Evidence for a Supergalactic Structure of Magnetic Deflection Multiplets of Ultra-high-energy Cosmic Rays

The distance between two points on the surface of a sphere, the great circle angular distance, is shown in Equation (2) in terms of vectors normal to the FOV. Correlations between event energy and angular distance from a grid point are found using a ranked correlation, Kendall's τ, that measures the strength of monotonic dependence Kendall (1945),

$$\begin{eqnarray}&&{\delta }_{{ij}}=\arctan \displaystyle \frac{| {\hat{{\boldsymbol{n}}}}_{{\boldsymbol{i}}}\times {\hat{{\boldsymbol{n}}}}_{{\boldsymbol{j}}}| }{{\hat{{\boldsymbol{n}}}}_{{\boldsymbol{i}}}\cdot {\hat{{\boldsymbol{n}}}}_{{\boldsymbol{j}}}}.\end{eqnarray} \tag{ 2 }$$

The Kendall correlation is generally more robust against noise than the other common ranked correlation—Spearman's ρ (Croux & Dehon 2010). Ranked correlation minimizes the effects on correlation strength by magnetic model (such as higher order terms of Equation 1(a)), composition assumption, energy reconstruction systematics, and detector exposure variation.

Kendall's τ ranked correlation is the linear correlation between the separate ordering of the two variables of interest (variable x sorted ranks: first, second, third, etc. versus variable y ranks: fifth, first, fourth, etc.), with n pairs of values, and is shown in Equation (3):

$$\begin{eqnarray}&&\tau =\displaystyle \frac{2}{n(n-1)}\displaystyle \sum _{j\lt k}\mathrm{sign}\left[\displaystyle \frac{{x}_{j}-{x}_{k}}{{y}_{j}-{y}_{k}}\right].\end{eqnarray} \tag{ 3 }$$

This correlation can be considered simply as the normalized sum of the sign of the slopes between all pairs of data points. A small correction is made for the rare occurrence of duplicate values and can be found in Kendall (1945).

The correlation coefficient τ has a range from −1 to +1, and a value of 0 means that there is no measured relationship between the variables. For +1, an increase (decrease) of x always follows an increase (decrease) of y. If $\tau =-1$ an increase (decrease) of x always follows a decrease (increase) in y (in this analysis, x and y are energy and angular distance). A negative correlation is consistent with the expectation for magnetic field deflected events—as energy decreases, deflection increases, as can be seen from Equation (1).

Any monotonic function (x^b, ${\mathrm{log}}_{10}(x)$, e^x, etc.) of distance, energy, or both will always return a τ coefficient with the same magnitude but not necessarily the same sign (±). The sign of the resulting τ would be the original τ multiplied by the signs of the first derivatives of the applied functions.

The pretrial two-sided significance of a correlation, z (probability of τ = 0), is a function of correlation strength and sample size n. This significance is found by counting permutations of the sample ranks with greater τ, or in the large n sample limit, Equation (4) (for n ≥ 50), which follows the standard normal distribution. Further details can be found in Kendall (1945),

$$\begin{eqnarray}&&z=\displaystyle \frac{\tau 3n(n-1)/2}{\sqrt{n(n-1)(2n+5)/2}}.\end{eqnarray} \tag{ 4 }$$

With the drift-diffusion picture of Figure 1 in mind, possible UHECR deflections from grid-point "sources" were found by a scanned maximization of the significance of energy–angle correlations inside spherical cap sections, or "wedges," using seven years of TA data (Lundquist & Sokolsky 2019). This scan was done at each point on an approximately equal 2° spaced grid of 6553 points on the FOV shown in Figure 2 (Teanby 2006).

