Mean free path of photons - NTNUweb.phys.ntnu.no/~mika/gausdal2.pdf · Mean free path of photons 10...
Transcript of Mean free path of photons - NTNUweb.phys.ntnu.no/~mika/gausdal2.pdf · Mean free path of photons 10...
![Page 1: Mean free path of photons - NTNUweb.phys.ntnu.no/~mika/gausdal2.pdf · Mean free path of photons 10 12 14 16 18 20 22 kpc 10kpc 100kpc Mpc 10Mpc 100Mpc Gpc log10(E/eV) radio photon](https://reader031.fdocument.pub/reader031/viewer/2022021816/5a79e53b7f8b9a3d058b7534/html5/thumbnails/1.jpg)
Mean free path of photons
10
12
14
16
18
20
22
Gpc100Mpc10MpcMpc100kpc10kpckpc
log1
0(E
/eV
)radio
photon horizon γγ → e+e− CMB
IR
Virgo ⇓
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 1 / 30
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Origin of cascade photons:
UHECRs:◮ Photon and neutrino production relatively tight connected:
⋆ protons:
p + γ3K →
p + π0→ p + 2γ
n + π+→ p + 2e± + 4ν
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 2 / 30
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Origin of cascade photons:
UHECRs:◮ Photon and neutrino production relatively tight connected:
⋆ protons:
p + γ3K →
p + π0→ p + 2γ
n + π+→ p + 2e± + 4ν
⋆ nuclei: A + γ3K → (A − 1) + n → (A − 1) + p + e− + νe
⋆ connection to UHECRs looser
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 2 / 30
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Origin of cascade photons:
UHECRs:◮ Photon and neutrino production relatively tight connected:
⋆ protons:
p + γ3K →
p + π0→ p + 2γ
n + π+→ p + 2e± + 4ν
⋆ nuclei: A + γ3K → (A − 1) + n → (A − 1) + p + e− + νe
HE and VHE photons from AGNs
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 2 / 30
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Origin of cascade photons:
UHECRs:◮ Photon and neutrino production relatively tight connected:
⋆ protons:
p + γ3K →
p + π0→ p + 2γ
n + π+→ p + 2e± + 4ν
⋆ nuclei: A + γ3K → (A − 1) + n → (A − 1) + p + e− + νe
HE and VHE photons from AGNs
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 2 / 30
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Diffuse cascade flux:
analytical estimate: [Berezinsky, Smirnov ’75 ]
Jγ(E) =
K(E/εX)−3/2 at E ≤ εX
K(E/εX)−2 at εX ≤ E ≤ εa
0 at E > εa
three regimes:
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 3 / 30
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Diffuse cascade flux:
analytical estimate: [Berezinsky, Smirnov ’75 ]
Jγ(E) =
K(E/εX)−3/2 at E ≤ εX
K(E/εX)−2 at εX ≤ E ≤ εa
0 at E > εa
three regimes:
◮ Thomson cooling:
Eγ =4
3
εbbE2e
m2e
≈ 100 MeV
(
Ee
1TeV
)2
◮ plateau region
◮ above pair-creation treshold smin = 4Eγεbb = 4m2e:
flux exponentially suppressed
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 3 / 30
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Diffuse flux, analytical estimate for low-energy part:
qi(E): # particles crossing energy E
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 4 / 30
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Diffuse flux, analytical estimate for low-energy part:
qi(E): # particles crossing energy E
cooling regime:
no generation of electrons for ε < εa/2: qe(Ee) = q0
Eγ ∝ E2
e ⇒ dEγ ∝ EedEe
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 4 / 30
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Diffuse flux, analytical estimate for low-energy part:
qi(E): # particles crossing energy E
cooling regime:
no generation of electrons for ε < εa/2: qe(Ee) = q0
Eγ ∝ E2
e ⇒ dEγ ∝ EedEe
inserting in energy conservation,
Eγdnγ = qe(Ee)dEe ,
givesJ(Eγ) ∝ E−3/2
γ
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 4 / 30
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Diffuse flux, analytical estimate for plateau region:
energy conservation and Ne/Nγ = const.
