High-Energy Phenomena and Dark Matter Searches in Galaxy ... · High-Energy Phenomena and Dark...

Introduction to dark matterIndirect dark matter searches

High-energy phenomena

High-Energy Phenomena and Dark MatterSearches in Galaxy Clusters

Christoph Pfrommer1

in collaboration with

Anders Pinzke2, Lars Bergström2,Torsten Enßlin3, Volker Springel3

1Canadian Institute for Theoretical Astrophysics, Canada2Stockholm University, Sweden

3Max-Planck Institute for Astrophysics, Germany

Nov 25, 2009 / McMaster University

Christoph Pfrommer Dark Matter Searches in Galaxy Clusters



Outline

1 Introduction to dark matterProperties of dark matterTheory and observationsRecent exciting developments

2 Indirect dark matter searchesOur approachGamma-ray signaturesImplications for cosmological structure formation

3 High-energy phenomenaCosmological simulations with cosmic raysShocks and particle accelerationNon-thermal emission from clusters




Properties of dark matterTheory and observationsRecent exciting developments

Outline








Properties of dark matterWhat have galaxy clusters taught us about dark matter?

Dark matter exists and is . . .

. . . gravitationally interacting: most of the matter in the Universeis dark (galaxy clusters, galactic rotation curves, gravitationallensing, CMB, . . . )

. . . cold: it was non-relativistic at the time of decoupling fromweak interactions – otherwise structure formation would haveproceeded ‘top-down’, in contrast to our hierarchical structureformation scenario (Ly-α forest, structure formation)

. . . non-baryonic: it does not interact electro-magnetically withordinary matter (CMB + structure formation)

. . . collisionless: it has a very weak self-interaction cross section(galaxy clusters, relic density)

→ dark matter is a weakly interacting massive particle (WIMP), butwhat is it really (SUSY, extra dimensions, . . . )?





The matter content of the Universe – 2009

WMAP Five-Year: Komatsu et al. (2009)





Zwicky: first evidence for dark matter

Zwicky (1933) applied the virial theorem to theComa cluster of galaxies:

Egrav + 2Ekin = 0

Ekin: energy based on galaxies motions,Egrav: estimated gravitational energy

Zwicky found about 400 times more estimatedmass than visually observable

“missing mass problem”: the gravity of the visible galaxies in thecluster would be far too small for such fast orbits:

→ there must be some non-visible form of matter that providesenough of the mass and gravity to hold the cluster together





Evidence for non-baryonic dark mattertemperature fluctuations the epoch of recombination (z ' 1100)have an amplitude δT/T ' 10−5

linear regime of structure formation: δ(z) ∝ 1/(1 + z)

if dark matter were baryonic, we would only have time to growfluctuations of size δ ' 10−2

since we observe galaxies with δ & 102, dark matter fluctuationsmust have started to grow before recombination which is onlypossible if dark matter does not interact electro-magneticallywith matter

NASA/WMAP: 5 year CMB sky





Evidence for collisionless dark matter

1E 0657-56 (“Bullet cluster”)(X-ray: NASA/CXC/CfA/M.Markevitch et al.; Optical:NASA/STScI; Magellan/U.Arizona/D.Clowe et al.; Lensing:NASA/STScI; ESO WFI; Magellan/U.Arizona/D.Clowe et al.)

red: X-ray emittingcluster plasma

blue: weak lensing(dark matter) map

yellow: galaxies

the (collisionless) galaxiesfollow the dark matter distri-bution, the bulk of the plasmalags behind

→ dark matter is collisionless





The WIMP miracle

Fermi introduced a new mass scaleof mweak ∼ 100 GeV to describe thebeta decay: n → p e− ν

assuming a new (heavy) particle X ,initially in thermal equilibrium, with arelic density

ΩX ∼1

mPlT0 〈συ〉∼

m2X

mPlT0 g4X

mx ∼ mweak ∼ 100 GeVgx ∼ gweak ∼ 0.6

ΩX ∼ 0.1

Remarkable coincidence: particle physics independentlypredicts particles with the right density to be dark matter





