Diffusion of UltraHigh Energy Particles in Expanding Universe

1
Diffusion of UltraHigh Energy Particles in
Expanding Universe

• A.Z. Gazizov
• LNGS, Italy

Based on works with V. Berezinsky and R. Aloisio
2
UHECR Spectrum
3
Unknown Features

• HECR are observed, but
• the nature of particles is unknown protons,
  nuclei (chemical composition ltAgt(E)?)
• sources of UHECR are unknown are they galactic
  or extragalactic (AGN, GRB, Hypernovae?)
• mechanism of acceleration up to Egt1020 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.

4
Protons as Extragalactic UHECR

• HiRes claims that CR spectrum at
• Egt1018 eV is dominated by protons.

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Energy Losses on CMB/IR
Proton mean energy loss length in CMBR

• 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 ?

z0
Red-shift, dE/dtH(t)E
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Universal Spectrum

• In case of homogeneously distributed sources with
  luminosity L0
• and power-law generation spectrum index gg

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 L0 L0 ns(0) is the emissivity, L0 and E
are in GeV, mab and ns(z) ns(0) (1z)b
describes hypothetical evolution of sources.
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Characteristic Lines
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Propagation Signatures in the Universal Spectrum
Propagation of protons in the intergalactic space
leaves imprints on the spectrum

• GZK cutoff
• Bump (washed out in the diffuse
  spectrum)
• Dip ( E 11018 41019eV )

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Rectilinear Propagation of Protons
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 L0 L0 ns(0), E01 GeV, Eg(E,z) is
energy on characteristic. Comoving distance to a
source is defined by coordinates i, j, k0,?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 (1z)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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Extragalactic Magnetic Fields
Configuration (distributed as charged baryonic
plasma?) and strength (10-3 ? B ? 100 nG) of
magnetic fields are basically unknown.
The only information comes from observations of
Faraday rotation in the core of cluster of
galaxies.
Hydrodynamical MC simulations of large scale
structure formation with B amplitude in the end
rescaled to those observed in cores of galaxies,

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

give different results for protons with Egt1020
eV the deflection angles are

• Dolag et al. lt 1? weak
  magnetic fields
• Sigl et al. 10? 20? strong
  magnetic fields

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Homogeneous Magnetic Field
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 310-3 10 nG.
Critical energy Ec0.9261018(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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Examples of Trajectories
N. Globus, D. Allard E. Parizot,
arXiv0709.1541 astro-ph
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Propagation of High-Energy Particles in
Magnetic Fields
Propagation of UHECR in turbulent magnetic fields
may be described by the following differential
equation
source generation function
space density
diffusion coefficient
energy loss
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)
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Syrovatsky Solution
Density of particles with energy E at distance r
from a source
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)
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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Static Universe Model
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
z0, 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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Transition from Galactic to Extragalactic Spectrum
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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Maximum distance to a source
R. Aloisio V. Berezinsky, Astrophys. J. 625,
249 (2005)
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Expanding Universe
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),
has been obtained
Here xc is the comoving distance between a
detector and a source, Eg(E,z) is the solution
to an ordinary differential equation
which defines the characteristic line E(E,t).
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Solution in the Expanding Universe
The variable
is the analog of the Syrovatskys solution
variable. It is the squared distance the particle
emitted at epoch z travels from a source to a
detector (z0) the integral is to be taken along
the characteristic line.
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Magnetic Field in Expanding Universe
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 (1z)2 describes the diminishing of Bc with
time due to magnetic flux conservation, and
(1z)-m is due to MHD amplification.
The critical energy derived from rL(Ec) lc(z)
is
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Diffuse Flux
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, arXiv0805.1867
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Convergence to Universal Spectrum
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Diffusion Solutions with Bc1 nG
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Diffusion Solutions with Bc0.1 nG
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Diffusion Solutions with Bc0.01 nG
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Conclusions

• GZK cutoff at E ? 51019 eV and dip at 1018 lt E lt
  41019 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 (EgtEc) part of the
  spectrum and suppresses its low-energy part
  (Elt1018 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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Conclusions cont.

• 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 lt 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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Thank you