1. Diffusion of UHECRs in the Expanding Universe A.Z. Gazizov LNGS, INFN, Italy R. Aloisio V. Berezinsky Based on works with R. Aloisio and V. Berezinsky SOCoR, Trondheim, June 2009

2. Assumptions UHECRs (E ≥ 10 18 eV) are mostly extragalactic protons. They are produced in yet unknown powerful distant sources (AGN ?) isotropically distributed in space at d ~ 40 – 60 Mpc at z=0. Production of CRs simultaneously started at some z max ~ 2 – 5 and CRs are accelerated up to E max = 10 21 – 10 23 eV. Source generation function is power-law decreasing, Q(E,z)  (1+z) m E -  g, with indices  g = 2.1 – 2.7; m = 0 – 4 accounts for possible evolution. The continuous energy loss (CEL) approximation due to red-shift and collisions with CMBR, p +  e +  e - + p; p +    (K   ) + X, is assumed: b(E,t) = dE/dt = E H(t) + b ee (E,t) + b  (E,t). b int (E,t) = b ee (E,t) + b  (E,t) is calculated using known differential cross-sections of p  –scattering off CMB photons.

3. Rectilinear Propagation of Protons If InterGalactic Magnetic Field (IGMF) is absent, HECRs move rectilinearly. For sources situated in knots of an imaginary cubic lattice with edge length d, the observed flux is E (E,z) is the solution to the ordinary differential equation -dE/dt = EH + b ee (E,t) + b int (E,t) with initial condition E (E,0) =E. V. Berezinsky, A.G., S.I. Grigorieva, Phys. Rev. D74, 043005 (2006) SOCoR, Trondheim, June 2009 Comoving distance to a source is defined by coordinates {i, j, k} = 0,  1,  2…

4. Characteristic Lines at High Energies SOCoR, Trondheim, June 2009

5. Intergalactic Magnetic Fields Space configuration (  charged baryonic plasma?), strength (10 -3  B  100 nG) and time evolution of IGMF are basically unknown.. Some information comes from observations of Faraday Rotation in cores of clusters of galaxies. K. Dolag, D. Grasso, V. Springel & I. Tkachev, JKAS 37,427 (2004); JCAP 1, 9 (2005);  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 Magnetohydrodynamic simulations of large scale structure formation with B amplitude in the end rescaled to the observed in cores of galaxies, SOCoR, Trondheim, June 2009 Recently J. Lee et al. arXiv:0906.1631v1 [astro-ph.CO] explained the enhancement of RM in high density regions at r ≥ 1h -1 Mpc from the locations of background radio sources by IGMF coherent over 1h -1 Mpc with mean field strength B ≈ 30 nG. G. Sigl, F. Miniati & T. A. Enßlin, Phys. Rev. D 70, 043007 (2004); E. Armengaud, G. Sigl, F. Miniati, Phys. Rev. D 72, 043009 (2005)

6. Homogeneous Magnetic Field Let protons propagate in homogeneous turbulent magnetized plasma. On the basic scale of turbulence l c = 1 Mpc the coherent magnetic field B c lies in the range 3×10 -3 — 30 nG. Characteristic diffusion length for protons with energy E, l d (E), determines the diffusion coefficient D(E)  c l d (E)/3. At E « E c, the diffusion length depends on the spectrum of turbulence: l d (E) = l c (E/E c ) 1/3 for Kolmogorov diffusion l d (E) = l c (E/E c ) for Bohm diffusion SOCoR, Trondheim, June 2009 The critical energy E c  EeV may be determined from r L (E c ) = l c. BcBc 1 nG If E » E c, i.e. r L (E c ) » l c, l d (E) = 1.2× Mpc. E EeV 2 B nG

7. Propagation in Magnetic Fields source generation function Propagation of UHECRs in turbulent magnetic fields may be described by differential equation: space densitydiffusion coefficientenergy loss SOCoR, Trondheim, June 2009 In 1959 S.I. Syrovatsky solved this equation for the case of D(E) and b(E) independent of t and r (e.g. for CRs in Galaxy).  S. I. Syrovatsky, Sov. Astron. J. 3, 22 (1959) [Astron. Zh. 36, 17 (1959) ]

