A Phenomenological Approach to the Study of the Cosmic Ray Galactic-Extragalactic Transition

A phenomenological approach to the study of the cosmic ray Galactic-extragalactic transition

C. De Donato

November 21, 2008

Ai miei genitori

“... fatti non foste a viver come bruti, ma per seguir virtute e canoscenza”

Dante Alighieri

Divina Commedia Inferno XXVI vv 118-120

Acknowledgments

I would like to thank my tutor Dr. Battistoni and all the people of the Univer- sit`adegli Studi di Milano which, despite not taking part in the Auger project, demonstrated a lot of interest in my studies with stimulating discussions. I am grateful to the Pierre Auger Collaboration; I have beneﬁted from the interaction with several colleagues and from a very stimulating professional environment. Special thanks are reserved to Prof. Gianni Navarra for his availability and the time he dedicated to my thesis work. I would like to thank also the Instituto de Ciencias Nucleares (UNAM), for its hospitality during my extended stays, and all the ICN group with whom I have spent a very productive and pleasant period of my PhD studies. A special mention is reserved to Dr. Gustavo Medina Tanco who guided and supported me along my PhD studies not only as a co-tutor but also as a friend. The knowledge and the passion he transmits in his work is a rare and special gift and I feel honored by the trust he put in my capabilities and work. The challenges he presented me every day made my work very stimulating and productive (and my night rests very short!), feeding my enthusiasm in my re- searches. Finally, but not less important, I want to thank all my family and friends, who always believed in me. There are not enough words to describe the importance of my parents, grandparents, brothers and sisters in my life. Their encourage- ment during my studies and their support in my professional choices have been fundamental. The values my parents have transmitted to me have been, and are still, decisive in my life and made me the person I am proud to be. I hope they will always perceive the love I feel for all of them. Special thanks are reserved to my best friends Rossella and Eleonora who always have been close to me anywhere I were. The experiences we lived and shared together made our friendship deep and unique; they will have a special place in my heart and life forever.

Abstract

The cosmic ray (CR) energy spectrum extends for many orders of magnitude with a power law index 2.7. Along this range of energies, three spectral features are known: the first≈ knee at E 3 P eV , the second knee at E 0.5 EeV and the ankle, a depression extending≈ from the second knee to beyond≈ 10 EeV . The nature of the second knee and of the ankle is still uncertain; a possible interpretation of the two features is the transition between the Galactic and extragalactic components. At energies between 1017 1018 eV the Galactic supernova remnants (SNR) are expected to become inefficient− as particle accelerators. This fact, combined with magnetic deconfinement, should lead to the end of the observation of a Galactic component of cosmic rays, although the picture could be confused by the existence of additional Galactic accelerators at higher energy. On the other hand, at energies above the second knee, extragalactic particles are able to travel from the nearest extragalactic sources in less than a Hubble time. Since Galactic deconfinement at these energies would also allow them to penetrate the Galactic field, the spectrum may present above 1017.5 eV a growing extragalactic component that becomes dominant above 1019 eV . Therefore, the region encompassing the second knee and the ankle could be the transition region between the Galactic and extragalactic components. The energy region spanning from 1017 eV to . 1019 eV is critical for understanding both, the Galactic and∼ the extragalactic cosmic ray fluxes. The detailed knowledge of the way in which this transition takes place, i.e. the exact spectral shapes of the involved components and the evolution of their composition as a function of energy, is essential in order to understand the origin and propagation of CRs, since important information can be deduced on the astrophysical sources, the conditions of propagation in the extragalactic space and the cosmological evolution of the sources of the most energetic CRs. The study of the transition region can provide important information able, in principle, to break the degeneracy between astrophysical models existing at the highest energies due to the low available statistics.

The present PhD thesis deals with theoretical and phenomenological aspects of ultra-high energy cosmic ray physics, with specific emphasis on the numerical modeling of the transition between the diffusive and ballistic propagation regimes in the 1017 to 1019 eV energy range, i.e., the end of the Galactic confinement and the mixing of the Galactic and extragalactic components of the cosmic ray flux. The Galactic spectrum from regular Supernova remnants was calculated using a numerical diffusive propagation code in a realistic realization of the interstellar medium, properly accounting for in-flight spallation and de-

iii iv cay. The calculated diffusive Galactic spectrum is used to analyze the end of the Galactic cosmic ray spectrum and its mixing with the extragalactic cosmic ray flux by comparing alternative theoretical scenarios with data from the most relevant experiments in the relevant energy region. In particular, the transition region is analyzed combining the diffusive Galactic spectrum from SNRs with two different models of extragalactic spectrum, one in which only protons are injected at the sources and another one in which a mixed composition of nuclei is injected instead. In order to discriminate between possible astrophysical scenarios, the experimental parameter Xmax, a composition indicator, was inferred from the different theoretical extragalactic (EG) models for the several hadronic interaction models currently in use in the literature. A comparison of these results with experimental data was performed with special attention to Auger and HiRes results. Spectral and composition data (in the form of Xmax) provided by HiRes and Auger experiments, which favor an EG mixed composition, were used to set constraints on the Galactic and extragalactic CR fluxes and on their change in composition as a function of energy. The conditions that must be met by the Galactic and extragalactic fluxes in order to reproduce simultaneously the total spectrum and elongation rate observed by either Auger or HiRes, were studied in detail. The present analysis favors a mixed extragalactic spectrum, combined with a Galactic spectrum enhanced by two additional high energy components of mixed composition, extending beyond the maximum energies expected from regular supernova remnants. The evolution of composition inside each component and their relative composition suggest two contributions, the main one consistent with acceleration in different populations of SNRs immersed in differing environments and a minor contribution from another acceleration mechanisms, without a rigidity cut-off, operating at the highest energies, possibly associated with inductors such as compact objects like pulsars and magnetars. The potential impact on the astrophysical analysis of the assumed hadronic interaction model was also assessed.

As a by-product of the diﬀusive propagation of lower energy CR, the diﬀuse neutrino background produced by our Galaxy from the decay of pions originated in p-p interactions between cosmic rays and the interstellar medium has been numerically calculated. Contents

1 UHECR physics 1 1.1 Thecosmicrayspectrum ...... 1 1.2 Transitionmodels...... 4 1.3 Sources ...... 7 1.3.1 “Bottom-up”models: cosmicaccelerators ...... 9 1.3.2 “Top-down” models: alternative models ...... 12 1.3.3 Z-burstmodelsandothers...... 13 1.4 InteractionwithCBR ...... 13 1.4.1 Protons ...... 14 1.4.2 Nuclei ...... 17 1.4.3 Photons...... 18 1.5 Magneticfields ...... 18 1.5.1 Galacticmagneticfield...... 18 1.5.2 Extragalacticmagneticfield ...... 19

2 Detection of UHECRs 23 2.1 EAS ...... 23 2.2 Detectiontechniques ...... 29 2.2.1 Surfacedetectors: GroundArrays ...... 30 2.2.2 Fluorescencedetectors ...... 32 2.2.3 Radiodetectors...... 37 2.3 Experiments...... 37 2.3.1 VolcanoRanchArray ...... 37 2.3.2 HaverahParkArray ...... 39 2.3.3 YakutskArray ...... 40 2.3.4 Akeno Giant Air-Shower Array (AGASA) ...... 42 2.3.5 Fly’sEyeDetector ...... 43 2.3.6 HiRes ...... 45 2.3.7 KASCADEandKASCADE-Grande ...... 46 2.3.8 PierreAugerObservatory ...... 47 2.3.9 AMIGAandHEAT ...... 49 2.3.10 TelescopeArray(TA)andTALE ...... 51

3 Results 53 3.1 Spectrum ...... 53 3.1.1 Dipcalibration ...... 57 3.1.2 SpectrumandEGastrophysicalmodels ...... 58 3.2 Composition ...... 61

v vi CONTENTS

3.2.1 Compositioninthetransitionregion ...... 62 3.2.2 Primaryphotons ...... 63 3.2.3 Neutrinos ...... 65 3.3 Anisotropy ...... 65 3.3.1 Large-scaleanisotropy ...... 65 3.3.2 Intermediatescaleanisotropy ...... 67 3.3.3 Smallscaleanisotropy ...... 67 3.4 Crosssection ...... 71 3.5 Summaryoftheresults ...... 72 3.6 Importanceofthetransitionregion ...... 73

4 The transition region: spectrum 75 4.1 SNRGalacticspectrum ...... 75 4.1.1 DiﬀusionGalacticmodel...... 75 4.1.2 DiﬀusiveGalacticspectrum ...... 79 4.2 EGspectrum ...... 82 4.3 Combinedspectrum ...... 84 4.4 Discussion...... 87

5 Thetransitionregion:composition 91 5.1 Xmax andHIM...... 91 5.2 Xmax energyproﬁles ...... 93 5.3 Compositionevolution ...... 98 5.3.1 Augerdata ...... 99 5.3.2 HiResdata ...... 102 5.4 Discussion...... 104

6 Galactic neutrino background 107 6.1 Pionproduction ...... 107 6.1.1 Isobaricmodel ...... 108 6.1.2 Scalingmodel...... 108 6.1.3 Pionspectrum ...... 109 6.2 Muondecay...... 111 6.3 Neutrinoproduction ...... 111 6.4 Neutrinooscillation...... 113 6.5 Neutrinoﬂux ...... 116

7 Conclusions 121

A Shock acceleration 125 A.1 Fermiacceleration ...... 125 A.2 SNRmaximumenergy ...... 129

B Electromagnetic and hadronic showers 133 B.1 Electromagneticshowers ...... 133 B.2 Hadronicshowers...... 134

Bibliography 137 Chapter 1

UHECR physics

Since their discovery in 1912 by the physicist Victor Hess [1], cosmic rays (CRs) have assumed an important role in astronomy and particle physics, especially after the discovery by Pierre Auger [2] in 1938 of extensive air showers that extended the energy scale known to science above 1015 eV . From one side, cosmic rays could represent an alternative to extend our knowledge of fundamental interaction physics at energies not reachable by man-made accelerators; on the other side, especially at energies above 1018 eV , ultra high energy cosmic rays (UHECRs) open a challenging window for astronomy, as they carry important information on their sources and on the Galactic and extragalactic medium they have to cross. The understanding of the origin of cosmic rays is a complex puzzle; in order to solve it many question has to be resolved: which is the nature of CRs and their composition, how the CR energy spectrum evolves, which are the sources of Galactic and extragalactic CRs and which are the acceleration mechanism able to provide such a huge energy, how CRs interact and propagate through interstellar and intergalactic medium and how is their distribution, if it follows the matter distribution of Galaxy or the distribution of nearby extragalactic sources. In this chapter the main topics regarding CR study are presented; the observed spectrum and its features are discussed along with the models describing the transition between the Galactic and extragalactic components. The main source and acceleration models as well as the propagation mechanisms and interactions are presented.

1.1 The cosmic ray spectrum

The cosmic ray (CR) spectrum extends for many orders of magnitude up till 1021 eV . From energies above 10 GeV 1, the spectrum follows a power law γ dN/dE E− with a spectral index γ 2.7 (Fig. 1.1(a)). Therefore, the CR ∼ 3 2 1∼ 2 intensity flux goes from 10 m− sec− at E GeV to 1 km− per century at the highest energies. This∼ great decrease in flux∼ divides∼ the wide energy range in two experimental regions, slightly overlapping around 1014 1015 eV . In the − 1At lower energies, the CR flux is affected by Sun modulation and by the geomagnetic local field.

1 2 CHAPTER 1. UHECR PHYSICS lower energy region, given the high flux, direct detections of cosmic ray flux and composition are possible through balloon or satellite experiments, while in the higher energy one only indirect measurements in large arrays are available. In fact at high energies, because of the low CR flux, extensive areas of detection are required to collect statistics, possible only on Earth surface. In this case, the energy and composition of cosmic rays are deduced from the detection of the cascade of particle, extensive air shower (EAS), generated by the interaction of the primary CR with the molecules of the atmosphere.

Along the wide range of energies, the only spectral features are the first knee at E 3 P eV , the second knee at E 0.5 EeV , the ankle, a dip extending from the≈ second knee to beyond 10 EeV ,≈ and the GZK cut-off. The nature of the knee, consisting of a steepening of the spectrum from a power low index γ =2.7 to γ =3.1, is well explained by the rigidity-models (rigidity- acceleration model and rigidity-confinement model), in which the maximum energy achievable by nuclei is rigidity dependent. In these models, the knees of the spectrum of nuclei of charge Z are related to the proton knee energy Z p p 15 Ekn = ZEkn, where Ekn = 2.5 10 eV . Beyond the highest energy knee F e 16 × Ekn =6.5 10 eV the total Galactic flux, dominated by the Iron component, must be steeper.× While the nature of the knee in the contest of the rigidity models seems to be confirmed by KASCADE data [3], the physical interpretation of the second knee and of the ankle is still uncertain [4]. A possible interpretation of the two features is the transition between the Galac- tic and extragalactic components. At energies between 1017 1018 eV the Galactic supernova remnants (SNR) are expected to become in−efficient as accelerators. This fact, combined with magnetic deconfinement, should mark the end of the Galactic component of cosmic rays, although the picture could be confused by the existence of additional Galactic accelerators at higher energy. On the other hand, at energies above the second knee, extragalactic particles are able to travel from the nearest extragalactic sources in less than a Hubble time. Consequently, the spectrum may present above 1017.5 eV a growing extragalactic component that becomes dominant above 1019 eV . The region between the second knee and the ankle could be the transition region between the Galactic and extragalactic components. The knowledge of the way this transition takes place, i.e. the exact spectral shapes and compositions, is essential for understanding CRs origin and propagation as important information can be deduced on the astrophysical sources, the conditions of propagation in the extragalactic space and the cosmological evolution of the sources of the most energetic CRs. Many experiments have observed both the two spectral characteristic (even if at slightly different energies) Akeno [5], Flys Eye stereo [6, 7, 8], Yakutsk [9, 10] and HiRes [11], while the ankle has been also confirmed by Haverah Park [12] and AGASA [13] experiments (Fig. 1.1(b)). The last spectral feature is known as the GZK cut-off, predicted by Greisen [14], Zatzepin and Kuz’min [15] in 1966 after the discovery of the cosmic microwave background (CMB) by Penzias and Wilson [16]. The interaction of protons with the CMB photons through photo-pion production would lead to a suppression in the CR proton flux at energies above 4 1019 eV . Flux deficit above 4 1019 has been observed by HiRes [17, 18]∼ and× Auger [19] with × 1.1. THE COSMIC RAY SPECTRUM 3

(a)

(b)

Figure 1.1: Cosmic ray spectrum. a) Cosmic ray energy spectrum with its spectral features and experimental regions. b) The knee, second knee and the ankle are evident in the Flux E3 spectrum. Experimental data are also shown. × 4 CHAPTER 1. UHECR PHYSICS

Figure 1.2: Transition models: dip model (left) and ankle model (right). In the left panel the extragalactic proton spectrum and the Galactic component (dominated by Iron nuclei above EF e) are shown, as well as the transition energy Etr at the second knee. Below Eb the extragalactic proton spectrum is calculated for diffusive propagation [21]. In the right panel the extragalactic proton spectrum and the Galactic component are shown. The extragalactic 2 spectrum is calculated for a injection spectrum E− and the Galactic component is inferred subtracting the EG spectrum from the observed total spectrum. The 19 transition energy Etr is around 10 eV . Data of KASCADE [3] and HiResI, HiResII monocular spectra [22] are shown. Adapted from [23]. . a significance of 4.8σ and 6σ, respectively, while AGASA observed an excess at E > 1020 eV [13]. These results are still inconclusive as the absolute energy scale is not completely understood, as well as chemical composition [20] (see 3). More statistics and understanding of the systematics, as well as precise measure-§ ments of composition above 4 1019 eV are essential to claim the GZK effect. ∼ ×

1.2 Galactic extragalactic cosmic rays: transition models

Although the region between the second knee and the ankle is naturally considered as the transition between the Galactic and extragalactic components, the way the transition takes place is very model dependent. The main models describing the transition between the Galactic and extragalactic components are the ankle model, the dip model and the mixed composition model.

In the ankle model the dip in the CR spectrum is interpreted in terms of a two 1.2. TRANSITION MODELS 5 components model, as the intersection of a flat extragalactic component with the steep Galactic component (Fig. 1.2, right panel). The transition between the Galactic cosmic rays (GCR) and extragalactic cosmic rays (EGCR) occurs at the 19 ankle Ea 1 10 eV [24, 25, 26, 27, 28, 29, 30, 31], where the two components have equal≈ fluxes.× The extragalactic spectrum is assumed to have a pure pro- 2 ton composition and a flat generation spectrum E− , valid for non-relativistic shocks acceleration. According to this model, energy losses are insignificant at E . 4 1019 eV and the dip is explained by the Hill-Scramm mechanism [31]. While× this model has the advantage of a flat generation spectrum that provides reasonable luminosities of the sources, it presents some problems. In fact, as it predicts a transition energy around the ankle, the ankle model doesn’t work well in the framework of the rigidity-model. Even if the model has to assume that at the highest energies the Galactic spectrum is composed of Iron nuclei, other additional Galactic mechanisms are required to accelerate particles at energies F e 16 beyond the Iron knee Ekn =6.5 10 eV , in contrast with the Standard Model of CR acceleration (see Fig. 1.2,× right panel).

A different explication of the dip and a lower transition energy comes from the dip model [32, 21, 23] (Fig. 1.2, left panel). In this model two spectral features are caused by the interaction of extragalactic protons with the CMB, the GZK + cutoff and the dip. The dip is predicted as the result of e e− pair production by the extragalactic protons with the CMB photons. The shape and position of the dip is analyzed in terms of modification factor, defined as the ratio between the spectrum calculated with all energy losses and the unmodified spectrum, the spectrum calculated with only adiabatic losses [23]. Including in sequence energy losses due to pair-production and pion production, the features of the dip and of the GZK cut-off appears in the modification factor, as well as an excess at E < 1018 eV (for definition, the modification factor must be 1), correspondent to a new component, i.e. the Galactic CR contribute. In this≤ model, the transition energy between the Galactic and extragalactic components is determined by the energy at which adiabatic energy losses are equal to pair production energy losses. This transition takes place at lower energies, around the second knee, in agreement with the rigidity-model (see Fig. 1.2, left panel). The calculated position and shape of the dip, which presence is confirmed by Akeno-AGASA [13], HiRes [11], Yakutsk [9, 10] and Fly’s Eye [33] experimental data2, is “universal” as it doesn’t depend on the type of propagation (recti- linear/diffusive), on the source density and separation (for separation distance d< 60 Mpc) and on the evolutionary model of the sources. The only factor that affect the dip is a heavy component of the EG spectrum [35, 36]. The shape and position of the dip is in agreement with observations for an EG pure proton composition with a maximum contamination of Helium nuclei of the order of 10 15%. Therefore, the position and shape of the dip is fixed by interaction with− CMB and can be used for energy calibration of the detectors (see 3.1.1) [23, 21]. § 2.7 On the other hand, as the dip model requires a generation spectrum E− , a solution is needed to prevent the too high emissivity needed at lower energies

2Auger data don’t contradict the high energy part of dip but an extension of Auger to lower energies it’s needed to conﬁrm the feature of the dip [34]. 6 CHAPTER 1. UHECR PHYSICS

Figure 1.3: EG mixed composition model. Spectrum for mixed composition 18 model [39] calculated with γ = 2.3, Ea = 3 10 eV , cosmological source 3 × 19 evolution (1 + z) at z 1.3 and a set of parameters xi. At E > 4 10 eV ≤ × 18 the spectrum is characterized by GZK cut-oﬀ. At energy Ea = 3 10 eV the transition to pure extragalactic component is completed. HiResI,× HiResII monocular spectra [22] are shown. Taken from [23].

[32, 37, 21]. One natural solution is to assume that the source spectrum has a standard shape with γ = 2.0 for non-relativistic shocks and γ = 2.2 2.3 for relativistic shocks. The steepening of the energy spectrum of generation− is the result of the natural distribution of the sources in luminosity or in maximum energy [36, 38].

An alternative and intermediate model is the mixed composition model [40, 41, 39]. In this model the EG cosmic ray composition is assumed to be mixed, in analogy with the Galactic component. It assumes an injection with a A- dependent regime, instead of a rigidity dependence,

γ 1 γ J(E) A − E− , (1.1) ∝ and a mixed composition spectrum proportional to the injection spectrum,

γ 1 γ Qi(E)= xiAi − KE− , (1.2) where i is the type of nucleus, K is a normalization constant, γ is the spectral index and xi are free parameters describing the composition at the sources. 1.3. SOURCES 7

Different models for the cosmological evolution of the sources are taken into account in the calculation. The calculated spectrum, fitted with the observed data, presents a spectral index γ 2.1 2.3, which, as in the ankle model, provides a reasonable luminosity. ≈ − Even if the model predicts a heavier composition, it predicts the GZK cutoff as the process of photo-disintegration is very efficient at E > 1019 eV and protons become the dominant component at E > 3 1019 eV . As in the ankle model, the intersection of the× Galactic and extragalactic components gives origin to the the dip structure but with the advantage of a lower transition energy (around E 3 1018 eV ), which softens the requirement of additional acceleration mechanisms.≈ × The inferred Galactic spectrum, found by subtraction of the extragalactic spectrum from the observed total spectrum, is shown in Fig. 1.3. On the other hand, the spectrum and mass composition depends on several parameters (cosmological and describing the source composition), making the model very flexible and able to reproduce many composition profiles.

The three model can be experimentally distinguished through measurements of the spectrum and of anisotropies, although the most discriminant feature is the chemical composition. Since in the ankle model the transitions take place around 1019 eV , the Galactic heavy component dominates up to the ankle energy. At higher energies the EG component begins to dominate and the composition becomes proton dominated. Consequently, the composition in the dip region is dominated by heavy nuclei. In contrast to the ankle model, in the dip model, as the transition is completed at energy around 1 1018 eV , the composition in the ankle region is proton dominated resulting× in a steeper increase of the elongation rate. The composition in the ankle region (Iron/proton) is a strong discriminant between the two models. In the case of the mixed model, the transition between Galactic and EG (mixed composition) component occurs at E 3 1018 eV . Consequently in the dip region the chemical composition is mixed,≈ × while at lower energies the Galactic heavy component dominates. This model predicts a slower decrease of the Fe component and a slower increase of the proton fraction in the transition energy range. At higher energies, due to photo-disintegration of the nuclei, the composition get lighter until at E 3 1019 eV becomes strongly proton-dominated. ≥ × The composition predictions obtained by the three model will be compared with experimental results in 3.2. §

1.3 Sources

The origin of cosmic rays, their sources and the acceleration mechanisms, is still an unsolved problem. The astrophysical models, or bottom-up models, are based on the existence of cosmological active objects which are able to create and accelerate cosmic rays. The acceleration mechanism is mainly based on electromagnetic processes, through which particles gain energy. The kind of acceleration may be direct (as in rotating compact object) or stochastic (as in the case of a moving magnetized plasma). The current paradigm for Galactic cosmic ray acceleration is the Fermi accel- 8 CHAPTER 1. UHECR PHYSICS eration mechanism by shock waves of Supernova Remnants (SNRs) [42] (see appendix A). SNRs are§ considered the main sources and acceleration mechanism of GCRs at energies below the knee. This suggestion was first made by Ginzburg e Syrovatskii [43] which identified the explosion of Supernovae as the most probable sources and acceleration sites of cosmic rays on the basis of energetic estimations. The density of cosmic rays in our Galaxy can be estimated from the differential flux assuming a uniform and isotropic distribution of CRs in the Galaxy volume:

dN dE 12 erg 3 ρ =4π E 10− 1 eV/cm . (1.3) E,CR dE βc ≃ cm3 ≃ Z The luminosity of cosmic rays in our Galaxy is given by V erg L = ρ g 1040 , (1.4) CR E,CR τ ∼ s where Vg is the volume of the Galactic disk (R = 15 kpc, h = 200 pc)

V = πR2d 4 1066 cm3 (1.5) g ∼ ∗ and τ 107 yr is the residence time of CRs (or confinement time) in our Galaxy. This luminosity∼ can be compared with estimates of the energy of astrophysical object produced during their life. Considering the typical supernova remnants, radio observations give an estimate of the kinetic energy of the accelerated electrons of 1047 erg, corresponding to a proton energy of 1049 erg3. Consequently, the∼ energy released by a SN explosion in the form of∼ kinetic energy is of the order of 1049 1051 erg. Taking 1 ∼ − a supernova rate of the order of (30 y)− , as deduced from the observation of 40 42 different galaxies, the corresponding luminosity is LSN 10 10 erg/s, comparable with the total luminosity of CRs in our Galaxy≈ disk.− For a typical SN of mass equal to 10 solar mass and with an acceleration time 103 yr in a magnetic field B 3 µG, the maximum energy reachable by a ∼particle of charge Ze is given by:≈

E 30Z TeV. (1.6) ≈ Supernovae are consequently the possible source of cosmic ray, able to accelerate particles through diﬀusive shocks up to 1015 eV (see A.2). This maximum energy is reachable only by the heaviest nuclei, according w§ith the observed knee.

At energies above the knee, shock acceleration by SNRs is not any more suﬃ- cient. Other mechanisms and sources able to accelerate cosmic rays to higher energies are needed, not only for UHECRs but also for the higher energy galactic cosmic rays that require supernova remnants in special environments [44]. In fact, the acceleration process may extends to higher energies, up to 1018 eV if the SN explosion takes places in an environment diﬀerent from the standard one, as in the case of particle accelerated by interaction with multiple supernova remnants as they move through the interstellar medium or in the case of SNRs in high density and magnetize medium as those occurring in the central region

3Assuming a density e/p ≈ 1% in analogy with the CR densities. 1.3. SOURCES 9 of the Galaxy. Other theories propose new acceleration models in which shock magnetic fields amplification by cosmic rays leads to higher acceleration energy and flatter energy spectrum [45]. In the following section a brief description of some of the models for UHECR acceleration is given.

1.3.1 “Bottom-up” models: cosmic accelerators The possible astrophysical objects able to accelerate particle to the highest energies were determined by Greisen and Cavallo [46, 47] considering the dimension of the acceleration region and its magnetic field. Basically, the acceleration region have to be comparable with the Larmor radius of the particle in a magnetic field B; on the other hand, the magnetic field must be low enough in order to have energy losses for synchrotron radiation lower than the energy gain. For an acceleration region of radius R and magnetic field B, the total magnetic energy of the source is given by

B2 4 W = πR3 (1.7) 4π 3 and the maximum acceleration energy is

E ZeBRβc (1.8) max ∝ where Ze is the charge of the particle. The map of the possible sources able to accelerate particle at energies higher than 1018 eV obtained is shown in Fig. 1.4. Possible sources are astrophysical compact objects with high magnetic ﬁeld, as neutron stars or pulsars (B 1013 Gauss, R = 10 km), or large acceleration regions with low magnetic ﬁeld,∼ as clusters of galaxies (GC, B 1 µGauss, R = 0.1 1 Mpc). The main possible candidates as sources of∼ the highest energy CRs− [50] are described below. A complete discussion on the sources is given in the review paper [51].

Pulsars (B 1013 Gauss, R = 10 km). • In young magnetized∼ neutrons stars the acceleration process is magneto- hydrodynamic rather than stochastic; the maximum energy achievable by a nucleus of charge Ze is Emax = ZeBRω/c, where ω is the the angular velocity (ω 200 Hz for the Crab pulsar), B is the surface magnetic field and R is the∼ radius of the neutron star. The acceleration spectrum is flat, 1 proportional to E− . Recent studies [52] show how relativistic winds of magnetars (pulsars with dipole magnetic fields approaching 1015 Gauss) can provide sources of ul- 1 tra relativistic light nuclei with a spectral index at injection E− steep- 2 ∝ 21 22 ening at higher energies to a spectrum E− with a cut-off at 10 − eV . ∝ Active Galactic Nuclei (AGN, B 103 Gauss, R 1010 km). • AGN are one of the most favored∼ sources for cosmic∼ rays at the highest energies [48, 53]. AGN are powered by the accretion of matter onto a super massive black hole of 106 108 solar masses. Typical values in the 2 ÷ central engine are R 10− pc, and B 5 G, which make possible the containment of protons∼ up to 1020 eV .∼ The main problem here is the large energy loss in a region of high field density, which would limit the 10 CHAPTER 1. UHECR PHYSICS

Figure 1.4: Hillas diagram: map of the astrophysical objects retained to be possible sources of UHECRs [48]. Size and magnetic ﬁeld strength of the acceleration regions are shown. Lines represent the energetic limit of 1020 eV for protons and Iron nuclei. Adapted from [49].

maximum energy achievable for protons and forbid the escape for heavy nuclei. Neutrons produced in p-p interactions could eventually escape the central region and then decay to protons with maximum energy around 1018 eV . Another solution is that the acceleration occurs in AGN jets where particles are injected with Lorentz factors larger than 10, and where energy losses are less signicant.

