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Lorentz invariance violation and chemical composition of UHECRs
A. Saveliev, L. Maccione, G. Sigland in collaboration with several other people
TAUP - Munich - 05.09.2011
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The problem of QG
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The problem of QG
Why do we need quantum gravity?Philosophical intuition: reductio ad unumLack of predictability by GR (singularities, BH entropy)
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The problem of QG
Why do we need quantum gravity?Philosophical intuition: reductio ad unumLack of predictability by GR (singularities, BH entropy)
What do we need to understand quantum gravity?Observe phenomena that “have to do” with QGExtract testable predictions from the theory(ies)
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Why LV?
Homogeneity
Principle of relativity
Isotropy
Pre-causality
Lorentz invariance
Implies linearity of coordinate transformations
Implies the group structure
Implies reciprocity togetherwith Principle of Relativity
Implies a notion of past and future
Known theories of gravity rest on Einstein’s equivalence principle local Lorentz invariance
von Ignatowski (1910-1911)
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Challenging Lorentz invariance
lPl =�
�GN/c3 ⇥ 1.6� 10�35 mMPl =
��c/GN ⇥ 1.22� 1019 GeV
Lorentz invariance relates, through homogeneity, short to long distances.
What happens if we have a minimum length scale?
Homogeneity likely to be broken
Boost invariance likely to be broken
-
Modified dispersion relations
€
M ≡ spacetime structure scale, generally assumed ≈ MPlanck =1019 GeV
Assuming rotation invariance we can expand this as
From a purely phenomenological point of view, the general form of Lorentz invariance violation (LIV) is encoded into the dispersion relations
Many of these QG models have led to modified dispersion relations
E2 = p2 + m2 + �(p, M)
…
5
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UHECRs can be probes of Lorentz symmetry violation, e.g. induced by Quantum Gravity
Lorentz violation is expressed through modified dispersion relations
An estimate of the critical momentum at which LV effects are important is
Exotic physics and UHECRs
E2 = c2p2�
1 +m2c2
p2+
⇤
n
�(n)pn�2
Mn�2Pl
⇥
€
m2
p2≈pn−2
Mn−2⇒ pcrit ≈ m
2Mn−2n
-
UHECRs can be probes of Lorentz symmetry violation, e.g. induced by Quantum Gravity
Lorentz violation is expressed through modified dispersion relations
An estimate of the critical momentum at which LV effects are important is
Exotic physics and UHECRs
E2 = c2p2�
1 +m2c2
p2+
⇤
n
�(n)pn�2
Mn�2Pl
⇥
€
m2
p2≈pn−2
Mn−2⇒ pcrit ≈ m
2Mn−2n
n pcrit for νe pcrit for e- pcrit for p+
2 p ≈ mν~1 eV p≈me=0.5 MeV p≈mp=0.938 GeV
3 ~1 GeV ~10 TeV ~1 PeV
4 ~100 TeV ~100 PeV ~3 EeV
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CR Spectrum
A View of the All Particle SpectrumKASCADE-Grande collaboration, arXiv:1009.4716
Ultra-High Energy Cosmic Rays
2Freitag, 3. Juni 2011
KASKADE-Grande collaboration arXiv:1009.4716
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LIV effects on UHECR spectra
Propagated (simulated) LIV spectra LM, Taylor, Mattingly, Liberati, JCAP 0904 (2009)
modified GZK effect
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LV in the spectrum
Final constraints in case n=4
red/blue regions:allowed by absence of VC up to ~1020 eV
green points/black crosses:in agreement with observed spectrum within 95% and 99% CL resp.
LM, Taylor, Mattingly, Liberati, JCAP 0904 (2009)
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Also nuclei?
E [eV]
1810
1910
]2
> [
g/c
mm
ax
<X
650
700
750
800
850 proton
iron
QGSJET01QGSJETIISibyll2.1EPOSv1.99
E [eV]
1810
1910
]2
) [g
/cm
ma
xR
MS
(X
0
10
20
30
40
50
60
70 proton
iron
Figure 1.6: Recent results for the chemical composition of cosmic rays at ultra high ener-gies, showing ⇥Xmax⇤ (left) and RMS (Xmax) (= �x) (right) for measurements (filled datapoints) and simulations (lines and unfilled data points). Upper panel: Pierre Auger Col-laboration; Lower panel: High Resolution Fly’s Eye Collaboration; taken from [25] and[26]
These cascades start by the hadronic interaction of a cosmic ray nucleuswith atoms of the atmosphere. In the first order only pions, either chargedor neutral, are produced. The neutral pions almost instantly decay into twophotons which induce an electromagnetic cascade. Charged pions for theirpart either decay (at low energies) into muons and muon neutrinos or re-interact (at high energies), producing both kinds of pions again (for furtherreference see e.g. [24]).
