Evidence for a Supergalactic Structure of Magnetic Deflection Multiplets of Ultra-High Energy Cosmic Rays
Telescope Array Collaboration, R.U. Abbasi, M. Abe, T. Abu-Zayyad, M. Allen, R. Azuma, E. Barcikowski, J.W. Belz, D.R. Bergman, S.A. Blake, R. Cady, B.G. Cheon, J. Chiba, M. Chikawa, A. di Matteo, T. Fujii, K. Fujisue, K. Fujita, R. Fujiwara, M. Fukushima, G. Furlich, W. Hanlon, M. Hayashi, N. Hayashida, K. Hibino, R. Higuchi, K. Honda, D. Ikeda, T. Inadomi, N. Inoue, T. Ishii, R. Ishimori, H. Ito, D. Ivanov, H. Iwakura, H.M. Jeong, S. Jeong, C.C.H. Jui, K. Kadota, F. Kakimoto, O. Kalashev, K. Kasahara, S. Kasami, H. Kawai, S. Kawakami, S. Kawana, K. Kawata, E. Kido, H.B. Kim, J.H. Kim, J.H. Kim, M.H. Kim, S.W. Kim, S. Kishigami, V. Kuzmin, M. Kuznetsov, Y.J. Kwon, K.H. Lee, B. Lubsandorzhiev, J.P. Lundquist, K. Machida, H. Matsumiya, T. Matsuyama, J.N. Matthews, R. Mayta, M. Minamino, K. Mukai, I. Myers, S. Nagataki, K. Nakai, R. Nakamura, T. Nakamura, Y. Nakamura, T. Nonaka, H. Oda, S. Ogio, M. Ohnishi, H. Ohoka, Y. Oku, T. Okuda, Y. Omura, M. Ono, R. Onogi, A. Oshima, S. Ozawa, I.H. Park, M.S. Pshirkov, J. Remington, D.C. Rodriguez, G. Rubtsov, D. Ryu, H. Sagawa, R. Sahara, Y. Saito, N. Sakaki, T. Sako, N. Sakurai, K. Sano, T. Seki, K. Sekino, et al. (44 additional authors not shown)
PP RE - PRINT J ULY
3, 2020Typeset using L A TEX twocolumn style in AASTeX62
Evidence for a Supergalactic Structure of Magnetic Deflection Multiplets of Ultra-High Energy Cosmic Rays
R.U. A
BBASI , M. A BE , T. A BU -Z AYYAD , M. A
LLEN , R. A
ZUMA , E. B
ARCIKOWSKI , J.W. B
ELZ , D.R. B
ERGMAN , S.A. B
LAKE , R. C
ADY , B.G. C
HEON , J. C
HIBA , M. C
HIKAWA , A. DI M ATTEO , ∗ T. F
UJII , K. F
UJISUE , K. F
UJITA , R. F
UJIWARA , M. F
UKUSHIMA ,
7, 11
G. F
URLICH , W. H
ANLON , M. H
AYASHI , N. H
AYASHIDA , K. H
IBINO , R. H
IGUCHI , K. H
ONDA , D. I
KEDA , T. I
NADOMI , N. I
NOUE , T. I
SHII , R. I
SHIMORI , H. I TO , D. I
VANOV , H. I
WAKURA , H.M. J
EONG , S. J
EONG , C.C.H. J UI , K. K
ADOTA , F. K
AKIMOTO , O. K
ALASHEV , K. K
ASAHARA , S. K
ASAMI , H. K
AWAI , S. K
AWAKAMI , S. K
AWANA , K. K
AWATA , E. K
IDO , H.B. K IM , J.H. K IM , J.H. K IM , M.H. K IM , S.W. K IM , S. K
ISHIGAMI , V. K
UZMIN , † M. K
UZNETSOV ,
20, 24
Y.J. K
WON , K.H. L EE , B. L
UBSANDORZHIEV , J.P. L
UNDQUIST ,
3, 26
K. M
ACHIDA , H. M
ATSUMIYA , T. M
ATSUYAMA , J.N. M
ATTHEWS , R. M
AYTA , M. M
INAMINO , K. M
UKAI , I. M
YERS , S. N
AGATAKI , K. N
AKAI , R. N
AKAMURA , T. N
AKAMURA , Y. N
AKAMURA , T. N
ONAKA , H. O DA , S. O
GIO ,
10, 28
M. O
HNISHI , H. O
HOKA , Y. O KU , T. O
KUDA , Y. O
MURA , M. O NO , R. O
NOGI , A. O
SHIMA , S. O
ZAWA , I.H. P
ARK , M.S. P
SHIRKOV ,
20, 31
J. R
EMINGTON , D.C. R
ODRIGUEZ , G. R
UBTSOV , D. R YU , H. S
AGAWA , R. S
AHARA , Y. S
AITO , N. S
AKAKI , T. S
AKO , N. S
AKURAI , K. S
ANO , T. S
EKI , K. S
EKINO , P.D. S
HAH , F. S
HIBATA , T. S
HIBATA , H. S
HIMODAIRA , B.K. S
HIN , H.S. S
HIN , J.D. S
MITH , P. S
OKOLSKY , N. S
ONE , B.T. S
TOKES , T.A. S
TROMAN , T. S
UZAWA , Y. T
AKAGI , Y. T
AKAHASHI , M. T
AKAMURA , R. T
AKEISHI , A. T
AKETA , M. T
AKITA , Y. T
AMEDA , H. T
ANAKA , K. T
ANAKA , M. T
ANAKA , Y. T
ANOUE , S.B. T
HOMAS , G.B. T
HOMSON , P. T
INYAKOV ,
20, 24
I. T
KACHEV , H. T
OKUNO , T. T
OMIDA , S. T
ROITSKY , Y. T
SUNESADA ,
10, 28
Y. U
CHIHORI , S. U DO , T. U
EHAMA , F. U
RBAN , T. W
ONG , K. Y
ADA , M. Y
AMAMOTO , K. Y
AMAZAKI , J. Y
ANG , K. Y
ASHIRO , M. Y
OSEI , Y. Z
HEZHER ,
7, 20
AND
Z. Z
UNDEL Department of Physics, Loyola University Chicago, Chicago, Illinois, USA The Graduate School of Science and Engineering, Saitama University, Saitama, Saitama, Japan High Energy Astrophysics Institute and Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah, USA Graduate School of Science and Engineering, Tokyo Institute of Technology, Meguro, Tokyo, Japan Department of Physics and The Research Institute of Natural Science, Hanyang University, Seongdong-gu, Seoul, Korea Department of Physics, Tokyo University of Science, Noda, Chiba, Japan Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba, Japan Service de Physique Th´eorique, Universit´e Libre de Bruxelles, Brussels, Belgium The Hakubi Center for Advanced Research and Graduate School of Science, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto, Japan Graduate School of Science, Osaka City University, Osaka, Osaka, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, University of Tokyo, Kashiwa, Chiba, Japan Information Engineering Graduate School of Science and Technology, Shinshu University, Nagano, Nagano, Japan Faculty of Engineering, Kanagawa University, Yokohama, Kanagawa, Japan Interdisciplinary Graduate School of Medicine and Engineering, University of Yamanashi, Kofu, Yamanashi, Japan Earthquake Research Institute, University of Tokyo, Bunkyo-ku, Tokyo, Japan Academic Assembly School of Science and Technology Institute of Engineering, Shinshu University, Nagano, Nagano, Japan Astrophysical Big Bang Laboratory, RIKEN, Wako, Saitama, Japan Department of Physics, Sungkyunkwan University, Jang-an-gu, Suwon, Korea Department of Physics, Tokyo City University, Setagaya-ku, Tokyo, Japan Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia Faculty of Systems Engineering and Science, Shibaura Institute of Technology, Minato-ku, Tokyo, Japan Department of Engineering Science, Faculty of Engineering, Osaka Electro-Communication University, Neyagawa-shi, Osaka, Japan Department of Physics, Chiba University, Chiba, Chiba, Japan Service de Physique Th´eorique, Universit´eLibre de Bruxelles, Brussels, Belgium Department of Physics, Yonsei University, Seodaemun-gu, Seoul, Korea Center for Astrophysics and Cosmology, University of Nova Gorica, Ajdovˇsˇcina, Slovenia Faculty of Science, Kochi University, Kochi, Kochi, Japan Nambu Yoichiro Institute of Theoretical and Experimental Physics, Osaka City University, Osaka, Osaka, Japan Department of Physical Sciences, Ritsumeikan University, Kusatsu, Shiga, Japan Quantum ICT Advanced Development Center, National Institute for Information and Communications Technology, Koganei, Tokyo, Japan Sternberg Astronomical Institute, Moscow M.V. Lomonosov State University, Moscow, Russia
