Stochastic cosmic ray sources and the TeV break in the all-electron spectrum arXiv:1809.????? Philipp Mertsch TeVPA 2018, Berlin 30 August 2018 Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 1 / 21
Cosmic ray e ± and cosmic ray origin What are the sources of cosmic rays? No source of local cosmic rays has been unambigously identified. Nuclei Electrons and positron • Far away and old sources • Only young nearby sources can contribute contribute at high energies due to energy losses → Features from individual sources averaged out → Can observe features from individual sources • Use anisotropies? Find sources with high-energy e ± Difficult! Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 2 / 21
Green’s function • Solve simplified transport equation for e ± spectral density ψ ∂ψ ∂ t − ∇ · κ · ∇ ψ + ∂ ∂ p ( b ( p ) ψ ) = δ ( r − r 0 ) δ ( t − t 0 ) Q ( p ) • For spatially independent κ and b ( p ), energy becomes pseudo time → Solve heat equation: b ( p ) � � − 3 / 2 e −| r − r 0 | 2 /ℓ 2 ( p , t ) πℓ 2 ( p , t ) ψ ( r , t , p ) = b ( p 0 ( p , t )) Q ( p 0 ( p , t )) where � p d p ′ D xx ( p ′ ) ℓ 2 ( p , t ) = 4 b ( p ′ ) p 0 ( p , t ) • Boundary condition in z -direction can be treated by method of mirror sources Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 3 / 21
Green’s function b ( p ) � � − 3 / 2 e −| r − r 0 | 2 /ℓ 2 ( p , t ) πℓ 2 ( p , t ) ψ ( r , t , p ) = b ( p 0 ( p , t )) Q ( p 0 ( p , t )) 10 5 10 4 t = 1000 yr 10 3 t = 10 4 yr s=0.3 kpc E 3 F [a.u.] 10 2 s=1.0 kpc t = 10 5 yr 10 1 s=3.0 kpc t = 10 6 yr 10 0 10 1 10 2 10 0 10 1 10 2 10 3 10 4 10 5 10 6 E [GeV] with D xx = D 0 p δ , Q ( p 0 ) ∝ p − Γ exp[ − p 0 / p c ] 0 Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 4 / 21
Green’s function b ( p ) � � − 3 / 2 e −| r − r 0 | 2 /ℓ 2 ( p , t ) πℓ 2 ( p , t ) ψ ( r , t , p ) = b ( p 0 ( p , t )) Q ( p 0 ( p , t )) 10 5 10 4 t = 1000 yr 10 3 t = 10 4 yr s=0.3 kpc E 3 F [a.u.] 10 2 s=1.0 kpc t = 10 5 yr 10 1 s=3.0 kpc t = 10 6 yr 10 0 10 1 10 2 10 0 10 1 10 2 10 3 10 4 10 5 10 6 E [GeV] with D xx = D 0 p δ , Q ( p 0 ) ∝ p − Γ exp[ − p 0 / p c ] 0 Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 4 / 21
Flux from a population of sources consider ensemble of sources at distances r i and with ages t i observer s i source ( b 0 t a ) 1 ( b 0 t c ) 1 1 1 ( b 0 t b ) ( b 0 t d ) Ignorance of r i and t i ⇒ cannot predict e ± spectrum E 3 F r a r b measure e ± spectrum ⇒ learn r d about r i and t i r c Energy E Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 5 / 21
The TeV break Kerszberg et al. , ICRC 2017 Ambrosi et al. (2017) ] -1 sr Preliminary ⋅ -1 250 s ⋅ E 3 × Flux [m − 2 s − 1 sr − 1 GeV 2 ] -2 m 2 10 ⋅ 2 [GeV 200 dN dE 150 3 Flux E 100 DAMPE H.E.S.S. (2008) H.E.S.S. (2009) 50 AMS-02 (2014) 10 Fermi-LAT (2017) HESS HE (2008) HESS LE (2009) 0 MAGIC (2011) AMS-02 (2014) 10 100 1,000 10,000 Energy [GeV] VERITAS (2015) Fermi-LAT HE (2017) Is the TeV break compatible with a HESS (2017) HESS Fit (2017) random ensemble of sources? 1 1 10 Energy [TeV] Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 6 / 21
A statistical model observer s i source • Contribution from source i to φ k depends on distance s i and age t i i φ i = ( φ 1 , φ 2 , . . . φ N ) T → Spectrum is a random vector: φ = � • Statistically characterised by joint distribution f ( φ 1 , φ 2 , . . . φ N ) Applications 1 Likelihood of a model: evaluate f (ˆ φ 1 , ˆ φ 2 , . . . ˆ φ N ) for measured ˆ φ 2 Extrapolate to higher energies: f ( φ M +1 , . . . φ N | φ 1 , φ 2 , . . . φ M ) 3 Quickly generate samples from model, e.g. for forecasting Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 7 / 21
Marginals and copula What is the joint distribution? • Non-parametric, e.g. kernel-density estimators? → curse of dimensionality • Multi-variate Gaussian? → would give Gaussian marginals (see below) Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 8 / 21