The two main quantities to deduct chemical composition from the cascadeare the average value and the fluctuation of Xmax (the (longitudinal) depthof the shower at which the number of secondary particles is maximal), called⇥Xmax⇤ and Xmax, respectively. Using a simplified air shower model based onthe ideas of Heitler, one can show that ⇥Xmax⇤ is smaller for heavier nucleiwith the same energy, approximately given by [7]
⇥Xmax⇤ = � (ln E � ⇥ln A⇤ + ⇥) , (1.21)
� and ⇥ being some hadronic, interaction-specific parameters, E the energyof the nucleus and A its mass number. On the other hand, RMS (Xmax) ispredicted to decrease with A.
12
PAO Coll, PRL 104 (2010)
HiReS Coll, PRL 104 (2010)
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LIV in heavy nuclei
Transformation of a nucleus (A,Z) into one or more nuclei (A’,Z’).
Dispersion relation
2 new processes: - emission of Cherenkov radiation in vacuum- spontaneous decay
1 existing process: photodisintegration
LIV for UHECR Nuclei
Ultra High Energy Cosmic Rays (UHECRs) are the particles withthe highest energies ever observed æ Candidates for observing LIVe�ects
I The main reaction for UHECR nuclei is photodisintegration, inthe simplest case:
A
Z
N + “ æ AÕZ
ÕN Õ + BW
N ÕÕ
with AÕ = A ≠ B and Z Õ = Z ≠ W .However, due to LIV and the MDR for composite particles[Jacobson et al., 2003],
E 2A,Z = p2A,Z + m2A,Z +
÷
A2pn+2
A,Z
MnPl,
two new reactions may appear:
14 A. Saveliev
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Lorentz invariance violations and UHE nuclei
A. Saveliev, LM, G. Sigl, JCAP 2011LIV for UHECR Nuclei
-102 -100 -10-2 -10-4 -10-6 -10-818.5
19.0
19.5
20.0
20.5
21.0
21.5-102 -100 -10-2 -10-4 -10-6 -10-8
h
logp thrdec êeV
4He
16O
56Fe
10-8 10-6 10-4 10-2 100 10218.5
19.0
19.5
20.0
20.5
21.0
21.510-8 10-6 10-4 10-2 100 102
h
logp thrVCêeV
4He
16O
56Fe
Constraints from Spontaneous Decay and VC radiation
Emax
= 1019.6 eV Emax
= 1020 eV4He ≠3 ◊ 10≠3 . ÷ . 4 ◊ 10≠3 ≠7 ◊ 10≠5 . ÷ . 1 ◊ 10≠416O ≠7 ◊ 10≠2 . ÷ . 1 ≠2 ◊ 10≠3 . ÷ . 3 ◊ 10≠256Fe ≠1 . ÷ . 200 ≠3 ◊ 10≠2 . ÷ . 4
[Saveliev et al., 2011]
Constraints from spontaneous decay and VC emission
LIV for UHECR Nuclei
-102 -100 -10-2 -10-4 -10-6 -10-818.5
19.0
19.5
20.0
20.5
21.0
21.5-102 -100 -10-2 -10-4 -10-6 -10-8
h
logp thrdec êeV
4He
16O
56Fe
10-8 10-6 10-4 10-2 100 10218.5
19.0
19.5
20.0
20.5
21.0
21.510-8 10-6 10-4 10-2 100 102
h
logp thrVCêeV
4He
16O
56Fe
Constraints from Spontaneous Decay and VC radiation
Emax
= 1019.6 eV Emax
= 1020 eV4He ≠3 ◊ 10≠3 . ÷ . 4 ◊ 10≠3 ≠7 ◊ 10≠5 . ÷ . 1 ◊ 10≠416O ≠7 ◊ 10≠2 . ÷ . 1 ≠2 ◊ 10≠3 . ÷ . 3 ◊ 10≠256Fe ≠1 . ÷ . 200 ≠3 ◊ 10≠2 . ÷ . 4
[Saveliev et al., 2011]
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Lorentz invariance violations and UHE nuclei
18.5 19.0 19.5 20.0 20.5 21.0log pêeV
0.01
0.1
1
10
100
1000
10000
105lMFPêMpc
h=-10-2
h=10-2
h=-10-4
h=10-4
h=0
A. Saveliev, LM, G. Sigl, JCAP 2011
Fe mean-free-path
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Conclusions
UHECRs can serve as probes for Planck scale physics: spectrum and chemical composition
Constraints coming from protons are very strong, BUT they might be invalid if heavy nuclei are present at UHE
Constraints from nuclei are becoming interesting and can be made stronger by computing photodisintegrated spectra (to be done!!)
