Thermal decoupling of WIMPs - PPC 2010

PPC 2010, Torino, 12-15 July 2010
Thermal decoupling of WIMPs
A link between particle physics properties
and the small-scale structure of (dark) matter
Torsten Bringmann, University of Hamburg

Outlook
Chemical vs kinetic decoupling of WIMPs
Kinetic decoupling from first principles
The size of the first protohalos
Observational prospects
Conclusions
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
2

Dark matter
Existence by now (almost)
impossible to challenge!
Ω CDM =0.233 ± 0.013 (WMAP)
electrically neutral (dark!)
non-baryonic (BBN)
cold ‒ dissipationless and negligible
free-streaming effects
collisionless (bullet cluster)
(structure formation)
credit: WMAP
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
3

Dark matter
Existence by now (almost)
impossible to challenge!
Ω CDM =0.233 ± 0.013 (WMAP)
electrically neutral (dark!)
non-baryonic (BBN)
cold ‒ dissipationless and negligible
free-streaming effects
collisionless (bullet cluster)
(structure formation)
credit: WMAP
WIMPS are particularly
good candidates:
well-motivated from particle physics
[SUSY, EDs, little Higgs, ...]
thermal production “automatically”
leads to the right relic abundance
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
3

The WIMP “miracle”
iracle”
IMP
ed by
)
2
eq
a 3 nχ
ls bee
unithe
relic
The number density of Weakly Interacting Massive
Particles in the early universe:
n χ eq
increasing〈σv〉
time
Fig.: Jungman, Kamionkowski & Griest, PR’96
Jungman, Kamionkowski & Griest, PR ’96
dn χ
dt
〈σv〉:
+3Hn χ = −〈σv〉
χχ → SM SM
( )
n 2 χ − n 2 χeq
(thermal average)
1) [for interaction strengths of the weak type]
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
4

The WIMP “miracle”
iracle”
IMP
ed by
)
2
eq
a 3 nχ
ls bee
unithe
relic
The number density of Weakly Interacting Massive
Particles in the early universe:
n χ eq
increasing〈σv〉
time
Fig.: Jungman, Kamionkowski & Griest, PR’96
Jungman, Kamionkowski & Griest, PR ’96
Relic density (today):
1) [for interaction strengths of the weak type]
dn χ
dt
〈σv〉:
+3Hn χ = −〈σv〉
χχ → SM SM
( )
n 2 χ − n 2 χeq
(thermal average)
“Freeze-out” when annihilation
rate falls behind expansion rate
(→ a 3 n χ ∼ const.)
Ω χ h 2 ∼ 3 · 10−27 cm 3 /s
〈σv〉
for weak-scale
interactions!
∼ O(0.1)
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
4

Freeze-out = decoupling !
WIMP interactions with heat bath of SM particles:
χ
SM
χ χ
χ
(annihilation)
SM SM (scattering) SM
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
5

Freeze-out = decoupling !
WIMP interactions with heat bath of SM particles:
χ
SM
χ χ
χ
(annihilation)
SM SM (scattering) SM
Boltzmann suppression of n χ
scattering processes much more frequent
continue even after chemical decoupling (“freeze-out”) at T cd ∼ m χ /25
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
5

Freeze-out = decoupling !
WIMP interactions with heat bath of SM particles:
χ
SM
χ χ
χ
(annihilation)
SM SM (scattering) SM
Boltzmann suppression of n χ
scattering processes much more frequent
continue even after chemical decoupling (“freeze-out”) at T cd ∼ m χ /25
Kinetic decoupling much later:
τ r (T kd ) ≡ N coll /Γ el ∼ H −1 (T kd )
Random walk in
momentum space
N coll ∼ m χ /T
Schmid, Schwarz, & Widerin, PRD ’99; Green, Hofmann & Schwarz, JCAP ’05, ...
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
5

Kinetic decoupling in detail
Evolution of phase-space density f χ given by the full
Boltzmann equation in FRW spacetime:
E(∂ t − Hp · ∇ p )f χ = C[f χ ]
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
6

