Cosmic-ray propagation in the light of the Myriad model Yoann - PowerPoint PPT Presentation

Cosmic-ray propagation in the light of the Myriad model Yoann Genolini In collaboration with: Pasquale Serpico, Pierre Salati & Richard Taillet Annecy-Le-Vieux, FRANCE BUSAN July 17th, 2017 1

A break in Galactic cosmic rays nuclei S.Ting presentation, CERN, december 2016 ⇒ A universal kinck at R ≥ 200 GV ? ∆ kinck ≈ 0 . 12 − 0 . 13 2

A break in Galactic cosmic rays nuclei Yang presentation, XSCR, march 2017 ⇒ A universal kinck at R ≥ 200 GV ? ∆ kinck ≈ 0 . 12 − 0 . 13 2

Origin of the break ? Standard expectation ? 3

Origin of the break ? ⇒ Explanation 1 : break in the diffusion coefficient ? -Blasi, P., Amato, E., Serpico, P. D. (2012). PRL, 109(6), 061101. -Tomassetti, N. (2012). The Astrophysical Journal Letters, 752(1), L13. 3

Origin of the break ? ⇒ Explanation 2 : break in the source spectrum ? -Ptuskin, V., Zirakashvili, V., Seo, E. S. (2013). APJ, 763(1), 47. -Tomassetti, N., Donato, F. (2015). APJL, 803(2), L15. 3

Origin of the break ? ⇒ Explanation 3 : local source contribution ? -Kachelriess, M., Neronov, A., Semikoz, D. V. (2015). PRL, 115(18). -Erlykin, A. D., Wolfendale, A. W. (2015). JPG, 42(12), 125201. 3

Origin of the break ? ⇒ Explanation 3 : local source contribution ? Genolini, Y., Salati, P., Serpico, P. D., & Taillet, R. (2017). Stable laws and cosmic ray physics. A&A, 600, A68. ⇒ Explanation 1 : break in the diffusion coefficient ? Indications for a high-rigidity break in the cosmic-ray diffusion coefficient, Preprint arxiv: 1706.09812 4

Origin of the break ? ⇒ Explanation 3 : local source contribution ? Genolini, Y., Salati, P., Serpico, P. D., & Taillet, R. (2017). Stable laws and cosmic ray physics. A&A, 600, A68. ⇒ Explanation 1 : break in the diffusion coefficient ? Indications for a high-rigidity break in the cosmic-ray diffusion coefficient, Preprint arxiv: 1706.09812 4

Explanation 3 one question would be.. May a particular configuration of the sources explain break features ? 1.4 10 4 10 4 Φ p .Ek 2 . 7 6 10 3 Fiducial 2 10 3 CREAM 2005 AMS 2015 10 2 10 3 10 4 10 5 1 10 Kinetic Energy Ek [GeV] ⇒ P ( Ψ ) ? 5

The Pure diffusive regime ∂ Ψ ∂t − ∇ r . ( K ∇ r Ψ) = Q ( r , E ) ⇒ Time independent equation ! 6

The Pure diffusive regime − ∇ r . ( K ∇ r Ψ) = Q ( r , E ) ⇒ Time independent equation ! ⇒ Continuous production in space and time ! 6

The Pure diffusive regime − ∇ r . ( K ∇ r Ψ) = Q ( r , E ) ⇒ Time independent equation ! ⇒ Continuous production in space and time ! Q ( r , t ) = � N q i δ ( r i − r ) δ ( t i − t ) i 6

The Pure diffusive regime − ∇ r . ( K ∇ r Ψ) = Q ( r , E ) ⇒ Time independent equation ! ⇒ Continuous production in space and time ! Q ( r , t ) = � N q i δ ( r i − r ) δ ( t i − t ) i �� N � � Q ( r , t ) � = q i δ ( r i − r ) δ ( t i − t ) i 6

The Pure diffusive regime − ∇ r . ( K ∇ r Ψ) = Q ( r , E ) ⇒ Time independent equation ! ⇒ Continuous production in space and time ! Q ( r , t ) = � N q i δ ( r i − r ) δ ( t i − t ) i q ν � Q ( r , t ) � ≃ V MW Θ( h − | z | ) Θ( R gal − r ) V MW = 2 h πR 2 and ν ≈ 3 SNs/century 6

The Pure diffusive regime V MW = 2 h πR 2 ν ≈ 3 SNs/century 7

Sources are discrete in space and time! Büsching, I., Kopp, A., Pohl, M., Schlickeiser, R., Perrot, C., & Grenier, I. (2005). The Astrophysical Journal, 619(1), 314. → Stochastic behaviour ! 8

Statistical treatment of the flux The flux from N sources writes : N N � � Ψ = ψ i ⇒ � Ψ � = � ψ � = N � ψ � i =1 i =1 One can expect to compute � ψ � from p ( ψ ) : � ∞ � ψ � = dψ ψ p ( ψ ) 0 With : � p ( ψ ) = D ( r s , t s ) d r s d t s (1) � �� � V ψ Normalized distribution in space and time for one source Integration over the domain of space and time that gives a flux between ψ and ψ + d ψ . 9

