1. A.Z. Gazizov LNGS, Italy Based on works with V. Berezinsky and R. Aloisio UHECR-08

3. HECR are observed, but the nature of particles is unknown: protons, nuclei (chemical composition (E)=? ); sources of UHECR are unknown: are they galactic or extragalactic ( AGN, GRB, Hypernovae…? ); mechanism of acceleration up to E>10 20 eV is unknown; the way the charged particles propagate in intergalactic space, either 1. rectilinearly or 2. via diffusion in ExtraGalactic Magnetic Fields is unknown; strength, time evolution and space distribution of these magnetic fields, B(E,t,r), are unknown. UHECR-08

4. In accordance with HiRes data, CR at E>10 18 eV are mostly (extragalactic) protons. UHECR-08

5. Protons p +  CMB → p + e + + e - p +  CMB → N + pions Nuclei Z +  CMB → Z + e + + e - A +  CMB → (A-1) + N A +  CMB → A’ + N + pions Photons  +  CMB → e + + e - e +  CMB → e +  Proton mean energy loss length in CMBR z=0 UHECR-08 Red-shift, dE/dt=H(t)E

6. In case of homogeneously distributed sources with luminosity L 0 and power-law generation spectrum index  g the diffuse universal spectrum arises. V. Berezinsky, A.G., S. I. Grigorieva, Phys. Rev. D 74, 043005 (2006); hep-ph/0204357 The diffuse flux where L 0 = L 0 n s (0) is the emissivity, L 0 and E are in GeV, m=  +  and n s (z) = n s (0) (1+z)  describes hypothetical evolution of sources., UHECR-08

7. Propagation of protons in the intergalactic space leaves imprints on the spectrum  GZK cutoff  Bump (washed out in the diffuse spectrum)  Dip ( E ~1×10 18 ÷ 4×10 19 eV ) UHECR-08

8. In a more realistic model sources are situated in the vertices of imaginary cubic lattice with edge length d ~ 30 ÷ 60 Mpc (AGN?) Here again L 0 = L 0 n s (0), E 0 =1 GeV, E g (E,z) is energy on characteristic. Comoving distance to a source is defined by coordinates {i, j, k}=0,  1,  2… It is assumed that all sources have the same luminosity, power-law generation index and maximum acceleration energy. Evolution may be included using (1+z) m factor for z  z c ~ 1.2. UHECR-08 Maximum distance between the detector and a source is defined either by z max or by E max.

9. Configuration (distributed as charged baryonic plasma?) and strength (10 -3  B  100 nG) of magnetic fields is basically unknown. 1.K. Dolag, D. Grasso, V. Springel & I. Tkachev, JKAS 37,427 (2004); 2.G. Sigl, F. Miniati & T. A. Enßlin, Phys. Rev. D 70, 043007 (2004); 3.K. Dolag, D. Grasso, V. Springel & I. Tkachev, Journal of Cosmology and Astro-Particle Physics 1, 9 (2005) 4.H. Kang, S. Das, D. Ryu & J. Sho, Proc. of 30 th ICRC, Merida, Mexico The only information comes from observations of Faraday rotation in the core of cluster of galaxies.  Dolag et al. : < 1  — weak magnetic fields  Sigl et al. : ~ 10  ÷20  — strong magnetic fields give different results: for protons with E>10 20 eV the deflection angle is Hydrodynamical MC simulations of large scale structure formation with B amplitude in the end rescaled to those observed in cores of galaxies, UHECR-08

10. Assume protons propagate through a turbulent magnetized plasma. On the basic scale of turbulence (assume, l c = 1 Mpc ) the r.m.s. of coherent magnetic field B c lies in the range 3×10 -3 ÷ 10 nG. Critical energy E c =0.926×10 18 (B c /1 nG) eV is determined by r L (E c ) = l c. Characteristic diffusion length for protons with energy E, l d (E), determines the diffusion coefficient D(E) = c l d (E)/3. If E » E c, i.e. r L (E c ) » l c, At E « E c, the diffusion length depends on the spectrum of turbulence: l d (E) = l c (E/E c ) 1/3 for the Kolmogorov diffusion; l d (E) = l c (E/E c ) for the Bohm regime. UHECR-08

11. In 1959 S.I. Syrovatsky solved this equation for the case of D(E) and b(E) being independent of t and (e.g. Galaxy). S. I. Syrovatsky, Sov. Astron. J. 3, 22 (1959) [Astron. Zh. 36, 17 (1959) ] source generation function Propagation of UHECR in turbulent magnetic fields may be described by the following differential equation: space densitydiffusion coefficientenergy loss UHECR-08

12. Hereis the squared distance a particle passes from a source in the direction of observer, while its energy diminishes from E g to E; (magnetic horizon) Density of particles with energy E at distance r from a source UHECR-08 b(E)=dE/dt is the total rate of energy loss due to interactions with CMB (red-shift + pair-production +  -photoproduction).