Cluster of Galaxies (GC, B 1 µGauss, R =0.1 1 Mpc). • Galaxy clusters are reasonable∼ sites for ultra-high− energy cosmic rays acceleration, since particles with energy up to 1020 eV can be contained by cluster ﬁelds ( 5 µG) in a region of size up to 500 kpc. Acceleration in clusters of galaxies∼ could be originated by the large scale motions and the related shock waves resulting from structure formation in the Universe 1.3. SOURCES 11

such as accretion ﬂow onto galaxy clusters and cluster mergers. How- ever, losses due to interactions with the microwave background during the propagation inside the clusters limit UHECRs in cluster shocks to reach at most 10 EeV .

Radio Galaxies Hot spots (RGH, B 0.1 1 mGauss, R = 1 kpc) and • Radio Galaxies Lobes (RGL, B 0.1∼µGauss,− R = 100 kpc). Fanaroff-Riley II galaxies are the∼ largest known dissipative objects (non- thermal sources) in the Universe. Regions of intense synchrotron emission observed within their lobes, known as “hot spots”, are produced when the jet ejected by a central super massive black hole, surrounded by an accretion disk, interacts with the intergalactic medium, generating turbulent fields. The result is the formation of a strong collision-less shock, which is responsible for particle re-acceleration and magnetic field amplification [54]. The acceleration of particles up to ultra relativistic energies in the hot spots is achieved by repeated scattering through the shock front, similar to the Fermi acceleration mechanism. In this case, the particle deflection is produced by Alfven waves in the turbulent magnetic field [55, 50]. For typical hot-spot conditions (B 300µG and βjet 0.3) and assuming that the magnetic field of the hot∼ spot is limited∼ to the observable region ( 1 kpc), one obtains a maximum acceleration energy for protons E ∼< 5 1020 eV . max ×

Gamma Ray Burst (GRB, B 109 Gauss, R = 104 105 km) [50]. • The origin of the detected gamma∼ ray bursts can be− explained by the collapse of massive stars or mergers of black holes or neutron stars. A relativistic shock is caused by a relativistic ﬁreball in a pre-existing gas, such as a stellar wind, producing/accelerating electrons/positrons to very high energy. The observed gamma-rays are emitted by relativistic electrons via synchrotron radiation and inverse Compton scattering. The high Lorentz factor (γ 102 103) allows for energy loss time larger than the acceleration time∼ but at− the same time limits the possible energy gain of a particle, as the particle distributions are extremely anisotropic in shock with large γ, limiting the probability of re-entering the shock. The bursts present an amazing variety of temporal proﬁles, spectra, and timescales. The duration of the detected GRB signal extends over 5 orders 3 2 51 of magnitude (10− 10 s) with an energy release up to 10 erg/s. This energy would account÷ for the luminosity required for cosmic rays above 1019 eV if the GRBs are uniformly distributed (independent of redshift). However, recent studies indicate that their redshift distribution seems to follow the average star formation rate of the Universe and that GRBs are more numerous at high redshifts.

It has to be noted that all this discussion is based on energetic estimates. Prob- lems as particle injection and the dynamics of the acceleration are still unsolved, as well as propagation processes and the structure of the magnetic ﬁelds involved. 12 CHAPTER 1. UHECR PHYSICS

1.3.2 “Top-down” models: alternative models Besides the astrophysical models of acceleration, other alternative models where developed in order to explain the origin of extreme high energy cosmic rays (EHECRs). As seen in 1.3.1, the bottom-up models, although can provide a number of sources that account§ for the maximum energy acceleration or for the luminosity of CRs, present many unsolved problems as the mechanism itself of acceleration, energy losses, the injection of particles in the acceleration region and the escape from it. These alternative scenarios resolve the problem of acceleration of particles up to 1020 eV and beyond with the so called top-down mechanisms, in which very massive (GUT scale) X particles decay and the resulting fragmentation process downgrades the energy to generate the observed EHECRs. Since the X particle mass (between 1022 and 1025 eV 4) is many order of magnitude higher than the observed cosmic ray energy, there are no problems with achieving the necessary energy scale. Moreover, as the luminosity of EHECRs is not very high, the amount of X particles needed is a small fraction of the dark matter. The top-down models can be divided in two categories, Topological Defect (TD) models and Super-Heavy Dark Matter (SHDM) models.

In the Topological Defect models, X particles are emitted by topological defects, as magnetic monopoles [56, 57], cosmic strings and necklaces (a closed loop of cosmic string including monopoles) [58], formed in the early stages of the Uni- verse evolution as a result of symmetry breaking phase transition predicted in the Grand Uniﬁed Theory (GUT). The X particles typically decay to quarks and leptons. The quarks hadronize, i.e., produce jets of hadrons containing mainly light mesons (pions) with a small percentage of baryons (mainly nucleons). The pions decay to photons, neutrinos/antineutrinos and electrons/positrons. The ﬁnal result is the production of energetic photons, neutrinos and charged leptons, together with a small fraction of nucleons, with energies up to the X mass without any acceleration mechanism. For a complete description of TD models see [59].

In the Super-Heavy Dark Matter models the X particles themselves are remnants of the early Universe. EHECRs may be produced from decay of some 12 metastable superheavy relic particles (MSRPs) of mass mX > 10 GeV that could also be a signiﬁcant part of the cold dark matter [60]. Their lifetime should be larger than or comparable to the age of the Universe. Since these particles are superheavy, they would be gravitationally attracted to the Galaxy and to the local supercluster, where their density could well exceed the average density in the Universe.

The main diﬀerences between bottom-up and top-down models are the composition of EHECRs and their generation spectrum. In astrophysical scenarios charge nuclei are accelerated, while in the top-down models, neutrinos and gamma ray are generated, with a small number of protons. Besides, the standard acceleration energy spectrum index (as in shock acceleration) is equal to 2 while in the top down models the decay of X particles generate a ﬂatter CR∼ spectrum, with a power law spectral index 1.5. ∼ 4c=1. 1.4. INTERACTION WITH CBR 13

1.3.3 Z-burst models and others Besides bottom-up and top-down models, there are also hybrid models that com- bine elements of both groups. The most successful of those is the Z-burst model [61, 62, 63]. According to this model, ultra high energy neutrinos are generated from remote sources somewhere in the Universe. These neutrinos annihilate with the relic neutrinos in the extended Galactic halo of 50 Mpc radius, which are remnants of the Big ∼ Bang, generating Z0 bosons. The Z0 boson decays and generates ﬂux of nucleons, pions, photons and neutrinos. The resonant energy for Z0 production is 2 1 MZ mν − 21 Eν = =4 10 eV. (1.9) 2mν eV × where mν is the mass of the cosmological neutrinos; consequently the energy requirement for the UHE neutrinos depends on the mass of the cosmological neutrinos. If the neutrino masses are low, of order of the mass diﬀerences derived from neutrino oscillations, the energy of the high energy neutrinos should increase. The problem of this model is that no astrophysical source is yet known to meet the requirements for the Z-burst hypothesis. As for the top-down models, the composition of EHECRs would be dominated by photons and neutrinos.

Besides the Z-burst model, other theories have been formulated to avoid energy loss with the microwave background radiation at 2.7oK. These theories assume on one side the existence of a stable, massive and supersymmetric hadron S0 with a lower cross section for the production of resonant particles at the observed energies [64, 65], on the other side a Lorentz invariance violation that suppresses the cross section for inelastic collision between nucleons and microwave background photons [66, 67, 68, 69].

1.4 Propagation: interactions with the cosmic background radiation

The particles produced in our Galaxy have to travel through the interstellar medium (ISM) to arrive at Earth5. The interstellar medium is composed of neutral and ionized gas, mostly hydrogen, and of the cosmic background radiation, photons left over from the Big Bang. The propagation of cosmic rays is consequently aﬀected by the interactions with the ISM which degrade their energy and change their composition. At energies around 1019 eV the Universe is not transparent for particles as photons, protons and nuclei; during their propagation they lose energy through the interaction with the cosmic background radiation, microwave (CMB), optic/infrared (O/IR) and radio. These interactions limit the distance of propagation of a CR before it loses most of its energy. The maximum distance of propagation depends on the particle and on its initial energy (Figs. 1.5, 1.6). The maximum propagation distance, known as the GZK radius, was calculated by Greisen, Zatsepin and Kuz’min [14, 15] after the discovery of the cosmic background radiation at 2.7oK by A. Penzias e R. Wilson in 1965. The GZK

5If they have extragalactic origin they have to cross also the ISM of their original galaxy and the intergalactic medium. 14 CHAPTER 1. UHECR PHYSICS

Figure 1.5: Attenuation length for photons, protons and Iron. The upper line (red shift limit) is the distance limit due to the age of the Universe.

radius (equal to 50 100 Mpc for protons) corresponds to an energy cut- oﬀ, the GZK cut-oﬀ,∼ in− the spectrum of protons around 6 1019 eV , due to photo-pion production in the interaction with the CMB. ×

1.4.1 Protons The main interactions between protons and cosmic background photons are the photo-production of neutral and charged pions (with a relative branching ratio 2/3 and 1/3) and photo-production of electron-positron pairs

o p + γ o p + π , (1.10) 2.7 K → + p + γ o n + π , (1.11) 2.7 K → + p + γ o p + e + e−. (1.12) 2.7 K → 18 The pair production process has an energy threshold Eth 10 eV and a mean free path of 1 Mpc but the energy loss is only about 0≃.1%. The attenuation length for this∼ process is of order of 1 Gpc around 1019 eV , making the universe optically thin to energetic protons. The structure of the ankle can be explained exclusively as a result of pair photo-production by nucleons traveling cosmological distances between the source and the Earth [21] (see “dip transition model” 1.4. INTERACTION WITH CBR 15

Figure 1.6: Panorama of the interactions of primary cosmic rays with + the CMB radiation. Curves “p + γCMB e e− + p” and “Fe + γCMB + → → e e− + Fe” indicate the energy loss length for protons and Iron nuclei for + 0 pair production. Curve “p + γCMB π n / π p” indicates the mean free path for photo-pion production in interaction→ of protons with the CMB. Curve “Fe + γCMB nucleus + n / 2n” indicates the mean free path for photo → + disintegration (single and double). Curve “γ + γCMB e e−” is the mean free path for the interaction of high energy photons with→ the CMB. As a reference, the mean decay length for neutrons is shown by the curve “n p e ν”. → in 1.2). § The photo-pion production is the dominant process at energies above 1019.6 eV , with an energy loss of 20%. The photo-pion production goes through∼ the ∆+ resonance; if the proton∼ energy is higher than few times 1019 eV , the photon energy at 2.7oK is enough to excite the resonance ∆+, which decays in pions and nucleons with a coincidence production of ultra high energy photons and neutrinos. The photo-pion production reduces drastically the mean free path of protons to 6 Mpc making the universe optically thick to the UHECR pro- ∼ tons. This determines the a GZK radius, RGZK 50 100 Mpc, the maximum distance to the sources that are able to contribute≃ apprecia− bly to the highest energy detected ﬂux.

In Fig. 1.7 the proton energy as a function of the distance of propagation is shown for diﬀerent initial energies [70]. Due to the interaction with the CMB, above 100 Mpc the energy of the proton decreases below 1020 eV independently∼ on its initial energy. If the primary spectrum extends above 16 CHAPTER 1. UHECR PHYSICS

Figure 1.7: Proton energy as a function of the propagation distance for diﬀerent initial energies: after 100 Mpc the interaction with the CMB photons reduces the initial energy∼ below the threshold energy for photo-pion production [70]. 1.4. INTERACTION WITH CBR 17

Figure 1.8: Energy loss time of heavy nuclei (He, O, Si, Fe) in the microwave background radiation: 1014 s corresponds to an energy loss length 1 Mpc. Adapted form [71]. ∼

1020 eV and the sources have a uniform distribution in the universe, photo- pion production should produce a pile-up in the spectrum followed by a strong decrease of ﬂux above 4 1019 eV , known as the GZK cut-oﬀ. ∼ × 1.4.2 Nuclei Heavy nuclei, in the interaction with the cosmic background radiation, undergo single and double photo-disintegration and photo-production of electron- positron pairs: A + γ o (A 1) + N, (1.13) 2.7 K → − A + γ o (A 2)+2N, (1.14) 2.7 K → − + A + γ o A + e + e−, (1.15) 2.7 K → where A is the atomic number and N is a nucleon (proton or neutron). In Fig. 1.8 the energy loss time of heavy nuclei in interaction with the microwave background radiation is shown. The energy loss length for He nuclei in photo- disintegration is about 10 Mpc at energy of 1020 eV , while heavier nuclei reach that distance at higher total energy. Since the threshold for photo-pion production for nuclei increases as A, at the observed energies pion production is relevant only for nuclei not heavier than He. At energies below 5 1019 eV , the energy losses occur mainly through the interaction with the IR× photons while, at higher energies, nuclei interact with both the IR and CMB radiation until interaction with CMB radiation becomes 18 CHAPTER 1. UHECR PHYSICS dominant above 2 1020 eV . For 5 1019 eV

1.4.3 Photons Photons of energy up to 1022 eV interact with the CMB and the cosmic IR/O and radio background producing electron-positron pairs that undergo inverse scattering Compton giving back energy to photons (Fig. 1.5). The result is an electromagnetic shower that produces a pile-up of gamma around 1012 eV . The pair production through interaction with the CMB is the most important process in a wide energy range (4 1014 eV

1.5 Propagation: interaction with magnetic ﬁelds

Besides the interactions with the cosmic background radiation, during their propagation CRs interact with the Galactic and extragalactic magnetic fields. CRs are bent and scattered by the regular and chaotic fields, producing a diffusive motion of the particles in the Galaxy. Knowledge of the diffusion mechanism and of the structure and magnitude of magnetic fields is of fundamental importance for studies of possible anisotropies in the distribution of the CR arrival directions.

1.5.1 Galactic magnetic field The Galaxy behaves as a magnetized volume with a magnetic field structured on scales of kpc and typical intensities of some micro Gauss. The magnetic field of our Galaxy can be described as the superposition of two components, one regular and one chaotic. The regular component has an intensity of some few micro Gauss and lies on the galactic plane with a direction towards galactic latitude 90 degrees. The chaotic component has an intensity of the same order as that of the regular ones but it’s produced from magnetic clouds generated from the motion of ionized gas. Cosmic rays are bent and scattered by the magnetic fields, chaotic and regular; 1.5. MAGNETIC FIELDS 19 the result is a diffusive motion of the particles in our Galaxy.

Let’s consider the regular component of the Galactic magnetic ﬁeld. The characteristic length for magnetic deﬂection is given by the Larmor radius (in units of kpc): 1 EEeV RL,kpc , (1.17) ≈ Z BµG

18 where EEeV is the particle energy in units of 10 eV , Z is the charge of the particle and BµG is the magnetic field intensity in micro Gauss. Given the typical intensity of the magnetic field in the ISM, the type of propagation depends on the energy and charge of the nucleus. If the Larmor radius of nuclei is much smaller than the transversal dimension of the magnetized Galactic disk, nuclei propagate diffusively inside the ISM and the Galaxy behaves as a confinement region disk of radius 15 20 kpc and thickness of the order of 200 pc. Given a nucleus of charge∼ Z, as− the energy increases, the gyroradius of the nucleus becomes comparable or larger than the transversal dimension of the confinement region and consequently the nucleus can escape from the Galaxy. As a consequence, at energies above 1017 eV protons experience a change in their propagation regime from diffusive to ballistic while Iron nuclei are confined inside the magnetized ISM at least up to energies 1019 eV . ∼ Besides the Galactic regular magnetic field, a chaotic, turbulent component (with a correlation length Lc 100 pc) exists, whose spectrum seems to be of Kolmogorov type. ∼ For wavelengths of the order of the Larmor radius, λ RL, the wave-particle interactions between cosmic rays and MHD turbulence are∼ resonant. The critical energy of a nucleus of charge Z (in units of 1018 eV ), below which resonant modes with the particle gyroradius exist, is defined by R L 102 pc, i.e: L ≈ c ≈ E 0.5 Z. (1.18) c,EeV ≈ ×

Consequently, for energies below Ec, the diffusion coefficient for the turbulent component is small and the propagation is diffusive while above this critical energy the propagation is essentially ballistic. Due to the interaction with the turbulent magnetic field, protons and nuclei propagate in the ISM in a different way inside the ankle region; protons at E & 3 1017 eV propagate ballistically while Iron nuclei propagate diffusively even at×E & 1019 eV . Therefore, in the region between 3 1017 eV and 1019 eV nuclei, starting from protons and going up to Iron∼ nuclei,× experience∼ a change in their propagation regime, from diffusive to ballistic, as the energy increases.

1.5.2 Extragalactic magnetic field At high energies, as the Galactic cosmic rays are able to escape from the Galaxy, in the same way extragalactic particles are able to penetrate in the Galactic confinement region. This is possible if extragalactic particles are able to reach our Galaxy in less than a Hubble time. This effect is known as magnetic horizon and can be estimated as follows. Measurements of Faraday rotation impose to 20 CHAPTER 1. UHECR PHYSICS the extragalactic magnetic field the limit

B L 1/2 c 1, (1.19) nG × Mpc ≤ where B is the EG magnetic field and Lc is the correlation length of the field, assumed of order of 1 Mpc [74]. Assuming a diffusion coefficient given by Bohm approximation 1 D R c, (1.20) ≈ 3 L where RL is given by eq. (1.17), the diffusion coefficient becomes:

0.1 E (Mpc)2 D EeV . (1.21) ≈ Z B Myr nG The diﬀusive propagation time from an EG source at a distance d is estimated as d2 τ . (1.22) ≈ D Using eq. (1.21), the diﬀusive propagation time gives the magnetic horizon

B τ 10 d2 Z nG . (1.23) Myr ≈ × Mpc × × E EeV Basically, given an EG source at distance d and a a nucleus of charges Z, there is a lower energy limit above which the nucleus can reach our Galaxy. If we consider a minimum distance d = 10 Mpc6, only protons with energy above 2 1017 eV or Fe nuclei with energy above 5 1018 eV are able to reach our Galaxy× in less than a Hubble time. In the energy× range between the second knee and the ankle, all kind of nuclei, starting from protons up to Iron nuclei, are able to arrive from the local universe. At the same time, at these energies the magnetic shielding of the Galaxy becomes permeable to these nuclei, allowing them to enter the ISM.

The propagation of the cosmic rays in the EG region is aﬀected by the magnetic ﬁeld. As for the interstellar medium, the intergalactic medium has a strong magnetic turbulent component [75, 76, 77, 78]. From the measurements of Faraday rotation, the correlation length is estimated to be of order of 1 Mpc, consistent with a maximum wavelength for the MHD turbulence determined by the largest kinetic energy injection scales in the intergalactic medium:

L L 1 Mpc. (1.24) max ∼ c ∼ As for the ISM, the MHD turbulence are resonant for wavelength of the order of the Larmor radius of the nucleus:

1 EEeV RL,Mpc Lmax 1 Mpc (1.25) ≈ Z BnG ≈ ≈ 6This distance is smaller than the distance from the Virgo cluster and deﬁne a region completely internal to the supergalactic plane. 1.5. MAGNETIC FIELDS 21

Figure 1.9: EG spectrum for different types of turbulence in the intergalactic medium: Kolmogorov (D E1/3), Bohm (D E) and an arbitrary case (D E2). The spectra are calculated∝ for a separation∝ distance of the sources d∝= 50 Mpc and an injection spectral index γ = 2.7. The AGASA experimental data with the “universal spectrum” (i.e. the spectrum not distorted by energy attenuation and diffusion) are also shown. The arrows indicates the transition energies of the propagation regime from diffusive to ballistic for protons and Iron nuclei. Between these two energies, different nuclei experience the same transition in the propagation regime. Adapted from [79].

Consequently for each nucleus there is a critical energy below/above which the propagation is diﬀusive/ballistic

E 1.0 Z EeV. (1.26) c ≈ × The propagation for nuclei in the intergalactic medium changes from diffusive to ballistic at 1018 eV , while for Fe nuclei the propagation remains diffusive up to energies∼ around 3 1019 eV . Besides, the type of turbulence present in the intergalactic medium× affect the spectrum and it is observable in the lower energy part of the EG spectrum, where the flux is suppressed by the magnetic horizon effect (Fig. 1.9).

Chapter 2

Detection of UHECRs

At energies higher than 1015 eV the integrated flux of cosmic rays decreases to less than one particle per square meter per year making impracticable direct measurements outside of the atmosphere. The primary cosmic rays, as they enter the atmosphere, interact with the air molecules producing an extended shower of particles (called extensive air shower, EAS) that can extend transversely up to several kilometers. The number of particles produced in the shower increases, at the expense of the average energy, until the critical energy is reached, energy at which the energy loss processes dominate the production of particles. The number of particles, after having reached the maximum value, begins to decrease as the observation altitude decreases. The shower development (i.e. the shower size) as a function of the traversed amount of atmosphere is called longitudinal profile of the shower and provides information on the shower energy and on the composition of the primary CR. The same information can be deduced from the lateral distribution of particles at the ground level, i.e. the spatial distribution of the different shower components (muonic and electromagnetic) at the observation level. The study of ultra high energy cosmic rays is based on the detection of the secondaries produced by the interaction of the primary with the atmosphere. The main techniques used are the sampling of particles at ground level through an array of detectors (Ground Array), which allows to reconstruct the lateral distribution of a shower, and the detection of the energy dissipated from particles traversing the atmosphere through detectors of fluorescence or Cˇerenkov light, which allows to reconstruct the longitudinal profile of the shower. In this chapter, after a description of the general characteristics of EAS, the two detection techniques and the main EAS experiments are presented.

2.1 Extensive Air Shower (EAS)

The primary cosmic rays, as they interact with the air molecules of the atmosphere, produce a cascade of particles with a shower front that propagates approximately at the speed of light with an extension at ground of several kilometers. The atmosphere behaves as a huge calorimeter whose depth is equal to approximately 13 interaction lengths for protons and 27 radiation lengths for

23 24 CHAPTER 2. DETECTION OF UHECRS photons. The average mass of the atmosphere, composed mainly by nitrogen (78%) and oxygen (21%), is 14.7 amu. Under isothermal approximation1, the density decreases exponentially at increasing altitude.

Figure 2.1: EAS diagram: the primary nucleon interacts with the nuclei of the atmosphere producing a cascade of particles. The EAS can be schematically thought as a sum of three components, namely the electromagnetic, the muonic and the hadronic component. The lateral extension of the shower depends on the transverse momentum of the hadronic component.

The atmospheric depth is deﬁned as the amount of matter seen by a primary cosmic ray entering vertically the atmosphere

∞ Xv = ρ(h)dh, (2.1) Z0 where ρ(h) is the density at altitude h. The atmospheric depth varies exponentially following the equation X (h)= X exp( h/h ), (2.2) v 0 − 0 1The troposphere temperature decreases approximately at increasing altitudes with a rate ∼−6.5o K/km; nevertheless the isothermal approximation is valid for the description of some features. 2.1. EAS 25

2 where X0 is the atmospheric depth at sea level (1030 g/cm ), h is the altitude above the sea level and h0 6.4 km. For inclined showers it’s deﬁned the slant atmospheric depth ∼ X(h)= Xv/ cos θzenith, (2.3) where θzenith is the zenithal angle of the shower axis.

The core of the shower consists of high-energy hadrons, which feed the electromagnetic and muonic components. The particles produced in hadronic interactions are mainly pions and kaons; charged pions (kaons) decay mainly in muons and neutrinos (the most penetrating component of the shower) while the neutral mesons decay into 2γ which initiate electromagnetic cascades composed of photons and electron-positron pairs via pair production and bremsstrahlung. The number of particles increases at the expense of the average energy until the energy loss processes dominate the production of particles. At this critical energy the size (i.e. number of particles) of the shower has reached its maximum and begins to decrease at increasing atmospheric depth. In Fig. 2.1 the structure of a cascade is shown: within few meters from the axis the survived hadronic component is concentrated, while moving away from the core the region is dominated by muons, neutrinos and by the electromagnetic component, which can extend to several hundreds of meters (the e.m. component is strongly diﬀused through multiple Coulomb scattering).

The perturbative QCD provides the calculation of a minimum part of the hadronic processes, in particular the ones at high transverse momentum (the central region) which are studied at high energy colliders. Nevertheless the energetic range studied in colliders is limited; the center of mass energy in a nucleon-nucleon collision is given by

√s 2m E. (2.4) ∼ n The energetic limit foreseen for LHC (p 14 T eV ) corresponds to a nucleon energy of 1017 eV . ∼ The altitude at which interactions occur and the transverse momentum of hadrons are not determined. Consequently there are large fluctuations on the altitude of the maximum development of the shower and on its lateral distribution, properties related to the primary nature and energy. In particular, the largest fluctuations regard the first interaction with the atmosphere which occurs at the highest energies unexplored by colliders. Nevertheless, simple models describing the development of showers exist (see appendix B). The toy model suggested by Heitler [80] (described in B.1) provides the§ macroscopic characteristics of an electromagnetic showers,§ while the development of hadronic showers induced by protons is described by a similar model [81] (described in B.2). Even if they cannot replace detailed simulations, these simple models predict§ the most important features of the cascades.

The nucleus-air interactions can be described applying the superposition model, considering a nucleus with atomic number A and total energy E0 as A nucleons 2 of energy E0/A interacting independently with the atmosphere . This model

2The binding energy of nucleons is ∼ 8 MeV , consequently at high energies they can be considered as free nucleons. 26 CHAPTER 2. DETECTION OF UHECRS consider the development of the shower as the superposition of A hadronic cascades initiated by protons, that can be approximated to electromagnetic cascades as, at growing energy, most of the energy is deposited in the electromagnetic component.

Given a shower induced by a nucleus with atomic number A and total primary energy E0, the depth of the shower maximum can be written in a general form, valid in the assumption of the superposition model [82]:

E X = (1 B)λ ln 0 < lnA > , (2.5) max − r E − c where Ec is the critical energy (Ec 81 MeV in air), λr is the radiation length 2 ∼ (λr 37.1 g cm− in air) and the parameter B is related to the nature of the shower,∼ B = 0 for a pure electromagnetic shower and B < 1 depending on the hadronic interaction model.

The muon and electromagnetic components at the shower maximum are functions of the primary energy and composition. The muonic component of a nucleus induced shower is related to the atomic number A by

N A = N p A0.15, (2.6) µ µ × p where Nµ is the muonic component of a proton induced shower of the same primary energy. The primary energy can be written as the sum of e.m. and hadronic energy and can be calculated from the number of electrons or muons at the shower maximum E 1.5 GeV N 0.97, (2.7) 0 ≈ × e E 19.7 GeV N 1.18. (2.8) 0 ≈ × µ The cascades initiated by Iron nuclei will develop at higher altitudes and will be less penetrating; moreover the muonic component will be larger with respect to a proton induced shower of the same primary energy. Besides, the ﬂuctuations on the maximum depth of a shower should be smaller for heavy nuclei. In fact, if we consider a shower induced by an Iron nuclei of energy E as the superposition of 56 cascades induced by A nucleons of energy E/56, the ﬂuctuation on the altitude of maximum development of the shower Xmax will be smaller with respect to a proton induced shower of the same primary energy E since Xmax is averaged on 56 cascades.

In Fig. 2.2 a simulation of the longitudinal proﬁles of proton and Iron induced showers of primary energy 1019 eV is shown. In the case of proton induced show- 2 2 ers, simulations show that the average Xmax is of order of 750 g/cm 53 g/cm while for Iron induced shower it is of order of 700 g/cm2 22 g/cm±2. ± Another important quantity, sensitive to the primary composition, is the elongation rate. It is deﬁned as the rate of increase of Xmax with the primary energy E0 [83]: dX dX D = max = ln(10) max . (2.9) d(log10 E) d(ln E) 2.1. EAS 27

Figure 2.2: Longitudinal proﬁle. Simulation of longitudinal proﬁles for showers of energy 1019 eV induced by protons (red) and Iron nuclei (blue).

Using the general equation for the depth of the shower maximum (eq. (2.5)), the elongation rate can be written as

dX D =2.3 max =2.3(1 B)λ . (2.10) d(ln E) − r

The elongation rate is a good measurement of the rate of change with energy of CR composition.

