# Predictions of Ultra-High Energy Cosmic Ray Propagation in the Context of Homogeneously Modified Special Relativity

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## Abstract

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## 1. Introduction

## 2. Homogeneously Modified Special Relativity

#### 2.1. Geometric Structure: Hamilton/Finsler Geometry

#### 2.2. HMSR Standard Model Extension

#### 2.3. HMSR Modified Kinematics

## 3. Ultra High Energy Cosmic Rays Propagation and GZK Cut-Off Effect

## 4. LIV Introduced Phenomenology in UHECR Propagation

#### 4.1. $\mathsf{\Delta}$ Resonance Creation Constraint

#### 4.2. Reduced Phase Space and Modified Inelasticity

## 5. Simulations on LIV Modified Propagation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

UHECR | Ultra High Energy Ccosmic Ray |

CR | Cosmic Rays |

LIV | Lorentz Invariance Violation |

HMSR | Homogeneously Modified Special Relativity |

SM | Standard Model |

SME | Standard Model extension |

VSR | Very Special Relativity |

DR | Dispersion Relation |

MDR | Modified Dispersion Relation |

MLT | Modified Lorentz Transformation |

## Appendix A. Derivative of the 0 Degree Homogeneous Function

## References

- Pierre Auger Collaboration. Observation of a Large-scale Anisotropy in the Arrival Directions of Cosmic Rays above 8 × 10
^{18}eV. Science**2017**, 357, 1266–1270. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pierre Auger Collaboration. Constraints on the origin of cosmic rays above 10
^{18}eV from large scale anisotropy searches in data of the Pierre Auger Observatory. Astrophys. J. Lett.**2012**, 762, L13. [Google Scholar] - Tinyakov, P.G.; Urban, F.R.; Ivanov, D.; Thomson, G.B.; Tirone, A.H. A signature of EeV protons of Galactic origin. Mon. Not. R. Astron. Soc.
**2016**, 460, 3479–3487. [Google Scholar] [CrossRef] [Green Version] - Abbasi, R.U.; Abe, M.; Abu-Zayyad, T.; Allen, M.; Azuma, R.; Barcikowski, E.; Belz, J.W.; Bergman, D.R.; Blake, S.A.; Cady, R.; et al. Search for EeV Protons of Galactic Origin. Astropart. Phys.
**2017**, 86, 21–26. [Google Scholar] [CrossRef] [Green Version] - Blasi, P. Origin of very high- and ultra-high-energy cosmic rays. C. R. Physique
**2014**, 15, 329–338. [Google Scholar] [CrossRef] [Green Version] - Greisen, K. End to the cosmic ray spectrum? Phys. Rev. Lett.
**1966**, 16, 748–750. [Google Scholar] [CrossRef] - Zatsepin, G.; Kuzmin, V. Upper limit of the spectrum of cosmic rays. JETP Lett.
**1966**, 4, 78–80. [Google Scholar] - Coleman, S.R.; Glashow, S.L. High-energy tests of Lorentz invariance. Phys. Rev. D
**1999**, 59, 116008. [Google Scholar] [CrossRef] [Green Version] - Stecker, F.W.; Scully, S.T. Searching for New Physics with Ultrahigh Energy Cosmic Rays. New J. Phys.
**2009**, 11, 085003. [Google Scholar] [CrossRef] - Scully, S.; Stecker, F. Lorentz Invariance Violation and the Observed Spectrum of Ultrahigh Energy Cosmic Rays. Astropart. Phys.
**2009**, 31, 220–225. [Google Scholar] [CrossRef] [Green Version] - Torri, M.D.C.; Bertini, S.; Giammarchi, M.; Miramonti, L. Lorentz Invariance Violation effects on UHECR propagation: A geometrized approach. J. High Energy Astrophys.
**2018**, 18, 5–14. [Google Scholar] [CrossRef] - Torri, M.D.C. Lorentz Invariance Violation Effects on Ultra High Energy Cosmic Rays Propagation, a Geometrical Approach. Ph.D. Thesis, Milan University, Milan, Italy, 2019. [Google Scholar]
- Colladay, D.; Kostelecký, V.A. Lorentz violating extension of the standard model. Phys. Rev. D