Corresponding author: J.P. [email protected] a r X i v : . [ a s t r o - ph . H E ] J u l A BBASI ET AL . Department of Physics, School of Natural Sciences, Ulsan National Institute of Science and Technology, UNIST-gil, Ulsan, Korea Graduate School of Information Sciences, Hiroshima City University, Hiroshima, Hiroshima, Japan Institute of Particle and Nuclear Studies, KEK, Tsukuba, Ibaraki, Japan Department of Research Planning and Promotion, Quantum Medical Science Directorate, National Institutes for Quantum and Radiological Science andTechnology, Chiba, Chiba, Japan CEICO, Institute of Physics, Czech Academy of Sciences, Prague, Czech Republic Department of Physics and Institute for the Early Universe, Ewha Womans University, Seodaaemun-gu, Seoul, Korea
Submitted to ApJABSTRACTEvidence for a large-scale supergalactic cosmic ray multiplet (arrival directions correlated with energy) struc-ture is reported for ultra-high energy cosmic ray (UHECR) energies above 10 eV using seven years of datafrom the Telescope Array (TA) surface detector and updated to 10 years. Previous energy-position correlationstudies have made assumptions regarding magnetic field shapes and strength, and UHECR composition. Herethe assumption tested is that, since the supergalactic plane is a fit to the average matter density of the localLarge Scale Structure (LSS), UHECR sources and intervening extragalactic magnetic fields are correlated withthis plane. This supergalactic deflection hypothesis is tested by the entire field-of-view (FOV) behavior of thestrength of intermediate-scale energy-angle correlations. These multiplets are measured in spherical cap sectionbins (wedges) of the FOV to account for coherent and random magnetic fields. The structure found is con-sistent with supergalactic deflection, the previously published energy spectrum anisotropy results of TA (thehotspot and coldspot), and toy-model simulations of a supergalactic magnetic sheet. The seven year data post-trial significance of this supergalactic structure of multiplets appearing by chance, on an isotropic sky, is foundby Monte Carlo simulation to be 4.2 σ . The ten years of data post-trial significance is 4.1 σ . Furthermore, thestarburst galaxy M82 is shown to be a possible source of the TA Hotspot, and an estimate of the supergalacticmagnetic field using UHECR measurements is presented. Keywords: astroparticle physics, cosmic rays, UHECR, supergalactic plane, multiplets, magnetic deflection,large-scale structure of universe INTRODUCTIONThe supergalactic plane (SGP) is the average matter dis-tribution of the local universe up to a distance of ∼
200 Mpc(a large percentage of its sources are within the GZK hori-zon of 100 Mpc) de Vaucouleurs (1975). Large scale mag-netic fields have been measured between clusters of galaxies,which make up the supergalactic plane, including the ComaCluster, and a ∼ ∼
90% of the baryonic mass ofthe universe is contained between galaxies, of which ∼ ∗ Currently at INFN, sezione di Torino, Turin, Italy † Deceased
Previous UHECR energy-position correlation (multiplet)searches for small scale galactic magnetic deflections havenot had significant results (Abreu et al. (2012), Aab et al.(2015), Bretz (2011), Wirtz et al. (2019)). These multipletsearches used linear correlations of angular distance versus /energy and included scanned parameters chosen by as-sumed magnetic field models and compositions. The presentanalysis uses intermediate-scale energy-position correlations(multiplets) to look for significant large scale magnetic, andsource, structure with minimal assumptions regarding partic-ular magnetic field models or UHECR composition.In this paper the oversampled multiplets are found at gridpoints evenly covering the field-of-view sky (FOV), eachhaving their own parameters of size, shape, pointing direc-tion, and energy threshold. The structure of these multi-plets is consistent with supergalactic deflection, the previ-ously published energy spectrum anisotropy results of TA(the hotspot and coldspot) (Abbasi et al. (2018a)), and toy-model simulations of a supergalactic magnetic sheet Bier-mann et al. (1997). Here we report the significance usingseven years of Telescope Array (TA) data (as in Lundquist &Sokolsky (2019)) and update it to ten years of data. UPERGALACTIC S TRUCTURE OF M ULTIPLETS OF
UHECR 3 ENERGY-ANGLE CORRELATIONSIt is assumed that UHECR are deflected as they travelthrough coherent magnetic fields according to Equation 1a,with a deflection variance by random fields as approximatedby Equation 1b ( Z is mass number, B is field strength, S is distance traveled in the field, E is particle energy, and L c is mean magnetic field coherence length). These deflectionequations are from Roulet (2004) in units more relevant tothe extragalactic case. The end effect of these fields is thatlower energy cosmic ray events are deflected to larger an-gular distances from their source than higher energy eventsin both lateral and transverse directions Roulet (2004). Thisdrift-diffusion process is diagrammed in Figure 1. δ ≈ . ◦ Z B nG S Mpc 10 eV E (1a) δ rms ≈ . ◦ Z B rms nG 10 eV E (cid:115) S Mpc (cid:115) L c Mpc (1b)2.1.
