Scuola di Dottorato in Fisica, Astrofisica e Fisica Applicata Dipartimento di Fisica Corso di Dottorato in Fisica, Astrofisica e Fisica Applicata Ciclo XXXI Lorentz invariance violation effects on ultra high energy cosmic rays propagation, a geometrical approach Settore Scientifico Disciplinare FIS/02 Supervisore: Professor Lino MIRAMONTI Coordinatore: Professor Francesco RAGUSA Tesi di Dottorato di: Marco Danilo Claudio TORRI Anno Accademico 2017/2018 Commission of the final examination: External Referees: Giovanni AmelinoCamelia Fernando Mendez External Members: Antonio Insolia Nicolao Fornengo Internal Member: Lino Miramonti Final examination: February 13th, 2019 Universita` degli Studi di Milano, Dipartimento di Fisica, Milano, Italy To my family MIUR subjects: FIS/02 Contents List of Figures ix Introduction xi Motivation xi Part I : Introduction to UHECR GZK puzzle 3 1 Cosmic Rays: basic properties 3 1.1 Cosmic Rays acceleration mechanisms4 1.2 Energy spectrum9 1.3 Composition and detection 20 2 GZK effect 25 2.1 Importance of UHECR for observational astronomy 25 2.2 Cosmic rays propagation 28 2.3 Electromagnetic radiation propagation 35 2.4 Cosmological red shift 37 2.5 Modifying GZK effect by LIV 43 Part II : An isotropic LIV perspective 49 3 A brief introduction to LIV 49 3.1 Introduction 49 3.2 String Theory 49 3.3 Loop Quantum Gravity 50 3.4 Space-time noncommutativity 51 3.5 Alternative proposed scenarios 52 v vi Contents 3.6 LIV 52 3.7 Maximum attainable velocities and Von Ignatowsi theory 53 3.8 Very Special Relativity 57 3.9 Standard Model Extension 60 3.10 Horava-Lifshitzˆ Gravity 67 3.11 New Special Relativity theories 68 4 LIV 77 4.1 Introduction 77 4.2 Modified Dispersion Relations 78 4.3 The Finsler geometric structure of space-time 81 4.4 Modified Lorentz Transformations 85 4.5 Relativistic Invariant Energy (Mandelstam variables) 87 4.6 Standard Model modifications 90 4.7 Allowed symmetries 98 4.8 Coleman-Mandula theorem generalization 99 4.9 VSR correspondence 101 4.10 DSR correspondence 103 4.11 SME correspondence 105 5 De Sitter projective relativity 107 5.1 Introduction 107 5.2 Possible kinematical groups 108 5.3 DeSitter projective Relativity 110 5.4 Modified dispersion relations in deSitter Relativity 112 5.5 Projective Anti-deSitter Relativity 115 5.6 Modified dispersion relations in Anti-deSitter Relativity 116 5.7 Massive particles kinematics in deSitters/Anti-deSitter Relativity 118 Part III : LIV Phenomenology on Cosmic Messengers 123 6 Cosmic messengers LIV phenomenology 123 6.1 Introduction 123 6.2 Effects induced on protons UHECR phenomenology 123 6.3 Effects induced on heavy UHECR phenomenology 127 7 LIV and Neutrino oscillations 135 7.1 LIV and Neutrino oscillations in an Hamiltonian approach 135 7.2 LIV and Neutrino oscillations in the isotropic scenario 137 7.3 Phenomenological analysis of the LIV effects on neutrino oscillations140 Conclusion 153 Appendices 155 A Finsler Geometry 159 A.1 Introduction 159 A.2 Minkowski norm 159 A.3 Legendre transformation 161 A.4 Finsler manifold 163 A.5 Global Legendre transformation 165 Bibliography 169 List of Publications 183 Acknowledgments 186 vii List of Figures 1.1 Charged particle interaction with a dust cloud in II◦ order Fermi mechanism.5 1.2 Charged particle interaction with a shock wave front in I◦ order Fermi mechanism.7 1.3 The differential distribution of cosmic rays energy observed on Earth by several experiments. All the changes in steepness are underlined. Figure taken from [1]. 13 1.4 Modification in spectrum steepness justified by the change of cos- mic rays mass composition, with the energy increase even the mean CRs mass grows up. 15 1.5 Hillas plot of the astrophysical objects candidates to be the cos- mic rays sources. On the x-axis the dimension of every object and on the y-axis the magnetic field strenght are reported. A way to evaluate the maximum attainable energy is explained in the main text, and the result is Emax = ZRBβ, with Z the charge num- ber of the particle, R the extension of the astrophysical object, B the magnetic field strenght, β the shock velocity in speed of light units. The diagonal lines represent the maximal energies for dif- ferent types of cosmic rays and different values of the shock wave propagation velocities. It is possible to accelerate to the same en- ergy inside very large low field regions or in compact structures with large magentic field values. 21 1.6 Developing of an Extended Air Shower in atmosphere of a pri- mary cosmic ray with an initial energy E0. 