To measure ψ from one source � p ( ψ ) = D ( r s , t s ) d r s d t s (2) V ψ V ψ : domain of space and time that gives a flux between ψ and ψ + d ψ . Surface equation in pure diffusive regime : � � q r 2 ψ = (4 π K t ) 3 / 2 exp 4 K t D ( r s , t s ) can assume two limiting behaviours, 2D or 3D ! � ψ − 8 / 3 3 D For : ψ ≫ � ψ � we have, p ( ψ ) ∝ ψ − 7 / 3 2 D 10

Variance of the total flux N � Ψ = ψ i ⇒ p ( ψ ) → P (Ψ) i =1 Central limit theorem ? ψ = N � ψ 2 � − � Ψ � 2 Ψ = � Ψ 2 � − � Ψ � 2 = N σ 2 σ 2 N � � � ∞ ψ 1 / 3 � ∞ cte = ∞ 3 D ψ 2 p ( ψ ) dψ ∝ � ψ 2 � = � ψ 2 / 3 � ∞ cte = ∞ 2 D 0 The variance diverges 11

But actually the pdf exists! Generalised central limit theorem ? The heavy tail behaviour conditioned the stable law limit ! � ∞ p ( ψ ′ ) d ψ ′ → ψ →∞ ψ α C ( ψ ) = η > 0 ∀ ψ � 0 , C ( ψ ) ≡ lim ψ For N sufficiently large : � Ψ − � Ψ � � 1 P (Ψ) → S [ α, 1 , 1 , 0; 1] σ N σ N � � � 1 /α 5 / 3 3 D η π N With : α = 2 D and σ N = 2Γ( α ) sin ( α π/ 2) 4 / 3 12

But actually the pdf exists! σ N = 1 , α = 5 / 3 → 3 D , α = 4 / 3 → 2 D ⇒ So one can define confidence intervals, pvalues... 13

Simulation check For example at 1TeV : 10 0 10 0 10 1 Stable law α = 4 / 3 10 − 1 Stable law α = 5 / 3 10 − 1 10 0 Gaussian law σ sim 10 − 2 10 − 2 Simulations 10 − 1 p value p value f 10 − 3 10 − 3 pd 10 − 2 E=1TeV E=1TeV E=1TeV 10 − 4 10 − 4 10 − 3 10 − 5 10 − 5 10 − 4 10 − 6 10 − 6 50 50 50 ∆[%] 0 0 0 − 50 − 50 − 50 − 0 . 06 − 0 . 04 − 0 . 02 0 . 00 0 . 02 0 . 04 0 . 06 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 log 10 (Ψ / � Ψ � sim ) log 10 (Ψ / � Ψ � sim ) log 10 (Ψ / � Ψ � sim ) Simulation generated 10 6 configurations of galaxies. Transition from the 2D to the 3D regime ! ⇒ Genolini, Y., Salati, P., Serpico, P. D., & Taillet, R. (2017). Stable laws and cosmic ray physics. A&A, 600, A68. 14

On the form of p ( ψ ) The diffusive propagator is not causal for some region in space and time... ◮ Loophole : reevaluation of � p ( ψ ) = D ( r s , t s ) d r s d t s V causal ψ � ψ − 8 / 3 for : ψ < ψ c ◮ p ( ψ ) ∝ ψ − 11 / 3 for : ψ > ψ c The variance converges again! Shall we use the central limit theorem ?... Not really if ψ c is very large. Simulations : Stable law is a very good approximation till 10TeV ! 15

Application of the theory! May a particular configuration of the sources explain break features ? 1.4 10 4 10 4 Φ p .Ek 2 . 7 6 10 3 Fiducial 2 10 3 CREAM 2005 AMS 2015 10 2 10 3 10 4 10 5 1 10 Kinetic Energy Ek [GeV] ⇒ P ( Ψ ) ? 16

Probability of such an excess We compute an upper value of the probability that a particular configuration of the sources gives a flux Ψ at 12.8TeV : � ∞ � + ∞ p value = dψ exp dψ th p ( ψ exp | ψ th ) p ( ψ th | Model ) , Ψ exp 0 Example for the benchmark models : Models MIN MED MAX Probabilities(Stable law 4/3) 0.031 0.0082 0.0013 17

A theory of stochasticity for CRs More generally.. Effect of stochasticity @ a given energy Could be great to include energy correlations Theoretical uncertainty that should be taken into account 18

Questions ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? More details in : Genolini, Y., Salati, P., Serpico, P. D., & Taillet, R. (2017). Stable laws and cosmic ray physics. A&A, 600, A68. 19

BACKUP 20

The break feature of the protons [Kachelrieß et al., 2015] • Local flux dominated by a 2Myr old SNR in order to explain the knee.. • D ⊥ ≪ D � • At E = 1 TeV, Ψ ≈ 2 . 86 � Ψ � ⇒ P (Ψ) can be used for an homogeneous diffusion model. Models MIN MED MAX Probabilities 0.0072 0.0012 0.00016 21

The break feature of the protons [Tomassetti et al., 2015] • Two component model, without prior on their number of sources • Homogeneous diffusion • At E = 10 GeV, Ψ ≈ 3 . 3 � Ψ � ⇒ P (Ψ) can be used ! The probability @10GeV is 8 . 6 × 10 − 5 ! 22