13. In R. Aloisio & V. Berezinsky, 2004, APJ 612, 900 the Syrovatsky solution was applied to calculation of UHECR spectrum in ‘static’ universe; it was assumed that the energy losses due to interactions with CMB are the same as at z=0, but the red-shift energy loss was included as well. The model assumed rectilinear particle propagation at very high energies and weak magnetic fields and diffusive one at low energies and strong magnetic fields. Interpolation was done at intermediate energies. The calculated spectrum is similar to the ‘universal’ one: GZK-cutoff, dip and fall down at low energies. There was also proved the ’propagation theorem’. …When the distance between sources, d, decreases, getting smaller than all scales involved (attenuation and diffusion lengths), the Syrovatsky solutions converge to the ‘universal’ spectrum. UHECR-08

14. In M. Lemoine, Phys. Rev. D 71, 083007 (2005) ; R. Aloisio & V. Berezinsky, Astrophys. J. 625, 249 (2005) the transition from Galactic to extragalactic UHECR spectrum at the second knee ( E~(3-7)×10 17 eV ) was proposed. UHECR-08

15. R. Aloisio & V. Berezinsky, Astrophys. J. 625, 249 (2005)

16. In V. Berezinsky & A.G. Astrophys. J. 643, 8 (2006) a solution to the diffusion equation in the expanding universe with time-dependent diffusion in turbulent magnetic fields coefficient D(E,t) and energy losses due to interactions with CMB, b(E,t), Here x c is the comoving distance between a detector and a source, E g (E,z) is the solution to an ordinary differential equation has been obtained: which defines the characteristic line E(E,t). UHECR-08

17. is the analog of the Syrovatsky’s solution variable. It is the squared distance the particle emitted at epoch z travels from a source to a detector (z=0); the integral is to be taken along the characteristic line. The variable UHECR-08

18. In the expanding universe evolution of magnetic fields should be taken into account. At epoch z parameters characterizing the magnetic filed (l c, B c ) become where (1+z) 2 describes the diminishing of B c with time due to magnetic flux conservation, and (1+z) -m is due to MHD amplification. The critical energy derived from r L (E c ) = l c (z) is UHECR-08

19. For power-law source generation function and sources being in the vertices of 3D cubic lattice, the diffuse flux is In the case of rectilinear proton propagation the flux is At intermediate energies a ( smooth ?) interpolation between these two solutions is to be used. An example is given in UHECR-08 R. Aloisio, V. Berezinsky & A.G, arXiv:0805.1867

20. UHECR-08

24.  GZK cutoff at E  5×10 19 eV and dip at 10 18 < E < 4×10 19 eV are (the observed) signatures of rectilinear UHE protons propagation in CMB from extragalactic sources.  The account for diffusion in reasonable EGMF does not influence the high-energy (E>E c ) part of the spectrum and suppresses its low-energy part ( E<10 18 eV ), thus allowing for the smooth transition from galactic to extragalactic spectrum at the second knee.  The successful in case of Galaxy Syrovatsky solution may be generalized to the description of diffusive UHE extragalactic particles propagation in the expanding universe with time and energy dependent energy losses b(E,t) and diffusion coefficient D(E,t). UHECR-08

25.  To describe the full extragalactic CR spectrum one should interpolate between the rectilinear and diffusive propagation regimes.  The calculated spectra in case of reasonable EGMF and reasonable assumptions about granularity of sources ( d < 50 Mpc ) retains the GZK-cutoff and dip features,  and converges to the universal spectrum with d  0.  The generalized Syrovatsky solution may be applied as well to the description of diffusive propagation of nuclei. The photodisintegration term may be taken easily into account. UHECR-08