Kinetic decoupling in detail
Evolution of phase-space density f χ given by the full
Boltzmann equation in FRW spacetime:
∫
d 3 p
E(∂ t − Hp · ∇ p )f χ = C[f χ ]
recovers the familiar
dn χ
dt
+3Hn χ = −〈σv〉
(
n 2 χ − n 2 χeq)
...
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
6

Kinetic decoupling in detail
Evolution of phase-space density f χ given by the full
Boltzmann equation in FRW spacetime:
∫
d 3 p
E(∂ t − Hp · ∇ p )f χ = C[f χ ]
recovers the familiar
T χ n χ ≡
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
dn χ
dt
d 3 p
(2π) 3 p2 f χ (p)
+3Hn χ = −〈σv〉
∫
(
n 2 χ − n 2 χeq)
...
Idea: consider instead the 2 nd moment ( d 3 p p 2 )
∫
and introduce
analytic treatment possible
no assumptions about f χ (p) necessary
Allows highly accurate treatment, to order O(T/m χ ) 10 −3
Bertschinger, PRD ’06; TB & Hofmann, JCAP ’07; TB, NJP ’09
6

The collision term
χ p
SM
k
˜p χ
˜k
SM
C =
∫
d 3 k
(2π) 3 2ω
∫
d 3˜k
(2π) 3 2˜ω
∫
d 3 ˜p
(2π) 3 2Ẽ (2π)4 δ (4) (˜p + ˜k − p − k)|M| 2
× g SM
[(
1 ∓ g ± (ω) ) g ± (˜ω)f(˜p ) − ( 1 ∓ g ± (˜ω) ) g ± (ω)f(p) ]
g ± : thermal distribution
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
7

The collision term
χ p
SM
k
˜p χ
˜k
SM
C =
∫
d 3 k
(2π) 3 2ω
∫
d 3˜k
(2π) 3 2˜ω
∫
d 3 ˜p
(2π) 3 2Ẽ (2π)4 δ (4) (˜p + ˜k − p − k)|M| 2
× g SM
[(
1 ∓ g ± (ω) ) g ± (˜ω)f(˜p ) − ( 1 ∓ g ± (˜ω) ) g ± (ω)f(p) ]
g ± : thermal distribution
Expansion in ω/m χ ∼ T/m χ
[
]
C ≃ c(T )m 2 χ m χ T ∆ p + p · ∇ p +3 f(p)
c(T )= ∑ ∫
g SM
6(2π) 3 m 4 dk k 5 ω −1 g ± ( 1 ∓ g ±) |M| 2 t=0
i
χT
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
7

The collision term
χ p
SM
k
˜p χ
˜k
SM
C =
∫
d 3 k
(2π) 3 2ω
∫
d 3˜k
(2π) 3 2˜ω
∫
d 3 ˜p
(2π) 3 2Ẽ (2π)4 δ (4) (˜p + ˜k − p − k)|M| 2
× g SM
[(
1 ∓ g ± (ω) ) g ± (˜ω)f(˜p ) − ( 1 ∓ g ± (˜ω) ) g ± (ω)f(p) ]
g ± : thermal distribution
Expansion in ω/m χ ∼ T/m χ
[
]
C ≃ c(T )m 2 χ m χ T ∆ p + p · ∇ p +3 f(p)
c(T )= ∑ ∫
g SM
6(2π) 3 m 4 dk k 5 ω −1 g ± ( 1 ∓ g ±) |M| 2 t=0
i
χT
Analytic solution if |M| 2 = |M| 2 0 (ω/m χ) 2
∫
generic situation for
dk k 5 ... = N × |M| 2 0 m−2 χ T 7 m SM ≪ ω ≪ ω res
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
7