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These wedge bins are defined by a maximum angular distance ${\delta }_{j}$ from the grid point, i, defined by Equation (2) and the boundaries of two azimuths defined by Equation (5), where B is latitude and L is longitude:

$$\begin{eqnarray}&&{\phi }_{{ij}}=\arctan \displaystyle \frac{\cos {B}_{i}\sin ({L}_{i}-{L}_{j})}{\cos {B}_{j}\sin {B}_{i}-\sin {B}_{j}\cos {B}_{i}\cos ({L}_{i}-{L}_{j})}.\end{eqnarray} \tag{ 5 }$$

The azimuths increase clockwise, and a great circle section, or wedge, pointed toward 90° supergalactic latitude (SGB) has an azimuth, ϕ, of 0°, while one pointed toward −90° SGB has a ϕ of 180°. The azimuthal angle difference, ${\rm{\Delta }}{\phi }_{{ij}}$, between the wedge pointing direction, ${\phi }_{i}$, and the azimuth of an event, ${\phi }_{{ij}}$, is shown in Equation (6). An example wedge is shown in Figure 3(a):

$$\begin{eqnarray}&&{\rm{\Delta }}{\phi }_{{ij}}=\mathrm{mod}(| {\phi }_{{ij}}-{\phi }_{i}| +180,360)-180\end{eqnarray} \tag{ 6 }$$

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This oversampling bin shape means that four parameters must be scanned at every grid point to maximize the pretrial correlation significance. Even though negative correlations are physically expected by a magnetic field drift-diffusion process, both the sign of the correlation, and its strength are not explicitly scanned for nor restricted. The limits on these parameters are large to account for most conceivable extragalactic magnetic deflection scenarios. The scans are all combinations of the following:

• 1.  
  Energy threshold, E_i: 10 to 80 EeV in 5 EeV steps.
• 2.  
  Wedge distance, D_i = $\max ({\delta }_{{ij}})$: 15° to 90° in steps of 5°.
• 3.  
  Wedge direction, ${\phi }_{i}$: 0° to 355°, 5° steps.
• 4.  
  Wedge width, ${W}_{i}=2* \max (| {\rm{\Delta }}{\phi }_{{ij}}| )$: 10° to 90°, 10° steps (5° on each side of ${\phi }_{i}$).

Events are inside the wedge if E_j ≥ E_i and ${\delta }_{{ij}}$ $\leqslant$ D_i and $-{W}_{i}/2\leqslant {\rm{\Delta }}{\phi }_{{ij}}\leqslant {W}_{i}/2$, where i is the index of the grid point. The energy−angle correlation is calculated inside the wedge, $\tau ({\delta }_{{ij}},{E}_{j})$, and the parameters (${E}_{i},{D}_{i},{\phi }_{i}$, 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 the 10 year data set to test the result.

The wedge parameters needed to maximize the correlation significance at each grid point were scanned for using seven years of TA data (Lundquist & Sokolsky 2019). For the seven year data set, the supergalactic coordinates of the most significant correlation of all the grid points is 183 SGB (latitude), −129 SGL (longitude). This wedge and the events inside are shown in Figure 3(a). There are 29 events with energies E ≥ 30 EeV. The pretrial one-sided significance of τ = −0.675 with a sample size of 29 events is 5.5σ. A scatter plot of energy versus angular distance from the grid point within this wedge is shown in Figure 3(b). A linear fit (Equation 1(a) with Z = 1) results in an estimate of B × $S=49$ nG∗Mpc. If the source is assumed to be at the distance to M82 (3.7 Mpc) with a pure proton emission, the average coherent magnetic field, perpendicular to the FOV, required to cause this deflection would be B = 13 nG.

Note, however, that the posttrial significance of any single correlation is not expected to be large, as the wedge scan parameter space is large. An individual correlation is not the test of a supergalactic structure.

Each Monte Carlo (MC), and data event, is defined by their energy, zenith angle, azimuthal angle, and trigger time. The latitude and longitude are defined from the center of the TA SD at 393 Lat., 1129 Long. These horizontal coordinates are used to calculate the longitude (SGL) and latitude (SGB) in supergalactic coordinates (Vallado 2007). The MC event sets have a zenith angle distribution of g(θ) = sin(θ)cos(θ) due to the event sampling response of a two-dimensional SD array, a uniform azimuth distribution, and the detection efficiency ∼100% for UHECRs E ≥ 10^19.0 eV. The event trigger times are approximated as a uniform distribution of modified Julian dates from the beginning to the end of the run time due to the approximately ∼100% SD on-time. All of these MC parameter distributions are in good agreement with the data set described in Section 5.

