Post on 09-Oct-2020
Dark matter distribution around massive blackholes and in phase-space
Francesc Ferrer
Washington University in St. Louis
Universite Libre de Bruxelles. Oct 4, 2013.
Outline
Introduction: detecting dark matter
The DM distribution around a supermassive BH
DM annihilation in phase-space
Phase space distribution: Eddington’s inversion
Conclusions
Laleh Sadeghian, FF & Clifford M. Will, PRD88, 063522 (2013) [arXiv:1305.2619]
FF & Daniel Hunter, JCAP 1309, 005 (2013) [arXiv:1306.6586]
Outline
Introduction: detecting dark matter
The DM distribution around a supermassive BH
DM annihilation in phase-space
Phase space distribution: Eddington’s inversion
Conclusions
Laleh Sadeghian, FF & Clifford M. Will, PRD88, 063522 (2013) [arXiv:1305.2619]
FF & Daniel Hunter, JCAP 1309, 005 (2013) [arXiv:1306.6586]
Outline
Introduction: detecting dark matter
The DM distribution around a supermassive BH
DM annihilation in phase-space
Phase space distribution: Eddington’s inversion
Conclusions
Laleh Sadeghian, FF & Clifford M. Will, PRD88, 063522 (2013) [arXiv:1305.2619]
FF & Daniel Hunter, JCAP 1309, 005 (2013) [arXiv:1306.6586]
Outline
Introduction: detecting dark matter
The DM distribution around a supermassive BH
DM annihilation in phase-space
Phase space distribution: Eddington’s inversion
Conclusions
Laleh Sadeghian, FF & Clifford M. Will, PRD88, 063522 (2013) [arXiv:1305.2619]
FF & Daniel Hunter, JCAP 1309, 005 (2013) [arXiv:1306.6586]
Outline
Introduction: detecting dark matter
The DM distribution around a supermassive BH
DM annihilation in phase-space
Phase space distribution: Eddington’s inversion
Conclusions
Laleh Sadeghian, FF & Clifford M. Will, PRD88, 063522 (2013) [arXiv:1305.2619]
FF & Daniel Hunter, JCAP 1309, 005 (2013) [arXiv:1306.6586]
Outline
Introduction: detecting dark matter
The DM distribution around a supermassive BH
DM annihilation in phase-space
Phase space distribution: Eddington’s inversion
Conclusions
Laleh Sadeghian, FF & Clifford M. Will, PRD88, 063522 (2013) [arXiv:1305.2619]
FF & Daniel Hunter, JCAP 1309, 005 (2013) [arXiv:1306.6586]
The case for dark matter
Most economical explanation of:I The rate of expansion of the universe.I The formation of large scale structure.I The dynamics of galaxies, clusters, . . .
Expected in natural extensions of the SM.
The case for dark matter
Most economical explanation of:I The rate of expansion of the universe.I The formation of large scale structure.I The dynamics of galaxies, clusters, . . .
Expected in natural extensions of the SM.
An example: WIMPs
Similar to a heavy neutrino, mχ ≈ 100 GeV, weak-scaleinteractions produce observed abundance from thermaldecoupling:
⇒< σv >≈ 3× 10−26 cm3 s−1
The same interactions make it potentially detectable:I χχ→ γγ, π0, e±, . . .I χN → χN
Other examples include axions, MeV particles, . . .
An example: WIMPs
Similar to a heavy neutrino, mχ ≈ 100 GeV, weak-scaleinteractions produce observed abundance from thermaldecoupling:
⇒< σv >≈ 3× 10−26 cm3 s−1
The same interactions make it potentially detectable:I χχ→ γγ, π0, e±, . . .I χN → χN
Other examples include axions, MeV particles, . . .
An example: WIMPs
Similar to a heavy neutrino, mχ ≈ 100 GeV, weak-scaleinteractions produce observed abundance from thermaldecoupling:
⇒< σv >≈ 3× 10−26 cm3 s−1
The same interactions make it potentially detectable:I χχ→ γγ, π0, e±, . . .I χN → χN
Other examples include axions, MeV particles, . . .
Indirect detection
Flux =< σv >4πm2
dm
dNγ
dEγ︸ ︷︷ ︸Number of SM particles
×∫ ∞
0ρ2(r)dl
︸ ︷︷ ︸Amount of DM2
I Astrophysical factor suggests looking at GC, dwarfspheroidals, . . .
I Photons and neutrinos point back to the source, whilecharged particles diffuse.