**1998**, 58, 116002. [Google Scholar] [CrossRef] [Green Version] - Cohen, A.G.; Glashow, S.L. Very special relativity. Phys. Rev. Lett.
**2006**, 97, 021601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Amelino-Camelia, G. Doubly special relativity. Nature
**2002**, 418, 34–35. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Amelino-Camelia, G. Doubly special relativity: First, results and key open problems. Int. J. Mod. Phys. D
**2002**, 11, 1643–1669. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L. The principle of relative locality. Phys. Rev. D
**2011**, 84, 084010. [Google Scholar] [CrossRef] [Green Version] - Amelino-Camelia, G.; Bianco, S.; Rosati, G. Planck-Scale-Deformed Relativistic Symmetries and Diffeomorphisms on Momentum Space. Phys. Rev. D
**2020**, 101, 026018. [Google Scholar] [CrossRef] [Green Version] - Torri, M.; Antonelli, V.; Miramonti, L. Homogeneously Modified Special relativity (HMSR). Eur. Phys. J. C
**2019**, 79, 808. [Google Scholar] [CrossRef] [Green Version] - Aloisio, R.; Boncioli, D.; Grillo, A.; Petrera, S.; Salamida, F. SimProp: A Simulation Code for Ultra High Energy Cosmic Ray Propagation. J. Cosmol. Astropart. Phys.
**2012**, 10, 7. [Google Scholar] [CrossRef] [Green Version] - Magueijo, J.; Smolin, L. Gravity’s rainbow. Class. Quant. Grav.
**2004**, 21, 1725. [Google Scholar] [CrossRef] [Green Version] - Magueijo, J.; Smolin, L. Lorentz invariance with an invariant energy scale. Phys. Rev. Lett.
**2002**, 88, 190403. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pfeifer, C.; Wohlfarth, M.N.R. Finsler geometric extension of Einstein gravity. Phys. Rev. D
**2012**, 85, 064009. [Google Scholar] [CrossRef] [Green Version] - Hohmann, M.; Pfeifer, C.; Voicu, N. Finsler gravity action from variational completion. Phys. Rev. D
**2019**, 100, 064035. [Google Scholar] [CrossRef] [Green Version] - Pfeifer, C. Finsler spacetime geometry in Physics. Int. J. Geom. Meth. Mod. Phys.
**2019**, 16, 1941004. [Google Scholar] [CrossRef] [Green Version] - Javaloyes, M.A.; Sánchez, M. On the definition and examples of cones and Finsler spacetimes. arXiv
**2018**, arXiv:1805.06978. [Google Scholar] [CrossRef] [Green Version] - Bernal, A.; Javaloyes, M.Á.; Sánchez, M. Foundations of Finsler spacetimes from the Observers’ Viewpoint. Universe
**2020**, 6, 55. [Google Scholar] [CrossRef] [Green Version] - Greenberg, O. Why is CPT fundamental? Found. Phys.
**2006**, 36, 1535–1553. [Google Scholar] [CrossRef] [Green Version] - Greenberg, O. CPT violation implies violation of Lorentz invariance. Phys. Rev. Lett.
**2002**, 89, 231602. [Google Scholar] [CrossRef] [Green Version] - Antonelli, V.; Miramonti, L.; Torri, M.D.C. Phenomenological Effects of CPT and Lorentz Invariance Violation in Particle and Astroparticle Physics. Symmetry
**2020**, 12, 1821. [Google Scholar] [CrossRef] - Chaichian, M.; Dolgov, A.D.; Novikov, V.A.; Tureanu, A. CPT Violation Does Not Lead to Violation of Lorentz Invariance and Vice Versa. Phys. Lett. B
**2011**, 699, 177–180. [Google Scholar] [CrossRef] [Green Version] - Tureanu, A. CPT and Lorentz Invariance: Their Relation and Violation. J. Phys. Conf. Ser.