Detector acceptance and bias, in the energy spectrum, are taken into account by interpolation sampling of a large set of MC events reconstructed through a surface detector simulation thrown with the average HiRes/TA spectrum (Abbasi et al. 2008). The same cuts applied to the data are applied to these fully simulated events, and there are ∼4 × 10⁵ with energies E ≥ 10^19.0 eV. The number of events in each isotropic MC event set is the same as the data in each 5 EeV bin of the parameter scan of Section 2.2. This simulated data has been shown to reproduce all measured geometric and photoelectric distributions accurately Ivanov (2012).

The result is that each set of these isotropic MC events simulates the expected data, given the detector configuration and on-time, with no energy anisotropies. These MC sets are used to calculate the posttrial probability of any potential anisotropy signal in the data.

A simple toy-model simulation of an intervening supergalactic magnetic sheet, between the galaxy and UHECR sources, is made by taking the isotropic event sets of Section 3.1 and embedding event deflections (assigning distance correlated energies) in supergalactic latitude (SGB), proportional to $1/\mathrm{energy}$, for a fraction of events. The coordinates of the MC events are isotropic and unchanged in the procedure. The approximate apparent deflection from the source of a charged particle in a coherent magnetic field is from Equation 1(a). An example of the resulting simulation is shown in Figure 4.

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The event deflections, ${\delta }_{B}$, from supergalactic latitude $\mathrm{SGB}=0^\circ$ are calculated for each MC event energy in the set, assuming a proton composition (Z = 1) and a particular B × S, according to Equation 1(a). Additionally, some random field noise was added by smearing the ${\delta }_{B}$ with a 5° standard deviation Gaussian. Then each event is assigned an energy based on its angular distance from the supergalactic plane (min[${\delta }_{B}$-${\delta }_{\mathrm{SGB}=0}$]). The beginning, and final, simulation is isotropic 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 than 10° are added to the isotropic proportion. This threshold adds additional random field noise in the simulation. This cut also results in a harder spectrum for the deflected events (red event in Figure 4), i.e., higher energy events on average closer to the supergalactic plane. This supergalactic energy bias is due to the lower number of high energy events, resulting in a better fit to a supergalactic magnetic deflection at higher energies (due to the boundary conditions of the energy spectrum and position isotropy).

A supergalactic sheet simulation, with an F = $65.7 \%$ isotropic fraction and B × S = 18.47 nG∗Mpc, is shown in Figure 4. These parameters are the result of selecting a random MC that looks similar to the data result and the choice of a proton composition. Note again that this is only an anisotropy of energies in supergalactic latitude as event positions are isotropic, and the total energy spectrum is unchanged.

The intent of this toy-model simulation is to show that the analysis method is sensitive to an energy symmetry caused by some kind of magnetic deflection structure correlated with the supergalactic plane. It is not intended to reproduce all aspects of actual data.

Though a test for a supergalactic structure of energy−angle correlations is not necessarily a priori obvious, the supergalactic sheet toy model leads to a reasonable answer. The mean $\langle \tau \rangle$ inside equal solid angle bins of angular distance (SGB_i) from the supergalactic plane (SGP) shows that three features are relevant for the supergalactic hypothesis—the minimum average τ, the minimum location being near the SGP, and the symmetry of τ around the SGP. Using all three features to calculate the data significance would be overfitting the problem. One test statistic is preferable though it should be correlated with these three supergalactic structure features. The single parameter chosen to test the supergalactic structure hypothesis is the curvature parameter, "a," of a parabolic fit (y = $a{(x-{x}_{0})}^{2}+{y}_{0}$) to the 〈τ〉.

The curvature, "a," is simply the lowest-order Taylor expansion term that can describe the symmetry around the SGP shown in the simulation (Figure 5(b)). Due to the boundaries of $| \tau |$ $\leqslant 1$ and $| {SGB}|$ < 90°, a greater correlation curvature, a, corresponds to a minimum, x₀, closer to the supergalactic plane, as shown in Figure 6(a). A larger curvature a also means that the minimum negative correlation averages are greater in magnitude, y₀, as shown in Figure 6(b). The parabola minimum y₀ has no correlation with the minimum supergalactic latitude (SGB). These relationships justify the use of the parabola curvature "a" as the single test statistic for a conservative estimate of the significance of supergalactic energy–angle correlations.