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Backup slides
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LV in the spectrum
Propagated (simulated) LIV spectra
• Effect of LIV: modify absorption of protons on the CMB (increases/decreases the photon energy needed to interact for ηp < 0/>0)
• Recovery of flux at high energy, due to reduced inelasticity
• Effect of sources: where is the closest UHECR source? We don’t know, but the effect is different from the one due to LIV
LM, Taylor, Mattingly, Liberati, arXiv:0902.1756
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LV in the photon spectrumGalaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)
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LV in the photon spectrumGalaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)GZK photons are produced by the decay of π0s due to pion production
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LV in the photon spectrumGalaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)GZK photons are produced by the decay of π0s due to pion production
In LI theory they are attenuated mainly by pair production onto CMB and URB leading to a theoretically expected photon fraction < 1% at 1019 eV and < 10% at 1020 eV.
-
LV in the photon spectrumGalaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)GZK photons are produced by the decay of π0s due to pion production
In LI theory they are attenuated mainly by pair production onto CMB and URB leading to a theoretically expected photon fraction < 1% at 1019 eV and < 10% at 1020 eV.
Present limits on photon fraction: 2.0%, 5.1%, 31%, 36% (95% CL) at 10, 20, 40, 100 EeV
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LV in the photon spectrumGalaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)GZK photons are produced by the decay of π0s due to pion production
In LI theory they are attenuated mainly by pair production onto CMB and URB leading to a theoretically expected photon fraction < 1% at 1019 eV and < 10% at 1020 eV.
Present limits on photon fraction: 2.0%, 5.1%, 31%, 36% (95% CL) at 10, 20, 40, 100 EeV
LIV strongly affects the threshold of this process: lower and also upper thresholds.
-
LV in the photon spectrumGalaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)GZK photons are produced by the decay of π0s due to pion production
In LI theory they are attenuated mainly by pair production onto CMB and URB leading to a theoretically expected photon fraction < 1% at 1019 eV and < 10% at 1020 eV.
Present limits on photon fraction: 2.0%, 5.1%, 31%, 36% (95% CL) at 10, 20, 40, 100 EeV
LIV strongly affects the threshold of this process: lower and also upper thresholds.
If kup < 1020 eV then photon fraction in UHECR much larger than present upper limits
-18-16-14-12-10-8-6-4-2 -18 -16 -14 -12 -10 -8 -6 -4 -2
-18-16-14-12-10-8-6-4-2
-18-16-14-12-10-8-6-4-2
Biref.
log
log
Crab
n=3
-10-8-6-4-202 -10 -8 -6 -4 -2 0 2
-10
-8
-6
-4
-2
0
2
-10
-8
-6
-4
-2
0
2
log
log
n=4
-
LV in the photon spectrumGalaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)GZK photons are produced by the decay of π0s due to pion production
In LI theory they are attenuated mainly by pair production onto CMB and URB leading to a theoretically expected photon fraction < 1% at 1019 eV and < 10% at 1020 eV.
Present limits on photon fraction: 2.0%, 5.1%, 31%, 36% (95% CL) at 10, 20, 40, 100 EeV
LIV strongly affects the threshold of this process: lower and also upper thresholds.
If kup < 1020 eV then photon fraction in UHECR much larger than present upper limits
LIV also introduces competitive processes: γ-decay
-18-16-14-12-10-8-6-4-2 -18 -16 -14 -12 -10 -8 -6 -4 -2
-18-16-14-12-10-8-6-4-2
-18-16-14-12-10-8-6-4-2
Biref.
log
log
Crab
n=3
-10-8-6-4-202 -10 -8 -6 -4 -2 0 2
-10
-8
-6
-4
-2
0
2
-10
-8
-6
-4
-2
0
2
log
log
n=4
-
LV in the photon spectrumGalaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)GZK photons are produced by the decay of π0s due to pion production
In LI theory they are attenuated mainly by pair production onto CMB and URB leading to a theoretically expected photon fraction < 1% at 1019 eV and < 10% at 1020 eV.