The distribution of DM: simulations
1 billion 4,100 M� particles. 0.5 kpc in the host halo.
The distribution of DM: observations
Jeans’ equation shows that M/L ∼ 1000. Clean systems.
The central supermassive black hole
I Will focus on the super-massive BH at the center of theGalaxy.
I Similar effects will occur in the cores of AGNs, or in IMBHs.
The central supermassive black holeWe will assume that a black hole of mass 4× 106M� growsadiabatically over ∼ 1010yr.
The central supermassive black holeWe will assume that a black hole of mass 4× 106M� growsadiabatically over ∼ 1010yr.
Is the growth adiabatic?
I Growth timeI Dynamical time
rh
σ≤ m
mEdd
rh ≈Gmσ2 → tdyn ≈ 104yr ≤ tSalpeter ≈ 5× 107yr
Caveats: Hierarchical mergers, initial BH seed off-center, kineticheating of DM by stars, . . .
Is the growth adiabatic?
I Growth timeI Dynamical time
rh
σ≤ m
mEdd
rh ≈Gmσ2 → tdyn ≈ 104yr ≤ tSalpeter ≈ 5× 107yr
Caveats: Hierarchical mergers, initial BH seed off-center, kineticheating of DM by stars, . . .
Is the growth adiabatic?
I Growth timeI Dynamical time
rh
σ≤ m
mEdd
rh ≈Gmσ2 → tdyn ≈ 104yr ≤ tSalpeter ≈ 5× 107yr
Caveats: Hierarchical mergers, initial BH seed off-center, kineticheating of DM by stars, . . .
Is the growth adiabatic?
I Growth timeI Dynamical time
rh
σ≤ m
mEdd
rh ≈Gmσ2 → tdyn ≈ 104yr ≤ tSalpeter ≈ 5× 107yr
Caveats: Hierarchical mergers, initial BH seed off-center, kineticheating of DM by stars, . . .
Growing a BH: Newtonian analysis
We are interested in the DM density:
ρ =
∫f (E ,L)d3v
= 4∫
dE∫
LdL∫
dLzf (E ,L)
r4|vr ||vθ| sin θ
= 4π∫
dE∫
LdLf (E ,L)
r2|vr |
The limits of integration are set by the requirements:I |vr | = (2E −2Φ−L2/r2)1/2 real⇒ 0 ≤ L ≤ [2r2(E −Φ)]1/2.I DM particle is bound to the halo⇒ Φ(r) ≤ E ≤ 0.
Take into account particles trapped inside the event horizon bymodifying boundary conditions in an ad hoc manner: L ≥ 2cRS.
Growing a BH: Newtonian analysis
We are interested in the DM density:
ρ =
∫f (E ,L)d3v
= 4∫
dE∫
LdL∫
dLzf (E ,L)
r4|vr ||vθ| sin θ
= 4π∫
dE∫
LdLf (E ,L)
r2|vr |
The limits of integration are set by the requirements:I |vr | = (2E −2Φ−L2/r2)1/2 real⇒ 0 ≤ L ≤ [2r2(E −Φ)]1/2.I DM particle is bound to the halo⇒ Φ(r) ≤ E ≤ 0.
Take into account particles trapped inside the event horizon bymodifying boundary conditions in an ad hoc manner: L ≥ 2cRS.
Growing a BH: Newtonian analysis
We are interested in the DM density:
ρ =
∫f (E ,L)d3v
= 4∫
dE∫
LdL∫
dLzf (E ,L)
r4|vr ||vθ| sin θ
= 4π∫
dE∫
LdLf (E ,L)
r2|vr |
The limits of integration are set by the requirements:I |vr | = (2E −2Φ−L2/r2)1/2 real⇒ 0 ≤ L ≤ [2r2(E −Φ)]1/2.I DM particle is bound to the halo⇒ Φ(r) ≤ E ≤ 0.
Take into account particles trapped inside the event horizon bymodifying boundary conditions in an ad hoc manner: L ≥ 2cRS.
Growing a BH: Newtonian analysis
Each particle in an initial DM distribution f (E ,L), will react tothe change in Φ caused by the growth of the BH by altering itsE , L and Lz . However, the adiabatic invariants remain fixed:
Ir (E ,L) ≡∮
vr dr =
∮dr√
2E − 2Φ− L2/r2 ,
Iθ(L,Lz) ≡∮
vθdθ =
∮dθ√
L2 − L2z sin−2 θ = 2π(L− Lz) ,
Iφ(Lz) ≡∮
vφdφ =
∮Lzdφ = 2πLz . (1)
The shape of the distribution function is also invariant,f (E ,L) = f ′(E ′(E ,L),L).
