Diffusion of UHECRs in the Expanding Universe A.Z. Gazizov LNGS, INFN, Italy R. Aloisio V....
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Transcript of Diffusion of UHECRs in the Expanding Universe A.Z. Gazizov LNGS, INFN, Italy R. Aloisio V....
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
SOCoR, Trondheim, June 2009
• 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…
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,
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)
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
SOCoR, Trondheim, June 2009
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
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).
)(),,(]),,([),,(),,( 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
SOCoR, Trondheim, June 2009
.)],(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:
SOCoR, Trondheim, June 2009
)()(
),(
)(
),(),()(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.
SOCoR, Trondheim, June 2009
,]'),',([
)'('
'')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
SOCoR, Trondheim, June 2009
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
SOCoR, Trondheim, June 2009
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
SOCoR, Trondheim, June 2009
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
SOCoR, Trondheim, June 2009
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
SOCoR, Trondheim, June 2009
V. Berezinsky & A.G. ApJ 669, 684 (2007)
PhenomenologicalPhenomenological Approach 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
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
SOCoR, Trondheim, June 2009
,])(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
SOCoR, Trondheim, June 2009
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
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.
,),(),(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
SOCoR, Trondheim, June 2009
.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
SOCoR, Trondheim, June 2009
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
Jüttner vs. Interpolation with BJüttner vs. Interpolation with Bcc= 0.01 nG= 0.01 nG
SOCoR, Trondheim, June 2009
Jüttner vs. Interpolation with BJüttner vs. Interpolation with Bcc= 0.1 nG= 0.1 nG
SOCoR, Trondheim, June 2009
Jüttner vs. Interpolation with BJüttner vs. Interpolation with Bc c = 1 nG= 1 nG
SOCoR, Trondheim, June 2009
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. .
SOCoR, Trondheim, June 2009
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.
SOCoR, Trondheim, June 2009
.),(2
)(),(
2
zE
zxzE m