A.Z. GazizovLNGS, Italy

A.Z. GazizovLNGS, Italy

Based on works with V. Berezinsky and R. Aloisio

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HECR are observed, butthe nature of particles is unknown: protons, nuclei (chemical composition <A>(E)=?);sources of UHECR are unknown: are they galactic or extragalactic (AGN, GRB, Hypernovae…?);mechanism of acceleration up to E>1020 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.

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HiRes claims that CR spectrum atE>1018 eV is dominated by protons.

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Protonsp + CMB → p + e+ + e-

p + CMB → N + pions

NucleiZ + 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

Red-shift, dE/dt=H(t)E

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In case of homogeneously distributed sources with luminosity L0

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/0204357The diffuse flux

where 0= L0 ns(0) is the emissivity, L0 and E are in GeV, m=+ and ns(z) = ns(0) (1+z) describes hypothetical evolution of sources.

,

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Propagation of protons in the intergalactic space leaves imprints on the spectrum:

GZK cutoff

Bump (washed out in the diffuse spectrum)

Dip ( E ~1×1018 ÷ 4×1019eV )

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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 0= L0 ns(0), E0=1 GeV, Eg(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 (probably, for z zc ~ 1.2).

Maximum distance between a detector and a source is defined either by zmax or by Emax.

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Configuration (distributed as charged baryonic plasma?) and strength (10-3 B 100 nG) of magnetic fields are 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 30th 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>1020 eV the deflection angles are

Hydrodynamical MC simulations of large scale structure formation with B amplitude in the end rescaled to those observed in cores of galaxies,

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Assume protons propagate through a turbulent magnetized plasma.

On the basic scale of turbulence (assume, lc = 1 Mpc ) the r.m.s. of coherent magnetic field Bc lies in the range 3×10-3 ÷ 10 nG.

Critical energy Ec=0.926×1018(Bc /1 nG) eV is determined by rL(Ec) = lc.

Characteristic diffusion length ld(E ) for protons with energy E determines the diffusion coefficient D(E) = c ld(E)/3.

If E » Ec, i.e. rL(Ec) » lc ,

At E « Ec , the diffusion length depends on the spectrum of turbulence:

ld(E) = lc(E/Ec)1/3 for the Kolmogorov diffusion;

ld(E) = lc(E/Ec) for the Bohm regime.

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N. Globus, D. Allard & E. Parizot, arXiv:0709.1541 [astro-ph]

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In 1959 S.I. Syrovatsky solved this equation for the case of D(E) and

b(E) being independent of t and r (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 density diffusion coefficient energy loss

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Here is the squared distance a particle

passes from a source in the direction of observer, while its energy diminishes from Eg to E; (magnetic horizon)

Density of particles with energy E at distance r from a source

b(E)=dE/dt is the total rate of energy loss due to interactions with CMB (red-shift + pair-production + -photoproduction).

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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.

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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)×1017 eV ) was proposed.

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)(max Er rect

)(max Erdiff

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

)(max Er rect

)(max Erdiff

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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 xc is the comoving distance between a detector and a source, Eg(E,z) is the solution to an ordinary differential equation

has been obtained:

)],,(),()([ tEbtEbEtHdt

dE pee

which defines the characteristic line (E,t).Quarks-08

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

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In the expanding universe evolution of magnetic fields should be taken into account. At epoch z parameters characterizing the magnetic filed (lc, Bc) become

where (1+z)2 describes the diminishing of Bc with time due to magnetic flux conservation, and (1+z)-m is due to MHD amplification.

The critical energy derived from rL(Ec) = lc(z) is

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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 using the Jüttner propagator is given in

R. Aloisio, V. Berezinsky & A.G, arXiv:0805.1867

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GZK cutoff at E 5×1019 eV and dip at 1018 < E < 4×1019 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>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 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).

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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.

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