The WIMP temperature
T χ
Resulting ODE for :
4.5
(
dy
dx =2m χc(T )
1 − T χ
H˜g −1/2 T
T. Bringmann, 2009
)
y = mχg −1/2
log 10
(
eff
Tχ/T 2 )
4.0
3.5
3.0
2.5
2.0
1.5
T χ ∝ a −2
T χ = T
x kd =m χ /T kd
(T < T kd )
(T > T kd ) Example:
m χ = 100 GeV
|M| 2 ∼ g 4 Y (E χ /m χ ) 2
1.0
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
log 10 (x = m χ /T )
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
8

The WIMP temperature
T χ
Resulting ODE for :
4.5
(
dy
dx =2m χc(T )
1 − T χ
H˜g −1/2 T
T. Bringmann, 2009
)
y = mχg −1/2
log 10
(
eff
Tχ/T 2 )
4.0
3.5
3.0
2.5
2.0
1.5
T χ ∝ a −2
T χ = T
x kd =m χ /T kd
(T < T kd )
(T > T kd ) Example:
m χ = 100 GeV
|M| 2 ∼ g 4 Y (E χ /m χ ) 2
1.0
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
log 10 (x = m χ /T )
Fast transition allows straight-forward definition of T kd
TB & Hofmann, JCAP ’07; TB, NJP ’09
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
8

T kd
in SUSY
Implement all SM-neutralino scattering amplitudes
Scan MSSM and mSUGRA parameter space
(~10 6 models, 3 σ WMAP, all collider bounds OK)
T. Bringmann, 2009
TB, NJP ’09
T. Bringmann, 2009
Tkd [MeV]
10 3
10 2
10
10 4 50 100 500 1000 5000
Higgsino (Z g < 0.05)
mixed (0.05 ≤ Z g ≤ 0.95)
Gaugino (Z g > 0.95)
⨯
K
⨯
′
⨯
QCD
I ′ J ∗ F ∗
⨯
xkd = mχ/Tkd
10 4
10 3
Higgsino (Z g < 0.05)
10 5 20 22 24 26 28 30
mixed (0.05 ≤ Z g ≤ 0.95)
Gaugino (Z g > 0.95)
m χ [GeV]
x cd = m χ /T cd
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
10 2
9

The smallest protohalos
Free streaming of WIMPs
after washes out density
contrasts on small scales
t kd
Loeb & Zaldarriaga, PRD ’05
e.g. Green, Hofmann & Schwarz, JCAP ’05
Similar effect from
baryonic oscillations
Bertschinger, PRD ’06
Cutoff in power spectrum
corresponds to smallest
gravitationally bound
objects in the universe
⎛
M fs =2.9×10 −6 ⎜
⎝
(
1 + ln
(
m χ
100 GeV
1
g
4
eff T kd
30 MeV
M ao =3.4 × 10 −6 (
T kd g 1 4
eff
50 MeV
)
/18.6
) 1
2
g 1 4
eff
( Tkd
30 MeV
) 1
2
) −3
M ⊙
⎞
3
⎟
⎠
M ⊙
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
10

The smallest protohalos
Free streaming of WIMPs
after t kd washes out density
contrasts on small scales
e.g. Green, Hofmann & Schwarz, JCAP ’05
Similar effect from
baryonic oscillations
Loeb & Zaldarriaga, PRD ’05
Bertschinger, PRD ’06
Cutoff in power spectrum
corresponds to smallest
gravitationally bound
objects in the universe
Strong dependence on particle physics properties,
no “typical” value of !
Mcut/M⊙
10 −4
10 −6
10 −8
10 −10
10 −12
M cut ∼ 10 −6 M ⊙
⨯
⨯
K
′
⨯
Higgsino (Z g < 0.05)
mixed (0.05 ≤ Z g ≤ 0.95)
Gaugino (Z g > 0.95)
I ′ J ∗ F ∗
m χ [GeV]
T. Bringmann, 2009
⨯
50 100 500 1000 5000
(see also Profumo, Sigurdson
& Kamionkowski, PRL ’06)
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
10