Besides the longitudinal development of the shower, the lateral distribution of particles at ground provides information on the primary CR. In Fig. 2.3 the lateral distributions of particles for photons, proton and Iron induced showers are shown. The main components are photons, electrons and muons. The hadronic component is concentrated near the core of the shower but its contribute is negligible at distances larger than few hundreds of meters. The photon density is prac- tically the same for distances from the core between 500 m and 1000 m in the case of protons and Iron induced showers. This is due to the fact that, although electrons and photons are produced higher in the atmosphere (with a conse- quential larger distribution) in Iron induced showers with respect to a proton induced shower, they suﬀer more atmospheric attenuation since they have to travel for larger distances. Consequently, these two eﬀects balance each other out and the distribution of the e.m. component is similar for protons and Iron primaries. On the other side, the muon density is higher in the case of Iron 28 CHAPTER 2. DETECTION OF UHECRS

Figure 2.3: Lateral distribution. Simulation of the lateral distributions of shower particles. Curves are obtained averaging on many showers. The photon, electron and muon densities in the plane perpendicular to the shower axis are shown for shower induced by protons (red line), Iron nuclei (blue line) and gamma rays (dashed line) of energy 1019 eV . 2.2. DETECTION TECHNIQUES 29 induced showers. Even if the diﬀerence is small, the ratio between the numbers of muons and the number of electrons is sensible to the primary composition; the ratio Nµ grows with the atomic number of the primary nucleus. Ne In the case of photons induced showers, the electron and photons distributions have a bigger slope with respect to the case of the proton and Iron nuclei primaries. The number of muons is absolutely smaller, of a factor & 5 with respect to the case of Iron induced showers of the same energy. Consequently, the number of muons and the ratio Nµ can be used as a composition parameter [84, 85]. Ne

As the surface arrays have a duty cycle much larger than fluorescence detectors, observables from the array are required for composition analysis. At present, besides the muon content of the shower, other ground parameters depending on it are under studies, all relying on the time structure of the shower front, the signal rise time in water Cˇerenkov tanks, the curvature of the shower front and the azimuthal asymmetry in the signal rise times. All these parameters are sensitive to primary mass as they depend on the height of the shower development and on the muon content itself [86]; the higher the shower develops in the atmosphere, the smaller will be the time spread between particles and consequently the time rise, while the shower front curvature will get flatter. Larger distance of Xmax from the array yields larger radii of curvature, while deeply penetrating showers produce shower fronts with small radii of curvature. Moreover, muons tend to travel in straight lines while electrons are more scattered. A shower with large number of muons will produce station triggers compact in time, resulting in a large radius of curvature. At the same time, azimuthal asymmetry in the signal rise times within the shower plane is an effect of varying the muon-to- electromagnetic component. All these parameters, that can be used in combination, are actually under studies [87]; limitations in the use of these parameters are due to the finite accuracy with which can be measured as measurement uncertainties have to be small enough to detect mass difference.

2.2 Detection techniques

The study of ultra high energy cosmic rays is based on the detection of the secondaries produced by the interaction of the primary with the atmosphere. The main techniques used are the sampling of particles at ground level through an array of detectors which measure their energy and density, and the detection of the energy dissipated from particles traversing the atmosphere through detectors of ﬂuorescence or Cˇerenkov light. The use of these two techniques allows to reconstruct the lateral distribution and the longitudinal proﬁle of the shower, respectively.

The older and more used experimental method consists in the use of particle detectors deployed on a large surface (Ground Array). The idea is to measure the total number of particles survived at the detection level to determine the lateral distribution of the shower from which its dimension and energy can be inferred. Historically the first surface detectors were Geiger-Muller counters, through which the physic Pierre Auger detected for the first time in 1983 particles pro- 30 CHAPTER 2. DETECTION OF UHECRS duced in an EAS of energy 1015 eV . As these detectors cannot provide arrival direction of the shower, they were replaced by water Cˇerenkov detectors and scintillators. In 1953 the MIT group [88] developed a reconstruction technique based on the arrival time of signals in scintillators that allow to determine the arrival direction of the shower. Analyzing the density distribution in the detector array, it’s possible to localize the zone of highest intensity correspondent to the impact point of the shower axis at ground, i.e the core. Therefore, large arrays of scintillators or water Cˇerenkov detectors have been built, as Volcano Ranch Array (built by Linsley, Scarsi and Rossi [89]), which has been the first to detect a cosmic ray of energy higher than 1020 eV [90], and the Pierre Auger Observatory, the biggest array ever built.

The second technique used in the study of EAS was born in 1962 [91, 46] following the idea to use our atmosphere as a huge scintillator. The fluorescence detectors detect, with photomultiplier, the fluorescence light emitted in the range 300 400 nm by nitrogen molecules excited by the shower particles. This technique− allows the reconstruction of the longitudinal profile of the shower. With the fluorescence detector Fly’s Eye, which has detected one of the most energetic showers, E 3 1020 eV [7], a new generation of experiments as HiRes and Auger began.∼ × The longitudinal profile of a shower can be also reconstructed detecting the Cˇerenkov light emitted from particles crossing the atmosphere. Although the emission efficiency of Cˇerenkov light is higher than the fluorescence one, this method can be used only for study of cosmic rays of energy lower than 1014 eV . In fact as the direct Cˇerenkov light is emitted in a cone of maximum∼ aperture 6o, its detection is possible only in the case of small angles between the shower axis∼ and the telescope. Consequently, this technique is indicated for studies of CRs arriving from a fixed astrophysical source and at energies at which the flux is not too low.

2.2.1 Surface detectors: Ground Arrays As seen in 2.1, an UHECR can induce a particle shower that, at ground level, can extends§ on a large surface, up to several km2, depending on the primary energy. The shower particles are detected, together with their arrival time, through several detectors deployed on a surface array (Ground Array).

The disposition of the detectors depends on the observation altitude and on the energy range of CRs under study. For UHECRs the separation distance of the detectors is of order of several hundreds of meters. The dimension of the detector is related to the shower component being studied (charged particles, muons, high energy photons, Cˇerenkov photons). While for charged particles the size of the detector range from 1 m2 to 20 m2, for muons detector it should be larger. The observation level also depends on the shower energy under study. In fact, the investigation of shower is more eﬃcient near and beyond the shower maximum; as the average depth of primaries of energy higher than 1019 eV is 750 g/cm2, the surface arrays designed for EHECR have to been built at ∼an atmospheric depth larger than 800 g/cm2. 2.2. DETECTION TECHNIQUES 31

Figure 2.4: Fluctuations of the relative densities of S(r). The relative density is deﬁned as the deviation of the density from the average density divided by the the average density. S(r) is the density measured in a scintillator at distance r (in meters) from the core. The simulations are done for diﬀerent primary masses at energy 1017 eV using COSMOS model [92].

In Ground Array detectors the particle density (or the deposited energy density) is measured with the particle arrival time. Assuming in ﬁrst approximation a plane shower front, the direction of the shower (and consequently of the primary) is reconstructed using the relative arrival times of signals at a minimum of three non collinear detectors. The resolution in the determination of the direction is limited by the accuracy of the time measurements and by the detector size. Giant arrays as Auger reaches an uncertainty of less than 1o [93]. Once the direction of the shower has been determined, the impact point of the shower axis at ground (the core) is deduced by the lateral distribution of the shower obtained ﬁtting the measured densities as a function of the distance from the core. For scintillators, a reasonable approximation of the lateral distribution function (LDF) is the generalized function of Nishimura-Kamata-Greisen (NKG) [94] proposed by Linsley, Scarsi and Rossi in 1961 [89],

α (η α) r − r − − S(r)= k 1+ , (2.11) r r 0 0 where S(r) is the particle density that hit a scintillator at distance r from the core, r0 is the Moliére unit (the product of one radiation length and the rms deflection of a particle of critical energy traversing one radiation length), α and η are constants determined empirically from the data and k is a constant proportional to the shower size. The LDF is used to determine the core of the shower through computational techniques based on the best fit between observed and 32 CHAPTER 2. DETECTION OF UHECRS expected densities. Usually the core is searched in the plane perpendicular to the shower axis using χ2 minimization or maximum likelihood method. Once the position of the core has been determined, the plane front approximation is replaced by the assumption of a spherical front [95, 96] and the size of the shower is determined from the fitted LDF. However, for large arrays, when the separation distance of detectors d >> r0, the measure of densities at distances r < r0 is unlikely. The measured size is consequently dependent on the extrap- olation of the LDF in this region. Moreover the size of a shower induced by a primary CR of a fixed energy is subjected to fluctuations due to the stochastic development of the cascade.

An alternative method is the measurement of the density of one shower component at a fixed large distance from the axis, S(r0). This density is related to the energy of the primary cosmic ray [97] and is assumed as a shower parameter in Ground Arrays, even if it is not very sensible to composition and quite independent on the shower maximum [98]. The relation between the primary energy and S(r0) is of the form E0 = k α × S(r0) where k, α are parameters determined by simulations. This fixed distance r0, at which the density is assumed as a parameter, is chosen in the range of distances where the fluctuations of the relative densities of the LDF are small. In Fig. 2.4 the fluctuations of the relative densities of S(r) are shown, where the relative density is defined as the deviation of the density from the average density divided by the average density [92].

2.2.2 Fluorescence detectors The idea of using the fluorescence light to detect an EAS traversing the atmosphere was developed independently by Greisen in 1960 [99] and by Delvaille [100], Suga and Chudakov [91] in 1962. The technique is based on excitation + of the 2P ans 1N band system of N2 and N2 molecules of the atmosphere by ionizing particles. The excited molecules can emit isotropically fluorescence photons in a typical time window of 10 50 ns. Most of the light is emitted in the wavelength range 300 400 nm, which− results a band selectable by detectors through opportune optical− filters. The atmosphere is quite transparent to the light in this wavelength range, with an attenuation length of the order of 15 km for vertical beams. In Fig. 2.5(a) the fluorescence spectrum is shown [101, 102, 103, 104]. The fluorescence yield per ionizing particle (Fig. 2.5(b)) is low, typically of order of 4.5 UV photons/m. Nevertheless, as an EAS with primary energy 0.1 EeV produces at the shower maximum more than 108 electrons, the number of fluorescence photons emitted is high enough to make this technique reliable even for a fluorescence efficiency equal to 0.5%. The fluorescence yield depends on the atmospheric pressure and temperature [101] and it’s the source of the largest uncertainty in energy reconstruction. Precise measurements of the fluorescence yield are crucial to increase the accuracy in the determination of the primary energy.

The ﬂuorescence detectors follow the trajectory of the shower particles and measure the energy dissipated in the atmosphere. The ﬂuorescence light, emitted isotropically along the shower trajectory, is collected by mirrors and detected by 2.2. DETECTION TECHNIQUES 33

100

0 0.25 0.3 0.35 0.4 0.45 0.5

(a) Nitrogen ﬂuorescence spectrum

(b) Fluorescence yield

Figure 2.5: Fluorescence spectrum and yield. a) Nitrogen ﬂuorescence spectrum near the ultraviolet. Most of the photons are emitted in the wavelength band 300 400 nm. b) Fluorescence photons emitted in the range 300 400 nm produced− by electrons of energy 80 MeV as a function of altitude− by Kakimoto et al. [103]. The calculation uses two typical atmospheric models: summer atmospheric model, with a surface temperature of 296oK and winter• atmospheric model, with a surface temperature of 273oK. Taken from◦ [103]. 34 CHAPTER 2. DETECTION OF UHECRS

Figure 2.6: Geometry of an EAS trajectory. The shower-detector plane (SDP) contains the extended cascade and the ﬂuorescence detector center. The angle ψ and the impact parameter Rp are determined ﬁtting the observed angles χi and the relative times ti [105].

photomultiplier tubes (PMT) as a time sequence of light signals. The shower- detector plane (SDP), containing the shower and the detector center, is reconstructed using this sequence of hit PMTs [105, 106]. The light reaching a PMT i with an emission angle θi from the shower axis is delayed from the time t0, deﬁned as the time at which the shower front plane passes in the nearest point to the detector (i.e. at distance Rp). The expected delayed time is given by:

Rp Rp Rp θi ti,exp t0 = = tan , (2.12) − c sin θi − c tan θi c 2 where c is the light speed, Rp is the distance of the detector from the shower axis (called impact parameter), ti,exp is the expected trigger time of PMT i. The angle θi is related to the elevation angle χi of the PMT and to the incidence 2.2. DETECTION TECHNIQUES 35 angle ψ of the shower in the SDP by the equation

θ = π ψ χ . (2.13) i − − i

The impact parameter Rp, the time parameter t0 and the incidence angle ψ in the SDP (Fig. 2.6) is obtained by the fit of the time sequence of signals with the function (2.12) minimizing the differences between the expected and measured 2 signal times i(ti,meas ti,exp) . In the case the shower− is seen by more detectors (stereo detection), once the SDP has beenP determined for each detector, the shower axis can be find as the intersection of the determined planes without the time information of the signals [106]. A further technique, used for the first time by the Pierre Auger Observatory, is the hybrid reconstruction of the shower geometry, which consists of the use of the information provided by both the fluorescence and surface detector [107]. The use of the time information provided by one or more tanks (the arrival time tGND of the shower front at ground level in a tank located at distance R~ GND) allows to determine the parameter t0, reducing the parameters in the fit of the time sequence and consequently breaking a possible degeneracy in the reconstruction of the shower axis.

Once the geometry of the shower is known, the signals recorded by each PMT is converted in number of fluorescence photons emitted at the track element seen by each PMT. This requires corrections for extinction due to the Rayleigh scattering, Mie (aerosol) scattering and ozone absorption. While the attenuation coefficient due to scattering Rayleigh is evaluable from the shower geometry, once known the atmosphere structure, the Mie scattering is very variable and its evaluation requires measurements through atmospheric monitoring [108]. The amount of fluorescence light emitted at the track element is proportional to the energy dissipated by the charged particles in that track. In order to reconstruct the shower energy, the following step is the elimination of contribute of the direct and scattered Cˇerenkov light; the first one is present mainly in the first time of the light curve, while the second one is significant in the tail of the signal. Even if the number of Cˇerenkov photons are larger than the fluorescence ones, they are collimated around the shower axis and their contribute is important in the case of angles between the shower axis and the telescope axis lower than 30o. Once the Cˇerenkov contribute has been subtracted, the numbers of photo-electrons are directly proportional to the number of charged particles of the shower in the angular bin. A typical light curve as a function of time is shown in Fig. 2.7 in which the total signal and the different light contributes are indicated.

The longitudinal development of the shower, given by the number of ionizing particles as a function of atmospheric depth, is well approximated by the function proposed by Gaisser and Hillas [109]

(Xmax X0)/λ X X − N(X)= N − 0 exp [(X X)/λ], (2.14) max X X max − max − 0 where X0 is the atmospheric depth of the ﬁrst interaction, Xmax is the depth of the shower maximum, Nmax is the number of particles at the shower maximum 36 CHAPTER 2. DETECTION OF UHECRS

Figure 2.7: Typical light curve as a function of time measured by Auger ﬂuorescence telescopes. The point with he relative errors are the number of photons/(m2 100 ns) observed entering the telescope. The total signal and the contributes× of direct Cˇerenkov light (dotted) and scattered Cˇerenkov light (dashed) are shown. and λ is the interaction length for protons ( 70 g/cm2). ∼ ∞ The total track length is estimated from the integration 0 N(X)dX, making some assumptions on the shape of the cascade beyond the observation level. R The energy of the electromagnetic component Eem is estimated from the total track length

critical energy of e− in air ∞ Eem = radiation length in air 0 N(X)dX =

MeV ∞ R = 2.18 g cm−2 0 N(X)dX, (2.15) where the total track length is expressed inRg/cm2. This energy represents a fraction of the total energy which varies with energy and primary mass for hadronic cascades; it represent about 80 90% of the total energy for a shower of 1019 eV . − Taking into account the “invisible energy” carried by high energy muons and neutrinos, the primary energy is estimated by Hillas [110] as

MeV ∞ E = 2.65 N(X)dX. (2.16) g cm 2 − Z0 This correction introduces a slight dependence on the hadronic interaction model since it requires some assumption on the primary mass. The energetic resolution is determined by various factors as the ﬂuorescence eﬃciency in air, the 2.3. EXPERIMENTS 37 determination of the direct and scattered Cˇerenkov light, the attenuation and scattering of photons and the uncertainty in the reconstruction of the shower geometry.

2.2.3 Radio detectors An alternative technique is the detection of EAS making use of the radio frequency (RF) pulses generated by particles. Radio pulses in the range 200 100 MHz induced by EAS were first measured in 1966 [111] but only recently− this technique has been proposed as the next generation of detection. The dominant process involved in the induction of radio signals is the coherent synchrotron emission by electron-positron pairs that propagates in the Earth magnetic field. This technique has some advantages with respect to the clas- sic ones; besides the duty cycle, which in principle should be 100% guaranting large statistics, the radio signals are not absorbed nor deflected on their path and their amplitude is proportional to the primary energy of the particle. Radio detectors consist basely of dipoles antennas displaced on the surface. The number of antennas employed determines the CR energy range for which a sat- isfactory signal-to-noise ratio and a reasonable event rate are achievable. The array configuration of the antennas which optimizes the detector performances has been studied by KASCADE experiment in the prototype phase since 2003 [112]. The radio technique has started a new generation of experiments, as the Low- Frequency Array (LOFAR) [113]. LOFAR, with its 100 stations of 100 dipoles antennas distributed on an area of radius 400 km, predicts to observe EAS up to 1020 eV at a rate of 1 event per year.∼ Several other experiments are incor- porating this new technique, as KASCADE, Icecube [114] and the Pierre Auger experiment. An array of about 20 km2 to be deployed at the Pierre Auger site, is currently in R&D phase [115].

2.3 UHECR experiments

In this section the main UHECR experiments are brieﬂy presented. The main experimental results will be presented in chapter 3. In Table 2.1 the data relative to the main UHECR§ experiments are reported.

2.3.1 Volcano Ranch Array The ﬁrst giant array was constructed at Volcano Ranch (New Mexico) from Linsley, Scarsi and Rossi [89]. Its conﬁguration, adopted from its prototype phase (1953) up to 1963, is shown in Fig. 2.8 together with the core position and the densities of its most energetic recorded event [90].

The Volcano Ranch array is constituted of 19 plastic scintillators for charged particle detection of size 3.3 m2, each one seen by 5 photomultiplier tubes. The separation distance of the detectors changed from 442 m, adopted in the ﬁrst period, to 884 increasing the total extension to 8.1 km2. The amplitude of the detected signals and their arrival time are measured through oscilloscopes. 38 CHAPTER 2. DETECTION OF UHECRS

Table 2.1: Experimental site for the study of ultra high energy cosmic rays (“op.” stands for operative). ∗Yakutsk array was modernized in 1993.

Experiment Begin End/ Latit. Longit. Alt. Atm. State m depth g/cm2 o o Volcano Ranch 1959 1963 35 09′N 106 47′ W 1770 834 o o SUGAR 1968 1979 30 32′S 149 43′ E 250 1015 o o Haverah Park 1968 1987 53 58′N 1 38′ W 200 1016 o o Yakutsk 1974 1993∗ 61 36′N 129 24′ E 105 1020 Fly’s Eye 1981 1993 40oN 113o W 1370 860 o o AGASA 1990 2004 35 47′N 138 30′ E 900 920 KASCADE 1996 op. 49.1o N 8.4o E 110 HiRes 1997(I) 2006 40oN 113o W 1370 860 1999(II) KASCADE 2003 op. 49.1o N 8.4o E 110 -Grande Auger 2004 op. 35.5o S 69.3o W 1350 880 Telescope Array 2007 op. 39.3o N 112.9 W 1400

Figure 2.8: Layout of the detectors of Volcano Ranch array. The densities of the ﬁrst event of energy higher than 1020 eV are shown. The shower core is indicated by letter A [90].

Besides the charge particle detectors, the array uses scintillators covered with 2.3. EXPERIMENTS 39

lead designed for measurements of muon density (Eµ > 220 MeV ). The signal measured in scintillators is the average energy lost by particles in the detector, expressed in energy lost by muons penetrating the detector vertically. The distribution of the measured energies, as a function of the distance r from the core, is ﬁtted with the following lateral distribution function [116]:

Ne α (η α) S(r)= CeR− (1 + R)− − , (2.17) r0 where R = r/r0, r0 is the Moli´ere unit (typically 100 m at Volcano Ranch), Ce is a normalization factor, α =1.23 and η is given by

N η = (3.88 0.05) (0.64 0.07)(sec θ 1)+(0.07 0.03)log e , (2.18) ± − ± − ± 108 with θ the zenithal angle of the shower. The signal at 600 m from the core, S(600) is taken as the energy estimator.

The data collected at Volcano Ranch provided the first measurements of the CR spectrum of energy higher than 1018 eV , giving the first clue of a flattening of the spectrum [117]. The most energetic event recorded at Volcano Ranch, shown in Fig. 2.8, with a primary energy equal to 1.4 1020 eV [118], was detected before the discover of the cosmic background radia× tion at 2.7oK [16] and the following prediction of the GZK cut-off [14, 15], opening a new issue in the cosmic ray physics.

2.3.2 Haverah Park Array The Haverah Park (UK) array, operative from 1967 to 1987, consists of water Cˇerenkov detectors of diﬀerent dimensions, deployed on an area of 12 km2. The conﬁguration adopted is shown in Fig. 2.9 together with the∼ densities recorded of one of the most extended events, with energy equal to 1020 eV [119, 120].

The array is composed of 4 central water Cˇerenkov detectors, separated by 500 m and of others 6 sub-arrays placed at distances of 50 150 m for an overall area of about 10 km2. The water Cˇerenkov detectors are− water tanks, each one seen by a photomultiplier tube, which dimension varies from 1 m2 to 34 m2 [119, 120]. The water Cˇerenkov detectors, differently from plastic scintillators, respond very efficiently to photons; since the detectors present 3.2 radiation lengths to the photon flux, photons of low energy (typically of order∼ of 10 MeV ) are totally absorbed. Similarly, at distances larger than 100 m, most of the electrons are completely absorbed while muons with energy higher than 250 MeV cross the detectors. Consequently, this kind of detector measures the energy flux of the shower disk with a good efficiency. The lateral distribution of water-Cˇerenkov signal densities ρ(r), in units of vertical equivalent muons (VEM) per m2, is fitted with the function [121]

(η+r/r0) ρ(r)= k r− , (2.19) × where r is the distance from the shower core in meters, r0 = 4000 m, k is a 40 CHAPTER 2. DETECTION OF UHECRS

Figure 2.9: Haverah Park layout. Densities recorded in each detector for an event of energy 1020 eV observed at Haverah Park. The area delimited by the dashed square in the left panel is reported enlarged on the right panel; the detectors not marked by letters are water Cˇerenkov detectors. The core is indicated by an asterisk [120].

normalization factor and η is given by

E η =3.49 1.29 sec θ +0.0165 ln , (2.20) − 1017eV where E is the shower energy and θ its zenithal angle. The function (2.19) is valid in the range 50 m

β 1 (η+r/r0)+β ρ(r)= k r− , (2.21) 800 × where β =1.03 0.05. The energy of the± primary CR is related to the density at 600 m, ρ(600), through the equation E [eV ]=7.04 1017 ρ(600)1.018 (2.22) 0 × × calculated by Hillas using simulations [98].

2.3.3 Yakutsk Array The most complex array is the one constructed by the Institute of Cosmophys- ical Research and Aereonomy at Yakutsk in Siberia. Its ﬁnal conﬁguration, developed from 1970 to 1974, is shown in Fig. 2.10 together with the densities 20 o of the most energetic event observed (E = (1.4 0.4) 10 eV , θzenith = 58.9 ) [122]. ± × 2.3. EXPERIMENTS 41

Figure 2.10: Layout of the detectors of Yakutsk array. The densities recorded in each detector for the most energetic event observed (E = (1.4 0.4) 1020 eV ) are shown. The area delimited by the dashed hexagon is shown± enlarged× below. The muon detectors are marked by squares [122].

The detectors are scintillators deployed in concentric hexagonal blocks. The dimensions and separation distances of the detectors vary from 0.25 m2 and 500 m in the internal block to 2 m2 and 1 km in the outer one. At one km from the center, muon detectors with an energy threshold of 0.5 GeV are deployed. One important characteristic of the array is the installation of a system of phototubes for the detection of the Cˇerenkov radiation in air associated with showers, useful for deducing information on the longitudinal development of cascades and for providing a calorimetric approach for energy calibration. The density lateral distribution is ﬁtted with the function [123, 10]

g α (η α) r − S(r)= N C R− (1 + R)− − 1.0+ , (2.23) e e 2000 h i where R = r/r0, r is the distance from the shower core, r0 is the Moli´ere unit (equal to 70 m at Yakutsk), Ce is a normalization factor, α = 1.3 and g is an energy dependent parameter equal to 1.6 at E = 1018.1 eV ,2.3 at E = 1018.3 eV and 3.5 for E > 1018.7 eV . The parameter η is given by

η =1.38+2.16 cos θ +0.15 log [Sθ(600)], (2.24) 42 CHAPTER 2. DETECTION OF UHECRS

where Sθ(600) is the density at 600 m for a shower with zenithal angle θ. The primary energy is related to the estimators S(600) and S(300) by the equa- tions [9, 124]

16 0.94 0.02 E0(eV ) = (6.5 1.6) 10 S0o (300) ± , (2.25) ± × 17 × 0.98 0.02 E (eV ) = (4.6 1.2) 10 S o (600) ± , (2.26) 0 ± × × 0 where S0o (300)/S0o (600) are the densities at 300 m/600 m for vertical showers, related to Sθ(300)/Sθ(600) (for details see [9, 124]).

Detailed studies of the spectrum have been concentrated in the region around 1019 eV restricting the area of the array to 10 km2 [125].

2.3.4 Akeno Giant Air-Shower Array (AGASA) The AGASA array, the biggest one before Auger construction, has been operative at Akeno (Japan) from 1990 to 2004. It consists of 111 scintillators of size 2.2 m2 deployed at a separation distance of 1 km on a overall surface of 100 km2 [126, 127]. The AGASA conﬁguration is shown in Fig. 2.11 together with the densities of one of the most energetic event detected (E =1.5 1020 eV , o × θzenith = 44.2 ) and its lateral distribution [128].

Figure 2.11: AGASA. Layout of the detectors of AGASA array (right) and density lateral distribution of one of the most energetic event observed (left) [128]. In the right panel, the detectors are identiﬁed by point with circle of radius correspondent to the logarithm of the charged particle density. The densities recorded in each detector for the observed event are shown in the left panel. 2.3. EXPERIMENTS 43

The scintillators measure exclusively the electromagnetic component of the shower, the sensibility to the photon component is lower than 10%. For the detection of the muon component (whose contribute at a distance of 600 m is about 10 14%), 27 muon detectors of diﬀerent size (2.4 10 m2) are installed. The array− includes 3 detectors, composed of 2 overlapping− scintillators separated by a lead layer, for the investigation of the relative fractions of electrons, photons and muons far from the core [129]. The lateral distribution is ﬁtted with the function [130, 13]

2 0.6 α (η α) r − S(r)= CR− (1 + R)− − 1.0+ , (2.27) 2000 where R = r/r0, r is the distance from the shower core, r0 is the Moli´ere unit (equal to 91.6 m at the altitude of two radiation length above the Akeno level), C is a normalization factor, α =1.2 and η is given by

η =3.97 1.79 (sec θ 1), (2.28) − × − where θ is the zenithal angle of the shower. The function (2.27) is valid for distances 500 m

17 1.02 0.02 E [eV ]=2.03 10 S (600) ± . (2.29) 0 × × 0 The systematic uncertainty in energy determination is of order of 18% above 1019 eV [13]; the primary energy resolution obtained is of order of 30% at 3 1019 eV and 25% at 1020 eV [13]. The× arrival directions∼ are determined with an angular resolution equal to 1.8o at 3 1019 eV which decreases with the shower energy to the value 1.3o at 1020 ×eV [132, 13]. Through the measurements provided by muon detectors, the muon density at 1000 m, ρ(1000), is used as a primary mass estimator. This parameter is determined with an error of order of 40% for hadronic showers with zenithal angle below 36o [133].

In the AGASA array the inelastic cross section proton-air has been determined in the energetic range 3 1014 eV 3 1018 eV [134], as well as the energy spectrum of CRs [5, 135,× 136, 131] and− the× muon energy spectrum derived from horizontal showers [137].

2.3.5 Fly’s Eye Detector Fly’s Eye has been the first fluorescence detector, built in Utah and operative from 1981 to 1993. For a detailed description of the experiment see ref. [105]. It consists of two fluorescence detectors (FE I and FE II) separated by a distance of 3.4 km. FE I is composed of 67 mirrors which cover the whole sky and 880 photomultiplier tubes. On the other side, FE II is composed of 36 mirrors and 464 PMTs, with a field of view covering only half of the sky in direction of FE I. The mirror diameter is equal to 1.6 m and the angular size of the PMTs 44 CHAPTER 2. DETECTION OF UHECRS is 5.5o [105]. The shower reconstruction can be monocular (if only seen by FE I) and stereo, i.e. seen by both the detectors (called also “eyes”). Although the integrated monocular exposition is much larger than the stereo one, the resolution of the latter is higher, not depending on the time information of the signals. The fluorescence detector follows the longitudinal development of the shower in the atmosphere observing the fluorescence light emitted by the nitrogen molecules excited at the passage of particles. The longitudinal profile recorded by FE I for the most energetic shower detected by Fly’s Eye (E =3 1020 eV [7]) is shown in Fig. 2.12. × To estimate the number of particles at a fixed atmospheric depth, it’s necessary to know the fluorescence efficiency in air, the attenuation of light from the source to the observation level and to calibrate the optical system. To this aim, the efficiency of the whole system is monitored at each mirror by a optic pulse [33], while nitrogen lasers and vertical flashers are used to investigate the light scattering in the atmosphere and to monitor the atmospheric conditions [138, 139].

