The author has suggested that the knee phenomenon in the cosmic ray energy spectrum at 3 PeV can be explained as a split between a radiation-dominated expansion and a matter-dominated expansion of an expanding heat bath. The model proposed in 1985, in fact, predicted that high energy cosmic rays are emitted from AGN, massive black holes, in agreement with recent data from the Pierre Auger Observatory. Similarly, the ankle phenomenon at 3 EeV is shown to be explained by a split between inflational expansion and ordinary material expansion of the expanding heat bath, not unlike that in the expansion of the universe. All the spectral indicies in the respective regions of the energy spectra agree with the theoretical calculation from the respective expansion rates. It is shown that the ankle energy is approximately equal to the threshold energy of cosmic ray production of an electron positron pair on the cmb photon.
Cite this paper
Y. Tomozawa, "Ankle Phenomenon in the Cosmic Ray Energy Spectrum," Journal of Modern Physics
, Vol. 4 No. 3, 2013, pp. 385-389. doi: 10.4236/jmp.2013.43054
 A. Angelis, “The Mysteries of Cosmic Rays, Cern Courier,” 1999.
 T. Wibig and A. W. Wolfendale, “Cosmic Ray Composition at the Ankle in the Energy Spectrum,” Astrophysics and Space Sciences Transactions, Vol. 7, 2011, pp. 275-278. doi:10.5194/astra-7-275-2011
 M. Ichimura, et al., 29th International Cosmic Ray Conference, Vol. 101, 2005.
 Y. Tomozawa, “Magnetic Monopoles, Cosmic Rays and Quantum Gravity,” In: H. Terazawa and M. Yasue, Eds., Proceedings of INS International Symposium on Composite Models of Quarks and Leptons, Tokyo, 1985, p. 386.
 Y. Tomozawa, “The Origins of Cosmic Rays and Quantum Effects of Gravity,” In: F. Mancini, Ed., Quantum Field Theory, Ersever Science Publishers B. V., Amsterdam, 1986, p. 241.
 Y. Tomozawa, “Cosmic Rays, Quantum Effects on Gravity and Gravitational Collapse,” Second Workshop on Fundamental Physics, University of Puerto Rico, Humacao, 24-28 March 1986.
 Y. Tomozawa, “Gravitational Waves, Supernova and Quantum Gravity, in Symmetry in Nature,” Scuola Normale Superiore, Pisa, 1989, p. 779.
 Y. Tomozawa, “Exact Solution of the Quantum Einstein Equation and the Nature of Singularity,” In: D. Blair, et al., Eds., Proceedings of the 5th Marcel Grossman Meeting on General Relativity, Perth, 1988, p. 527.
 Y. Tomozawa, “Black Hole Oscillation,” In: D. Blair, et al., Eds., Proceedings of the 5th Marcel Grossman Meeting on General Relativity, Perth, 1988, p. 629.
 A. Majumdar and Y. Tomozawa, “Source Abundance of Cosmic Rays,” Progress of Theoretical Physics (Kyoto), Vol. 82, No. 3, 1989, pp. 555-562.
 A. Majumdar and Y. Tomozawa, “Dynamical Behavior in Gravitational Collapse,” Nuovo Cimento, Vol. 197B, No. 8, 1992, pp. 923-939. doi:10.1007/BF02899295
 Y. Tomozawa, Astronomy and Astrophysics Supplement Series, Vol. 97, 1993, p. 117.
 The Pierre Auger Collaboration, “Correlation of the Highest-Energy Cosmic Rays with the Positions of Nearby Active Galactic Nuclei,” Science, Vol. 318, No. 5852, 2007, pp. 938-943. doi:10.1126/science.1151124
 Y. Tomozawa, “High Energy Cosmic Rays from AGN and the GZK Cutoff,” Modern Physics Letters A, Vol. 23, 2008, p. 1991.
 G. F. Giudice, M. A. Luty, H. Murayama and H. R. Rattazzi, Journal of High Energy Physics, Vol. 12, 1998, p. 027.
 L. Randall and R. Sundrum, “Out of This World Super-symmetry Breaking,” Nuclear Physics B, Vol. 557, No. 1-2, 1999, pp. 79-118.
 F. Aharonian, et al., “HESS Very-High-Energy Gamma-Ray Sources without Identified Counterparts,” Astronomy & Astrophysics, Vol. 477, No. 1, 2008, pp. 353-363.
 Y. Tomozawa, “Evidence for a Dark Matter Particle,” 2010. arXiv:1002.1938.
 Y. Tomozawa, “Cosmic Rays from AGN, the Knee Energy Mass Scale and Dark Matter Particles,” 2010.
 Y. Tomozawa, “Search for a Dark Matter Particle Family,” 2012. arXiv:1207.2465
 K. Greisen, “End to the Cosmic-Ray Spectrum?” Physical Review Letters, Vol. 16, No. 7, 1966, pp. 748-750.
 G. T. Zatsepin, V. A. Kuzmin and Z. Pisma, Journal of Experimental and Theoretical Physics, Vol. 4, 1966, p. 114.
 V. F. Hess, “100 Years Cosmic Rays—Anniversary of Their Discovery.”
 E. Kolb and M. S. Turner, “The Early Universe,” Addison-Wesley, Redwood City, 1990.