Estimation and reduction of the biases by the galactic magnetic ﬁeld on the UHECR correlation studies

. The correlation studies between ultrahigh-energy cosmic ray (UHECR) anisotropy and source candidates are keys to understand the origin of UHECRs. Especially, studies with consideration of source models, magnetic ﬁelds, and mass compositions are necessary. We estimated the biases caused by the coherent deﬂection due to the galactic magnetic ﬁeld (GMF) in the previous maximum-likelihood analysis for searching the UHECR sources (Aab et al. 2018, Abbasi et al. 2018). In our work with simulated mock datasets, we ﬁnd that the anisotropic fraction f ani is estimated systematically lower than the true value when we ignore the e ﬀ ect caused by the GMF. We also develop the maximum-likelihood method which includes the GMF model and conﬁrm that the estimated parameters would be improved. We apply the method to the observational datasets obtained from the Telescope Array and Auger experiments.


Introduction
Understanding the origin of ultrahigh-energy cosmic rays (UHECRs) is one of the important problems in physics today. Due to large statistics of the Telescope Array experiment [1] and Auger experiment [2], intermediatescale anisotropies of UHECRs are found. These UHECR anisotropies are suggested to correlate with cosmic ray sources. Many studies have investigated the correlation between the UHECR anisotropy and source candidates [3][4][5][6]. In [5], the Auger collaboration reported the maximum likelihood analysis between UHECRs in south-sky and cosmic ray (CR) flux nearby starburst galaxies (SBGs). The current dataset obtained by the TA experiment does not contradict to the Auger results [6]. On the other hand, however, the UHECR anisotropy is determined by the combination of the source distribution, mass composition in the source, and magnetic fields. It is important to conduct a correlation study with taking into account of magnetic fields and mass compositions.
In [7], we investigated the biases caused by the Galactic magnetic field (GMF) with simulated UHECR datasets (mock datasets). We showed the contribution of SBGs is estimated to be lower with ignorance of GMF. We also suggested that we can reduce the bias when we conduct the maximum-likelihood analysis with consideration of the deflections in the GMF model.
In this study, we apply the method we proposed in [7] to the currently accessible datasets from the TA and Auger experiments. We discuss the parameter space of the SBG source model. * e-mail: ryo.higuchi@riken.jp 2 Method

Models
In this study, we assume three models for the source, magnetic fields, and mass composition. For a source model, we adapt the SBG source model suggested by [6]. The SBG source model is based on the 23 nearby SBGs (see Table 1 in [5]). The total CR flux from each SBG i is described as a sum of the von Mises -Fisher function [8] as follows: where n, n i , and θ are direction in the sky, direction of each SBG i, and separation angular scale, respectively. The relative flux weight of each SBG i is shown as f i (see Table 1 in [5] for a value). The separation angular scale θ presents a Gaussian-like scattering around the SBG. The anisotropic fraction parameter f ani is introduced as a fraction of UHECR events which can be explained by the assumed source model. The other component 1 − f ani is assumed as isotropic background: where F iso = 1/4π is isotropic CR flux. With the maximum-likelihood analysis in Section 2.2, the parameter set ( f ani , θ) can be estimated. For a Galactic magnetic field model, we assume the regular component of Jansson & Farrar 2012 (JF12) model [9,10]. We assume the separation angular scale θ presents the extragalactic magnetic fields and random magnetic fields. As a mass-composition model, we assume the Heinze & Fedynitch 2019 (HF19) model [11].

Maximum-likelihood analysis
We describe the maximum-likelihood analysis in [5,6]. The test statistics T S is determined with a likelihood ratio between the SBG source model and the isotropic background: The likelihood of each model is written as: where n CR and ω(n) indicate the arrival direction of each CR event, and the sky coverage of the experiments, respectively. For the sky coverage ω(n), we adapt the equations in [12] (see also [7]).

Maximum-likelihood analysis with GMF model
The CR flux introduced in equations 1-2 does not reflect the coherent CR deflection by the GMF structure. In this study, we calculate the CR flux after passing through the GMF as follows: First, we define the corresponding function A BT (n, R) between arrival directions on earth and the outside the Galaxy. The function is determined based on the CR trajectory in backtracking calculation with CR-Propa3 [13]. Note that each trajectory of CR depends on the arrival direction on the earth and its rigidity R = E/Ze. Because the CR flux conserves along the CR trajectory, we can rewrite the CR flux F org (n, θ) to that on the earth F ′ earth (n CR , f ani , θ, R CR ): With a calculation of L(F) from CR flux on the earth F ′ earth (n CR , f ani , θ, R CR ), one can estimate the parameter set ( f ani , θ) with reduction of GMF bias.