Figure 2.12: Longitudinal proﬁle of the most energetic event observed at Fly’s Eye. The reconstructed energy is equal to 3 1020 eV [7]. ×

One of the advantages of the ﬂuorescence technique is the capability to measure directly the atmospheric depth of the shower maximum (Xmax). The Xmax distributions provided by Fly’s Eye have given important results regarding composition of primary CRs [140, 141] and useful information on the inelastic cross- section proton-air deduced from the exponential tail of the distributions [142]. 2.3. EXPERIMENTS 45

2.3.6 HiRes The successor of Fly’s Eye, HiRes (High Resolution Fly’s Eye), after a prototype phase (HiRes-MIA) begun in 1993, began to take data in monocular mode in 1997 [143], and in stereo mode in 1999. The detector has been designed by the universities of Utah, Adelaide, Illinois, Colombia and New Mexico. The detector is a stereo system, as Fly’s Eye, constituted of two fluorescence telescopes placed at a distance of 12.6 km, HiRes I and HiRes II, composed of 21 and 42 mirrors respectively. Each mirror is seen by 256 PMTs. The field of view of the HiRes I is 360o in azimuth and 17o in elevation while HiRes II has a field of view of 360o in azimuth and 31o in elevation. The aim of HiRes was to increase by a factor 10 the sensibility in the spectral region of energy higher than 1019 eV ; the achieved resolution in the determi- 2 nation of Xmax is of about 30 g/cm [144], lower enough than the difference in Xmax of showers induced by protons and Iron. The improvement in aperture and in Xmax resolution of HiRes has been obtained reducing the aperture of each PMTs from 5.5o to 1o and increasing the diameters of the mirrors from 1.5 to 2 m. For the accurate reconstruction of the longitudinal profile of the shower and in order to define the aperture of the detector (which grows up with energy), a precise atmospheric monitoring is fundamental. Horizontal attenuation monitors and LIDARs are used in order to estimate the attenuation coefficient for Rayleigh and Mie scattering [108].

Figure 2.13: HiRes aperture. The apertures (deﬁned as the product of col- lection area and solid angle) of the HiRes I and HiRes II detectors operating in monocular mode are shown. Taken from [145].

The monocular reconstruction is described in detailed in references [146, 147], while the stereo one is described in [144]. Diﬀerently from the angular resolution 46 CHAPTER 2. DETECTION OF UHECRS achieved in the monocular reconstruction [148, 149], the stereo one is essentially constant in azimuthal and zenithal angle (of order of 0.6o for showers of energy 1019 eV ), with deviations lower than 0.1o for zenithal angles lower than 70o [150]. The biggest uncertainty corresponds to the case in which the two detectors lie on the same shower detector plane (SDP). Most of the systematic errors in the determination of the arrival direction of CRs are due to the uncertainty of the pointing direction of the telescopes, inferior to 0.2o [150]. The energy resolution achieved in monocular and stereo reconstruction are of order of 17% and 13% respectively [18, 144] at energies above 1018.5 eV . The apertures of HiRes I and HiRes II operating in monocular mode are shown in Fig. 2.13 [145, 18]. Over the period of stereo operation (December 1999 to April 2006), HiRes reached a stereo exposure of 3200 km2 sr yr at energies above 1020 eV [18, 144].

HiRes experiment provided important result on composition and spectrum of CRs, as well as on the proton-air cross section. These results will be presented in details in chapter 3. § 2.3.7 KASCADE and KASCADE-Grande The KASCADE-Grande experiment [151], located at Forschungszentrum Karl- sruhe (Germany), is a multiarray consisting of the KASCADE experiment, a trigger array called Piccolo and a scintillator array called Grande. The layout of KASCADE-Grande is shown in Fig. 2.14. Besides these three arrays, the experiment includes an array of digital antennas (LOPES) to study the radio emission in EAS at energy higher than 1016 eV .

Figure 2.14: Layout of KASCADE-Grande. Taken from [151]. 2.3. EXPERIMENTS 47

The main goal of KASCADE-Grande is to provide conclusive results on the nature of the knee (observing the Iron knee at the expected energy of 100 P eV following the observation of KASCADE [152] and EAS-TOP [153]) and∼ to measure the composition in the transition region between the Galactic and extragalactic components, studying cosmic rays in the energy range 1014 1018 eV . − The KASCADE experiment is a multiple detector setup consisting mainly of an array of 252 scintillators, a muon tracking detector (MTD) [154] and a central detector. The KASCADE array is structured in 16 clusters. The detector stations of 12 clusters (192 stations) are composed of two separated detectors, an unshielded liquid scintillator (for the electromagnetic component) and a shielded plastic scintillator (for the muonic component). These clusters allow the reconstruction of the lateral distributions of muons and electrons separately for each event, important for composition studies. The stations of the other 4 clusters house only the liquid scintillator for the electromagnetic component. The Central Detector of KASCADE allow to investigate the muon component of EAS at four diﬀerent threshold energies (including timing and arrival directions). A hadron calorimeter with more than 44 000 electronic channels of liquid ionization chambers in 9 layers reconstructs the hadronic content of air showers. The Grande array is composed of 37 station of plastic scintillator detectors deployed on a surface of about 0.5 km2 with a separation distance of about 137 m. Each scintillator detector is divided into 16 individual scintillators, each one viewed by a high-gain photomultiplier for timing and low particle density measurements. The 4 central scintillators are additionally viewed by a low-gain PMT in order to avoid saturation problems in the case of high particle densities. The Piccolo array, close to the center of KASCADE-Grande and constituted by plastic scintillators detectors, ensures the joint measurements of KASCADE- Grande providing a fast triggering of all the detectors.

2.3.8 Pierre Auger Observatory The Pierre Auger Observatory is the largest cosmic ray detector even built, conceived to study UHECRs with energy above 1018 eV with full efficiency above 1019 eV . ∼ In order to increase the collecting power of the previous experiments with a full sky coverage, the project foresees two large experimental sites, one already in operation in the southern hemisphere (located in Malargüe, Argentina) and the other in the northern hemisphere (in Colorado). The distinctive and innovative feature of the experiment is the simultaneous detection of showers with the two traditional techniques, the fluorescence and ground detections. The combined use of two independent techniques, called “hybrid detection”, allows the reduction of the systematic errors thanks to the cross-calibration of the detectors [107, 155]. In this section only the Southern site of the observatory is briefly described. For a full description of the South Observatory and an overlook of the Northern site project see [156, 157, 158] and [159] respectively.

The Southern site comprises a Surface Detector (SD), a ground array of 1600 water-Cˇerenkov tanks (of dimension 10 m2 1.2 m) deployed over a 3000 km2 area and spaced each other by 1.5 km (Fig.× 2.15); each tank contains∼ 12 tonnes 48 CHAPTER 2. DETECTION OF UHECRS

Figure 2.15: Layout of the southern site of the Pierre Auger Observa- tory. Dots indicate the 1600 water-Cˇerenkov tanks constituting the SD. At the periphery of the array the 4 optical stations of the FD are shown together with the FOV of each telescope (lines).

of pure water and three 9” hemispherical PMTs that collect the Cˇerenkov light. The SD area is overlooked by four optical stations at the periphery of the array, each one composed of 6 fluorescence telescopes constituting the Fluorescence Detector (FD). Each telescope detect the fluorescence light emitted at the passage of an EAS within a field of view of 30o in azimuth and 28.6o in elevation using a spherical mirror of 3 m2 effective area that focuses light onto a camera composed by 440 hexagonal PMTs, each one of angular diameter of 1.5o. The FD telescopes are completed by optical calibration systems [160]. Several systems for atmospheric monitoring, for corrections of light attenuation due the Rayleigh and Mie scattering, complete the Observatory. For Rayleigh scattering, measurements of air temperature, pressure and humidity are provided by meteorological stations, while the knowledge of the aerosol content for Mie corrections needs several atmospheric systems [161], as the central laser facility (CLF), the aerosol phase function monitors (APFs), the horizontal attenuation monitors (HAMs), the elastic back-scatter LIDAR stations and the Raman lidar system. For the future, two enhancements, AMIGA (Auger Muons and Infill for the Ground Array) [162] and HEAT (High Elevation Auger Telescopes) [163], are planned in order to extend the baseline energy range ( 2.3.9) and an array of about 20 km2 of radio detectors to be deployed at the§ Pierre Auger site, is currently in R&D phase [115].

After a ﬁrst engineering phase, the southern site began to take data in January 2004 and, completed the construction of the SD and FD sites in summer 2008, it reached its ﬁnal aperture of 7850 km2. ∼ 2.3. EXPERIMENTS 49

The ground array provides a huge collecting area with an easily calculable aperture and a 100% duty cycle and a detection eﬃciency of 100% for energies above 1018.5/1018.8 eV at zenith angles below/above 60o. It measures the primary energy in relation to the particle density at 1000 m from the shower core, S(1000) 3 [165]. This parameter is estimated using as lateral distribution function the modiﬁed NKG (eq. (2.11)) function

β r β r + 700 − S(r)= S(1000) − , (2.30) 1000 1700 where r is the distance from the core in meters, S(r) is the signal size at a core distance r and β is the LDF slope.

The fluorescence detector operates in clear moonless nights, for a total duty cycle of the order of 10%. The fluorescence measurements determine the shower energy in an independent way, integrating the size or energy deposit profile in the atmosphere. The longitudinal profile of the shower is obtained using a fit with the Gaisser-Hillas function (eq. (2.14)). The FD provides the conversion between S(1000) and the cosmic ray primary energy, which otherwise should be determined using simulation affected by uncertainties related to hadronic interaction models. The total systematic uncertainty achieved in energy determination in hybrid mode is equal to 22% [166]. Besides a high energy resolution, the hybrid detection has advantages∼ also in the determination of CR arrival directions and composition. In fact, in anisotropy studies, hybrid data provide high-precision shower arrival directions (resolution better than 0.5o) which are used to cross-check SD-derived directions and to directly measure the SD angular resolution. In mass composition studies, the FD measures the depth of shower maximum Xmax with a total uncertainty around 2 20 g cm− [166], the least indirect and most confident of all mass indicators [167]. Meanwhile, hybrid data are being used to calibrate and cross-check several promising mass sensitive parameters measured by the SD alone [87].

The integrate exposure, achieved by SD from 1 January 2004 up to 28 February 2008, mounts up to about 5165 km2 sr yr for events with zenith angle below 60o, about 3 times AGASA exposure [165]. The integrated SD exposure for events with zenith angle above 60o, achieved in the same period, amounts to about 1510 km2 sr yr ( 29%) [168]. Results from the Auger∼ experiments will be presented in details in chapter 3. § 2.3.9 AMIGA and HEAT The Southern site of the Pierre Auger Observatory, in its original design, has a full efficiency energy threshold of 3 1018 eV and . 1018 eV for surface and fluorescence events, respectively.∼ Two× enhancements, AMIGA (Auger Muons and Infill for the Ground Array) [162] and HEAT (High Elevation Auger Tele- scopes) [163], are planned in order to extend the observational energy range for high quality data down to 1017 eV .

3Actually this parameter is used for showers with zenith angle below 60. At larger zenith angle, due to deﬂection of particles in the geomagnetic ﬁled, another parameter (N19) is used [164]. 50 CHAPTER 2. DETECTION OF UHECRS

The baseline design of Auger is optimized for energies corresponding to the middle of the ankle and upwards to the highest energies, in order to solve the GZK controversy. The extension of the energy range lowering the threshold energy to 1017 eV , achievable with these enhancements, will allow the complete inclusion of the ankle and of the second knee inside the observation range of Auger, region where the Galactic-extragalactic transition takes place. Moreover, it would take the advantage of an overlapping of Auger range with KASCADE-Grande [169], fundamental in order to validate results.

Figure 2.16: Layout of Auger enhancements, AMIGA and HEAT. White lines show the six original telescopes FOVs while black lines indicate the FOVs of the three telescopes of HEAT. The SD tanks are indicated by dots: black dots are the original SD tanks (at a separation distance of 1500 m), slashed and white dots indicate AMIGA SD tanks plus buried muon counters placed at 433 m and 750 m respectively. Adapted from [162].

AMIGA will consist of 85 pairs of Cˇerenkov detectors and 30 m2 muon counters buried 2.5 m underground, placed in a graded infill of triangular grids of 433 m and∼ 750 m separation distance. The resultant AMIGA infill area will be constituted by two hexagons of 5.9 km2 and 23.5 km2 corresponding to the 433 m and 750 m arrays, respectively. The energy threshold of the two arrays will be lowered to 1017 eV and 1017.6 eV , respectively [170]. The analogous enhancement∼ of the∼ fluoresce detector is HEAT. In order to lower the energy threshold for CR detection, HEAT will consists of three additional telescopes with higher elevation angle, from 30o to 58o, located next to the FD optical station of Coihueco, in front of the AMIGA infills. These additional telescopes will be used in combination with the existing telescopes of Coihueco sites and with the AMIGA infills for hybrid detection. 2.3. EXPERIMENTS 51

The aim of the two enhancements is not limited to energy spectrum measurements at energies below Auger baseline range. These enhancements will also allow detailed composition studies, thanks to the muon counters; the combined measurements of the atmospheric depth of maximum shower development, Xmax, and the shower muon content [171], which are parameters very sensible to primary mass composition, will provide important information on the composition in the transition region, essential for understanding the second knee and ankle features and in order to discriminate the possible existing astrophysical scenarios. Other mass sensitive parameters, like the slope of the lateral distribution function, rise-time of the signals in the surface detectors, curvature radius, etc. depend strongly on these two parameters [86].

2.3.10 Telescope Array (TA) and TALE Telescope Array (TA) is situated in Utah and started full operation in autumn 2007. It consists of 512 plastic scintillators of 3 m2 size, installed in a 750 km2 array at a separation distance of 1.2 km, which measure distribution of charged particles at the ground, and three air fluorescence stations. The stations are located on a triangle overlooking the array and consist of 12-14 telescopes. Each fluorescence station has a field of view 3o 33o in elevation and 108o in azimuth. Hybrid detection is possible observing− the longitudinal development∼ and the lateral distribution of the shower simultaneously. The acceptance of the ground array, limited to 9 times that of AGASA, is incremented to 23 times AGASA acceptance thanks∼ to the air fluorescence telescopes.

The TA project is complementary to the Auger South Observatory and will provide anisotropy, composition and spectral measurements in the energy range 1018.5 1020.5 eV in the northern hemisphere. As it’s located in the northern hemisphere,− it will allow comparisons between TA data and AGASA and HiRes measurements. Moreover the comparison with the South Auger Observatory results will allow a search for possible north-south asymmetry in the energy spectrum and in the distribution of arrival directions. An extension to lower energies (TALE) is planned to provide detailed measurements in the transition region covering the energy range 1016.5 1018.5 eV . TALE will consist of the surface infill and muon infill-arrays together− with an extension of the field of view of the fluorescence stations to higher elevation angles. 52 CHAPTER 2. DETECTION OF UHECRS

Figure 2.17: TA layout: SD station are indicated by diamonds, FD stations by pentagons. Taken from [172]. Chapter 3

Results

In this chapter the main results on UHECRs from the mayor air shower experiments (AGASA, HiRes and Auger) are presented. Spectrum and composition results are shown and discussed with particular attention to the transition region. Regarding composition, an important result from Auger on photon primary ﬂux, which exclude most of the top-down models, is presented. An overview of the anisotropy results is given with particular attention to the correlation of EHECRs with AGN provided by Auger.

3.1 Spectrum

In Fig. 3.1 an overview of several experimental results on the spectrum over many decades in energy is presented. In the lower energy range, where direct measurements are possible, data from SOKOL [173], JACEE [174, 175], Grig- orov [176], Proton Satellite [177] and Runjob [178] are reported, while at higher energy data are available from air shower experiments as HiRes [22], HiRes-MIA [8], Fly’s Eye [33], KASCADE [3], Yakutsk [9], Akeno [5, 135], AGASA [13], Haverah Park [120, 12], Auger [34, 165], BLANCA [179], Tibet [180, 181] and Mt. Norikura [182]. Over the whole energy interval there is considerable dispersion of data, which can be limited to ﬂux normalization below the knee energy. At higher energies the situation worsen, especially at energies around 1017 eV ; in this region, poorly explored by experiments, not only the discrepancies between experiments data are large but also the statistical errors are larger, due to lack of statistics. At growing energies the discrepancies are still large, even if the ankle feature is clear and an energy renormalization seems to be able to reconcile several experiments (see 3.1.1). Around the GZK energy, statistic is still too poor to claim for the GZK§ eﬀect. Results from the mayor air shower experiments are presented in detail in the following.

The AGASA spectrum above 1018.5 eV is shown in Fig. 3.2. The data were collected from February 1990 to January 2004, for an integrated exposure of 1838 km2 yr sr. The total energy uncertainty is 18%. The mayor systematic ∼error sources in energy estimation are reported in [131, 13]. The dotted line

53 54 CHAPTER 3. RESULTS

Figure 3.1: Experimental spectra from air shower experiments (HiRes [22], HiRes-MIA [8], Fly’s Eye [33], KASCADE [3], Yakutsk [9], Akeno [5, 135], AGASA [13], Haverah Park [120, 12], Auger [34, 165], BLANCA [179], Tibet [180, 181], Mt. Norikura [182]) as well as from direct measurements (SOKOL [173], JACEE [174, 175], Grigorov [176], Proton Satellite [177], Runjob [178]).

Figure 3.2: AGASA energy spectrum. The dotted curve represents the expected ﬂux for a GZK model with uniform distribution of the sources. The number of events observed in the highest energy bins are indicated. Upper limits are put at 90% CL. Taken from [131]. 3.1. SPECTRUM 55

(a) HiRes Mono energy spectrum (b) HiRes Stereo energy spectrum

Figure 3.3: HiRes spectra: a) HiRes monocular energy spectrum [18]; b) HiRes stereo energy spectrum [184]. The ﬁtted spectral indexes and their errors are indicated. The dip and cut-oﬀ features are observed. .

represents the expected ﬂux for a GZK model with uniform distribution of the sources [183]. The number of events observed in the highest energy bins are indicated. Eleven events above 1020 eV were detected against the expected 1.9 events for a GZK model with a uniform distribution of sources, corresponding∼ to a deviation of 4σ1.

In Fig. 3.3(a) the monocular energy spectra from HiRes-I and HiRes-II determined by HiRes experiment are shown [18]. The HiRes-I spectrum includes data taken between May 1997 and June 2005. The HiRes-II spectrum includes data taken between December 1999 and August 2004 [145]. The overall systematic uncertainty in the energy determination is about 17%, corresponding to a flux uncertainty of 30% for a spectral slope equal to 2.8 (for details see ref. [18]). The cut-off∼ feature above 4 1019 eV is observed− with a significance of × 19.75 0.04 4.8σ. The spectral slope obtained fitting monocular data above 10 ± eV is equal to 5.1 0.7. The dip structure around (3 4) 1018 eV is also seen, − ± − × 18.65 0.05 with a change in slope from 3.26 0.02 to 2.81 0.03 at E = 10 ± eV . The HiRes stereo energy spectrum− ± (using data− collected± from December 1999 to March 2006), shown in Fig. 3.3(b), is consistent with the monocular one. The fitted spectrum slope below and above the ankle is 3.33 0.18 and 2.75 0.25, respectively, in agreement with the monocular values− [184]± . − ± The stereo spectrum is obtained using a fiducial volume cut in order to select events seen with 100% efficiency. This procedure reduces the systematic errors in aperture estimation to the expense of statistic. On the other side it provides a more reliable energy spectrum than monocular data thanks to a better geometrical reconstruction and to redundant energy estimations (thanks to information from two eyes).

1A preliminary revision of the reported energy spectrum has been presented by Techie at the RICAP meeting in June 2007. The new analysis reduces the eleven events to 5/6 events above 1020 eV . 56 CHAPTER 3. RESULTS

(a) Auger spectra

(b) Auger combined spectrum

Figure 3.4: Auger energy spectrum. a) Auger energy spectra determined using SD vertical events (θ < 60o) [165], SD inclined events (60o <θ< 80o) [168] and hybrid events [34]. b) Auger combined energy spectrum determined using the three spectra shown in (a); adapted from [19]. 3.1. SPECTRUM 57

The Auger experiment determined three independent energy spectra, shown in Fig. 3.4(a), using SD data with zenith angle below 60o (vertical SD data) [165], SD data with large zenith angle, 60o <θ< 80o, (inclined SD data) [168] and Hybrid data (SD vertical + FD) [34]. The data used in the analysis are those collected from 1 January 2004 to 28 February 2007, for a SD integrated exposure equal to 5165 km2 sr yr2, about three times that achieved by AGASA and similar to monocular HiRes [165]. As the energy calibration is based on the FD measurements, all spectra are affected by the uncertainty in the FD energy scale 22% [166], (the largest contribution, 14%, is given by the absolute fluorescence∼ yield used [185]). The systematic∼ uncertainty in the SD spectrum is given by two contributions, the conversion of S(1000) to FD energy (< 10%) and the exposure calculation (3%) while the systematics uncertainty in the hybrid spectrum is dominated by the uncertainty in exposure calculation (16%) [19]. The three spectra are all consistent within the statistical errors and can be combined with proper statistical weights [19] (see Fig. 3.4(b)) with normalization factors less than 3%. The combined energy spectrum shows the dip and cutoff features. At E 4.5 1018 eV a flattening of the spectrum slope from 3.30 0.06 to 2.62 ∼0.02× is observed while the spectral slope obtained fitting− all the± data above− 3.6 ±1019 eV is equal to 4.1 0.4 [186, 19]. The flux suppression above 4 1019 ×eV is observed with− a statistical± significance of 6σ. Nevertheless the Auger× collaboration is cautious to interpret the dip and cut-off features as a GZK effect [186, 187], as such a claim should be supported by composition measurements ( 3.2). §

3.1.1 Dip calibration As seen in 1.2, two different interpretations of the ankle exist. One possibility is that§ the ankle is naturally formed as the intersection of the Galactic and extragalactic CR components, while the other interpretation is that the ankle is the dip structure of CR spectrum produced by pair creation in interactions of CR protons with the cosmic microwave background radiation. In any case experimental observations confirm the dip feature in the spectrum around (3 5) 1018 eV . In− Fig.× 3.5 (left panel) the Auger combined energy spectrum [19] and the HiResI and HiResII monocular spectra [11, 145] are shown together with those determined by AGASA [131], Akeno [134] and Yakutsk [10]. The systematic errors in energy measurements are high, from 18% in AGASA to 22% in Auger. The flux suppression above the GZK cut-off energy is observed with different significance by all the experiments except AGASA. Even if the fluxes differs from each other (the Auger and AGASA fluxes at 1019 eV differs by a factor 2.5), the dip structure is observed in all the spectra.∼ ∼ Berezinsky et al. [21] proposed to use the dip feature for energy calibration of the different experiments. As they assume that the dip structure is due to the pair creation dip of the cosmological CR protons, its position in the energy

2The exposure is calculated for SD events of energy above 3 × 1018 eV and with zenith angle below 60. 58 CHAPTER 3. RESULTS

Figure 3.5: Dip calibration. Fluxes of AGASA, HiRes and Yakutsk before (left panel) and after (right panel) dip-calibration. Auger data are shown for comparison after and before energy recalibration of a shift factor equal to the maximum shift allowed by systematic energy errors. Adapted from [23]. spectrum can be “universally” calculated and the absolute energy scale defined [21]. In Fig. 3.5 the fluxed measured by AGASA, Yakutsk and HiRes are shown after dip-calibration of the energy scale [23]. The spectra are scaled by factors of 0.9 for AGASA, 0.75 for Yakutsk and 1.2 for HiRes. After an energy recalibration of the experimental spectra with the calculated dip position, the fluxes measured by AGASA, Yakutsk and HiRes agree with high precision without any renormalization of the fluxes [23] (the fluxes in figure are not modified). The same happens with the Auger data, even if an energy-shift of a factor 1.5 is necessary, well above the maximum shift (equal to 1.2) allowed by its systematic energy errors [20]. Even if the Auger recalibration can be discussable, the shift factors calculated for the different experiments suggest that the measured energy is systematically higher for ground detectors than for fluorescence detectors; in particular, if this were the case, AGASA and Yakutsk energy shall be scaled down by 10% and 25%, while HiRes and Auger shall be scaled up by 20% and 40 50%. If this were the case, the systematic errors in deriving the energy spectra− shall be due to the different energy scales of the experiments, defined by the air fluorescence yield in the case of HiRes and Auger, by MC for AGASA and by calibration with Cˇerenkov light for Yakutsk. The difference between HiRes and Auger should be related to the different air fluorescence yield assumed, measured by Kakimoto et al. [103] in the case of HiRes and by Nagano et al. [185] in the case of Auger (the difference between the two yield amounts to (11 15)%). −

3.1.2 Spectrum and EG astrophysical models In Fig. 3.6 Auger data are compared with some astrophysical models [41, 188]. The total ﬂuxes predicted by models are normalized to the data at energy 1019 eV . The models assume an injection spectral index, an exponential cut-oﬀ 3.1. SPECTRUM 59

Figure 3.6: Auger combined energy spectrum, together with a ﬁt (black line) and the predictions of two astrophysical models (blue and red lines). The parameters of the models (mass composition at sources, source distribution, spectral index and maximum energy) are indicated. Taken from [19].

at an energy Emax Z, where Z is the nucleus charge, a mass composition at the acceleration site× and a distribution of sources. The parameters of each models are indicated in figure. Basically, two composition models are considered, an EG spectrum with mixed composition at the sources, with nuclear abundances similar to those of the low-energy CRs, and an EG spectrum with pure proton composition. In the cases of mixed composition model, an uniform distribution of sources and a spectral index γ = 2.2 are assumed, while two different values of the maximum energy are considered (1020, 1021 eV ). In these cases, there is good agreement between data and the predicted flux down to the ankle energy, where the requirement of another component (maybe Galactic) emerges. In the case of a pure proton model, the maximum energy is fixed to 1021 eV , while two spectral indexes and different kind of source distribution are considered (uniform and strong); in one case a uniform distribution of sources is considered with a spectral power index equal to 2.2; in the other case, it was assumed a redshift evolution of sources proportional to the star formation rate (SFR) and a spectral index 2.3. In the case of the EG proton model , while the agreement at low energies is good, reproducing well the ankle features, at higher energies a stronger evolution of the sources is needed to reproduce Auger data.

In Figure 3.7 the HiRes monocular spectra [22] are compared with astrophysical models, proton EG model and mixed composition model. For each of the two models, diﬀerent sources evolution are considered, uniform, strong and SFR. 60 CHAPTER 3. RESULTS

(a) HiRes and proton model

(b) HiRes and mixed composition model

Figure 3.7: Calculated spectra for pure proton model (a) and mixed composition model (b) compared with HiRes monocular data [22]. Diﬀerent source evolution models are considered; the corresponding Galactic components are inferred from the overall spectrum by subtracting the EG component in the case of uniform and SFR source evolution models. Taken from [39]. 3.2. COMPOSITION 61

Figure 3.8: data from Fly’s Eye [6], HiRes/MIA [8], HiRes [189] and Auger [167].

In each case, the value of the spectral index providing the best fit with data was determined and the Galactic component was inferred by subtracting the propagated EG spectrum from the measured flux. In the case of the proton model, the transition energy between Galactic and EG CRs, depends strongly on the source evolution, as well as, consequently, the energy requirement of the Galactic component. For the strong evolution case, the EG component can account for the whole CR flux down to energy 4 1017 eV , lower than the confinement limit of charged nuclei in the Galaxy. In∼ the× different cases of mixed composition model, the end of the transition between Galactic and EG cosmic rays roughly coincides with the ankle. Above 1018.5 eV the predicted spectrum is quite insensitive to the source evolution model. Differently from the proton model, the inferred Galactic component depends weakly on the source evolution model considered. On the other side, its energy requirements are higher; the Galactic component has to extend up to 1018.5 eV . ∼

3.2 Composition

Composition measurements are of fundamental importance to understand the ankle feature and to confirm the GZK effect. As seen in 2.1, one of the most confident parameter to determine primary composition is§ the depth of the shower maximum Xmax. In Fig. 3.8, the average Xmax as a function of energy observed by fluorescence experiments is shown. HiRes [189, 8] measurements suggest a proton dominant component above 1018 eV by comparing their data with Monte Carlo simulations with CORSICA-QGSJet model. While at energies below 3 1018 eV , × 62 CHAPTER 3. RESULTS

Figure 3.9: Composition predictions for dip (left) and ankle (right) models: the calculated elongation rates are shown for QGSJET [190] model of interaction, QGSJET-II v03 [191, 192] and SIBYLL [193]. measurements of Fly’s Eye [6], HiRes-Mia [8], HiRes [189] and Auger [167] are shown. Adapted from [23].