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The fit on the large-scale behavior of the correlation strength, τ, is used because it is not explicitly scanned for and contains more information by its sign (±) than the pretrial significance. The pretrial significance of the correlations is not used in this analysis so that the significance test is independent of the wedge scan for the maximum significance of individual energy–angle correlations.

To calculate the data significance of a supergalactic structure of energy–angle correlations, the analysis described above was applied to the data and the isotropic MC sets. The number of MC sets with a correlation curvature a greater than the data gives the probability of the measured supergalactic structure of energy–angle correlations if there actually is not such a structure, i.e., if it is a statistical fluctuation in the data.

By applying this analysis to isotropic MC sets, as described in Section 3.1, and counting the number of MC with an a parameter larger than data (Figure 7(b)), the posttrial significance of the supergalactic structure of multiplets can be found. The resulting a distribution of 1,000,000 MC sets is shown in Figure 9 for the seven year data statistics and energy–angle correlation significance scan.

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For the seven year data analysis, there are 14 MC sets with a larger curvature than the data, which result in the significance of a supergalactic structure of multiplets of ∼4.2σ.

For the 10 years of data with no updated wedge correlation significance scan, the resulting a distribution of 1,000,000 MC sets is shown in Figure 10. The distribution has a smaller standard deviation due to no new scans for energy–angle correlation significances. The result is a smaller τ on average. There are 22 MC sets with a larger curvature than the data, shown in Figure 8(b), which results in the significance of a supergalactic structure of multiplets of ∼4.1σ.

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The total number of MC sets that were used to calculate the significance was limited by the computing time necessary for each simulation. Overall, the number of correlations calculated was 4 × 10¹⁴, and this took more than 200 years of equivalent CPU computing time.

Clues about UHECR sources, and intervening fields, may be found from the maximum significance wedge scan parameters of the apparent magnetic deflection multiplets. Due to the significance maximization, there is a bias toward greater statistics, as can be seen in Equation (4), so the data are compared to isotropic MC by taking the ratio of the parameter probability distribution functions (PDFs; normalized histograms of data divided by MC). The PDF ratio shows how many times more likely a scan parameter value is to be found in data than isotropic MC. PDF ratio plots for wedge pointing direction and energy threshold parameters are shown in Figure 11.

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These ratios are done for negative energy–angle correlations at grid-point positions $| \mathrm{SGB}| \leqslant 40^\circ$ (about the boundary where the average correlation is zero as shown in Figure 8(a)) and have a linear fit to $1/E$ versus angular distance with R² > 0 (Figure 3(b)). An R² > 0 is a better fit than a horizontal line, and the δ ∝ $1/E$ model explains some of the variance of the data inside the wedge. For data, there are 2045 correlations used and greater than 3.99 × 10⁸ for MC.

The data distribution of wedge pointing directions, Figure 11(a), provides further indication of a supergalactic structure with four deviations seemingly correlated with the supergalactic plane (SGP). Two larger peaks are approximately perpendicular to the SGP (∼195° and ∼345°), and two smaller peaks are close to parallel (∼90° and ∼285°). These peaks suggest an overall diffusion of low energy events away from the supergalactic plane, similar to the supergalactic magnetic sheet simulation of Section 3.2.

The data distribution of the energy threshold parameters may provide information regarding UHECR sources and intervening fields. The median energy threshold is 30 EeV, and the three largest deviations from the isotropic distribution are at 35 EeV, 45 EeV, and 60 EeV. The 60 EeV peak appears to correspond to the 57 EeV threshold of the TA Hotspot analysis (Abbasi et al. 2014).

The median energy threshold of 30 EeV is above the significant Pierre Auger Observatory (PAO) large-scale dipole measurements in Aab et al. (2018a) at 8 EeV, which is consistent with the localized intermediate-scale energy–angle magnetic deflections in this analysis.

The 39 EeV cutoff for maximum event correlation with starburst galaxies, reported by PAO in Aab et al. (2018b), may be related to the 35 EeV and 45 EeV peaks.

These threshold deviations from isotropy are also consistent with the result using AGASA data that showed a possible large-scale cross-correlation between UHECR and the supergalactic plane between 50 and 80 EeV energy bins (Burgett & O'Malley 2003). Adjusting the AGASA energy scale to the TA energy scale by multiplying by 0.75, this becomes 38 and 60 EeV energy bins.

