Strict constrains on cosmic ray propagation and

abundancesAaron VincentIPPP Durham

MultiDark - IFT Madrid - Nov. 23 2015

Stanford/SLACIgor Moskalenko

Troy Porter Elena Orlando

IFIC (Valencia)Roberto Ruiz de

Austri

MPE GarchingAndrew Strong

IPPP DurhamAaron Vincent

U. ReykjavíkGulli Johannesson

ImperialRoberto Trotta

NASA GoddardPhil Graff

The Galbayes collaboration

CambridgeMike Hobson Farhan Feroz

Galbayes in a nutshell

Fantastic four issue 1

Most sophisticated ever Bayesian determination of the cosmic ray injection abundances and propagation parameters in the Galaxy using the GALPROP numerical package.

Follow-up to Trotta et al 2011

We use MultiNest nested sampling algorithm and SkyNet neural network machine learning tools.

Cosmic rays and dark matter Local

p, p, e+, e�, ...direct observation

Galactic centre

• Inverse compton scattering of CRs with Starlight, IR and CMB

• Bremsstralung

p, p, e+, e�, ...

secondary gammas

For DM hunters, cosmic ray propagation tells us:

What the backgrounds look like

What the DM signal looks like

(Standard, intragalactic) cosmic ray production

Primaries Secondaries

Supernovae, SNRs, pulsars, stellar winds accelerate particles to relativistic energies

Particles interact with the Interstellar Medium (ISM) or

decay, producing new particles along the way

Charged particles in the turbulent ISM

Energy is injected into plasma on large scales Diffuses to smaller scales

“Big whirls have little whirls, which feed on their velocity,

and little whirls have lesser whirls, and so on to viscosity”

Lewis Fry Richardson

Energy in plasma

Energy in B field fluctuations

Particle-plasma interaction

� ~B

�~B

�~ B

� ~B

�~B

� ~B

�~B

CR diffusion in the turbulent ISM is a resonant process. Particles will scatter predominantly on B irregularities with a projected scale equal to the gyroradius:

Example: 1 GeV proton in microgauss field:

rg ⇠ 10�6pc

1

k⇠ rg =

pc

ZeB⌘ R

B

Looks like a random walk (diffusion) with coefficient

Dxx

'✓�B

B

◆�2

vrg

⇠ R1/3

If E(k)dk ⇠ k�5/3dk(“Kolmogorov turbulence”)

DiffusionTheory tells us D(R) ~ power law of the rigidity, but given all

the unknowns about the ISM, we can’t get the normalization or power from first principles

R =pc

ZeDxx

= D0

✓R

R0

◆�

@ (~x, t)

@t

= rD

xx

(R)r (~x, t) + q(R, ~x, t)

free parameter

Other effects

•Diffusive reacceleration: scattering on bulk plasma (transverse B) waves, governed by Alfvén speed vAlf.

•Spallation/fragmentation: heavier elements probe a smaller (closer) region of the ISM.

•Nuclear decay: radioactive elements must have been produced nearby.

•Energy loss: Inverse-Compton, Bremsstrahlung, synchrotron

•Convection

Model: the galaxy as a cylinder

2zh

R ' 20kpc

Axially-symmetric, 2D galactic propagation zone

diffusion zone

gas

R� = 8.5kpc

@

@t= q(~r, p, t) +r(D

xx

r � ~V ) +@

@pp2D

pp

@

@p

1

p2

+@

@p

hp � p

3r · ~V )

i� 1

⌧f � 1

⌧r

source diffusion reacceleration

energy loss fragmentation decay

Solve for steady state@

@t= 0

Secondary/Primary ratiosRatio of secondary (e.g. B) to primary (e.g. C) is a measure of the column density of ISM

“stuff” traversed by the cosmic ray on its way from a source to the earth. This is given by

D0xx (diffusion coeff ~ 1/diffusion time) zh (height of the diffusion zone)

To break this degeneracy, one looks at radioactive secondaries,

e.g. 10Be, 26Al, which limits the distance from which we observe particles (i.e. sensitive only to D0xx, not zh)

Long distance few bounces

Short distance many bounces

Degeneracy:

GALPROP: http://galprop.stanford.edu

2zh

R ' 20kpc

diffusion zone

R� = 8.5kpc

•We use an updated version of the publicly available code

•Fully Numerical

• 3d grid in r, z and p

• Position-dependent source, gas and radiation distributions based on observations

•Can simultaneously compute CRs, gammas and synchrotron

gas

Free parameters

Propagation

Abundances: H, He, C, N, O, Ne, Na, Mg, Al, Si

20 Free parameters!

Nuisance parameters

Modulation by solar magnetic field

account for possible inconsistencies between

data sets

20 Free parameters + 10 nuisance parameters

Bayesian Inference

Few elements: fast Linear: very fast

Slow

Since small Z probes a different distance scale, it makes sense to separate out Z = 1,2 from heavier (Z > 4)

Nested Sampling: MultiNest

• Technique for construction of “isocontours” in the parameter volume

• These allow the transformation of the likelihood volume integral to a one-dimensional one

• Allows for a ~ order of magnitude speedup with respect to standard MCMC

838 Nested Sampling for General Bayesian Computation

In principle, such a point could be obtained by sampling Xi uniformly from within thecorresponding restricted range (0, Xi−1), then interrogating the original likelihood-sorting todiscover what its θi would have been. In practice, it would naturally be obtained directly asθi, by sampling within the equivalent constraint L(θ) > Li−1 (with L0 = 0 to ensure completeinitial coverage), in proportion to the prior density π(θ). This too finds a random point,distributed just the same. The second method is equivalent to the first, but bypasses the useof X. So we don’t need to do the sorting after all! That’s the key.

Successive points are illustrated in Figure 3, in which prior mass is represented by area.Thus, point 2 is found by sampling over the prior within the box defined by L > L1, and soon. Such points will usually be found by some MCMC approximation, starting at some pointθ∗ known to obey the constraint (if available), or at worst starting at θi−1 which lies on anddefines the current likelihood boundary.

0 1X X X

L

L

L

L

L

LL

Parameter space 3 2 1

1

2

3

1

23

Figure 3: Nested likelihood contours are sorted to enclosed prior mass X.

It is not the purpose of this introductory paper to develop the technology of navigationwithin such a volume. We merely note that exploring a hard-edged likelihood-constrained do-main should prove to be neither more nor less demanding than exploring a likelihood-weightedspace. For example, consider a uniform prior weighted by a C-dimensional unit Gaussian like-lihood L(θ) = exp(− 1

2 |θ|2). Conventional Metropolis-Hastings exploration is simply accom-

plished with trial moves of arbitrary direction having step-length |δθ| around 1 for efficiency.Most points have |θ| ≈

√C, so the relaxation time is about C steps.

In nested sampling, the corresponding hard constraint is the ball |θ| <√

C (or thereabouts).The typical point has |θ| ≈

√C − 1/

√C, that being the median radius of the ball. Again, the

efficient trial step-length is |δθ| ≈ 1, so the relaxation time per iterate is much the same asbefore. There are well-developed methods, such as Hamiltonian (or “hybrid”) Monte Carlo(Duane et al. (1987), Neal (1993)), slice sampling (Neal (2003)) and more, which learn aboutmore general shapes of L in order to explore the likelihood-weighted space more efficiently.Similar methods ought to work for exploring likelihood-constrained domains, but have not yetbeen developed.

In terms of prior mass, successive intervals w scan the prior range from X = 1 down to

Neural Networks• Machine learning tool to train a set of “hidden” nodes to replace the likelihood

evaluation (in this case GALPROP) with a set of simpler nonlinear operations

• SkyNet: NN training tool

• BAMBI: implementation for acceleration of bayesian searches

• 20-50% speedup

D0 (1028 cm2 s!1)2 4 6 8 10 12 14 16 18

Post

erio

rpro

bability

0

0.2

0.4

0.6

0.8

1

/0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Post

erio

rpro

bability

0

0.2

0.4

0.6

0.8

1

80

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

Post

erio

rpro

bability

0

0.2

0.4

0.6

0.8

1

Galprop onlyBAMBI: low tol.BAMBI: med tol.

Data2

TABLE I. Data sets to be used for all three scans; Scan 2 will additionally use data presented in Table V. AV: We haveremoved the TRACER data since the high-energy bins have a very large width which is not compatible with our computation ofthe �2.