Auger measurements [167] agree with HiRes, at higher energy data indicates a heavier composition compatible with a mixed composition model if compared with prediction from Monte Carlo simulations with CORSICA-QGSJet model. Moreover at energies above 1019 eV , HiRes and Auger present very diﬀerent ∼ < Xmax > trends, with a composition getting lighter/heavier respectively. On the other side, Fly’s Eye data [6] favor a heavier composition in the lower energy region, getting lighter above 3 1018 eV . As shown by these results, it’s∼ not× possible at the moment to take any conclusion on the composition of UHECRs. Further investigations and more precise measurements are needed.

3.2.1 Composition in the transition region Composition measurements are essential for the interpretation of the dip feature in the spectrum, related to the transition of the Galactic-extragalactic components. The three model described in 1.2 (dip model, ankle model and mixed composition model) can be experimentally§ distinguished through measurements of the spectrum and of anisotropies, although the most discriminant feature is the chemical composition.

In Fig. 3.9 experimental data are compared to composition predictions for the dip and ankle models. Model predictions are calculated for diﬀerent hadronic interaction models (QGSJET [190], QGSJET-II v03 [191, 192] and SIBYLL [193]). In the ankle model the Galactic heavy component dominates up to the ankle energy, resulting in a great discrepancy in the energy range 1.5 5 1018 between the calculated elongation rate and the experimental ones (Fig.− × 3.9, right panel). In contrast to the ankle model, as in the dip model the transition is completed at energy around 1 1018 eV where the composition is proton dominated, the calculated elongation× rate presents a steeper increase, in agree- 3.2. COMPOSITION 63

850

Mixed composition

SFR source evolution

QGSJet-II

800

Protons )

2 750

> ( g/cm ( >

700 max

Iron

650

HiRes Stereo

Fly's Eye Stereo+13 g/cm

HiRes-Mia

600

17.5 18.0 18.5 19.0 19.5

log E (eV)

Figure 3.10: Composition predictions for the EG mixed composition model: the elongation rate has been calculated for an EG mixed composition model [39] with the parameters given in Fig. 1.3. The mass composition evolves from almost pure iron composition at E 3 1017 eV to a strongly proton- 19 ≈ × 18 dominated one at E 3 10 eV . At energy Ea =3 10 eV the transition to pure extragalactic≥ component× is completed and the× composition is mixed. ment with HiRes and HiRes-MIA data (Fig. 3.9, left panel).

In Fig. 3.10 the composition predictions for mixed model are compared with experimental data. The mass composition evolves from almost pure iron composition at E 3 1017 eV to a lighter composition due to enrichment by protons ≈ × 18 and light nuclei of extragalactic origin. At energy Ea = 3 10 eV the transition to pure extragalactic component is completed and che×mical composition evolution proceeds further due to photo-disintegration of the nuclei. At energy E 1.3 1019eV , seen in the plot, all nuclei are disappearing faster than before and≈ composition× becomes strongly proton-dominated at E 3 1019 eV . The calculated elongation rate agrees well with Fly’s Eye (stereo)≥ and× HiRes (stereo) data but above 1019 eV doesn’t ﬁt the decreasing elongation rate of Auger data (Fig. 3.10).

3.2.2 Primary photons Another important result in composition studies is the fraction of gamma ray in UHECRs. As seen in 1.3.2, 1.3.3, the Top-Down models and the Z-burst model predict a higher fraction§ § of gamma rays. The fraction of gamma rays in UHECRs is an important parameter in order to set signiﬁcant constrains on 64 CHAPTER 3. RESULTS

(a) Upper limits on the integral ﬂux of photons

(b) Upper limits on the fraction of photons in the integral CR ﬂux

Figure 3.11: Photon limits. a) The upper limits on the integral flux of photons calculated by Auger using SD data (“SD”, black arrows) are shown [194]. A flux limit derived indirectly by AGASA (“A”) is shown for comparison [195]. b) The Auger upper limits on the fraction of photons in the integral CR flux are shown along with previous experimental limits (HP: Haverah Park [196]; A, A2: AGASA [195, 197]; AY: AGASA-Yakutsk [198]; Y: Yakutsk [199]; FD: Auger Hybrid limit [200, 201], SD: Auger SD limit [194]). Predictions from top-down models (SHDM, TD and ZB from ref. [202], SHDM’ from ref. [203]) and GZK photon flux/fraction [202] are also shown in both the figures. Figures adapted from [194]. 3.3. ANISOTROPY 65 top-down models and acceleration models. Besides the information on acceleration model, the detection of UHE photons is important in order to prove the GZK effect. In fact, UHE photons can be tracers of GZK process. Photon fluxes originated in resonant photo-pion production are sensitive to source features, as the type of primary, the distance of the sources and injection spectrum, and to propagation parameters as EG magnetic fields and EG background radiation. The cosmogenic gamma rays produced by GZK 2 4 mechanism is estimated to be 10− 10− of the total flux of UHECRs. In the 20 years of operation, Auger may be− able to detect cosmogenic gamma. Auger reported new results on the 95%CL upper limit of gamma ray flux at high energies [194]. The results are shown in Fig. 3.11(a) along with the gamma ray fluxes predicted by top down models (Super-Heavy Dark Matter models and Topological Defect models) and Z-burst model [202, 203]. The predicted GZK photon flux is also shown. Auger upper limits on the fraction of photons in UHECR flux are shown in Fig. 3.11(b) (from SD [194] and FD [200, 201] data) along with the results reported by Haverah Park [196], Yakutsk [199], AGASA [195, 197] and AGASA-Yakutsk [198] and theoretical predictions. Most of the model of SHDM and topological defects are excluded by Auger observation.

3.2.3 Neutrinos The detection of ultra high energy cosmic neutrinos at energies around 1018 eV is an important goal to test fundamental particle physics at energies unexplored by accelerators. All models of UHECR origin predict the existence of UHE cosmic neutrinos from the decay of charged pions produced in the interaction of CRs in their sources or in the interaction with the cosmic background radiation during propagation. The interaction of UHECR protons with the cosmic microwave background give origin to the “cosmogenic” or GZK neutrinos.

The Auger experiments has carried out a study of tau neutrinos searching Earth- skimming events at energies > 1017 eV . Such neutrinos are expected to generate tau leptons in Earth of enough energy to escape and produce showers that can be detected by the SD. The data collected between 1 January 2004 and 31 August 2007 do not present any candidate event and limits have been set [214]. Limits on the tau neutrino flux are show in Fig. 3.12 together with other experiments [204, 205, 206, 207, 208, 209, 210, 211]3. In Fig. 3.13 the upper limit on the neutrino flux νe and ντ from the analysis of HiRes data [215] are shown, along with theoretical curves [216, 217, 218] and to calculated flux limits from other experiments [219, 220, 206, 205, 221, 214]

3.3 Anisotropy 3.3.1 Large-scale anisotropy Measurements of large-scale anisotropy are related to the Galactic/extragalactic origin of CRs of energy 1018 eV . If their origin is Galactic, the sky distribution of 1018 eV CRs may∼ be not completely isotropised due to the eﬃciency of

3Limits from other experiments are converted to a single avor assuming a 1:1:1 ratio of the 3 neutrino ﬂavors and scaled to 90% CL where needed. 66 CHAPTER 3. RESULTS

Figure 3.12: Tau neutrino limits. Limits at 90% CL for a diffuse flux of ντ from the Pierre Auger Observatory along with limits from other experiments are shown [204, 205, 206, 207, 208, 209, 210, 211]. Differential (squares) and integrated (constant lines) limits are used. The expected fluxes of GZK neutrinos Refs. [212, 213] are indicted by the shaded area. Adapted from [214].

Figure 3.13: Neutrino limits. HiRes-II neutrino flux limit: νe limit [215], ντ limit [211], νe and ντ combined flux limit [215]. Dotted line: cosmogenic per flavor neutrino flux limit from fits to HiRes cosmic-ray data [216]. Dashed line: cosmogenic per flavor neutrino flux limit derived from fits to existing cosmic and gamma-ray data [217]. Dot-dashed line: cosmogenic per flavor neutrino flux from fits to HiRes and AGASA cosmic-ray data [218]. Also shown are calculated neutrino flux limits from Flys Eye [219, 220], ANITA-lite [206], RICE [205], AGASA [221] and Auger [214] experiments. Adapted from [215]. 3.3. ANISOTROPY 67 their escape from the Galaxy. On the other end, if their origin is extragalactic, their sources should be cosmologically distributed (for energies below the GZK) resulting in a isotropised distribution. The predictions for the shape and amplitude of anisotropy, in case of Galactic origin, are very model dependent, but can be searched as modulation in RA. Searches for such modulation have been done by AGASA and Auger. AGASA found a 4% amplitude modulation in RA in the energy range 1 EeV

3.3.2 Intermediate scale anisotropy The excess of events in the direction of the Galactic center (GC) and the deﬁcit in the direction of the anti-Galactic center claimed by AGASA [222] (Fig. 3.14(a)), was examined by Auger [225] and HiRes [224]. Auger searched for anisotropy in the direction of the GC in two energy range, 0.1 EeV

3.3.3 Small scale anisotropy Clustering of high energy events was claimed by AGASA collaboration: they observed excess of doublets and one triplet in the arrival direction of UHECRs above 4 1019 eV [132]. An additional event was observed by HiRes [228] in the direction× of the AGASA triplet. The direction of this triplet/quartet lies on the super-galactic plane, where merging galaxies and AGN are situated.

4AGASA deﬁcit was observed integrating the sky plot into 20o circles in the energy range 1017.8 eV

(a) AGASA

(b) HiRes

Figure 3.14: a) Plot of the statistical signiﬁcance of the deviations of the Akeno and AGASA data, integrated over 20o circles. b) HiRes sky plot, with data integrated over 20o circles. The deﬁcit near the galactic anti-center appears at right ascension 95o, declination 10o. Taken from [224]. 3.3. ANISOTROPY 69

Searches for correlation with BL Lacertae (BL Lacs) objects have been performed in the past, giving results not consistent; the correlation observed by AGASA, HiRes and Yakutsk [229] (operating in the Northern hemisphere) has not been observed by Auger experiment in the southern experiment [230]. Again, the northern hemisphere correlation has to be conﬁrmed by Telescope Array experiment.

An important result comes from the Pierre Auger experiment [231, 232]. Studies of small scale anisotropy of the arrival distribution of UHECRs above 1019 eV were made by Auger, evaluating the correlation between the arrival direction of the observed UHECRs with AGN by scanning the threshold energy Eth, the maximum separation angle ψ and the maximum redshift zmax that minimize the chance probability P . Using data acquired between 1 January 2004 and 26 May 2006, they found 12 events among 15 correlated with AGN. The parameters that o minimize P are Eth = 56 EeV , ψ = 3.1 and zmax = 0.018 (corresponding to distance D 75 Mpc). A search protocol was designed and tests were applied to independent≤ data, taken from 27 May 2006 up to 31 August 2007. With the o same parameters (Eth = 56 EeV , ψ = 3.1 and zmax = 0.018, D 75 Mpc) they found 8 arrival directions among 13 correlated to AGN, with≤ a chance 3 probability P = 1.7 10− . The hypothesis of isotropy in the distribution of the arrival direction× of CRS at the highest energy was rejected with at least a 99% conﬁdence level. The analysis was repeated for the overall period with new reconstruction and calibration algorithms, lowering the number of correlation to 27 events but without changing substantially the result (Eth = 57 EeV , o ψ = 3.2 and zmax = 0.017, D 71 Mpc). In Fig. 3.15 a sky map of the 27 correlated events of energy E >≤57 EeV within 3.2o along with the position of 442 AGN within D < 71 Mpc (292 in the ﬁeld of view of the Pierre Auger Observatory) is shown.

Given the Auger result, HiRes made the same kind of search analyzing the correlation between the pointing directions of the observed UHECRs and the position of AGN visible in the northern hemisphere [234]. They used data collected in stereo mode (which have an angular resolution of 0.8o) collected from 1997 to 2006, for a total number of events equal to 6636 in the energy range [1017.4, 1020.1]. They performed three kind of search for correlation decreasing their energy scale of 10% to account for the difference between the energy scale of Auger and HiRes; in the first search the parameters prescribed by Auger (ψ = o 3.1 , Eth = 56 EeV, zmax =0.018) were used while in the second one the data where divided in two sets, one for determining the parameters which provide o the best correlation (ψ = 1.7 , Eth = 15.8 EeV, zmax = 0.020) and the other set to test with these optimum parameters. In the third search, the complete set of data was analyzed using the statistical prescription described by Finley and Westerhoff [235]. The three searches gave results consistent with isotropy, with the most significant correlation with a chance probability of 24% given by the third search. Moreover they analyzed the degrees of auto-correlation in the stereo data at all angles and energies; data result consistent with isotropy with a probability of 97%. 70 CHAPTER 3. RESULTS

Figure 3.15: Aitoff projection of the celestial sphere in galactic coordinates with circles of ψ = 3.2o centered at the arrival directions of 27 cosmic rays detected by the Pierre Auger Observatory with reconstructed energy higher than Eth = 57 EeV . The positions of AGN (292 over 442 in the field of view of the Southern Auger Observatory) with zmax = 0.017 (Dmax = 71 Mpc) are indicated by asterisks. The catalog for AGN used is the 12th edition of the cataloger of quasars and active nuclei [233]. Centaurus A, one of the closest AGN, is marked in white. The field of view of the observatory (with zenith angles smaller than 60o) is limited by the solid line while the super-Galactic plane is indicated by the dashed line. Integrated exposure is represented by colors (darker color indicates larger relative exposure). Taken from [232]. 3.4. CROSS SECTION 71

Figure 3.16: Proton-air particle production cross sections [236, 237]. Data [238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248] and predictions [193, 249, 190, 250, 251, 191, 192, 252] are shown along with new results from HiRes [253], EAS-TOP [254] and ARGO-YBJ [255]. Taken from [236].

3.4 Cross section

The development of an air-shower is determined by the proton-air cross section, not only regarding the depth of the first interaction but also the following interactions. Several updated and new attempts to measure this cross section are reported in [237]. In cosmic ray physics, measurements of the inelastic cross sections p-air can be done studying the correlation between the overall shower profile and the depth of the first interaction. An improved cross section analysis method, that has been put forward, accounts for the cross section dependence on the correlation between the first interaction point and the depth of shower maximum [256, 236]. In particular, the part of proton-air cross section that can be measured in EAS experiments is the particle production cross section. In proton-nucleus collisions, the total cross section can be written as

σtot = σela + σinel

= σela + σq ela + σprod, (3.1) − where σela and σinel denote the elastic and inelastic cross sections, respectively. The inelastic cross section is the sum of the cross section for quasi-elastic scat- 5 tering (σq ela) and that of particle production σprod . The quasi-elastic cross − 5Quasi-elastic processes are interactions, in which the incoming proton is elastically scattered on at least one of the nucleons of the target nucleus, leading to a nucleus breakup without production of “new” particles such as pions or kaons. 72 CHAPTER 3. RESULTS section amounts to about 10% of the inelastic cross section. In Monte Carlo simulations of hadron-air interactions,∼ only the production cross section is used6 and consequently, it is this cross section that is derived in the analysis of data from cosmic ray experiments. Fig. 3.16 shows a compilation of proton-air cross section measurements [236]. The low-energy data are obtained from analysis of the hadron flux in the atmosphere and high energy data are from air shower observations. The uncertainties in the measurements of p-air cross section are related to the selection cuts for identifying proton-induced showers and to the simulation models used to describe the correlation between the depth of the first interaction and the shower observables as the maximum shower depth. A complete discussion of the uncertainties in the measurement of p-air cross section and of the different cross section analysis method is reported in ref. [237].

3.5 Summary of the results

Significant results have been reached in the last years on the GZK cut-off; flux deficit above 4 1019 has been observed by HiRes [17, 18] and Auger [19, 187] with a significance× of 4.8σ and 6σ, respectively. Nevertheless, these results are still inconclusive as the absolute energy scale is not completely understood, as well as chemical composition [20]. In fact, a calibration of the energy scale using the predicted ankle position takes to a great agreement of the observed fluxes, suggesting the existence of systematics related to the detection technique [23, 21] (see 3.1.1). More statistics and understanding of the systematics are essential to claim§ the GZK effect. The next years of operation of Auger, now that it reached its full designed aperture, will be crucial in this sense. Regarding chemical composition, the GZK cut-off is a signature of a proton composition of the extragalactic (EG) CRs. Auger measurements of the elongation rate indicate a composition compatible with an EG mixed composition model getting heavier at the highest energies [167]. On the other side, search for anisotropies has given important results about correlation with AGN, compatible with a proton dominant composition [231]. Above 1019 eV precise measurements of composition are of mandatory to make sure that the cut-off observed by HiRes and Auger is the GZK cut-off for UHECR protons [186].

The problem of the second knee and of the ankle is still unsolved, unless it is clear that their nature is related to the transition between Galactic and extragalactic cosmic rays. The importance of this region is crucial for understanding cosmic ray physics. The existing transition models, and consequently diﬀerent astrophysical scenarios, can be discriminated by composition and spectral measurements in this region. Nevertheless there is the paucity of data involved in the determination of the energy spectrum in the region encompassing the second knee and the ankle and divergence between Xmax measurements by diﬀerent experiments below 3 1018 eV [6, 8, 189, 167].h i A proper experimental× characterization of this very important region will likely have to wait until the release of the KASCADE-Grande [169] and Auger enhancements data [257].

6In case of primary nuclei and their fragments, the inelastic cross section is simulated. 3.6. IMPORTANCE OF THE TRANSITION REGION 73

Precise data and high statistics are required not only for composition but also for flux. Moreover, the overlapping region between Auger enhancements and KASCADE-Grande is important in order to validate the results. The flux and composition measured by the two experiments, in a quite unexplored region, will allow to check the existence of possible systematic errors in energy determination as well as in composition parameters. This is an important unsolved problem that could account for different experimental fluxes and the different positions of the dip observed by Akeno-AGASA [13], HiRes [11], Yakutsk [9, 10] and Fly’s Eye [33]. The dip-calibration proposed by Berezinsky et al. [21, 23], after which the fluxes measured by AGASA, Yakutsk and HiRes agree with high precision, indicates in fact a systematic in the energy determination related to the detection technique and calibration.

Regarding the distribution of arrival directions of cosmic rays, intermediate scale anisotropy studies by HiRes [224] and Auger [225] didn’t confirm the excess and deficit detected by AGASA [222] in the direction of the Galactic center and anti-Galactic center respectively. Moreover the modulation in RA found by [222] AGASA was not confirmed by Auger results [223], even if the sky regions covered by the two experiments are not the same. HiRes found no evidence of declination dependence that could account for the different in power law index between HiRes and Auger [224]. Regarding small scale anisotropy, an important result is provided by Auger, who reported a significant correlation between CRs of energy & 57 EeV and AGN within a distance of 71 Mpc [231, 232]. This small scale anisotropy is not confirmed by HiRes analysis∼ [234], but in any case it opens the possibility of new astronomy with ultra high energy particles. The reports of anisotropy in the northern sky have to wait for confirmation or rejection by the Telescope Array experiment [172], that will allow not only a comparison between TA data and AGASA and HiRes measurements but also a search for possible north-south asymmetry in the energy spectra and in the distribution of arrival directions through a comparison with Auger data.

Other interesting results were provided by Auger experiments regarding upper limit on neutrino ﬂux [214] and photon primaries [194]. This last result excludes most of the model of SHDM and topological defects.

3.6 Importance of the transition region

In the last years the interest in the transition region between Galactic and extragalactic components of UHECRs has increased significantly. Several efforts in ultra high energy cosmic ray have been directed to the study of the highest energy region of the spectrum in order to solve the GZK controversy [14, 15]. With the knowledge acquired in the last decades, it became clear that the only observation of the total spectrum is not sufficient to discriminate the existent theoretical models of UHECRs; the variation of the composition as a function of energy turns then into the key to discriminate both fluxes and to select among a variety of theoretical options. Each single result (spectrum, anisotropy, composition) is not sufficient in order to explain the origin of ultra high energy cosmic rays (UHECRs) and the pro- 74 CHAPTER 3. RESULTS cesses involved in their acceleration and propagation to Earth. On the other hand, the combined analysis of all the information provided by data is a powerful tool to discriminate and set constrains on the existing theoretical models. Nevertheless an additional problem presented itself at very high energy in the region of the predicted GZK cut-off. When the statistics are low, different astrophysical model can produce total energy spectra experimentally indistinguishable at very high energy within the actual energy resolution [258]. This has been shown for example in 3.1.2 where HiRes and Auger data were compared to different EG astrophysical§ model. This degeneracy, beyond few times 1019 eV, can only be broken with supple- mentary information coming either from higher energy neutrinos or from composition measurements at lower energies, in the region of the second knee and ankle, i.e. the transition region between Galactic and extragalactic components of cosmic rays. This is a theoretically challenging region where the smooth matching of the two rapidly varying spectra has yet to be explained.

The region between 1017 eV and 1019 eV includes several important astrophysical characteristics: the end of Galactic acceleration mechanisms and of the Galactic magnetic confinement, the possible manifestation of particles from pulsars, magnetars and compact SNRs, the entry of the EGCR flux in the interstellar medium and the appearance of observable anisotropies in the CR flux, the progressive transition of propagation from a diffusive regime to a ballistic one. The detailed knowledge of the way this transition takes place, i.e. the exact spectral shapes of the involved components and the evolution of compositions as a function of energy, is essential for understanding CRs origin and propagation. These information on the transition region encode information on the propagation conditions in the extragalactic medium and in the Galactic Halo, as well as on the injection spectrum at the sources and their cosmological distribution, helping to break the degeneracy between astrophysical models existing at the highest energies due to the low available statistics. In fact, the energy at which the transition take place and the way the Galactic and extragalactic components mix together depend not only on the EG composition but also on the evolution (as seen in 3.1.2) of the sources and on the diffusion of EG cosmic rays in EG magnetic fields§ (see Fig. 1.9 in 1.5.2) as well as on the distribution of sources. On the other hand, the knowledge§ of the exact energy range, shape and composition of the Galactic component will allow to discriminate eventual acceleration mechanisms and sources in our Galaxy besides the shock acceleration by SNRs.

Nevertheless, the transition region has been poorly explored by previous CR experiments and the data available lack in statistics. Given the acquired knowledge of the importance of this region, the Auger collaboration designed two enhancements (AMIGA and HEAT) which have the aim to explain the nature of the second knee and ankle through detailed measurements of CR ﬂux and composition in this region in order to discriminate possible astrophysical scenarios. Working in conjunction with the baseline Auger surface and ﬂuorescence detectors (SD, FD), the enhancements will aid in a fundamental way to our understanding of ultra-high energy cosmic rays in its astrophysical context. Chapter 4

The transition region: spectral analysis

In section 1.2 the three main models describing the transition between Galac- tic and extragalactic§ CRs have been presented and their predictions on spectral features and composition have been discussed and compared with experimental data ( 3). In this§ chapter the transition region between Galactic and extragalactic CRs is analyzed in order to put some constraints on the existing astrophysical models. This is done analyzing the matching conditions of the Galactic and EG spectrum in the context of two transition models, the dip model and the mixed composition model. As SNRs are considered the main sources and acceleration mechanism of GCRs, we calculate the Galactic diffusive spectrum from regular SNRs using the numerical diffusive propagation code GALPROP [259, 260]. The calculated Galactic spectrum from SNRs is then combined with two different models of EG spectrum, one in which only protons [21] are injected at the sources and another in which a mixed composition containing heavy nuclei [39] is injected instead. The transition region between Galactic and extragalactic components is analyzed in the two different EG scenarios, and the combined total spectrum is compared with the available experimental data.

4.1 Diﬀusive Galactic spectrum from SNRs

The current paradigm for Galactic cosmic ray acceleration is the Fermi acceleration mechanism by shock waves of SuperNova Remnants (SNRs) [42]. In this section the Galactic diﬀusive spectrum from SNRs is calculated using the numerical diﬀusive propagation code GALPROP [259, 260].

4.1.1 Diffusion Galactic model The numerical diffusive propagation code GALPROP [259, 260] is used to reproduce the Galactic spectrum from SNRs. The diffusive model assumes cylindrical

75 76 CHAPTER 4. THE TRANSITION REGION: SPECTRUM symmetry in the Galaxy, with coordinates R and z equal to the Galactocentric radius and to the distance from the Galactic plane, respectively. The propagation region is, in cylindrical coordinates, bounded by R = Rh = 30 kpc and z = zh =4 kpc, beyond which free escape is assumed. The diffusion equation is:| | ∂ψ ∂ 1 1 = q(~r, p)+ ~ (Dxx ~ ψ) (pψ ˙ ) ψ ψ (4.1) ∂t ∇× ∇ − ∂p − τf − τr where ψ(~r, p, t) is the density per unit of total particle momentum, q(~r, p) is the source term, Dxx is the spatial diffusion coefficient,p ˙ = dp/dt is the momentum loss rate and τf and τr are the time scale of fragmentation and the time scale of radioactive decay respectively. The GALPROP code solves the diffusion equation for all cosmic-ray species starting from the heaviest nucleus and then proceeding to lighter nuclei using the computed secondary source functions. The numerical solution of the transport equation is based on a Crank-Nicholson [261, 259] implicit second-order scheme. Nuclear fragmentation and production of secondary isotopes are included in the code. All kind of reactions producing stable or radioactive intermediate states are considered. The code uses cross-section measurements and energy dependent fitting functions. Cross section are calculated using different algorithms, modern nuclear codes and phenomenological approximations, described in [262, 263, 260, 264, 265, 266].

δ The diffusion coefficient is taken as βD0(ρ/ρD) , where ρ is the particle rigidity, D0 is the diffusion coefficient at a reference rigidity ρD and δ = 0.6. The diffusion coefficient can be inferred from the abundances of light nuclei, produced mainly through spallation of heavy elements, as Li, Be and B, which give an estimation of the time of residence of CRs in the galaxy of 1.5 107 yr [267]. ≈ × 28 2 1 In this calculation the diffusion coefficient is taken as D =5.75 10 cm s− 0 × at the reference rigidity ρD =4 GV . 0.6 The assumption of a diffusion coefficient with an energy dependence E− is not universally accepted [268]. In fact, the turbulence in the interstellar medium seems to follow closely a Kolmogorov spectrum, which should lead to an energy dependence of E1/3 for the diffusion coefficient. In this scenario, the difference between primary and secondary cosmic ray energy spectra could be explained by the trapping of primary cosmic rays in high density regions with Kolmogorov turbulence near their acceleration sites, where secondary CRs would be mainly produced, and the diffusion of the latter inside this region and the interstellar medium. However, since at high energy both models produce similar spectra and composition profiles, we think our analysis is to a large extent independent of these assumptions.

The assumed distribution of cosmic rays sources is the one reproducing the cosmic-ray distribution determined by the analysis of EGRET gamma-ray data, which has the same parameterization of the R-dependence as that used for SNRs [259]:

f(R,z)= f(R)exp( z /z ) (4.2) −| | scale where zscale =0.2 kpc and: 4.1. SNR GALACTIC SPECTRUM 77

19.3 R 4 kpc, 21.9 4 kpc≤ < R 8 kpc,  15.8 8 kpc < R ≤ 10 kpc, f(R)=  ≤  18.3 10 kpc < R 12 kpc,   13.3 12 kpc < R ≤ 15 kpc, ≤  7.4 R> 15 kpc.   This choice is due to the fact that the SNR distribution [269] produces CR distribution distinctly diﬀerent from the measurements. A solution to the apparent discrepancy between the radial gradient in the diﬀuse Galactic gamma- ray emissivity and the distribution of SNRs was proposed by Strong et al. [270].

According to shock acceleration models [271, 272], the injection spectrum is a power law function in rigidity with a break at rigidity ρ0, beyond which it falls exponentially with a rigidity scale ρc:

α ρ − ρ<ρ0, I(ρ)= ρ0  exp ρ 1 / ρc ) ρ>ρ ,  − ρ0 − ρ0 0 h i where ρ0 = 1.8 P V , ρc = 1.26 P V and α = 2.05, which is the case of strong shock waves (M >> 1) [273, 274]. Stable nuclei with Z 26 are injected, with energy independent isotopic abundances derived from low≤ energy CR measurements [260, 275, 276, 277, 278, 279, 280].

The interstellar hydrogen distribution, molecular, atomic and ionized (H2, HI, HII), is derived from radio HI and CO surveys in 9 Galactocentric rings and information on the ionized component [264] (Fig. 4.1). The helium fraction of the gas is taken equal to 0.11. The distribution of molecular hydrogen is derived indirectly from CO radio- emission. The assumption that the conversion factor H2/CO is the same for the whole Galaxy [281] is replaced with a conversion factor that assumes diﬀerent values in each Galactocentric ring (but constant inside it) [270]. The atomic hydrogen (HI) distribution is taken from [282], with a z-dependence calculated using two approximation at diﬀerent Galactocentric distance R [283, 284]. The ionized component HII is calculated using a cylindrically symmetrical model [285].