Present limits on photon fraction: 2.0%, 5.1%, 31%, 36% (95% CL) at 10, 20, 40, 100 EeV
LIV strongly affects the threshold of this process: lower and also upper thresholds.
If kup < 1020 eV then photon fraction in UHECR much larger than present upper limits
LIV also introduces competitive processes: γ-decay
If photons above 1019 eV are detected then γ-decay threshold > 1019 eV
-18-16-14-12-10-8-6-4-2 -18 -16 -14 -12 -10 -8 -6 -4 -2
-18-16-14-12-10-8-6-4-2
-18-16-14-12-10-8-6-4-2
Biref.
log
log
Crab
n=3
-10-8-6-4-202 -10 -8 -6 -4 -2 0 2
-10
-8
-6
-4
-2
0
2
-10
-8
-6
-4
-2
0
2
log
log
n=4
-
LV in the photon spectrum
-10-8-6-4-202 -10 -8 -6 -4 -2 0 2
-10
-8
-6
-4
-2
0
2
-10
-8
-6
-4
-2
0
2
log
log
-18-16-14-12-10-8-6-4-2 -18 -16 -14 -12 -10 -8 -6 -4 -2
-18-16-14-12-10-8-6-4-2
-18-16-14-12-10-8-6-4-2
Biref.
log
log
Crab
Galaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)GZK photons are produced by the decay of π0s due to pion production
In LI theory they are attenuated mainly by pair production onto CMB and URB leading to a theoretically expected photon fraction < 1% at 1019 eV and < 10% at 1020 eV.
Present limits on photon fraction: 2.0%, 5.1%, 31%, 36% (95% CL) at 10, 20, 40, 100 EeV
LIV strongly affects the threshold of this process: lower and also upper thresholds.
If kup < 1020 eV then photon fraction in UHECR much larger than present upper limits
LIV also introduces competitive processes: γ-decay
If photons above 1019 eV are detected then γ-decay threshold > 1019 eVn=3 n=4
-
LV in the neutrino sector
Effects on oscillations
Ecr ⇡ MPl✓
�m2
M2Pl⌘n⌫
◆1/n 0.2 GeV (n=3)20 TeV (n=4)
Strong constraints already from neutrino experiments!
Oscillations: �c/c . 10�27
Also quantum decoherence effects alter oscillation patterns
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LIV: prospects for the UHE neutrino sector
Neutrino Cherenkov emissionvery low rate, irrelevant on Hubble scales
Neutrino splittingvery fast rate above 1019 eV
Neutrino decaycan mimic a Z-burst effect --> effect at only 5% on the UHECR spectrum, not visible yet
Mattingly, LM, Galaverni, Liberati, Sigl, JCAP 1002:007,2010
⇥ ! ⇥�(⇥g)
⌫ ! ⌫⌫⌫̄
� ! �qq̄
Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation 10
This width can be turned into a decay length as
Lννν̄ =c
Γννν̄∼ 1.7× 10−3 Mpc η−4ν
(
E
1019 eV
)−13
. (18)
This makes clear we need to push the required energies above 1018.5 eV (with ην = 1)for the rate to be appreciable. As a final remark, we notice that the decay length in
Eq. (18) strongly depends on both the energy and ην . Therefore, the error about its
actual magnitude we might have made in our estimate will reflect in very small errors
in the determination of the energy at which LV effects start to be relevant as well as of
the constraint on ην .
4.1.2. Z boson resonance At such high energies the Z could be real - i.e. there is a
resonance in the matrix element. Even in this regime, however, the neutrino decay
time can be computed easily, as the only hypothesis one has to relax is that the Z 4-
momentum r satisfies r2 # M2Z . The magnitude of r2 can be easily computed exploitingthe kinematic equations. We obtain r2 = 16/27ηνE4ν/M
2Pl. The final decay length is then
Lννν̄ ∼ 1.7× 10−3 Mpc η−4ν(
Eν1019 eV
)−13
×
[
(
1−16
27ην
E4
M2Pl M2Z
)2
+
(
ΓZMZ
)2]
. (19)
A comparison between the two different decay lengths, Eq. (18) and Eq. (19), can be
found in Fig. 1. The Z resonance is hit at E4 = 27/16 (η−1ν M2PlM
2Z). We notice that even
(E/eV)10
log18 18.5 19 19.5 20 20.5 21 21.5 22
(L/M
pc)
10lo
g
−20
−15
−10
−5
0
5
10 Full computation
Approximate computation
Figure 1. Comparison between computations of the decay length without(Eq. (18)/red dashed line), and with (Eq. (19)/black solid line) the Z boson resonance.
though the two computations lead to very different results above the resonance, they
will not lead to any appreciable effect in the neutrino spectra, as at such energies the
decay lengths are anyway much smaller than the propagation distance of cosmological
neutrinos.