Growing a BH: Newtonian analysis
Each particle in an initial DM distribution f (E ,L), will react tothe change in Φ caused by the growth of the BH by altering itsE , L and Lz . However, the adiabatic invariants remain fixed:
Ir (E ,L) ≡∮
vr dr =
∮dr√
2E − 2Φ− L2/r2 ,
Iθ(L,Lz) ≡∮
vθdθ =
∮dθ√
L2 − L2z sin−2 θ = 2π(L− Lz) ,
Iφ(Lz) ≡∮
vφdφ =
∮Lzdφ = 2πLz . (1)
The shape of the distribution function is also invariant,f (E ,L) = f ′(E ′(E ,L),L).
Newtonian BH
For a Newtonian point mass,
Ir (E , L) = 2π(−L +
Gm√−2E
)
And we can find the final DM density in the form:
ρ(r) =4πr2
∫ 0
−Gm/rdE∫ Lmax
0LdL
f ′(E ′(E ,L),L)√2E + 2Gm/r − L2/r2
Young 80, Quinlan et al. 95, Gondolo & Silk 95
Growing a BH: Relativistic analysis1. Generalize the definition of density:
Jµ(x) ≡∫
f (4)(p)pµ
µ
√−g d4p ,
2. Use relativistic expressions to write it in terms of theinvariants of motion (energy, angular momentum, . . . ).
3. Use relativistic expressions for the actions.
For Kerr:
E ≡ −u0 = −g00u0 − g0φuφ , (2)Lz ≡ uφ = g0φu0 + gφφuφ , (3)
C ≡ Σ4(uθ)2
+ sin−2 θL2z + a2 cos2 θ(1− E2) , (4)
gµνpµpν = −µ2 . (5)
And need to calculate the jacobian
d4p = |J|−1dEdCdLzdµ
Growing a BH: Relativistic analysis1. Generalize the definition of density:
Jµ(x) ≡∫
f (4)(p)pµ
µ
√−g d4p ,
2. Use relativistic expressions to write it in terms of theinvariants of motion (energy, angular momentum, . . . ).
3. Use relativistic expressions for the actions.
For Kerr:
E ≡ −u0 = −g00u0 − g0φuφ , (2)Lz ≡ uφ = g0φu0 + gφφuφ , (3)
C ≡ Σ4(uθ)2
+ sin−2 θL2z + a2 cos2 θ(1− E2) , (4)
gµνpµpν = −µ2 . (5)
And need to calculate the jacobian
d4p = |J|−1dEdCdLzdµ
Example: Schwarzschild BH
The positivity of the radial action determines the boundaryconditions, including the effects of the horizon.
12 16 20 24 28
(L/Gm)2
8/9
1
E2
r = 5Gm
r = 6Gm
r = 12Gm
r = 20Gm
Example: Schwarzschild BH
For a constant phase-space distribution:
0 4 8 20 40 60 80
r/Gm
0.0
0.2
0.4
0.6
0.8
1.0ρ[ 10
8G
eV/c
m3]
Fully relativistic
Gondolo & Silk
Example: Schwarzschild BH
For a, more realistic, cuspy DM distribution:
0 1 2 3 4 5 6 7
log10 (r/RS)
10
12
14
16
18
log
10
( ρ[G
eV/c
m3])
Relativistic
Non-relativistic
DM annihilation
Initial Hernquist profile
Consequences
The gravitational potential is still dominated by the BH:
0 1 2 3 4
log10 (r/RS)
-12
-10
-8
-6
-4
-2
0
2
4lo
g10
(m(r
)/M�
)
no annihilation
with annihilation
Consequences
No big changes for DM annihilation, but precession rates couldbe observable.
0.1 1 10
Semi-major axis [mpc]
10−4
0.01
1
100
104
Pre
cess
ion
rate
[arc
min/y
r] ADM,non−ann
ADM,ann
AS
AJ
AQ
Indirect detection
Flux =< σv >4πm2
dm
dNγ
dEγ︸ ︷︷ ︸Number of SM particles
×∫ ∞
0ρ2(r)dl
︸ ︷︷ ︸Amount of DM2
I Astrophysical factor suggests looking at GC, dwarfspheroidals, . . .
I Photons and neutrinos point back to the source, whilecharged particles diffuse.
