diffusion of uhecrs in the expanding universe

Diffusion of UHECRs Diffusion of UHECRs in the Expanding Universein the Expanding Universe

A.Z. GazizovA.Z. GazizovLNGS, INFN, ItalyLNGS, INFN, Italy

A.Z. GazizovA.Z. GazizovLNGS, INFN, ItalyLNGS, INFN, Italy

Based on works with R. Aloisio R. Aloisio and V. BerezinskyV. Berezinsky

SOCoR, Trondheim, June 2009

AssumptionsAssumptions


• 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-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) + bee (E,t) + b (E,t).

bint(E,t) = bee (E,t) + b (E,t) is calculated using known differential cross-sections of p–scattering off CMB photons.

Rectilinear Propagation of ProtonsRectilinear Propagation of ProtonsIf 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)

with initial condition E (E,0) =E .

V. Berezinsky, A.G., S.I. Grigorieva, Phys. Rev. D74, 043005 (2006)SOCoR, Trondheim, June 2009

,)21()21()21( 222 kjidxz ijkijk

.]'),',([

'

'')1(),( int

0

E

E zzEb

dz

dtdzzzE

dE

dE zg exp

).,()1]()21()21()21[(

)],([

4

1)(

,,2222 ijk

g

kji ijk

ijkp zE

dE

dE

zkji

zEQ

dEJ

E

Comoving distance to a source is defined by coordinates {i, j, k} = 0, 1, 2…

Characteristic Lines at High Energies


Intergalactic Magnetic FieldsIntergalactic Magnetic FieldsSpace 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 > 1020 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,


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)

Homogeneous Magnetic FieldHomogeneous Magnetic Field

Let protons propagate in homogeneous turbulent magnetized plasma. On the basic scale of turbulence lc = 1 Mpc the coherent magnetic field Bc lies in the range 3×10-3 — 30 nG.

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

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

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

ld(E) = lc(E/Ec) for Bohm diffusion


The critical energy Ec EeV may be determined from rL(Ec) = lc.Bc

1 nG

If E » Ec , i.e. rL(Ec) » lc , ld(E) = 1.2× Mpc .EEeV2

BnG

Propagation in Magnetic FieldsPropagation in Magnetic Fields

source generation function

Propagation of UHECRs in turbulent magnetic fields may be described by differential equation:

space density diffusion coefficient energy loss


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).

)(),,(]),,([),,(),,( 3sppp rrtrEQntrEb

EntrEDtrEn

t

-][ div

S. I. Syrovatsky, Sov. Astron. J. 3, 22 (1959) [Astron. Zh. 36, 17 (1959) ]

Syrovatsky SolutionSyrovatsky SolutionThe space density of protons np(E,r) with energy E at distance r from a source


.)],(4[

),(4exp

)()(

1),(

23

2

g

g

g

E

gp

EE

EEr

EQdEEb

rEn

2323

2

4);,(

)],(4[

),(4exp

);,(tccr

tErtP

EE

EEr

ErEP rect

g

g

gdiff

The probabilities to find the particle at distance r in volume dV at time t (or when its energy reduces from Eg to E ), i.e. propagators, are

)'E(b)'E(D

'dE)E,E(gE

E

g

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 Eg to E;

Diffusion in the Expanding UniverseDiffusion in the Expanding UniverseIt was shown in V. Berezinsky & A.G. ApJ 643, 8 (2006) that the solution to the diffusion equation in the expanding Universe

Xs 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:


)()(

),(

)(

),(),()(3),( 3

32

2 ss

x xxta

tEQn

ta

tED

E

tEbnntH

E

ntEb

t

n

Pdiff(Eg,E,z)

.),(),(4

)],(4[]),,([)(),( 23

2

0

max

zEdE

dE

zE

zExzzEQz

dz

dtdzExn gs

s

z

s

exp

E

zmax is determined either by red-shift of epoch when UHECR generation startedor by maximum acceleration energy Emax = E (E,zmax) .

Terms in the Solution 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.


,]'),',([

)'('

'')1(),( int

0

g

zg

E

zzEbz

dz

dtdzzzE

dE

dE Eexp

,)1()1(

1)(

30

zzH

zdz

dt

m

)'(

]'),',([)'(

'

''),(

20 za

zzEDz

dz

dtdzzE

z E

with m = 0.3, = 0.7, H0 = 72 km/sMpc .

Magnetic Field in the Expanding UniverseMagnetic 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 (lc , Bc)

(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, rL [Ec(z) ] lc(z) determines the critical energy of protons at epoch z


lc(z) = lc /(1+z), Bc(z) = Bc × (1+z)2-m .

Bc

1 nGEc(z) ≈ 1 × 1018 (1+z)1-m eV .

Superluminal SignalSuperluminal Signal

Since v ≤ c, for all xs there exists zmin (minimum red-shift), given by

,)1(

min

03

0

z

m

sz

dz

H

cx


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).

Problem: 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 SS to DD (no energy losses!).