Correlation
The distance between two points on the surface of a sphere,the great circle angular distance, is shown in Equation 2 interms of vectors normal to the field-of-view. Correlations be-tween event energy and angular distance from a grid point arefound using a ranked correlation, Kendall’s τ , that measuresthe strength of monotonic dependence Kendall (1945). δ ij = arctan | ˆn i × ˆn j | ˆn i · ˆn j (2)The Kendall correlation is generally more robust againstnoise than the other common ranked correlation - Spear-man’s ρ Croux & Dehon (2010). Ranked correlation mini-mizes the effects on correlation strength by magnetic model(such as higher-order terms of Equation 1a), compositionassumption, energy reconstruction systematics, and detectorexposure variation.Kendall’s τ ranked correlation is the linear correlation be-tween the separate ordering of the two variables of interest(variable x sorted ranks: 1st, 2nd, 3rd, etc. versus variable y ranks: 5th, 1st, 4th, etc.), with n pairs of values, and is shownin Equation 3. τ = 2 n ( n − (cid:88) j Pictograph of UHECR Drift-diffusion deflection “wedge”bins (spherical cap sections) displayed on a flat space. (a) Two dif-ferent energy events having traveled through coherent and randommagnetic fields. The purple vector represents the low energy eventspherical arc, and the red vector is a higher energy event. Coherentand random magnetic field components describe the average per-pendicular to the field-of-view (FOV) sphere. Dashed circles rep-resent possible random field RMS deflections. (b) A spherical capsection (wedge) is a simple shape that best encompasses the like-liest positions. Pointing direction is the spherical arc φ , ∆ φ is thewedge width, and D is the maximum angular distance (sphericalcap radius). The correlation coefficient τ has a range from − to +1 ,and a value of zero means that there is no measured relation-ship between the variables. For +1 , an increase ( decrease )of x always follows an increase ( decrease ) of y . If τ = − an increase ( decrease ) of x always follows a decrease( increase ) in y (in this analysis x and y are energy and an-gular distance). A negative correlation is consistent with theexpectation for magnetic field deflected events - as energy de-creases, deflection increases, as can be seen from Equation 1.Any monotonic function ( x b , log ( x ) , e x , etc.) of dis-tance, energy, or both will always return a τ coefficient withthe same magnitude but not necessarily the same sign ( ± ). A BBASI ET AL . Figure 2. A supergalactic Hammer-Aitoff projection of the equaldistance oversampling grid. This is a grid of 6553 points with amean spacing of 2.1 ± . ◦ . The grid boundary is defined by theequatorial edge of the field of view at Dec. = -16 ◦ . The red diamondis the location of the Hotspot (Abbasi et al. (2014)), and the greendiamond is the location of the energy spectrum anisotropy Abbasiet al. (2018a). The red line is the supergalactic plane (SGP) and theblue line is the galactic plane (GP). The sign of the resulting τ would be the original τ multipliedby the signs of the first derivatives of the applied functions.The pre-trial two-sided significance of a correlation, z ,(probability of τ =0 ) is a function of correlation strength andsample size n . This significance is found by counting per-mutations of the sample ranks with greater τ , or in the large n sample limit, Equation 4 (for n ≥ z = τ n ( n − / (cid:112) n ( n − n + 5) / (4)2.2. Correlation Binning With the drift-diffusion picture of Figure 1 in mind, pos-sible UHECR deflections from grid point “sources” werefound by a scanned maximization of the significance ofenergy-angle correlations inside spherical cap sections, or“wedges,” using seven years of TA data Lundquist & Sokol-sky (2019). This scan was done at each point on an ap-proximately equal 2 ◦ spaced grid of 6553 points on the FOVshown in Figure 2 Teanby (2006).These wedge bins are defined by a maximum angular dis-tance δ j from the grid point, i , defined by Equation 2 and theboundaries of two azimuths defined by Equation 5 where B is latitude and L is longitude. φ ij = arctan cos B i sin ( L i − L j )cos B j sin B i − sin B j cos B i cos ( L i − L j ) (5)The azimuths increase clockwise and a great circle section,or wedge, pointed towards 90 ◦ supergalactic latitude (SGB)has an azimuth, φ , of zero. While one pointed towards -90 ◦ SGB has a φ of 180 ◦ . The azimuthal angle difference, ∆ φ ij , between the wedge pointing direction, φ i , and the azimuth ofan event, φ ij , is shown in Equation 6. An example wedge isshown in Figure 3(a). ∆ φ ij = mod( | φ ij − φ i | + 180 , − (6)This oversampling bin shape means that four parametersmust be scanned at every grid point to maximize the pre-trialcorrelation significance. Even though negative correlationsare physically expected by a magnetic field drift-diffusionprocess; the sign of the correlation, and its strength, are notexplicitly scanned for nor restricted. The limits on theseparameters are large to account for most conceivable ex-tragalactic magnetic deflection scenarios. The scans are allcombinations of the following:1. Energy Threshold, E i : 10 to 80 EeV in 5 EeV steps.2. Wedge Distance, D i =max( δ ij ) : 15 ◦ to 90 ◦ in steps of5 ◦ .3. Wedge Direction, φ i : 0 ◦ to 355 ◦ , 5 ◦ steps.4. Wedge Width, W i = 2 ∗ max( | ∆ φ ij | ) : 10 ◦ to 90 ◦ , 10 ◦ steps (5 ◦ on each side of φ i ).Events are inside the wedge if E j ≥ E i & δ ij ≤ D i & − W i / ≤ ∆ φ ij ≤ W i / , where i is the index of the gridpoint. The energy-angle correlation is calculated inside thewedge, τ ( δ ij , E j ) , and the parameters ( E i , D i , φ i , and W i )are chosen such that the correlation has the minimum p-value(Equation 4). This scan was done using seven years of data.The same bin parameters at each grid point were used for theten years of data set to test the result.2.3. Correlation Example The wedge parameters needed to maximize the correla-tion significance at each grid point were scanned for usingseven years of TA data Lundquist & Sokolsky (2019). Forthe seven-year data set, the supergalactic coordinates of themost significant correlation of all the grid points is 18.3 ◦ SGB (latitude), -12.9 ◦ SGL (longitude). This wedge, andthe events inside, are shown in Figure 3(a). There are 29events with energies E ≥ 30 EeV. The pre-trial one-sided sig-nificance of a τ = − . with a sample size of 29 events is5.5 σ . A scatter plot of energy versus angular distance fromthe grid point within this wedge is shown in Figure 3(b). Alinear fit (Equation 1a with Z =1 ) results in an estimate of B × S = 49 nG*Mpc. If the source is assumed to be at thedistance to M82 (3.7 Mpc) with a pure proton emission, theaverage coherent magnetic field, perpendicular to the FOV,required to cause this deflection would be B =13 nG.Note however, that the post-trial significance of any singlecorrelation is not expected to be large, as the wedge scanparameter space is large. An individual correlation is not thetest of a supergalactic structure. UPERGALACTIC S TRUCTURE OF M ULTIPLETS OF UHECR 5 (a) (b) Figure 3. (a) A supergalactic Hammer-Aitoff projection of the seven years data spherical cap section, or “wedge”, with the maximum sig-nificance at 18.3 ◦ SGB, -12.9 ◦ SGL. The correlation strength is τ = − . , and with 29 data events has a pre-trial one-sided significance of5.5 σ . The energy threshold is E i ≥ EeV, wedge width W i =30 ◦ , distance D i =80 ◦ , and direction φ i =90 ◦ . (b) A scatter plot of /E j versusdistance δ ij for events in the wedge. A linear fit (by Equation 1a with Z =1 ) results in an estimate of B × S = 49 nG*Mpc. If the sourceis assumed to be the same distance as M82 (3.7 Mpc) with a pure proton emission, then the coherent magnetic field required to cause thisdeflection would be B =13 nG.3. SIMULATIONSThe same analysis is applied to isotropic simulations inorder to calculate the significance of any anisotropy (as de-scribed further in Section 4). This is a simulation of data withthe TA SD configuration while assuming no specific sourcesor correlation with the supergalactic (or galactic) plane.A second simulation is used to demonstrate that the