WIMP detection

Correct relic density → DM annihilation in the Early Universe

q

indiect detection

DM

pro

duct

ion

now

:pa

rtic

le c

ollid

ers

DM scattering now:direct detection

χ χ

q

DM

annihilation now:





Indirect detection of dark matter

νeν_µ

νe

e+

Springel et al. 2008

Supersymmetric

χµ−

µ+

e−χneutralinos

νµ

_





Indirect detection of dark matter

νeν_µ

νe

e+

H.E.S.S.

neutralinos

νµ

_

Supersymmetric

PAMELA

Fermi

χµ−

µ+

e−χ

and cooling:limiting horizon ~ 1 kpc

Diffusive transport





Imaging air Cerenkov telescopes – the technique (1)

high-energy γ-ray impacts the Earth’s atmosphere and sets offan electro-magnetic cascade in the vicinity of a nucleus

e+/e− travel faster than the speed of light in the atmosphere →emission of a cone of blue Cerenkov light





Imaging air Cerenkov telescopes – the technique (2)

primary γ-rays and hadrons cause different showercharacteristics → separation of γ-rays from ‘background’ events

opening angle and shower location in the shower image allowsreconstructing the initial energy and direction of the γ-ray





PAMELA and HESS data on electrons and positrons

Energy (GeV)

1 10 100

))

-(e!

)+

+(e!

) /

(+

(e!

Po

sit

ron

fra

cti

on

0.01

0.02

0.1

0.2

0.3

PAMELA

PAMELA: (Adriani et al. 2009)

rising positron fraction with energy→ e−/e+ pair acceleration source

Energy (GeV)210 310

)-1

sr

-1 s

-2 m2

dN

/dE

(G

eV3

E

210

15%± E ∆

ATICPPB-BETSKobayashi

H.E.S.S.H.E.S.S. - low-energy analysisSystematic errorSystematic error - low-energy analysis Broken power-law fit

HESS: (Aharonian et al. 2009)

break in the e−/e+ spectrum→ maximum voltage of acceleratoror DM particle mass





Combining recent electron and positron dataFermi: excess number of leptons compared to background model (Abdo et al. 2009)

Bergström, Edsjö & Zaharijas 2009

MDM = 1.6 TeV, 100% μ+μ-, EF=1100

FermiHESS (×0.85)HESS LE (×0.85)TotalBackground (×0.85)DM signal

E3 Φ [

GeV

2 m

-2 s

-1 s

r-1]

10

100

Positron energy, Ee+ [GeV]100 1000

PAMELA

Posi

tro

n fr

acti

on

0.01

0.1

Ee+ [GeV]10 100





Interpretations of recent electron and positron data

excess number of leptons comparedto background (Fermi/HESS)

break in the e−/e+ spectrumindicates special energy scale(HESS)

rising positron fraction with energy(PAMELA)

Bergström, Edsjö & Zaharijas 2009

MDM = 1.6 TeV, 100% μ+μ-, EF=1100

FermiHESS (×0.85)HESS LE (×0.85)TotalBackground (×0.85)DM signal

E3 Φ [

GeV

2 m

-2 s

-1 s

r-1]

10

100

Positron energy, Ee+ [GeV]100 1000

PAMELA

Posi

tro

n fr

acti

on

0.01

0.1

Ee+ [GeV]10 100

1.) nearby pulsars:energetics convincing but smoothness of Fermi data was claimed tobe difficult to model (Harding & Ramaty 1987, Aharonian et al 1995, Malyshev et al. 2009)

2.) DM annihilations:excellent fit to data but enhancement of cross-section over standardvalue and muon decay channel necessary (Bergström et al. 2009)

→ Sommerfeld enhancement: 〈συ〉 ∼ c/υ (Arkani-Hamed et al. 2009)




Our approachGamma-ray signaturesImplications for cosmological structure formation

Outline








The key questions

How can we test this scenario?

Which are the most promising objects to target?

What are the cosmological implications of such an effective darkmatter annihilation?

