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The Power Spectrum of Cosmic Ray
Arrival Directions
Markus Ahlers
UW-Madison & WIPAC
Vulcano Workshop 2016
“Frontier Objects in Astrophysics and Particle Physics”
Vulcano, May 24, 2016
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 1
Map of CR Arrival Directions
Cosmic ray anisotropies up to the level of one-per-mille have been observed at
various energies [Super-Kamiokande’07; Milagro’08; ARGO-YBJ’09,’13;EAS-TOP’09]
[Tibet AS-γ’05,’06,’15;IceCube’10,’11,’16; HAWC’13,’14]
ECR ≃ 10 TeV, NCR ∼ 3.2 × 1011[IceCube (IC59-IC86-IV)’16]
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 2
Map of CR Arrival Directions
Cosmic ray anisotropies up to the level of one-per-mille have been observed at
various energies [Super-Kamiokande’07; Milagro’08; ARGO-YBJ’09,’13;EAS-TOP’09]
[Tibet AS-γ’05,’06,’15;IceCube’10,’11,’16; HAWC’13,’14]
ECR ≃ 1 TeV, NCR ∼ 4.9 × 1010[HAWC’14]
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 3
Time-Dependence of Anisotropy
(a) Period 1 (b) Period 2 (c) Period 3 (d) Period 4
(e) Period 5 (f) Period 6 (g) Period 7 (h) Period 8
IceCube Preliminary
(i) Period 9 (j) Period 10 (k) Period 11 (l) Period 12
20
[AMANDA (7 yrs) & IceCube (5 yrs)’13]]
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 4
Expected CR Anisotropies
• turbulent motion of charged particles in Galactic
environment described by diffusion
• diffusion tensor K (in general anisotropic):
Kij =BiBj
3ν‖+
δij − BiBj
3ν⊥+
ǫijkBk
3νA
• expected dipole anisotropy is small and depends
on:
• (local) source distribution [Erlykin & Wolfendale’06]
[Blasi & Amato’12; Sveshnikova et al.’13; Pohl & Eichler’13]
• (local) ordered magnetic field B
[e.g. Schwadron et al.’14; Mertsch & Funk’14]
• rigidity dependence of diffusion, K ∝ ρ 0.3−0.6
• relative velocity of the medium [Compton & Getting’35]
observational limitations
rnCR
−KrnCR
density gradient
dipole anisotropy
relative intensity
deficitexcess
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 5
Observational Limitations
Equatorial
60120180300 240
60
30
−30
−60
North
East
South
West
Zenith
anisotropy
-0.0008 0.0008
[MA et al.’16]
• ground-based detectors are calibrated by CR data reduces anisotropies
• true CR dipole defined by amplitude D, and orientation (RA,DEC) = (α,δ)
observable only projected dipole with amplitude D cos δ and orientation (α,0)
further problems by limited field of view (cross-talk with small-scale structure)
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 6
Observed Dipole Amplitude and Phase
naive expectation (∝ E1/3 )
103 104 105 106 107
energy [GeV]
10−4
10−3
10−2
dip
ole
amp
litu
de
naive expectation (Galactic Center)
−250
−200
−150
−100
−50
0
50
dip
ole
ph
ase
[deg
ree]
Baksan’87
Kamiokande’97
Super-K’07
MACRO’03
IceCube’10
IceCube’12
IceCube’15
IceTop’15
KASCADE’15
Tibet’15
Baksan’81
Baksan’09
Mt.Norikura’89
Musala Peak’75
EAS-TOP’95
Tibet’05
Milagro’09
EAS-TOP’96
EAS-TOP’09
ARGO-YBJ’11
ARGO-YBJ’15
IceTop’13
[MA’16]
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 7
Local Magnetic Field and Dipole Anisotropy
• strong ordered magnetic fields in the
local environment
diffusion tensor reduces to projector:
Kij →BiBj
3ν‖
• TeV–PeV dipole data consistent with
magnetic field direction inferred by
IBEX data [McComas et al.’09]
• 1–100 TeV phase indicates a local
gradient within longitudes:
120. l . 300
phase flip induces by Vela supernova
remnant? [MA’16]
315
135
0
90
180
27
0
loca
l mag
netic
field
Gal
acti
cC
ente
r
Com
pto
n-G
etti
ng
−1 0 1
reconstructed δ ⋆0h [10−3]
−3
−2
−1
0
1
reco
nst
ruct
edδ⋆ 6h
[10−
3]
1.1
1.4
1.6
1.8
2.1
2.4
2.8
3.1
3.2
0.0 0.2
0.4
0.6
0.91.1
1.5
1.2
1.7
2.0
2.5
3.0
2.0
2.6
2.7
IceCube
IceTop
ARGO-YBJ
Tibet-ASγ
EAS-TOP
[MA’16]
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 8
Local Magnetic Field and Dipole Anisotropy
• strong ordered magnetic fields in the
local environment
diffusion tensor reduces to projector:
Kij →BiBj
3ν‖
• TeV–PeV dipole data consistent with
magnetic field direction inferred by
IBEX data [McComas et al.’09]
• 1–100 TeV phase indicates a local
gradient within longitudes:
120. l . 300
phase flip induces by Vela supernova
remnant? [MA’16]
135
225
180
Vela
SNR
−6 −4 −2 0 2
predicted δ ⋆0h [10−3]
−8
−6
−4
−2
0
2
pre
dic
ted
δ⋆ 6h
[10−
3]
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
projected
not projected
1 10 102 103
energy [TeV]
0.1
1
10
amplitude [10−3]
[MA’16]
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 9
Suggested Origin of Small-Scale Anisotropy
• CR acceleration in nearby SNRs [Salvati & Sacco’08]
• local magnetic field structure with energy-dependent magnetic mirror leakage
[Drury & Aharonian’08]
• preferred CR transport directions [Malkov, Diamond, Drury & Sagdeev’10]
• magnetic reconnections in the heliotail [Lazarian & Desiati’10]
• non-isotropic particle transport in the heliosheath [Desiati & Lazarian’11]
• heliospheric electric field structure [Drury’13]
• magnetized outflow from old SNRs [Biermann, Becker, Seo & Mandelartz’12]
• strangelet production in molecular clouds or neutron stars
[Kotera, Perez-Garcia & Silk ’13]
• interplay between regular and turbulent magnetic fields
[Battaner, Castellano & Masip’14; Mertsch & Funk’14; Schwadron et al.’14; MA’16]
small-scale anisotropies from local magnetic field mapping of a global dipole
[Giacinti & Sigl’12; MA’14; MA & Mertsch’15]
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 10
Analogy to Gravitational Lensing
CMB temperature fluctuations Cosmic Ray Gradient
Local M
agnetic Tu
rbulen
ceLar
ge S
cale
Str
uct
ure
small scale temperature fluctuations small scale anisotropies
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 11
Gedankenexperiment
• Idea: local realization of magnetic turbulence introduces small-scale structure
[Giacinti & Sigl’11]
• Particle transport in (static) magnetic fields is governed by Liouville’s equation of
the CR’s phase-space distribution f :
d
dtf (t, x, p) = 0
• “trivial” solution:
f (0, 0, p) = f (−T, x(−T), p(−T))
• Gedankenexperiment:
Assume that at look-back time −T initial condition is homogenous, but not
isotropic:
f (0, 0, p) = f (p(−T))
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 12
Angular Power Spectrum
• Every smooth function g(θ, φ) on a sphere can be decomposed in terms of
spherical harmonics Yℓm:
g(θ, φ) =∞∑
ℓ=0
ℓ∑
m=−ℓ
aℓmYmℓ (θ, φ) ↔ aℓm =
∫dΩ(Ym
ℓ )∗(θ, φ)g(θ, φ)
• angular power spectrum:
Cℓ =1
2ℓ+ 1
ℓ∑
m=−ℓ
|aℓm|2
• related to the two-point auto-correlation function: (n1/2 : unit vectors, n1 · n2 = cos η)
ξ(η) =1
8π2
∫dn1
∫dn2δ(n1n2 − cos η)g(n1)g(n2) =
1
4π
∑
ℓ
(2ℓ+ 1)CℓPℓ(cos η)
Note that individual Cℓ’s are independent of coordinate system (assuming full
sky coverage).