Maximum-likelihood analysis with GMF model for the observational datasets
In the method introduced in Section 2.3, we need to know the mass A of each CR event. It is difficult to analyze with current UHECR datasets knowing all the event-by-event mass A. In this study, we assume mass probability function p A (E) and calculate CR flux F ′ earth weighted with the probability. The mass probability function p A (E) is described with an CR energy spectrum of each mass A (J A ), Regarding J A , we follow the formulas and best-fit parameters in [11] (see also [7]). We include p A (E) into equation 2 and normalize the CR flux as follows: .
We substitute F norm in equation 7 into equations 4 and 3, and calculate T S .

Dataset
We apply the maximum-likelihood analysis in Section 2.4 to observational datasets obtained by TA and Auger experiments (TA-11yr dataset and Auger 2015 dataset). For Auger 2015 dataset, we refer [2]. TA-11yr dataset (Auger 2015 dataset) is taken in 2008 May -2019 May (2004 January-2014 March) and includes 279 (225) events above 43 (52) EeV. We follow the energy thresholds for the TA-11yr dataset and Auger 2015 dataset in [6] and [2], respectively. We also apply the mock datasets calculated from the convolution of the SBG source model, JF12 model, and HF19 model. See [7] for details.

Estimation and reduction of the GMF bias
In this section, we briefly summarize the results of the simulated mock datasets in [7]. The left panel in Figure 1 shows an example of the best-fit parameters for 1000 mock datasets in the same manner as previous studies [5,6]. We see the difference between true parameters ( f true ani , θ true ) (which are used to generate mock datasets) and estimated parameters. Each best-fit parameter set is estimated to maximize T S value with ignorance of coherent deflection by GMF (Section 2.2). In many true parameters ( f true ani , θ true ), estimated anisotropic fraction f ani in north-/south-sky are smaller than the true one.
The right panel in Figure 1 shows the same ones in the left panel, but for T S value calculated with consideration for GMF (Section 2.3). When the separation angular scale θ is small, parameter set ( f ani , θ) is well estimated in 1σ uncertainty.

Application for TA and Auger datasets
We show T S distribution for observational datasets in , θ Auger ) = (5%, 34 deg), respectively. Although T S -value is small to determine the source model, we search the parameter space ( f ani , θ) in which mock datasets can reproduce the T S . Regarding the mock datasets, we assume the same number of events as observational datasets for each dataset. The model assumptions for the generation of the mock dataset are the same as in Section 4.1. We show the parameter space ( f ani , θ) which is consistent with T S from observational datasets in Figure 3. Figure 3 suggests that we can exclude the parameter space θ < 12 (37) deg θ < 20 (45) deg at f ani = 20 % (100 %) from the TA 11-yr & Auger 2015 datasets with 95% C.L..

Summary
Deflections of UHECRs by magnetic fields and mass composition of UHECRs affect the correlation study between UHCR anisotropies and source candidates. Especially Figure 1. Distributions of the best-fit parameters for the 1000 mock event datasets in case of ( f true ani , θ true ) = (40 %, 10 deg). The left panels show the results with ignorance of coherent deflection by GMF (Section 2.2). The right panels show the same but with consideration for GMF (Section 2.3). In both panels, true parameters ( f true ani , θ true ) are shown by the grey stars. The black, blue, and red contours indicate the 68 % and 95 % tile containment for all, north and south-sky datasets, respectively. The black cross, blue circle, and red triangle show the most frequent values for all, north and south-sky datasets, respectively. The distributions of best-fit parameters are smoothed with a kernel-Gaussian distribution. In the left panel, best-fit parameter ( f Auger ani , θ Auger ) = (9.7 %, 12.9 deg) in [5] is shown as a black triangle. when we ignore the coherent deflections by GMF, the contribution from nearby SBGs should be estimated lower [7]. Thus we need the analysis with the assumption of both magnetic fields & mass composition models. In our work [7], we propose the maximum-likelihood analysis technique with consideration of these models. In this study, we apply the technique to current available observational datasets obtained from the TA and Auger experiments. Although the T S -value is small to determine the source mode for current observational datasets, we search the parameter space ( f ani , θ) which mock datasets can reproduce the T S . It is suggested that the parameter space θ < 12 (37) deg θ < 20 (45) deg at f ani = 20 % (100 %) are excluded from the TA 11-yr & Auger 2015 datasets.