I will argue in favor of gamma-ray observations of galaxy clustersbeing able to scrutinize the DM interpretation ofFermi/HESS/PAMELA data and will end with a surprisingcosmological result.

Pinzke, CP, Bergström, Phys. Rev. Lett., 2009, 103, 181302





Dissecting the dark matter flux

Nγ =

[∫LOS

ρ2χ dlχ

]〈συ〉2M2

χ

[ ∫ Mχ

Eth

(dNγ

dE

)SUSY

Aeff(E) dE]

∆Ω

4πτexp

astrophysics: contains the uncertainty about the DM profile withits central behavior and the substructure distribution

particle physics: assuming DM is supersymmetric, there is theuncertainty about the cross section, neutralino mass, and decaychannels

detector properties: energy dependent effective area, detectorresponse, scanning strategy, . . .





Boost factors





Galaxy clusters vs. dwarf galaxiesWhy look for clusters in the γ-ray band?

1 The DM annihilation flux of the smooth halo component scalesas F ∼

∫dVρ2/D2 ∼ M/D2 assuming a universal density

scaling1: the smooth component of dwarfs and galaxy clustersare equally bright!

2 Substructure in dark matter halos is less concentrated comparedto the smooth halo component (dynamical friction, tidal heatingand disruption): the DM luminosity is dominated by substructureat the virial radius, IF present!→ these regions are tidally stripped in dwarf galaxies→ galaxy clusters are dynamically ‘young’ and their subhalopopulation can boost the DM luminosity by up to 200(Springel et al. 2008).

1A more refined argument that takes into account the different haloformation epochs breaking scale invariance yields the same result.





Indirect detection of DM through gamma-rays

νeν_µ

νe

e+

χµ−

µ+

e−χneutralinos

νµ

_

Supersymmetric






νeν_µ

νe

e+

radiation

χ

γ

µ−

Final state

µ+

e−χneutralinos

νµ

_

Supersymmetric






νeν_µ

νe

e+

radiation

_

Supersymmetric

χ

γ

µ−

Final state

µ+

e−

γ Compton

χCMBneutralinos

Inverse νµ





Gamma-ray spectrum from DM annihilations

10-1 100 101 102 103

E γ [GeV]

10-11

10-10

10-9

E γ

2 d

F/dE

[ G

eV c

m-2

s-1

]

FornaxDM-IC-SFEDM-FSR-SFE





Gamma-ray spectrum from DM vs. CR interactions

10-1 100 101 102 103

E γ [GeV]

10-11

10-10

10-9

E γ

2 d

F/dE

[ G

eV c

m-2

s-1

]

FornaxDM-IC-SFEDM-FSR-SFECR- π 0





Gamma-ray spectrum for various galaxy clusters

10-1 100 101 102 103

E γ [GeV]

10-11

10-10

10-9

E γ

2 d

F/dE

[ G

eV c

m-2

s-1

]

FornaxComaVirgo

DM-IC-SFEDM-FSR-SFECR- π 0





DM gamma-rays: without substructure

10-1 100 101 102 103

E γ [GeV]

10-12

10-11

10-10

10-9

10-8

E γ

2 d

F/dE

[ G

eV c

m-2

s-1

]

Fornax:

Cl. w/o SubB

DM-SFE





DM gamma-rays: with substructure

10-1 100 101 102 103

E γ [GeV]

10-12

10-11

10-10

10-9

10-8

E γ

2 d

F/dE

[ G

eV c

m-2

s-1

]

Fornax:

Cl. w/ SubBCl. w/o SubB

DM-SFE





DM gamma-rays: with substructure and Milky Way

10-1 100 101 102 103

E γ [GeV]

10-12

10-11

10-10

10-9

10-8

E γ

2 d

F/dE

[ G

eV c

m-2

s-1

]

Fornax:Cl. w/ SubB + MWCl. w/ SubBCl. w/o SubB

DM-SFE





Probing small scales with gamma-rays

10-1 100 101 102 103

E γ [GeV]

10-14

10-13

10-12

10-11

10-10

10-9

10-8

F (

> E

γ )

[ p

h cm

-2 s

-1]