8. Syrovatsky Solution The space density of protons n p (E,r) with energy E at distance r from a source SOCoR, Trondheim, June 2009 The probabilities to find the particle at distance r in volume dV at time t (or when its energy reduces from E g to E ), i.e. propagators, are b(E) = dE/dt is the total rate of energy loss. is the squared distance a particle passes from a source while its energy diminishes from E g to E;

9. Diffusion in the Expanding Universe It was shown in V. Berezinsky & A.G. ApJ 643, 8 (2006) that the solution to the diffusion equation in the expanding Universe X s is the comoving distance between the detector and a source. with time-dependent D(E,t) and b(E,t) and scale parameter a(z) = (1+z) -1 is: SOCoR, Trondheim, June 2009 P diff (E g,E,z) z max is determined either by red-shift of epoch when UHECR generation started or by maximum acceleration energy E max = E (E,z max ).

10. Terms in the Solution is the analog of the Syrovatsky variable, i.e. the squared distance a particle emitted at epoch z travels from a source to the detector. SOCoR, Trondheim, June 2009 with  m = 0.3,  = 0.7, H 0 = 72 km/sMpc.

11. Magnetic Field in the Expanding Universe In the expanding Universe a possible evolution of average magnetic fields is to be taken into account. At epoch z parameters characterizing the magnetic filed, basic scale of turbulence and strength (l c, B c ) (1+z) 2 describes the diminishing of B with time due to magnetic flux conservation; (1+z) -m is due to MHD amplification. Equating the Larmor radius to the basic scale of turbulence, r L [E c (z) ]  l c (z) determines the critical energy of protons at epoch z SOCoR, Trondheim, June 2009 l c (z) = l c /(1+z), B c (z) = B c × (1+z) 2-m. BcBc 1 nG E c (z) ≈ 1 × 10 18 (1+z) 1-m eV.

12. Superluminal Signal Since v ≤ c, for all x s there exists z min (minimum red-shift), given by SOCoR, Trondheim, June 2009 such that only particles emitted at z ≥ z min (x s ) reach the detector. And for any observed energy E there exists E min [E,z min (x s )]  E. CRs generated with E g < E min (E,z min ) do not contribute to the observed flux J p (E). Problem: Problem: Diffusion equation is the parabolic (relativistic non-covariant) one.  n/  t and  2 n/  r 2 enter on the same foot. It does not know c. Hence: the superluminal propagation is possible. A generated proton can SD immediately arrive from S to D (no energy losses!).

13. Superluminal Range of Energies SOCoR, Trondheim, June 2009 The hatched region corresponds to superluminal velocities. The integrand of Syrovatsky solution as a function of E g for fixed E. On the other hand, the exact solution to the diffusion equation implies E min = E. Contribution of the energy range [E,E min ] results in the superluminal signal. At low energies E and high B diffusion is good solution. However, the problem arises with energy increasing and B decreasing.

14. Diffusion & Rectilinear SOCoR, Trondheim, June 2009 At low energies E, high B and large x s the diffusion approach is correct. At high energies E, low B and small x s the rectilinear solution is valid. For each B, E and x s the type of propagation is uncertain. Moreover, it changes during the propagation due to energy losses b(E,z) and varying magnetic field B(z). The diffusion coefficient D(E,z) varies. Can the interpolation between these regimes solve the problem? In case of Quantum Mechanics, relativization of parabolic Schrödinger equation brought to the Quantum Field Theory. Can the diffusion equation be modified so that to avoid the superluminal signal? In spite of many attempts, the covariant differential equation describing diffusive propagation is still unknown.

15. Diffusion & Rectilinear Solutions with B=1 nG SOCoR, Trondheim, June 2009 V. Berezinsky & A.G. ApJ 669, 684 (2007)

16. Phenomenological Approach SOCoR, Trondheim, June 2009 J. Dunkel, P. Talkner & P. Hänggi, arXiv:cond-mat/0608023v2; J. Dunkel, P. Talkner & P. Hänggi, Phys. Rev. D75, 043001 (2007) pointed out an analogy between the Maxwellian velocity distribution of particles with mass m and temperature T and the Green function of the solution to diffusion equation with constant diffusion coefficient D Transformation may be done changing v  r and kT/m  2Dt.