23 −23 6.1 Inelasticity for perturbation fpπ ' 9 · 10 128 6.2 Inelasticity for perturbation fpπ ' 0 129 ix x List of Figures 6.3 Optical depth as function of energy and LIV parameters, respec- −23 −23 −23 tively: 1) fpπ ' 9 · 10 , 2) fpπ ' 6 · 10 , 3) fpπ ' 3 · 10 , −24 −24 −24 4) fpπ ' 9 · 10 , 5) fpπ ' 6 · 10 , 6) fpπ ' 3 · 10 , 7) fpπ ' 0 130 6.4 Optical depth as function of energy and LIV parameters, respec- −23 −23 −23 tively: 1) fpπ ' 9 · 10 , 2) fpπ ' 6 · 10 , 3) fpπ ' 3 · 10 , −24 −24 −24 4) fpπ ' 9 · 10 , 5) fpπ ' 6 · 10 , 6) fpπ ' 3 · 10 , 7) fpπ ' 0 131 7.1 Comparison of the oscillation probability νµ ! νe, computed, as function of baseline L, for neutrino energy E = 1 GeV, “stan- dard theory” (red curve) and LIV (blue curve), for LIV parameters −23 δf32 = δf21 = 1 × 10 . 141 7.2 Same analysis of fig.7.1, but for the oscillation νµ ! ντ . 142 7.3 Same analysis of fig.7.1, but for the oscillation νe ! ντ . 143 −25 7.4 Same analysis of fig.7.1, but for LIV parameters δfkj ' 10 . 144 7.5 Same analysis of fig.7.1, but for LIV parameters δf32 = δf21 = 4:5 × 10−27 and for neutrino energy E = 100 GeV. 145 7.6 Same of fig.7.5, but for the oscillation probability Pνµντ . 146 7.7 Same of fig.7.5, but for Pνeντ . 147 7.8 Survival probability for muonic neutrino, for the same conditions of fig.7.5 147 7.9 Percentage variations induced in neutrino oscillation probabilities by LIV corrections. On the vertical axis, as function of the base- line L, the percentage differences between the oscillation probabil- ities, for a 100 GeV neutrino, in presence and in absence of LIV, normalized respect to their average value. The 3 different curves correspond to the percentage differences for the 3 oscillation prob- abilities: Pνeνµ (blue), Pνµντ (violet) and Pνeντ (green curve). 148 7.10 Comparison of the Pνµνe oscillation probability, as function of base- line L, for neutrino energy E = 1 TeV, for ”classical theory”, pre- serving Lorentz Invariance (orange curve) and for LIV including models, with parameters equal, respectively, to δf32 = δf21 = −27 −28 4:5 × 10 (blue), δf32 = δf21 = 4:5 × 10 (red) and δf32 = −29 δf21 = 4:5 × 10 (green curve). 149 7.11 Same analysis of fig.7.10, but for the case of Pνµντ . 150 7.12 Same analysis of fig.7.10, but for Pνeντ 151 7.13 Comparison of the results for the muonic neutrino survival prob- ability in a classical theory and in models with LIV corrections, corresponding to three different values of the δfkj parameters, as illustrated in fig.7.10. Also the color code is the same adopted in figs.7.10-7.12. 152 Introduction xi Introduction Motivation In scientific literature with the term ”Cosmic rays” (CRs) are indicated highly en- ergetic particles of extraterrestrial origin [1, 2, 3]. They were originally discov- ered at the beginning of the twentieth century and were originally thought as composed mainly by electromagnetic radiation, because of their great penetrat- ing power. The study of CRs developed through various stages, till the con- firm of the existence of the Extended Air Showers (EAS) by Pierre Auger in 1938 [4]. The EAS are cascades of secondary products generated by the interaction of charged particles, that constitutes the original CRs, with the atmosphere. In fact it is well known that cosmic rays are of different nature, made by bare atomic nuclei, principally of two types: very light ones (principally protons or alpha particles) or heavier ones (mainly atomic nuclei close to F e element) [5, 6]. Their energy spans many decades and their energetic spectrum ranges from 108 eV to 1021 eV . Because they include the most energetic particles, accessible nowadays, it is very interesting to study this kind of radiation, that can furnish new data about the production of particles in highly energetic environments [7, 8]. For ex- ample, a cosmic proton, with an energy of 1020 eV , colliding with a stationary identical particle of the atmosphere, can originate a reaction with a free energy (energy in the center of momentum reference frame) of about 400 T eV . This quantity must be confronted with the 14 T eV of maximum free energy available at LHC, motivating the interest in this kind of highly energetic collisions. Another important motivation to study cosmic rays is due to the necessity of understand their origin, intended as their creation and acceleration mechanism. Gravity is supposed to be the ultimate engine of CRs acceleration, even if till now the process is not fully understood, it is possible only modeling part of it. There- fore it is supposed that CRs are generated in gigantic gravitational collapses, such as in supernovae explosions, the accretion disks of black holes, gamma ray xii Thesis overview bursts (GBR) or active galactic nuclei (AGN). Solving the problem of CR origin and composition can provide an insight of the accelerating structures and can so be useful in conducting astrophysical observations.