The cosmic ray shadow of the Moon observed with the ANTARES neutrino telescope

One of the main objectives of the ANTARES telescope is the search for point-like neutrino sources. Both the pointing accuracy and the angular resolution of the detector are important in this context and a reliable way to evaluate this performance is needed. In order to measure the pointing accuracy of the detector, one possibility is to study the shadow of the Moon, i.e. the deficit of the atmospheric muon flux from the direction of the Moon induced by the absorption of cosmic rays. Analysing the data taken between 2007 and 2016, the Moon shadow is observed with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3.5\sigma $$\end{document}3.5σ statistical significance. The detector angular resolution for downward-going muons is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.73^{\circ }\pm 0.14^{\circ }.$$\end{document}0.73∘±0.14∘. The resulting pointing performance is consistent with the expectations. An independent check of the telescope pointing accuracy is realised with the data collected by a shower array detector onboard of a ship temporarily moving around the ANTARES location.


Introduction
The detection of cosmic neutrinos is a new and unique method to study the Universe. The weakly interacting nature of neutrinos makes them a complementary cosmic probe to other messengers such as the electromagnetic radiation, γ -rays, gravitational waves and charged cosmic rays. Neutrinos can travel cosmological distances, crossing regions with high matter or radiation field densities, without being absorbed. They allow the observation of the distant Universe and the interior of the astrophysical sources.
A milestone has been set with the first evidence of a cosmic signal of high-energy neutrinos [1] by the IceCube detector [2,3]. The ANTARES telescope [4], although much smaller a e-mail: tommaso.chiarusi@bo.infn.it b e-mail: matteo.sanguineti@ge.infn.it than the IceCube detector, is the largest undersea neutrino telescope currently in operation. One of its main goals is the search for astrophysical point-like sources of neutrinos. To this aim, the pointing accuracy of the detector is important and an evaluation of this performance is required.
The interaction of cosmic rays in the atmosphere produces downward-going muons that can be recorded by underground, underice or underwater experiments. Atmospher-ic muons represent a large source of background for cosmic neutrino detection, but at the same time they can be used to calibrate the detector. Due to absorption effects of cosmic rays by the Moon, a deficit in the atmospheric muon event density (expressed as number of events per square degrees) in the direction of the Moon, the so-called Moon shadow, is expected. With this approach, the Moon shadow has been already measured and reported by MACRO [5], SOUDAN [6], L3+Cosmics [7] and by IceCube [8] Collaborations. It is worthy to mention here that other experiments, like CYGNUS [9], TIBET [10], CASA [11], ARGO-YBJ [12], and recently also HAWC [13] measured the Moon shadow by exploiting surface arrays detectors.
This work presents the first measurement of ANTARES angular resolution with atmospheric downward-going muons and the detector pointing performance making use of a celestial source for calibrations. A complementary estimation of the telescope pointing accuracy has been performed by means of a surface array of particle detectors arranged onboard a ship deck. The ship was temporarily routing above the ANTARES detector, allowing to correlate the signals from the detection of atmospheric showers with the signals induced by downward-going muons in the underwater telescope. This paper is organized as follows: in Sect. 2 the ANTARES detector is introduced together with the motivations of the present analysis; in Sect. 3 the Moon shadow analysis is described; the surface array analysis is presented in Sect. 4 and the conclusions are reported in Sect. 5.