anal-ysis is able to find the hypothesized supergalactic structure.This is a simple toy-model simulation of a supergalactic mag-netic sheet that results in an energy-dependent diffusion ofevents away from the supergalactic plane. This sheet simu-lation is used to motivate the test statistic that tests the hy-pothesis of supergalactic sources and magnetic fields; this isfurther described in Section 4. This simulation can also beused to estimate the average coherent field strength betweenour galaxy and supergalactic sources.3.1. Isotropic Simulation Each Monte Carlo (MC), and data event, is defined by theirenergy, zenith angle, azimuthal angle, and trigger time. Thelatitude and longitude are defined from the center of TA at39.3 ◦ Lat., 112.9 ◦ Long. These horizontal coordinates areused to calculate the longitude (SGL) and latitude (SGB) insupergalactic coordinates Vallado (2007). The MC event setshave a zenith angle distribution of g( θ ) = sin ( θ ) cos ( θ ) due tothe event sampling response of a two-dimensional SD array,a uniform azimuth distribution, and the detection efficiency ∼ ≥ . eV. The event trigger timesare approximated as a uniform distribution of modified Ju- lian dates from the beginning to the end of the run time dueto the approximately ∼ ∼ × with en-ergies E ≥ . eV. The number of events in each isotropicMC event set is the same as data in each 5 EeV bin of theparameter scan of Section 2.2. This simulated data has beenshown to reproduce all measured geometric and photoelectricdistributions accurately Ivanov (2012).The result is that each set of these isotropic MC eventssimulates the expected data, given the detector configurationand on-time, with no energy anisotropies. These MC sets areused to calculate the post-trial probability of any potentialanisotropy signal in the data.3.2. Supergalactic Magnetic Sheet A simple toy-model simulation of an intervening super-galactic magnetic sheet, between the galaxy and UHECRsources, is made by taking the isotropic event sets of Sec-tion 3.1 and embedding event deflections (assigning distancecorrelated energies) in supergalactic latitude (SGB), propor-tional to /energy , for a fraction of events. The coordinatesof the MC events are isotropic and unchanged in the proce-dure. The approximate apparent deflection from the source of A BBASI ET AL .a charged particle in a coherent magnetic field is from Equa-tion 1a. An example of the resulting simulation is shown inFigure 4.The event deflections, δ B , from supergalactic latitude SGB = 0 ◦ are calculated for each MC event energy in theset, assuming a proton composition ( Z =1 ) and a particular B × S , according to Equation 1a. Additionally, some randomfield noise was added by smearing the δ B with a 5 ◦ standarddeviation Gaussian. Then each event is assigned an energybased on its angular distance from the supergalactic plane( min [ δ B - δ SGB =0 ]). The beginning, and final, simulation isisotropic with respect to the supergalactic longitude (SGL).After the assignment of an energy to each event position,those with an assigned position-deflection error greater than10 ◦ are added to the isotropic proportion. This threshold addsadditional random field noise in the simulation. This cut alsoresults in a harder spectrum for the deflected events (red eventin Figure 4) i.e. higher energy events on average closer to thesupergalactic plane. This supergalactic energy bias is due tothe lower number of high energy events resulting in a betterfit to a supergalactic magnetic deflection at higher energies(due to the boundary conditions of the energy spectrum andposition isotropy).A supergalactic sheet simulation, with an F =65 . isotropic fraction and B × S =18 . nG*Mpc, is shown inFigure 4. These parameters are the result of selecting arandom MC that looks similar to the data result and thechoice of a proton composition. Note again that this is onlyan anisotropy of energies in supergalactic latitude as eventpositions are isotropic, and the total energy spectrum is un-changed.The intent of this toy-model simulation is to show that theanalysis method is sensitive to an energy symmetry caused bysome kind of magnetic deflection structure correlated withthe supergalactic plane. It is not intended to reproduce allaspects of actual data. SUPERGALACTIC STRUCTURENo single correlation tests the hypothesis that sources andmagnetic fields have a relation to local large scale structure.And no single correlation can be significant considering theaverage ∼ F =65 . isotropy and B × S =18 . nG*Mpc).It can be seen via this simple model in the projection of τ that if there are magnetically induced energy-angle corre-lations clustered in the supergalactic plane, negative correla-tion wedges will be close to the supergalactic plane. Further-more, since negative correlations viewed from the opposite Figure 4. Toy-model supergalactic magnetic sheet simulation. Bluecircles are the F = 65 . isotropic fraction of MC events. Redsquares are the anisotropic MC events magnetically diffused awayfrom the supergalactic plane with B × S =18 . nG*Mpc. Overall,event positions are isotropic, and the energy spectrum is createdaccording to the published HiRes/TA result. direction appear as positive correlations (as can been seen byEquation 3 for [ x = E, y = D ] → [ x = E, y = -D ]) , positive corre-lations are expected at large distances from the supergalacticplane. 4.1. Significance Test Though a test for a supergalactic structure of energy-anglecorrelations is not necessarily a priori obvious, the super-galactic sheet toy-model leads to a reasonable answer. Themean <τ > inside equal solid angle bins of angular distance( SGB i ) from the supergalactic plane (SGP) shows that threefeatures are relevant for the supergalactic hypothesis - theminimum average τ , the minimum location being near theSGP, and the symmetry of τ around the SGP. Using all threefeatures to calculate the data significance would be overfit-ting the problem. One test statistic is preferable though itshould be correlated with these three supergalactic structurefeatures. The single parameter chosen to test the supergalac-tic structure hypothesis is the curvature parameter, “ a ,” of aparabolic fit (y = a ( x − x ) + y ) to the <τ > .The curvature, “ a ”, is simply the lowest order Taylor ex-pansion term that can describe the symmetry around the SGPshown in simulation (Figure 5(b)). Due to the boundaries of | τ |≤ | SGB | < ◦ , greater correlation curvature, a , cor-responds to a minimum, x , closer to the supergalactic plane,as shown in Figure 6(a). A larger curvature a also meansthat the minimum negative correlation averages are greater inmagnitude, y , as shown in Figure 6(b). The parabola min-imum y has no correlation with the minimum supergalac-tic latitude (SGB). These relationships justify the use of theparabola curvature “a” as the single test statistic for a conser- UPERGALACTIC S TRUCTURE OF M ULTIPLETS OF UHECR 7 (a)(b) Figure 5. A supergalactic magnetic sheet simulation. (a) Projectionof the correlation strength τ for all grid points. Solid curves indicatethe galactic plane (GP) in blue and supergalactic plane (SGP) in red.