Sensitivity of Cross-Correlation Studies by Using X_max Information

Abstract

The investigation about cosmic-ray sources through the study of the arrival direction of the particles is completely dependent on primary mass compositions. This is due to heavier particles having an average larger deflection during propagation in galactic and extragalactic magnetic fields. Recent result from the Auger Collaboration made it possible to obtain information on the chemical composition of the cosmic rays on an event-by-event basis in addition to data from the surface detector. Here, we have investigated the gain in detection efficiency of standard cross-correlation using the Landy–Szalay estimator. Our sample contained only events with the highest $X_{\max }$ values (corresponding to the lightest charged particles, for which the deflections in magnetic fields are smaller on average). We obtained results showing that that gain depends on the proton fraction ranging from a factor of ∼2 to ∼4.

1. Introduction

Ultra-high energy cosmic rays (UHECRs), particles with energies above ${10}^{18}$ eV (1 EeV = ${10}^{18}$ eV), are messengers of astrophysical sources carrying fundamental information about the universe. Although these high energy particles have been detected since the 1960s, their sources, chemical composition, mechanism of acceleration at the sources, and how they propagate to Earth are still open questions [1,2,3,4,5,6]. The study of anisotropies through the cross-correlation between the arrival directions of ultra high energy cosmic rays (UHECRs) and the positions of astrophysical objects is an important ingredient to determine the sources of these cosmic particles [7].

Since most cosmic rays are charged particles, they are deflected along their path through the Galactic and extragalactic magnetic fields. The limited knowledge of the magnitude of these fields and the chemical composition of cosmic rays makes the identification of the sources of the ultra-energetic cosmic rays a very difficult task. Part of this difficulty arises from the fact that information about the chemical composition of the primary cosmic rays is obtained statistically from the distribution of the atmospheric depth at which the maximum deposition of energy by the extensive air shower, $X_{\max }$, occurs [4]. At the Pierre Auger Observatory, this information is measured by fluorescence telescopes, which operate only on clear nights, without moonlight, corresponding to ∼13% of the operating time of the experiment. The Pierre Auger Collaboration has recently shown that it is possible to obtain the same information in an event-by-event basis by using deep neural networks and data from the surface detectors (which operate almost $100\%$ of the time) of the Pierre Auger Observatory; see [8] for details. This opens a very interesting window for anisotropy studies. The main goal of the present paper is to investigate the gain in detection power of the Landy–Szalay anisotropy cross-correlation estimator [9] when we select samples of the highest $X_{\max }$ values, corresponding to the lightest elements. Since the magnetic deflection in the cosmic-ray trajectory is proportional to their electrical charge, the selection of such samples, consisting of lighter nuclei, must have a higher correlation with the direction of the sources that produced them, which is quite promising for detecting anisotropies in the cosmic-ray arrival directions and, possibly, unveil their origins. The article is organized as follows. We begin in Section 2 with an overview of the Landy–Szalay cross-correlation estimator method. In Section 3, we describe detection sensitivity, and the article concludes with some conclusions in Section 4.

2. Landy–Szalay Cross-Correlation Estimator

The two-point cross-correlation function is a method designed to measure the degree of similarities between the distributions in the sky of two sets of astrophysical objects. The Landy–Szalay [9] cross-correlation method is based on the estimator:

$$\omega \left(\theta \right)=\frac{N_{D_1{D}_2}\left(\theta \right)-N_{D_1{R}_2}\left(\theta \right)-N_{R_1{D}_2}\left(\theta \right)+N_{R_1{R}_2}\left(\theta \right)}{N_{R_1{R}_2}\left(\theta \right)},$$

(1)

where $D_1$ denotes the data set of interest (the arrival directions measured by the Pierre Auger Observatory, in our case), $D_2$ is a reference catalog of possible astrophysical sources, and $R_1$ and $R_2$ are isotropic simulated sky maps with the same exposures of $D_1$ and $D_2$, respectively. Notation $N_{X_1{X}_2}\left(\theta \right)$ indicates the number of pairs between the two samples, $X_1$ and $X_2$, as a function of angular distance $\theta$. For the simplicity of notation, we will refer to $N_{X_1{X}_2}\left(\theta \right)$ only by $N_{X_1{X}_2}$.

It is interesting to note that this estimator fluctuates around zero if the data set $D_1$ or the catalog $D_2$ is isotropic. In this case, $D_1=R_1$ (or $D_2=R_2$ ) by definition, and Equation (1) is

$$\omega \left(\theta \right)=\frac{\stackrel{\approx 0}{{\displaystyle \overbrace{N_{R_1{D}_2}-N_{R_1{D}_2}}}}-\stackrel{\approx 0}{{\displaystyle \overbrace{N_{D_1{R}_2}+N_{D_1{R}_2}}}}}{N_{R_1{R}_2}}\approx 0.$$

(2)

For better comprehension, the distributions of the number of pairs obtained from one specific simulation of an isotropic cosmic-ray sky contained 1000 events ($D_1$) generated by taking into account the exposure of the Pierre Auger Observatory. To test the cross-correlation, we used the positions of starburst galaxies (SBG) ($D_2$) [1]. The isotropic mock samples ($R_1$ and $R_2$) were generated as explained above with 10,000 directions sampled for each one. Figure 1 shows the resulting $\omega \left(\theta \right)$ obtained for this specific simulation. We can see that it fluctuates around zero as expected.