The ANTARES neutrino telescope
The ANTARES detector is deployed 40 km offshore from Toulon, France (42 • 48 N, 6 • 10 E) anchored at a depth of about 2475 m. The telescope detects the Cherenkov light stimulated in the medium by relativistic particles by means of a three dimensional grid of optical modules (OMs), pressure resistant glass spheres each containing one 10 photomultiplier tube (PMT). The OMs are arranged in triplets, forming a storey, along twelve vertical lines, for a total of 885 OMs [4]. The lines are anchored on the sea bottom and kept taut by a buoy at the top. Each PMT is nominally oriented 45 • downward with respect to the vertical direction. This orientation enhances the efficiency for the reconstruction of upward-going tracks, but still allows the detection of downward-going muons with smaller efficiency. A titanium cylinder in each storey houses the electronics for readout and control, together with compasses and tiltmeters. The total length of each line is 450 m, without any instrument along the lower 100 m. The distance between storeys is 14.5 m and the distance between two lines ranges between 60 and 75 m. The lines are connected to a central junction box which, in turn, is connected to shore via an electro-optical cable. Due to sea currents, a positioning system comprising hydrophones, compasses and tiltmeters is used to monitor the detector geometry [14]. Finally the absolute orientation is provided by the triangulation of acoustic signals between lines and the deployment vessel at the sea surface using GPS [14,15]. The first detection line was deployed in 2006; the detector was completed in 2008.
The recorded information of each photon detected on a PMT is referred to as hit, and consists of the detection time, the amount of electric charge measured on the PMT anode and the PMT identification. The ensemble of hits contained in a certain time-window, identified after some trigger condition, is called event. Muon candidates are identified by requiring spacetime causality between the hits of one event [16,17]. The quality of the reconstruction of muon trajectories depends on the goodness of such spacetime correlation.

The Moon shadow analysis
Atmospheric muons are a valuable resource for validating the detector performance and characterising some of the possible systematics associated to the experimental setup. The muons produced in the interactions of primary cosmic rays in the upper layers of the atmosphere can traverse several kilometres of water equivalent; for this reason only downward-going atmospheric muons can be measured [18][19][20]. For those primary cosmic rays absorbed by the Moon, a deficit in the flux of the secondary muons can be measured, being directly correlated to the position of the Moon in the sky.
The energy threshold for muons detectable at the depth of the ANTARES telescope is about 500 GeV when they are at the sea surface level, most of them with energy above 1 TeV. Primaries which are progenitors of such highly energetic muons are practically not affected by the Earth geomagnetic field. This assumption of large rigidity holds also for the secondary muons detected by the ANTARES detector, thus they can be exploited in the study of the Moon shadow without introducing any bias. The smearing of muon direction with respect to the primary cosmic rays due to pion transverse momentum and pion decay is limited by the large Lorentz factor [21]. The analysis presented in this paper covers the data-taking period spanning from 2007 to 2016, corresponding to a total live-time of 3128 days. Figure 1 shows the position of the Moon in the horizontal coordinate system of the detector for such a period. The Moon altitude ranges above the horizon up to about 75 • .
The analysis is performed in three steps, described in Sect. 3.1, 3.2 and 3.3. First, quality cuts are defined to reduce the number of candidate atmospheric muon events to a sample which provides the best sensitivity for this search. The second part concerns the estimation of the telescope angular resolution for atmospheric muons by studying the monodimensional profile of the Moon shadow. In the third part, the pointing precision is determined evaluating a possible shift of the measured direction of the Moon with respect to the nominal values provided by astronomical libraries [24].