(b) Mean τ inside equal solid angle bins of supergalactic latitude(SGB). The parabolic fit ( y = a ( x − x ) + y ) shows the curvatureparameter, a , chosen as the test statistic. a =2 . × − . vative estimate of the significance of supergalactic energy-angle correlations.The fit on the large scale behavior of the correlationstrength, τ , is used because it is not explicitly scanned for andcontains more information by its sign ( ± ) than the pre-trialsignificance. The pre-trial significance of the correlations isnot used in this analysis so that the significance test is inde-pendent of the wedge scan for the maximum significance ofindividual energy-angle correlations.To calculate the data significance of a supergalactic struc-ture of energy-angle correlations, the analysis describedabove was applied to the data and the isotropic MC sets. Thenumber of MC sets with a correlation curvature a greater thanthe data gives the probability of the measured supergalacticstructure of energy-angle correlations if there actually isn’tsuch structure i.e. if it is a statistical fluctuation in the data. (a)(b) Figure 6. The behavior of the three mean τ parabola fit param-eters (Figure 5(b)) with respect to each other in random isotropicMC simulations. (a) The parabola fit curvature, a , versus minimumsupergalactic latitude (SGB), x , shows that a high curvature tendsto a minimum near the supergalactic plane. (b) The parabola fitcurvature, a , versus the fit minimum value, y , shows that a highcurvature tends to a higher magnitude negative mean τ .5. DATA SETFor this analysis, Surface Detector (SD) data recorded be-tween May 11 of 2008 and 2019 is used. Data from 2016 isexcluded due to issues with SD communication towers thatcreated a significant day to day change of the trigger delayvariance within each day of the year. This introduced non-physical equatorial anisotropies that are non-trivial to com-pensate for.The reconstruction method used for these events is thesame as the “Hotspot” and energy spectrum anisotropy re- A BBASI ET AL .sults (Abbasi et al. (2014), Abbasi et al. (2018a)). The energyof reconstructed events is determined by the SD array andrenormalized by 1/1.27 to match the calorimetrically deter-mined fluorescence detector energy scale (Abu-Zayyad et al.(2013)).Due to the inclusion of lower energy events, down to10 . eV, tighter data cuts than the hotspot analysis are re-quired for good zenith angle and energy resolutions. Aftercuts, there were 3018 events in the seven-year data set, andthere is a total of 4321 events using ten years of data. Eventsin the data set match the following criteria:1. E ≥ . eV (where detection efficiency is ∼ < ◦ .4. Shower lateral distribution fit χ /dof < < ◦ .6. Shower core > θ ) = sin ( θ ) cos ( θ ) . The azimuthal angle distribution isin very good agreement with the theoretical uniform distri-bution. The geometrical zenith angle distribution is due tothe flat detector plane, the uniform azimuthal angle distribu-tion, and the detection efficiency ∼ ≥ . eV. The energy spectrum is also in goodagreement with the published spectrum (Abu-Zayyad et al.(2013),Abbasi et al. (2015)). And finally, the event triggertimes are in good agreement with the uniform time distribu-tion used for the isotropic MC of Section 3.1.The energy resolution and pointing direction resolution ofevents in the data set range from ∼ 10 to 15% and ∼ ◦ to1.5 ◦ , respectively, depending on core distance from the arrayboundary and improve with increasing energy. These resolu-tions are sufficient to search for large-scale and intermediate-scale UHECR energy anisotropies. RESULTSThe resulting data energy-angle correlations for sevenyears of data are shown in Figure 7(a) and ten years of data isshown in Figure 8(a). Individual correlations with the high-est pre-trial significance are negative, which means that thereis a trend for the angular distance to increase with decreasingenergy. This trend is the expectation for a grid point that hap-pens to be near a source of magnetically scattered UHECRevents. It can be seen that the negative τ correlations appearthemselves well correlated with the supergalactic plane.Figure 7(b) shows the seven-year data result of the mean τ correlation inside equal solid angle bins parallel to thesupergalactic plane (SGP). The parabolic fit curvature is a =(2 . ± . × − with a minimum at − . ◦ SGB. Ac-cording to the R (Coefficient of Determination) goodness-of-fit the model predicts 88 % of the variance of the data. Thedata correlations have a very similar form to that of the su-pergalactic magnetic sheet simulation, shown in Figure 5(a),that has a =2 . × − with a minimum at − . ◦ SGB.Previously, by applying this analysis to isotropic MC sets(using data positions and random energies) the number ofMC with an a parameter greater than data was two out of200,000 trials which resulted in a post-trial significance ofthe supergalactic structure of multiplets of ∼ σ Lundquist &Sokolsky (2019). (a)(b) Figure 7. Seven year-data result. (a) Projection of the correlationstrength τ for all grid points. Negative correlations expected formagnetic deflections are apparent around the supergalactic plane.Solid curves indicate the galactic plane (GP) in blue and super-galactic plane (SGP) in red. White and grey hexagrams indicatethe galactic center (GC) and anti-galactic center (Anti-GC) respec-tively. (b) Mean τ inside equal solid angle bins of supergalactic lat-itude (SGB). The correlation curvature is a =(2 . ± . × − . Figure 8(b) shows the ten years of data mean τ corre-lation with no new scan of wedge parameters for maxi- UPERGALACTIC S TRUCTURE OF M ULTIPLETS OF UHECR 9mum correlation significances. The parabola curvature is a =(1 . ± . × − , and the minimum is at . ◦ SGB.According to the R = 0 . goodness-of-fit the model pre-dicts 91 % of the variance of the data. It can be seen that thecorrelations are similar to the seven-year result, though thesupergalactic structure may not be quite as significant. (a)(b) Figure 8. Ten years of data result. (a) Projection of the correlationstrength τ for all grid points. Negative correlations expected formagnetic deflections are apparent around the supergalactic plane.(b) Mean τ inside equal solid angle bins of supergalactic latitude(SGB). The correlation curvature is a =(1 . ± . × − . Significance of Supergalactic Structure By applying this analysis to isotropic MC sets, as describedin Section 3.1, and counting the number of MC with an a parameter larger than data (Figure 7(b)), the post-trial sig-nificance of the supergalactic structure of multiplets can befound. The resulting a distribution of 1,000,000 MC setsis shown in Figure 9 for the seven-year data statistics andenergy-angle correlation significance scan. Figure 9. The distribution of the curvature parameter a of the mean τ parabola chosen as the supergalactic structure of multiplets teststatistic for 1,000,000 isotropic MC sets. The purple bars are theMC probability distribution function (PDF). The red line is a Gaus-sian distribution fit to the MC distribution. The curvature for thedata is a =2 . × − shown as a blue vertical line. There are 14MC with a larger curvature than data, which gives a significance of4.2 σ . For the seven-year data analysis there are 14 MC sets witha larger curvature than data, which results in the significanceof a supergalactic structure of multiplets of ∼ σ .For the ten years of data with no updated wedge correlationsignificance scan, the resulting a distribution of 1,000,000MC sets is shown in Figure 10. The distribution has a smallerstandard deviation due to no new scan for energy-angle corre-lation significances. The result is smaller τ on average. Thereare 22 MC sets with a larger curvature than data, shown inFigure 8(b), which results in the significance of a supergalac-tic structure of multiplets of ∼ σ .The total number of MC sets that were used to calculatethe significance was limited by the computing time neces-sary for each simulation. Overall, the number of correlationscalculated was 4 × and this took more than 200 years ofequivalent CPU computing time.6.2. Scan Parameter Distributions Clues about UHECR sources, and intervening fields, maybe found from the maximum significance wedge scan param-eters of the apparent magnetic deflection multiplets. Due tothe significance maximization, there is a bias towards greaterstatistics, as can be seen in Equation 4, so the data is com-pared to isotropic MC by taking the ratio of the parame-ter probability distribution functions (PDF) (normalized his-tograms of data divided by MC). The PDF ratio shows howmany times more likely a scan parameter value is to be foundin data than isotropic MC. PDF ratio plots for wedge point-0 A BBASI ET AL .. BBASI ET AL .. Figure 10. The distribution of the curvature parameter a of themean τ parabola chosen as the supergalactic structure of multipletstest statistic for 900,000 isotropic MC sets. The purple bars arethe MC PDF. The red line is a Gaussian distribution fit to the MCdistribution. The curvature for the data is a =1 . × − shown asa blue vertical line. There are 22 MC with a larger curvature thandata, which gives a significance of 4.1 σ . ing direction