⇒ qi(Ei)Ei = const ⇒ qe(Ee) ∝ 1/Ee
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 5 / 30
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Diffuse flux, analytical estimate for plateau region:
energy conservation and Ne/Nγ = const.
⇒ qi(Ei)Ei = const ⇒ qe(Ee) ∝ 1/Ee
IC regime:
Eγ =4Ee
3 ln(2Eeεbb/m2e)
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 5 / 30
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Diffuse flux, analytical estimate for plateau region:
energy conservation and Ne/Nγ = const.
⇒ qi(Ei)Ei = const ⇒ qe(Ee) ∝ 1/Ee
IC regime:
Eγ =4Ee
3 ln(2Eeεbb/m2e)
to log. accuracyJ(Eγ) ∝ E−2
γ
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 5 / 30
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Monte Carlo vs. analytical estimate: single source
1e+11
1e+12
1e+13
1e+14
1e+15
1e+06 1e+07 1e+08 1e+09 1e+10 1e+11 1e+12 1e+13 1e+14 1e+15
E2 *J
(E)
E/eV
100MpcE-3/2
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 6 / 30
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Monte Carlo vs. analytical estimate: single source
1e+11
1e+12
1e+13
1e+14
1e+15
1e+06 1e+07 1e+08 1e+09 1e+10 1e+11 1e+12 1e+13 1e+14 1e+15
E2 *J
(E)
E/eV
100Mpc1000Mpc
E-3/2
E-1.9
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 6 / 30
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Fermi-LAT vs. UHECR data: no evolution [Berezinsky et al. ’10 ]
106 108 1010 1012 1014 1016 1018 1020 102210-2
100
102
104
E2 J(E)
, eV
cm-2s-1
sr-1
E, eV
Fermi LAT
p
zmax=2, Emax=1021eVHiRes
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 7 / 30
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Fermi-LAT vs. UHECR data: no evolution [Berezinsky et al. ’10 ]
106 108 1010 1012 1014 1016 1018 1020 102210-2
100
102
104
E2 J(E)
, eV
cm-2s-1
sr-1
E, eV
Fermi LAT
p
zmax=2, Emax=1021eVHiRes
integrating EJ(E) gives bound ωcas <∼ 6 · 10−7 eV/cm3
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 7 / 30
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Cascade limit for cosmogenic neutrinos [Berezinsky et al. ’10 ]
1016 1017 1018 1019 1020 1021 102210-1
100
101
102
103
104
105
106
1021
1020
IceCube 5yr tilted
2.0
RICE
Auger diff.
E-2 cascade
ANITA
E2 J(
E), e
Vcm
-2s-1
sr-1
E, eV
ANITA-liteAuger
2.6
nadirJEM-EUSOBAIKAL
e =
1022
m=3
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 8 / 30
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Cascade limit for cosmogenic neutrinos
1016 1017 1018 1019 1020 1021 102210-1
100
101
102
103
104
105
106
1021
1020
IceCube 5yr tilted
2.0
RICE
Auger diff.
E-2 cascade
ANITA
E2 J(
E), e
Vcm
-2s-1
sr-1
E, eV
ANITA-liteAuger
2.6
nadirJEM-EUSOBAIKAL
e =
1022
m=3
Assumes proton primaries
for nuclei reduced neutrino fluxes. . .