Other DM candidates
Formalism applicable to any DM candidate that is nonrelativistic
before kinetic decoupling
Many WIMPs have
smaller spread in
than neutralinos, e.g.
Kaluza-Klein DM
(Number of free parameters
in the theory…)
M cut
Mcut/M⊙
10 −4
10 −5
Formalism does not allow to compute
m LKP [GeV]
T. Bringmann, 2009
KK dark matter
(mUED, ΛR = 20, 30, 40)
WMAP 3σ
10 −6 500 600 700 800 900 1000
M cut
if DM has never been in thermal equilibrium, like the axion
for hot or warm DM
decaying DM
e.g.
10
20
30
Tkd [MeV]
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
11

Survival of microhalos
N-body simulations can
follow evolution until z~26
(for field halos and adopting a special multi-scale
technique)
Diemand, Moore & Stadel, Nature ’05
General expectation afterwards: tidal disruption
important, but compact core should survive...
Berezinsky et al., PRD ’03, PRD ’08; Moore ’05, Diemand, Kuhlen & Madau ApJ
’06; Green & Goodwin, MNRAS ’07, Goerdt et al., MNRAS ’07; ...
...though prospects might be much worse.
Zhao et al., ApJ ’07
Details not well understood and still under debate,
more input from simulations needed!
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
12

Indirect detection of WIMPs
DM indirect detection:
DM
ark Matter Candidates 5
e
+
_
p
"
!
DM
!
e +
Total flux:
Φ SM ∝〈ρ 2 χ〉 = (1 + BF)〈ρ χ 〉 2
Fig.: Bergström, NJP ’09
Figure 1. Illustration of the volumes in the solar neigbourhood entering the
calculation of the average boost factor in the dark matter halo. Here we have in
mind a dark matter particle of mass around 100 GeV annihilating into, from left to
right, positrons, antiprotons, and gamma-rays. The difference in size for antiprotons
and positrons depends on the different energy loss properties, as positrons at these
energies radiate through synchrotron and inverse Compton emission much faster than
do antiprotons.
e influence Torsten of baryons Bringmann, could University give an enhanced of Hamburg density through adiabatic contraction
Thermal decoupling of WIMPs ‒
13

Indirect detection of WIMPs
DM indirect detection:
DM
ark Matter Candidates 5
e
+
_
p
"
!
DM
!
e +
Total flux:
Φ SM ∝〈ρ 2 χ〉 = (1 + BF)〈ρ χ 〉 2
“Boost factor”
Fig.: Bergström, NJP ’09
each decade in Msubhalo contributes about the same
Figure 1. Illustration of the volumes in the solar neigbourhood entering the
calculation of the average boost factor in the dark matter halo. Here we have in
mind a dark matter particle of mass around 100 GeV annihilating into, from left to
right, positrons, antiprotons, and gamma-rays. The difference in size for antiprotons
and positrons depends on the different energy loss properties, as positrons at these
(large extrapolations necessary!)
energies radiate through synchrotron and inverse Compton emission much faster than
do antiprotons.
depends on uncertain form of microhalo profile (
(still) important to include realistic value for !
e.g. Diemand, Kuhlen & Madau, ApJ ’07
c v ...) and dN/dM
M cut
e influence Torsten of baryons Bringmann, could University give an enhanced of Hamburg density through adiabatic contraction
Thermal decoupling of WIMPs ‒ 13

Observational prospects
Is there a way to directly probe M cut
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
14

Observational prospects
Is there a way to directly probe M cut
Point sources
sources rather dim; difficult to resolve
strong limits from background
Pieri, Branchini & Hofmann, PRL ’05
Pieri, Bertone & Branchini, MNRAS ’08
Kuhlen, Diemand & Madau, ApJ ’08
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
14

Observational prospects
Is there a way to directly probe M cut
Point sources
sources rather dim; difficult to resolve
strong limits from background
Proper motion
strong limits from background
only for rather large masses
Pieri, Branchini & Hofmann, PRL ’05
Pieri, Bertone & Branchini, MNRAS ’08
Kuhlen, Diemand & Madau, ApJ ’08
Koushiappas, PRL ’06
Ando et al., PRD ’08
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
14

