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■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

❙❡❧❢✲❛♥♥✐❤✐❧❛t✐♥❣ ❉▼ ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❧✐❣❤t ❢♦r❝❡ ♠❡❞✐❛t♦r

▼✐❤❛❡❧ P❡t❛↔

■♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ P✐❡r♦ ❯❧❧✐♦ ❛♥❞ ▼❛✉r♦ ❱❛❧❧✐

❆str♦♣❛rt✐❝❧❡ ♣❤②s✐❝s ❣r♦✉♣ ❙■❙❙❆✱ ❚r✐❡st❡

❏✉♥❡ ✶✸✱ ✷✵✶✽

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥

❙❡❛r❝❤ ❢♦r ❉▼ ❛♥♥✐❤✐❧❛t✐♦♥✴❞❡❝❛② s✐❣♥❛t✉r❡s ✐♥ t❤❡ s❦② ❯s❡ ❣❛❧❛①✐❡s ❛s ♣❛rt✐❝❧❡ ♣❤②s✐❝s ❧❛❜♦r❛t♦r✐❡s ❙tr♦♥❣ ❝♦♥str❛✐♥ts ♦♥ t❤❡r♠❛❧ r❡❧✐❝s ❘❛♣✐❞ ✐♠♣r♦✈❡♠❡♥ts ✐♥ ♦❜s❡r✈❛t✐♦♥❛❧ ❞❛t❛ ⇒ ◆❡❡❞ ❢♦r ❛❝❝✉r❛t❡ ♠♦❞❡❧✐♥❣ ♦❢ ❉▼ r❡❧❛t❡❞ s✐❣♥❛❧s

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❈✉rr❡♥t ❝♦♥str❛✐♥ts

DOI:1 0. 1 03 8/ NP HYS 4049

1 01 1 02 1 0−

3

1

− 2

1

− 1

1

2

1

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1

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1

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1

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1

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1

8

1 03 1 04 m

DM (GeV)

1 00 1 01 〈σ v 〉 (×1 0−

26 cm 3s− 1

) Icecube 201 6/ANTARES 201 5 Planck 201 5 AMS 201 5 HESS: 201 6 Ferm i-LAT 201 6 Galactic centre bulge em ission Ferm i-LAT 201 5 Dwarf galaxies

❋✐❣✉r❡✿ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ❝♦♥str❛✐♥ts ♦♥ ❉▼ ❛♥♥✐❤✐❧❛t✐♦♥ ❝r♦ss✲s❡❝t✐♦♥✳

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

Pr♦❥❡❝t ♦✉t❧✐♥❡ ✲ ❛r❳✐✈✿✶✽✵✹✳✵✺✵✺✷

▼♦t✐✈❛t✐♦♥✿

• ❡♥❡r✐❝ ❡♥❤❛♥❝❡♠❡♥t ♦❢ s✐❣♥❛❧ ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❧✐❣❤t ♠❡❞✐❛t♦rs ✶

Pr♦❜❡ ❡✛❡❝t ♦❢ ❉▼ ✈❡❧♦❝✐t② ❛♥✐s♦tr♦♣② ♦♥ ❛♥♥✐❤✐❧❛t✐♦♥ s✐❣♥❛❧ ❊r❛ ♦❢ ❤✐❣❤ ♣r❡❝✐s✐♦♥ ❝♦s♠♦❧♦❣② ❛♥❞ ❛str♦♥♦♠② ✭❋❡r♠✐✲▲❆❚✱ ❛❝❝✉r❛t❡ ♠❡❛s✉r❡♠❡♥ts ♦❢ st❡❧❧❛r ❦✐♥❡♠❛t✐❝s✱ ✳✳✳✮ ❖✉t❧✐♥❡✿ ▼♦❞❡❧ t❤❡ ❉▼ ❤❛❧♦ ♣❤❛s❡✲s♣❛❝❡ ❞✐str✐❜✉t✐♦♥ ❙t✉❞② t❤❡ ❛♥♥✐❤✐❧❛t✐♦♥ r❛t❡s ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❙♦♠♠❡r❢❡❧❞ ❡♥❤❛♥❝❡♠❡♥t ❆♣♣❧② t❤❡ ❛♥❛❧②s✐s t♦ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