Data Experiment Energy Range

p PAMELA 0.44 – 1000 GeV/nCREAM 3 — 200 TeV/n

p PAMELA 0.28–128 GeV/n

He PAMELA 0.13 – 504 GeV/nCREAM 0.8 – 50 TeV/n

B/CACE-CRIS (’97-’98) 72–170 MeV/n

HEAO-3 0.62–35 GeV/nCREAM 1.4–1450 GeV/n

10Be/9Be ACE-CRIS (’97-’99) 81–132 MeV/nISOMAX 0.51–1.51 GeV/n

B HEAO-3 0.62 — 35 GeV/n

CHEAO-3 0.62– 35 GeV/nCREAM 86–7415 GeV/n

N HEAO-3 0.62– 35 GeV/nCREAM 95–826 GeV/n

O HEAO-3 0.62 –35 GeV/nCREAM 64 –7287 GeV/n

TABLE II. Abundances to be used in scan 1

Element Variable ValueH iso_abundance_01_001 1.06e+06

iso_abundance_01_002 0He iso_abundance_02_003 0

iso_abundance_02_004 7.199e+04Li iso_abundance_03_006 0

iso_abundance_03_007 0Be iso_abundance_04_009 0B iso_abundance_05_010 0

iso_abundance_05_011 0C iso_abundance_06_012 2819

iso_abundance_06_013 0N iso_abundance_07_014 182.8

iso_abundance_07_015 0O iso_abundance_08_016 3822

iso_abundance_08_017 0iso_abundance_08_018 1.286

The 1D posterior distributions for the cosmic-ray parameters resulting from BAMBI runs with � = 0.5, 0.7, 1.0are shown in Fig. 1. For comparison, results for a MultiNest run are shown in black. The b.f. parameter values areshown on the plots as circled crosses. This is of interest, since the best-fit point is usually found towards the end ofthe scan and thus is expected to have been computed by the network. As can be seen, results for � = 0.5, 0.7 closelyresemble results for the MN only run, while results for sigma = 1.0 display small discrepancies. For sigma = 0.5 (0.7,1.0), 25% (36%, 45%) of all likelihood evaluations were computed using the network.

9

TABLE V. Data sets to be used for Scan 2 (over abundances), in addition to the data presented in Table I. Note that forTRACER, we only use the three highest-energy bins.

Data Experiment Energy Range

NeACE-CRIS 85 – 240 MeV/nHEAO-3 0.62 – 35 GeV/nCREAM 47– 4150 GeV/n

Na ACE-CRIS 100 – 285 MeV/nHEAO-3 0.8 – 35 GeV/n

MgACE-CRIS 100–285 MeV/nHEAO-3 0.8–35 GeV/nCREAM 27–4215 GeV/n

Al ACE-CRIS 100–285 MeV/nHEAO-3 0.8–35 GeV/n

Si

ACE-CRIS 120–285 MeV/nHEAO-3 0.8– 35 GeV/nCREAM 27-2418 GeV/nTRACER 138 – 2088 GeV/n

TABLE VI. List of model parameters, for Scan 2, for fixing the source injection abundances. Priors are flat. The nuisanceparameters are the same modulation parameters as in Table IX

Element Prior Rangeproton_norm_flux [2, 8] ⇥10

�9

He [0.1, 2] ⇥10

5

C [0.1,6] ⇥10

3

N [0.1, 5] ⇥10

2

O [0.1,10] ⇥10

3

Ne [0, 1] ⇥10

3

Na [0, 5] ⇥10

2

Mg [0, 1.5] ⇥10

3

Al [0, 5] ⇥10

2

Si [0 1.5] ⇥10

3

TABLE VII. The four diffusion propagation models for Scan 3, and the additional parameters required by GALPROP.