Observational prospects
Is there a way to directly probe M cut
Point sources
sources rather dim; difficult to resolve
strong limits from background
Proper motion
strong limits from background
only for rather large masses
Pieri, Branchini & Hofmann, PRL ’05
Pieri, Bertone & Branchini, MNRAS ’08
Kuhlen, Diemand & Madau, ApJ ’08
Koushiappas, PRL ’06
Ando et al., PRD ’08
Gravitational lensing
virial radius much larger than Einstein radius
multiple images of time-varying sources in
strong lensing systems! Moustakas et al., 0902.3219
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
14

Observational prospects
Is there a way to directly probe M cut
Point sources
sources rather dim; difficult to resolve
strong limits from background
Proper motion
strong limits from background
only for rather large masses
Pieri, Branchini & Hofmann, PRL ’05
Pieri, Bertone & Branchini, MNRAS ’08
Kuhlen, Diemand & Madau, ApJ ’08
Koushiappas, PRL ’06
Ando et al., PRD ’08
Gravitational lensing
virial radius much larger than Einstein radius
multiple images of time-varying sources in
strong lensing systems! Moustakas et al., 0902.3219
Anisotropy probes
angular correlations in EGRB
[again mostly large masses]
γ -ray flux (one-point) probability function
Ando et al., PRD ’06+’07
Fornasa et al., PRD ’09
Lee, Ando, & Kamionkowski, JCAP ’09
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
14

Observational prospects
Is there a way to directly probe M cut
Point sources
sources rather dim; difficult to resolve
strong limits from background
Proper motion
strong limits from background
only for rather large masses
Waiting for clever ideas!
Pieri, Branchini & Hofmann, PRL ’05
Pieri, Bertone & Branchini, MNRAS ’08
Kuhlen, Diemand & Madau, ApJ ’08
Koushiappas, PRL ’06
Ando et al., PRD ’08
Gravitational lensing
virial radius much larger than Einstein radius
multiple images of time-varying sources in
strong lensing systems! Moustakas et al., 0902.3219
Anisotropy probes
angular correlations in EGRB
[again mostly large masses]
γ -ray flux (one-point) probability function
Ando et al., PRD ’06+’07
Fornasa et al., PRD ’09
Lee, Ando, & Kamionkowski, JCAP ’09
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
14

Conclusions
Dark Matter decouples in two stages:
chemical decoupling
kinetic decoupling
For WIMPs,
relic density
size of smallest mini-clumps
10 −11 M ⊙ M cut 10 −3 M ⊙
strong dependence on particle physics properties!
An analytic treatment from first principles allows to
determine the cutoff with a precision of O(T/m χ ) 10 −3
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
15

Conclusions
Dark Matter decouples in two stages:
chemical decoupling
kinetic decoupling
For WIMPs,
relic density
size of smallest mini-clumps
10 −11 M ⊙ M cut 10 −3 M ⊙
strong dependence on particle physics properties!
An analytic treatment from first principles allows to
determine the cutoff with a precision of O(T/m χ ) 10 −3
Observational consequences
determination of “boost factor” for indirect DM detection
direct measurement of cutoff: challenging but not impossible
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
15

Conclusions
Dark Matter decouples in two stages:
chemical decoupling
kinetic decoupling
For WIMPs,
relic density
size of smallest mini-clumps
10 −11 M ⊙ M cut 10 −3 M ⊙
strong dependence on particle physics properties!
An analytic treatment from first principles allows to
determine the cutoff with a precision of O(T/m χ ) 10 −3
Observational consequences
determination of “boost factor” for indirect DM detection
direct measurement of cutoff: challenging but not impossible
A new window into the particle
nature of dark matter!
Torsten Bringmann, University of Hamburg Thermal decoupling of WIMPs ‒
15