✶❇♦❞❞② ❡t ❛❧✳ ❛r❳✐✈✿✶✼✵✷✳✵✵✹✵✽✱ ❇❡r❣strö♠ ❡t ❛❧✳ ❛r❳✐✈✿✶✼✶✷✳✵✸✶✽✽ ▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥

❋♦❝✉s ♦♥ ❣❛♠♠❛ r❛②s ❢r♦♠ ❉▼ ❛♥♥✐❤✐❧❛t✐♦♥s✳ ❉✐✛❡r❡♥t✐❛❧ ♣❤♦t♦♥ ✢✉① ♣r♦❞✉❝❡❞ ❜② ❛♥♥✐❤✐❧❛t✐♦♥s✿ ❞Φ ❞Eγ = ✶ ✽π ❞N ❞Eγ

• ❞Ω
• ❞ℓ
• ❞✸v✶

f ( r, v✶) mχ

• ❞✸v✷

f ( r, v✷) mχ · (σvr❡❧) ◆♦♥✲st❛♥❞❛r❞ ✐♥❣r❡❞✐❡♥ts✿ ❆♥♥✐❤✐❧❛t✐♦♥ ❝r♦ss✲s❡❝t✐♦♥ ❜♦♦st❡❞ ❜② r❡❧❛t✐✈❡ ✈❡❧♦❝✐t② ❞❡♣❡♥❞❡♥t ❢❛❝t♦r ✲ ❙♦♠♠❡r❢❡❧❞ ❡♥❤❛♥❝❡♠❡♥t✿ σvr❡❧✵ → σvr❡❧✵ · S(vr❡❧) ❈♦♠♣✉t❡ ♣❤❛s❡✲s♣❛❝❡ ❞✐str✐❜✉t✐♦♥ ❢♦r ✈❛r✐♦✉s ❉▼ ❞❡♥s✐t② ♣r♦✜❧❡s ✭◆❋❲✱ ❇✉r❦❡rt ❛♥❞ ♥♦♥✲♣❛r❛♠❡tr✐❝ ♣r♦✜❧❡✮

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❙♦♠♠❡r❢❡❧❞ ❡♥❤❛♥❝❡♠❡♥t

• ❡♥❡r✐❝ ❜♦♦st ♦❢ ❝r♦ss✲s❡❝t✐♦♥ ❢♦r ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ♣❛rt✐❝❧❡s ❢♦r

✐♥t❡r❛❝t✐♦♥s ♠❡❞✐❛t❡❞ ❧✐❣❤t s❝❛❧❛r ♦r ✈❡❝t♦r ❢♦r❝❡ ♠❡❞✐❛t♦r φ ✭✐✳❡✳ r❡q✉✐r❡s Eχ ≈ mχ ❛♥❞ mφ ≪ αχmχ✮✳ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣ ❣✐✈❡s r✐s❡ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦t❡♥t✐❛❧✿ V (r) = ∓αχ r exp (−mφr) ❙♦❧✈❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ t♦ ♦❜t❛✐♥ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥ ❞✐st♦rt✐♦♥ ❞✉❡ t♦ t❤❡ ♠❡❞✐❛t♦r ❡①❝❤❛♥❣❡✿ χ′′(x) + v✷

r❡❧

α✷

χ

+ V (x)

• χ(x) = ✵

⇒ S(vr❡❧; ξ) =

• χ(✵)

χ(∞)

• ✷

∝ ✶ vα

r❡❧

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❉▼ ♣❤❛s❡✲s♣❛❝❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥

❉✐✛❡r❡♥t ✐s♦tr♦♣✐❝ ♠♦❞❡❧✐♥❣s ♣r❡s❡♥t ✐♥ ❧✐t❡r❛t✉r❡ ▼❛①✇❡❧❧✲❇♦❧t③♠❛♥♥ ❛♣♣r♦①✐♠❛t✐♦♥✿ f (r, v) = ρ❉▼(r) (✷πσ✷(r))✸/✷ · ❡①♣

• v✷

✷σ✷(r)

• ❊❞❞✐♥❣t♦♥✬s ✐♥✈❡rs✐♦♥ ✭❜❛s❡❞ ♦♥ s♣❤❡r✐❝❛❧ ❏❡❛♥s ❡q✉❛t✐♦♥✮✿

f (E) = ✶ √ ✽π✷ ❞ ❞E E

✵

❞Ψ √ E − Ψ ❞ρ ❞Ψ , E = Ψ(r) − v✷ ✷ ❊❞❞✐♥❣t♦♥✬s ✐♥✈❡rs✐♦♥ ❣✐✈❡s ✉♥✐q✉❡ s♦❧✉t✐♦♥ ❢♦r s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ ❛♥❞ ❡r❣♦❞✐❝ ✭❤❡♥❝❡ ✐s♦tr♦♣✐❝✮ s②st❡♠✳