Virgo: 5.8deg, 5x10 -3MO •

FERMI

EGRET

DM-SFECR- π 0





Implications for cosmological structure formationProbing the linear power spectrum on the smallest scales

−2 −6

log

10

CDM prediction(Hofmann et al. 2002)

limit obtained by EGRETnon−detection of Virgo(Pinzke, CP, Bergstrom 2009)

upper limit byLy− forest(Seljak et al. 2006)

2 x 108

lower limit possiblewith Fermi sensitivity

M / Msol

k x kpc

P ~ k−3P

α

630300.60.001

103

10





Conclusions on dark matter searches

Gamma-ray observations of galaxy clusters by Fermi will test theDM interpretation of the Fermi/HESS/PAMELA data in the nextyears.

If the DM interpretation is correct, then we either live in a warmdark matter Universe or there is a new dynamical effect duringnon-linear structure formation that wipes out the smalleststructures.

Gamma-ray observations might be the most sensitive probes ofthe smallest cosmological structures.




Cosmological simulations with cosmic raysShocks and particle accelerationNon-thermal emission from clusters

Outline








Shocks in galaxy clusters

1E 0657-56 (“Bullet cluster”)(X-ray: NASA/CXC/CfA/M.Markevitch et al.; Optical:NASA/STScI; Magellan/U.Arizona/D.Clowe et al.; Lensing:NASA/STScI; ESO WFI; Magellan/U.Arizona/D.Clowe et al.)

Abell 3667(radio: Johnston-Hollitt. X-ray: ROSAT/PSPC.)





Radiative cool core cluster simulation: gas density

10-1

100

101

102

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15x [ h-1 Mpc ]

-15

-10

-5

0

5

10

15

y [

h-1 M

pc ]

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

〈1+δ

gas〉





Mass weighted temperature

105

106

107

108

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15x [ h-1 Mpc ]

-15

-10

-5

0

5

10

15

y [

h-1 M

pc ]

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

〈Tρ

gas〉/〈ρ

gas〉

[K]





Mach number distribution weighted by εdiss

1

10

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15x [ h-1 Mpc ]

-15

-10

-5

0

5

10

15

y [ h

-1 M

pc ]

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

〈Mε

diss/〈ε

diss〉





Radiative simulations – flowchart

CP, Enßlin, Springel (2008)





Collisionless shocks at supernova remnants

Astrophysical collisionless shocks can:

accelerate particles (electrons and ions)

amplify magnetic fields (or generate them from scratch)

exchange energy between electrons and ions

SN 1006 X-rays (CXC/Hughes) G347.3 HESS TeV(Aharonian et al. 2006)

Tycho X-rays (CXC)





Diffusive shock acceleration – Fermi 1 mechanism

Spectral index depends on the Mach number of the shock,M = υshock/cs:

log p

strong shock

10 GeV

weak shock

keV

log f





Radiative simulations with cosmic ray (CR) physics






Radiative simulations with extended CR physics






Hadronic cosmic ray proton interaction

π0

π+

µ+

νµ

ν_µ

νe

p

CRp

e+





Hadronic cosmic ray proton interaction





Mach number distribution weighted by εdiss

1

10

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15x [ h-1 Mpc ]

-15

-10

-5

0

5

10

15

y [ h

-1 M

pc ]

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

〈Mε

diss/〈ε

diss〉





Mach number distribution weighted by εCR,inj

1

10

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15x [ h-1 Mpc ]

-15

-10

-5

0

5

10

15

y [ h

-1 M

pc ]

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

〈Mε

CR,in

j〉/〈ε

CR,in

j〉





CR pressure PCR

10-17

10-16

10-15

10-14

10-13

10-12

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15x [ h-1 Mpc ]

-15

-10

-5

0

5

10

15

y [

h-1 M

pc ]

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

〈PC

Rρ

gas〉/〈ρ

gas〉

[erg

cm−

3h2 70

]





Multi messenger approach for non-thermal processes





Hadronic γ-ray emission, Eγ > 100 GeV

10-12

10-11

10-10

10-9

10-8

S γ (

Eγ >

100

GeV

) [

ph c

m-2

s-1

]