The Interstellar Radiation Field (ISRF) is calculated using emissivities based on stellar populations and dust emission [286, 287]. Data of COBE/DIRBE [288, 289, 290, 291, 292] are used for the infrared emissivities per atom of HI and H2, for spectral shape and for distribution of the old stellar disk component. Local stellar emissivity is calculated using stellar luminosity function, local stellar densities and absolute magnitudes taken from [293]. In Fig. 4.2 the energy density distributions of the Optical, far-infrared and microwave components of the ISRF are shown. The Optical (OR) and far- infrared (FIR) components are shown for diﬀerent altitudes z above the galactic plane (0, 1, 2, 3, 4 kpc). 78 CHAPTER 4. THE TRANSITION REGION: SPECTRUM

Figure 4.1: Distribution of atomic (HI), ionized (HII) and molecular (H2) hydrogen as a function of the Galactocentric distance R. The curves show the distributions at diﬀerent altitudes z above the galactic plane (0.0, 0.1, 0.2 kpc).

Figure 4.2: Distribution of the ISRF energy density for the optical (OR), far- infrared (FIR) and microwave (CMB) components as a function of the Galacto- centric distance; the optical and far-infrared components are shown for diﬀerent altitudes z above the galactic plane (0, 1, 2, 3, 4 kpc). 4.1. SNR GALACTIC SPECTRUM 79

Figure 4.3: Mean value of the Galactic turbulent magnetic ﬁeld component perpendicular to the CR propagation. The curves show the mean value as a function of the Galactocentric distance R at diﬀerent altitudes z above the galactic plane.

The Galactic turbulent magnetic ﬁeld is assumed to have a mean value of the component perpendicular to the CR propagation given by

R R z B = B exp − 0 exp | | , (4.3) 0 − R −z scale scale where B0 = 6 µG, Rscale = 10 kpc, zscale = 2 kpc and R0 = 8.5 kpc the Sun Galactocentric distance (Fig. 4.3).

4.1.2 Diffusive Galactic spectrum The calculated diffusive Galactic spectrum is shown in Fig. 4.4 superimposed to several experimental data results from the air shower experiments (HiRes [22], HiResMIA [8], Fly’s Eye [33], KASCADE [3], Yakutsk [9], Akeno [5, 135], AGASA [13], Haverah Park [120, 12], Auger [34, 165], BLANCA [179], Tibet [180, 181], Mt. Norikura [182]) as well as from direct measurements (SOKOL [173], JACEE [174, 175], Grigorov [176], Proton Satellite [177], Runjob [178]). Besides the calculated total Galactic spectrum, the different Z-grouped nuclei components are shown. It can be seen that there is considerable dispersion over the whole energy interval, which highlights the inherent difficulty in CR spectral measurements over so many decades in energy. Furthermore, it can be seen that, below the first knee the dispersion is mainly limited to flux normalization, while the spectral indexes seem to be fairly consistent for different experiments. The situation worsens at 80 CHAPTER 4. THE TRANSITION REGION: SPECTRUM

Figure 4.4: Diffusive Galactic spectrum, ΦG. The different Z-grouped nuclei components are shown. higher energies until large discrepancies are apparent above 1017 eV. However, even at these high energies, as seen in 3.1.1, a careful renormalization in energy [23] seems to be able to reconcile the§ several experimental data sets, giving a rather clear and consistent picture for the ankle. Unfortunately, normalization flux differences among spectra still remain between major experiments after the previous energy scale shifting, and they represent a concern to be experimentally addressed.

The diffusive Galactic spectrum ΦG is normalized to match KASCADE data at 3 106 GeV [3]. At this energy the differential flux value of the all particle ∼energy× spectrum and its uncertainty are the same for QGSJet 01 and Sibyll 2.1 based analysis of the KASCADE experiment. While, with this renormalization, our spectrum agrees with data of various experiments at lower energies (JACEE, SOKOL, Tibet, KASCADE, Haverah Park and Akeno), beyond the knee the diffusive spectrum presents a strong deficit of flux (Fig. 4.4).

Since at E > 107 GeV the composition is dominated by intermediate (Z :6 12, i.e the CNO group) and heavier (Z : 19 26) nuclei, we renormalize these− CNO H − components (ΦG , ΦG ) in order to resolve the flux difference between the calculated spectrum and experimental data. In Fig. 4.5 the unmodified total galactic spectrum ΦG (solid red line) is shown together with the spectra obtained with different values of renormalization of the heavier nuclei, while the CNO group renormalization is kept constant (dashed lines). A renormalization of the two groups by a factor 2 produces a good agreement with the experimental data (see Fig. 4.5).

The renormalization of the CNO group and of the heavy nuclei group by a factor of 2 is acceptable since it corresponds to a renormalization of the injected abundances into the acceleration mechanism. The CNO nuclei of Galactic CRs are over abundant with respect to Iron by several orders of magnitude (Fig. 4.6). Therefore, the Iron contribution to CNO ﬂux resulting from spallation is small. 4.1. SNR GALACTIC SPECTRUM 81

Figure 4.5: Diﬀusive Galactic spectrum, ΦG, with renormalization of the CNO group (Z : 6 12), ΦCNO, and of the heavy component (Z : 19 26), ΦH . − G − G Agreement with experimental data is obtained for k′ 1 and k 0.8 1.2. ≈ ≈ −

Figure 4.6: The cosmic ray elemental abundances measured at Earth compared to the solar system abundances, all relative to Silicon. Adapted from [294]. 82 CHAPTER 4. THE TRANSITION REGION: SPECTRUM

Consequently, to a good approximation, the observed Galactic abundances of Iron and of the CNO group at the Solar circle can be varied independently by modifying their relative abundances at injection. This amount of relative renormalization is well inside the present uncertainties regarding the detailed workings of the injection mechanism.

The renormalized diﬀusive Galactic spectrum reproduces well the data up to E 108 GeV , beyond which the spectrum falls steeply because of the end of the≈ SNR acceleration eﬃciency.

4.2 Extragalactic spectrum

In order to study how the transition between the Galactic and extragalactic components takes place, the Galactic spectrum originated in SNRs is compared with two diﬀerent possible scenarios for the extragalactic component. The two cases are chosen in the context of two diﬀerent transition models, the dip model by Berezinsky [21] and the mixed composition model by Allard [39].

In the first model [21], a pure proton extragalactic spectrum is considered. The author analyzes different cases of EG proton spectrum, resulting in a model dependent GZK shape. The shape of the GZK feature is affected by many parameters as variation of the maximum acceleration energy, cosmological evolution of sources, distance and distribution of the sources. In the case considered, the author calculates a pure proton extragalactic spectrum, accelerated by a homogeneous distribution of cosmic sources. The power law generation spectrum index is 2.7 and the parameter describing the cosmological evolution of the sources is taken as m = 0 (non-evolutionary case). The spectrum is calculated without a maximum acceleration energy limit. Within this model, Berezinsky considers different cases of local overdensity/deficit of sources (Fig. 4.7(a)):

1. universal spectrum: n/n0 = 1;

2. overdensity of sources: n/n0 = 2, R = 30 Mpc;

3. overdensity of sources: n/n0 = 3, R = 30 Mpc;

4. deﬁcit of sources: n/n0 = 0, R = 10 Mpc;

5. deficit of sources: n/n0 = 0, R = 30 Mpc; where n0 is the mean extragalactic source density and n is the local overdensity/deficit in regions of size R. Local overdensities/deficits of UHECR sources affect the shape of the GZK modulation, but do not affect the low energy region where the matching with the Galactic spectrum occurs (see Fig. 4.7(a)).

In the second model [39], the extragalactic spectrum is calculated for a mixed composition at injection typical of low energy cosmic rays and a uniform distribution of sources. The maximum acceleration energy of a nucleus i, with charge Z , is taken as Ei = Z Ep , where Ep = 1020.5 eV is the maximum i max i × max max 4.2. EG SPECTRUM 83

(a) EG proton model

(b) EG mixed composition model

Figure 4.7: Extragalactic proton (a) and mixed composition (b) models. Figure (a) shows the pure proton EG spectrum calculated by Berezinsky [21] for different cases of local overdensity/deﬁcit of sources. Figure (b) shows the mixed composition EG spectrum calculated by Allard [39] for diﬀerent source evolution models. Spectra are normalized to HiRes data (see text for details). 84 CHAPTER 4. THE TRANSITION REGION: SPECTRUM proton acceleration energy. Assuming a power law source spectrum of index β, the number of nuclei i injected with energy [E, E + dE] is given by:

β 1 β ni(E)dE = xiAi − kE− dE (4.4) with k a normalization constant, Ai the atomic weight and xi parameters describing the composition at the source. The source spectral index β is determined ﬁtting the high energy CR data. Three diﬀerent source evolution models in red shift are considered by Allard (Fig. 4.7(b)):

a) strong evolution model: the injection rate is proportional to (1 + z)4 for z < 1 and constant for 1

b) SFR model: the EGCR injection power is proportional to the star formation rate which correspond to a redshift evolution (1 + z)3 for z < 1.3 and a constant injection rate for 1.3

c) uniform source distribution model: no evolution, β =2.3.

In both cases, the various parameters of the models are tuned to ﬁt the available CR data at UHE and are, in that highest energy regime, experimentally indistinguishable at present. The spectra of both the models, used in the next sections, are renormalized to HiRes data.

4.3 Combined spectrum: matching Galactic and extragalactic components

It has been seen in section 3.1.2 (Fig. 3.7) that the energy requirement of the inferred Galactic component§ varies in function of the source evolution assumed in the case of an EG pure proton model, while for the EG mixed composition model is quite independent on the source evolution. The calculated diffusive Galactic spectrum falls very steeply at energies above 5 1016 eV , while the EG components have a low energy cut-off due to the magnetic× horizon at energy 1017 eV . It is obvious, that the two EG spectra (especially in the case of the∼ mixed composition model) and the calculated Galactic spectrum would not match the observed total spectrum. In order to study how the transition between the Galactic and extragalactic components takes place, the combined theoretical (Galactic ΦG plus extragalactic ΦEG) spectrum is subtracted from the available data. Two different approaches are used.

First, we try to match the experimental data by varying the normalization of H the heavy Galactic component ΦG , while keeping constant the previous renor- CNO malization of the CNO group ΦG ( 4.1.2). The best reproduced spectrum for the two extragalactic models are shown§ in Figs. 4.8, 4.9 and 4.10. In the case of the EG proton model the CNO group is renormalized by a factor 2.2 while the renormalization factor of the heavier nuclei is varied between 1.8 and 2.2 (Figs. 4.3. COMBINED SPECTRUM 85

Figure 4.8: Galactic and extragalactic spectrum matching for the proton model for an EG lower energy limit of 108 GeV . The sum of the renormalized diffusive Galactic spectrum and of the extragalactic spectrum (ΦEG) for different cases of local overdensity/deficit of sources ( 4.2) is shown for different renormalizations H § CNO of the heavy component (ΦG ). The CNO group (ΦG ) of the diffusive Galactic spectrum (ΦG) has been renormalized by a factor 2.2.

4.8, 4.9). In the case of the EG mixed composition model, the CNO group is renormalized by a factor 2 while the renormalization factor of the heavier nuclei varies between 2.0 and 2.4 (Fig. 4.10).

In the case of the proton model, a discontinuity appears when the two spectra are added, regardless of the lower limit adopted for the extragalactic component: 108 GeV (Fig. 4.8) or 5 107 GeV (Fig. 4.9). The latter corresponds to cosmic accelerators operating for× the entire Hubble time. For both, proton and mixed-composition models, there is a flux deficit above 108 GeV. The problem is much stronger for the mixed-composition model where, regardless of the luminosity evolution of EG CR sources, the total spectrum presents a large deficit of flux between 108 GeV and 3 109 GeV (Fig. 4.10). ≈ × In order to solve this flux deficit, the only way out seems to be the introduction of an additional Galactic component. This component is estimated by subtracting the sum of the diffusive Galactic and extragalactic fluxes from a smooth fit to the world data. This method confirms us the need of the renormalization of the CNO group by a factor 2.2 and 2 for the proton and mixed-composition models, respectively. In≈ the case of≈ the proton models (Figs. 4.11, 4.12), the observed deficit can be resolved with the introduction of an additional cosmic ray flux component (the first additional Galactic component, GA1) ΦGA1. For the mixed-composition models this is not enough and one more additional cosmic ray flux component must be introduced (the second additional Galactic component, GA2) ΦGA2 (Fig. 4.13). The additional component ΦGA1 common to both families of models is obtained with a shift in energy of a factor 3 of the diffusive Galactic heavy component, renormalized by a factor 0.8≈ in the case of the mixed-composition models and . 0.6 for the proton models≈ respectively. 86 CHAPTER 4. THE TRANSITION REGION: SPECTRUM

Figure 4.9: Idem to Fig. 4.8 but for the EG proton model with a lower energy limit of 5 107 GeV. ×

Figure 4.10: Idem to Fig. 4.9 but for the Galactic and extragalactic spectrum CNO for the mixed-composition model. The CNO group (ΦG ) of the diﬀusive Galactic spectrum (ΦG) has been renormalized by a factor 2. Three source evolution models are considered ( 4.2): (a) strong, (b) SFR and (c) uniform. § 4.4. DISCUSSION 87

Figure 4.11: Galactic and extragalactic spectrum for the proton model for an 8 tot EG lower limit of 10 GeV : the total galactic spectrum (φG ) and the total tot i spectrum (φG + φEG) for diﬀerent cases i of local overdensity/deﬁcit of sources (see 4.2 for details) are shown. The additional Galactic component φGA1 and § H the total high energy Galactic component (φG + φGA1) are also shown.

The second additional component ΦGA2 is obtained in an analogous way but with an energy-shift factor of 10 and a renormalization by a factor 0.2. The corresponding spectra are shown≈ in Figs. 4.11, 4.12 and 4.13. In each≈ ﬁgure, the Galactic diﬀusive spectrum (ΦG), each additional Galactic component (ΦGA1, tot tot ΦGA2), the total Galactic spectrum (ΦG =ΦG +ΦGA1 or ΦG =ΦG +ΦGA1 + tot ΦGA2) and the total spectrum (ΦG +ΦEG) are shown. The sum of the Galactic H H heavier nuclei component (ΦG ) and the additional Galactic components (ΦG + H ΦGA1 or ΦG +ΦGA1 +ΦGA2) is also shown.

4.4 Discussion on the spectrum

The matching conditions of the Galactic and extragalactic components of cosmic rays along the second knee and the ankle have been analyzed in two different extragalactic scenarios. It seems clear that an acceptable matching of the Galactic and extragalactic fluxes can only be achieved if the Galaxy has additional accelerators, besides the fiducial SNRs assumed here, operating in the interstellar medium. In the particular case of the proton model, only one additional component is required. This could well represent the contribution from compact and highly magnetized SNRs, like those occurring in the central, high density regions of the Galactic bulge, inside the dense cores of molecular clouds or those expanding into the circumstellar winds of their progenitors. Actually, this component does not need to originate in a particular type of source in itself, but could be the result of a non-homogeneous SNR population drawn from a spectrum of progenitor masses and evolving in different environments corresponding to the various gas phases that fill the interstellar medium [295]. From the point of view of CR luminosity this should not be a problem, since the Fe component and the 88 CHAPTER 4. THE TRANSITION REGION: SPECTRUM

Figure 4.12: Idem to Fig. 4.11 but for the EG proton model with a lower energy 7 tot limit of 5 10 GeV : the total galactic spectrum (φG ) and the total spectrum (φtot + φi ×) for diﬀerent cases i of local overdensity/deﬁcit of sources (see 4.2 G EG § for details) are shown. The additional Galactic component φGA1 and the total H high energy Galactic component (φG + φGA1) are also shown.

Figure 4.13: Galactic and extragalactic spectrum for the mixed-composition tot tot i model: the total galactic spectrum (φG ) and the total spectrum (φG + φEG) for different cases i of source evolution models (see 4.2 for details) are shown. § The two additional components φGA1, φGA2 and the total high energy Galactic H component (φG + φGA1 + φGA2) are also shown. 4.4. DISCUSSION 89 additional high energy component amount to 9% and . 1%, respectively, of the total Galactic diffusive spectrum. ∼ In the specific case of the proton model, it is apparent that a perfectly smooth match between the Galactic and the extragalactic spectra is not possible. Some discontinuity or wiggle seems unavoidable in the energy spectrum between 3 1016 and 1017 eV (see figures 4.8, 4.9, 4.11 and 4.12). This is due to the∼ fact× that the∼ extragalactic spectrum should have a rather abrupt low energy cut-off due to the finite distance to the nearest extragalactic sources and their limited age. Of course, whether such spectral feature is actually observable is strongly dependent in practice on the magnitude of the experimental errors in the determination of the primary energy from shower measurements and on the available statistics.

The matching of the mixed-composition model, on the other hand, has wider astrophysical implications for the Galaxy. The Galactic CR production has to be extended up to at least the middle of the dip and this requires, besides the previous additional component, another high energy Galactic component. The origin of these cosmic rays pushes even further the acceleration requirements imposed on the Galaxy. It is likely that a different population of Galactic accelerators must be invoked. Viable candidates could include, for example, rapidly spinning inductors, like pulsars or magnetars, or even very high energy episodic events like Galactic gamma ray bursts [296]. If this were the case, it is very likely that photon emission at TeV energies should uncover the sources. Changes in propagation regime inside the Galaxy at these high energies should manifest as an anisotropic component embedded in the isotropic incoming extragalactic background. Again, considerable statistics might be necessary to make such effect observable. The energy requirements involved in the production of the second additional 7 34 component are rather modest: 10− of the SNR CR component, or 10 erg/sec pumped into particles between∼ 1017 and 1018 eV. Therefore,∼ few, or even a single source, could be responsible∼ for this∼ component. This carries attached the potential problem of undesirable fine-tuning because of the requirement that, at this precise moment in time, the CR luminosity of these few (or this single) sources is such that it allows for a smooth spectral matching along the ankle.

It is important to note that the region between some few times 1017 eV and approximately 1019 eV is a transition region from the point of view of propagation [4, 257]. In fact, Larmor radius (in pc) of a CR nucleus of charge Z can be conveniently parametrized as:

10 E r EeV (4.5) L,pc ∼ Z × B µG where EEeV is its energy and BµG is some appropriate average of the Galactic magnetic ﬁeld inside the propagation region. Since the transversal dimensions of the Galactic disk are on the order of a few times 102 pc, we see that the diﬀusion approximation for protons should start to be broken somewhere between the second knee and 1 EeV. The same should happen to heavier nuclei at progressively higher total energies: 3 EeV for the CNO group and 10 ∼ ∼ 90 CHAPTER 4. THE TRANSITION REGION: SPECTRUM

EeV for Fe. The corresponding transition for each nuclei should be gradual, with the propagation eventually becoming ballistic at higher energies. This transition results in a complicated picture in which the end of the Galactic confinement spans almost 1.5 decade in energy, depending on the particular nuclei considered. Our interpretation of the additional Galactic components (but not their necessity) is affected to some extent by the implicit assumption that the particles propagate diffusively. This assumption is increasingly questionable for protons as we approach 1 EeV, but should be valid for heavier nuclei in the energy range considered here. A work on a detailed analysis of these effect is in progress. Chapter 5

The transition region: composition analysis

In the previous chapter ( 4) different scenarios for the combination of the Galac- tic flux with alternative extragalactic§ models have been analyzed from the point of view of the energy spectrum. As seen in 4.3, an acceptable matching of the Galactic and extragalactic fluxes can only be§ achieved if the Galaxy has additional accelerators besides the regular SNRs assumed in 4.1. In the case of pure proton EG spectrum, this can be achieved by one additional§ Galactic component, while in the mixed-composition EG spectrum two additional component are required, as the Galactic spectrum has to extend up to middle of the ankle. Nevertheless the different extragalactic models are able to produce total spectra that are indistinguishable within the current experimental resolution. The composition of UHECRs is essential to understand the transition between Galactic and extragalactic cosmic rays and to discriminate the different astrophysical scenarios. In this chapter the transition region is analyzed from the composition point of view, using one of the most reliable parameter, the atmospheric depth of the maximum longitudinal development of a shower (Xmax). In the following sections different parameterizations of Xmax , deduced from the hadronic interaction models currently in use, will beh usedi in order to infer the composition energy profile for the different cases of Galactic-extragalactic combined spectrum described in section 4.3. An analysis of the composition§ energy profiles of the extragalactic spectrum and of the two additional Galactic components (determined in the previous chapter) implied by Auger and HiRes Xmax data is performed.

5.1 Xmax and hadronic interaction models Since UHECR experiments do not directly measure the composition, the primary CR composition has to be to inferred from parameters characterizing the shower development proﬁle. One of the most reliable parameters is Xmax, the atmospheric depth of the maximum longitudinal development of a shower [171]. The variation of Xmax with energy gives information about the change in composition of theh CR ﬂux.i

91 92 CHAPTER 5. THE TRANSITION REGION: COMPOSITION

Table 5.1: Coeﬃcients of the parameterization of Xmax as a function of primary energy and composition (eq. (5.1)) for diﬀerenth hadroni ic interaction models. Model a b c p α β QGSJetII-v03 −0.033 −0.010 58.313 184.315 9.60 × 10−4 −2.41 × 10−3 QGSJetII-v02 0.184 −5.930 115.111 −15.895 2.00 × 10−3 6.50 × 10−2 QGSJet01 0.074 −3.695 103.804 −19.123 6.85 × 10−4 2.14 × 10−2 EPOS 1.6 −0.011 1.831 29.212 265.847 −1.47 × 10−4 4.69 × 10−4 Sibyll 2.1 0.300 −8.422 134.960 −72.051 3.38 × 10−4 2.62 × 10−3

Diﬀerent hadronic interaction models (HIMs) can be used to interpret the Xmax dependence on energy and on primary composition, and this is the main uncertainty associated with this parameter. The composition predictions depend on the HIM used. For the sake of completeness, all the HIMs currently used in the literature are taken into account in what follows:

EPOS 1.6, • QGSJet 01, • QGSJetII v02, • QGSJetII v03, • Sibyll 2.1. •

For each hadronic interaction model, a parameterization of Xmax as a function of the primary energy and composition is calculated: h i

E 2 E X (E,A,i) = a Log + b Log + c (1 + α A) (5.1) h maxi i Aǫ i Aǫ i i " # +pi(1 + βiA), where E is the primary energy, A is the atomic mass of the primary CR, i is the hadronic interaction model, ǫ 81 MeV is the critical energy in air and ∼ ai, bi, ci, pi, αi, βi are coefficients determined for each HIM by fitting equation (5.1) to shower simulations (see Table 5.1). Parameterization (5.1) is a good approximation in the energy range [1017 eV , 1020 eV ]. Our parameterization for p and Fe, for several HIMs, is shown in Fig. 5.1 together with the experimental results of HiRes [189], HiResMIA [8], Fly’s Eye [6] and Auger [167]. This parameterization allows us to compute the average Xmax for showers generated by any nucleus of interest and for all observed primary energies above the second knee. With this parameterization, in the following sections we reproduce Xmax energy profiles for different composition scenarios in the context ofh the variousi cases of the combined Galactic-extragalactic spectrum described in 4.3. § 5.2. XMAX ENERGY PROFILES 93

Figure 5.1: X parameterization as a function of primary energy and com- h maxi position. For each HIM, the Xmax dependence on energy for proton and Iron primary is shown. Experimentalh datai from HiRes [189], HiResMIA [8], Fly’s Eye [6] and Auger [167] are also shown.

5.2 Galactic-extragalactic combined spectra: Xmax energy proﬁles

Heretofore, different scenarios for the combination of the Galactic flux with alternative extragalactic models have been analyzed from the point of view of the energy spectrum. For each of these scenarios, and under different assumptions for the composition of individual flux components, the Xmax energy profile along the transition region is estimated. As seen in 4.3, an acceptable matching of the Galactic and extragalactic fluxes can only be§ achieved if the Galaxy has additional accelerators besides the regular SNRs assumed in 4.1.1. This can be achieved by including either one (pure proton EG spectrum)§ or two (mixed-composition EG spectrum) additional Galactic components. In order to infer Xmax for the combined flux, some assumptions on the composition of the additional Galactic components have to be made. The first additional Galactic component (GA1), which is needed in both of the EG scenarios previously considered, is probably contributed by compact, highly magnetized SNRs. Consequently this component is likely to be dominated by heavy elements, say, Iron, at the highest energies. The second additional Galactic component (GA2), on the other hand, might be associated with a minor population of still more powerful SNR accelerators or with a completely different population of particle accelerators, e.g., rapidly spinning inductors, like pulsars or magnetars. In the first case, GA2 would 94 CHAPTER 5. THE TRANSITION REGION: COMPOSITION

Figure 5.2: Mixed-composition EG scenario of Allard and co-workers. Con- tribution to the extragalactic flux of different nuclei in the “uniform source distribution model”. (Adapted from: [41]). be Fe dominated, while in the latter case, GA2 could be proton dominated. Consequently, two different possibilities for GA2 are considered: pure proton and pure Fe. The Xmax values as a function of energy are calculated in the energy range [1017 eV, 1020 eV ] for all the possible combination of Galactic and extragalactic spectra previously discussed in 4.3 from the point of view of the shape of the energy spectrum1: § pure proton EG model: the combination of the calculated Galactic spec- • trum from SNRs with the EG “universal” (see 4.2) spectrum in the two cases of lower EG energy limit (5 1016 eV and§ 1017 eV ) is considered; the composition of the Galactic× additional component∼ is assumed to be heavy (Iron); mixed EG composition model: the combination of the calculated Galactic • spectrum from SNRs with the mixed EG spectrum in its “uniform source distribution model” variant (see 4.2) is considered; GA1 is assumed to be composed by pure Fe, while two§ opposite scenarios for GA2 are analyzed: pure proton and pure Fe. The EG composition is taken from Fig. 5.2 [41]. For all these scenarios with different component spectra and composition as- 17 20 sumptions, Xmax values are calculated in the energy range [10 eV, 10 eV ] using the parameterizationh i given in 5.1 for the various HIMs in current use. § 1For the diffusive Galactic spectrum from SNRs we kept the renormalization of the CNO group by a factor ∼ 2 determined in the previous analysis (§4.3) 5.2. XMAX ENERGY PROFILES 95

Figure 5.3: Xmax, EPOS 1.6: the Xmax proﬁles calculated for all the possible combined spectra are shown superimposed on the experimental data. The curves represent: (i) the combined spectra for an EG proton model with lower energy limit 1017 eV (cyan blue) and 5 1017 eV (green) respectively; (ii) the combined∼ spectra for an EG mixed-composition× model with pure proton (purple) and pure Fe (brown-orange) GA2. Xmax values are calculated using the EPOS 1.6 HIM. h i 96 CHAPTER 5. THE TRANSITION REGION: COMPOSITION

Figure 5.4: Xmax, QGSJet01: idem to Fig. 5.3 but using the QGSJet01 HIM.

Figure 5.5: Xmax, QGSJetII v02: idem to Fig. 5.3 but using the QGSJetII v02 HIM. 5.2. XMAX ENERGY PROFILES 97

Figure 5.6: Xmax, QGSJetII v03: idem to Fig. 5.3 but using the QGSJetII v03 HIM.

Figure 5.7: Xmax, Sibyll 2.1: idem to Fig. 5.3 but using the Sibyll 2.1 HIM. 98 CHAPTER 5. THE TRANSITION REGION: COMPOSITION

The results are shown for each HIM in Figs. 5.3 (EPOS 1.6), 5.4 (QGSJet01), 5.5 (QGSJetII v02), 5.6 (GQSJetII v03) and 5.7 (Sibyll 2.1), together with the experimental results of HiRes [189], HiResMIA [8], Fly’s Eye [6] and Auger [167].

First of all we notice that, as expected, for all the HIMs the theoretical Xmax values in the case of EG proton model are indistinguishable for the two different EG lower energy limits (green and cyan blue curves are superimposed on the figures). On the other hand, the model is very much compatible with HiRes- 18 MIA data below 10 eV while, at higher energies, Xmax is too large when compared to any of the available data sets. h i As expected, much more variability in the calculated elongation rates is observed for the lower energy regime in the case of EG mixed-composition models, depending on the assumptions made about the composition of GA2 (purple and brown-orange curves). However, at energies larger than 3 1018 eV, all the solutions converge to the same profile since the effects of∼ GA2× become progressively negligible on the combined composition profile. All these results are qualitatively independent of the assumed HIM. Nevertheless, even if all the HIMs give basically the same trend at all energies, actually what data set is quantitatively compatible with the mixed composition theoretical models at energies beyond 3 1018 eV does depend on the assumed HIM. Furthermore, at the highest energies,× E > 1019 eV, there could be an indication that the models give a systematically lighter composition than what the data suggests. Unfor- tunately, at present, statistics at these energies are too low for all experiments as to render any solid conclusion. The most critical energy region for the understanding of the Galactic-extragalactic transition and disentangling its flux components, is 1017 - 3 1018 eV. Inside this region, different experimental results, and the∼y are different∼ × indeed, imply very different astrophysical scenarios, in particular with regard to the highest accelerators present in our own Galaxy. In the following sections we will center on two of the most important experimental results today, those of Auger and HiRes, and test what, if any, further modifications should be applied to the composition profile in order to make the data and the theoretical models as compatible as possible.