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 8 / 30
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Gamma-rays and extragalactic magnetic fields (EGMF)
Observations only in clusters,◮ synchrotron halo: ⇒ B ∼ (0.1 − 1)µG◮ Faraday rotation: ⇒ B ∼ (1 − 10)µG
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 9 / 30
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Gamma-rays and extragalactic magnetic fields (EGMF)
Observations only in clusters,◮ synchrotron halo: ⇒ B ∼ (0.1 − 1)µG◮ Faraday rotation: ⇒ B ∼ (1 − 10)µG
Origin of seed for EGMF is mysterious
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 9 / 30
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Gamma-rays and extragalactic magnetic fields (EGMF)
Observations only in clusters,◮ synchrotron halo: ⇒ B ∼ (0.1 − 1)µG◮ Faraday rotation: ⇒ B ∼ (1 − 10)µG
Origin of seed for EGMF is mysterious
Seed required as input for EGMF simulations
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 9 / 30
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Gamma-rays and extragalactic magnetic fields (EGMF)
Observations only in clusters,◮ synchrotron halo: ⇒ B ∼ (0.1 − 1)µG◮ Faraday rotation: ⇒ B ∼ (1 − 10)µG
Origin of seed for EGMF is mysterious
Seed required as input for EGMF simulations
Aharonian, Coppi, Volk ’94: Pair halos around AGNs
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 9 / 30
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Gamma-rays and extragalactic magnetic fields (EGMF)
Observations only in clusters,◮ synchrotron halo: ⇒ B ∼ (0.1 − 1)µG◮ Faraday rotation: ⇒ B ∼ (1 − 10)µG
Origin of seed for EGMF is mysterious
Seed required as input for EGMF simulations
Aharonian, Coppi, Volk ’94: Pair halos around AGNs
Plaga ’95: EGMFs deflect and delay cascade electrons
⇒ search for delayed “echoes” of multi-TeV AGN flares/GRBs
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 9 / 30
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Influence of EGMF on flux from single source: deflections
deflection of electrons:
ϑ ∼lcool
RL∝ E−2
e
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 10 / 30
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Influence of EGMF on flux from single source: deflections
deflection of electrons:
ϑ ∼lcool
RL∝ E−2
e
⇒ flux within angle ϑ reduced by factor E2
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 10 / 30
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Influence of EGMF on flux from single source: deflections
deflection of electrons:
ϑ ∼lcool
RL∝ E−2
e
⇒ flux within angle ϑ reduced by factor E2
⇒ cooling regime: transition from
J(E) ∝ E−1.5 → E0.5 → E−1.5
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 10 / 30
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Influence of EGMF on flux from single source: deflections
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 11 / 30
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Influence of EGMF on flux from single source: time
ext0
O
Oe+
+ jet
obse−
e−e
probability for misalignement p ∝ ϑobs ⇒ most blazars viewed withϑobs ∼ ϑjet
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 12 / 30
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Influence of EGMF on flux from single source: time
ext0
O
Oe+
+ jet
obse−
e−e
probability for misalignement p ∝ ϑobs ⇒ most blazars viewed withϑobs ∼ ϑjet
⇒ halos are not symmetric
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 12 / 30
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Influence of EGMF on flux from single source: time
ext0
O
Oe+
+ jet
obse−
e−e
probability for misalignement p ∝ ϑobs ⇒ most blazars viewed withϑobs ∼ ϑjet
⇒ halos are not symmetric
⇒ time-delay is function of ϑ,
Tdelay(ϑ) ∼ 3 × 106yr
[
(ϑobs + Θjet)
5◦
] [
ϑ
5◦
]
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 12 / 30
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Observer misaligned with jet: [Neronov et al. ’10 ]