**2013**, 474, 2031. [Google Scholar] [CrossRef] [Green Version] - Chaichian, M.; Fujikawa, K.; Tureanu, A. Electromagnetic interaction in theory with Lorentz invariant CPT violation. Phys. Lett. B
**2013**, 718, 1500–1504. [Google Scholar] [CrossRef] [Green Version] - Duetsch, M.; Gracia-Bondia, J.M. On the assertion that PCT violation implies Lorentz non-invariance. Phys. Lett. B
**2012**, 711, 428–433. [Google Scholar] [CrossRef] [Green Version] - Greenberg, O. Remarks on a Challenge to the Relation between CPT and Lorentz Violation. arXiv
**2011**, arXiv:1105.0927. [Google Scholar] - Antonelli, V.; Miramonti, L.; Torri, M.D.C. Neutrino oscillations and Lorentz Invariance Violation in a Finslerian Geometrical model. Eur. Phys. J. C
**2018**, 78, 667. [Google Scholar] [CrossRef] [Green Version] - Torri, M.D.C. Neutrino Oscillations and Lorentz Invariance Violation. Universe
**2020**, 6, 37. [Google Scholar] [CrossRef] [Green Version] - Kostelecky, A. Riemann-Finsler geometry and Lorentz-violating kinematics. Phys. Lett. B
**2011**, 701, 137. [Google Scholar] [CrossRef] [Green Version] - Girelli, F.; Liberati, S.; Sindoni, L. Planck-scale modified dispersion relations and Finsler geometry. Phys. Rev. D
**2007**, 75, 064015. [Google Scholar] [CrossRef] [Green Version] - Edwards, B.R.; Kostelecky, V.A. Riemann-Finsler geometry and Lorentz-violating scalar fields. Phys. Lett. B
**2018**, 786, 319. [Google Scholar] [CrossRef] - Lämmerzahl, C.; Perlick, V. Finsler geometry as a model for relativistic gravity. Int. J. Geom. Meth. Mod. Phys.
**2018**, 15, 1850166. [Google Scholar] [CrossRef] [Green Version] - Lämmerzahl, C.; Perlick, V. Axiomatic formulations of modied gravity theories with nonlinear dispersion relations and Finsler Lagrange Hamilton geometry. Eur. Phys. J. C
**2018**, 78, 969. [Google Scholar] - Schreck, M. Classical Lagrangians and Finsler structures for the nonminimal fermion sector of the Standard-Model Extension. Phys. Rev. D
**2016**, 93, 105017. [Google Scholar] [CrossRef] [Green Version] - Stecker, F.W. Photodisintegration of ultrahigh-energy cosmic rays by the universal radiation field. Phys. Rev.
**1969**, 180, 1264–1266. [Google Scholar] [CrossRef] - Boncioli, D.; di Matteo, A.; Salamida, F.; Aloisio, R.; Blasi, P.; Ghia, P.L.; Grillo, A.; Petrera, S.; Pierog, T. Future prospects of testing Lorentz invariance with UHECRs. arXiv
**2015**, arXiv:1509.01046. [Google Scholar] - Boncioli, D.; Pierre Auger Collaboration. Probing Lorentz symmetry with the Pierre Auger Observatory. In Proceedings of the ICRC17, Busan, Korea, 10–20 July 2017; Volume 561. [Google Scholar]
- Lang, R.G.; Pierre Auger Collaboration. Testing Lorentz Invariance Violation at the Pierre Auger Observatory. In Proceedings of the ICRC19, Madison, WI, USA, 24 July–1 August 2019; p. 327. [Google Scholar]
- Bietenholz, W. Cosmic Rays and the Search for a Lorentz Invariance Violation. Phys. Rep.
**2011**, 505, 145–185. [Google Scholar] [CrossRef] [Green Version] - Mattingly, D. Modern tests of Lorentz invariance. Living Rev. Rel.
**2005**, 8, 5. [Google Scholar] [CrossRef] [Green Version] - Kostelecky, V.A.; Russell, N. Data Tables for Lorentz and CPT Violation. Rev. Mod. Phys.
**2011**, 83, 11–31. [Google Scholar] [CrossRef] [Green Version] - Pierre Auger Collaboration. Depth of maximum of air-shower profiles at the Pierre Auger Observatory. II. Composition implications. Phys. Rev. D
**2014**, 90, 122006. [Google Scholar] [CrossRef] [Green Version] - Diaz, J.S.; Klinkhamer, F.R.; Risse, M. Changes in extensive air showers from isotropic Lorentz violation in the photon sector. Phys. Rev. D
**2016**, 94, 085025. [Google Scholar] [CrossRef] [Green Version] - Klinkhamer, F.R.; Niechciol, M.; Risse, M. Improved bound on isotropic Lorentz violation in the photon sector from extensive air showers. Phys. Rev. D
**2017**, 96, 116011. [Google Scholar] [CrossRef] [Green Version] - Lang, R.G.; Martínez-Huerta, H.; de Souza, V. Limits on the Lorentz Invariance Violation from UHECR astrophysics. Astrophys. J.
**2018**, 853, 23. [Google Scholar] [CrossRef]