Optimisation of quality cuts
A dedicated Monte Carlo (MC) production is used to optimize the selection criteria applied to the reconstructed muon tracks. The MC generation of the atmospheric muon sample is performed with the MUPAGE code [25], which was proved to reliably reproduce this background source in several ANTARES analyses, such as reported in [26][27][28] . MUPAGE implements parametric formulas for the flux, the radial distribution, the multiplicity and the energy spectrum of muons at a given depth, allowing for a fast production of both single and bundle muon events. Muons are generated on the surface of a cylinder-shaped volume of water, 650 m high, with a radius of 290 m, containing the detector. This volume is larger than the instrumented volume and corresponds to the region in which muons can produce detectable signals. The generation of the MC sample is subdivided in different batches corresponding to the periods of data-taking, referred to as runs. The simulation reproduces the effective data taking conditions of the ANTARES detector, which can vary on a run-by-run basis [29]. The simulation includes the generation of Cherenkov light stimulated by the muon and its propagation up to the PMTs on the basis of the measured characteristics of light propagation [30]. Optical background, caused by bioluminescence and radioactive isotopes (mainly 40 K) present in sea water, is also added according to the measured rate. This technique allows to correlate the actual time of each run to the position of the Moon in the sky. In particular, it is possible to assign an absolute time-stamp, generated randomly within the period of each considered run, to each MC event reconstructed as a downward-going muon. A detailed production compliant with the actual live time is used to generate, reconstruct and select the MC sample of events within the restricted area of 10 • around the nominal position of the Moon at the time of each event. In order to evaluate the contamination of mis-reconstructed events in the proximity of the Moon, a smaller MC sample, with 1/3 of the actual live time, is generated over the whole visible sky.
The detector response is then simulated taking into account the main features of the PMTs and of the electronics [31,32]. Finally, the PMT signals are processed to reconstruct the atmospheric muon tracks with the standard ANTARES algorithm for track-like events. This is a robust track-fitting procedure based on a likelihood maximisation [17]. Figures of merit are determined by means of two quality parameters: Λ, which varies linearly with the logarithm of the recon-structed track likelihood, and β, the angular error associated to the reconstructed direction.
Two different MC simulation sets are prepared: the sample S 1 considering the shadowing effect of the Moon and the sample S 0 without this effect. In the sample S 1 , the Moon shadow is obtained by removing the muons generated within the Moon disk, assuming a radius of 0.26 • . The information from all the considered simulated runs is combined to obtain statistical evidence of the Moon shadow. For each of the two MC samples, S 1 and S 0 , a one dimensional histogram is built with the distribution of events as a function of the angular distance δ with respect to the Moon, up to 10 • . Such a histogram is subdivided into 25 bins, each one sized Δδ = 0.4 • and corresponding to an annulus of increasing radius centered on the Moon. The content of each bin is normalised to the corresponding annulus area, resulting in an event density.
The cuts on the quality parameters Λ and β are chosen to achieve the best sensitivity for the Moon shadow detection. The approach of the hypothesis test is used: the null hypothesis H 0 relates to the case of atmospheric muons without the Moon shadow, while the alternative hypothesis H 1 corresponds to the presence of the Moon. The used test statistic is defined as λ = −2 log L H 1 L H 0 , with L H 0 and L H 1 the likelihoods obtained under the H 0 and H 1 hypotheses. Assuming that the event population in each bin follows a Poisson probability distribution, using the χ 2 definition in [33], the chosen Fig. 2 The test statistics λ distribution for the "Moon shadow" hypothesis H 1 (dotted curve) and the "no Moon shadow" hypothesis H 0 (smooth curve). The dashed area corresponds to the 50% of the pseudoexperiments where the Moon shadow hypothesis is correctly identified. The shaded area quantifies the expected median significance (here 3.4 σ ) to observe the Moon shadow test statistic can be conveniently written as: with where n i stands for the measurement in the i-th bin to be compared with the expectations N i,H under the H 0 and H 1 hypotheses. The following reduced expression for λ is used: where for simplicity the expected counts N i,H 0 and N i,H 1 are renamed as ν i and μ i , respectively. The two possible distributions of λ, f (λ|H 0 ) and f (λ|H 1 ), valid separately under the hypotheses H 0 and H 1 , respectively, are obtained by means of pseudo-experiments (PEs). The number of events in the i-th bin n i is determined by extracting 10 6 random values generated according to a Poisson distribution with expectation values equal to ν i and μ i . Several hypothesis tests are performed assuming different selection criteria for Λ and β. For each set of values, the distributions f (λ|H 0 ) and f (λ|H 1 ) are compared. The median of f (λ|H 1 ) is taken as the critical value for λ, i.e. as the threshold to separate the two hypothesis. The set of best cut values of Λ and β corresponds to that for which the two f (λ|H ) distributions have the minimal overlap. Figure 2 shows the distribution f (λ|H 0 ) (black curve) and f (λ|H 1 ) (red curve) for the optimised quality cuts Λ cut = −5.9, β cut = 0.8 • , and the critical value is λ = −6.15. The dashed area below f (λ|H 1 ) represents the fraction of PEs where the Moon shadow hypothesis is correctly identified; the filledcoloured area below f (λ|H 0 ) corresponds to a p-value equal to 3.6 × 10 −4 , or equivalently 3.4 σ . This is the expected median significance of the Moon shadow effect with the MC data set.