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❆♥✐s♦tr♦♣✐❝ ❉▼ ♣❤❛s❡✲s♣❛❝❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥

❱❡❧♦❝✐t② ❛♥✐s♦tr♦♣② ❝❤❛r❛❝t❡r✐③❡❞ ❜②✿ β(r) ≡ ✶ − σ✷

t

✷σ✷

r

❙❧♦♣❡✲❛♥✐s♦tr♦♣② ✐♥❡q✉❛❧✐t②✷✿ ✲d ❧♥ ρ

d ❧♥ r ≥ ✷β

❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s✿ ❖s✐♣❦♦✈✲▼❡rr✐tt ♠♦❞❡❧✿ β(r) =

r✷ r✷+r✷

a

E → Q = E − L✷ ✷r✷

a

, ρ(r) → ρ(r) ·

• ✶ + r✷

r✷

a

• ❈♦♥st❛♥t ♦r❜✐t❛❧ ❛♥✐s♦tr♦♣②✿ β(r) = βc

f (E, L) = L−✷βc · fβc(E)

✷❆♥✫❊✈❛♥s ❛r❳✐✈✿❛str♦✲♣❤✴✵✺✶✶✻✽✻✈✹✱ ❈✐♦tt✐✫▼♦r❣❛♥t✐ ❛r❳✐✈✿✶✵✵✻✳✷✸✹✹ ▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❉▼ ✈❡❧♦❝✐t② ❞✐str✐❜✉t✐♦♥

0.2 0.4 0.6 0.8 1 v/vesc 0.0 0.5 1.0 1.5 2.0 2.5 3.0 P(v)

Eddington M.B. O.M. (ra/rs = 1) βDM = − 1/2

0.2 0.4 0.6 0.8 1 v/vesc 1 2 3 4 P(v)

Eddington M.B. O.M. (ra/rs = 1) βDM = − 1/2

❋✐❣✉r❡✿ ❱❡❧♦❝✐t② ❞✐str✐❜✉t✐♦♥s ❝♦♠♣✉t❡❞ ✉♥❞❡r ❞✐✛❡r❡♥t ❛ss✉♠♣t✐♦♥s ❢♦r ◆❋❲ ✭❧❡❢t✮ ❛♥❞ ❇✉r❦❡rt ✭r✐❣❤t✮ ❞❡♥s✐t② ♣r♦✜❧❡s✳

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❱❡❧♦❝✐t② ❛✈❡r❛❣❡❞ ❡♥❤❛♥❝❡♠❡♥t ❛♥❞ J✲❢❛❝t♦rs

❚❤❡ ❡✛❡❝t ♦❢ ❙♦♠♠❡r❢❡❧❞ ❡♥❤❛♥❝❡♠❡♥t ❝❛♥ ❜❡ ❡♥r❛♣t✉r❡❞ ✐♥ ✈❡❧♦❝✐t②✲❛✈❡r❛❣❡❞ ❜♦♦st ❢❛❝t♦r✿ S(vr❡❧)(r) = ✶ ρ✷(r)

• ❞✸v✶f (r,

v✶)

• ❞✸v✷ f (r,

v✷)S(vr❡❧) ❈♦rr❡s♣♦♥❞✐♥❣ ❛str♦♣❤②s✐❝❛❧ J✲❢❛❝t♦r ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❛s ❢♦❧❧♦✇s✿ J =

• ❞Ω
• ❞ℓ ρ✷(r) · S(vr❡❧)(r)

❉✐✛❡r❡♥t✐❛❧ ❛♥♥✐❤✐❧❛t✐♦♥ ✢✉① ♣r♦♣♦rt✐♦♥❛❧ t♦ J✲❢❛❝t♦r✿ dΦ dEγ = ✶ ✽π σvr❡❧✵ m✷

χ

dN dEγ · J

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❱❡❧♦❝✐t② ❛✈❡r❛❣❡❞ ❡♥❤❛♥❝❡♠❡♥t