and energy threshold parameters are shown inFigure 11.These ratios are done for negative energy-angle correla-tions at grid point positions | SGB | ≤ ◦ (about the bound-ary where the average correlation is zero as shown in Figure8(a)) and have a linear fit to /E versus angular distance withan R > (Figure 3(b)). An R > is a better fit than a hori-zontal line, and the δ ∝ /E model explains some of the vari-ance of the data inside the wedge. For data, there are 2045correlations used and greater than 3.99 × for MC.The data distribution of wedge pointing directions, Fig-ure 11(a), provides further indication of a supergalactic struc-ture with four deviations seemingly correlated with the su-pergalactic plane (SGP). Two larger peaks are approximatelyperpendicular to the SGP ( ∼ ◦ and ∼ ◦ ), and twosmaller peaks close to parallel ( ∼ ◦ and ∼ ◦ ). Thesepeaks suggest an overall diffusion of low energy events awayfrom the supergalactic plane, similar to the supergalacticmagnetic sheet simulation of Section 3.2.The data distribution of the energy threshold parametersmay provide information regarding UHECR sources and in-tervening fields. The median energy threshold is 30 EeV, andthe three largest deviations from the isotropic distribution areat 35 EeV, 45 EeV, and 60 EeV. The 60 EeV peak appears tocorrespond to the 57 EeV threshold of the TA hotspot analy-sis Abbasi et al. (2014).The median energy threshold of 30 EeV is above the sig-nificant Pierre Auger Observatory (PAO) large scale dipolemeasurements in Aab et al. (2018a) at 8 EeV, which is consis- tent with the localized intermediate-scale energy-angle mag-netic deflections in this analysis.The 39 EeV cutoff for maximum event correlation withstarburst galaxies, reported by the PAO in Aab et al. (2018b),may be related to the 35 EeV and 45 EeV peaks.These threshold deviations from isotropy are also consis-tent with the result using AGASA data that showed a possiblelarge scale cross-correlation between UHECR and the super-galactic plane between 50 and 80 EeV energy bins Burgett &O’Malley (2003). Adjusting the AGASA energy scale to theTA energy scale by multiplying by 0.75, this becomes 38 and60 EeV energy bins.The data distributions of wedge angular distance, D , andwidth, W , do not show any significant deviations fromisotropy. 6.3. M82 Galaxy as Anisotropy Source The most significant single correlation using ten years ofSD data is at 30.3 ◦ SGB, -3.2 ◦ SGL, and shown in Fig-ure 12(a). With 75 events (E ≥ 35 EeV) and τ = − . , ithas a pre-trial significance of 5.10 σ . This significance is anincrease from 4.58 σ at this grid point, with seven years ofdata using the same wedge and energy threshold parameters.Figure 12(b) shows a scatter plot of energy versus angulardistance. A linear fit (Equation 1a with Z =1 ) results in anestimate of B × S = 41 nG*Mpc.Recently, the Pierre Auger Observatory (PAO) has statedthat the likeliest source of events with E > 39 EeV are star-burst galaxies Aab et al. (2018b). The most significant cor-relation reported here is 11.3 ◦ from M82 (as shown by theblue diamond in Figure 12(a)), pointing directly over the TAHotspot (Figure 2). M82 is the closest starburst galaxy to ourgalaxy.If the source is assumed to be at the same distance to M82(3.7 Mpc) with a pure proton emission, the average coherentmagnetic field perpendicular to the FOV required to causethis deflection is B =11 nG. The large deviations from thelinear fit of Figure 12(b) imply that, in this region, the randomfield deflections have a large correlation length scale ( L c ),and the random field ( B rms ) of Equation 1b is on the sameorder of magnitude as the coherent field deflection.Random variations of the data are created to estimate theuncertainty on the location of the source of the maximumsignificance energy-angle correlation. The energies of eventsoutside the wedge are scrambled with other events outsidethe wedge. Inside, the energies of wedge events with an en-ergy less than the wedge threshold, E < 35 EeV, are random-ized within the wedge. The locations of the 75 data eventsin the wedge with E ≥ 35 EeV are not changed. This ensuresthat the spectrum is not changed, inside or outside the wedge,and that the number of events E ≥ 35 EeV does not dramati-cally increase due to the Coldspot (Abbasi et al. (2018a)). UPERGALACTIC S TRUCTURE OF M ULTIPLETS OF UHECR 11 (a)(b) Figure 11. PDF ratio plot of scanned parameters. (a) Wedge point-ing direction parameter, φ . This distribution provides further indi-cation of the supergalactic structure of multiplets. The blue ver-tical lines are parallel to the SGP. The red lines are perpendicularto the SGP. Two significant peaks can be seen nearly perpendic-ular, and two smaller peaks near parallel, to the SGP. (b) Energythreshold, E . The three largest deviations are at 35 EeV, 45 EeV,and 60 EeV. This distribution may provide information regardingUHECR sources and intervening fields. The analysis, including scanning for maximum significancewedge correlations at all grid points, was repeated for 5000of these random variations on the data.The estimated location of each randomized data sets sourceis the most significant negative correlation near the knownsource grid point. The maximum distance searched, within aspherical cap centered on the known grid point, is the dis-tance that minimizes the average τ inside the cap (corre-lations are more positive outside). A spherical cap limiteris necessary due to the fact that an entirely different set ofevents, than the wedge of interest, say on the other side of (a)(b) Figure 12. (a) Supergalactic projection of the ten years data mostsignificant “wedge” multiplet at 30.3 ◦ SGB, -3.2 ◦ SGL. The cor-relation τ = − . with 75 data events has a pre-trial one-sidedsignificance of 5.10 σ . This significance is an increase from 4.58 σ at this grid point, with seven years of data. The energy threshold is E i ≥ EeV, wedge width W i =90 ◦ , angular distance D i =70 ◦ , anddirection φ i =120 ◦ . The blue diamond is the location of the star-burst galaxy M82. (b) Scatter plot of /E j versus angular distance δ j in the wedge. A linear fit (by Equation 1a with Z =1 ) results inan estimate of B × S = 41 nG*Mpc. If the source is assumed tobe at the same distance to M82 (3.7 Mpc) with a pure proton emis-sion, then the average coherent magnetic field required to cause thisdeflection would be B =11 nG. the FOV, can easily have a more significant correlation dueto the number of scans done at each grid point.The result is that the apparent sources have a median dis-tance of 2.4 ◦ from the original source with a +1 σ quantileof 6.8 ◦ and 6.2 % are greater than or equal to 11.3 ◦ away (theangular distance from M82 to the maximum significance gridpoint). The distribution of distances is shown in Figure 13.This distribution means that M82 is not excluded as a possi-ble source of the events in this energy-angle correlation andthe TA UHECR Hotspot/Coldspot Abbasi et al. (2018a).2 A BBASI ET AL .. PDF ratio plot of scanned parameters. (a) Wedge point-ing direction parameter, φ . This distribution provides further indi-cation of the supergalactic structure of multiplets. The blue ver-tical lines are parallel to the SGP. The red lines are perpendicularto the SGP. Two significant peaks can be seen nearly perpendic-ular, and two smaller peaks near parallel, to the SGP. (b) Energythreshold, E . The three largest deviations are at 35 EeV, 45 EeV,and 60 EeV. This distribution may provide information regardingUHECR sources and intervening fields. The analysis, including scanning for maximum significancewedge correlations at all grid points, was repeated for 5000of these random variations on the data.The estimated location of each randomized data sets sourceis the most significant negative correlation near the knownsource grid point. The maximum distance searched, within aspherical cap centered on the known grid point, is the dis-tance that minimizes the average τ inside the cap (corre-lations are more positive outside). A spherical cap limiteris necessary due to the fact that an entirely different set ofevents, than the wedge of interest, say on the other side of (a)(b) Figure 