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 13
Gedankenexperiment (continued)
• Initial configuration has power spectrum Cℓ.
• For small correlation angles η flow remains
correlated even beyond scattering sphere.
• Correlation function for η = 0:
ξ(0) =1
4π
∫dp1 f
2(p1(−T))
scattering length
p1
p2
p2(-T)
p1(-T)
• On average, the rotation in an isotropic random rotation in the turbulent magnetic
field leaves an isotropic distribution on a sphere invariant:
〈ξ(0)〉 =1
4π
∫dp1 f
2(p1)
The weighted sum of 〈Cℓ〉’s remains constant:
1
4π
∑
ℓ≥0
(2ℓ+ 1)Cℓ =1
4π
∑
ℓ≥0
(2ℓ+ 1) 〈Cℓ(T)〉
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 14
Evolution Model
• Diffusion theory motivates that each 〈Cℓ〉 decays exponentially with an effective
relaxation rate [Yosida’49]
νℓ ∝ L2 ∝ ℓ(ℓ+ 1)
• A linear 〈Cℓ〉 evolution equation with generation rates νℓ→ℓ′ requires:
∂t〈Cℓ〉 = −νℓ〈Cℓ〉+∑
ℓ′≥0
νℓ′→ℓ
2ℓ′ + 1
2ℓ+ 1〈Cℓ′〉 with νℓ =
∑
ℓ′≥0
νℓ→ℓ′
• For νℓ ≃ νℓ→ℓ+1 and Cℓ = 0 for l ≥ 2 this has the analytic solution:
〈Cℓ〉(T) ≃3C1
2ℓ+ 1
ℓ−1∏
m=1
νm
∑
n
ℓ∏
p=1( 6=n)
e−Tνn
νp − νn
• For νℓ ≃ ℓ(ℓ+ 1)ν we arrive at a finite asymptotic ratio:
limT→∞
〈Cℓ〉(T)
〈C1〉(T)≃
18
(2ℓ+ 1)(ℓ+ 2)(ℓ+ 1)
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 15
Comparison with CR Data
5 10 15 20 25 30multipole moment ℓ
10−12
10−11
10−10
10−9
10−8
10−7
10−6
pow
ersp
ectr
um
Cℓ
Ahlers 2013
HAWC 2014
IceCube 2013
[MA’14; updated with HAWC data]
limT→∞
〈Cℓ〉(T)
〈C1〉(T)≃
18
(2ℓ+ 1)(ℓ+ 2)(ℓ+ 1)
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 16
Simulation via Backtracking• Consider a local (quasi-)stationary solution of the diffusion approximation:
4π〈 f 〉 ≃ n + r∇n − 3 p K∇n︸ ︷︷ ︸1st linear correction
• backtracking over long time-scales T and evaluating in mean field
4πfi ≃ 4π δf (−T, ri(−T), pi(−T))︸ ︷︷ ︸deviation from 〈 f〉
+n + [ri(−T)− 3pi(−T)K]∇n
• Ensemble-averaged Cℓ’s (ℓ ≥ 1): [MA & Mertsch’15]
〈Cℓ〉
4π≃
∫dp1
4π
∫dp2
4πPℓ(p1p2) lim
T→∞〈r1i(−T)r2j(−T)〉︸ ︷︷ ︸
relative diffusion
∂in∂jn
n2
• Sum of 〈Cℓ〉 related to diffusion tensor:
1
4π
∑
ℓ≥0
(2ℓ+ 1)〈Cℓ〉 ≃ 2TKsij
∂in∂jn
n2
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 17
Simulation via Backtracking
• CR arrival direction determined by backtracking of CRs towards a homogeneous
initial dipole anisotropy (ballistic laminar turbulent)
-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 -0.10 -0.05 0.00 0.05 0.10 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -1.0 -0.5 0.0 0.5 1.0
• Kolmogorov turbulence with energy density comparable to B0
• two cases: dipole vector aligned with or perpendicular to B0
• asymptotically limited by simulation noise:
N ≃4π
Npix
2TKsij
∂in∂jn
n2
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 18
Comparison to CR Data
5 10 15 20
multipole moment ℓ
10−4
10−3
10−2
0.1
1re
lati
ve
pow
ersp
ectr
um
Cℓ/
C1
IceCube noise level
HAWC noise level