-5 0 5

-5

0

5

-8 -6 -4 -2 0 2 4 6 8x [ Mpc ]

-8

-6

-4

-2

0

2

4

6

8

y [

Mpc

]

-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8





Inverse Compton emission, EIC > 100 GeV

10-12

10-11

10-10

10-9

10-8

S IC

,tota

l (E

γ > 1

00 G

eV)

[ ph

cm

-2 s

-1 ]

-5 0 5

-5

0

5

-8 -6 -4 -2 0 2 4 6 8x [ Mpc ]

-8

-6

-4

-2

0

2

4

6

8

y [

Mpc

]

-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8





Total γ-ray emission, Eγ > 100 GeV

10-12

10-11

10-10

10-9

10-8

S tot

al (

Eγ >

100

GeV

) [

ph c

m-2

s-1

]

-5 0 5

-5

0

5

-8 -6 -4 -2 0 2 4 6 8x [ Mpc ]

-8

-6

-4

-2

0

2

4

6

8

y [

Mpc

]

-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8





CR proton and γ-ray spectrum (Pinzke & CP 2009)

10-2 100 102 104 106 108

p = Pp / (mp c)

10-3

10-2

10-1

1.0

f(p)

p2

10-2 100 102 104 106 108 1010

Eγ [ GeV ]

10-12

10-11

10-10

10-9

Fγ(>

Eγ)Eγ

[GeV

cm−

2s−

1]

sIC

pIC

Pion decay





Universal CR spectrum in clusters

GLAST: ~ 2.4

IACT: ~ 2.2αp

pα

10-4 10-2 100 102 104 106 108 1010

p

0.001

0.01

0.1

1

10

< f(

p) p

2 >

Normalized CR spectrum shows universal concave shape→ governed byhierarchical structure formation and the implied distribution of Mach numbersthat a fluid element had to pass through in cosmic history (Pinzke & CP 2009).





Gamma-ray scaling relations

1014 1015

Mvir [ MO • ]

1041

1042

1043

1044

1045

1046L

γ (E

γ > 1

00 M

eV)

[ ph

s-1

]total γ-ray emission, excl. galaxies, R<Rvirpion-decay γ-ray emissionprimary IC emissionsecondary IC emission

Scaling relation + complete sample of the brightest X-ray clusters(HIFLUGCS)→ predictions for Fermi and IACT’s





Predicted cluster sample for Fermi and IACT’s

0 20 40 60 80 100

10-10

10-9

10-8

0 20 40 60 80 100

10-13

10-12

10-10

10-9

10-8

OPHIUCHU

COMA

FORNAX

A3627

PERSEUS

A3526

A1060

3C129

AWM7

M49

A0754

CRs with galaxiesCRs witout galaxies

Fγ(Eγ>

100

GeV

)

Fγ(Eγ>

100

MeV

)

extended HIFLUGCS cluster ID

black: optimistic model, including galactic ‘point sources’ that bias γ-ray fluxhigh; red: realistic model, excluding galactic ‘point sources’





Predicted cluster sample for Fermi – brightest objects

10-10 10-9 10-80

2

4

6

8

10

12

14 excl. galaxies

Oph

iuch

us

Coma

A36

27, P

erse

us

Forn

ax

A36

26, A

1060

3C12

9

Ncl

uste

rs

Fγ [ph cm−2 s−1]





MAGIC observations of Perseus





Upper limit on the TeV γ-ray emission from Perseus

100 1000E [GeV]

10-14

10-13

10-12

10-11

10-10F

γ ( >

E )

[ph

s-1

cm

-2]

radiative physics w/ gal. x 3.5

radiative physics w/ gal.

radiative physics w/o gal.