Deficit significance and angular resolution
The optimized quality cuts reported above are applied to the data sample collected in the period 2007-2016, selecting 9.6 × 10 5 events out of the pre-cuts collection about three times larger. Figure 3 shows the resulting distribution for the muon density as a function of δ in the range of 0 • ≤ δ ≤ 10 • with bin size of Δδ = 0.4 • . A clear deficit of events is evident in the region around the Moon position (δ < 1.2 • ).
For the estimation of the angular resolution, the Moon shadowing effect is assumed to follow a Gaussian distribution with standard deviation σ res , which corresponds to the detector angular resolution itself. This is motivated by the fact that the apparent size of the Moon in the sky is sufficiently small compared to the expected value of the detector angular resolution, affecting the estimation by less than a few percents. A similar approach has already been followed by [5,8,34]. The number of expected events is evaluated by fitting the distribution in Fig. 3 with the following function [6]: The two free parameters are k, the average muon event density in the H 0 scenario, and σ res . The Moon radius R Moon is fixed to 0.26 • .
The angular resolution for downward-going atmospheric muons resulting from the fit is σ res = 0.73 • ± 0.14 • , with the fitted value of k = 2376 ± 3 events per square degrees. The goodness of the fit is found to be χ 2 /dof = 23.5/23.
The significance of the shadowing is evaluated using a χ 2 test comparing the measured event density with the flat distribution dn dδ 2 = k. Such χ 2 test leads to a p-value equal to 4.3 × 10 −4 corresponding to a significance of the Moon shadow effect of 3.3 σ . This value is compatible with the expected significance of the Monte Carlo previously described.

Absolute pointing
The procedure for evaluating the pointing accuracy of the Moon shadow is partially inspired by [5]; it is based on determining the statistical significance of the selected data set under the assumption of the Moon in a given direction. All possible placements are considered within a field of view (FoV) centered on the nominal position of the Moon. This work differs from [5] in the way the significance of the results is evaluated.
The event distribution of the detected muons, compliant to the determined quality cuts, is represented as function of The expectations under H 0 are obtained parameterising the event distribution of the measured atmospheric muons which fall in the FoV relative to the position of the Moon four hours before the timestamp of each event. The parameterisation is done with a second degree polynomial of the form: The expectations under H 1 are then obtained by subtracting from p 2 (x, y, k|H 0 ) a bi-dimensional Gaussian point spread function: In Eq. (6) the same spread is assumed in both dimensions, so that σ x = σ y ≡ σ res . The σ res is fixed to value of the angular resolution found in the previous sub-section. The array of free parameters θ is composed of the amplitude of the Moon deficit A and the assumed position of the Moon (x s , y s ) in the FoV. For each bin in the FoV, the value of the test statistic λ is minimised finding the best estimation of A. The smallest value λ min is found equal to −17.05, for the fitted deficit amplitude A min = 20 ± 5, in the bin with center in x = 0.5 • and y = 0.1 • . Such coordinates are taken as the best esti- The interval corresponding to a desired confidence level (C L) is obtained for λ ≤ λ cut = λ min + Q, where Q is the quantile accounting for two degrees of freedom and confidence level C L [35].
An additional strategy is used to cross-check the confidence intervals found with the method reported above. This is done by exploiting the PE technique. In each bin of the FoV, a reference number of events {n i } re f is computed using the superposition of Eqs. (5) and (6). For this purpose the Moon is assumed to be in O, σ res = 0.73 • and A = A O . For each PE, a corresponding data set {n i } P E is extracted as Possionian fluctuations of the reference set {n i } re f . Using 10 5 PEs, the distribution of the best value of λ j is determined at the j-th bin of the FoV.
For each λ j distribution, the range −∞, λ C L j is considered, where λ C L j is the value of λ j such that its cumulative distribution is F(λ C L j ) = C L; the j-th bin is included into the confidence interval if λ m j ≤ λ C L j . Figure 6 shows the estimation of the confidence regions for C L ≡ {68.27%, 95.45%, 99.73%} using both the meth- The statistical significance of the apparent shift with respect to the Moon nominal position is determined using PEs. The method relies on the probability density function of the test statistics Θ = λ O −λ min , with λ O and λ min defined as before. The Θ test statistic is interpreted as a profile likelihood whose distribution asymptotically tends to that one of a χ 2 with two degrees of freedom.
In Fig. 7 the normalised distribution of the Θ test statistic is shown, where the measured value of the test statistic Θ meas = 3.68 is indicated for reference by the red-dashed line. Integrating the Θ distribution for values larger than Θ meas , a p-value = 0.23 is obtained, corresponding to a significance of 1.2 σ . This indicates that the shift is compatible with a statistical fluctuation.