Ahlers & Mertsch (2015)
σ2= 1, rL/Lc = 0.1, λmin/Lc = 0.01, λmax/Lc = 100, ΩT = 100
model (p = 2/3)
model (p = 1/2)
model (p = 1/3)
simulation (B0 ‖ ∇n)
simulation (B0 ⊥ ∇n)
IceCube (rescaled)
HAWC (rescaled)
[MA & Mertsch’15]
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 19
Local Description: Relative Scattering
• evolution of Cℓ’s: [MA & Mertsch’15]
∂t〈Cℓ〉 = −1
2π
∫dp1
∫dp2Pℓ(p1p2)〈(p1∇f1 + iωLf1) f2〉
• large-scale dipole anisotropy gives an effective “source term”:
−1
2π
∫dp1
∫dp2Pℓ(p1p2)〈(p1∇f1) f2〉 → Q1δℓ1
• BGK-like Ansatz for scattering term (〈iωLf 〉 → − ν2
L2〈f 〉) [Bhatnagaer, Gross & Krook’54]
−1
2π
∫dp1
∫dp2Pℓ(p1p2)〈(iωLf1) f2〉 →
1
2π
∫dp1
∫dp2Pℓ(p1p2)ν(p1p2)L
2〈f1f2〉
• Note that ν(1) = 0 for vanishing regular magnetic field.
ν(x) ≃ ν0(1 − x)p
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 20
Summary
• Large- and small-scale anisotropy can be understood in the context of standard
diffusion theory.
• Local magnetic turbulence is a source of small-scale anisotropies.
• Analogous to induced high-ℓ multipoles in CMB temperature power spectrum
from gravitational lensing in small scale structure.
• Ensemble-averaged Cℓ’s (ℓ ≥ 1) related to relative diffusion:
〈Cℓ〉
4π≃
∫dp1
4π
∫dp2
4πPℓ(p1p2) lim
T→∞〈r1i(−T)r2j(−T)〉
∂in∂jn
n2
• Microscopic description of 〈Cℓ〉 evolution via modified BGK-like Ansatz.
• Open issues: damping effects for CR rigidity distributions, partial sky coverage of
experiments, angular resolution, . . .
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 21
Thank you for your attention!
Markus Ahlers (UW-Madison) The Power Spectrum of CR Arrival Directions May 24, 2016 slide 22
Appendix
Appendix
Simulated Turbulence
• 3D-isotropic turbulence: [Giacalone & Jokipii’99]
δB(x) =N∑
n=1
A(kn)(an cosαn + bn sinαn) cos(knx + βn)
• αn and βn are random phases in [0, 2π), unit vectors an ∝ kn × ez and bn ∝ kn × an
• with amplitude
A2(kn) =
2σ2B20G(kn)∑N
n=1G(kn)
with G(kn) = 4πk2n
kn∆lnk
1 + (knLc)γ
• Kolmogorov-type turbulence: γ = 11/3
• N = 160 wavevectors kn with |kn| = kmine(n−1)∆lnk and ∆lnk = ln(kmax/kmin)/N
• λmin = 0.01Lc and λmax = 100Lc [Fraschetti & Giacalone’12]
• rigidity: rL = 0.1Lc
• turbulence level: σ2 = B20/〈δB2〉 = 1
Appendix
Powerspectrum of CR Arrival Directions Cosmic ray anisotropies up to the level of one-per-mille have been observed at
various energies
[Tibet AS-γ’05,’06; Super-Kamiokande’07; Milagro’08; ARGO-YBJ’09,’13;EAS-TOP’09]
[IceCube’10,’11; HAWC’13,’14]
[IceCube [arXiv:1603.01227]]
Appendix
Powerspectrum of CR Arrival Directions Cosmic ray anisotropies up to the level of one-per-mille have been observed at
various energies
[Tibet AS-γ’05,’06; Super-Kamiokande’07; Milagro’08; ARGO-YBJ’09,’13;EAS-TOP’09]
[IceCube’10,’11; HAWC’13,’14]
[HAWC [arXiv:1408.4805]; note: low−ℓ power under-estimated]
Appendix
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