ext0
O
Oe+
+ jet
obse−
e−e
probability for misalignement p ∝ ϑobs ⇒ most blazars viewed withϑobs ∼ ϑjet
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 13 / 30
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Observer misaligned with jet: [Neronov et al. ’10 ]
ext0
O
Oe+
+ jet
obse−
e−e
probability for misalignement p ∝ ϑobs ⇒ most blazars viewed withϑobs ∼ ϑjet
⇒ halos are not symmetric
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 13 / 30
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Observer misaligned with jet: [Neronov et al. ’10 ]
ext0
O
Oe+
+ jet
obse−
e−e
probability for misalignement p ∝ ϑobs ⇒ most blazars viewed withϑobs ∼ ϑjet
⇒ halos are not symmetric
⇒ time-delay is function of ϑ,
Tdelay(ϑ) ∼ 3 × 106yr
[
(ϑobs + Θjet)
5◦
] [
ϑ
5◦
]
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 13 / 30
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Asymmetric halos around TeV blazars (“GeV jets”):
10.10.010.001
ϑobs = 0◦ ϑobs = 3◦ ϑobs = 6◦ ϑobs = 9◦
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 14 / 30
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“GeV jets”: B dependence
10.10.010.001
10−17 G 10−16 G 10−15 G 10−14 G
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 15 / 30
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“GeV jets”: time dependence of flares
10.10.010.001
0 < Tdelay < 105 yr 3 × 106 yr< Tdelay < 107 yr
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 16 / 30
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Lower limit on EGMF: [A. Neronov, I. Vovk ’10, F. Tavecchio et al. ’10 ]
choose blazar: large z, stationary, low GeV, high multi-TeV emission
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 17 / 30
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Lower limit on EGMF: [A. Neronov, I. Vovk ’10, F. Tavecchio et al. ’10 ]
choose blazar: large z, stationary, low GeV, high multi-TeV emission
TeV photons cascade down:◮ small EGMF: fill up GeV range
◮ “large” EGMF: deflected outside, isotropized
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 17 / 30
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Lower limit on EGMF: [A. Neronov, I. Vovk ’10, F. Tavecchio et al. ’10 ]
choose blazar: large z, stationary, low GeV, high multi-TeV emission
TeV photons cascade down:◮ small EGMF: fill up GeV range
◮ “large” EGMF: deflected outside, isotropized
open questions:◮ influence of EGMF structure?
◮ time-dependence for flaring sources?
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 17 / 30
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Lower limit on EGMF: [A. Neronov, I. Vovk ’10, F. Tavecchio et al. ’10 ]
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 18 / 30
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Lower limit on EGMF: [A. Neronov, I. Vovk ’10, F. Tavecchio et al. ’10 ]
B >∼ 10−15 G
some dependence on ϑjet
no simulation of elmag. cascade with B
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 18 / 30
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Lower limit on EGMF: [A. Neronov, I. Vovk ’10, F. Tavecchio et al. ’10 ]
B >∼ 10−15 G
some dependence on ϑjet
no simulation of elmag. cascade with B
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 18 / 30
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Lower limit on EGMF: uniform field [Dolag et al. ’10 ]
10-16 G10-15 G
10-14 G
10-14 Gdirect
10-13 G
5×10-15 GEmax=100 TeV
HESS
Fermi
8 9 10 11 12 13 14-14
-13
-12
-11
log 10
(E
2 F/e
rg c
m-2
s-1
)
log10(E/eV)Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 19 / 30
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Lower limit on EGMF: uniform field [Dolag et al. ’10 ]
10-16 G10-15 G
10-14 G
10-14 Gdirect
10-13 G
5×10-15 GEmax=100 TeV
HESS
Fermi
8 9 10 11 12 13 14-14
-13
-12
-11
log 10
(E
2 F/e
rg c
m-2
s-1
)
log10(E/eV)