Superluminal Range of EnergiesSuperluminal Range of Energies


The hatched region corresponds to superluminal velocities.

The integrand of Syrovatsky solution as a function of Eg for fixed E.

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.

At low energies E and high B diffusion is good solution.However, the problem arises with energy increasing and B decreasing.

Diffusion & RectilinearDiffusion & Rectilinear


At low energies E, high B and large xs the diffusion approach is correct. At high energies E, low B and small xs the rectilinear solution is valid.

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 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?Can the diffusion equation be modified so that to avoid the superluminal signal?

In spite of many attempts, the covariant differential equation In spite of many attempts, the covariant differential equation describing diffusive propagation is still unknown. describing diffusive propagation is still unknown.

Diffusion & Rectilinear Solutions with B=1 nGDiffusion & Rectilinear Solutions with B=1 nG


V. Berezinsky & A.G. ApJ 669, 684 (2007)

PhenomenologicalPhenomenological Approach 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

kT

mv

kT

mvPM 2

exp2

)(223

.4

exp)4(

1),(

2

23

tD

r

tDtrP

diff

Transformation may be done changing v r and kT/m 2Dt .

Jüttner‘s RelativizationJüttner‘s Relativization


,])(exp[)(

)(||1)(

5

vZ

vvvf

;)1()( 212 vvwhere = 0,1;

;/)(4)(;/)(4)( 1120 KZKZ

Non-relativistic limit implies the dimensionless temperature parameter

K1() and K2() 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:

,2

exp2

),;(223

kT

vm

kT

mTmvfM

F. Jüttner, Ann. Phys. (Leipzig) 34, 856 (1911)

Generalization to DiffusionGeneralization to Diffusion


J. Dunkel et al. extended this relativization to the propagator of the solution to the diffusion equation.

,2

,2

0

D

tc

tc

xxv

Using a formal substitution

.

1

1

2exp

12

|)|();,(

2

0

2

2

52

02

3

00

tcxxD

tc

tcxx

Dtc

Zt

xxtcxxtPJ

This propagator reproduces rectilinear propagation for and looks like the diffusion propagator for

,||,0 0 tcxx

.||, 0 tcxx

The superluminal signal is impossible in this approach.The superluminal signal is impossible in this approach.

Jüttner’s Propagator in the Expanding UniverseJüttner’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.

,),(),(2

)(

)'(]'),',([

)'('

'2

)(

22

2

20

2222

zEzE

zx

zazzED

zdzdt

dz

zx

tD

tc

D

tc m

z

m

E

.1),;,( maxmaxmin EzExs;)(

),(zx

xzx

m

ss

z

m

mz

dz

H

czx

03

0 )'1(

')( is the maximum comoving distance the

particle would pass moving rectilinearly.

Generalized Jüttner’s PropagatorGeneralized Jüttner’s Propagator


.1

),(exp

)(4

),(

)1(

1)1(),,(

21

223

zE

K

zE

xxtEP

msgJ

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 xs from a source with energy E :

,),(1

),(exp

)(

),(

)1(

)],([

)(14

1),(

22

252

1

2

min

zEdE

dEzE

K

zEEEQ

zd

xcxEn gg

ss

For = 0

and for =1.),(

1

),(exp

)(

),(

)1(

)],([

)(14

1),(

21

22

1

2

min

zEdE

dEzE

K

zEEEQ

zd

xcxEn gg

ss

.)()(

1,2

)()(1 3

2

2

1

23

21

KKe

KKandforandFor

Three Length Scales: Three Length Scales: xxss, , xxmm and and ½½

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 line E (Eg ,zg ; E) passing through {Eg, zg} gives the red-shift z of epoch when energy diminishes to E.

gz

z

mm dzzHcx '))'1(( 21310 - max distance (rectilinear

propagation)

]'),';,([)'(''');,( 2 zzzEDzadzdtdzEzE gg

z

z

gg

g

E - squared diffusion distance


2If ½ « xs « xm ( = xm /2 » 1) and «1 pure diffusion; PJ

If ½ » xm ( « 1 ) and xs /xm 1 PJ 0 . Just for xs /xm 1, PJ This is the pure rectilinear propagation.

23

2

)4(

4exp

x

PS

s

34

)1(

srect xP

Diffusion vs. Diffusion vs. RectilinearRectilinear


Jüttner vs. Interpolation with BJüttner vs. Interpolation with Bcc= 0.01 nG= 0.01 nG


Jüttner vs. Interpolation with BJüttner vs. Interpolation with Bcc= 0.1 nG= 0.1 nG


Jüttner vs. Interpolation with BJüttner vs. Interpolation with Bc c = 1 nG= 1 nG


ConclusionsConclusionsDiffusion in turbulent IGMF 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 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. .


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.Conclusions cont.


.),(2

)(),(

2

zE

zxzE m

Thank you.


diffusion of uhecrs in the expanding universe

Documents

uhecrs e

t b e

energy e

e ht bee e

initial condition e

t bint e

p g e e p p g p

zm egg