17. Jüttner‘s Relativization SOCoR, Trondheim, June 2009 where  = 0,1; Non-relativistic limit implies the dimensionless temperature parameter K 1 (  ) and K 2 (  ) are modified Bessel functions of second kind.  = m/kT. In 1911 Ferencz Jüttner, starting from the maximum entropy principle, proposed for 3-d Maxwell’s PDF the following relativistic generalizations: F. Jüttner, Ann. Phys. (Leipzig) 34, 856 (1911)

18. Generalization to Diffusion SOCoR, Trondheim, June 2009 J. Dunkel et al. extended this relativization to the propagator of the solution to the diffusion equation. Using a formal substitution This propagator reproduces rectilinear propagation for and looks like the diffusion propagator for The superluminal signal is impossible in this approach.

19. Jüttner’s Propagator in the Expanding Universe SOCoR, Trondheim, June 2009 “Jüttner‘s“ propagator of Dunkel et al. is not valid in the important case of t and E dependent D(E,t) and when energy losses b(E,t) are taken into account. On the other hand, the solution to the diffusion equation in the expanding Universe with account for D(E,t) and b(E,t) is already known V. Berezinsky & A.G., ApJ 643, 8 (2006). In R. Aloisio, V. Berezinsky & A.G., ApJ 693, 1275 (2009) the approach of Dunkel at al. is generalized to this case. is the maximum comoving distance the particle would pass moving rectilinearly.

20. Generalized Jüttner’s Propagator SOCoR, Trondheim, June 2009 In terms of  and  we arrive at generalized Jüttner’s propagator for particles ‘diffusing‘ in the expanding Universe filled with turbulent IGMF and losing their energy both adiabatically and in collisions with CMB. For  = 1 Actually, there are two solutions for space density of particles at distance x s from a source with energy E : For  = 0 and for  =1

21. Three Length Scales: x s, x m and ½ Assume the source S has emitted a proton at epoch z g with energy E g. What is the probability to find this particle at distance x s from a source with energy E in volume dV ? Characteristic line E (E g,z g ; E) passing through {E g, z g } gives the red-shift z of epoch when energy diminishes to E. - max distance (rectilinear propagation) - squared diffusion distance SOCoR, Trondheim, June 2009 2 If ½ « x s « x m (  = x m /2 » 1) and  «1 pure diffusion; P J  If ½ » x m (  « 1 ) and x s /x m  1 P J  0. Just for  x s /x m  1, P J  This is the pure rectilinear propagation. s

22. Diffusion vs. Rectilinear SOCoR, Trondheim, June 2009

23. Jüttner vs. Interpolation with B c = 0.01 nG SOCoR, Trondheim, June 2009

24. Jüttner vs. Interpolation with B c = 0.1 nG SOCoR, Trondheim, June 2009

25. Jüttner vs. Interpolation with B c = 1 nG SOCoR, Trondheim, June 2009

26. Conclusions Diffusion in turbulent IGMF 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 Syrovatsky solution may be generalized to the diffusive propagation of extragalactic CRs in the expanding Universe with time and energy dependent b(E,t) and D(E,t). Superluminal propagation Superluminal propagation is inherent to (parabolic) diffusion equation. It distorts the calculated spectrum of UHECRs. The formal analogy between Maxwell’s velocity distribution and of the propagator of diffusion equation solution allows the relativization of the latter (as it was done by F. Jüttner for the velocity distribution) see J. Dunkel et al.. SOCoR, Trondheim, June 2009

27. It is possible to generalize the Jüttner’s propagator to the diffusion in the expanding Universe with energy and time dependent energy losses and diffusion coefficient. Generalized Jüttner‘s propagator eliminates the superluminal signal and smoothly interpolates between the rectilinear and diffusion motion. Spectra calculated using this propagator have no peculiarities. A natural parameter describing the measure of ‘diffusivity’ of the propagator is Conclusions cont. SOCoR, Trondheim, June 2009

28. Thank you. SOCoR, Trondheim, June 2009