-
Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation 12
we have to resort to full MonteCarlo simulations of the UHECR propagation from
sources to the Earth.
We simulated then the propagation of UHECR protons in the Inter Galactic
Medium using the Monte Carlo package CRPropa [65], suitably modified to take into
account LV in the neutrino sector. The simulation parameters are the following: we
simulated unidimensional UHECR proton propagation, with source energy spectrumdN/dE ∝ E−2.2, from a spatially uniform distribution of sources located at redshiftz < 3 according to the Waxman & Bahcall (WB) distribution used in [66]. The injection
proton spectrum was tuned to fit AUGER data [59].
5.1. Results for flavor bind LV scenario
Figure 2 shows the outcome of the simulations for different values of the LV parameter ηνin the best case scenario, together with experimental sensitivities from some existing and
planned observatories, as well as the Waxman & Bahcall bound [68, 69] for reference.+
Results are in agreement with qualitative expectations previously discussed: Above a
(E / eV)10
log13 14 15 16 17 18 19 20 21
)−1
sr
−1 s
−2dN
/dE
(eV
cm2 E
−210
−110
1
10
210
(E / eV)10
log13 14 15 16 17 18 19 20 21
)−1
sr
−1 s
−2dN
/dE
(eV
cm2 E
−210
−110
1
10
210
Experimental sensitivitiesARIANNA − A phaseARIANNA − fullANITAWB limitAUGER
= 0η = 1η
−1 = 10η−2 = 10η−3 = 10η−4 = 10η
Figure 2. Evolution of the predicted LV neutrino spectra varying ην in the “best casescenario”. Sensitivities of main UHE neutrino operating and planned experiments areshown, as found in [51, 47, 67]. The Waxman & Bahcall limit [68, 69] in the interestingenergy range is shown for reference.
+ This limit is in fact an estimate of the neutrino luminosity of sources of UHE Cosmic Rays andγ-rays, in the hypothesis that the sources are optically thin to the escape of UHE particles and thatboth γ-rays and neutrinos are originated from UHECR interactions with radiation backgrounds. It isworth mentioning that this bound might be strongly affected by QG effects, as shown in [70].
�(4)� .✓
Eobs
6⇥ 1018 eV
◆�13/4
LIV: prospects for the UHE neutrino sector
Mattingly, LM, Galaverni, Liberati, Sigl, JCAP 1002:007,2010
-
The Greisen-Zatsepin-Kuzmin effect
p + � ! N + ⇡ Eth =2mpm⇡ + m2⇡
4✏⇠ 4 · 1019 eV on CMB
9:1 L51321+PY*62143+PZ/[)3+ HLYZK$1;;176
F73"-0,1 3%, ;$0573- ;(0,180,8#*- 3012(3 2(3$06%P- 9%3_)$07,5
,73"-0, !
"&$-10,%,3-
27"#(&;(0, ;$0573#(0,
;%($8;$0573#(0,8-,-$). "011
;(0,8;$0573#(0,8-,-$). "011
;(0,8;$0573#(0,8$%#-
!107$3-18271#89- (,830120"0)(3%" 9%3_.%$5?,". W0$-,#e81.22-#$. 9$-%_(,) %# !dSRSS
307"5 %P0(5 #*(1 30,3"71(0,I
!"#$%%
& #'&
()#$
!"
%&& !!!" #
9:1 L51321+PY*62143+PZ/[)3+ HLYZK$1;;176
F73"-0,1 3%, ;$0573- ;(0,180,8#*- 3012(3 2(3$06%P- 9%3_)$07,5
,73"-0, !
"&$-10,%,3-
27"#(&;(0, ;$0573#(0,
;%($8;$0573#(0,8-,-$). "011
;(0,8;$0573#(0,8-,-$). "011
;(0,8;$0573#(0,8$%#-
!107$3-18271#89- (,830120"0)(3%" 9%3_.%$5?,". W0$-,#e81.22-#$. 9$-%_(,) %# !dSRSS
307"5 %P0(5 #*(1 30,3"71(0,I
!"#$%%
& #'&
()#$
!"
%&& !!!" #
UHECRs at E>1020 eV must be produced within 100 Mpc from us. OR there is violation of Lorentz symmetry for boosts > 1011.
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