1 2 3 4 5 r / rs 0.50 0.75 1.00 1.25 1.50

• S(vrel)
• (r)/
• S(vrel)
• E(r)

Maxwell-Boltzmann Osipkov-Merrit (ra/rs = 5) Osipkov-Merrit (ra/rs = 1) βDM = − 1/2 10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10 10

1 rs J1 − E · dJ1 − E dr

α = 0.25◦ α = 0.5◦ α = 1◦

❋✐❣✉r❡✿ ❱❡❧♦❝✐t② ❛✈❡r❛❣❡❞ ❡♥❤❛♥❝❡♠❡♥t ❢❛❝t♦r r❛t✐♦ ❛s ❢✉♥❝t✐♦♥ ♦❢ r/rs✳

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❆♣♣❧✐❝❛t✐♦♥✿ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

❉✇❛r❢ s♣❤❡r♦✐❞❛❧ ❣❛❧❛①✐❡s ✭❞❙♣❤✮ ♣r❡s❡♥t ♦♥❡ ♦❢ t❤❡ ♣r✐♠❡ t❛r❣❡ts ❢♦r ❞❡t❡❝t✐♦♥ ♦❢ ❉▼ ❛♥♥✐❤✐❧❛t✐♦♥ ❡✈❡♥ts✿ ❉▼ ❞♦♠✐♥❛t❡❞ ♦❜❥❡❝ts❀ ✶✵ − ✶✵✵× ❤✐❣❤❡r ♠❛ss t♦ ❧✉♠✐♥♦s✐t② r❛t✐♦ t❤❡♥ ✐♥ r❡❣✉❧❛r ❣❛❧❛①✐❡s ❘❡❧❛t✐✈❡ ♣r♦①✐♠✐t② ♦❢ ▼❲ ❞✇❛r❢s ❙♠❛❧❧ ❉▼ ✈❡❧♦❝✐t✐❡s ❡①♣❡❝t❡❞ ❙tr♦♥❣ ❋❡r♠✐✲▲❆❚ ❝♦♥str❛✐♥t ♦♥ ❣❛♠♠❛ r❛② ✢✉① ❙t❡❧❧❛r ❞✐str✐❜✉t✐♦♥ ❛♥❞ ✈❡❧♦❝✐t② ❞✐s♣❡rs✐♦♥ ♠❡❛s✉r❡♠❡♥ts ❛❧❧♦✇ ❢♦r r❡❝♦♥str✉❝t✐♦♥ ♦❢ ❣r❛✈✐t❛t✐♦♥❛❧ ♣♦t❡♥t✐❛❧ → ❉▼ ❞❡♥s✐t② ♣r♦✜❧❡ ❙♣❡❝✐❛❧ t❤❛♥❦s t♦ ▼✳ ❲❛❧❦❡r ❢♦r ♣r♦✈✐❞✐♥❣ ✉s ✇✐t❤ t❤❡ ♣r✉♥❡❞ ❞❛t❛

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❆♣♣❧✐❝❛t✐♦♥✿ ❇❛②❡s✐❛♥ ❛♥❛❧②s✐s ♦❢ ❞❙♣❤

❖❜s❡r✈❛t✐♦♥s ♣r♦✈✐❞❡ ✉s ✇✐t❤ ♣r♦❥❡❝t❡❞ s✉r❢❛❝❡ ❜r✐❣❤t♥❡ss Σ⋆ ❛♥❞ ❧✐♥❡✲♦❢✲s✐❣❤t ✈❡❧♦❝✐t② ❞✐s♣❡rs✐♦♥ σ❧♦s✳ ❯s✐♥❣ ❏❡❛♥s ❛♥❛❧②s✐s ♦♥❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ σ❧♦s ❢♦r ❛ ❣✐✈❡♥ ♠♦❞❡❧✿ σ✷

❧♦s(R) =

✶ Σ⋆(R) ∞

R✷

dr✷ √ r✷ − R✷

• ✶ − β⋆(r)R✷

r✷

• pr ⋆(r)

pr ⋆(r) =GN ∞

r

dx ρ⋆(x)Mt♦t(x) x✷ ❡①♣

• ✷

x

r

dy β⋆(y) y

• ❯s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐❦❡❧✐❤♦♦❞ ❢♦r r❛❞✐❛❧❧② ❜✐♥♥❡❞ ❞❛t❛✿