5.3 Composition evolution along the transition region

If our understanding of Xmax in terms of composition is reasonably correct, Auger and HiRes elongationh ratei data suggest a composition profile along the ankle compatible with a mixed extragalactic composition, despite differences in the energy dependence of Xmax for both experiments. The agreement between Auger Xmax data [167] and our predictions is remark- able at energies around 3 1018 eV for all HIMs. This agreement extends to higher energies, even beyond× 2 1019 eV, for QGSJet01 and QGSJetII v02 and v03, while EPOS and SIBYLL× display different trends. The picture is more complicated at lower energy where, under the present assumptions, there is no agreement regardless of the assumed HIM. The situation almost reverses in the case of the HiRes-MIA combined data set, 5.3. COMPOSITION EVOLUTION 99 for which a good agreement can be obtained at low energies with the EG proton model, but there is no clear fit at higher energies. It must be noted, however, that the composition profiles shown in Figs. 5.3, 5.4, 5.5, 5.6 and 5.7, result simply from the combination of our previous solution to the total energy spectrum for the mixed model and the unmodified mixed model compositions as determined by [41], plus fixed compositions for GA1 and GA2. Therefore, in this section we analyze whether the composition profile of the two additional Galactic components and of the extragalactic flux can be modified in a suitable way in order to satisfy, simultaneously, the existing spectral and elongation rate data along the transition region. We use to this end a simplified model in which the shape of the extragalactic energy spectrum is kept, to first order, identical to Allard’s mixed composition model [39], but the composition is limited to just two nuclei, proton and Iron, whose relative abundances can change appropriately in order to reproduce the behavior of Xmax as a function of energy. The admitted lack of consistency in this approachh is,i we believe, compensated by the insight gained into the phenomenological constraints imposed by the data on the astrophysical models at play2. On the other hand, the spectral shapes of both additional components and their normalizations are preserved, but their compositions as a function of energy are now functions to be determined during the fitting process. A binary mixture of p and Fe is also assumed. It is further assumed, a priori, that the diffusive Galactic spectrum and its composition are the ones determined previously in 4.3 with the calculated renormalization of the CNO group. The diffusive Galactic§ flux from SNRs, spectrum as well as composition, is kept constant afterward during the fitting procedure. This procedure is applied to both, Auger and HiRes data in order to gain an insight on how present experimental uncertainties can affect our astrophysical understanding of the transition region.

5.3.1 Auger data Fig. 5.8 shows the total spectrum and Galactic components that are used in this section. For each source evolution model (see 4.2), the EG spectrum is normalized to the surface detector (SD) Auger flux§ at 1019.05 eV [165], where one can expect a negligible contribution from the Galactic CR flux. Afterward, the GA1 and GA2 components are renormalized in order to match the total spectrum with the Auger spectrum data; the hybrid Auger spectrum at lower energy, 1018 1018.3 eV [34], is used to this end. The lowest energy branch of the ankle is approximated− by a smooth interpolation between the highest energy KASCADE spectrum and the lowest energy hybrid Auger data. It must be noted that, regardless of the renormalization, the extragalactic theoretical spectral models are too soft to allow a good fit to the Auger data at the highest energies. This inconsistency have also effects at lower energies, where the position of the ankle is artificially pushed down from the position determined by Auger at 1018.6 eV. Therefore, as a compromise, we consider here the model ∼ 2An alternative analysis to the spectral fit is the use of slanted shower data which, for Akeno data was performed by [295]. 100 CHAPTER 5. THE TRANSITION REGION: COMPOSITION

tot a,b,c Figure 5.8: Combined total spectrum ΦG +ΦEG renormalized to Auger spectrum data [165, 34] for different EG evolution models ( 4.2). The Galactic § spectrum ΦG, the Galactic additional components ΦGA1, ΦGA2 and the total tot Galactic spectrum ΦG are also shown. The normalization of the second Galac- tic additional component is the one determined for the EG “uniform source distribution” model. See the text for details. corresponding to a “uniform source distribution”, which is the hardest one and provides the best visual fit to the highest energy data. From the estimated combined Galactic-extragalactic total spectrum and using the Xmax parameterization given in 5.1, eq. (5.1), we compute the relative abundances of proton and Fe nuclei of the§ EG spectrum and of the two additional Galactic components (GA1 and GA2) that match the Auger Xmax data [167]. The EG, GA1 and GA2 compositions are determined in theh energyi range of 17 19 available Auger Xmax data, 3 10 to 5 10 , for all the hadronic interaction models. ∼ × ∼ × The corresponding Iron and proton fluxes for the EG spectrum and for GA1 and GA2 are shown in Fig. 5.9. The different curves for each nucleus correspond to inferences obtained with different HIMs. The corresponding average atomic weights, , for each HIM are shown in Fig. 5.12. 5.3. COMPOSITION EVOLUTION 101

(a) EG composition: proton (left) and Iron (right) components.

(b) GA1 and GA2 composition: proton (left) and Iron (right) components.

Figure 5.9: Auger data. (a) EG proton and Iron composition; (b) GA1 and GA2 proton and Iron compositions. The ﬂuxes are calculated, for each HIM, in order to reproduce the Auger X data [167]. The Galactic additional h maxi components (ΦGA1, ΦGA2), the EG spectrum (ΦEG) and the total spectrum (Φtot = ΦG + ΦGA1 + ΦGA2 + ΦEG) are also shown, as well as the Auger spectrum data [165, 34]. 102 CHAPTER 5. THE TRANSITION REGION: COMPOSITION

tot a,b,c Figure 5.10: Combined total spectrum ΦG + ΦEG renormalized to HiRes spectrum data [22, 8] for diﬀerent EG evolution models ( 4.2). The Galactic § spectrum ΦG, the Galactic additional components ΦGA1, ΦGA2 and the total tot Galactic spectrum ΦG are also shown. The normalizations of the two Galactic additional component are the ones determined for the EG “SFR” model. See the text for details.

5.3.2 HiRes data

In order to assess the astrophysical implications of the present experimental uncertainties in the total spectrum and elongation rate, the same procedure as in the previous section is also applied to the HiRes data [189, 22, 8]. Fig. 5.10 shows the total spectrum, for diﬀerent assumptions regarding the redshift evolution of the sources, and the Galactic components under considera- tion. Since the HiRes spectrum extends to lower energies than those of Auger, in this case both the additional Galactic components are renormalized in order to match the HiResII data in the energy range 1017.3 1018.3 eV (Fig. 5.10). The EG spectra, on the other hand, are already normalized− to HiRes data ( 4.2). § As in the Auger case, the position of the ankle measured by HiRes (6 1018 eV) is not accurately reproduced with any of the combined spectra if a× reasonable agreement at higher energies is required. Consequently, we select for subsequent analysis the EG “SFR” model, which is the one that provides the best visual ﬁt 5.3. COMPOSITION EVOLUTION 103

(a) EG composition: proton (left) and Iron (right) components.

(b) GA1 and GA2 composition: proton (left) and Iron (right) components.

Figure 5.11: HiRes data. (a) EG proton and Iron composition; (b) GA1 and GA2 proton and Iron compositions. The ﬂuxes are calculated for each HIM in order to reproduce the HiRes and HiResMIA X data [189, 8]. The Galactic h maxi additional components (ΦGA1, ΦGA2), the EG spectrum (ΦEG) and the total spectrum (Φtot = ΦG +ΦGA1 +ΦGA2 +ΦEG) are also shown, as well as the HiRes spectrum data [22, 8]. 104 CHAPTER 5. THE TRANSITION REGION: COMPOSITION

Figure 5.12: Average atomic weight of the combined cosmic ray flux: the values were obtained from the calculated EG, GA1 and GA2 compositions in order to reproduce the Auger (Fig. 5.9) and HiRes (Fig. 5.11) Xmax data, under different HIM assumptions. h i to the experimental data. It is assumed, as in 5.3.1, that the composition of the EG spectrum and of the two Galactic additional§ components is a binary mixture of proton and Iron nuclei and compute their relative abundances in order to match the Xmax energy dependence of the HiRes data. h i The compositions as a function of energy of EG, GA1 and GA2 are determined for all the HIM in the energy range of HiRes and HiResMIA Xmax data [189, 8], from 1017 eV to 2 1019 eV . ∼ ∼ × The results for the calculated Iron and proton fluxes for GA1, GA2 and EG are shown in Fig. 5.11. The average atomic weights, A , are shown in Fig. 5.12, for each HIM. h i

5.4 Discussion on composition

In section 5.2 the Xmax dependence on energy was calculated for all the possible scenarios obtained§ by combining the calculated SNR Galactic spectrum with different extragalactic models ( 4.3). Several HIMs were used. § Simple assumptions were made on the composition of the two additional Galac- tic components. The first one, which is required by both EG models, was assumed to be Iron dominated, as it is probably contributed by compact and highly magnetized SNRs. The origin and properties of GA2, required in the case of a mixed composition EG spectrum, are more uncertain. Consequently, as an exploratory test, in section 5.2 two cases for GA2 were considered: a pure proton and a pure Iron composition.§ For all the possible combined spectra, different parameterizations of Xmax obtained for the HIMs used in literature were used to calculate the behaviorh i 17 20 of Xmax in the energy range [10 eV, 10 eV ]. The calculated Xmax energy profiles were compared with the available experimental data from Fly’s Eye, HiRes, HiRes-MIA and Auger. The EG proton model is very much compatible with HiRes-MIA data at energies below 1018 eV for any HIM. However, if the current interpretation of 5.4. DISCUSSION 105

Xmax in terms of shower composition is correct, the later model is completely incompatible with higher energy data which points to a mixed composition. In the case of the EG mixed-composition model, also two energy regimes are clearly distinguishable. At energies above 3 1018 the slope of the elongation rate seems to be consistent with that∼ of the× data up to, possibly, 2 1019 eV. The effect of different HIMs is basically to change the normalization∼ × of the curves, changing at the same time the experimental data set with which the models are more 18 compatible. For energies smaller than 3 10 eV, the calculated Xmax profiles are very dependent on the assumption∼ × made regarding the compositionh i of GA2. A heavy GA2 can fit reasonably well the Fly’s Eye data, while a light GA1 is more consistent with HiRes-MIA in all the energy interval and with Auger between 4 6 1017 eV. Between 6 1017 and 2 1018 eV Auger data is more or less− in-between× both solutions.∼ × These result∼ s× are qualitatively independent of the HIM. Even if the assumption of a pure chemical composition for GA1 and GA2 is artificial, it is clear that the present discrepancies between the various experimental results are of astrophysical significance, since they have quite different implications with respect to the nature of most powerful Galactic accelerators. However, and surprisingly enough, the uncertainty associated with the several HIM currently in use in the literature does not pose a too large qualitative problem from the astrophysical point of view; that is, unless our present understanding of the hadronic interaction processes is not, somehow, fundamentally wrong. In sections 5.3.1 and 5.3.2 we determined the composition, as a function of energy, of the§ extragalactic§ spectrum (EG) and the two additional Galactic components (GA1, GA2) that fits Xmax data of Auger and HiRes, under the simplifying assumption of a binary mixture of p and Fe for the cosmic ray flux. The main result is that, regardless of the experimental data set considered, the composition has to be mixed to some extent along all the spectrum. The assumed HIM plays a very important quantitative role in the inferred composition of any individual component. An important point to note regarding GA1 and GA2 is that, even if they have mixed composition, the composition profile inside each one of them is similar in the sense that the individual fluxes are lighter at lower energies and then become heavier as the energy increases. This is a systematic effect that is quantitatively, but not qualitatively, affected by the HIM used. Profiles such of those we obtained for GA1 and GA2 are similar to what can be expected from different populations of SNRs immersed in differing environments. If this is correct, then probably only SNRs are required in order to explain the main part of the Galactic flux up to energies . 3 1018 eV. In any case, GA2 is lighter than GA1, which could be an evidence for the× existence of a minor contribution coming from a different source like, for example, inductors associated with compact objects. The estimated average atomic weight (see, Fig. 5.12) varies widely depending on the assumed HIM. However, it can be seen that, for a given HIM, HiRes always implies heavier compositions on the Galactic side of the transition region than Auger. Despite this systematic difference, both data sets show the same trend of a decreasing value of A as a function of energy. The opposite happens ath higheri energies on the extragalactic side of the flux, where Auger implies an increasing A as a function of energy regardless of the HIM, and HiRes predicts eitherh ai rather constant or a much more slowly 106 CHAPTER 5. THE TRANSITION REGION: COMPOSITION changing composition. Also, for the latter data set, the uncertainties introduced by the HIM are qualitatively more important, since the high energy slope of the profile can change sign depending on the adopted interaction model. It must be noted that, for both data sets, there seems to be a break in the average energy composition profile at an energy around 1018 eV or slightly higher, which may further highlight the physical associati∼on between the ankle feature in the cosmic ray spectrum an the Galactic-extragalactic flux transition. Chapter 6

Galactic neutrino background from cosmic ray interaction with the ISM content

As a by-product of the diffusive propagation of lower energy CR, the production of a diffuse neutrino background by our Galaxy, due to p-p interactions between cosmic rays and the interstellar medium, has been numerically calculated [297]. The neutrino flux is calculated as the product of the decay of charged pions which, in turn, are created in collisions of cosmic-ray particles with interstellar gas. From the diffusive Galactic spectrum from SNRs calculated in 4.1 using the propagation code GALPROP, we estimated the charged pion pro§duction in interactions of proton and helium CR nuclei with protons and helium of the interstellar medium. From this estimation we calculated the flux of neutrinos coming from the decay of charged pions, in the energy range 1 MeV 1 P eV . −

6.1 Pion production

Pion production in pp-collisions is calculated following a method developed by Dermer [298, 299], which combines isobaric and scaling models of the reaction [300]. The two models work well at low and high energy respectively (for a proton momentum Pp < 3 GeV/c and Pp > 7 GeV/c respectively). In the region between P1 =3 GeV and P2 =7 GeV (corresponding to energy E1, E2), the following linear connection is used:

P P F (E , E ) = [F (E , E ) F (E , E ) ] p − 1 π π p π π 1 isobaric − π π 2 scaling × P P 2 − 1 + Fπ(Eπ, Ep)isobaric (6.1) where Fπ(Eπ, Ep) is the energy distribution of pions of energy Eπ produced by a CR proton of energy Ep and momentum Pp.

107 108 CHAPTER 6. GALACTIC NEUTRINO BACKGROUND

6.1.1 Isobaric model

In the isobaric model [301] it is assumed that the ∆-isobar of mass m∆ produced in pp-collisions has either the same direction (+) or the opposite direction (-) of the colliding proton in the center of mass system (CMS). The Lorentz factors of the forward (+) and backward (-) moving isobars are γ± = γ γ∗ (1 β β∗ ), ∆ c ∆ ± c ∆ where γc = √s/2mp is the Lorentz factor of the CMS in the laboratory system 2 (LS) and γ∗ = (s + m m )/2√sm is the Lorentz factor of the isobar in the ∆ ∆ − p ∆ CMS. The CMS energy is given by √s = 2mp(Ep + mp). p The produced isobar decays isotropically producing a pion with the distribution:

+ + 1 H[γπ; a ,b ] H[γπ; a−,b−] fπ(Eπ, Ep; m∆)= + + + (6.2) 4m γ β × γ β γ−β− π π′ π′ ∆ ∆ ∆ ∆

2 where Eπ and Ep are the pion and proton energy in the LS and γπ′ = (m∆ + 2 2 mπ mp)/(2m∆mπ) is the pion Lorentz factor in the rest frame of the ∆-isobar. The− function H[x; a,b] is deﬁned by

1, a x b H[x; a,b]= 0, otherwise≤ ≤ and a± =γ±γ′ (1 β±β′ ), b± = γ±γ′ (1 + β±β′ ). ∆ π − ∆ π ∆ π ∆ π In this model the distribution of pions is calculated by the integration over the isobar mass spectrum (M = m + m , M = √s m ): 1 p π 2 − p

M2 f (E , E ; m ) F (E , E ) = dm π π p ∆ π π p ∆ (m m0 )2 +Γ2 ZM1 ∆ − ∆ Γ 0 0 (6.3) M2 m M1 m × tan 1 − ∆ tan 1 − ∆ − Γ − − Γ where Eπ, pπ and Ep, p are the pion and proton energy and momentum in the 0 LS, m∆ is the average mass of the ∆-isobar, Γ is the width of the Breight- Wigner distribution and √s is the CMS energy.

6.1.2 Scaling model

The scaling model [302, 303] gives the Lorentz invariant cross section for pion production in pp-collisions as:

3 d σ Q B p E = A G (E )(1 x˜ ) exp · ⊥ (6.4) π d3p · π p − π × −1+4m2/s π p 6.1. PION PRODUCTION 109

where Eπ, pπ and Ep, p are the pion and proton energy and momentum in the LS, √s is the CMS energy. The quantity in eq. (6.4) are:

2 R Gπ± (Ep)=(1+4mp/s)− , (6.5) 2.6 2 R G 0 (E )=(1+23E− )(1 4m /s) , (6.6) π p p − p 2 2 Q = (C1 C2 p + C3 p )/ 1+4mp/s, (6.7) − · ⊥ · ⊥ 2 q x˜π = x∗ + (4/s)(p + mπ), (6.8) k ⊥ q 2mπ√sγcγπ(βπcosθ βc) x∗ = 2 − 2 , (6.9) k [(s m2 m )2 4m2 m ]1/2 − π − X − π X where θ is the pion polar angle in LS, γc, γπ the Lorentz factor of the CMS and of the pion in the LS, A, B, C1,2,3, R are positive constants and mX is the X channel of the reaction (pp π± + X). → The energy distribution of pions can be obtained integrating over the polar angle θ

2πp 1 d3σ F (E , E )= π dcosθ E , (6.10) π π p ησ(E ) × π d3p h p ism Zcosθmin π where ησ(Ep) sm is the inclusive cross section of pion production in the scaling modelh and i

2 2 s m +m γ E − X π c π − 2√s cosθmin = . (6.11) βcγcpπ

6.1.3 Pion spectrum The pion production cross section is given by

dσ(Eπ, Ep) = ησπ(Ep) F (Eπ, Ep) (6.12) dEπ h i where F (Eπ, Ep) is the distribution of pions produced in pp-collisions, given by the isobaric (eq. (6.3)) or scaling (eq. (6.10)) model (depending on the CR proton energy) and ησπ(Ep) is the inclusive cross section of pion production which parameterizationh is giveni by Badhwar, Stephens, & Golden in [302], and Dermer in [298] for diﬀerent channels.

In Fig. 6.1 the calculated π+ flux is shown for fixed values of the CR proton energy (Ep 3 10 100 1000 GeV ), taking into account the shape of the CR proton spectrum.∼ − − − In order to validate our calculation, the total pion flux has been calculated also with the code Pythia 8.1 [304] (a standard tool for the generation of high-energy collisions) from a simulated input proton spectrum of the same shape constituted of a sample of 106 protons. The total spectrum represents well the sum of each contribution calculated with GALPROP for fixed values of the proton energy, validating our result. 110 CHAPTER 6. GALACTIC NEUTRINO BACKGROUND

Figure 6.1: Pion number as a function of energy calculated with GALPROP and Pythia codes (arbitrary units). The π+ flux (in arbitrary units) is shown for fixed values of the CR proton energy (Ep 3 10 100 1000 GeV ), taking ∼ − − − + into account the shape of the CR proton spectrum. The total pion flux (π , π− and πtotal) has been calculated also with the code Pythia 8.1 from a simulated input proton spectrum of the same shape (curve CR protons). 6.2. MUON DECAY 111

Table 6.1: Functions for neutrinos from muon decay

f0(x) f1(x) 2 2 νµ 2x (3 2x) 2x (1 2x) ν 12x2(1− x) 12x2(1− x) e − −

6.2 Muon decay

Since the muons originated from pion decay are produced fully polarized, the energy distribution of the neutrinos/antineutrinos in the muon rest frame is given by [271] dn 1 = [f (x) f (x)cosθ] (6.13) dxdΩ 4π 0 ∓ 1 where x = Eν′ /mµ with Eν′ the neutrino energy in the muon rest frame and θ the polar angle between the neutrino and the muon spin. The functions f0(x) and f1(x) are given in table 6.1.

Integrating over the polar angle and transforming the distribution to the LS, the energy distribution of neutrinos in the LS becomes: dn 1 = [g0(y,βµ) P olµg1(y,βµ)] (6.14) dy βµ − where y = Eν /Eµ, Eµ and Eν the muon and neutrino energy in the LS and βµ is the muon velocity/c in the LS. P olµ is the muon polarization given by: 1 2E r 1+ r P ol = π (6.15) µ β E (1 r) − 1 r µ µ − − 2 2 with r = mµ/Mπ and Eπ the pion energy in the LS. The functions g0(y,βµ), g1(y,βµ) are given by xmax dx g (y,β )= f (x) , (6.16) 0 µ 0 x Zxmin xmax 2y/x 1 dx g (y,β )= f (x) − (6.17) 1 µ 1 β x Zxmin µ with x =2y/(1 + β ) and x = min[1, 2y/(1 β )]. min µ max − µ

6.3 Neutrino production

The main decay channel for charged pions is

π± µ± + ν (ν ). (6.18) → µ µ The distributions of muons and muon neutrinos in the LS from the direct decay of mesons (pions or kaons) is given by dn dn BR = = (6.19) dE dE (1 m2 /M 2)P ν µ − µ M 112 CHAPTER 6. GALACTIC NEUTRINO BACKGROUND

where M is the meson (pion/kaon) mass, PM is the meson momentum in the LS and BR is the branching ratio of the decay.

Taking into account the meson production, the distribution of muon neutrinos produced directly by the decay of pions produced in a pp-collision is given by

max Eπ dn F (Eν , Ep) = dEπFπ(Eπ, Ep) = (6.20) min dE Eπ ν Z max Eπ BR = dEπFπ(Eπ, Ep) 2 2 , min (1 m /M )Pπ ZEπ − µ π with

Emin = E /(1 r)+ M 2(1 r)/(4E ), (6.21) π ν − π − ν Emax = (s M 2 + M 2)/(2√s), (6.22) π − X π

Eν the neutrino energy in the LS, Eπ, Pπ the pion energy and momentum in 2 2 the LS, Ep is the proton energy in the LS, r = mµ/Mπ and MX is the mass of the X channel of the reaction (pp π± + X). → In eq. (6.20) Fπ(Eπ, Ep) is the distribution of pions produced in pp-collisions, given by the isobaric (eq. (6.3)) or scaling (eq. (6.10)) model (or their interpolation) depending on the CR proton energy.

Besides the muon neutrinos produced directly from the pion (or kaon) decay, muon and electron neutrinos derive from the decay of muons coming from meson decays: µ± e± + ν (ν ) + ν (ν ) (6.23) → e e µ µ dn Given the neutrino distribution from muon decay dy (eq. (6.14)), the energy distribution of neutrinos from the decay of muons produced in the decay of a pion of energy Eπ in the LS is given by:

max Eµ dn dn ymax dn 1 dn F (Eν , Eπ)= dEµ = dy , (6.24) min dEµ dEν dEµ y dy ZEµ Zymin where y is the ratio between the muon and neutrino energies in the LS (y = max min Eν /Eµ) and ymin = Eν /Eµ , ymax = Eν /Eµ . The muon minimum and maximum energy are:

min Eµ = max[mµ,γπ(Eµ∗ βπPµ∗)], (6.25) max − Eµ = max[mµ,γπ(Eµ∗ + βπPµ∗)], (6.26) where γπ the pion Lorentz factor in the LS and Eµ∗, Pµ∗ the muon energy and momentum in the pion rest frame:

2 2 Mπ + mµ Eµ∗ = , (6.27) 2Mπ 2 2 Mπ mµ Pµ∗ = − . (6.28) 2Mπ 6.4. NEUTRINO OSCILLATION 113

Taking into account pion production through pp-collisions we have:

max Eπ ymax dn 1 dn F (Eν , Ep)= dEπ dyFπ(Eπ, Ep) , (6.29) min dEµ y dy ZEπ Zymin where Ep is the proton energy, Fπ(Eπ, Ep) the distribution of pions produced in pp-collisions (eqs. (6.3), (6.10)) and

E E∗ + P ∗ Emin = M max 1, ν + µ µ , (6.30) π π × E + P 4E µ∗ µ∗ ν s M 2 + M 2 Emax = − X π . (6.31) π 2√s are the minimum and maximum pion energy in the LS.

The neutrino production cross section is given by

dσ(Eν , Ep) = ησπ(Ep) F (Eν , Ep) (6.32) dEν h i where ησπ(Ep) is the inclusive cross section of pion production which parameterizationh is giveni by Badhwar, Stephens, & Golden in [302], and Dermer in [298] for diﬀerent channels.

Fig. 6.2 shows the neutrino production cross section from the decay of negative and positive pions. For a given proton energy, we calculated the cross section for the muon neutrinos produced directly from the pion decay and for the electron and muon neutrinos produced from the muon decay.

6.4 Neutrino oscillation

During their travel to Earth, neutrinos are subjected to oscillations. As the Interstellar density is very low1, we can use the approximation of oscillation in vacuum. Given a neutrino of ﬂavor a, the probability of oscillation in a neutrino of ﬂavor b in vacuum is given by:

2 2 ∆mjkx P = δ 4 Re[U U ∗ U ∗ U ] sin ab ab − aj bj ak bk 4E k>jX 2 2 ∆mjkx + 2 Im[U U ∗ U ∗ U ] sin (6.33) aj bj ak bk 2E k>jX where Uij are the matrix elements responsible of the mixing of neutrinos ν = U ∗νm, with νm = (ν1, ν2, ν3) the mass eigenstates m1, m2, m3, ν = (νe, νµ, ντ )

1The density of the interstellar gas covers the range between n ≈ 3 × 10−3 cm−3 (hot gas) to n ≈ 104 cm−3 (molecular clouds). The average density of hydrogen nuclei is n ≈ 1 cm−3 in the galactic disk. 114 CHAPTER 6. GALACTIC NEUTRINO BACKGROUND

Figure 6.2: Calculated cross section for neutrinos from negative (a) and positive (b) pions decay produced in pp-collisions. The cross sections have been calculated for ﬁxed values of the cosmic ray proton energy. 6.4. NEUTRINO OSCILLATION 115 and ∆m2 = m2 m2, j, k =1, 2, 3. In case of CP conservation, the matrix U jk k − j is real (U † = U ⊤):

Ue1 Ue2 Ue3 U = U U U = (6.34)  µ1 µ2 µ3 Uτ1 Uτ2 Uτ3   c13c12 s12c13 s13 = s12c23 s23s13c12 c12c23 s23s13s12 s23c13 , −s s −c s c c s − c s s c c  12 23 − 23 13 12 − 12 23 − 23 13 12 23 13   where cij = cos(θij ) and sij = sin(θij ), and (6.33) becomes

∆m2 x P = δ 4 [U U U U ] sin2 jk (6.35) ab ab − aj bj ak bk 4E Xk>j The oscillation lengths are given by:

4πE E eV 2 Lij = 2 =2.48 km 2 . (6.36) ∆mij GeV ∆mij Assuming a normal hierarchy of neutrinos masses, in the approximation

m2 + m2 ∆m2 ∆m2 ∆m2 = m2 1 2 , δm2 = m2 m2, (6.37) 13 ∼ 23 ∼ 3 − 2 2 − 1 the oscillation lengths become 4πE E L = 3.1 104 km , (6.38) 12 δm2 ≈ × GeV 4πE E L = 1.033 103 km . (6.39) 13 ∆m2 ≈ × GeV where the actual values of the mixing angles and of the mass diﬀerences were used (see table 6.2 [305]).

