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Diffusion of UHECRs in the Expanding Universe

Diffusion of UHECRs in the Expanding Universe. A.Z. Gazizov LNGS, INFN, Italy. Based on works with R. Aloisio and V . Berezinsky. Assumptions. UHECRs ( E ≥ 10 18 eV ) are mostly extragalactic protons . They are produced in yet unknown powerful distant sources ( AGN ? )

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Diffusion of UHECRs in the Expanding Universe

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

  2. Assumptions • UHECRs (E ≥ 1018eV) 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 zmax~ 2 – 5 and CRs are accelerated up to Emax = 1021 – 1023 eV. • Source generation function is power-law decreasing, Q(E,z)  (1+z)m E-gg, • with indicesgg = 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 + g  e+ + e- + p; p + g  p,0(K ,0) + X, is assumed:b(E,t) = dE/dt = E H(t) + bee (E,t) + b (E,t). • bint(E,t) = bee (E,t) + b (E,t)is calculated using known differential cross-sections • of pg –scattering off CMB photons. SOCoR, Trondheim, June 2009

  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 + bee(E,t) + bint(E,t) withinitial conditionE (E,0) =E . Comoving distance to a source is defined by coordinates {i, j, k} = 0, 1, 2… V. Berezinsky, A.G., S.I. Grigorieva, Phys. Rev. D74, 043005 (2006) SOCoR, Trondheim, June 2009

  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. Recently J. Lee et al. arXiv:0906.1631v1 [astro-ph.CO] explained the enhancement of RM in high density regions at r ≥ 1h-1Mpc from the locations of background radio sources by IGMF coherent over 1h-1Mpc with mean field strength B ≈ 30 nG. Magnetohydrodynamic simulationsof large scale structure formation with B amplitude in the end rescaled to the observed in cores of galaxies, K. Dolag, D. Grasso, V. Springel & I. Tkachev, JKAS 37,427 (2004); JCAP 1, 9 (2005); 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) give different results: for protons with E > 1020eVthe deflection angle is • Dolag et al. : < 1 — weak magnetic fields • Sigl et al. : ~ 10 ÷20 — strong magnetic fields SOCoR, Trondheim, June 2009

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

  7. Propagation in Magnetic Fields Propagation of UHECRs in turbulent magnetic fields may be described by differential equation: space density diffusion coefficient energy loss source generation function 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) ] SOCoR, Trondheim, June 2009

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

  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 with time-dependent D(E,t) and b(E,t)and scale parameter a(z) = (1+z)-1 is: Pdiff(Eg,E,z) Xs is the comoving distance between the detector and a source. zmax is determined either by red-shift of epoch when UHECR generation started or by maximum acceleration energy Emax = E (E,zmax) . SOCoR, Trondheim, June 2009

  10. Terms in the Solution with Wm = 0.3, L = 0.7,H0 = 72 km/sMpc. 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

  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 epochzparameters characterizing the magnetic filed, basic scale of turbulence and strength(lc , Bc) lc(z) = lc/(1+z), Bc(z) = Bc × (1+z)2-m . (1+z)2describes the diminishing of B with time due to magnetic flux conservation; (1+z)-mis due to MHD amplification. Equating the Larmor radius to the basic scale of turbulence, rL [Ec(z) ] lc(z) determines the critical energy of protons at epoch z Bc Ec(z) ≈ 1 × 1018 (1+z)1-meV . 1 nG SOCoR, Trondheim, June 2009

  12. Superluminal Signal Problem: Diffusion equation is the parabolic (relativistic non-covariant) one. n/t and 2n/r2 enter on the same foot. It does not know c. Hence: the superluminal propagation is possible. A generated proton can immediately arrive from S to D(no energy losses!). Since v ≤ c, for all xsthere existszmin (minimum red-shift), given by such that only particles emitted at z ≥ zmin(xs)reach the detector. And for any observed energy E there exists Emin[E,zmin(xs)]  E. CRs generated with Eg < Emin(E,zmin)do not contribute to the observed flux Jp(E). SOCoR, Trondheim, June 2009

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

  14. Diffusion & Rectilinear • At low energies E, highB and large xs the diffusion approach is correct. • At high energies E,lowB and small xs the rectilinear solution is valid. Can the interpolation between these regimes solve the problem? For each B, E and xs 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 diffusion equation be modified so that to avoid the superluminal signal? In case of Quantum Mechanics, relativization of parabolic Schrödinger equation brought to the Quantum Field Theory. In spite of many attempts, the covariant differential equation describing diffusive propagation is still unknown. SOCoR, Trondheim, June 2009

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

  16. Phenomenological Approach 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 . SOCoR, Trondheim, June 2009

  17. Jüttner‘s Relativization 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) where  = 0,1; Non-relativistic limit implies the dimensionless temperature parameter  = m/kT. K1() and K2() are modified Bessel functions of second kind. SOCoR, Trondheim, June 2009

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

  19. Jüttner’s Propagator in the Expanding Universe “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. SOCoR, Trondheim, June 2009

  20. Generalized Jüttner’s Propagator In terms of a andx 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 xsfrom a source with energy E : For  = 0 and for  =1 SOCoR, Trondheim, June 2009

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

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

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

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

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

  26. Conclusions • Diffusion in turbulentIGMFdoes not influence the high-energy(E>Ec) part of the spectrum and suppresses its low-energy part (E<1018 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 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. Conclusions cont. • 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 andsmoothly 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 SOCoR, Trondheim, June 2009

  28. Thank you. SOCoR, Trondheim, June 2009

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