Marginals and copula What is the joint distribution? • Non-parametric, e.g. kernel-density estimators? → curse of dimensionality • Multi-variate Gaussian? → would give Gaussian marginals (see below) → Use copulas to factorise problem: ◮ Multi-variate PDF on unit hypercube ◮ Have uniform marginals ◮ Encode correlations Sklar’s theorem f ( φ 1 , φ 2 , . . . φ N ) = f 1 ( φ 1 ) f 2 ( φ 2 ) . . . f N ( φ N ) c ( F 1 ( φ 1 ) , F 2 ( φ 2 ) , . . . F N ( φ N )) marginals copula CDFs (= 1D PDFs) Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 8 / 21
Marginals: analytical approach observer source • Total flux is sum of fluxes from individual sources N N J i ( E ) = c � � J ( E ) = G ( E , t i , r i ) 4 π i =1 i =1 • r i and t i are random variables ⇒ Z i = G ( E , t i , r i ) is a random variable • What is f Z ( z )? Central limit theorem? c • � J � = 4 π � Z � is the flux from smooth distribution of sources. Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 9 / 21
Marginals: analytical approach observer • Total flux is sum of fluxes from individual sources N N J i ( E ) = c � � J ( E ) = G ( E , t i , r i ) 4 π i =1 i =1 • r i and t i are random variables ⇒ Z i = G ( E , t i , r i ) is a random variable • What is f Z ( z )? Central limit theorem? c • � J � = 4 π � Z � is the flux from smooth distribution of sources. Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 9 / 21
Diverging variance Lee (1979), Lagutin & Nikulin (1995), Ptuskin et al. (2006), PM (2010), Genolini et al. (2016) • � Z 2 � diverges: (4 D 0 ) 1 − 3 2 m Q m � Z m � = 1 1 2 m E − 2+ δ + 3 0 2 m (1 − δ ) − m Γ r 2 2 − r 2 ( b 0 (1 − δ )) 2 − 3 3 t max 2 m m π 1 � 1 e − m Λ 2 /λ 2 � ρ 2 2 m � m (Γ − 2)+ δ 1 ( λ 2 ) 1 − 3 d λ 2 (1 − λ 2 ) × 1 − δ ρ 2 0 2 where Λ 2 = b 0 (1 − δ ) i = b 0 (1 − δ ) E 1 − δ L 2 ρ 2 E 1 − δ r 2 and i 4 D 0 4 D 0 • Z m with m ≥ 2 increases faster with r → 0 than density of sources falls off • cannot apply central limit theorem • introduce minimum distance r min ?! Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 10 / 21
Diverging variance Lee (1979), Lagutin & Nikulin (1995), Ptuskin et al. (2006), PM (2010), Genolini et al. (2016) • � Z 2 � diverges: (4 D 0 ) 1 − 3 2 m Q m � Z m � = 1 1 2 m E − 2+ δ + 3 0 2 m (1 − δ ) − m Γ r 2 2 − r 2 ( b 0 (1 − δ )) 2 − 3 3 t max 2 m m π 1 � 1 e − m Λ 2 /λ 2 � ρ 2 2 m � m (Γ − 2)+ δ 1 ( λ 2 ) 1 − 3 d λ 2 (1 − λ 2 ) × 1 − δ ρ 2 0 2 where Λ 2 = b 0 (1 − δ ) i = b 0 (1 − δ ) E 1 − δ L 2 ρ 2 E 1 − δ r 2 and i 4 D 0 4 D 0 • Z m with m ≥ 2 increases faster with r → 0 than density of sources falls off • cannot apply central limit theorem • introduce minimum distance r min ?! Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 10 / 21
Stable law PM (2010) • for z → ∞ , f Z ( z ) has a power law tail: 1 1 1 E − δ − 4 3 Γ Q 4 / 3 z − α − 1 f Z ( z ) ≃ 0 r 2 8 π 2 D 0 t max max � �� � ≡ c + • Generalised central limit theorem for distributions with power law tail Gendenko & Kolmogorov (1949) N � Fluxes distributed as N →∞ Z i − → a N + b N S ( α, 1 , 1 , 0 , 1) stable law i =1 with a N = N � Z � � � α π c + N α b N = 2Γ(1 /α ) sin( π/ 2 α Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 11 / 21
Numerical result PM (2018) 2.5 2.0 1 ] ) 1.5 2 sr 2 s 1.0 lg ( E 3 [GeV 2 cm 0.5 0.0 0.5 median 95% range 1.0 68% range 10 1 10 2 10 3 10 4 10 5 E [GeV] SN rate = 10 − 4 yr − 1 , Γ = 2 . 2, E cut = 10 5 GeV, z max = 4 kpc, full KN-losses, 10 4 realisations of source distribution Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 12 / 21
Semi-analytical model: copula • Analytical computation of c ( F 1 , F 2 , . . . F N ) seems intractable • Compute a large ensemble of random samples in a Monte Carlo approach • Parametrise likelihood by pair copula construction Pair copula construction Idea: factorise joint PDF into product of (conditional) bi-variate PDFs, e.g.: f ( x 1 , x 2 , x 3 ) = f 1 ( x 1 ) f 2 ( x 2 ) f 3 ( x 3 ) c 12 ( F 1 ( x 1 ) , F 2 ( x 2 )) c 23 ( F 2 ( x 2 ) , F 3 ( x 3 )) × c 13 | 2 ( F 1 | 2 ( x 1 | x 2 ) , F 3 | 2 ( x 3 | x 2 )) pair copulas Technical details • Used regular D-vine • Tried various copulas, but Normal pair copula fits best • Determine (conditional) correlation coefficients by fitting Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 13 / 21
Energy-energy correlations PM (2018) Philipp Mertsch (RWTH Aachen) Stochastic sources and the TeV electron break 30 August 2018 14 / 21
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