(DR) (DC) (DCR) (BR)Diffusion- Diffusion- Convection and Break in D

xx

Reacceleration Convection Reaccelerationv_alfven v_alfven D_rigid_br

z0 z0 D_g_2

v0 v0

+ abundance scans:

Results

2.1 million likelihood evaluations (~20% by neural nets) 5.5 CPU years

1.9 million likelihood evaluations (~20% by neural nets) 35 CPU years

A few days (no big deal)

Results: abundances

Atomic number A12 14 16 18 20 22 24 26 28 30

Abundance

100X

i=X

Si

100

101

102

103

104

CN

ONe

NaMg

AlSi

previous (ACE-only results)

this work (to appear)

Solar (photosphere)

Meteoritic

Volatiles (C,N,O) depleted with respect to stellar abundances since CRs are preferentially accelerated

from refractory-rich dust grains

Results: propagation parameters

5 10 150

0.2

0.4

0.6

0.8

1

1.2 • Posterior mean: 9.0296× Best fit: 8.7602

• Posterior mean: 6.102× Best fit: 6.3302

D0 (1028 cm2 s−1)

Posteriorprobability

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

1.2 • Posterior mean: 0.38032× Best fit: 0.41109

• Posterior mean: 0.46065× Best fit: 0.46588

δ

Posteriorprobability

Light (B … Si) elements

10 20 30 400

0.2

0.4

0.6

0.8

1

1.2 • Posterior mean: 30.0165× Best fit: 31.4466

• Posterior mean: 8.9699× Best fit: 8.9223

vAlf (km/s)

Posteriorprobability

p, p,He

Trotta et al 2011

Consistent; more free parameters

= wider posteriors

Posterior bands

E/nuc (GeV)10-1 100 101

10Be/

9Be

10-2

10-1

100

E/nuc (GeV)10-1 100 101 102 103

B/C

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4E/nuc (GeV)

10-1 100 101 102 103E

2:3#

[GeV

m2

ssr

]!1

0

1

2

3

4

5

6

7

8O

E/nuc (GeV)10-1 100 101 102 103

E2:3#

[GeV

m2

ssr

]!1

0

1

2

3

4

5

6

7

8

9C

primaries

secondaries

C O

B/C 10Be/9Be

Hydrogen and Helium

Ekin/nuc (GeV)10-1 100 101 102 103

E2:7#

[GeV

m2

ssr

]!1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

57p

Ekin/nuc (GeV)10-1 100 101 102 103

7p=p

#10-4

0

0.5

1

1.5

2

2.57p=p

Ekin/nuc (GeV)10-1 100 101 102 103

E2:7#

[GeV

m2

ssr

]!1

0

500

1000

1500He

Ekin/nuc (GeV)10-1 100 101 102 103

E2:7#

[GeV

m2

ssr

]!1

0

2000

4000

6000

8000

10000

12000

14000p

p

He

p

p/p

2D posteriors

D0,xx

zh

vAlf

�

vAlf�

Light (B … Si) elementsp, p,He

Z ~ 1 are probing larger scales in the ISM (don’t

disintegrate)Propagation properties are

significantly different from the local ones responsible for

B/C + heavier elements!!

Antiproton ratio

E/nuc (GeV)10-1 100 101 102 103

7p=p

#10-4

0

0.5

1

1.5

2

2.5

Ekin/nuc (GeV)10-1 100 101 102 103

7p=p

#10-4

0

0.5

1

1.5

2

2.57p=p

Fit to B, C, N, O, Ne Na, Al, Mg, Si

PAMELAAMS-02 (unpublished)

Fit to H, He

Simultaneously fitting everything leads to significant tension between datasets,

bad fit

Summary

• Largest ever scan over the propagation parameters in a fully numeric implementation of the cosmic ray diffusion equation.

• Sensitivity of small vs large Z to propagation distance is very important. If you’re looking for an excess, make sure you’re consistently using the CR propagation parameters.

• Paper out very soon

• Also: GALDEF files with best parameters for GALPROP

• Future: extension to more models, nuclear cross sections, heavier elements

Extras

AMS-02

E/nuc (GeV)10-1 100 101 102 103

7p=p

#10-4

0

0.5

1

1.5

2

2.5

Ekin/nuc (GeV)10-1 100 101 102 103

7p=p

#10-4

0

0.5

1

1.5

2

2.57p=p

PAMELAAMS-02

Ekin/nuc (GeV)10-1 100 101 102 103

E2:7#

[GeV

m2

ssr

]!1

0

2000

4000

6000

8000

10000

12000

14000p

More propagation parameters

Modulation Parameters

Rescaling parameters

Numbers

More numbers