Table 6.2: Actual values provided by experiments [305]. Actual values 2 5 2 δm 7.92(1 0.09) 10− eV 2 +0± .14 · 3 2 ∆m 2.6(1 0.15) 10− eV 2 − ·+0.18 sin θ12 0.314(1 0.15) 2 +0− .35 sin θ23 0.45(1 0.20) 2 +2.3− 2 sin θ13 0.8 0.8 10− − · As the distances from which neutrinos come from are many orders of magnitude bigger than the oscillation lengths for neutrinos in the energy range considered2, the oscillation probability are averaged:

P = δ 2 [U U U U ] (6.40) ab ab − aj bj ak bk Xk>j 2We will consider the energy range 1 MeV − 1 P eV 116 CHAPTER 6. GALACTIC NEUTRINO BACKGROUND and consequently:

P = 1 2U 2 U 2 2[U 2 (U 2 + U 2 )] (6.41) ee − e1 e2 − e3 e1 e2 P = 1 2U 2 U 2 2[U 2 (U 2 + U 2 )] µµ − µ1 µ2 − µ3 µ1 µ2 P = 1 2U 2 U 2 2[U 2 (U 2 + U 2 )] ττ − τ1 τ2 − τ3 τ1 τ2 P = 2U U U U 2[U U (U U + U U )] eµ − e1 µ1 e2 µ2 − e3 µ3 e1 µ1 e2 µ2 P = 2U U U U 2[U U (U U + U U )] eτ − e1 τ1 e2 τ2 − e3 τ3 e1 τ1 e2 τ2 P = 2U U U U 2[U U (U U + U U )]. µτ − µ1 τ1 µ2 τ2 − µ3 τ3 µ1 τ1 µ2 τ2 The neutrino ﬂuxes at Earth after oscillation (at the top of the atmosphere) are so given by:

N = N 0 P + N 0 P (6.42) νe νe · ee νµ · eµ N = N 0 P + N 0 P (6.43) νµ νµ · µµ νe · eµ N = N 0 P + N 0 P (6.44) ντ νe · eτ νµ · µτ 0 0 where Nνi is the number of neutrinos at the production point (Nντ = 0).

6.5 Neutrino ﬂux

Using the diﬀusive galactic model described in 4.1.1, we calculated the diﬀusive galactic spectrum from SNRs. From the proton§ and helium spectra and from the ISM gas distributions, we estimated the charged pion production. Basically, the interactions considered are

p p π± + X, (6.45) →

p p π+ + X, X = p + n, P =1.65 GeV/c, (6.46) → th p p π+ + X, X = d, P =0.791 GeV/c, (6.47) → th + p p π− + X, X =2p + π , P =0.80 GeV/c (6.48) → th (6.49) with Pth the threshold momentum of the CR proton, where pions decay in muon neutrinos and muons π± µ± + ν (ν ), (6.50) → µ µ the latter decaying in neutrinos and electrons

µ± e± + ν (ν ) + ν (ν ) . (6.51) → e e µ µ Using the calculated cross section for neutrino production, we calculated the ﬂux of neutrinos at Earth for neutrino energy in the range 1 MeV 1 P eV , taking into account neutrino oscillations (Fig. 6.4). For comparison,− in Figs. 6.3(a), 6.3(b), the total neutrino ﬂux and the contribution of each kind of neutrino are shown in absence of neutrino oscillations.

The integrated total energy ﬂux of neutrinos is estimated as 1.003 105 eV 2 1 × cm− s− and the contribution of the galaxy bulge is about 20% of the total 6.5. NEUTRINO FLUX 117

(a)

(b)

Figure 6.3: Flux of neutrinos produced by pion decay through the interaction of cosmic rays with the ISM gas in absence of oscillations. The curves indicates the total neutrino/antineutrino ﬂux (ν + ν), the electron (νe + νe) and muon (νµ + νµ) contributions. In panel a) the diﬀerent muon neutrino components, (π) (π) neutrinos coming directly from the pion decay (νµ +νµ ) and neutrinos coming (µ) (µ) from the muon decay (νµ + νµ ) are also shown. 118 CHAPTER 6. GALACTIC NEUTRINO BACKGROUND

Figure 6.4: Flux of neutrinos produced by pion decay through the interaction of cosmic rays with the ISM gas including neutrino oscillations. The curves indicates the total neutrino/antineutrino ﬂux (ν + ν), the electron (νe + νe), muon (νµ + νµ) and tau (ντ + ντ ) contributions. 6.5. NEUTRINO FLUX 119

Figure 6.5: Sky map of the total neutrino ﬂux produced by pion decay through the interaction of galactic cosmic rays with the ISM gas. Flux (in 2 1 1 cm− s− sr− ) is in logarithmic scale.

ﬂux. The calculated Galactic luminosity of neutrinos produced by Galactic protons Gal 38 1 from SNR interacting with the interstellar gas is L 8.7 10 erg s− . ν ≈ × In Fig. 6.5, the skymap of the total neutrino ﬂux in Galactic latitude and lon- gitude coordinates is shown.

The calculated total neutrino ﬂux produced by pion decay through the interaction of galactic cosmic rays with the ISM gas is compared in Fig. 6.6 with spectra of other origins, from the lowest energy neutrinos produced in the big bang (CνB) to the highest energies associated with gamma ray bursts (GRB) and active galaxy nuclei (AGN) as well as GZK neutrino ﬂux. At intermediate energies (0.1 MeV

Figure 6.6: Calculated total neutrino ﬂux produced by pion decay through the interaction of galactic cosmic rays with the ISM gas compared with spectra of neutrinos of other origins, from the lowest energy neutrinos produced in the big bang to the highest energies associated with the sources of the cosmic rays (gamma ray bursts or active galaxies). Neutrinos at intermediate energies, produced in the sun (pp, 8B, hep), reactors and supernovae, as well as atmospheric neutrinos (produced in collisions of cosmic rays in the atmosphere) are also shown. Adapted from [306] and [307]. Chapter 7

Conclusions

The matching conditions of the Galactic and extragalactic components of cosmic rays along the second knee and the ankle have been analyzed in two diﬀerent EG scenarios.

The diffusive Galactic spectrum from regular SNRs has been calculated using the numerical diffusive propagation code GALPROP. The calculated spectrum was first compared to experimental data at energies below 1017 eV . The good agreement with the experimental data readily obtained up to the first knee, can be extended beyond it up to 102 P eV only if the abundances of nuclei of intermediate and heavy masses are renormalized by a factor 2 with respect to the cosmic abundances at the acceleration source. This result implies that injection of nuclei heavier than protons can be twice as efficient for first order Fermi mechanisms operating in the vicinity of shock waves. The latter empha- sizes our present theoretical uncertainties regarding the physical mechanisms through which supra-thermal particles are injected and accelerated in the prox- imity of shock fronts.

This first assessment of the Galactic cosmic ray flux was used to analyze the matching conditions with two alternative models for the extragalactic flux: a pure proton model by Berezinsky [21] and a mixed composition model by Allard et al. [39].

The ﬁrst step was the matching of the observed energy spectrum by combining the Galactic and extragalactic ﬂuxes. From this process, it becomes clear that additional Galactic components are needed to account for the observations. The minimum amount of additional components that allows us to satisfactorily reproduce the spectrum is either one or two. The pure proton model is the less expensive one requiring only one additional component, which we name GA1. The mixed composition model on the other hand requires, besides GA1, another component at still higher energies, GA2. The luminosities requirements associated with these proposed additional components were estimated, demonstrating their plausibility from the energetic point of view, which also highlights their conceptual importance.

If no additional information are incorporated in the analysis, both theoretical

121 122 CHAPTER 7. CONCLUSIONS models remain indistinguishable from the experimental point of view. Therefore, the eﬀect of including composition information, in the form of elongation rate data, Xmax , was also analyzed. Xmax is the atmospheric depth at which the electromagnetich i component of cosmich rayi showers attain its maximum development and is, at present, one of the most trusted tracers of primary composition and, in particular, of composition changes as a function of energy. Spectral and composition data provided by HiRes and Auger experiments, which favor an EG mixed composition, were used to set constraints on the Galactic and extragalactic CR ﬂuxes and on their change in composition as a function of energy.

For this study, the energy spectral shape of each one of the components is determined from the matching of the observed total energy spectrum. The elongation rate of the combined ﬂuxes is then ﬁtted to the HiRes and Auger data sets by changing appropriately the composition as a function of energy of each one of the components, Galactic and extragalactic. Due to practical limitations imposed by the present experimental uncertainties and in order to make the analysis simpler, only a binary mixture of p and Fe is considered.

The main result is that the additional Galactic components, GA1 and GA2, must have a mixed composition. Furthermore, inside each one of these components there is a progressive evolution of the composition from lighter to heavier as the energy increases. This is consistent with this components being originated in different populations of compact SNRs. If this is correct, then SNRs, probably in a variety of environments, wold be responsible for the main part of the Galactic flux up to energies . 3 1018 eV. × It must be noted, however, that GA2 is globally lighter than GA1 implying, very likely, a sizable proton contribution at the largest energies in the Galac- tic flux. The later, in turn, indicates the possible existence of a quantitatively minor, but qualitatively very important contribution from another acceleration mechanisms, able to accelerate protons to the highest energies. Pulsars and/or magnetars are likely candidates.

Unless our knowledge of hadronic interactions is radically wrong at energies of 102 PeV, which should be clariﬁed soon by the LHC, the uncertainties introduced by the HIM might be important from the quantitative point of view but not so much in qualitative terms. In fact, there are some results that seem rather independent of the HIM, like the diminution in average atomic mass along the low energy branch of the ankle and the existence of a discontinuity in the slope of the energy proﬁle of A around the mid ankle. These results seem also to be supported by both, Augerh i and HiRes.

Our present results may be certainly considered as preliminary due to experimental uncertainties and simpliﬁcations in the numerical approach. First, there is the paucity of data involved in the determination of the energy spectrum in the region encompassing the second knee and the ankle and the scatter of Xmax measurements when diﬀerent experiments are compared among them- selvesh belowi 3 1018 eV. A proper experimental characterization of this very important region× will likely have to wait until the release of the KASCADE- 123

Grande [169] and Auger enhancement data [257].

Second, there are arguable simplifying assumptions related with our diffusive treatment of the Galactic component at the highest energies which is, very likely, undergoing a progressive change in propagation to a full ballistic regime. This can be somehow mitigated by the fact that Fe and most intermediate nuclei should still be diffusive inside this energy interval, while protons would only deviate importantly from the diffusive approximation at the highest energies considered for the Galactic flux.

In any case, the importance of the transition region as a play ground for disentangling the Galactic and extragalactic cosmic ray ﬂuxes is unquestionable and considerable eﬀort should be invested in its full experimental characterization and theoretical modeling.

Appendix A

Shock acceleration: Fermi mechanism

The current paradigm for Galactic cosmic ray acceleration is the Fermi acceleration mechanism by shock waves of Supernova Remnants (SNRs) [42]. SNRs are considered the main sources and acceleration mechanism of GCRs. The basic principle is that acceleration is reached through transfer of macroscopic kinetic energy from a magnetized cloud in movement to elementary particles or nuclei of the interstellar medium (ISM). The energy gain is a consequence of relativistic boosts and allows an increase of the particle energy many times its original value. This kind of mechanism, known as 2nd order Fermi acceleration, has been proposed by Fermi in 1949 [308], which considered a charged particle acceleration through the interaction with a moving magnetized cloud, or plasma. The original theory has been adapted and developed in the contest of shock waves originated from the explosion of Supernovae and it’s known as 1st order Fermi acceleration [309, 310, 311, 312, 313].

A.1 First and second order Fermi acceleration

In the original work [308], Fermi considered the energy gain of particles through diﬀusion in turbulent magnetic ﬁelds of a moving plasma (Fig. A.1(a)).

Let’s consider a particle of energy E1 entering a moving magnetized cloud with angle θ1 with respect to the cloud velocity V . The particle diﬀuses in the cloud scattering on the magnetic irregularities until the average motion coincides with that of the gas cloud. In the cloud rest frame the particle total energy is1

E′ = γE (1 βcosθ ), (A.1) 1 1 − 1 where γ and β = V/c are the cloud Lorentz factor and velocity/c. If the scattering is collision-less, in the cloud rest frame there isn’t energy change during the diﬀusion and the particle energy just before it escapes is E2′ = E1′ .

1We consider particles which are already suﬃciently relativistic (E ≈ pc) without considering the problem of injection in the acceleration region.

125 126 APPENDIX A. SHOCK ACCELERATION

(a) gas cloud

(b) shock front

Figure A.1: Scheme of particle acceleration in two cases: a) acceleration by a moving ionized cloud, b) acceleration at a plane shock front.

The energy gain is obtained through the Lorentz transformation to the laboratory frame. The ﬁnal energy of the particle escaping from the cloud with an angle θ2 is E2 = γE2′ (1 + βcosθ2). (A.2) and the energy gain:

∆E 1 βcosθ + βcosθ β2cosθ cosθ = − 1 2 − 1 2 1. (A.3) E 1 β2 − 1 − The original idea was developed in the contest of shock waves originated in SN explosions [309, 310, 311, 312, 313]. The situation is shown in Fig. A.1(b). Let’s consider a plane shock front2 moving with speed u ; the shocked gas −−→1 moves away from the shock with speed −→u2 relative to the shock front, with u < u . In the laboratory frame the gas behind the shock moves (to the | 2| | 1| left) at speed −→V = −→u1 + −→u2. The energy gain in this case is given by the same equation (A.3) with− β = V/c, the velocity of the shocked gas (downstream)

2We use the approximation of inﬁnite plane front shock. A.1. FERMI ACCELERATION 127 relative to the unshocked gas (upstream).

The diﬀerence between the two cases consists on the average fractional energy gain for each “encounter”. If for each “encounter” we have an energy gain ∆E/E = ξE, after n “encounters” the energy of the particle is given by:

n En = E0(1 + ξ) , (A.4) where E0 is the particle energy at injection. The probability of remaining in the acceleration region after n “encounters” is n (1 Pesc) , where Pesc is the escape probability from the acceleration region per− “encounter”. The number of “encounters” needed to reach a ﬁnal energy E is given by: E n = ln /ln(1 + ξ) (A.5) E 0 and the number of particles accelerated to energy greater than E is

γ (1 P )n 1 E − N( E) − esc . (A.6) ≥ ∝ P ∝ P E esc esc 0 The spectral index γ is given by

1 P 1 T γ = ln /ln(1 + ξ) esc = cycle , (A.7) 1 P ≈ ξ ξ T − esc esc where Tcycle and Tesc are the characteristic times for the acceleration cycle and for the escape from the acceleration region, respectively, related to the escape probability by the relation: Tcycle Pesc = (A.8) Tesc The maximum energy achievable by a particle depends on the number of “encounters”, i.e. on the time duration of the acceleration process. If the acceleration works for a time t, the maximum numbers of “encounters” is given by t/Tcycle and the maximum energy is:

t E E (1 + ξ) Tcycle . (A.9) ≤ 0 Higher energy particles take longer to accelerate than lower energy particle and if the accelerator has a life time TA, the maximum energy reachable through the Fermi mechanism is given by:

TA T Emax = E0(1 + ξ) cycle , (A.10)

The diﬀerence between the two cases consists on the average fractional energy gain for each “encounter”. In the case of the gas cloud the distribution of the numbers of “encounters” n as a function of θ2 is

dn = constant, 1 cos θ 1. d cos θ2 − ≤ 2 ≤ 128 APPENDIX A. SHOCK ACCELERATION

In the case of a plane front shock the distribution is the projection of an isotropic ﬂux onto a plane,

dn = 2cos θ , 0 cos θ 1. d cos θ2 2 ≤ 2 ≤ Averaging over the escape angle

< cos θ2 >=0, (cloud) < cos θ2 >=2/3, (shock) and <∆E>2 1 βcosθ1 − 2 E1 = 1 β 1, (cloud) − −2 2 2 <∆E>2 1 βcosθ1+ 3 β 3 β cos θ1 − −2 E1 = 1 β 1. (shock) − − For the entering angle θ1, in the cloud case, the collision probability is proportional to the relative speed between the cloud and the particle

dn c V cos θ1 = − , 1 cos θ 1, d cos θ1 2c − ≤ 1 ≤ while in the case of the plane shock front

dn = 2cos θ , 1 cos θ 0. d cos θ1 1 − ≤ 1 ≤ Consequently < cos θ1 >= V/3c, (cloud) < cos θ >= −2/3, (shock) 1 − and the average fractional energy gain for each “encounter” is:

1 2 <∆E> 1+ 3 β 4 2 2 ξ = E1 = 1 β 1 3 β , (cloud) −4 4− 2 ∼ <∆E> 1+ 3 β+ 9 β 4 4 u1 u2 2 ξ = E1 = 1 β 1 3 β = 3 −c . (shock) − − ∼ In the cloud case, even if in each “encounter” the particle can both gain or lose energy, on average the energy gain is proportional to β2 from which the name “second order Fermi acceleration”. On the other hand, in the shock front case, the particle can only gain energy but the average energy gain is proportional to β (“ﬁrst order Fermi acceleration”)3.

Another substantial diﬀerence between the two mechanism is the resulting spectrum. In the case of “second order Fermi acceleration” (cloud), the acceleration 7 region is the Galactic disc, characterized by Tesc 10 years. The acceleration rate is given by the collision rate between a cosmic∼ ray particle with speed c and clouds of spatial density ρc and cross section σc. In this case we have 1 Tcycle (A.11) ∼ cρcσc and equation (A.7) becomes

1 T 1 1 1 γ cycle . (A.12) 4 2 ≈ ξ Tesc ≈ 3 β cρcσc Tacc 3In both the case the approximation is valid for V non relativistic. A.2. SNR MAXIMUM ENERGY 129

The spectral index γ in the case of “second order Fermi acceleration” is not universal but depends on the cloud properties.

In the case of the plane shock front, the rate of “encounters” is given by the projection of the isotropic ﬂux of cosmic rays onto the plane shock front, while the rate of convection downstream is given by ρCRu2. Consequently, the probability of escape from the acceleration region is

ρCRu2 4u2 Pesc = = , (A.13) cρCR/4 c and the spectral index γ is P 3 γ esc = . (A.14) ≈ ξ u /u 1 1 2 − The spectral index in the case of “first order Fermi acceleration” is independent on the velocity of the plasma but depends on the ratio of the upstream and downstream velocities. Calling M = u1/c1 the Mach number, where c1 is the sound speed in the gas4, in the case of a strong shock (M >> 1) and for a mono-atomic gas, the spectral index becomes 4 γ 1+ . (A.15) ≈ M 2 The spectral index for “first order Fermi acceleration” is universal and its value is close to the observed one (without propagation effects)5.

A.2 Supernovae explosion: maximum energy

The approximation of a inﬁnite plane shock front for supernova blast waves is valid in the case the diﬀusion length λD is much less than the radius of curvature of the shock [82]. The time scale of acceleration (time in which the SN blast wave is active as accelerator) can be estimated as the time it takes the expanding shell to sweep an amount of matter of the interstellar medium (ISM) equal to its own mass. This time corresponds to the end of the free-expansion phase of the SNR and beginning of the adiabatic phase, time at which the shell begins to slow down. For a typical SN mass of the order of 10 solar masses, expanding at a velocity 5 108 cm/s in a ISM of density ρ =1 proton/cm3, the characteristic time of × acceleration is TA 1000 years. The escape from∼ the acceleration region is possible only in the downstream region, i.e. inside the shell. Cosmic rays are eventually injected in the Galactic disk after the beginning of the shock disintegration. The “active” life time of the SN blast wave determines the maximum acceleration energy. The acceleration rate is given by: dE ξE = . (A.16) dt Tcycle

4 A shock is possible only in the case u1 >c1. 5The diﬀerential spectral index needed for accelerators to describe the spectrum is ∼ 2.1, correspondent to γ ∼ 1.1. 130 APPENDIX A. SHOCK ACCELERATION

The particle current with convection is given by:

J~ = D∆N + ~uN (A.17) − and for a single “encounter” Tcycle is given by:

4 D D T = 1 + 2 , (A.18) cycle c u u 1 2 where D1 and D2 are the diffusion coefficient of the upstream and downstream region, respectively. An analysis by Lagage and Cesarsky [314, 315, 316] provides an estimate of the diffusion coefficient. In order to diffuse on the irregularities of the magnetic field, the diffusion length cannot be smaller than the Larmor radius of the particle: pc λ > R = , (A.19) D L ZeB where Z is the charge of the particle, p, v are the particle momentum and velocity and B is the magnetic field. From 1 D = λ v, (A.20) 3 D the minimum diffusion coefficient is vR cR 1 Ec D = L L . (A.21) min 3 ∼ 3 ∼ 3 ZeB

For a strong shock (u2 = u1/4) the fractional energy gain ξ is 4 4 u u u ξ β = 1 − 2 = 1 ; (A.22) ∼ 3 3 c c taking D1 = D2 = Dmin we have 20E Tcycle > . (A.23) 3ZeBu1 The acceleration rate obtained is independent on energy

dE ξE 3 u1 = u1ZeB , (A.24) dt Tcycle ≤ 20 c and the maximum energy achievable is limited by the active time of the shock TA dE 3 u E = T 1 ZeB(u T ), (A.25) max dt A ≤ 20 c 1 A where TA can be calculated from 4 π(u T )3ρ = M . (A.26) 3 1 A ISM ejecta In the case of a typical SN of mass of the order of 10 solar masses, expanding at velocity 5 108 cm/s in a medium of density ρ = 1 proton/cm3 and in a × A.2. SNR MAXIMUM ENERGY 131

magnetic ﬁeld B 3 µG, the characteristic time is TA 1000 year and the maximum energy∼ reachable by a particle of charge Ze is: ∼

E Z 3 104 GeV, (A.27) max ≤ × × This energy limit can be smaller due to energy losses by synchrotron radiation due to the magnetic ﬁeld. The energy loss rate for a relativistic particle of mass Am and charge Ze in a magnetic ﬁeld B is (in units cgs):

4 dE 3 erg Z me 2 2 1.6 10− E B , (A.28) − dt synchrotron ≈ × s A m where E is the total energy of the particle. Due to the dependence on me/m, the energy loss by synchrotron radiation is negligible for protons and nuclei in case of low magnetic ﬁeld. Consequently, the maximum energy achievable by protons is of the order of 100 T eV .

Due to the dependence of the maximum energy on the charge of the nucleus, if the steepening of the spectrum is due to the break of this acceleration mechanism, a change in composition towards heavier nuclei should be observed as energy increases in the region of the knee. On the other hand, the same change in composition can be the result of the dependence of the escape time from the Galaxy at increasing energies. In any case, the acceleration mechanism through SN explosion can explain the origin of GCR till the knee energy. For ultra high energy cosmic rays (UHECR) this mechanism is not more sufficient. The acceleration can be more efficient if the SN explosion takes place in different environments; in the case of an explosion in the circumstellar winds of their progenitors [317] or in the case of shock termination of stellar winds [318], the maximum energy can be increased of a factor 100 or 10 respectively. The combination of particular conditions of the medium and a quasi perpendicular configuration of the magnetic field, as in the shock termination of extragalactic winds [319, 320], could extend the maximum energy up to energies of order of 1020 eV .

Appendix B

Electromagnetic and hadronic showers

The determination of the longitudinal development of a shower and of its lateral distribution is subjected to large uncertainties due to the fact that perturbative QCD provides the calculation of a minimum part of the hadronic processes and that the energetic range studied in colliders is limited. Nevertheless, simple models describing the development of showers exist. The toy model suggested by Heitler [80] (described in B.1) provides the macroscopic characteristics of electromagnetic showers, while§ the development of hadronic showers can be described by a similar model [81] (described in B.2). Even if they cannot replace detailed simulations, these simple models predict§ the most important features of the cascades.

B.1 Electromagnetic showers

The development of electromagnetic cascades can be described using a toy + model, suggested by Heitler [80]. Heitler’s model consider e e− and photons un- + dergoing repeated two-body splitting via bremsstrahlung or e e− pair production (see Fig. B.1a). Let’s assume that the energy of the particle is equally di- 1 vided into the two outgoing particles. Deﬁning “splitting length” as d = λr ln 2 2 where λr is the radiation length (λr 37.1 g cm− in air), after each splitting length the number of particles is doubled∼ and the energy is then E/2. After n splitting lengths, corresponding to a distance x = nλr ln 2, the size of the n n shower is N = 2 = exp(x/λr) and the particles energy is E0/2 where E0 is the energy of the primary. The production stops when the particles reach the critical energy Ec (Ec 81 MeV for electrons in air), energy at which cross section for bremsstrahlung∼ falls. Then, energy is mainly lost by ionization and the number of particles decreases. The maximum number of particles is given by the ratio Nmax = E0/Ec. The atmospheric depth of the maximum longitudinal development of a shower, called Xmax, is determined by the number of

1d is the distance over which an electron loses on average half of its energy by radiation.

133 134 APPENDIX B. ELECTROMAGNETIC AND HADRONIC SHOWERS

Figure B.1: EAS models: schematic model of (a) an electromagnetic cascade and (b) a hadronic shower. In the electromagnetic cascade, following the Heitler model, at each splitting length the number of particles doubles. After n split- n ting the particle energy is E0/2 , with E0 the primary energy. In the hadron shower, dashed lines indicate neutral pions which do not re-interact, but quickly decay, yielding electromagnetic subshowers (not shown), while solid lines indicate charged pions (not all pion lines are shown after the n = 2 level). Taken from [81].

radiation length for which energy reduces to Ec:

em Xmax = λr ln (E0/Ec) (B.1)

The rate of increase of Xmax with the primary energy [83], named elongation rate, dX dX D = max = ln(10) max (B.2) d(log10 E) d(ln E) in the case of electromagnetic showers is given (using eq. (B.1)) by

em dXmax D = 2.3λr. (B.3) d(log10 E) ≃ Even if this model makes many assumptions, it provides the macroscopic characteristics of the electromagnetic dominated showers.

B.2 Hadronic showers

The development of an hadronic shower can be obtained in a similar way [81]. The atmosphere can be divided in layers of thickness dI = λI ln2, where λI is the interaction length for strongly interacting particles (see Fig. B.1b). As the inter- 2 action length for hadrons is bigger than the radiation length (λI 120 g cm− for pions in air at energies below 1014 eV ), the hadronic component∼ carries a larger fraction of the shower energy at deeper atmospheric depth with respect to the electromagnetic component. Let’s consider a proton of energy E0 which interacts with the atmosphere producing pions. The primary energy is divided B.2. HADRONIC SHOWERS 135 between the charged pions, which decay in muons, and the electromagnetic particles: E = E N + EπN 0.85 GeV (N + 23.5N ), (B.4) 0 c max c µ ≈ e µ π π 14 where Ec = 20 GeV is critical energy for pions (Ec 30 GeV at E = 10 eV , π 17 ∼ Ec 10 GeV at E = 10 eV ), the energy at which the decay length of charged pions∼ becomes less than the distance of the next interaction point. The number of electrons is taken as Ne = Nmax/g, g = 10 [81], and the number of muons n is equal to the number of charged pions Nµ = Nπ = (Nch) , where Nch is the multiplicity of charged particles produced in hadron interactions and n is the number of interactions (Nch = 10 [81]). The muon number is function of the primary energy:

E β E 0.85 N = 0 104 0 , (B.5) µ Eπ ≈ P eV c where β is given by ln Nµ β = π =0.85. (B.6) ln (E0/Ec ) The primary energy can be written as the sum of e.m. and hadronic energy:

π E0 = Eem + Eh = Eem + NµEc ; (B.7) consequently the electromagnetic fraction of a shower is

β 1 E E − em =1 N 0 . (B.8) E − µ Eπ 0 c The e.m. energy of the shower is 90% of the primary energy at E = 1017 eV . ∼ 0 The number of electrons is given by

1 E E α N = em 106 0 , (B.9) e g Ee ≈ P eV c where α is given by 1 β α =1+ − 1.03. (B.10) 105(1 β) 1 ≈ − − The primary energy can be calculated from the number of electrons or muons (at the shower maximum) inverting eqs. (B.5), (B.9) :

E 1.5GeV N 0.97, (B.11) 0 ≈ × e E 19.7GeV N 1.18. (B.12) 0 ≈ × µ If only the ﬁrst generation of neutral pions contributes to the electromagnetic component and if nπ is the number of charged pions produced in the ﬁrst interaction at depth X0 = λI ln 2, the atmospheric depth of the shower maximum induced by a proton is:

E Xp = X + λ ln 0 , (B.13) max 0 r 3n E π × c 136 APPENDIX B. ELECTROMAGNETIC AND HADRONIC SHOWERS

where X0 is function of the primary energy E0. The maximum depth of the shower increases with energy; more energetic is the primary proton, more penetrating will be the induced shower. The calculated estimation is an approximation underestimating Xmax since only the ﬁrst interaction is considered and other factors as the inelasticity factor are neglected. The elongation rate (eq. (B.2)) in the case of proton induced shower becomes (using. eq. (B.13))

p dXmax em d D = D + [X0 ln (3nπ)] , (B.14) d(log10 E) ≃ d(log10 E) − where Dem is given by eq. (B.3). The result obtained for a proton shower can be extended to a generic nucleus using the superposition model to describe the nucleus-air interactions. Follow- ing this model, a nucleus of atomic weight A and primary energy E0 can be seen 2 as A nucleons of energy E0/A interacting independently with the atmosphere . This model consider the development of the shower as the superposition of A hadronic cascades initiated by protons, that can be approximated to electromagnetic cascades as, at growing energy, most of the energy is deposited in the electromagnetic component. Given a nucleus with atomic number A and primary energy E0, the shower maximum is given by XA = Xp λ ln A (B.15) max max − r and the muonic component of a nucleus induced shower becomes:

N A = N p A0.15. (B.16) µ µ ×

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