for coherence lengths λ <∼ lint ∼ 5kpc:
⇒ bound improves as λ1/2
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 19 / 30
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Lower limit on filling factor: [Dolag et al. ’10 ]
model filaments by a top-hat:
10 Mpc
B = 0
B = 10−10 G
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 20 / 30
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Lower limit on filling factor: [Dolag et al. ’10 ]
f=0.10f=0.50f=0.70f=0.80f=0.90f=0.60,Emax=100 TeV
HESS
Fermi
8 9 10 11 12 13 14-14
-13
-12
-11
log 10
(E
2 F/e
rg c
m-2
s-1
)
log10(E/eV)Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 21 / 30
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Lower limit on filling factor: [Dolag et al. ’10 ]
f=0.10f=0.50f=0.70f=0.80f=0.90f=0.60,Emax=100 TeV
HESS
Fermi
8 9 10 11 12 13 14-14
-13
-12
-11
log 10
(E
2 F/e
rg c
m-2
s-1
)
log10(E/eV)
Linear filling factor >∼ 50%
mainly 3-step cascade: γ → e± → γ
photon mean free path Dγ(E) ∼ 1000–50 Mpc
electron mean free path De(E) ∼ few kpc
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 21 / 30
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Lower limit on filling factor: [Dolag et al. ’10 ]
f=0.10f=0.50f=0.70f=0.80f=0.90f=0.60,Emax=100 TeV
HESS
Fermi
8 9 10 11 12 13 14-14
-13
-12
-11
log 10
(E
2 F/e
rg c
m-2
s-1
)
log10(E/eV)
Linear filling factor >∼ 50%
mainly 3-step cascade: γ → e± → γ
photon mean free path Dγ(E) ∼ 1000–50 Mpc
electron mean free path De(E) ∼ few kpc
⇒ electrons are created “everywhere” and feel B only close tointeraction point
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 21 / 30
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Effect of time-delay [Dermer et al. ’10 ]
10-14
10-13
10-12
10-11
HESS
νF ν (e
rg c
m-2
s-1
)
1ES 0229+200, z = 0.14EBL: Finke et al. (2010)θ
beam = 0.1; E
max = 6 TeV
λcoh
= 1 Mpc
10-18
Fermi LAT IACTs
Kneiske & Dole (2010) EBL
10-17
10-16 10
-15
10-14 G
10-19
(Aharonian et al. 2007)
10-11
)
1ES 0229+200, z = 0.14EBL: Finke et al. (2010)θ
beam = 0.1; E
max = 6 TeV
λ = 1 Mpc
BIGMF
= 10-18 G
∞→t
∞→t
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 22 / 30
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Effect of time-delay [Dermer et al. ’10 ]
Fermi LAT IACTs
10-14
10-13
10-12
10-11
νF
ν (er
g cm
-2 s
-1)
1ES 0229+200, z = 0.14EBL: Finke et al. (2010)θ
beam = 0.1; E
max = 6 TeV
λcoh
= 1 Mpc
104
Fermi LAT IACTs
BIGMF
= 10-18 G
103100
10
1 yr
105 yr
∞→t
1ES 0229+200, z = 0.14EBL: Finke et al. (2010)θ = 0.1; E = 6 TeV
BIGMF
= 10-19 G
∞→t
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 23 / 30
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How to create EGMFs in voids?
primordial fields:◮ inflation
◮ phase transitions (QCD, electroweak)
◮ reionization
astrophysical (require seed fields):
+ outflows from AGNs, dwarf galaxies
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 24 / 30
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How to create EGMFs in voids?
primordial fields:◮ inflation
◮ phase transitions (QCD, electroweak)
◮ reionization too weak
astrophysical (require seed fields):
+ outflows from AGNs, dwarf galaxies
– outflows colliminated
– B > 0 and B = 0 plasma does not mix
– contamination with heavy elements
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 24 / 30
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Expectation for “causally” generated fields: [Caprini, Durer,... ]
1 Mpc
15 kpc
compression
dynamo
EW non�hel
QCD helical
QCD non�hel
EW helical
0.001 0.01 0.1 1 10 100 100010�26
10�20
10�14
10�8
0.01
104
Λ�Mpc
B�nanoGauss
Phase transitions, n!2
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 25 / 30
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Primordial magnetic fields:for a Gaussian field
〈Bi(k)B∗
j (k′)〉 = δ(k − k′)[ (
δij − kikj
)
S(k) + iεijlklH(k)
]
energy density ρ = 4π∫
∞
0k2S(k)