**Figure 2.**Inelasticity obtained in the case of LIV parameter ${f}_{p\pi}\simeq {f}_{p}=0$ as a function of the proton energy ${E}_{p}$ and of the photon energy ${\u03f5}^{\prime}$ defined in the proton rest frame.

**Figure 3.**Inelasticity obtained in the case of LIV parameter ${f}_{p\pi}\simeq {f}_{p}=9\times {10}^{-23}$ as a function of the proton energy ${E}_{p}$ and of the photon energy ${\u03f5}^{\prime}$ defined in the proton rest frame.

**Figure 4.**Inelasticity obtained in the case of LIV parameter ${f}_{p\pi}\simeq {f}_{p}=3\times {10}^{-24}$ as a function of the proton energy ${E}_{p}$ and of the photon energy ${\u03f5}^{\prime}$ defined in the proton rest frame.

**Figure 5.**GZK sphere radius as a function of production energy E simulated for 1000 protons generated randomly inside of a sphere with a radius of ∼$1270\phantom{\rule{0.166667em}{0ex}}\mathrm{Mpc}$, centered on Earth. The simulations for ten different values of the LIV parameter are shown in the energy range $0\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$–$2\times {10}^{20}\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$.

**Figure 6.**Attenuation length as a function of energy E, plotted for ten different values of the LIV parameter.

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**MDPI and ACS Style**

Torri, M.D.C.; Caccianiga, L.; di Matteo, A.; Maino, A.; Miramonti, L.
Predictions of Ultra-High Energy Cosmic Ray Propagation in the Context of Homogeneously Modified Special Relativity. *Symmetry* **2020**, *12*, 1961.
https://doi.org/10.3390/sym12121961

**AMA Style**

Torri MDC, Caccianiga L, di Matteo A, Maino A, Miramonti L.
Predictions of Ultra-High Energy Cosmic Ray Propagation in the Context of Homogeneously Modified Special Relativity. *Symmetry*. 2020; 12(12):1961.
https://doi.org/10.3390/sym12121961

**Chicago/Turabian Style**

Torri, Marco Danilo Claudio, Lorenzo Caccianiga, Armando di Matteo, Andrea Maino, and Lino Miramonti.
2020. "Predictions of Ultra-High Energy Cosmic Ray Propagation in the Context of Homogeneously Modified Special Relativity" *Symmetry* 12, no. 12: 1961.
https://doi.org/10.3390/sym12121961