Analysis of data collected with a surface array
The pointing performance of the ANTARES telescope is cross-checked in a completely independent way, exploiting the measurements made with a surface array detector. The device was temporarily onboard of a ship circulating around the position of the telescope, synchronised to a GPS reference. The surface array was composed of a set of 15 liquid scintillator detection units, designed for the measurement of Two different sea campaigns were performed: a first campaign of seven days in 2011 and a second campaign of six days in 2012. Given the area covered by the ship routing above the ANTARES telescope, the range of the muon zenith θ is limited to 2 • ≤ θ ≤ 27 • . Figure 8 shows the recorded positions on the sea surface of the ship during these two periods.
The shower array is used to trigger the possible timecorrelations with the ANTARES events. The typical trigger rate of the surface array is around 1 Hz requiring coincidences in at least 3 detection units in a 650 ns time window. The rate of reconstructed muons is ∼ 0.25 Hz when applying cuts on the quality parameters Λ ≥ −6 and β ≤ 0.6 • . The coincidence time-window between the surface array and the underwater telescope is set to 10 μs. The rate of coincidences is about 40 per day, with an expected rate of random coincidences of about 0.2 events per day.
The direction of the atmospheric shower is estimated by correlating the GPS position of the ship with the ANTARES barycentre. An uncertainty of 25 m, about one half of the ship deck hosting the shower array, is assumed for the shower position detected by the array. An error of 70 m is considered for the possible displacement of the muon/muon bundle axis inside the detector volume. Considering only the ship routes  The results of the two campaigns are shown in Fig. 9: the represented Δθ (left) and Δφ (right) are the differences between the directions of the shower axis and the reconstructed muon underwater.
According to the Gaussian fit of the two distributions in Fig. 9, the absolute pointing appears to be consistent with the nominal expectations, i.e. with a null systematic shift in both zenith and azimuth within the errors (see caption of Fig.  9). The large uncertainty in the azimuth estimation is due to the low zenith angle tested during the campaigns, as can be derived from Fig. 8. The results of the surface array analysis are in good agreement with the pointing performance found with the Moon shadow analysis.

Conclusions
This paper describes the estimations of the pointing performance of the ANTARES telescope using the Moon shadow effect and a dedicated surface array.
The selected events from the data recorded in the 2007-2016 period with altitude angles 0 • ≤ h ≤ 75 • , allowed the identification of the Moon shadow with 3.5 σ statistical significance. The corresponding detector angular resolution for downward-going atmospheric muons is 0.73 • ± 0.14 • .
The pointing accuracy of the detector is consistent with the expectations.