The data distributions of wedge angular distance, D, and width, W, do not show any significant deviations from isotropy.

The most significant single correlation using 10 years of SD data is at 303 SGB, −32 SGL, and shown in Figure 12(a). With 75 events (E ≥ 35 EeV) and τ = −0.412, it has a pretrial significance of 5.10σ. This significance is an increase from 4.58σ at this grid point, with seven years of data using the same wedge and energy threshold parameters. Figure 12(b) shows a scatter plot of energy versus angular distance. A linear fit (Equation 1(a) with Z = 1) results in an estimate of B × $S=41$ nG∗Mpc.

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Recently, PAO has stated that the likeliest source of events with E > 39 EeV are starburst galaxies (Aab et al. 2018b). The most significant correlation reported here is 113 from M82 (as shown by the blue diamond in Figure 12(a)), pointing directly over the TA Hotspot (Figure 2). M82 is the closest starburst galaxy to our galaxy.

If the source is assumed to be at the same distance to M82 (3.7 Mpc) with a pure proton emission, the average coherent magnetic field perpendicular to the FOV required to cause this deflection is B = 11 nG. The large deviations from the linear fit of Figure 12(b) imply that, in this region, the random field deflections have a large correlation length scale (L_c), and the random field (B_rms) of Equation 1(b) is on the same order of magnitude as the coherent field deflection.

Random variations of the data are created to estimate the uncertainty on the location of the source of the maximum significance energy–angle correlation. The energies of events outside the wedge are scrambled with other events outside the wedge. Inside, the energies of wedge events with energy less than the wedge threshold, E < 35 EeV, are randomized within the wedge. The locations of the 75 data events in the wedge with E ≥ 35 EeV are not changed. This ensures that the spectrum is not changed, inside or outside the wedge, and that the number of events E ≥ 35 EeV does not dramatically increase due to the Coldspot (Abbasi et al. 2018a). The analysis, including scanning for maximum significance wedge correlations at all grid points, was repeated for 5000 of these random variations on the data.

The estimated location of each randomized data set source is the most significant negative correlation near the known source grid point. The maximum distance searched, within a spherical cap centered on the known grid point, is the distance that minimizes the average τ inside the cap (correlations are more positive outside). A spherical cap limiter is necessary due to the fact that an entirely different set of events from the wedge of interest, say on the other side of the FOV, can easily have a more significant correlation due to the number of scans done at each grid point.

The result is that the apparent sources have a median distance of 24 from the original source with a $+1\sigma$ quantile of 68 and $6.2 \%$ are greater than or equal to 113 away (the angular distance from M82 to the maximum significance grid point). The distribution of distances is shown in Figure 13. This distribution means that M82 is not excluded as a possible source of the events in this energy–angle correlation and the TA UHECR Hotspot/Coldspot (Abbasi et al. 2018a).

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The result presented here appears to be consistent with the results of He et al. (2016), who used a Bayesian analysis of the relative deflection of TA Hotspot events in two energy bins (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 of M82's magnetic field in Jones et al. (2019), the integrated magnetic field angle is 351° in equatorial coordinates using the same definition as Section 2.2. Rotating into supergalactic coordinates results in an angle of 308°. The coherent magnetic field direction necessary to create the most significant multiplet is 120 ± 90°, so it is either 82° or 98° from M82's magnetic field direction. The circular standard deviation of the pointing direction of the wedge simulations shown in Figure 14 is 21°, which means the wedge magnetic field direction is  ≥4σ different from M82's magnetic field direction. This direction discrepancy implies that if M82 is the source, then magnetic fields outside M82 were the primary source of multiplet pattern deflections.

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The average linear fit to $1/E$ versus angular distance from the grid point, inside wedges with a negative correlation, can give an estimate of coherent magnetic field strength times the distance traveled through the field (see Equation 1(a) as shown in Figure 13(b)). These B × S values are independent of the ranked correlation pretrial significances, which were maximized to choose the wedge parameters.

If the coherent magnetic field in the vicinity of positive correlations is considered negligible and those correlations are set to B × S = 0, then the mean B × S in supergalactic latitude (SGB) bins appears as Figure 15. Given 〈B × S〉 =21 nG∗Mpc and if the composition is protonic, then the average coherent field component, perpendicular to the FOV, in the vicinity of the supergalactic plane ($| \mathrm{SGB}| \leqslant 40^\circ$) is 5.6 nG (assuming a source distance of 3.7 Mpc).