12. (a) Supergalactic projection of the ten years data mostsignificant “wedge” multiplet at 30.3 ◦ SGB, -3.2 ◦ SGL. The cor-relation τ = − . with 75 data events has a pre-trial one-sidedsignificance of 5.10 σ . This significance is an increase from 4.58 σ at this grid point, with seven years of data. The energy threshold is E i ≥ EeV, wedge width W i =90 ◦ , angular distance D i =70 ◦ , anddirection φ i =120 ◦ . The blue diamond is the location of the star-burst galaxy M82. (b) Scatter plot of /E j versus angular distance δ j in the wedge. A linear fit (by Equation 1a with Z =1 ) results inan estimate of B × S = 41 nG*Mpc. If the source is assumed tobe at the same distance to M82 (3.7 Mpc) with a pure proton emis-sion, then the average coherent magnetic field required to cause thisdeflection would be B =11 nG. the FOV, can easily have a more significant correlation dueto the number of scans done at each grid point.The result is that the apparent sources have a median dis-tance of 2.4 ◦ from the original source with a +1 σ quantileof 6.8 ◦ and 6.2 % are greater than or equal to 11.3 ◦ away (theangular distance from M82 to the maximum significance gridpoint). The distribution of distances is shown in Figure 13.This distribution means that M82 is not excluded as a possi-ble source of the events in this energy-angle correlation andthe TA UHECR Hotspot/Coldspot Abbasi et al. (2018a).2 A BBASI ET AL .. Figure 13. Distribution of distances from the actual most significantcorrelation grid point to those found in randomized data with thewedge embedded. The result presented here appears to be consistent with theresults of He et al. (2016) that used a Bayesian analysis ofthe relative deflection of TA Hotspot events in two energybins (E < 75 EeV and E > 75 EeV). Their result was a 99.8 % probability that M82 is the hotspot source.According to the recent light polarization measurement ofM82’s magnetic field in Jones et al. (2019), the integratedmagnetic field angle is 351 ◦ in equatorial coordinates usingthe same definition as Section 2.2. Rotating into supergalac-tic coordinates results in an angle of 308 ◦ . The coherent mag-netic field direction necessary to create the most significantmultiplet is 120 ± ◦ , so it is either 82 ◦ or 98 ◦ from M82’smagnetic field direction. The circular standard deviation ofthe pointing direction of the wedge simulations shown in Fig-ure 14 is 21 ◦ , which means the wedge magnetic field direc-tion is ≥ σ different from M82’s magnetic field direction.This direction discrepancy implies that if M82 is the sourcethen magnetic fields outside M82 were the primary source ofmultiplet pattern deflections.6.4. Supergalactic Field Estimate The average linear fit to /E versus angular distance fromthe grid point, inside wedges with a negative correlation, cangive an estimate of coherent magnetic field strength timesdistance traveled through the field (see Equation 1a as shownin Figure 3(b)). These B × S values are independent of theranked correlation pre-trial significances, which were maxi-mized to choose the wedge parameters.If the coherent magnetic field in the vicinity of posi-tive correlations is considered negligible, and those corre-lations are set to B × S = 0 , then the mean B × S in super-galactic latitude (SGB) bins appears as Figure 15. Given = 21 nG*Mpc and if the composition is protonic,then the average coherent field component, perpendicular to Figure 14. Distribution of pointing direction of wedges found inrandomized data with the most significant wedge embedded. the FOV, in the vicinity of the supergalactic plane ( | SGB | ≤ ◦ ) is 5.6 nG (assuming a source distance of 3.7 Mpc). Figure 15. Mean B × S inside equal solid angle SGB bins setting B × S = 0 for wedges with positive correlations. The fitted parabolaalso demonstrates the correlation between apparent magnetic de-flection multiplets with the supergalactic plane. These values areindependent of the ranked correlation pre-trial significances, whichwere maximized to choose the wedge parameters. The mean within | SGB | ≤ ◦ is = 21 nG*Mpc. If proton is the assumedcomposition, then the average coherent field component, perpendic-ular to the FOV, in the vicinity of the supergalactic plane assuminga distance of 3.7 Mpc is 5.6 nG. If the coherent magnetic field in the vicinity of positivecorrelations is considered to be unknown, and those correla-tions are ignored, then the mean B × S in supergalactic lat-itude (SGB) bins appears as Figure 16. If proton is the as-sumed composition, then the average coherent field compo- UPERGALACTIC S TRUCTURE OF M ULTIPLETS OF UHECR 13nent, perpendicular to the FOV, in the vicinity of the super-galactic plane assuming a distance of 3.7 Mpc is ∼ Figure 16. Mean B × S inside equal solid angle SGB bins notcounting wedges with positive correlations. These values are in-dependent of the ranked correlation pre-trial significances, whichwere maximized to choose the wedge parameters. The mean within | SGB | ≤ ◦ is = 32 nG*Mpc. If proton is the assumedcomposition, then the average coherent field component, perpendic-ular to the FOV, in the vicinity of the supergalactic plane assuminga distance of 3.7 Mpc is 8.6 nG. Recently in Globus et al. (2019), the best fit average ex-tragalactic field to the PAO dipole, assuming a local largescale structure (LSS) distribution of sources according toCosmicFlows-2 catalog, was estimated to be 0.6 nG us-ing the PAO mixed composition E ≥ S , necessary for agreement with PAO on the extragalacticfield strength is ∼ 50 Mpc. The mean distance of galaxiesin the CosmicFlows-2 catalog, within the GZK horizon of ∼ 100 Mpc, is 51 Mpc Tully et al. (2013). Given the modeland experimental uncertainties, TA and PAO seem to have agood order of magnitude agreement on the extragalactic fieldstrength. SYSTEMATIC CHECKSA test of variation of isotropic MC parameters was doneby calculating the significance of the supergalactic structurefor seven years of data using two different MC. The first MCused the actual positions of the data and randomized energiesaccording to the energy spectrum. This result had a 4.3 ± . σ significance (two out of 200,000 trials with an a parametergreater than data) and was reported in Lundquist & Sokolsky(2019). That significance is consistent with the current result,using completely isotropic position MC, of 4.2 σ with overfive times more MC sets used in the calculation. 7.1. Energy/Temperature Systematic Though ranked correlation is likely to decrease the effectof systematics, and temperature is taken into account for en-ergy reconstruction, there is a possibility of a residual amountof correlation between the two. To test for this, each eventtrigger time was assigned the closest in time temperaturemeasurement from three Delta, Utah stations taken from theNOAA databases NOAA National Centers for Environmen-tal Information (2019).Using the ten years of data set, the correlation betweenenergy and temperature is τ = 0 . (a small tendency forenergy to increase with increasing temperature) with a 0.9 % probability it is actually zero given enough samples. Addi-tionally, there may be a correlation between angular distancefrom a grid point, and temperature as the average temperaturein equatorial Right Ascension (R.A.) varies about 5 ◦ .To check the possibility that the supergalactic structurefound could be an artifact of temperature variations the par-tial Kendall correlation, τ xy.z , between energy and angulardistance is done, removing temperature as a possible con-founding variable. This is shown in Equation 7 ( x stands forenergy, y for angular distance, and z for temperature). τ xy.z = τ xy − τ xz τ yz (1 − τ xz )(1 − τ yz ) (7)The average τ xy.z in equal solid angle bins of supergalacticlatitude (SGB) results in a parabolic fit curvature decrease of0.8 % . This decrease is a very small difference and likely aneffect of noise in the temperature measurements used. There-fore, no evidence for a temperature anisotropy producing theresults is found.7.2. Galactic Field Influence The energy-angle correlation wedge parameter spaceshould minimize the number of exclusively galactic fieldcreated correlations that result from the correlation signif-icance scan. The minimum wedge distance is 15 ◦ (with aresulting mean of 63 ◦ for 10 years of data), and the mini-mum wedge width is 10 ◦ (with a mean of 26 ◦ ). The meangalactic magnetic field deflection expectation for UHECRprotons with energies E ≥ 26 EeV (the average data wedgeenergy threshold) is ≤∼ ◦ for the various models in Farrar& Sutherland (2019), and the expected dispersion around themean is < ◦ for E ≥ 10 EeV.The result of rotating into galactic coordinates and plottingthe τ at each grid point for the ten years of data is shownin Figure 17(a). The negative curvature of the average τ with respect to galactic latitude (Gb), shown in Figure 17(b),could suggest that possible magnetic deflections from ap-parent sources closer to the galactic plane are influenced bygalactic magnetic fields with different directions than the av-erage extragalactic fields. This behavior is consistent with the4 A BBASI ET AL .average widening of the wedge bins near the galactic planeshown in Figure 18. (a)(b) Figure 17. Ten years of data result shown in galactic coordinates.