The annihilation cross-section
Annihilations in the halo are non-relativistic, v ≈ 10−3.The amplitude is analytical for k → 0
M∝∫
eikxVBorn(x)
Including factors of k lY ml in a partial wave expansion, σ ∝ k2l−1
σv = a + bv2 + . . .
The annihilation cross-section
Annihilations in the halo are non-relativistic, v ≈ 10−3.The amplitude is analytical for k → 0
M∝∫
eikxVBorn(x)
Including factors of k lY ml in a partial wave expansion, σ ∝ k2l−1
σv = a + bv2 + . . .
The annihilation cross-section
Annihilations in the halo are non-relativistic, v ≈ 10−3.The amplitude is analytical for k → 0
M∝∫
eikxVBorn(x)
Including factors of k lY ml in a partial wave expansion, σ ∝ k2l−1
σv = a + bv2 + . . .
More complicated velocity dependence
If there are new light particles mediating long-range forcesbetween the dark matter, an enhancement occurs at lowvelocities:
σ → σ × πα
vIf annihilation proceeds near a resonant state,
vσ ∝ 1
(v2/4 + ∆)2
+ Γ2A(1−∆)/4m2
χ
Enhancements at low velocities, v ∼ 10−3, different than atdecoupling.
More complicated velocity dependence
If there are new light particles mediating long-range forcesbetween the dark matter, an enhancement occurs at lowvelocities:
σ → σ × πα
vIf annihilation proceeds near a resonant state,
vσ ∝ 1
(v2/4 + ∆)2
+ Γ2A(1−∆)/4m2
χ
Enhancements at low velocities, v ∼ 10−3, different than atdecoupling.
More complicated velocity dependence
If there are new light particles mediating long-range forcesbetween the dark matter, an enhancement occurs at lowvelocities:
σ → σ × πα
vIf annihilation proceeds near a resonant state,
vσ ∝ 1
(v2/4 + ∆)2
+ Γ2A(1−∆)/4m2
χ
Enhancements at low velocities, v ∼ 10−3, different than atdecoupling.
Substructure enhanced
Lattanzi & Silk
Substructure enhanced
Lattanzi & Silk
What went in calculating the flux?
The averaged cross-section
〈σv〉 → S(v)〈σv〉
But, the flux isΦ = Rate × vrel
We have to average this, using the dark matter velocitydistribution.
What went in calculating the flux?
The averaged cross-section
〈σv〉 → S(v)〈σv〉
But, the flux isΦ = Rate × vrel
We have to average this, using the dark matter velocitydistribution.
What went in calculating the flux?
The averaged cross-section
〈σv〉 → S(v)〈σv〉
But, the flux isΦ = Rate × vrel
We have to average this, using the dark matter velocitydistribution.
How should we calculate the fluxes?
Flux ∝∫
dvreldllosfpair (r , vrel)× σvrel
The usual approach assumes
fpair (r , vrel) = ρ2(r)× fMB(vrel)
See Robertson & Zentner for an approximate Jeans’ based analysis
How should we calculate the fluxes?
Flux ∝∫
dvreldllosfpair (r , vrel)× σvrel
The usual approach assumes
fpair (r , vrel) = ρ2(r)× fMB(vrel)
See Robertson & Zentner for an approximate Jeans’ based analysis
Which velocity distribution?
Since DM is assumed to be heavy, use Maxwell-Boltzmann?
Kuhlen et al.
Which velocity distribution?
Since DM is assumed to be heavy, use Maxwell-Boltzmann?
Kuhlen et al.
Obtaining the phase-space distribution
Assume that dark matter satisfies the colisionless Boltzmannequation,
dfdt
= 0
Very hard to solve! Only a few exact solutions known, foundfinding integrals of motion (singular isothermal sphere,Hernquist, Jaffe, . . . ).Taking velocity moments we obtain the Jeans’ equation:
v2c =
GM(r)
r= −v2
r
(d log νd log r
+d log v2
r
d log r+ 2β
).
Necessary condition, useful to obtain density profiles fromobservational data.
Obtaining the phase-space distribution
Assume that dark matter satisfies the colisionless Boltzmannequation,
dfdt
= 0
Very hard to solve! Only a few exact solutions known, foundfinding integrals of motion (singular isothermal sphere,Hernquist, Jaffe, . . . ).Taking velocity moments we obtain the Jeans’ equation:
v2c =
GM(r)
r= −v2
r
(d log νd log r
+d log v2
r
d log r+ 2β
).
Necessary condition, useful to obtain density profiles fromobservational data.