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If the coherent magnetic field in the vicinity of positive correlations is considered to be unknown, and those correlations are ignored, then the mean B × S in supergalactic latitude (SGB) bins appears as Figure 16. If proton is the assumed composition, then the average coherent field component, perpendicular to the FOV, in the vicinity of the supergalactic plane assuming a distance of 3.7 Mpc is ∼8.6 nG.

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Recently in Globus et al. (2019), the best-fit average extragalactic field to PAO dipole, assuming a local large-scale structure (LSS) distribution of sources according to the CosmicFlows-2 catalog, was estimated to be 0.6 nG using PAO mixed composition E ≥ 8 EeV in Aab et al. (2017). If the TA composition is largely protonic, as in Abbasi et al. (2018b), then the average distance traveled, S, necessary for agreement with PAO on the extragalactic field strength is ∼50 Mpc. The mean distance of galaxies in the CosmicFlows-2 catalog, within the GZK horizon of ∼100 Mpc, is 51 Mpc (Tully et al. 2013). Given the model and experimental uncertainties, TA and PAO seem to have a good order of magnitude agreement on the extragalactic field strength.

Though ranked correlation is likely to decrease the effect of systematics, and temperature is taken into account for energy reconstruction, there is a possibility of a residual amount of correlation between the two. To test for this, each event trigger time was assigned the closest in time temperature measurement from three Delta, Utah stations taken from the NOAA databases (NOAA National Centers for Environmental Information 2019).

Using the 10 year data set, the correlation between energy and temperature is $\tau =0.027$ (a small tendency for energy to increase with increasing temperature) with a $0.9 \%$ probability it is actually zero given enough samples. Additionally, there may be a correlation between angular distance from a grid point, and temperature as the average temperature in equatorial R.A. varies about 5°.

To check the possibility that the supergalactic structure found could be an artifact of temperature variations, the partial Kendall correlation, ${\tau }_{{xy}.z}$, between energy and angular distance is done, removing temperature as a possible confounding variable. This is shown in Equation (7) (x stands for energy, y for angular distance, and z for temperature):

$$\begin{eqnarray}&&{\tau }_{{xy}.z}=\displaystyle \frac{{\tau }_{{xy}}-{\tau }_{{xz}}{\tau }_{{yz}}}{(1-{\tau }_{{xz}}^{2})(1-{\tau }_{{yz}}^{2})}.\end{eqnarray} \tag{ 7 }$$

The average ${\tau }_{{xy}.z}$ in equal solid angle bins of supergalactic latitude (SGB) results in a parabolic fit curvature decrease of $0.8 \%$. This decrease is a very small difference and likely an effect of noise in the temperature measurements used. Therefore, no evidence for a temperature anisotropy producing the results is found.

The energy–angle correlation wedge parameter space should minimize the number of exclusively galactic field created correlations that result from the correlation significance scan. The minimum wedge distance is 15° (with a resulting mean of 63° for 10 years of data), and the minimum wedge width is 10° (with a mean of 26°). The mean galactic magnetic field deflection expectation for UHECR protons with energies E ≥ 26 EeV (the average data wedge energy threshold) is $\leqslant \sim$ 15° for the various models in Farrar & Sutherland (2019), and the expected dispersion around the mean is <10° for E ≥ 10 EeV.

The result of rotating into galactic coordinates and plotting the τ at each grid point for the 10 years of data is shown in 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 apparent sources closer to the galactic plane are influenced by galactic magnetic fields with different directions from the average extragalactic fields. This behavior is consistent with the average widening of the wedge bins near the galactic plane shown in Figure 18.

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Additionally, no apparent galactic structure of multiplets is found by the method in Section 6.1 when rotating the galactic coordinates by 90°. This is shown in Figure 19 by the average τ in equal solid angle bins of galactic longitude (Gl) centered on the intersection between the galactic plane (GP) and the supergalactic plane (SGP). This rotation is where the correlations appear to have the most galactic symmetry according to Figure 17(a) though the resulting correlation curvature a from the fit is 18% of the supergalactic curvature result.

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