L❦✐♥ ≡

◆

• k=✶

✶ √ ✷π ∆σlos (k)

• α(k)

❡①♣  −✶ ✷

• σlos (k) − σlos
• α(k)
• ∆σlos (k)
• α(k)
• ✷



▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❆♣♣❧✐❝❛t✐♦♥✿ ❇❛②❡s✐❛♥ ❛♥❛❧②s✐s ♦❢ ❞❙♣❤

❋✐❣✉r❡✿ ❉r❛❝♦ ❧✐♥❡✲♦❢✲s✐❣❤t ✈❡❧♦❝✐t② ❞✐s♣❡rs✐♦♥ ❞❛t❛ ❛♥❞ ✻✽✪ ❛♥❞ ✾✺✪ ❝r❡❞✐❜✐❧✐t② ✐♥t❡r✈❛❧s ❢♦r ✜ts ✉s✐♥❣ ◆❋❲ ✭❧❡❢t✮ ❛♥❞ ❇✉r❦❡rt ✭r✐❣❤t✮ ♣r♦✜❧❡✳

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❆♣♣❧✐❝❛t✐♦♥✿ ❇❛②❡s✐❛♥ ❛♥❛❧②s✐s ♦❢ ❞❙♣❤

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❆♣♣❧✐❝❛t✐♦♥✿ ❏✲❢❛❝t♦rs

10

• 3

10

• 1

10

1

ξ 10

18

10

19

10

20

10

21

10

22

10

23

J(ξ) [GeV2 / cm5] NFW Burkert Non-parametric

❋✐❣✉r❡✿ ✻✽✪ ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ❢♦rJ✲❢❛❝t♦rs ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ξ =

mφ αχmχ ❢♦r

❉r❛❝♦ ✭❧❡❢t✮ ❛♥❞ ❙❝✉❧♣t♦r ✭r✐❣❤t✮✳

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❆♣♣❧✐❝❛t✐♦♥✿ ❏✲❢❛❝t♦rs

C a r i n a D r a c

• F
• r

n a x L e

• I

L e

• I

I S c u l p t

• r

S e x t a n s U r s a M i n

• r

17.0 17.5 18.0 18.5 19.0 19.5 log10 J [GeV2 / cm5]

NFW Burkert β ≥ − ∞ Bonnivard et al. 2015 Ackermann et al. 2014 Geringer-Sameth et al. 2015

C a r i n a D r a c

• F
• r

n a x L e

• I

L e

• I

I S c u l p t

• r

S e x t a n s U r s a M i n

• r

20 21 22 23 24 25 log10 J [GeV2 / cm5]

NFW Edd NFW M.B. NFW O.M. NFW βc BUR Edd BUR M.B. BUR O.M. BUR βc β → − ∞

❋✐❣✉r❡✿ J✲❢❛❝t♦rs ❢♦r ✽ ❞❙♣❤ ✇✐t❤ t❤❡ ✻✽✪ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s✳

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❙✉♠♠❛r②

❉❡✈❡❧♦♣❡❞ ♥✉♠❡r✐❝❛❧ ❝♦❞❡ ❢♦r ❝♦♠♣✉t✐♥❣✿ ♣❤❛s❡✲s♣❛❝❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❉▼ ❢♦r ❛♥ ❛r❜✐tr❛r② ❞❡♥s✐t② ♣r♦✜❧❡ ❛♥❞ ❞✐✛❡r❡♥t ♦r❜✐t❛❧ ❛♥✐s♦tr♦♣② ❛ss✉♠♣t✐♦♥s J✲❢❛❝t♦rs ❢♦r ❛r❜✐tr❛r② ❝r♦ss✲s❡❝t✐♦♥ ✈❡❧♦❝✐t② ❞❡♣❡♥❞❡♥❝❡ ❜❛s❡❞ ♦♥ t❤❡ ♣❤❛s❡✲s♣❛❝❡ ❞✐str✐❜✉t✐♦♥ ❇❛②❡s✐❛♥ ✐♥❢❡r❡♥❝❡ ♦❢ t❤❡ ❞❙♣❤ ❉▼ ❤❛❧♦ ♣❛r❛♠❡t❡rs✳ ❈❛r❡❢✉❧ ❛♥❛❧②s✐s ♦❢ J✲❢❛❝t♦rs ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❙♦♠♠❡r❢❡❧❞ ❡♥❤❛♥❝❡♠❡♥t✳ ◆♦✈❡❧ r❡s✉❧ts ❢♦r ✈❛r✐♦✉s ❉▼ ♦r❜✐t❛❧ ❛♥✐s♦tr♦♣✐❡s✳

