka University of Technology, Nagaoka City, Niigata 940-2188, Japan  \n287 Tata Institute of Fundamental Research, Mumbai 400005, India  \n288 Eindhoven University of Technology, Postbus 513, 5600 MB Eindhoven, Netherlands  \nDepartment of Physics and Astronomy, Sejong University, Gwangjin-gu, Seoul 143-747, Republic of Korea  \n290 Concordia University Wisconsin, Mequon, WI 53097, USA  \n291 Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan  \n292 Department of Physics, Rikkyo University, Toshima-ku, Tokyo 171-8501, Japan  \n(compiled 14 December 2021)  \nThe ever-increasing number of detections of gravitational waves from compact binaries by the Advanced LIGO and Advanced Virgo detectors allows us to perform ever-more sensitive tests of general relativity (GR) in the dynamical and strong-field regime of gravity. We perform a suite of tests of GR using the compact binary signals observed during the second half of the third observing run of those detectors. We restrict our analysis to the 15 confident signals that have false alarm rates GLYPH<20> 10 GLYPH<0> 3 yr GLYPH<0> 1 . In addition to signals consistent with binary black hole mergers, the new events include GW200115 042309, a signal consistent with a neutron star-black hole merger. We find the residual power, after subtracting the best fit waveform from the data for each event, to be consistent with the detector noise. Additionally, we find all the post-Newtonian deformation coe GLYPH<14> cients to be consistent with the predictions from GR, with an improvement by a factor of GLYPH<24> 2 in the GLYPH<0> 1PN parameter. We also find that the spin-induced quadrupole moments of the binary black hole constituents are consistent with those of Kerr black holes in GR. We find no evidence for dispersion of gravitational waves, non-GR modes of polarization, or post-merger echoes in the events that were analyzed. We update the bound on the mass of the graviton, at 90% credibility, to m g GLYPH<20> 1 : 27 GLYPH<2> 10 GLYPH<0> 23 eV = c 2 . The final mass and final spin as inferred from the pre-merger and post-merger parts of the waveform are consistent with each other. The studies of the properties of the remnant black holes, including deviations of the quasi-normal mode frequencies and damping times, show consistency with the predictions of GR. In addition to considering signals individually, we also combine results from the catalog of gravitational waves signals to calculate more precise population constraints. We find no evidence in support of physics beyond general relativity.", 'I. INTRODUCTION': "The first three observing runs of Advanced LIGO [1] and Advanced Virgo [2] have led to detections of signals consistent with coming from the three canonical classes of compact binary systems: binary black holes (BBH) [3], binary neutron stars (BNS) [4], and neutron star-black holes (NSBH) [5]. 1 These observations had, in particular, a profound impact on fundamental physics as they allowed us to probe the properties of gravity in the highly nonlinear and dynamical regime. These detections subjected Einstein's general relativity (GR),  \nwhich had passed all previous experimental tests to date with flying colors, to scrutiny in an entirely new regime. These new gravitational-wave tests [6-11] complement existing laboratory and astrophysical tests of GR [12, 13].  \nGR has a well posed initial value formulation, making it possible to calculate the two-body evolution. Despite the progress made on the analytical fronts (see for example [14-23]) and numerical fronts (see for example [24-30]), modelling compact binaries in modified gravity theories is still in its infancy. We are thus not yet able to carry out tests of GR that rely on highaccuracy waveforms in alternative theories. Instead, it is possible to devise a strategy based on the currently best-understood theory of gravity (GR) and look for possible departures from its predictions [31, 32]. This approach enables constraints to be placed on potential deviations from GR, although it has been argued that without reference to specific alternatives, it is di GLYPH<14> cult to assess the ability of these methods to detect GR violations [33].  \nGiven the major advances in the compact binary dynamics  \nmodeling using analytical (see for instance [34, 35] for reviews) and numerical relativity techniques within GR, e.g. [36-39], such null tests of GR have been successful so far. However, the null tests are also sensitive to the physics within GR that is not accounted for in the waveform model that is used (such as eccentricity, presence of exotic compact objects, etc) in addition to beyond-GR physics. For example, some of the tests performed here, which involve the dynamics of the merger remnant, may be interpreted as tests of GR or as a test of the black hole nature of the remnant. However, for uniformity, they will be referred to as tests of GR in this paper. The tests are not completely independent of each other and vary in sensitivity to di GLYPH<11> erent types of deviations [40], and by considering them together we obtain the best overview of how well predictions and observations agree.  \nTheoretically, gravitational waves in a modified theory of gravity may di GLYPH<11> er from GR broadly in three di GLYPH<11> erent ways: generation, propagation, and polarization. Gravitational-wave generation relates the outgoing radiation to the properties of the source, a hard problem even in GR. Propagation of gravitational waves in a modified theory of gravity can di GLYPH<11> er from that in GR via e GLYPH<11> ects such as dispersion [41], birefringence [42], and amplitude damping [43, 44]. An e GLYPH<11> ective-field theoretic approach to the problem can be found in [45-48]. Tests based on propagation e GLYPH<11> ects target those modified theories which predict generation of gravitational waves to be very close to that of GR, but di GLYPH<11> er in the way the waves propogate. This can happen in theories like massive graviton theories where the generation e GLYPH<11> ects are suppressed by powers of r =GLYPH<21>g GLYPH<28> 1 where GLYPH<21>g is the Compton wavelength of the graviton and r the size the binary [49].  \nA general metric theory of gravity can allow up to six modes of polarization: two tensor, two vector, and two scalar modes [50, 51]. In GR, one only has the two tensor modes referred to as plus and cross . Hence, searching for non-GR modes of polarization is also an e GLYPH<11> ective method to search for violation of GR. Nevertheless, there are alternative theories which predict modified GW generation or propagation but do not predict additional non-GR modes of polarization, e.g. [52, 53]. Hence, confidently detecting signatures of one or more of these three e GLYPH<11> ects would strongly suggest a possible GR violation.  \nWhile these theoretical insights get reflected in our analysis strategies while searching for possible departures from GR, the organization of this paper instead classifies the tests of GR as consistency tests and parameterized tests . Consistency tests, as the name indicates, search for possible violations of GR by asking how consistent the observed signal is with that of GR and do not invoke any parametrization of the deviation from GR. Consistency tests may be tests of self-consistency of the signal or overall consistency of the signal with the data . We consider one of each type of test: The inspiral-merger-ringdown (IMR) consistency test checks for consistency between the low- and high-frequency parts of the signal, while the residuals test subtracts the best-fit GR waveform from the data and asks whether there is any statistically significant residual power. Parameterized tests, on the other hand, invoke a specific parametrization that is appropriate for searching for possible  \ndeviations from GR in terms of certain physical e GLYPH<11> ects. For example, the parameterized tests of post-Newtonian (PN) coe GLYPH<14> cients are sensitive to the physical e GLYPH<11> ects that appear at di GLYPH<11> erent PN orders. Similarly, by parameterizing the dispersion relation, one can look for possible imprints of non-GR propagation e GLYPH<11> ects in the gravitational waveforms. A variant of the null-stream based method allows us to probe non-GR modes of gravitational-wave polarization.  \nThe detected coalescences of massive compact objects may involve not only black holes of classical GR, but also di GLYPH<11> erent compact objects described by exotic physics, commonly referred to as exotic compact objects (ECOs). They include objects like firewalls [54], fuzzballs [55], gravastars [56], boson stars [57], AdS black bubbles [58] and dark matter stars [59]. What these objects have in common is the absence of a horizon, causing ingoing gravitational waves (e.g., resulting from merger) to reflect multiple times o GLYPH<11> e GLYPH<11> ective radial potential barriers, with wave packets leaking out to infinity, potentially, at regular times; these are called echoes [60-62].  \nWe also perform a series of tests that search for possible GR violation or non-Kerr nature of the merger remnant, specifically in the postinspiral part of the waveform. Ringdown tests [63-66] probe the consistency of the post-merger dynamics with the predictions for Kerr black holes in GR, while searches for echoes constrain the presence of repeating ringdown signals [60, 61, 67-70] expected in certain classes of exotic compact objects (ECOs).  \nTests of GR performed on the data of the previous observing runs have set increasingly stringent limits [10, 11]. The bounds on the parameterized post-Newtonian deformation coe GLYPH<14> cients have been used to constrain the parameter space of alternatives to GR [71, 72] relying on several assumptions about the underlying GR violation (critiqued in [40, 73]). The joint detection of gravitational waves and the gamma rays from the binary neutron star merger GW170817 has placed extremely stringent bounds on the speed of gravitational waves [74], which in turn has had a profound impact on constraining certain classes of modified theories from a cosmological standpoint [75-78]. This joint detection also has been used to constrain the number of space-time dimensions [9, 79]. Results on the measurement of the properties of the merger remnant from ringdown, bounds on spin-induced quadrupole moment of compact binary constituents and the outcome of the search for echoes have implications for models of black hole mimickers [80].  \nThis paper analyzes events reported during the second half of the third observing run of LIGO and Virgo (O3b) [81], extending our previous analysis [11] which reported the status of the bounds up to and including the first half of the third observing run (O3a). This paper also provides joint bounds combining the events that occurred during the first three observing runs whenever possible, in addition to deriving limits on possible departures from GR for individual events.  \nThe analysis methods used here are largely similar to those used in our O3a analysis [11], and there are only significant di GLYPH<11> erences for the polarization-based test of GR and the search for post-merger echoes. The polarization test was updated to probe mixed polarization content using a framework based upon the null stream. [82]. The morphology-dependent search  \nTABLE I. Summary of methods and results. This table summarizes the names of the tests performed, the corresponding sections, the parameters involved in the test, and the improvement with regard to our previous analysis. The analyses performed are: RT = residuals test; IMR = inspiral-merger-ringdown consistency test; PAR = parametrized tests of GW generation; SIM = spin-induced moments; MDR = modified GWdispersion relation; POL = polarization content; RD = ringdown; ECH = echoes searches. The last column provides the approximate improvement in the bounds over the previous analyses reported in [11]. This is defined as X GWTC GLYPH<0> 2 = X GWTC GLYPH<0> 3, where X denotes the width of the 90% credible interval for the parameters for each test, using the combined results on all events considered. For the MDR test, some of the bounds have worsened in comparison to GWTC-2. See the corresponding section for details. Note that the high improvement factor for p SEOB is due to the larger number of events from GWTC-2 analysed here compared to [11].  \nfor echoes is replaced by a wavelet-based [83] morphologyindependent search [84, 85]. Besides these changes, the waveform models employed in most analyses have been upgraded to more accurate and complete ones, accounting for more physics, the details of which are discussed in Sec. III. Table I summarizes the tests that are performed, the quantities that are used for the test, the fractional changes with regard to the previous analyses, and the section where details about each test can be found.  \nThe paper is organized as follows. Section II discusses the data used in this paper while Sec. III describes the method of extracting astrophysical information about events from the data. Section IV discusses two tests of consistency with GR: examining the residuals left in the data after subtracting the best-fit GR waveform (Sec. IV A) and looking at consistency of the inspiral and postinspiral portions of the waveform with GR (Sec. IV B). Section V discusses two tests of gravitational wave generation, the parameterized tests of GR (Sec. V A) and the test for BBH nature using the spin-induced quadrupole moment (Sec.V B). Section VI discusses tests of gravitational wave propagation looking for non-GR dispersion of gravitational waves. Section VII reports results from the searches for non-GR polarization. Section VIII discusses various tests using the merger remnants, specifically two analyses of ringdown (Sec. VIII A) and a search for the signatures of echoes (Sec. VIII B). Section IX discusses the conclusions.", 'II. DATA, EVENTS, AND SIGNIFICANCE': "The global network of gravitational wave detectors completed their third observing run in March 2020. O3b adds 35 candidate events with probability of being of astrophysical origin better than 0.5, including the first confident observations of neutron star-black hole systems [5, 81]. The analyses presented here are focused on events from O3b, though the joint  \nbounds that are reported also include events from previous observing runs. Following our O3a analysis [11], we consider only those events with false alarm rates lower than 10 GLYPH<0> 3 per year that were confidently observed in two or more detectors as determined by any search pipeline used in the catalog of O3b events [81]. Of the 14 binary black hole mergers and the one neutron star-black hole merger (GW200115 042309) that pass this threshold, nine events are observed with three detectors and six are observed with only two detectors. The median total masses in the detector frame of these analysed events range from GLYPH<24> 8-140 M GLYPH<12> :  \nThe LIGO interferometers maintained sensitivities comparable to that in O3a [86], and the Virgo interferometer achieved a GLYPH<24> 20% improvement during O3b. As with previous results, noise subtraction methods [83, 87-89] were applied to selected events in order to improve parameter estimation. The third gravitational wave transient catalog (GWTC-3) [81] includes details on instrument performance. Three events analysed here were identified in as requiring additional data quality mitigation. One, GW191109 010717 had data quality issues in both LIGO detectors, while the other two (GW200115 042309 and GW200129 065458) were each only a GLYPH<11> ected by noise transients in one interferometer. Details on the noise transient removal techniques can be found in GWTC-3's Table XIV. Appendix A below discusses how these data quality issues can a GLYPH<11> ect analyses in this article.  \nTable II shows selected source properties of events from the O3b observing run that are included in this paper. Similar tables in O2 [10] and O3a [11] analyses provide selected source properties for the other events included in the analyses presented here. Every test detailed in the following sections has its own sub-selection criteria of events from this table based on the physics it explores; these criteria are detailed in the respective sections for each analysis.  \nDetection significance is given by four search pipelines, three of which rely on GR-based templates (P y CBC [90-  \n92], MBTA [93], and G st LAL [94, 95]) and one that does not ( coherent W ave B urst ; c WB [96-98]). Details on these pipelines, including the two di GLYPH<11> erent P y CBC configurations used, is included in Appendix D of GWTC-3 [81]. While making significance determinations using searches based on GR could potentially lead to selection biases disfavoring events that deviate significantly from GR, the use of the minimally modeled search c WB helps alleviate this concern: c WB is sensitive to at least some of the potential chirp-like signals that deviate enough from GR that they would not be detected with high significance by the other three pipelines. However, we are unable to fully exclude the possibility of a hidden population of signals that show significant departures from GR thus evading detection by all of our pipelines. Of the O3b events which satisfy our thresholds, only one (GW200225 060421) was detected with more significance by the c WB pipeline, though the di GLYPH<11> erence in detection significance between the various pipelines for this event is negligible.", 'III. PARAMETER INFERENCE': "Many of the tests performed here build upon two BBH waveform families. Consistency tests, as well as tests of gravitational-wave generation and propagation are based on precessing phenomenological ( Phenom ) models. Phenomenological models are currently constructed following the twistingup technique [100-102], where the signal in a co-precessing frame is approximated by an aligned-spin model and is then transformed to the detector's inertial frame by means of frequency (or time) dependent transfer functions. Previous works made use of the frequency-domain P henom P family, which includes P henom P v 2 and P henom P v 3HM [103], where version numbers distinguish between di GLYPH<11> erent PN-based approximations of the precession dynamics, with v2 (v3) adopting a single (double) spin prescription. The su GLYPH<14> x 'HM' is appended to indicate that the model includes some higherorder multipole moments. We will employ these models only for spin-induced quadrupole moment tests and otherwise upgrade to the most recent frequency-domain P henom X family [104, 105], which includes IMRP henom XP and IMRP he -nom XPHM [106]. While the precession prescriptions adopted by P henom P v 3HM and IMRP henom XPHM are similar, the aligned-spin baseline of P henom X, including the higher-order multipoles (2 ; 1) ; (3 ; 3) ; (3 ; 2) and (4 ; 4), is calibrated to a larger set of numerical simulations with respect to its predecessors.  \nFor parameterized tests of PN theory (Sec. V A), as well as for the ringdown test of Sec. VIII A 2, we also employ the spinning e GLYPH<11> ective one-body family (SEOB) [107-110]. In the EOB framework, the two-body dynamics is mapped onto that of an e GLYPH<11> ective body moving in a deformed black-hole spacetime, the deformation being the symmetric mass ratio. The dynamics and gravitational radiation are obtained solving numerically the Hamilton equations, which are built from the EOB Hamiltonian and radiation-reaction force. The latter combines information from PN theory, gravitational self-force, perturbation theory, and NR simulations [34, 111-114]. In this work, we will employ approximants belonging to the latest generation of  \nSEOB models, SEOBNR v 4 [115], specifically the aligned-spin model SEOBNR v 4HM [116] and the frequency-domain model SEOBNR v 4 ROM, a reduced-order-model version of SEOBNR v 4, which allows a significant speed up with respect to the original time-domain version [115, 117]. On top of the leading (2 ; 2) multipole, the SEOBNR v 4HM model also includes the spherical harmonic multipoles (2 ; 1) ; (3 ; 3) ; (4 ; 4) ; (5 ; 5).  \nFor most tests, our choice of waveform model is dictated by the need for computational e GLYPH<14> ciency. However, for some tests, this is dictated by technical and physical considerations. For instance, we use time-domain waveform models for the ringdown tests, since such tests are most conveniently formulated in the time domain [118-120].  \nMost template-based tests presented here rely on the assumption that the signal was generated in a quasi-circular BBH coalescence. It has been shown that signatures of eccentricity may lead to biases in the estimated source parameters [121-124]. The issue could especially a GLYPH<11> ect the analysis of short-duration signals such as GW190521, where the complementarity between inspiral and merger-ringdown cannot be e GLYPH<14> ciently leveraged to break degeneracies in current templates [125-129].  \nComprehensive BBH waveform models including precession, higher-order multipole moments and eccentricity are not yet available, and a thorough assessment of the impact of waveform systematics is beyond the scope of this analysis. Work in this direction is crucial, as inaccuracies in waveform templates could lead to detecting false violations of GR [130]. As for higher-order multipole moments, their relative contribution can be estimated by computing their SNR distribution on a reference set of posterior samples [5, 131-133]. Based on the IMRP henom XPHM parameter estimation runs presented in GWTC-3 [81], we find that the SNR distributions for individual higher-order multipoles is consistent with Gaussian noise for all the events reported in Table II, though the sum of several subdominant harmonics does contribute a non-negligible amount of SNR for GW191109 010717. Therefore, we choose to neglect them in the most computationally expensive tests, restricting their inclusion to the residuals, IMR consistency, and ringdown tests.  \nOur event list includes the NSBH candidate GW200115 042309. We do not consider matter e GLYPH<11> ects in its analysis and expect them to be negligible: tidal imprints are suppressed by the relatively extreme mass ratio of the system and a higher SNR would be needed for them to be confidently detected [5, 134, 135].  \nWe perform Bayesian parameter estimation using either the python based package bilby [136, 137] or the software library LALInference [138], which belongs to the LIGO Scientific Collaboration Algorithm Library Suite [139]. Runs performed with bilby are automated through the package bilby pipe [137] and rely on static nested sampling [140] as available through the package dynesty [141], while LALInference runs use either nested or Markov-chain Monte Carlo sampling [142]. One of the ringdown tests presented here makes use of pyRing [143], which relies on cpnest [144]. Power spectral densities are generated through the software BAYESWAVE [83, 145] and match those used in GWTC-3  \nTABLE II. List of O3b events considered in this paper. The first block of columns gives the names of the events and lists the instruments (LIGO H anford, LIGO L ivingston, V irgo) involved in each detection, as well as some relevant properties obtained assuming GR: luminosity distance D L, redshifted total mass (1 + z ) M , redshifted chirp mass (1 + z ) M , redshifted final mass (1 + z ) M f, dimensionless final spin GLYPH<31> f = c j ~ S f j = ( GM 2 f ), and network signal-to-noise ratio SNR. Reported quantities correspond to the median and 90% symmetric credible intervals, as computed in Table IV in GWTC-3 [81]. The final mass and final spin quantities are inferred from analysis of the entire signal and are for the remnant long after the coalescence and ringdown are complete, as described in [99]. The last block of columns indicates which analyses are performed on a given event according to the selection criteria in Sec. II: RT = residuals test (Sec. IV A); IMR = inspiral-merger-ringdown consistency test (Sec. IV B); PAR = parametrized tests of GW generation (Sec. V A); SIM = spin-induced moments (Sec. V B); MDR = modified GW dispersion relation (Sec. VI); POL = polarization content (Sec. VII); RD = ringdown (Sec. VIII A); ECH = echoes searches (Sec. VIII B).", '[81], unless otherwise stated.': 'When combining results obtained from multiple events, we employ two methods. The first method relies on the multiplication of the individual likelihoods corresponding to the deformation parameters that are inferred from the data. This method assumes that the deformation parameters take the same value across events [146]. This is a restrictive assumption for all the tests we consider except for the modified dispersion test. In order to address this, wherever possible, we combine the information from the tests for di GLYPH<11> erent events hierarchically using a model that does not make this restrictive assumption and hence provides bounds which are qualitatively more robust [147].  \nWe quantify the agreement of our results with GR using several statistical indicators. In Section IV B we present GR quantiles Q 2D GR for joint distributions, which denote the fraction of the likelihood contained within the isoprobability contour passing through the GR value, with a smaller GR quantile indicating a better agreement with GR. In Sections V A and VI, we report instead quantiles on one-dimensional distributions Q GR. When error bars are reported, they denote 90% confidence intervals and, likewise, we show 90% credible regions (intervals) when presenting joint (individual) posterior distributions.', 'A. Residuals test': 'Measuring the remnant coherent power in the network data after the subtraction of the best-fit GR template can be used to quantify consistency of GR waveform model with the data. The random noise in di GLYPH<11> erent detectors can be taken to be incoherent. The presence of consistent noise in the network after removing the GW signal from the data indicates an inconsistency between the signal present in the data and the GR template used. The residual analysis is designed to detect such discrepancies of the data with GR [6, 10, 148, 149].  \nA residual data set is obtained by subtracting the waveform corresponding to maximum likelihood parameters from individual detector data with a window size of one second around the trigger time. The window size of one second is used due to the relatively short length of the signal. Then the residual SNR, or SNR90, is computed as the 90% credible upper limit on the remnant coherent network SNR in the residual data using the BAYESWAVE pipeline [83, 145, 150]. BAYESWAVE uses a template-independent model to characterize any excess power in the residual compared to the detector noise.  \nWe follow the method used in the previous analyses performed in GWTC-1 [10] and GWTC-2 [11]. However, we use the new phenomenological waveform model PhenomX-  \nPHM [104, 105] as the GR waveform model. For each gravitational wave event, in addition to calculating SNR90, additional BAYESWAVE runs are done on two hundred randomly selected time segments on a time window of 4096s symmetric around the event time. This allows us to calculate p -values of residual SNRs for individual events, which is equal to the probability of obtaining a background value of SNR90 higher than that of the event. We perform the analysis on all the events listed in Table II.  \nThe results from the residual analysis are summarized in Table III. For each event, we have presented the SNR of the best-fit waveform SNRGR, SNR90, fitting factor FF90 = SNRGR = (SNR 2 90 + SNR 2 GR ) 1 = 2 , and p -values calculated from the background analysis. To analyze the trends between SNR90 and SNRGR, in Fig. 1 we present the scatter of SNR90 and SNRGR. The absence of correlation between SNR90 and SNRGR in the figure indicates that data is consistent with GR templates and the values of SNR90 depend purely on the noise levels in the detectors at the detection of individual events. GW191222 033537 shows the highest p -value = 1 : 0 with SNR90 = 4 : 87 and FF90 = 0 : 93. Even though, GW200219 094415 has the lowest fitting factor FF90 = 0 : 74 with SNR90 = 10 : 23, its p -value = 0 : 1 is slightly above the lowest p -value = 0 : 05 which corresponds to the event GW200225 060421.  \nIf the left-over coherent network SNR were purely from detector noise, we should expect the SNR90 p -values to be uniformly distributed within [0 ; 1]. To demonstrate the consistency of the observed p -values with the noise (null) hypothesis, in Fig. 2, we present a probability-probability (PP) plot of the p -values 2 . To produce the PP plot, we have considered all the events in GWTC-3 that pass the FAR threshold. The measurement of p -values is subjected to uncertainty due to the finite size of background runs. If N is the total number of background trials around an event, and if n of them produce SNR90 greater than that of the event, then the likelihood of the estimated p -value ˆ p = n = N is a binomial function,  \nL ( ˆ p ) = N n ! p n (1 GLYPH<0> p ) N GLYPH<0> n ; (1)  \nwhere p is the true p -value [11]. Assuming uniform prior, we can obtain posterior distribution of p -value as a Beta distribution,  \nP( p j N ; n ) = Beta( n + 1 ; N GLYPH<0> n + 1) : (2)  \nIn Fig. 2, the light-blue band around the PP curve represents the 90% uncertainty region of the p -value posteriors. The diagonal dashed line denotes the prior hypothesis with surrounding light-gray band representing 90% uncertainty region of the null hypothesis due to the finite number of events [151, 152].  \nThe PP plot is well with in the 90% credible region of the null hypothesis indicating no significant deviation in the residual data from the expected incoherent noise distribution in the individual instruments.  \nFIG. 1. Results of the residuals analysis (Sec. IV A). Scatter plot of the maximum-likelihood template (SNRGR) and the upper limit on the residual network SNR (SNR90) for each event. The colorbar denotes the p -values of individual events. Solid (empty) circles represent the O3b (pre-O3b) events. The O3b events with highest (lowest) p -values are highlighted by green (purple) diamonds.  \n<!-- image -->  \nFIG. 2. Results of the residuals analysis (Sec. IV A). The blue curve shows the fraction of events with p -values of the residual SNR less than or equal to the abscissa (PP plot). The light-blue band represents the 90% credible interval of the observed p -values. The diagonal dashed line denotes the null hypothesis with the surrounding lightgrey area denoting the 90% uncertainty region of the null hypothesis due to the finite number of events.  \n<!-- image -->', 'B. Inspiral-merger-ringdown consistency test': 'The IMR consistency test checks the consistency of the mass and spin of the remnant BH inferred from the low- and highfrequency parts of the signal. To achieve this, we divide the GWsignal into two parts in the frequency domain at the cuto GLYPH<11> frequency f IMR c which is the dominant mode GW frequency  \nTABLE III. Results of the residuals analysis (Sec. IV A). For individual events we list the SNR of the best-fit waveform (SNRGR), 90% credible upper limit on the remnant coherent network SNR (SNR90), fitting factor FF90, and p -values calculate from the background analysis.  \nof the innermost stable circular orbit (ISCO) of the remnant Kerr BH [153, 154]. The mass and spin of the remnant BH are estimated by applying NR-calibrated fits [99, 155-158] to the median values of the redshifted component masses, dimensionless spins, and spin angles obtained using the full IMR signal and the waveform model IMRP henom XPHM. The lowand high-frequency regimes roughly correspond to the inspiral and postinspiral, respectively, of the dominant mode of the waveform. To make sure that the two regimes of the signal have enough information, we calculate the SNR of the inspiral and the postinspiral parts of the waveform for each event using their maximum a posteriori parameter values obtained from the full IMR signal.  \nWe analyze only those signals which have SNRs greater than 6 in both the inspiral and the postinspiral parts. This constraint was also used in previous studies [10, 11]. We also impose an extra mass constraint (1 + z ) M < 100 M GLYPH<12> as in our previous analysis of GWTC-2 events [11] to ensure enough inspiral signal for heavier BBHs. The SNRs for the inspiral and the postinspiral regimes of the events analyzed are given in Table IV.  \nWe independently estimate the posterior distributions of the mass M f and the dimensionless spin GLYPH<31> f of the remnant BH from both the inspiral and the postinspiral parts of the signal. To constrain possible deviations from GR, two fractional deviation parameters GLYPH<1> M f = ¯ M f and GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f are defined, where  \nGLYPH<1> M f ¯ M f = 2 M insp f GLYPH<0> M postinsp f M insp f + M postinsp f ; GLYPH<1> GLYPH<31> f ¯ GLYPH<31> f = 2 GLYPH<31> insp f GLYPH<0> GLYPH<31> postinsp f GLYPH<31> insp f + GLYPH<31> postinsp f ; (3)  \nand ¯ M f and ¯ GLYPH<31> f denote the mean values of final mass and final spin obtained from analyzing the inspiral and postinspiral parts of the signal, respectively. Here the superscripts denote the inspiral (insp) and the postinspiral (postinsp) portions of the signal. The 2D posterior distribution of these fractional de-  \nFIG. 3. Combined results of the IMR consistency test for BBH events which satisfy the selection criteria (see Table IV and Appendix B). The combined bounds are obtained assuming the same deviation for all events. The main panel shows the 90% credible regions of the 2D posteriors on ( GLYPH<1> M f = ¯ M f ; GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f ) assuming a uniform prior, with (0 ; 0) being the expected value for GR. The side panels show the marginalized posterior on GLYPH<1> M f = ¯ M f and GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f . The gray distributions correspond to posteriors obtained by combining individual results. The other colored traces correspond to the O3b events given in Table IV where the color encodes the median redshifted total mass.  \n<!-- image -->  \nTABLE IV. Results from the IMR consistency test (Sec. IV B). f IMR c denotes the cuto GLYPH<11> frequency between the inspiral and postinspiral regimes; GLYPH<26> IMR, GLYPH<26> insp, and GLYPH<26> postinsp are the SNR in the full signal, the inspiral part, and the postinspiral part respectively; and the GR quantile Q 2D GR denotes the fraction of the reweighted posterior enclosed by the isoprobability contour that passes through the GR value, with smaller values indicating better consistency with GR. The results are given only for O3b events which satisfy the selection criteria. See Appendix B for the updated results on GWTC-2 events.  \nation parameters should peak around (0 ; 0) when the test is applied to a signal from a quasi-circular BBH coalescence in GR, given that we use a waveform model for such signals to analyze the data.  \nThe parameter estimation runs employed the IMRP henom XPHM waveform with uniform priors on the redshifted com-  \nFIG. 4. Distributions on the remnant mass (blue) and spin (red) fractional deviation parameters obtained by hierarchically combining the GWTC-3 events (solid trace). For comparison, we also show the results obtained using GWTC-2 (dot dashed traces) and GWTC-1 (dashed) events. The vertical dashed line shows the GR prediction. Triangles mark the GWTC-3 medians, and vertical bars the symmetric 90%-credible intervals.  \n<!-- image -->  \nFIG. 5. Posteriors on the hyperparameters GLYPH<22> and GLYPH<27> of GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f distribution. The GWTC-2 and GWTC-3 posteriors on GLYPH<27> show a marked deviation from zero primarily due to GW190814 posterior on GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f peaking away from zero. The black trace shows the posteriors for GWTC-3 events excluding GW190814. See Appendix B for more information about these deviations. The corresponding hyperparameters of the GLYPH<1> M f = ¯ M f distribution do not show any such deviation.  \n<!-- image -->  \nponent masses and spins. These priors translate into nontrivial priors on GLYPH<1> M f = ¯ M f and GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f . Thus, as in the previous analysis [11], we reweight the posteriors to obtain uniform priors on the deviation parameters. We provide our results in Fig. 3, where we show the 90% credible regions of the twodimensional posteriors on the fractional deviation parameters for the O3b events which satisfy our selection criteria.  \nThe reweighted posteriors on the fractional deviation parameters GLYPH<1> M f = ¯ M f and GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f of individual events are interpolated on a grid with bounds [ GLYPH<0> 2 ; 2] for both the parameters, and  \nthe interpolated posteriors are then multiplied to obtain the combined posteriors. Here we assume the same deviation for all events to obtain the combined results. As shown in gray in Fig. 3, the combined posteriors on the fractional deviation parameters of GWTC-3 events are consistent with the GR prediction with GLYPH<1> M f = ¯ M f = GLYPH<0> 0 : 02 + 0 : 07 GLYPH<0> 0 : 06 and GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f = GLYPH<0> 0 : 06 + 0 : 10 GLYPH<0> 0 : 07 . The two-dimensional GR quantile values Q 2D GR for the events are given in Table IV. Q 2D GR is defined as the fraction of the posterior enclosed by the isoprobabilty contour that passes through (0 ; 0), the GR value. Smaller values indicate better consistency with GR. The GR quantile of the combined distribution is 79 : 6% which is similar to the value obtained for GWTC-2 (78 : 7%). Among the O3b events, GW200225 060421 has the lowest Q 2D GR value of 1 : 3% and GW200224 222234 has the highest value of 20 : 7%.  \nWe can also combine the results hierarchically, as discussed in Sec. III B of our previous analysis [11]. Fig. 4 presents the results where the fractional mass (blue) and spin (red) deviation parameters for events from multiple observing runs are plotted with GLYPH<1> M f = ¯ M f = 0 : 03 + 0 : 14 GLYPH<0> 0 : 13 and GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f = GLYPH<0> 0 : 05 + 0 : 37 GLYPH<0> 0 : 38 , which are consistent with the expected values in GR. Treating GLYPH<1> M f = ¯ M f and GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f independently, we find that the Gaussian model parameters are constrained to ( GLYPH<22>; GLYPH<27> ) = (0 : 04 + 0 : 08 GLYPH<0> 0 : 07 ; 0 : 05 + 0 : 10 GLYPH<0> 0 : 04 ) and ( GLYPH<0> 0 : 04 + 0 : 12 GLYPH<0> 0 : 12 ; 0 : 19 + 0 : 17 GLYPH<0> 0 : 13 ) for GLYPH<1> M f = ¯ M f and GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f respectively, with 90% credibility. These bounds are not significantly di GLYPH<11> erent from the ones reported in GWTC-2 [11], except for that of GLYPH<27> for GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f . It peaks significantly away from zero as shown in Fig. 5 due to GW190814 whose updated posteriors (see Appendix B for more details related to the updated GWTC-2 results) show marked deviation from GR. We also show the posteriors excluding GW190814 which peak at GLYPH<27> = 0.', 'A. Generic modifications': "Deviations from GR, such as additional fields or highercurvature corrections, may alter the binary's binding energy and angular momentum, and its energy and angular momentum flux [15, 16, 19, 20, 22, 23, 159-161]. This in turn would result in modifications to the binary motion and, hence, to the GW signal emitted by the system. A practical approach to quantifying such e GLYPH<11> ects entails introducing a finite number of parameters that encapsulate possible deviations of a waveform from its GR prediction. We will focus here on parametrizations of the frequency-domain GW phase evolution since observations are in general most sensitive to it (as opposed to changes in the amplitude).  \nSmall modifications to the GW phase could accumulate for events with many detectable GW cycles and thus parametrized tests initially focussed on the inspiral part of the waveform, whose duration in the detector band grows for low mass binaries. The inspiral can be treated perturbatively within the post-Newtonian framework [34, 162-172], which expands observables in powers of v= c , with each O ([ v= c ] 2 n ) being referred to as of n PN order. With the intrinsic parameters of the bi-  \nnary given, the coe GLYPH<14> cients at di GLYPH<11> erent orders of v= c in the PN series are uniquely determined, and so is the perturbative expansion of the early-inspiral phasing within GR. Treating such PN coe GLYPH<14> cients as measurable parameters of the waveform is therefore a sensible consistency test of GR [173-180]. While these parameterized waveforms could capture a wide variety of beyond-GR e GLYPH<11> ects, the abrupt onset of waveform modifications, possible when nonperturbative phenomena such as dynamical scalarization are at play, may not be fully captured by them [181, 182]. However, in the spirit of null tests, these di GLYPH<11> erences may still appear as apparent violations of GR.  \nThis approach can be applied by directly modifying coe GLYPH<14> cients in a specific waveform model that encodes PN information [32] or by adding corrections that correspond to deformations of a given inspiral PN coe GLYPH<14> cient at low frequencies and tapering the corrections to zero at a specific cuto GLYPH<11> frequency [9]. Corrections are applied in both cases at the level of the aligned-spin phasing; however, the first method can be leveraged to perform parametrized tests with precessing phenomenological templates, as these automatically inherit non-GR corrections introduced in the aligned-spin phase by virtue of the twisting-up construction [10, 11].  \nHere we present results obtained with the second method, which we apply to the frequency domain model SEOBNR v 4 ROM [115, 117], a reduced-order model of the timedomain aligned-spin approximant SEOBNR v 4. We do not include results obtained with the first method, for which an upgrade to the precessing IMRP henom XPmodel is under development. Due to time constraints, these results will be presented elsewhere. Past analyses [10, 11] showed good consistency between the two approaches, despite the di GLYPH<11> erences in the waveform models being used and the physics content included. Dedicated studies would be needed to thoroughly assess the e GLYPH<11> ect of waveform systematics on parametrized tests across parameter space and quantify the e GLYPH<11> ects of specific approximations, such as the omission of precession and higher-order multipole moments.  \nFractional deviations are applied to the full phase as corrections scaling with f ( GLYPH<0> 5 + n ) = 3 at each n = 2-th PN order. Following previous works [10, 11], we reweight the posteriors to reparametrize the results as fractional deviations applied to the nonspinning terms of a 3.5PN TaylorF2 phase [183], which is obtained by applying the stationary phase approximation [184] to time-domain post-Newtonian waveforms:  \n' PN( f ) = 2 GLYPH<25> f t c GLYPH<0> ' c GLYPH<0> GLYPH<25> 4 + 3 128 GLYPH<17> GLYPH<16> GLYPH<25> ˜ f GLYPH<17> GLYPH<0> 5 = 3 7 X i = 0 h ' i + ' i l log( GLYPH<25> ˜ f ) i GLYPH<16> GLYPH<25> ˜ f GLYPH<17> i = 3 : (4)  \nHere, ˜ f = GM (1 + z ) f = c 3 , with M (1 + z ) being the redshifted total mass of the binary, ' c ; tc are the coalescence phase and time, and GLYPH<17> the symmetric mass ratio. This parametrization has the advantage of avoiding potentially singular behavior of the deviation coe GLYPH<14> cients, which might occur as a result of cancellations between the nonspinning and spin-dependent phasing coe GLYPH<14> cients.  \nPhasing corrections to the inspiral phase are tapered o GLYPH<11> at the same cuto GLYPH<11> frequency used in previous analyses, i.e.,  \nTABLE V. Parametrized test event selection for all binaries meeting the FAR < 10 GLYPH<0> 3 yr GLYPH<0> 1 threshold. Here f PAR c denotes the cuto GLYPH<11> frequency at which non-GR corrections to the inspiral phase are tapered away (see main text for details); GLYPH<26> IMR and GLYPH<26> insp are the optimal SNRs computed on the full signal or on the region f GLYPH<20> f PAR c respectively. The last column denotes if the event is included in parametrized tests on the PN deviation coe GLYPH<14> cients.  \nf PAR c = 0 : 35 f 22 peak , where f 22 peak is the GW frequency of the (2,2)-mode at the peak of the amplitude as defined in the model SEOBNR v 4.  \nWe introduce the following parametric deviations to GW inspiral phasing:  \nf GLYPH<14> ˆ ' GLYPH<0> 2 ; GLYPH<14> ˆ ' 0 ; GLYPH<14> ˆ ' 1 ; GLYPH<14> ˆ ' 2 ; GLYPH<14> ˆ ' 3 ; GLYPH<14> ˆ ' 4 ; GLYPH<14> ˆ ' 5 l ; GLYPH<14> ˆ ' 6 ; GLYPH<14> ˆ ' 6 l ; GLYPH<14> ˆ ' 7 g ; (5)  \nwhere each GLYPH<14> ˆ ' i represents the fractional deviation from the GR PNcoe GLYPH<14> cient at the i = 2-th PN order, following the parametrization adopted in previous analyses [7, 9, 10, 185, 186]. The subscript l is used to denote coe GLYPH<14> cients of logarithmic-inf terms. We do not present bounds for the 2 : 5PN non-logarithmic term, as this is degenerate with the coalescence phase, as can be seen from Eq. (4). As predicted in GR, the coe GLYPH<14> cients corresponding to GLYPH<0> 1PN and 0 : 5PN are identically zero, so we parametrize GLYPH<14> ˆ ' GLYPH<0> 2 and GLYPH<14> ˆ ' 1 as absolute deviations, with a prefactor equal to the 0PN coe GLYPH<14> cient (3 = 128 GLYPH<17> ); all other coe GLYPH<14> cients represent fractional deviations from the GR value.  \nAs detailed in Sec. I, we consider all binaries that meet the significance threshold of FAR < 10 GLYPH<0> 3 yr GLYPH<0> 1 and impose the additional requirement that SNR GLYPH<21> 6 in the inspiral regime, as defined with respect to f PAR c . Eligible events are summarized in Table V. The parametrization presented above recovers GR in the limit GLYPH<14> ˆ ' i ! 0, thus consistency with GR can be claimed if 0 is included within a given confidence interval and, in what follows, we will report 90% credible intervals for the posteriors of GLYPH<14> ˆ ' i . We adopt uniform priors on GLYPH<14> ˆ ' i that are symmetric about zero and compute their posterior distributions using LALI nference .  \nAs in previous analyses, we only allow the coe GLYPH<14> cients GLYPH<14> ˆ ' i to vary one at a time. It has been shown that this procedure is effective at picking up the deviations from GR that modify more  \nthan one PN coe GLYPH<14> cient [187-189]. Allowing multiple deviations in the same template is bound to produce less informative posteriors, due to correlations among di GLYPH<11> erent parameters and also with GR coe GLYPH<14> cients. Such correlations can be e GLYPH<11> ectively reduced with alternative choices of the deformation parameters, which can be guided by a principal component analysis [190, 191], thus increasing the e GLYPH<11> ectiveness of multiparameter tests. It was also shown that synergies between third-generation detectors and space-based interferometry might dramatically improve the precision of multiparameter tests [73].  \nIn Fig. 6, we present the 90% upper bounds on the deviation coe GLYPH<14> cients obtained from the combined distribution of events from GWTC-3, under the assumption that deviations take the same value for all the events. While the combined bounds are fully consistent with GR, we do note that for a number of events GLYPH<14> ˆ ' i = 0 falls outside the 90% credible interval of the deviation coe GLYPH<14> cients distributions. We find that the addition of an extra degree of freedom can enhance the weight of secondary modes observed in GR parameter estimation runs, as can be seen for GW191216 213338 and GW200316 215756, where a secondary mode in mass ratio present in the GR parameter estimation runs dominates the results when inspiral deviation parameters are included. This is partly due to the singular behavior of the parametrization employed here, and indeed the application of reweighting alleviates the problem. Still, the presence of a secondary mode remains and so do the broad posteriors for the deviation coe GLYPH<14> cients. Appendix A contains a more extended discussion on the impact of noise properties and waveform systematics on our bounds.  \nWe find that there is no uniform improvement over the GWTC-2 results [11] across all deviation coe GLYPH<14> cients. This is consistent with the modest improvement factor due to the increased number of events, which can be estimated to be GLYPH<24> 1 : 2. We find that this factor is comparable to fluctuations in the final bounds determined by individual events. The most striking di GLYPH<11> erence with respect to the GWTC-2 analysis is the new constraint obtained for GLYPH<14> ˆ ' GLYPH<0> 2, for which we obtain an upper bound j GLYPH<14> ˆ ' GLYPH<0> 2 j GLYPH<20> 7 : 3 GLYPH<2> 10 GLYPH<0> 4 at the 90% credible level. This result improves upon the GWTC-2 bound by a factor GLYPH<24> 2. The improvement is driven by the inclusion of GW200115 042309, due to its long duration. The presence of a non-zero GLYPH<0> 1PN term can be associated to the emission of dipolar radiation, which is forbidden in GR but can be excited in alternative theories of gravity and is related to energy and momentum being transferred from the binary to additional fields. This result is less stringent than the one obtained with combined measurements of binary pulsars [192] or with the observation of GW170817 [9].  \nWe also computed hierarchically combined posteriors for all the deviation coe GLYPH<14> cients introduced above, where we allow the deviation coe GLYPH<14> cients to take independent values for each event. These are shown in Fig. 7, plotted against the combined posteriors employed to obtain the upper bounds of Fig. 6. All results are consistent with the GR prediction with at least 90% credibility. In Fig. 8, we show the joint distribution of the mean GLYPH<22> and standard deviation GLYPH<27> of the population-marginalized posterior for the deviation coe GLYPH<14> cients of Eq. (5). All distributions are consistent with the GR prediction for which GLYPH<22> = GLYPH<27> = 0,  \nwith deviations occurring at GLYPH<0> 1PN being the most tightly constrained.  \nIn Table VI, we report the medians, 90% credible intervals, and GR quantiles Q GR = P ( GLYPH<14> ˆ ' i < 0) for both the hierarchical and the joint-likelihood approaches. Values of Q GR significantly di GLYPH<11> erent from 50% indicate that the null hypothesis falls in the tails of the combined distribution. In the hierarchical analysis the most constrained parameter is GLYPH<14> ˆ ' GLYPH<0> 2 = GLYPH<0> 0 : 05 + 0 : 99 GLYPH<0> 1 : 25 GLYPH<2> 10 GLYPH<0> 3 at the 90% credible level, while deviations in the 3.5PN order coe GLYPH<14> cient are the least constrained, with GLYPH<14> ˆ ' 7 = 0 : 14 + 1 : 05 GLYPH<0> 1 : 16 . For the majority of the PN coe GLYPH<14> cients, the hierarchical analysis of GWTC-3 data obtains tighter constraints than the ones obtained with GWTC-2 events [11].  \nResults for the shifts to inspiral phase can also be mapped onto constraints on specific theories, in particular via the parametrized post-Einstein (ppE) framework [71, 177]. For instance, bounds on the coupling constant of Einstein-dilatonGauss-Bonnet and dynamical Chern-Simons gravity were obtained using GWTC-2 events [72, 193, 194]. However, the upper bounds reported here depend on the parametrization being used and on specific details of the analysis, such as the frequency at which non-GR corrections are being tapered o GLYPH<11> . The priors we impose on the deviation coe GLYPH<14> cients are not designed to suit any specific theory and, depending on the theory that is being considered, di GLYPH<11> erent sampling parameters and prior bounds might be preferable to the ones adopted here. Furthermore, in alternative theories of gravity multiple inspiral coe GLYPH<14> cients will be subject to deviations from GR; our bounds refer to single-coe GLYPH<14> cient deviations, that might capture at once deviations at several PN orders, so the mapping would be ambiguous. Likewise, one would also need a robust estimate of the error caused by neglecting currently unknown higher-order PN corrections and deviations in the merger-ringdown phase, which will also di GLYPH<11> er from the GR one [195, 196]. Finally, it is not clear whether some of these theories, such as Lovelock, Chern-Simons or Horndeski gravity, of which Einsteindilaton-Gauss-Bonnet gravity is a special case, would admit at all a well-posed initial value problem in their most general form [197], although well-posed formulations are possible in the weak-coupling limit [198-200].", 'B. Spin-induced quadrupole moment': "Spinning objects have quadrupole and higher contributions to the multipole decomposition of their gravitational field due to their rotational deformations. Following the no-hair conjecture, the spin-induced multipole moments take unique values for black holes given their mass and spin [201-203]. Gravitational waveforms describing spinning compact binary systems encode information about these spin-induced multipole moment e GLYPH<11> ects. The leading order term can be schematically represented as,  \nQ = GLYPH<0> GLYPH<20> GLYPH<31> 2 m 3 : (6)  \nHere Q is the quadrupole moment scalar and is the leading order term in the gravitational wave phase at 2 PN order. m and GLYPH<31> are the mass and the dimensionless spin of the compact object.  \n/circledot  \nM  \n<!-- image -->  \n-  \nFIG. 6. 90% upper bounds on the magnitude of the parametrized test coe GLYPH<14> cients discussed in Sec. V A. The bounds were obtained with a pipeline based on the model SEOBNR v 4 ROM, combining all eligible GWTC-3 events, under the assumption that deviations take the same value for all the events. Filled gray diamonds mark analogous results obtained with GWTC-2 data [11]; in this case, we also show bounds obtained with a pipeline based on IMRP henom P v 2, that are marked by unfilled black diamonds. Horizontal stripes indicate constraints obtained with individual events, with cold (warm) colors representing low (high) total mass events. The left and right panel show constraints on PN deformation coe GLYPH<14> cients, from GLYPH<0> 1PN to 3 : 5PN order. The best improvement with respect to the GWTC-2 bounds is achieved for the GLYPH<0> 1PN term, thanks ot the inclusion of the NSBH candidate GW200115 042309.FIG. 7. Combined GWTC-3 results for the parametrized deviation coe GLYPH<14> cients of Sec. V A. Filled distributions represent the results obtained hierarchically combining all events. This method allows the deviation coe GLYPH<14> cients to assume di GLYPH<11> erent values for di GLYPH<11> erent events. Unfilled black curves represent the distributions obtained in Fig. 6, by assuming the same value of the deviation parameters across all events. Horizontal ticks and dashed white lines mark the 90% credible intervals and median values obtained with the hierarchical analysis.  \n<!-- image -->  \nAlong with this leading-order e GLYPH<11> ect, we have included higherorder PN terms that appear through the inspiral phase [167, 204] of gravitational waveform.  \nWhile Kerr BHs have GLYPH<20> = 1 [201-203], compact stars have a value of GLYPH<20> that di GLYPH<11> ers from the BH value, determined by the star's mass and internal composition. Numerical simulations of spinning neutron stars show that the value of GLYPH<20> can vary between GLYPH<24> 2 and GLYPH<24> 14 for these systems [205-207]. Moreover, for currently available models of spinning boson stars, GLYPH<20> can have values GLYPH<24> 10-150 [208-211]. More exotic stars like gravastars can even take negative values for GLYPH<20> [212]. Hence, an independent measurement of GLYPH<20> from gravitational-wave observations can be used to distinguish black holes from other exotic ob-  \njects [213-216]. However, to fully understand the nature of compact objects, one may also include e GLYPH<11> ects such as the tidal deformations that arise due to the external gravitational field [217-220] and tidal heating [221-226] along with the spininduced deformations, an extensive study of these e GLYPH<11> ects is not in the scope of this paper.  \nFor a spinning compact binary system, the coe GLYPH<14> cients GLYPH<20> i , i = 1 ; 2 represent the primary and secondary components' spin-induced quadrupole moment parameters. The correlation of GLYPH<20> i with the masses and spin parameters of the binary are evident from Eq. (6), which makes the simultaneous estimation of GLYPH<20> 1 and GLYPH<20> 2 hard. The higher-order terms present at the 3PN order help break this degeneracy, but are not enough to  \nTABLE VI. Results from parametrized tests of GW generation (Sec. V A). Combined constraints on the deviation parameters GLYPH<14> ˆ ' i from the full set of GWTC-3 data using the the SEOBNR v 4 ROM waveform model. General (restricted) constraints are obtained under the assumption that deviation coe GLYPH<14> cients can (cannot) vary across the observed events. Q GR indicates the quantile corresponding to the GR value for the distributions plotted in Fig. 6. For general constraints, we also provide the mean GLYPH<22> and standard deviation GLYPH<27> of the inferred hyperdistribution. For GLYPH<14> ˆ ' i and GLYPH<22> , we report the median as well as the 90%-credible intervals, while for GLYPH<27> we only present upper bounds. For the restricted method, the null hypothesis can occasionally fall in the tail of the distribution, while the hierarchical analysis places it in the bulk of the inferred distribution.  \n<!-- image -->  \nFIG. 8. Joint distribution for the hyperparameters GLYPH<22> and GLYPH<27> of the GR deviation coe GLYPH<14> cients for the parametrized tests on generic GW modifications of Sec. V A. Contours mark 90% credible regions and they all include GLYPH<22> = GLYPH<27> = 0, which corresponds to the GR prediction. These results were obtained with a pipeline based on SEOBNR v 4 ROM. The GLYPH<30> GLYPH<0> 2-contour has been rescaled by a factor of 200 to improve visibility, as deviations for this term are the most tightly constrained. Warm (cold) colors refer to deviations occurring at higher (lower) PN orders.  \n<!-- image -->  \ngive reasonable constraints with our current detector sensitivities. However, a combination of these parameters can be measured [11, 213, 227, 228]. For this reason, we introduce the symmetric and anti-symmetric combinations of GLYPH<20> i ,  \nGLYPH<20> s = ( GLYPH<20> 1 + GLYPH<20> 2) = 2 ; (7)  \nGLYPH<20> a = ( GLYPH<20> 1 GLYPH<0> GLYPH<20> 2) = 2 : (8)  \nFor binary black holes, GLYPH<20> s = 1 and GLYPH<20> a = 0. We measure the parameterised deviations GLYPH<20> s = 1 + GLYPH<14>GLYPH<20> s assuming GLYPH<20> a = 0. This assumption restricts our study to binaries consisting of compact stars with identical spin-induced deformations. If the data supports GLYPH<20> 1 , GLYPH<20> 2, expect a significant o GLYPH<11> set away from zero in GLYPH<14>GLYPH<20> s measurements. Further studies are required for such situations.  \nThe method employed here is the same as in previous analysis [11]. We perform a Bayesian analysis with LALInference using a nested sampling algorithm to estimate the posteriors on GLYPH<14>GLYPH<20> s . The parametrized deviations GLYPH<14>GLYPH<20> s are introduced in the inspiral phase of the IMRP henom P v 2 waveform model.  \nWe analyse the events listed in Table II passing the selection criterion. Along with the FAR < 10 GLYPH<0> 3 yr GLYPH<0> 1 criteria, we consider two additional conditions for selecting the events for this test. First, we select inspiral-dominated events having an inspiral network SNR GLYPH<21> 6. Since the test relies on at least one of the binary's components having nonzero spin, we also drop events whose e GLYPH<11> ective inspiral spin parameter measurements include zero at the 68% credible level. Given the component masses mi and the dimensionless spins GLYPH<31> i = ~ Si GLYPH<1> ˆ L = m 2 i pointing parallel to the orbital angular angular momentum axis, the e GLYPH<11> ective inspiral spin parameter GLYPH<31> e GLYPH<11> is defined as, GLYPH<31> e GLYPH<11> = ( m 1 GLYPH<31> 1 + m 2 GLYPH<31> 2) = ( m 1 + m 2) [229]. To compute the combined bounds we include events reported in [11] satisfying both the above selection criteria. This gives us a total of 13 events considering the entire GWTC-3.  \nIn Fig. 9, the posterior distributions on GLYPH<14>GLYPH<20> s for the events listed in Table II are derived assuming a uniform prior on GLYPH<14>GLYPH<20> s ranging between [ GLYPH<0> 500 ; 500]. Individual events constrain positive values of GLYPH<14>GLYPH<20> s more strongly than negative ones. This is primarily because of how these parameters are correlated with the e GLYPH<11> ective inpsiral spin parameter of the binary system [228].  \nAs most of the events we observe have small but positive GLYPH<31> e GLYPH<11> ,  \nFIG. 9. The posterior probability distribution on the spin-induced quadrupole moment parameter, GLYPH<14>GLYPH<20> s from the events listed in the SIM column of Table II, passing the selection criteria described in Section V B. The black dashed vertical line indicates the BBH value ( GLYPH<14>GLYPH<20> s = 0). The colored vertical lines show the 90% symmetric bounds on GLYPH<14>GLYPH<20> s calculated from the individual events assuming a uniform prior ranging between [ GLYPH<0> 500 ; 500] on GLYPH<14>GLYPH<20> s .  \n<!-- image -->  \nFIG. 10. Joint posterior probability distribution on the spin-induced quadrupole moment parameter GLYPH<14>GLYPH<20> s from the GWTC-3 events. Bounds obtained by multiplying the likelihoods (restricted) and by hierarchically combining events (generic) are shown. The analysis is performed assuming uniform prior ranging between [-500, 500] on GLYPH<14>GLYPH<20> s .  \n<!-- image -->  \nthe combined posterior and the 90% bounds are expected to show this feature.  \nWe also consider a case where the analysis is restricted to only positive GLYPH<14>GLYPH<20> s as is well motivated in the case of neutron stars [205, 206, 215] and boson stars [208], in this case the event provides the tightest upper limits is GW191216 213338, with 90% credible bounds of GLYPH<14>GLYPH<20> s < 10 : 65.  \nWe show the combined posterior distribution on GLYPH<14>GLYPH<20> s from all the GW events passing the selection criteria in Fig. 10. The red curve draws the posterior distribution obtained by multiplying the likelihoods of each individual signal. In contrast, the population-marginalized posterior from the hierarchical analysis is shown in the blue curve. Dotted lines show the 90% symmetric credible intervals, and a dashed line marks the BBH value ( GLYPH<14>GLYPH<20> s = 0). We estimate the combined symmetric 90% bound on GLYPH<14>GLYPH<20> s considering GWTC-3 events to be GLYPH<14>GLYPH<20> s = GLYPH<0> 16 : 0 + 13 : 6 GLYPH<0> 16 : 7 and, conditional on positive values, GLYPH<14>GLYPH<20> s < 6 : 66 from the joint likelihood analysis. With 90% credibility, we find GLYPH<14>GLYPH<20> s = GLYPH<0> 26 : 3 + 45 : 8 GLYPH<0> 52 : 9 from the hierarchical analysis.  \nThe generic population results constrain GLYPH<14>GLYPH<20> s < 51 : 85 when we restrict to positive prior region. Also, we find the hyperparameters to be consistent with the Kerr BBH hypothesis with 90% credible bounds with GLYPH<22> = GLYPH<0> 26 : 8 + 26 : 3 GLYPH<0> 34 : 1 and GLYPH<27> < 41 : 8. Compared to the previous bounds reported in [11], GLYPH<22> = GLYPH<0> 24 : 6 + 30 : 7 GLYPH<0> 35 : 3 and GLYPH<27> < 52 : 7, the GLYPH<27> estimate improves, meaning tighter constraints on GLYPH<14>GLYPH<20> s , while the peak of the distribution is shifted more towards the negative prior region. The shift in the peak or GLYPH<22> omits the BBH value with the 90% credibility and can be associated to the poor GLYPH<14>GLYPH<20> s constraints on the negative side of the prior region from the individual events, emerging from waveform degeneracies at GLYPH<14>GLYPH<20> s < 0 with a certain region of the spin parameter space. A future study employing waveform models including higher harmonics may help break those degeneracies and hence to improve our overall parameter estimation [228, 230]. Moreover, a more generic approach has been recently proposed [230] that uses a hierarchical mixturelikelihood formalism to estimate the fraction of events in the population that deviated from BBH nature. With the increased number of detections in the future, it would be more natural to employ generic approaches that considers the population to be comprised of BBH and non-BBH subpopulations.  \nThe combined log Bayes factor of log 10 B Kerr GLYPH<14>GLYPH<20> s , 0 = 0 : 9 is obtained supporting the BBH hypothesis over the hypothesis of all events being non-BBH. This changes to log Bayes factor of log 10 B Kerr GLYPH<14>GLYPH<20> s > 0 = 2 : 2 if we only allow GLYPH<14>GLYPH<20> s GLYPH<21> 0. The findings here are all consistent with the results reported in GWTC-2 [11] although the combined constraints are not directly compatible due to the di GLYPH<11> erent selection of events.", 'VI. TESTS OF GRAVITATIONAL WAVE PROPAGATION': 'GR predicts that GWs propagate nondispersively and hence they are described by the dispersion relation E 2 = p 2 c 2 , where E and p are the energy and momentum of the wave. Detection of dispersion of GWs can be seen as a signature of modifications to GR. For example, some of the Lorentz violating theories of gravity predict a modified dispersion relation [45, 231234]. We use a parameterized model [41, 49] for dispersion of GWs that helps search for the presence of dispersion using the data without referring to the details of the modified theory.  \nOur parameterized dispersion relation reads [41]  \nE 2 = p 2 c 2 + A GLYPH<11> p GLYPH<11> c GLYPH<11> ; (9)  \nwhere A GLYPH<11> and GLYPH<11> are two phenomenological parameters characterizing dispersion. The modified dispersion relation causes frequency modes of GWs to propagate at di GLYPH<11> erent speeds, changing the overall phase morphology of the GW that are observed with respect to the GR predictions. This can be incorporated in the waveform as frequency-dependent corrections to its phase evolution [10, 41]. Here we assume that the waveform obtained in the local wave zone [235] of the system is consistent with GR [10].  \nFor di GLYPH<11> erent choices of GLYPH<11> , the modified dispersion leads to a deviation in the GR phasing formula. For example, GLYPH<11> = 0 with A GLYPH<11> > 0 corresponds to the dispersion e GLYPH<11> ect of a massive graviton with mass m g c 2 = p A 0 [49]. We choose to test  \nthe dispersion relation for a set of eight discrete values of GLYPH<11> between 0 and 4 with a step of 0 : 5 excluding GLYPH<11> = 2. When GLYPH<11> = 2, the speeds of all the frequency components are modified in the same way; therefore, the GW signal remains unchanged from the GR prediction except for an overall change in the time of arrival of the signal.  \nOur method is identical to the previous analyses performed in GWTC-1 and GWTC-2 [10, 11], except for the use of a more up-to-date IMRPhenomXP [105] waveform model as opposed to the PhenomPv2 [236] waveform employed in GWTC-1 and GWTC-2 [10, 11]. We perform parameter estimation using the nested sampling algorithm [140] as implemented in the LALInference package [138] and obtain bounds on the phenomenological parameters A GLYPH<11> for each event. As in the case of preceding analyses, we perform the sampling for A GLYPH<11> < 0 and A GLYPH<11> > 0 separately [10, 11], and then combine the posterior to produce the joint A GLYPH<11> posterior. We choose uniform priors for the phenomenological parameters A GLYPH<11> . However, while computing the bound on the graviton mass m g , which is derived from A 0, we re-weight the posteriors such that the prior on m g is uniform.  \nPropagation e GLYPH<11> ects are independent of the source properties. Therefore we can combine the results from individual events to compute overall constraints over the phenomenological dispersion parameters. We obtain the combined posterior distributions of A GLYPH<11> by multiplying the likelihoods from individual events and weighting the product with the prior.  \nWe perform the analysis on the 12 BBH candidate events in the catalog that are listed in Table II. Though we analyzed GW191109 010717, the posteriors obtained were too wide to be informative, and following the study regarding this event reported in Appendix A, which finds that nonstationarities in the detector noise could dominate over the signal, we exclude this event from further analysis. Analysis of another BBH event, GW200316 215756 has sampling issues and is thus excluded from the analysis. Further, we do not include NSBH event GW200115 042309 in this analysis due to the computational constraints. Nonetheless, this is among the closest events in the catalog and would have a negligible impact on the joint bounds.  \nFig. 11 shows the violin plots of joint posteriors on the phenomenological parameters A GLYPH<11> for various values of GLYPH<11> , which are obtained by combining posteriors from analysis of individual events. The red violin plots represent the posteriors obtained from all 43 selected events (31 events from pre-O3b and 12 O3b events). For some of the GLYPH<11> values, the posteriors show biases with respect to the previous results [11] due to the inclusion of O3b events. We have identified the events GW200219 094415 and GW200225 060421 as having the strongest impact in biasing the combined posterior. These are the events with the lowest residual SNR p -values among all the O3b events (see Table III). GW200225 060421 shows p -value of 0 : 05 with fitting factor FF90 = 0 : 86 and GW200219 094415 has p -value = 0 : 1 with FF90 = 0 : 74. These events require detailed analysis to understand the reasons for the observed deviations, which we leave for follow-up work. For comparison, in Fig. 11, we also plot the combined posteriors from all the events exclud-  \nFIG. 11. Results for the modified dispersion analysis (Sec. VI). The red violin plots show the combined posteriors of the parameter A GLYPH<11> calculated from the GWTC-3 events with the error bars denoting the 90% credible intervals. For comparison, we also present the combined posteriors after excluding the events GW200219 094415 and GW200225 060421 using blue violin plots. The gray plots in the background are the combined posteriors corresponding to GWTC2 [11].  \n<!-- image -->  \ning GW200219 094415 and GW200225 060421 (blue violin plots). These are consistent with the GWTC-2 [11] results (gray plots in the background) and show an average improvement of 1 : 3 over the previous results, which is in agreement with the Gaussian expectation for improvement from 41 events compared to 31 of GWTC-2 [11].  \nIn Fig. 12, we present the scatter plot of 90% credible upper bounds on j A GLYPH<11> j , for A GLYPH<11> > 0 and A GLYPH<11> < 0 separately. In the figure, red-filled diamond markers represent the GWTC-3 bounds. We also show the bounds from the analysis excluding the events GW200219 094415 and GW200225 060421 in the blue diamond markers. For quantitative comparison, we list j A GLYPH<11> j bounds, including bounds on the graviton mass m g , in Table VII. To demonstrate the level of bias in the posteriors with respect to the GR hypothesis, we included the GR quantiles Q GR = P ( A GLYPH<11> < 0) in Table VII.  \nThe updated 90% credible bound on the graviton mass obtained by combining posteriors of 43 GWTC-3 events is m g GLYPH<20> 1 : 27 GLYPH<2> 10 GLYPH<0> 23 eV = c 2 , which is 2 : 5 times better than the Solar System bound of 3 : 16 GLYPH<2> 10 GLYPH<0> 23 eV = c 2 [237]. Compared to the previous GWTC-2 bound 1 : 76 GLYPH<2> 10 GLYPH<0> 23 eV = c 2 [11], the improvement is a factor of 1 : 4.', 'VII. POLARIZATIONS': "Measuring the polarization content of GWs is a way of constraining possible deviations from GR, as the theory allows only two of the six polarization states predicted by generic metric theories of gravity [50, 51]. Assuming M generic polarization modes, the frequency-domain strain data ˜ d ( f ) measured by a network of D detectors can be written as the combination  \nα  \nTABLE VII. Results for the modified dispersion analysis (Sec. VI). The table shows 90%-credible upper bounds on the graviton mass m g and the absolute value of the dimensionless phenomenological parameter ¯ A GLYPH<11> = A GLYPH<11> = eV 2 GLYPH<0> GLYPH<11> . Q GR = P ( A GLYPH<11> < 0) denotes the quantiles corresponding to GR hypothesis. The < and > labels denote the bounds on j ¯ A GLYPH<11> j for A GLYPH<11> > 0 and A GLYPH<11> < 0 respectively. We also included bounds computed from GWTC-2 [10, 11] for comparison.  \n<!-- image -->  \nFIG. 12. Results for the modified dispersion analysis (Sec. VI). The scatter plot of 90% credible upper bounds on the modulus of deviation parameters A GLYPH<11> . The one-sided bounds are computed for positive and negative values of the parameters separately. Filled (open) diamond markers represent the GWTC-3 bounds including (excluding) the events GW200219 094415 and GW200225 060421. The gray markers in the background denoted the numbers obtained from the previous analysis [11].  \n<!-- image -->  \nof a signal ˜ s ( f ) and noise ˜ n ( f ), or alternatively, as  \n˜ d ( f ) = F ˜ h ( f ) + ˜ n ( f ) ; (10)  \nwhere ˜ s ( f ) = F ˜ h ( f ), F 2 R D GLYPH<2> M are the beam pattern functions of the detectors and ˜ h ( f ) 2 C M are the signal's polarization modes. We could interpret the gravitational-wave signal as a geometric projection on the subspace spanned by the basis vectors of F . By projecting the data on the subspace orthogonal to these vectors, one can then construct null streams, i.e., linear combinations of the data containing no information on the signal [238, 239]. Given D detectors, it is possible to construct at most D GLYPH<0> M null streams. The projection operation can be formalized through the introduction of a null operator P [240]  \nP = I GLYPH<0> F ( F y F ) GLYPH<0> 1 F y ; (11)  \nwhere I is the identity matrix and y denotes conjugate transpose. The quantities F depend on the sky location of the signal, as well on the polarization angle and event time and, by construction, P ˜ s ( f ) = 0 .  \nAt least M + 1 detectors are needed to apply the null stream method in the most generic case, although for specific sky  \nlocations less detectors will su GLYPH<14> ce to test certain polarization hypotheses [241, 242]. The beam pattern functions of the breathing and longitudinal scalar modes are not linearly independent, and thus the maximum number of independent polarization modes is five [243, 244]. Consequently, past analyses [7, 9, 11, 245] tested only pure polarization hypotheses, as these are fully characterized by two polarisation modes at most, and in this case it is possible to construct a null stream with the strain measured by three detectors.  \nIn this work, we use a method that allows tests of mixed polarization states with 2 and 3 detectors [246]. This enables all our events to be used to compute combined Bayes factors, while the previous analysis [11] was restricted to 3-detector events. The method builds upon an e GLYPH<11> ective antenna pattern function ¯ F 2 C D GLYPH<2> L that is constructed from a subset of L < M polarization modes. For each hypothesis to be tested, the relevant polarization state is projected into the chosen basis: thus, one orthogonalizes the data with respect to a smaller subspace spanned by the basis modes, rather than the assumed polarization modes. Each polarization mode ˜ hm can be rewritten as a linear combination of the basis modes, plus an additional orthogonal component  \n˜ hm ( f ) = L X k = 1 C k m ˜ h k ; k ( f ) + C ? m ˜ h ? ( f ) ; (12)  \nwith C k m ; C ? m 2 C . We perform the null projection with respect to the subspace spanned by the component of the beam pattern vectors parallel to the basis mode(s). Therefore, the method is sensitive to any component of a given polarization hypothesis that is parallel to the chosen basis modes(s). The subspace removed by the null projection does not need to coincide with the polarization subspace of the hypothesis being tested.  \nWe will conduct analyses employing either one ( L = 1) or two ( L = 2) basis modes. The L = 2 parameterization allows more freedom in the choice of the basis modes, but at the cost of a weaker distinguishability between di GLYPH<11> erent polarization hypotheses. The subspaces spanned by the beam pattern function vectors for di GLYPH<11> erent hypotheses, in fact, will generally have a larger overlap in the L = 2 than in L = 1 case. The polarization content is constrained to be a linear combination of the basis modes and, therefore, the L = 1 analysis is expected to produce more stringent results, due to the strongest constraints imposed on the signal. On the other hand, the L = 2 analysis will be able to capture orthogonal components missed by the L = 1 analysis.  \nRight ascension, declination and polarization angle are free  \nparameters shared by both analyses. Additionally, we marginalize over the relative amplitude and phase with respect to the basis mode(s). While this implies the L = 2 analysis has more degrees of freedom than the L = 1 one, the polarization content of GR can be represented by both methods. In fact, in the quadrupolar approximation, h + and h GLYPH<2> only di GLYPH<11> er in terms of a relative amplitude and phase, which are marginalized over in the computation of the evidence, and therefore only one basis mode is su GLYPH<14> cient to capture both.  \nIn the L = 2 analysis, we choose one tensorial mode and one non-tensorial mode as the basis, to better capture possible extra degrees of freedom predicted by alternative theories of gravity. All events are analyzed with both models, to check consistency between the two formulations. The nature of null projection prohibits the comparison of model evidences with di GLYPH<11> erent number of basis modes [246]; therefore, we conduct the analyses with one and two basis modes independently.  \nWe reanalyze here all of the events from the first three observing runs with a FAR below our 10 GLYPH<0> 3 yr GLYPH<0> 1 threshold. The pipeline employed here has been optimized to analyze the polarization content based on excess power in the data. Some events from the third observing run, namely GW190425 081805, GW190720 000836, GW190828 065509, GW191129 134029, GW200115 042309, GW200202 154313, and GW200316 215756, do not have a strong enough timefrequency track for the pipeline to capture them and we exclude them from our analysis.  \nWe test whether the data support a number of non-GR polarization hypotheses, which we denote by T (tensorial), V (vector) and S (scalar) or combinations of these terms. In the L = 1 case, we choose the tensorial plus mode h + as the basis for all hypotheses involving tensor modes, the vector mode x as the basis for the pure vector and vector-scalar hypotheses, and the scalar breathing mode as a basis for the scalar hypothesis. In the L = 2 analyses, following the notation introduced above, we project mixed polarizations of the type T GLYPH<24> on a basis including h + and the relevant non-tensorial GLYPH<24> mode, VS polarizations on a vector-scalar basis and TVS polarizations on the h + and scalar-breathing mode. The basis modes chosen for each polarization hypothesis and the corresponding free parameters are summarized in Table VIII.  \nWe present combined log 10 Bayes factors assuming the events are independent from each other. Table IX shows the combined log 10 Bayes factors of the 2-detector and 3-detector events for the L = 1 analysis, with the last row showing the combined log 10 Bayes factor for all eligible GWTC-3 events. The pure-scalar, pure-vector and vector-scalar hypotheses are significantly disfavored, while any mixed hypothesis involving tensor modes (i.e., tensor-scalar, tensor-vector, and tensor-vector-scalar) cannot be ruled out conclusively.  \nTable X shows the combined log 10 Bayes factors of the 3detector events for the L = 2 analysis. There are no available data for O1, due to the fact that only two interferometers were operational at the time (LIGO Hanford and LIGO Livingston). In this case, the last row shows the combined log 10 Bayes factor for O2 and O3 events. Mixed hypotheses in this case are more strongly disfavored than the pure vector hypothesis (the pure  \nscalar hypothesis cannot be tested here, due to the fact that the longitudinal and breathing modes for interferometers are not linearly independent). This is particularly evident for the TVS hypothesis, which has the largest number of free parameters. All mixed hypotheses will be penalized by a higher Occam factor, due to the increased prior volume in the Bayesian evidence integral, as each mode is characterized by a relative amplitude and phase with respect to the basis modes. The number of free parameters necessary to represent each hypothesis, depending on the number of basis modes employed, is reported in the last column of Table VIII.  \nIf the polarization content of the signals is purely tensorial, mixed-mode hypotheses including a tensorial component will be also able to represent the data, as the signal space of the pure tensorial hypothesis is a subspace of that of the mixedmode hypotheses including a tensorial component. In this case, we expect to see a larger penalization of mixed hypotheses in the L = 2 analysis, as the same polarization hypothesis will correspond to a larger parameter space than in the L = 1 analysis, as can be seen from Table VIII. This expectation is confirmed when comparing the results reported in Tables IX and X. On the other hand, if we consider the vector-scalar hypothesis, the additional parameters in the L = 2 analysis will help to fit the tensorial component of the data: this explains why we see a less negative log 10 Bayes factor in the L = 2 analysis than in the L = 1 one for this hypothesis. A similar argument applies to the purely scalar and vectorial hypotheses.  \nThese results support the conclusion that the population of events observed is consistent with the pure tensorial hypothesis, as predicted by GR, in line with the conclusion of past analyses of GWTC-1 and GWTC-2 data [10, 11].", 'A. Ringdown': 'The highly distorted BH remnant formed from the merger emits gravitational radiation which is referred to as ringdown. The late-ringdown waveform can be expressed as a superposition of quasi-normal modes (QNM) with a complex frequency [247, 248]. The real part of the complex frequency is the oscillation frequency and the imaginary part is the inverse damping time of the mode. According to GR, for astrophysical BHs, the frequency and damping times are completely determined by the mass and spin of the remnant BH [201203, 249]. In fact, at the current sensitivity, electric-like BH charges have been shown to not leave a detectable imprint on ringdown measurements [250]. The relationship between frequency and remnant parameters, detection of multi-mode ringdown signals o GLYPH<11> ers a unique test of the BH nature of the merger remnant [63, 251] and could be used to distinguish among di GLYPH<11> erent classes of ECOs [13].  \nThe study of the QNM spectrum, and the post-merger waveform in general, contains a wealth of information about the remnant BH. The spectrum of radiation emitted during the ringdown is usually expressed in terms of spheroidal harmonic basis functions with spin-weight GLYPH<0> 2 denoted by GLYPH<0> 2 S \' mn . The in-  \nTABLE VIII. The table summarizes the choices of basis used in the polarization test. x , GLYPH<2> , b , l , x , and y represent the plus mode, cross mode, scalar breathing mode, scalar longitudinal mode, vector x mode, and vector y mode respectively. The first column shows the polarization hypothesis being tested, the third column reports the number of basis modes, and the last column reports the number of free parameters that are marginalized over in the computation of the evidence.TABLE IX. Combined log 10 Bayes factors B for various polarization hypotheses against the tensor hypothesis, using both 2-detector and 3-detector events. Polarization states have been projected onto one basis-mode as detailed in Sec. VII. Positive (negative) values indicate that the hypothesis indicated in the superscript is favored (disfavored) with respect to the tensorial hypothesis. Error bars refer to 90% credible intervals.  \nTABLE X. Combined log 10 Bayes factor B for various polarization hypotheses against the tensor hypothesis, for 3-detector events. Polarization states been projected onto two basis-modes as explained in Sec. VII. Positive (negative) values indicate that the hypothesis indicated in the superscript is favored (disfavored) with respect to the tensorial hypothesis. Error bars refer to 90% credible intervals.  \ndices ( \'; m ) represent the angular decomposition of the modes, whereas the index n denotes various tones of the spectrum start-  \ning with n = 0. A schematic decomposition of the post-merger signal reads [11],  \nh + ( t ) GLYPH<0> ih GLYPH<2> ( t ) = + 1 X \' = 2 \' X m = GLYPH<0> \' + 1 X n = 0 A \' mn exp " GLYPH<0> t GLYPH<0> t 0 (1 + z ) GLYPH<28>\' mn # exp " GLYPH<0> 2 GLYPH<25> i f \' mn ( t GLYPH<0> t 0) 1 + z # GLYPH<0> 2 S \' mn ( GLYPH<18>; GLYPH<30>; GLYPH<31> f ) ; (13)  \nwhere A \' mn denotes the amplitude of the mode, t 0 is the start time of the ringdown model, and z is the redshift of the source. The frequency and the damping time of a mode characterized by the three indices are denoted by GLYPH<28>\' mn and f \' mn , respectively, while GLYPH<31> f is the final spin. The polar and azimuthal angles ( GLYPH<18>; GLYPH<30> ),  \nmeasured relative to the final spin axis, describe the direction to the observer. These coordinates assume the spin of the black hole to be along the GLYPH<18> = 0 direction. The contribution of counter-rotating perturbations is ignored, since it\'s expected to be negligible in the post-merger regime of the signals under  \nconsideration. We approximate the spheroidal harmonics with spherical harmonics, leading to a feature, called mode-mixing of QNMs [111, 252, 253], which mixes modes that have the same m index but di GLYPH<11> erent \' indices. Mode mixing, especially of \' = j m j = 2 mode, is expected to be negligible for the modes that we consider here [118].  \nWe present two analyses of ringdown: the time-domain ringdown analysis pyRing [254, 255], based on damped sinusoids, and the parametrized ringdown analysis p SEOB, based on the SEOBNR v 4HM waveform model [120].', '1. The py R ing analysis': "py R ing employs a time-domain approach. The three types of templates used in the analysis are Kerr220, Kerr221, and KerrHM. Kerr220 has only the fundamental modes, i.e., ( '; j m j ; n ) = (2 ; 2 ; 0), Kerr221 has the fundamental modes and its first overtones (both GLYPH<6> m , with the same damping time and frequencies of opposite sign). Instead, KerrHM includes the fundamental mode and higher moments (HMs) with ' GLYPH<20> 4, without overtones. The KerrHM template also takes into account the e GLYPH<11> ect of mode-mixing due to the use of spherical harmonics instead of spheroidal harmonics to describe the ringdown.  \nWhile in the Kerr220 and Kerr221 models the amplitudes and phases are left free to vary, in the KerrHM model, quasi-circular, aligned-spin NR fits are used to compute the mode amplitudes, frequencies, and damping times. The model calibration assumes that the remnant BH originated from the quasi-circular coalescence of progenitor BHs with spins aligned with the orbital angular momentum. As other templates account for precession, this assumption does not a GLYPH<11> ect the overall e GLYPH<14> ciency of the analysis. The progenitors' masses and spins are sampled with uniform priors. In contrast to the GWTC-2 results [11], where they were independently measured, the remnant mass and spin are now also obtained through NR fits from the progenitor parameters [157]. This increases the coherence of the measurement, leading to tighter constraints on the remnant parameters. The model amplitudes are calibrated against NR simulations at 20 M after the peak of the ' = j m j = 2 complex strain, where M is the total mass of the progenitor binary. The other two template types remain unchanged from previous analysis [11].  \npy R ing uses a reference time t 0, which is computed from an estimate of the peak of the strain ( h 2 + + h 2 GLYPH<2> ) from the full IMR analyses (with the approximant IMRPhenomXPHM ) assuming GR. The sky location is fixed to coincide with the maximum likelihood value inferred from the full IMR analysis. When assuming the Kerr221 template, we fit the data starting at t 0 [255, 256]. To take into account the NR calibration of the template, the start time of the analysis is instead set to 15 G ¯ M f (1 + z ) = c 3 after t 0 when employing the KerrHM template, where ¯ M f is the median IMR value of the remnant mass. This choice extrapolates the model outside its nominal validity region, however we have verified that this does not induce appreciable biases [257]. In the Kerr220 case, due to the greater flexibility of this template, the analysis starts 10 G ¯ M f (1 + z ) = c 3 after t 0, which is when a linearised ringdown description is  \nexpected to dominate the signal for our analysis [257, 258]. Both sky locations and start times at each detector, together with all configuration data required to reproduce the analysis, are released in [259].  \nWe report results where: i) the remnant parameters are constrained relative to the prior, and ii) the Bayesian evidence favors the presence of a signal over pure Gaussian noise when using our most sensitive template (Kerr221). Estimates of the remnant parameters obtained for the five events from O3b that pass the above criteria, are reported in Table XI.  \nThere is agreement between the remnant properties obtained from the three waveform templates and the IMR analysis. The contribution of overtones or HMs during ringdown is quantified by log Bayes factors (log 10 B HM 220 , log 10 B 221 220 ), which are reported in Table XI. (Note, however, that the Bayes factors depend sensitively upon the chosen priors [118].) When comparing a fit including all QNMs included in KerrHM, versus the same model including only the ' = j m j = 2 ; n = 0 mode, we observe no strong evidence for the presence of HMs. Additionally, we search for evidence of HMs augmenting the Kerr220 ; Kerr221 templates with each of the ( '; j m j ; n ) = (3 ; 3 ; 0) ; (3 ; 2 ; 0) ; (2 ; 1 ; 0) ; (2 ; 0 ; 0) modes separately. Following [260], to increase the sensitivity to these modes we assume the amplitudes of modes with opposite m signs to be related by the non-precessing symmetry A '; GLYPH<0>j m j ; n = ( GLYPH<0> 1) ' A GLYPH<3> '; j m j ; n , while also imposing an amplitude hierarchy on the higher angular modes A '; m ; n = A 2 ; 2 ; 0 < 0 : 9, as expected for quasi-circular binaries of moderate mass ratio. Under this set of assumptions, we find no significant evidence for the presence of higher modes in all events except for GW191109 010717, which shows weak evidence for the presence of the (3 ; 2 ; 0) and (2 ; 1 ; 0) modes in addition to the (2 ; 2 ; 0) one at times smaller than t 0 + 5 ¯ Mf . However, as discussed below, non-stationary noise around the event time does not allow to draw reliable inference from this signal. Moreover for the parameter space under consideration, the templates signalling such evidence are composed only of fundamental n = 0 modes and thus not reliable before GLYPH<24> t 0 + 10 ¯ Mf [257, 258]. Comparing the Kerr220 and Kerr221 analyses, with both templates starting at the peak, we observe weak evidence for the presence of overtones only for the loudest among these signals (for example GW200224 222234 shows such evidence). The estimates of the remnant parameters get closer to the full IMR waveform estimates when including overtones, in agreement with NR predictions.  \nThe exception to the above discussion is GW191109 010717, which shows an overestimation of the remnant mass and spin compared to IMR analyses. As discussed in Appendix A, such discrepancies can be attributed to possible nonstationarities in the detector noise and we will exclude this event from any combined statement made below.  \nTo investigate possible modifications to the ringdown spectrum of the remnant BH, we add parametrised deviations to the Kerr221 template in the frequency and damping time with respect to their GR values for the n = 1 mode. This parametrization can almost fully cover the two-tone parameter space and has a number of desirable features that facilitate both sampling and interpretation [118]. Parameter estimation is performed  \nTABLE XI. The median, and symmetric 90%-credible intervals, of the redshifted final mass and final spin, inferred from the full IMR analysis (IMR) and the py R ing analysis (Sec. VIII A 1) with three di GLYPH<11> erent waveform models (Kerr220, Kerr221, and KerrHM). A positive value of log 10 B HM 220 indicates support for HM in the data, and a positive value of log 10 B 221 220 shows support for the presence of the first overtone. A positive value of log 10 O modGR GR quantify the level of disagreement with GR. The catalog-combined (including GWTC-2 events) log odds ratio is negative ( GLYPH<0> 0 : 90 GLYPH<6> 0 : 45).  \nover the same set of parameters appearing in the GR template, with the addition of the deviation parameters on which we impose uniform priors in the [ GLYPH<0> 1 ; 1] range for the frequency GLYPH<14> ˆ f 221 and in the [ GLYPH<0> 0 : 9 ; 1] range for the damping time GLYPH<14> ˆ GLYPH<28> 221. The lower bound on GLYPH<14> ˆ GLYPH<28> 221 prevents issues due to the finite time resolution in the waveform sampling. [11]. If GR provides an accurate description of the ringdown emission, we expect to observe posterior distributions of the deviation parameters to be centered around zero, together with a Bayesian evidence disfavouring the addition of non-GR parameters.  \nThe inferred values of the frequency deviation parameters are consistent with GR for all events analysed, while weak constraints can be extracted on the damping times deviations from single events. The damping time estimation of low-SNR events is more sensitive to violations of the Gaussianity and stationarity hypotheses compared to the frequency estimation [11]. Additional studies investigating this behaviour will be required in the future to properly derive joint posteriors on this parameter when combining many weak events. The posterior distribution of GLYPH<14> ˆ GLYPH<28> 221 often tends to rail towards the lower prior bound GLYPH<0> 0 : 9 for events with low SNR in the ringdown regime, as the data show little evidence for the first overtone.  \nTo combine the set of measurements for all 21 available events we make use of a hierarchical analysis [11]. The single events posteriors used to derive this joint bound are the marginalised GLYPH<14> ˆ f 221 posteriors obtained when allowing both the frequency and the damping time of the 221 mode to deviate from the GR predictions. We obtain a constraint on the frequency deviation equal to GLYPH<14> ˆ f 221 = 0 : 01 + 0 : 27 GLYPH<0> 0 : 28 , overlapping with the GR predicted value for a Kerr BH, and show its posterior probability distribution in Fig. 13. The corresponding hyperparameter values are: GLYPH<22> = 0 : 01 + 0 : 18 GLYPH<0> 0 : 18 ; GLYPH<27> < 0 : 22. Although GW191109 010717 is excluded from the combined analysis, we note that even though the mass and spin estimates coming from this event show some tension with the ones coming from an IMR analyses, the parametrised deviations do not indicate preference for additional parameters required to describe the ringdown emission. We do not allow to obtain informative constraints on GLYPH<14> ˆ GLYPH<28> 221.  \nThe single event odds ratios log 10 O modGR GR values, computed following a procedure similar to our previous analysis [11], are reported in Table XI. The highest log 10 O modGR GR value among  \nFIG. 13. The posterior distribution of the fractional frequency deviation for the ' = j m j = 2 ; n = 1 mode, GLYPH<14> ˆ f 221, from the py R ing joint hierarchical analysis (triangles and small vertical bars indicate respectively median and 90% CLs). The measurements of GLYPH<14> ˆ f 221 from individual events, and its combined value using all available 21 GW events (red solid line), both show consistency with GR. Compared to the corresponding GWTC-2 constraint (dashed-dotted blue line), the hierarchically combined posterior on the frequency deviation shows a 90% CL shrinkage ratio of GLYPH<24> 8%. See Sec. VIII A 1 for details.  \n<!-- image -->  \nO3b events, GLYPH<0> 0 : 09, corresponds to GW200129 065458 and does not signal significant tension. By considering all the GWTC-3 events that passed our selection criteria (including previous GWTC-2 results), we find a combined log odds ratio of GLYPH<0> 0 : 90 GLYPH<6> 0 : 44, at 90% uncertainty, favouring the hypothesis that GR gives an accurate description of the observed ringdown signals.  \nFinally, as an agnostic test of the consistency of the ringdown emission with GR predictions, a single damped sinusoid (DS) template is used to fit the data. In this case we are not assuming an underlying Kerr metric, nor that the object emitting the signal is a BH, thus the frequency, damping time, and complex amplitude are considered as free parameters without imposing any predictions from GR. We adopt uniform priors on the frequency, damping time, log of the magnitude, and the phase of the complex amplitude. The fit starts at 10 G ¯ M f (1 + z ) = c 3 after  \nt 0. The results on the frequency and damping time obtained in this case are reported in Table XII.  \nThe resulting values agree with the measured values obtained from IMR analyses assuming GR and with those obtained from the additional pSEOB ringdown test discussed below, except for GW191109 010717, where we find an overestimation of the frequency and damping time from the py R ing analysis with respect to the full IMR analyses, which is compatible with the overestimation of the remnant mass and spin.", '2. The p SEOBNR v 4HM analysis': "The pSEOBNRv4HM ringdown analysis [11, 120], which uses parameterized spinning EOB waveforms with higher modes calibrated to non-precessing NR simulations [116, 261], measures QNM frequencies within the framework of a complete IMR binary black hole waveform model. It makes full use of gravitational wave signal modelling and hence the complete SNR of the signal. In this approach, the ringdown start-time is built into the model, based on calibrations to NR, and may hence be considered complementary to the post-merger timedomain ringdown analysis described in the previous section.  \nIn the SEOBNRv4HM model, starting from estimates of the initial binary's masses and spins, NR fits [156, 262] are used to predict the mass and spin of the remnant object, which are then used to predict the ringdown frequencies and damping times [64, 263]. Thus, the frequency and damping time of the ( '; GLYPH<6> m ; 0) QNM, ( f ' m 0 ; GLYPH<28>' m 0) are functions of the initial masses and spins:  \nf GR ' m 0 = f GR ' m 0 ( m 1 ; m 2 ; GLYPH<31> 1 ; GLYPH<31> 2) ; (14)  \nGLYPH<28> GR ' m 0 = GLYPH<28> GR ' m 0 ( m 1 ; m 2 ; GLYPH<31> 1 ; GLYPH<31> 2) : (15)  \nIn the parameterized version of the SEOBNRv4HM model used here ( pSEOBNRv4HM ), deviations in the frequency and damping time are described through fractional deviations, ( GLYPH<14> ˆ f ' m 0 ; GLYPH<14> ˆ GLYPH<28>' m 0), from the corresponding GR predictions as:  \nf ' m 0 = f GR ' m 0 (1 + GLYPH<14> ˆ f ' m 0) ; (16)  \nGLYPH<28>' m 0 = GLYPH<28> GR ' m 0 (1 + GLYPH<14> ˆ GLYPH<28>' m 0) : (17)  \nWe use LALInference [138] to stochastically sample over the parameter space of f GLYPH<14> ˆ f ' m 0 ; GLYPH<14> ˆ GLYPH<28>' m 0 g , along with the set of GR  \n.  \nTABLE XII. The median, and symmetric 90% CLs of the frequency and damping time from a single damped-sinusoid (DS) analysis, compared to the IMR predictions for the fundamental ' = j m j = 2 ; n = 0 mode from Sec. VIII A 1  \nparameters, and finally reconstruct ( f ' m 0 ; GLYPH<28>' m 0) using Eqs. (14)(17). Here, we keep our analysis identical to the one presented in previous analysis [11] and restrict ourselves to fractional deviations of the least-damped dominant QNM, i.e., ( GLYPH<14> ˆ f 220 ; GLYPH<14> ˆ GLYPH<28> 220), keeping the other QNMs fixed at their nominal GR values.  \nThe analysis is performed on events from Table II with an SNR GLYPH<21> 8 in the preand post-inspiral regimes, following Table IV. A reasonable SNR in the inspiral is required to break the degeneracy between the fundamental ringdown frequency deviation parameter and the remnant mass. The detector-frame total mass threshold criterion used (due to computational limitations) in the previous analysis [11] was relaxed in this analysis. In reporting joint constraints, we include events from past analyses. These include the 3 events from O3a [11], and five events from first two observing runs and O3a, reported for the first time in [120]. The final list of events that were used for this analysis is listed in Table XIII.  \nThe results of the analysis is summarised in Fig. 14. The left panel of Fig. 14 shows the 2D posteriors (along with the marginalised 1D posteriors) of the frequency and damping time for all the events listed in Table XIII. The contours are colored by the median redshifted total mass (1 + z ) M of the corresponding binary. We also show the 90% credible bounds on the fractional deviations from GR in the right panel color coded by the median redshifted mass of the binary. We specifically highlight the bounds from two events, GW150914 and GW200129 065458. GW150914 previously provided the strongest bounds with this method [120]. However, GW200129 065458, similar in source properties to GW150914, but with an SNR of 26.5 [81], provides currently provides the strongest single event bound, GLYPH<14> ˆ f 220 = GLYPH<0> 0 : 01 + 0 : 08 GLYPH<0> 0 : 08 ; GLYPH<14> ˆ GLYPH<28> 220 = 0 : 11 + 0 : 23 GLYPH<0> 0 : 23 . The joint constraints are reported using two methods: multiplying posteriors (given a flat prior on the deviation parameters) and hierarchically combining events. The joint bounds from these two methods read  \nGLYPH<14> ˆ f 220 = 0 : 02 + 0 : 03 GLYPH<0> 0 : 03 ; GLYPH<14> ˆ GLYPH<28> 220 = 0 : 13 + 0 : 11 GLYPH<0> 0 : 11 (18)  \nby multiplying the posteriors and  \nGLYPH<14> ˆ f 220 = 0 : 02 + 0 : 07 GLYPH<0> 0 : 07 [ GLYPH<22> = 0 : 02 + 0 : 04 GLYPH<0> 0 : 04 ; GLYPH<27> < 0 : 06] (19)  \nGLYPH<14> ˆ GLYPH<28> 220 = 0 : 13 + 0 : 21 GLYPH<0> 0 : 22 [ GLYPH<22> = 0 : 13 + 0 : 13 GLYPH<0> 0 : 13 ; GLYPH<27> < 0 : 19] (20)  \nby combining hierarchically. The numbers in the square brackets are the hyper-parameter estimates. There is a significant improvement from results previously published in [11] (filled / unfilled upward / downward triangles in the right plot of Fig. 14), which were: GLYPH<14> ˆ f 220 = 0 : 03 + 0 : 38 GLYPH<0> 0 : 35 ; GLYPH<14> ˆ GLYPH<28> 220 = 0 : 16 + 0 : 98 GLYPH<0> 0 : 98 using the hierarchical method and GLYPH<14> ˆ f 220 = 0 : 03 + 0 : 07 GLYPH<0> 0 : 06 ; GLYPH<14> ˆ GLYPH<28> 220 = 0 : 13 + 0 : 18 GLYPH<0> 0 18 using the restricted method.  \n: We note a couple of points about the joint posteriors. First, though we perform single-event analyses over GW191109 010717 and GW200208 130117, we do not include them in the computation of the joint bounds (hierarchical or simple combination). The posterior on GLYPH<14> ˆ f 220 show multimodalities for these two events. For GW200208 130117, the secondary of two modes is consistent with the GR prediction GLYPH<14> ˆ f 220 = 0, while for GW191109 010717, none of the modes  \nFIG. 14. Left panel : The 90% credible levels of the posterior probability distribution of the fractional deviations in the frequency and damping time of the (2 ; GLYPH<6> 2) QNM, ( GLYPH<14> f 220 ; GLYPH<14>GLYPH<28> 220) and their corresponding one-dimensional marginalized posterior distributions, for events from O1, O2 and O3 passing a SNR threshold of 8 in both the pre- and post-merger signal. Posteriors for GW150914 and GW200129 065458 are separately shown. The joint constraints on ( GLYPH<14> f 220 ; GLYPH<14>GLYPH<28> 220) obtained multiplying the posteriors (given a flat prior) from individual events are given by the filled grey contours, while the hierarchical method of combination yields the black dot dashed curves in the 1D marginalized posteriors. Right panel : 90% credible interval on the one-dimensional marginalised posteriors on GLYPH<14>GLYPH<27> i = ( GLYPH<14> f 220 ; GLYPH<14>GLYPH<28> 220), colored by the median redshifted total mass (1 + z ) M , inferred assuming GR. Filled gray (unfilled black) downward triangles mark the constraints obtained when all the events are combined by multiplying posteriors (hierarchically). For comparison, we mark the previously published bounds from [11] using filled / unfilled upward triangles. The bounds from GW200129 065458 (square) and GW150914 (diamond) are indicated by the separate markers. See Sec. VIII A 2 for details.  \n<!-- image -->  \nare. Follow-up investigations with synthetic signals in segments of data immediately adjacent to the event suggests the possibility of noise systematics not accounted for. The same study rules out, within our statistical uncertainties, any systematic bias due to missing physics in the SEOBNRv4HM waveform model.  \nWe also note that the joint posterior distribution on GLYPH<14> ˆ GLYPH<28> 220 in the left plot of Fig. 14 does not include the GR prediction at the 90% credible level. Although insu GLYPH<14> cient to claim a violation of GR, this apparent deviation definitely warrants further investigation. The trend of overestimating the combined damping time is consistent with what is observed on an event-by-event analysis, where the posterior on GLYPH<14> ˆ GLYPH<28> 220, although consistent with 0 is biased towards positive values. Hence a combination of information across multiple events is expected to reduce statistical uncertainties and make this bias more prominent. One possible reason might be a prior on ( GLYPH<14> ˆ f 220 ; GLYPH<14> ˆ GLYPH<28> 220) which is asymmetric around 0 with greater support for positive values. This is because, since ( f 220 ; GLYPH<28> 220) are strictly positive quantities, the priors on ( GLYPH<14> ˆ f 220 ; GLYPH<14> ˆ GLYPH<28> 220) are strictly greater than GLYPH<0> 1. However, the upper prior boundary  \nis free to be as large as is required for the posterior to not rail against it and it usually greater than 1. For events with moderately high SNRs analysed with this method, the e GLYPH<11> ect of the prior on the final posterior can be non-negligible. We also note that while the posteriors on the fractional deviation show more support towards positive values, the frequency and damping time reconstructed using Eqs. (16) and (17) are consistent with those predicted using estimates of initial masses and spins from [81] and NR fits [158]. This gives us more confidence in the measured QNMs, while also pointing to the possibility that correlations among the remnant parameters may be responsible for the apparent deviation. Further, as has been argued in [11], imperfect noise modelling can also lead to overestimation of damping time [120]. Finally, we can not rule out the statistical uncertainties of working with a sample of just 12 events.", 'B. Echoes': 'Mergers of certain classes of exotic compact objects that do not have a horizon can cause ingoing gravitational waves (e.g.,  \nTABLE XIII. Redshifted damping times and frequencies of the 220 mode as well as final redshifted mass and spin as inferred from the analysis for di GLYPH<11> erent events that are analysed using the p SEOBNR v 4HM method (Sec. VIII A 2).  \nresulting from merger) to reflect multiple times o GLYPH<11> e GLYPH<11> ective radial potential barriers, with wave packets leaking out to infinity at regular times; these are called echoes [60-62]. Previous analysis [11] employed a morphology-dependent method [264] using the waveform template proposed in [67] to search for GWechoes. Here, however, we employ a method that is independent of the morphology of the signal [84, 85].  \nWe make use of the BAYESWAVE pipeline [83, 145, 150] to search for echoes and compute the Bayes factors for the signal versus noise hypothesis. We further analyze the background around each event to quantify the significance of the Bayes factors.  \nOur method employs combs of decaying identical sine-Gaussians as the basis functions or the generalized wavelets [84, 85]. The sine-Gaussians are parameterized by an amplitude A , a central frequency f 0, a damping time GLYPH<28> , and a reference phase GLYPH<30> 0. In addition to these parameters, basis functions include three extra parameters which characterize the way in which the sine-Gaussians are arranged in the wavelet; the central time of the first echo t 0, the time separation between sine-Gaussians GLYPH<1> t , a phase di GLYPH<11> erence between them GLYPH<1> GLYPH<30> , an amplitude damping factor GLYPH<13> , and a widening factor w .  \nTo perform the search, we use uniform priors for all the wavelet parameters except the damping time GLYPH<28> and the amplitude A . The central frequency, f 0 is sampled uniformly from the interval between the lower cut-o GLYPH<11> frequency and half the sampling rate of the analysis, [30 ; 1024] Hz. Also, GLYPH<1> t , GLYPH<30> 0 , GLYPH<1> GLYPH<30> and GLYPH<13> are sampled uniformly with respective ranges, GLYPH<1> t 2 [0 ; 0 : 7] s, GLYPH<30> 0 2 [0 ; 2 GLYPH<25> ], GLYPH<1> GLYPH<30> 2 [0 ; 2 GLYPH<25> ], GLYPH<13> 2 [0 ; 1], and w 2 [1 ; 2]. The damping time GLYPH<28> is sampled such that the corresponding the quality factor Q = 2 GLYPH<25> f 0 GLYPH<28> is distributed uniformly in the interval [2 ; 40], which ensures GLYPH<28> to be within the time scales set by masses of the events ( GLYPH<24> [3 GLYPH<2> 10 GLYPH<0> 4 ; 0 : 2] s). The wavelet amplitude A is sampled based on signal-to-noise ratio as described in [83].  \nTo construct background distributions for the log Bayes factors log 10 B S N , we use stretches of data randomly picked in an asymmetric time window around each coalescence event. Back-  \ngrounds are composed by 100 trials in [ t event GLYPH<0> 3072s ; t event GLYPH<0> 300 s] and 100 trials in [ t event + 1024 s ; t event + 2048 s], , where t event is the coalescence time of the event, each trial analyzing an interval of 4 s. The start of the time window after coalescence is chosen in order to avoid the presence of putative echoes signals in the background trials.  \nThe foreground run is instead performed on a 4 s time interval starting at t event + 3 GLYPH<28> 220, since we want to start analyzing at a time that is safely beyond the plausible duration of the ringdown of the remnant object. GLYPH<28> 220 is a conservatively long estimate for the decay time of the 220 mode in the ringdown, obtained from the fitting formula for GLYPH<28> 220( M f ; GLYPH<31> f ) [64, 265]. We take the upper bound of 90% credible interval of GLYPH<28> 220 distribution computed from final mass and spin samples for the event. This is typically of the order of a few milliseconds.  \nThe complementary empirical cumulative distribution function of background distributions of log 10 B S N are used to quantify the search outcome via p -values for each event. We compute the p -value for log 10 B S N for each event, which is the fraction of background log 10 B S N above the log Bayes factor of the event.  \nFor all the events, the signal versus noise Bayes factor B S N are within the corresponding background distributions, and the corresponding p -values are tabulated in Table XIV. If echoes are not present in the data, we expect the p -values to follow a uniform distribution between [0 ; 1]. Fig. 15 plots the p -value versus the cumulative fraction of events and the dotted dash line shows the prediction if no signal is present with corresponding 90% uncertainty regions marked in light-color bands. We follows the recipe described in Sec. IV A to make the PP plot. As can be seen, the measurement is consistent with the absence of echoes within 90% credible region. We conclude that we find no statistically significant evidence for echoes from the morphology-independent search we carried out. As the methodology employed here is di GLYPH<11> erent from that of our previous analysis [11], and relies of the p -values, one cannot have a fair comparison of the results between the two.  \nTABLE XIV. Results of the echoes analysis (Sec. VIII B). List of p -values for signal to noise Bayes Factor B S N for the events that are analysed. In the absence of any echoes signal these should be uniformly distributed between [0 ; 1]. Fig. 15 shows the corresponding PP plot with 90% credible intervals superimposed on it. There is no evidence for the presence of echoes.', 'IX. CONCLUSIONS AND OUTLOOK': 'Gravitational-wave observations provide a unique tool to test fundamental physics. The strongly gravitating, highly dynamical and radiative spacetime associated with the late inspiral, merger and ringdown of compact binaries facilitates tests of general relativity in a regime that is unaccessible otherwise. Binary black holes and binary neutron star mergers observed in the past observing runs already set limits on possible deviations from GR [3, 6, 7, 9-11, 79, 99, 242, 259, 266-269]. Here we discuss a pool of tests aimed at unearthing deviations from GR using the events detected during the second part of the third observing run of advanced LIGO and advanced Virgo. We perform ten tests of GR on the 15 events that have a false alarm rate less than 10 GLYPH<0> 3 yr GLYPH<0> 1 . These tests are the same ones as in the previous analysis [11], except with the following updates. Our search for post-merger echoes is morphologyindependent in this paper and the method to test for non-GR polarization modes is refined to address mixed polarizations as opposed to scalar-only, vector-only, and tensor-only hypotheses as was the case in [11]. Furthermore, some of the tests rely on more up-to-date waveforms; in the residuals and inspiral-merger-consistency tests, we account for higher order multipole moments for all the events from the second part of the third observing run.  \nWe subtract the maximum-likelihood GR waveform from the data to verify the consistency of the residuals with detector noise, thereby showing the consistency of the signals in the data with GR. Independent estimates of the mass and spin of the merger remants, from the inspiral and postinspiral parts of the waveform for di GLYPH<11> erent events show mutual consistency. The fractional changes in the final mass and spin from this test, assuming they take the same values for all the events and combining all the events analyzed so far, are constrained  \nFIG. 15. Results of the echoes analysis (Sec. VIII B). Plot of fraction of events for which the echoes signal-to-noise p -value is less than or equal to the abscissa. The light-blue band represents the 90% credible interval of the observed p -values, while the diagonal dashed line is expectation from the null hypothesis. The light-gray band around the diagonal line represents the 90% uncertainty band of the null hypothesis.  \n<!-- image -->  \nto GLYPH<1> M f = ¯ M f = GLYPH<0> 0 : 02 + 0 : 07 GLYPH<0> 0 : 06 and GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f = GLYPH<0> 0 : 06 + 0 : 10 GLYPH<0> 0 : 07 at 90% credibility.  \nTests aimed at looking for parametrized departures from GR in the post-Newtonian phasing coe GLYPH<14> cients all find consistency with GR within the statistical uncertainties. The most wellconstrained parameter is the absolute value of the GLYPH<0> 1PN coe GLYPH<14> -cient, which is bound to GLYPH<20> 7 : 3 GLYPH<2> 10 GLYPH<0> 4 at 90% credibility, assuming its value is the same for all the events. As certain modified theories of gravity predict dispersion of GWs, we searched for this e GLYPH<11> ect and found no evidence for dispersion. The bound on the graviton mass is updated to m g GLYPH<20> 1 : 27 GLYPH<2> 10 GLYPH<0> 23 eV = c 2 , at 90% credibility. A general metric theory of gravity admits up to six modes of GW polarization. We searched for non-GR polarization modes and found no signature of such modes.  \nAnalyses to measure the spin-induced quadrupole moments of the binary components found no signatures of exotic compact objects. Further, tests for deviations from GR in the ringdown of the remnant black hole were carried out using two independent methods and the frequency deviation parameters are constrained to GLYPH<14> ˆ f 221 = 0 : 01 + 0 : 27 GLYPH<0> 0 : 28 and GLYPH<14> ˆ f 220 = 0 : 02 + 0 : 07 GLYPH<0> 0 : 07 , at 90% credibility, by hierarchically combining the results from the events that are analyzed. We also found no evidence for post-merger echoes from the merger remnant from our morphology-independent search.  \nFuture observing runs with improved detector sensitivities will provide a larger catalog of compact binary observations and events with larger SNR. These observations will enable us to carry out more stringent tests of GR in parts of the parameter  \nspace that are not covered here. Developing accurate waveform models that cover the diverse physics that will be revealed by the sources is an important step towards this goal. Finally, devising new tests or improving the existing ones, optimizing their sensitvity to predictions from specific modified theories of gravity, can play a very important role in constraining beyondGR physics using future GW observations.', 'ACKNOWLEDGMENTS': "Analyses in this paper made use of N um P y [270], S ci P y [271], A stropy [272, 273], IP ython [274], qnm [275], PES ummary [276], and GW py [277]; plots were produced with M atplotlib [278], and S eaborn [279]. Posteriors were sampled with S tan [280], CPN est [144], P y M ultinest [281, 282], B ilby [136, 137], and LALI nference [138]. Power spectral densities are generated through the software BAYESWAVE [83, 145]. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation. The authors also gratefully acknowledge the support of the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen / Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Netherlands Organization for Scientific Research (NWO), for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigaci'on (AEI), the Spanish Ministerio de Ciencia e Innovaci'on and Ministerio de Universidades, the Conselleria de Fons Europeus, Universitat i Cultura and the Direcci'o General de Pol'ıtica Universitaria i Recerca del Govern de les Illes Balears, the Conselleria d'Innovaci'o, Universitats, Ci'encia i Societat Digital de la Generalitat Valenciana and the CERCA Programme Generalitat de Catalunya, Spain, the National Science Centre of Poland and the European Union - European Regional Development Fund; Foundation for Polish Science (FNP), the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Social Funds (ESF), the European Regional Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the French Lyon Institute of Origins (LIO), the Belgian Fonds de la Recherche Scientifique (FRS-FNRS), Actions de Recherche Concert'ees (ARC) and Fonds Wetenschappelijk Onderzoek - Vlaanderen (FWO), Belgium, the Paris ˆ Ile-de-France Region, the National  \nResearch, Development and Innovation O GLYPH<14> ce Hungary (NKFIH), the National Research Foundation of Korea, the Natural Science and Engineering Research Council Canada, Canadian Foundation for Innovation (CFI), the Brazilian Ministry of Science, Technology, and Innovations, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the Ministry of Science and Technology (MOST), Taiwan, the United States Department of Energy, and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, INFN and CNRS for provision of computational resources.  \nThis work was supported by MEXT, JSPS Leading-edge Research Infrastructure Program, JSPS Grant-in-Aid for Specially Promoted Research 26000005, JSPS Grant-in-Aid for Scientific Research on Innovative Areas 2905: JP17H06358, JP17H06361 and JP17H06364, JSPS Core-to-Core Program A. Advanced Research Networks, JSPS Grant-in-Aid for Scientific Research (S) 17H06133 and 20H05639 , JSPS Grant-in-Aid for Transformative Research Areas (A) 20A203: JP20H05854, the joint research program of the Institute for Cosmic Ray Research, University of Tokyo, National Research Foundation (NRF) and Computing Infrastructure Project of KISTI-GSDC in Korea, Academia Sinica (AS), AS Grid Center (ASGC) and the Ministry of Science and Technology (MoST) in Taiwan under grants including AS-CDA-105-M06, Advanced Technology Center (ATC) of NAOJ, Mechanical Engineering Center of KEK.  \nWe would like to thank all of the essential workers who put their health at risk during the COVID-19 pandemic, without whom we would not have been able to complete this work.", 'Appendix A: E GLYPH<11> ect of waveform systematics and noise artifacts on the tests': 'In the context of null tests of GR, the posterior distributions on deformation parameters are sensitive to every assumption that goes into the analysis. These include assumptions about the relevant physics included in the waveforms employed and detector noise model, among others. Hence, any deviation from GR seen in the posteriors, statistically significant or not, could be due to the breakdown of one or more of the above assumptions. We discuss below some of these features and discuss the specific case of GW191109 010717 as an example.', '1. Waveform systematics and parameter degeneracies': 'The increasing number of events detected results in a large variety of systems observed, spanning regions of parameter space where di GLYPH<11> erent approximants lead to visibly di GLYPH<11> erent estimates of source parameters [81, 135, 283, 284]. Such discrepancies are expected for non-standard events, and are a direct  \nconsequence of specific approximations inherent to each waveform model (such as neglecting precessional e GLYPH<11> ects, higher order multipole moments, etc.) or lack of a uniform coverage in the parameter space of NR waveforms, which are used to train or calibrate the waveform models. Extreme spins, highly unequal masses and nearly edge-on inclinations are all elements that can enhance such di GLYPH<11> erences [285-287].  \nAnother aspect to consider in this context is the presence of waveform degeneracies, where two or more templates provide a good fit to the signal: in this case, posteriors can show a complicate bi(multi)-modal structure, with di GLYPH<11> erent modes clearly separated in parameter space. The addition of non-GR parameters complicates the picture, increasing the dimensionality of the likelihood surfaces to be explored and possibly enhancing such multimodalities. Multi-modal features might appear also in the distributions for deviation parameters, with some of the modes having consistent support away from zero. It has been pointed out that this can be a direct consequence of a known degeneracy between source parameters and deviation coe GLYPH<14> cients [120]. In this sense, one needs to be extremely cautious in classifying support for values away from GR as violations from GR, without thorough cross-comparisons of multiple analyses [40] which might be individually prone to systematic biases related to the partial inclusion of beyondGR e GLYPH<11> ects [288]. The pSEOBNRv4HM model, an IMR model where ringdown complex frequencies deviations are free to vary, is one of the analyses which commonly su GLYPH<11> ers from such degeneracies [11, 120]. One example of degeneracy is due to the strong correlation between the fundamental ringdown frequency and the mass of the remnant BH. Such degeneracy is broken when the mass is constrained independently from other phases of the coalescence. This considerations led to the requirement of considering only events with SNR > 8 in the pre-merger regime for this analysis. While additional selection criteria can help in minimizing these e GLYPH<11> ects, it is important to carefully interpret the results of each test keeping in mind the assumptions underlying the analysis of the signal.', '2. Non-stationarity and non-Gaussianity of the detector noise': 'An additional important aspect of data analysis is the impact of transient non-gaussian noise or glitches a GLYPH<11> ecting the data around the time of the event analysed. Glitches can lead to systematic deviations in the null parameter that is measured and mimic false deviations from GR [11, 289]. These issues will a GLYPH<11> ect both low and high mass systems in di GLYPH<11> erent ways, depending on the amplitude and duration of the noise transient [290], and might exacerbate waveform systematics e GLYPH<11> ects, which tend to be more severe for short signals. Furthermore, since the parameterized models we employ invoke parameters over and above the GR source parameters, these additional parameters can capture any residual noise artifacts that are left after methods such as deglitching [83, 291] are employed to mitigate non-stationarities or non-Gaussianities. As we detect more and more signals, including high-mass ones, these considerations become of increasing importance while testing for deviations from GR. Already in the first half of the third  \nobserving run, a few events have shown deviations which were understood to be an e GLYPH<11> ect of incomplete models of the noise background (e.g., in the ringdown regime) [11]. Gaussian noise fluctuations alone are expected to cause deviations from GR for GLYPH<24> 1 in ten events with our choice of credible intervals, though the results of our hierarchical analyses should be robust against these e GLYPH<11> ects. As an example, we next discuss how statistically significant deviations seen in some of the analyses, as mentioned in the main text, may be understood.', '3. Anomalies observed in the case of GW191109 010717': "Among the di GLYPH<11> erent events we analysed, GW191109 010717 is the only event which led to statistically significant deviations in three of our null tests (i.e., modified dispersion tests, Sec. VI, and two ringdown tests, Sec. VIII A).  \nGW191109 010717 is the heaviest system in our event list and hence has the shortest signal duration. It is also the only event considered in this work for which both Livingston and Hanford frames required mitigation, due to the presence of glitches overlapping with the inspiral track of the signal (see Table XIV of GWTC-3[81]). Such noise artifacts belong to the category of 'slow scattering' glitches [292], which are due to light scattering and lead to long duration arches ( GLYPH<24> 2 GLYPH<0> 2 : 5s) in time-frequency spectograms, as discussed in [293] (see also Sec. IIIB of GWTC-3 [81] for a discussion of data quality of O3b events). Data below 30(40) Hz were a GLYPH<11> ected in Livingston (Hanford), with noise directly overlapping the time of the trigger, and glitches were subtracted with BayesWave [83, 291]. The same glitch subtraction algorithm was applied only to another event in our selection list, GW200115 042309. However, GW200115 042309 has a long inspiral compared to GW191109 010717, which lasts for less than a second in detector band, meaning the glitch overlaps with a significant portion of the signal. GR parameter estimation runs [81] already show evidence of a multimodal likelihood surface for this event.  \nPrevious investigations have shown that BayesWave deglitching is not expected to a GLYPH<11> ect parameter estimation [291, 294] nor tests of GR [289]. However, these works did not consider situations in which two detectors are a GLYPH<11> ected by the type of glitch that impacted GW191109 010717. Tests of injections overlapping scattered light in one detector have been previously shown to mimic GR violations [289]. Injections performed around the trigger time in detector noise could partially reproduce the deviations seen in the actual analyses of the event, indicating noise properties might be the main explanation behind our results. Further studies would be required to assess the reliability of tests of GR in similar situations, as well as the sensitivity of di GLYPH<11> erent analyses to residual noise features. These investigations are outside the scope of this paper and, as a cautionary measure, we drop GW191109 010717 from combined statements in the context of ringdown and modified dispersion tests.  \nFIG. 16. Same as Fig. 3 but for GWTC-2 events (see Table XV). The gray distribution corresponds to the joint posterior of GWTC-2 events. O3a (O1 and O2) events are plotted with solid (dot-dashed) traces.  \n<!-- image -->  \nTABLE XV. Same as Table IV but here we include the updated results of GWTC-2 events which satisfy the inspiral-merger-ringdown consistency test selection criteria. Only the Q 2D GR values are updated compared to results given in Table IV of our previous analysis [11].", 'Appendix B: Revisiting inspiral-merger-ringdown consistency test results of GWTC-2 events': "In this section, we revisit the IMR consistency test results of GWTC-2 events which are summarized in Table IV of our previous analysis [11]. Here we describe the main reasons for this reanalysis.  \nFirst, for some events the parameter estimation analyses of the inspiral and the postinspiral parts used di GLYPH<11> erent prior  \nbounds. This is not necessarily problematic but it can lead to the 2D distribution of the prior on the fractional deviation parameters ( GLYPH<1> M f = ¯ M f ; GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f ) peaking away from (0 ; 0). Such priors are undesirable, since we do not want to prefer a GR deviation a priori . The prior distributions on the deviation parameters for GW170823 and GW190503 185404 peaked significantly away from (0 ; 0), so we reanalyzed these events using the same priors for masses and spins in the inspiral and postinspiral analyses. The GR quantile value for GW190503 185404 (94 : 3%) is significantly higher than its previous value (53 : 2%). This can be attributed to the fact that the new prior for this event peaks at zero whereas the old prior peaked close to the peak of the posterior (well away from zero). The new and old posteriors peak at almost the same place, causing the reweighted posterior to shift further away from (0 ; 0).  \nSecond, in our GWTC-1 analysis [10], the prior distributions on the fractional mass and spin parameters of O1 and O2 events were computed only using the prior range on component masses, and not accounting for the additional constraints on the mass priors. This was discontinued for O3a events in GWTC2 [11], where prior samples were generated considering the full set of priors. However, the GWTC-1 priors were used for O1 / O2 events. To maintain uniformity, we recomputed the priors for O1 / O2 events which were then used to reweight the posteriors. The old prior for the event GW170814 favored fractional mass deviation parameters further away from zero compared to the new prior which pushed the portion of the posterior with less probability closer to the origin. This is likely the reason why the GR quantile value of GW170814 in our previous analysis [11] is significantly higher (22 : 9%) than the current value (9 : 9%).  \nThird, we change the limits of the fractional deviation parameters, GLYPH<1> M f = ¯ M f and GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f . As can be seen from Fig. 3 of the GWTC-2 analysis [11], the 90% credible regions of the posteriors on ( GLYPH<1> M f = ¯ M f ; GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f ) for a few of the GWTC-2 events such as GW190814 were not closed within the range of the deviation parameters for which they were calculated. The ranges of deviation parameters were earlier set to [ GLYPH<0> 1 : 5 ; 1 : 5] for GLYPH<1> M f = ¯ M f and [ GLYPH<0> 1 ; 1] for GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f . We now set the ranges of both deviation parameters to [-2,2], which encloses all the 90% credible regions we find. This change in the ranges of deviation parameters has at most a small e GLYPH<11> ect on the GR quantiles, with the largest absolute di GLYPH<11> erence of 0.5 percentage points for GW190828 063405.  \nThe new results obtained with these three changes are given in Table XV and the posteriors are shown in Fig. 16. The three events whose contours do not enclose the origin are GW170823 (orange dot-dashed), GW190503 185404 (orange solid), and GW190814 (blue solid). Additionally, GW170814, GW170818, and GW190828 063405 show some small multimodal structures. The possible reasons for the high Q 2D GR values for GW170823 and GW190814 have already been discussed in our previous study [11]. Specifically, GW170823 is the event with the lowest SNR among the events in Table XV and has a relatively high redshifted mass, while GW190814 has a low SNR in its postinspiral part due to its low redshifted mass leading to significant bias in its final spin measurements compared to the relatively high SNR inspiral regime. This  \nbias leads to the marginalized GW190814 GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f posterior (blue solid) in the right side panel of Fig. 16 peaking significantly away from the GR value. GW190503 185404, and GW190814 also have prominent bimodal and multimodal posteriors which can be seen as additional peaks away from GR value in the marginalized GLYPH<1> M f = ¯ M f and GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f posteriors. This significant bias from the GR value for GW190814 is the reason why we see the GLYPH<27> posterior of GLYPH<1> GLYPH<31> f = ¯ GLYPH<31> f in Fig. 5 peaking away from zero for GWTC-2 and, in turn, for GWTC-3. Considering the change in Q 2D GR values with respect to our previous analysis [11], we have already explained why we see significant di GLYPH<11> erences for GW170814 and GW190503 185404 with absolute di GLYPH<11> erences of 13 : 0 and 41 : 1 percentage points, respectively. The next largest absolute change is 2 : 3 percentage  \n- [1] J. Aasi et al. (LIGO Scientific Collaboration), Class. Quantum Grav. 32 , 074001 (2015), arXiv:1411.4547 [gr-qc].\n- [2] F. Acernese et al. (Virgo Collaboration), Class. 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