Figure 1. Example of Landy–Szalay cross-correlation estimator between a simulated isotropic dataset and positions of starburst galaxies. The red line is the value of estimator $\omega \left(\theta \right)$ as a function of angular scale $\theta$ obtained for the data. The black dotted line shows the expected value in the isotropic case.

A possible choice of a global estimator for the cross-correlation between samples $D_1$ and $D_2$ that takes into account the statistical penalization for searches in different angular scales $\theta$ is provided by a reduced ${\chi }_{\nu }^2$ as follows:

$${\chi }_{\nu }^2=\frac{1}{N_{bins}}\sum _1^{N_{bins}}{\left(\frac{\omega \left(\theta \right)-\stackrel{=0}{{\displaystyle \overbrace{{⟨\omega \left(\theta \right)⟩}_{iso}}}}}{{\sigma }_{iso}\left(\theta \right)}\right)}^2,$$

(3)

where ${⟨\omega \left(\theta \right)⟩}_{iso}$ and ${\sigma }_{iso}$ are the expected value and standard deviation of $\omega \left(\theta \right)$ obtained from isotropic skies. $N_{bins}$ is the number of bins for searches in different angular scales $\theta$. For simplicity, from now on, we will only refer to ${\chi }_{\nu }^2$ as ${\chi }^2$.

The statistical test was performed by comparing the ${\chi }^2$ obtained for a given data set with ${\chi }_{iso,99\%}^2$, defined as ${\int }_{-\infty }^{{\chi }_{iso,99\%}^2}d{\chi }^{2}p\left({\chi }_{iso}^2\right)$ = 0.99, where $p\left({\chi }_{iso}^2\right)$ is the reduced ${\chi }^2$ probability density function. If ${\chi }^2$ obtained from the data is greater than ${\chi }_{iso,99\%}^2$, the isotropy hypothesis is rejected.

3. Detection Sensitivity

The goal of this section is to compute the gain in detection power when it is possible to use a subsample of events, selected by the highest $X_{\max }$ values and corresponding to the lightest nuclei. Since the magnetic deflection of the cosmic-ray trajectory is proportional to its electrical charge, we expect that such a selection presents a higher correlation with the direction of the sources that produced them in comparison with the cross-correlation considering the entire sample.

Our toy model consists of a mix of events coming from the SBG listed in [1] and contains the positions of the 23 starburst galaxies in Galactic coordinates and from the background. It has been pointed out that the diffusive shock acceleration inside superwinds of starbursts can accelerate cosmic rays to the maximum energies observed [10,11]. The events injected at the SBG are protons while the background is composed of an equal mix of nitrogen (N) and iron (Fe) nuclei. Particles are simulated with energies above 40 EeV following an energy spectrum $dN/dE\propto E^{-\gamma }$, with $\gamma =-5$, in accordance with Auger the energy spectrum [5,12] in the considered energy range. Energy losses are neglected during the proton’s propagation because the SBGs are very close to our Galaxy [1]. In addition, a smearing in the arrival direction of protons is performed by using a von Mises–Fischer distribution with an angular radius corresponding to ${13}^{\circ }$, according to [1]. Nitrogen and iron nuclei directions are isotropically distributed in the sky. The entire sample is generated by taking into account the exposure of the Pierre Auger Observatory, and the depth of shower maximum $X_{\max }$ for each particle is generated according to Gumbel parameterizations by considering the EPOS-LHC model [13].

Each simulated cosmic-ray sky is composed of 1000 events, a fraction f of which consists of protons coming from SBG and the remaining, $1-f$, of nitrogen and iron isotropically distributed. The sensitivity of the cross-correlation estimator will be studied as a function of the fraction of protons $f_p=f$ and the fraction of selected events $f_{X_{\max }}^{sel}$ corresponding to the highest $X_{\max }$ values. For each case, we performed 1000 cosmic-ray sky simulations. The detection power, P, is computed by the fraction of skies that have a ${\chi }^2$ larger that ${\chi }_{iso,99\%}$:

$$P=\frac{N\left({\chi }^2>{\chi }_{iso,99}^2\right)}{N_{total}},$$

(4)

where $N({\chi }^2>{\chi }_{iso,99}^2)$ is the number of cosmic-ray skies with ${\chi }^2$ larger that ${\chi }_{iso,99\%}^2$ and $N_{total}$ is the total number of simulated skies for a given scenario.

In Figure 2, we show the resulting detection power as a function of the proton fraction $f_p$ for different fractions of selected events $f_{X_{\max }}^{sel}$. The net gain when we use a small subset containing the highest $X_{\max }$ depends of the proton fraction, ranging from a factor of ∼2 to ∼4.

Figure 2. Detection power as a function of the proton fraction $f_p$ for different fractions of selected events $f_{X_{\max }}^{sel}$. The lines indicate the different values of the highest $X_{\max }$ considered.

4. Summary

The approach presented here shows us the increase in detection power of the Landy–Szalay cross-correlation estimator when using a small subset of data containing the highest $X_{\max }$ events. The gain depends on the proton fraction ranging from a factor of ∼2 to ∼4. Anisotropy studies in combination with $X_{\max }^{SD}$ measurements will continue to be explored in future works. This analysis opens an interesting window for future applications with respect to real data.