(a) Hammer-Aitoff galactic projection of the correlation strength τ for all grid points. Negative correlations expected for magnetic de-flections are not apparent around the galactic plane. (b) Mean τ inside equal solid angle bins of galactic latitude (Gb). The resultingcorrelation structure curvature is a = − . × − . Additionally, no apparent galactic structure of multiplets isfound by the method of Section 6.1 when rotating the galacticcoordinates by 90 ◦ . This is shown in Figure 19 by the average τ in equal solid angle bins of galactic longitude (Gl) centeredon the intersection between the galactic plane (GP) and thesupergalactic plane (SGP). This rotation is where the corre-lations appear to have the most galactic symmetry accordingto Figure 17(a) though the resulting correlation curvature a from the fit is 18 % of the supergalactic curvature result. SUMMARYIntermediate-scale energy-angle correlations inside spher-ical cap sections, or “wedges,” have been shown to have a ∼ σ correlation with the supergalactic plane. Seven years of Figure 18. The mean wedge width inside equal solid angle bins ofgalactic latitude (Gb) for the ten years of data result. Wider binsare consistent with larger random field deflections near the galacticplane. Figure 19. Ten years of data result shown in galactic coordinatesfor the mean τ inside equal solid angle bins of galactic longitude(Gl) centered on the intersection between the galactic plane (GP)and the supergalactic plane (SGP). Telescope Array (TA) data has a 4.2 σ post-trial significance,and the ten years of data significance is 4.1 σ post-trial. Theseresults may be evidence of large scale extragalactic magneticdiffusion of UHECR from sources within the local large scalestructure as there does not appear to be a galactic correlationstructure.Additionally, the highest significance single energy-anglecorrelation has increased from a pre-trial 4.6 σ significance(in the seven years of data) to 5.1 σ (in the ten years of data)with no new scan of wedge parameters. This correlation liesdirectly over the TA Hotspot, and its origin point is consis-tent with the starburst galaxy M82 being a source of theseevents. This result is consistent with other results assuming UPERGALACTIC S TRUCTURE OF M ULTIPLETS OF UHECR 15magnetic deflection such as He et al. (2016) and with thestarburst galaxy overdensity anisotropy study of Aab et al.(2018b).If M82 is the source of the most significant correlation,then the average coherent magnetic field component perpen-dicular to the FOV, within this section of the sky, is estimatedto be 11 nG assuming a purely proton composition.The average perpendicular magnetic field correlated withthe supergalactic plane is also estimated to be on the orderof 10 nG assuming a cosmic ray travel length of 3.7 Mpcand a proton composition. A mixed composition and/or alonger travel length results in a smaller magnetic field esti-mate. This result is consistent with other estimates of extra-galactic magnetic fields via theory, simulation and astrophys-ical measurements (Ryu et al. (1998), Globus et al. (2019),Kronberg (1994) for example).Confirmation of these results awaits sufficient data to becollected by the TA expansion to TAx4 Kido (2019).The Telescope Array experiment is supported by the JapanSociety for the Promotion of Science(JSPS) through Grants-in-Aid for Priority Area 431, Specially Promoted ResearchJP21000002, Scientific Research (S) JP19104006, SpeciallyPromoted Research JP15H05693, Scientific Research (S)JP15H05741, Science Research (A) JP18H03705, and forYoung Scientists (A) JPH26707011; by the joint researchprogram of the Institute for Cosmic Ray Research (ICRR),The University of Tokyo; by the U.S. National Science Foun- dation awards PHY-0601915, PHY-1404495, PHY-1404502,and PHY-1607727; by the National Research Foundation ofKorea (2016R1A2B4014967, 2016R1A5A1013277,2017K1A4A3015188, 2017R1A2A1A05071429); by theRussian Academy of Sciences, RFBR grant 20-02-00625a(INR), IISN project No. 4.4502.13, and Belgian SciencePolicy under IUAP VII/37 (ULB). The foundations of Dr.Ezekiel R. and Edna Wattis Dumke, Willard L. Eccles, andGeorge S. and Dolores Dor´e Eccles all helped with generousdonations. The State of Utah supported the project throughits Economic Development Board, and the University of Utahthrough the Office of the Vice President for Research. Theexperimental site became available through the cooperationof the Utah School and Institutional Trust Lands Adminis-tration (SITLA), U.S. Bureau of Land Management (BLM),and the U.S. Air Force. We appreciate the assistance of theState of Utah and Fillmore offices of the BLM in crafting thePlan of Development for the site. Patrick Shea assisted thecollaboration with valuable advice on a variety of topics. Thepeople and the officials of Millard County, Utah have been asource of steadfast and warm support for our work which wegreatly appreciate. We are indebted to the Millard CountyRoad Department for their efforts to maintain and clear theroads which get us to our sites. We gratefully acknowledgethe contribution from the technical staffs of our home insti-tutions. An allocation of computer time from the Center forHigh Performance Computing at the University of Utah isgratefully acknowledged.REFERENCES Aab, A., Abreu, P., Aglietta, M., et al. 2015, EPJ C, 75, 269.https://arxiv.org/abs/1410.0515Aab, A., et al. 2017, in The Pierre Auger Observatory:Contributions to the 35th ICRC (ICRC 2017)Aab, A., Abreu, P., Aglietta, M., et al. 2018a, The AstrophysicalJournal, 868, 4, doi: 10.3847/1538-4357/aae689Aab, A., et al. 2018b, Astrophys. J., 853, L29,doi: 10.3847/2041-8213/aaa66dAbbasi, R. U., et al. 2008, Phys. Rev. Lett., 100, 101101.https://arxiv.org/abs/astro-ph/0703099—. 2014, Astrophys. J., 790, L21. https://arxiv.org/abs/1404.5890—. 2015, Astropart. Phys., 68, 27 . https://arxiv.org/abs/1410.3151Abbasi, R. U., Abe, M., Abu-Zayyad, T., et al. 2018a, TheAstrophysical Journal, 862, 91—. 2018b, The Astrophysical Journal, 858, 76,doi: 10.3847/1538-4357/aabad7Abreu, P., Aglietta, M., Ahn, E. J., et al. 2012, Astropart. Phys., 35,354. https://arxiv.org/abs/1111.2472Abu-Zayyad, T., et al. 2013, Astrophys. J., 768, L1.https://arxiv.org/abs/1205.5067 Biermann, P. L., Kang, H., & Ryu, D. 1997, 9 p.https://arxiv.org/abs/astro-ph/9709250Bonafede, A., Feretti, L., Murgia, M., et al. 2010, Astron.Astrophys., 513, A30. https://arxiv.org/abs/1002.0594Bretz, H. P. 2011, PhD thesis, Rheinisch-Westphalian TechnicalUniversity of AachenBurgett, W. S., & O’Malley, M. R. 2003, Phys. Rev., D67, 092002.https://arxiv.org/abs/hep-ph/0301001Croux, C., & Dehon, C. 2010, SMA, 19, 497de Vaucouleurs, G. 1975, Astrophys. J., 202, 610,doi: 10.1086/154014Farrar, G. R., & Sutherland, M. S. 2019, Journal of Cosmology andAstroparticle Physics, 2019, 004,doi: 10.1088/1475-7516/2019/05/004Globus, N., Ding, C., & Farrar, G. R. 2019, in Proceedings, 36thICRC (ICRC 2019): Madison, Wisconsin, July 24-August 1,2019, Vol. ICRC2019Govoni, F., Orr`u, E., Bonafede, A., et al. 2019, Science, 364, 981,doi: 10.1126/science.aat7500 BBASI ET AL ..