Fγ, min (B0 = 10 µG, εB < εth / 3)

The MAGIC Collaboration: Aleksic et al. & CP et al. 2009





Results from the Perseus observation by MAGIC

assuming f ∝ p−α with α = 2.1, PCR ∝ Pth:ECR < 0.017Eth → most stringent constraint on CR pressure!

upper limits consistent with cosmological simulations:Fupper limits(100GeV ) = 3.5 Fsim (optimistic model)

simulation modeling of pressure constraint yields〈PCR〉/〈Pth〉 < 0.07 (0.14) for the core (entire cluster)

3 physical effects that resolve the apparent discrepancy:

concave curvature ‘hides’ CR pressure at GeV energiesgalactic ‘point sources’ bias γ-ray flux high and pressurelimits low (partly physical)relative CR pressure increases towards the outer parts(adiabatic compression and softer equation of state of CRs)





Minimum γ-ray flux in the hadronic model

0.1 1.0 10.010-3

10-2

10-1

100

101

B[µG]

j ν,I

C(B

)/j ν,I

C(B

CM

B)

jν(B)/ jν(BCMB)

jIC(B)/ jIC(BCMB)

For jν(B) :αν = 1αν = 1.15αν = 1.30αν = 1.45

Synchrotron emissivity of high-energy, steady state electrondistribution is independent of themagnetic field for B BCMB!

Synchrotron luminosity:

Lν = Aν

∫dV nCRngas

ε(αν+1)/2B

εCMB + εB

→ Aν

∫dV nCRngas (εB εCMB)

γ-ray luminosity:

Lγ = Aγ

∫dV nCRngas

→ minimum γ-ray flux:

Fγ,min =Aγ

Aν

Lν

4πD2





Minimum γ-ray flux in the hadronic model: Fermi

Minimum γ-ray flux (Eγ > 100 MeV) for the Coma cluster:

CR spectral index 2.0 2.3 2.6 2.9Fγ [10−10 ph cm−2s−1] 0.8 1.6 3.4 7.1

These limits can be made even tighter when considering energyconstraints, PB < Pgas/30 and B-fields derived from Faradayrotation studies, B0 = 3 µG:Fγ,COMA & (1.1 . . . 1.5)× 10−9γ cm−2s−1 . FFermi, 2yr

Non-detection by Fermi seriously challenges the hadronicmodel.

Potential of measuring the CR acceleration efficiency fordiffusive shock acceleration.





Conclusions on high-energy phenomena

In contrast to the thermal plasma, the non-equilibrium distributions ofCRs preserve the information about their injection and transportprocesses and provide thus a unique window of current and paststructure formation processes!

1 Cosmological hydrodynamical simulations are indispensable forunderstanding non-thermal processes in galaxy clusters→ illuminating the process of structure formation

2 Multi-messenger approach including radio synchrotron, hardX-ray IC, and HE γ-ray emission:

fundamental plasma physics: diffusive shock acceleration,large scale magnetic fields, and turbulencenature of dark mattergold sample of clusters for precision cosmology





Literature for the talk

Pinzke, Pfrommer, Bergström, Phys. Rev. Lett., 2009, 103, 181302, Gamma-raysfrom dark matter annihilations strongly constrain the substructure in halos

Pfrommer, 2008, MNRAS, 385, 1242 Simulating cosmic rays in clusters ofgalaxies – III. Non-thermal scaling relations and comparison to observations

Pfrommer, Enßlin, Springel, 2008, MNRAS, 385, 1211, Simulating cosmic rays inclusters of galaxies – II. A unified scheme for radio halos and relics withpredictions of the γ-ray emission

Pfrommer, Enßlin, Springel, Jubelgas, Dolag, 2007, MNRAS, 378, 385,Simulating cosmic rays in clusters of galaxies – I. Effects on theSunyaev-Zel’dovich effect and the X-ray emission

Pfrommer, Springel, Enßlin, Jubelgas, 2006, MNRAS, 367, 113,Detecting shock waves in cosmological smoothed particle hydrodynamicssimulations

Enßlin, Pfrommer, Springel, Jubelgas, 2007, A&A, 473, 41,Cosmic ray physics in calculations of cosmological structure formation

Jubelgas, Springel, Enßlin, Pfrommer, A&A, , 481, 33,Cosmic ray feedback in hydrodynamical simulations of galaxy formation


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