Eddington’s formula
Gives the phase space distribution, if we know the densityprofile:
f (E) =1√8π2
∫ E
0
dΨ√E −Ψ
d2ρ
dΨ2 . (6)
Given a profile (NFW, cored, . . . ) we obtain the phase spacedistribution, which provides a full description of the dark matterdistribution.Check that ρ(r) ≡
∫d3v f (E).
Eddington’s formula
Gives the phase space distribution, if we know the densityprofile:
f (E) =1√8π2
∫ E
0
dΨ√E −Ψ
d2ρ
dΨ2 . (6)
Given a profile (NFW, cored, . . . ) we obtain the phase spacedistribution, which provides a full description of the dark matterdistribution.Check that ρ(r) ≡
∫d3v f (E).
Eddington’s formula
Gives the phase space distribution, if we know the densityprofile:
f (E) =1√8π2
∫ E
0
dΨ√E −Ψ
d2ρ
dΨ2 . (6)
Given a profile (NFW, cored, . . . ) we obtain the phase spacedistribution, which provides a full description of the dark matterdistribution.Check that ρ(r) ≡
∫d3v f (E).
The phase-space distribution
0 2 4 6 8 10 12 14 16
ε
-2
0
2
4
6lo
g( f/g
(c))
NFW
Einasto
NFW + baryons
Deriving the velocity distribution
Now that we have the full distribution function, we can find thevelocity distribution at each point:
P(v) =f(ψ − v2/2
)
ρ(ψ)
Check that∫
P(v)dv = 1.
Deriving the velocity distribution
Now that we have the full distribution function, we can find thevelocity distribution at each point:
P(v) =f(ψ − v2/2
)
ρ(ψ)
Check that∫
P(v)dv = 1.
Velocity distribution
0 100 200 300 400 500
v [km/s]
0.00
0.01
0.02
Pr(v
)[(
km/s
)−1]
0.1 kpc1 kpc10 kpc
Obtaining the relative velocity distribution
We move to the CM, to obtain Prel (vrel):
fsp(~v1)fsp(~v2)d~v1d~v2 = fpair(~vcm, ~vrel)d~vcmd~vrel. (7)
frv(vrel) = 4πv2rel2π
∫ ∞
0dvcmv2
cm
∫ π
0dθ sin(θ)
· fsp
(√v2
rel/4 + v2cm + vrelvcm cos(θ)
)
· fsp
(√v2
rel/4 + v2cm − vrelvcm cos(θ)
).
(8)
Recover the standard results for a Maxwell-Boltzmanndistribution.
Obtaining the relative velocity distribution
We move to the CM, to obtain Prel (vrel):
fsp(~v1)fsp(~v2)d~v1d~v2 = fpair(~vcm, ~vrel)d~vcmd~vrel. (7)
frv(vrel) = 4πv2rel2π
∫ ∞
0dvcmv2
cm
∫ π
0dθ sin(θ)
· fsp
(√v2
rel/4 + v2cm + vrelvcm cos(θ)
)
· fsp
(√v2
rel/4 + v2cm − vrelvcm cos(θ)
).
(8)
Recover the standard results for a Maxwell-Boltzmanndistribution.
Relative velocity distribution
0 100 200 300 400 500 600
vrel [km/s]
0.000
0.003
0.006
0.009
0.012
Prel
(vrel
)[(
km/s
)−1]
0.1 kpc1 kpc10 kpc
Fluxes
We have all the ingredients to perform the los integration:
Flux ∝∫
dvreldllosfpair (r , vrel)× σvrel
Or a volume integration, if we are interested in e± yields in thecenter of the galaxy.
Fluxes
We have all the ingredients to perform the los integration:
Flux ∝∫
dvreldllosfpair (r , vrel)× σvrel
Or a volume integration, if we are interested in e± yields in thecenter of the galaxy.
Boost factor
0.0012 0.0014 0.0016 0.0018 0.0020
ξ
100
101
102
103F
EddingtonJeans
Enhancements up to 1000!
Conclusions
I A full general relativistic treatment shows significantdeviations of the DM distribution around a black hole. Theycould affect tests of no-hair theorems.
I Gamma-ray and neutrino fluxes might depend on thevelocity distribution, which generically deviates from thenaive Maxwell-Boltzmann approximation.
I Using the full phase space distribution from the Eddingtoninversion suggests that fluxes from the center of the haloare up to 103 times larger.
I Constraints on Sommerfeld enhanced models from IC,synchroton or diffuse backgrounds might have to bere-evaluated.
I The velocity distribution also affects direct detection rates.