helicity density h = 4π∫
∞
0kH(k)
characterized by Bλ and coherence length Lc
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 26 / 30
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Primordial magnetic fields:
for a Gaussian field
〈Bi(k)B∗
j (k′)〉 = δ(k − k′)[ (
δij − kikj
)
S(k) + iεijlklH(k)
]
characterized by Bλ and coherence length Lc
inflation: “acausal”, Lc ≫ H−10 possible
phase transitions:
◮ require 1./2.order transition
◮ “causal”, Lc(t∗) <∼ H(t∗)−1
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 26 / 30
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Primordial magnetic fields:
for a Gaussian field
〈Bi(k)B∗
j (k′)〉 = δ(k − k′)[ (
δij − kikj
)
S(k) + iεijlklH(k)
]
characterized by Bλ and coherence length Lc
inflation: “acausal”, Lc ≫ H−10 possible
phase transitions:
◮ require 1./2.order transition
◮ “causal”, Lc(t∗) <∼ H(t∗)−1
⇒ 〈Bi(x)Bj(y)〉 has compact support ⇒ [· · · ] is analytic function
◮ analyticity & finite ρ ⇒ Bλ ∼ B0(Lc/λ)5/2
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 26 / 30
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Primordial magnetic fields:
for a Gaussian field
〈Bi(k)B∗
j (k′)〉 = δ(k − k′)[ (
δij − kikj
)
S(k) + iεijlklH(k)
]
characterized by Bλ and coherence length Lc
inflation: “acausal”, Lc ≫ H−10 possible
phase transitions:
◮ require 1./2.order transition
◮ “causal”, Lc(t∗) <∼ H(t∗)−1
⇒ 〈Bi(x)Bj(y)〉 has compact support ⇒ [· · · ] is analytic function
◮ analyticity & finite ρ ⇒ Bλ ∼ B0(Lc/λ)5/2
how are fields evolving after creation?
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 26 / 30
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Evolution of primordial magnetic fields:
non-helical fields: damped above kD, below B ∝ 1/a2
0.1 1 10 100 100010
�5
10�4
0.001
0.01
0.1
K
EB
0.01 0.1 1 10 100 100010
5
104
0.001
0.01
0.1
K
B
)fifififin DDDD fifififin
180 MeV 22 MeV
kD(η)
2π/L(η)
= const
n+3= const
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 27 / 30
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Evolution of primordial magnetic fields:
helical fields: fluctuations are tranferred to larger scales 1/k
magnetic energy is transferred to larger scales
0.01 0.1 1 10 100 100010
�5
10�4
0.001
0.01
0.1
K
EB
)fifififin DDDD fifififin
180 MeV 22 MeV
kD(η)
= const
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 28 / 30
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Evolution of primordial magnetic fields:
helical fields: fluctuations are tranferred to larger scales 1/k
magnetic energy is transferred to larger scales
0.01 0.1 1 10 100 100010
�5
10�4
0.001
0.01
0.1
K
EB
)fifififin DDDD fifififin
180 MeV 22 MeV
kD(η)
= const
Picture assumes “static” large scale tail Bλ ∝ λ−5/2:
◮ interaction with turbulent fluid generates Bλ ∝ λ−3/2
⇒ fields from phase transitions could be strong enough
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 28 / 30
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Summary
Fermi non-observation of TeV blazars requires EGMF
⇒ quantitative conclusions:
◮ sure: large filling factor f >∼ 0.5
◮ bound on EGMF: depends on assumed ∆t, B >∼ 1018 G
can be improved by more/longer simultanous observations
limit ⇒ detection: CTA?
cascade limit from Fermi data reduced by factor ∼ 7
⇒ km3 neutrino telescope cannot detect cosmogenic neutrinos
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 29 / 30
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Summary
Fermi non-observation of TeV blazars requires EGMF
⇒ quantitative conclusions:
◮ sure: large filling factor f >∼ 0.5
◮ bound on EGMF: depends on assumed ∆t, B >∼ 1018 G
can be improved by more/longer simultanous observations
limit ⇒ detection: CTA?
cascade limit from Fermi data reduced by factor ∼ 7
⇒ km3 neutrino telescope cannot detect cosmogenic neutrinos
Michael Kachelrieß (NTNU, Trondheim) High Energy Astrophysics Nordic Winterschool, 2011 29 / 30