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

❈♦♥❝❧✉s✐♦♥s ❛♥❞ ♦✉t❧♦♦❦

❈♦♥❝❧✉s✐♦♥s✿ ■♥ ♣r❡s❡♥❝❡ ♦❢ ❙♦♠♠❡r❢❡❧❞ ❡♥❤❛♥❝❡♠❡♥t t❤❡ ❞❙♣❤ ❝♦♥str❛✐♥ts ♦♥ σv✵ str❡♥❣t❤❡♥ ❜② O(✶✵✸) ✲ O(✶✵✺) ❈✐r❝✉❧❛r❧② ✭r❛❞✐❛❧❧②✮ ❜✐❛s❡❞ ❉▼ ♦r❜✐ts ❧❡❛❞ ❡♥❤❛♥❝❡♠❡♥t ✭s✉♣♣r❡ss✐♦♥✮ ♦❢ ❛♥♥✐❤✐❧❛t✐♦♥ r❛t❡ ❙②st❡♠❛t✐❝ ✉♥❝❡rt❛✐♥t✐❡s ❞✉❡ t♦ ❉▼ ♦r❜✐t❛❧ ❛♥✐s♦tr♦♣② ❛t t❤❡ ❧❡✈❡❧ ♦❢ ♦❜s❡r✈❛t✐♦♥❛❧ ✉♥❝❡rt❛✐♥t✐❡s ❖✉t❧♦♦❦✿ ❯s❡ ❞❡t❛✐❧❡❞ ♣❤❛s❡✲s♣❛❝❡ ♠♦❞❡❧✐♥❣ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❞✐r❡❝t ❞❡t❡❝t✐♦♥ ❡①♣❡r✐♠❡♥ts ❖❜t❛✐♥ ❝♦♥s❡r✈❛t✐✈❡ ✭♥♦♥✲♣❛r❛♠❡tr✐❝✮ ❜♦✉♥❞s ♦♥ ▼✐❧❦② ❲❛② ❉▼ ❞❡♥s✐t② ❞✐str✐❜✉t✐♦♥

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s

■♥tr♦❞✉❝t✐♦♥ Pr♦❥❡❝t ❙✉♠♠❛r②

◆♦♥✲♣❛r❛♠❡tr✐❝ ❉▼ ♣r♦✜❧❡ ❢r♦♠ ❏❡❛♥✬s ❡q✉❛t✐♦♥

❏❡❛♥✬s ❡q✉❛t✐♦♥ ❛❧❧♦✇s t♦ r❡❝♦♥str✉❝t t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ♣♦t❡♥t✐❛❧ ❢r♦♠ t❤❡ st❡❧❧❛r ❞✐str✐❜✉t✐♦♥ ❛♥❞ ❦✐♥❡♠❛t✐❝s✿ ❞p ❞r + ✷β⋆(r) r p(r) = −ρ⋆(r)❞Φ ❞r , p(r) = ρ⋆(r)σ✷

r (r)

❉❡❣❡♥❡r❛❝② ❜❡t✇❡❡♥ st❡❧❧❛r ✈❡❧♦❝✐t② ❞✐s♣❡rs✐♦♥ ❛♥✐s♦tr♦♣② β⋆(r) ❛♥❞ ❣r❛✈✐t❛t✐♦♥❛❧ ♣♦t❡♥t✐❛❧ Φ(r)✳ ❆ss✉♠✐♥❣ P❧✉♠♠❡r st❡❧❧❛r ♣r♦✜❧❡✱ ✐s♦tr♦♣② ✭β⋆ = ✵✮ ❛♥❞ ❝♦♥st❛♥t σ❧♦s(R) ≡ σ❧♦s ♦♥❡ ✜♥❞s✿ ρ❉▼(r) = ✺σ✷

❧♦s

✹πG r✷ + ✸R✷

✶/✷

(r✷ + R✷

✶/✷)✷

▼✐❤❛❡❧ P❡t❛↔ ■♥❞✐r❡❝t ❞❡t❡❝t✐♦♥ ✐♥ ▼✐❧❦② ❲❛② s❛t❡❧❧✐t❡s