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On the quantum statistical physics
of dark matter freeze-out1,2
Mikko Laine
University of Bern, Switzerland
1Based partly on collaborations with Simone Biondini and Seyong Kim.
2Supported by the SNF under grant 200020-168988.
1
There is “dark matter”
2
3D. Clowe et al, A direct empirical proof of the existence of DM, astro-ph/0608407.
3
What is dark matter?
4
Yet to be discovered particles? Basic requirements:
• not visible ⇒ electrically neutral
• around long ago & still today ⇒ stable or very long-lived
• correct structure formation long ago ⇒ rather heavy
Known particles (well described by the Glashow-Weinberg-Salam
“Standard Model”) fail to satisfy these requirements.
(An extension by keV sterile neutrinos might help. An extension
by µeV axions is quite popular. Primordial black holes may help,
but they cannot be generated within the Standard Model.)
5
In this talk: “WIMP paradigm”
Postulate the existence of Weakly Interacting Massive Particles
(“heavy neutrinos”) which cannot decay and are thus stable.
“Indirect detection” from galactic center:
DM
DM
DM′
“Direct detection” by nuclear recoil:
DM DM′
“Collider search” through missing energy:
DM′
DM
6
Text-book WIMP is in trouble
7
Lee-Weinberg equation4 (n=number density, H=Hubble rate)
(∂t + 3H)n = −〈σv rel〉 (n2 − n
2eq) .
DM
DM
Entropy conservation (part of Einstein equations)
(∂t + 3H)s = 0 .
Combining the two we get
∂t
(n
s
)
= −〈σv rel〉(n2 − n2
eq)
s.
4B.W. Lee and S. Weinberg, Cosmological Lower Bound on Heavy Neutrino Masses,
Phys. Rev. Lett. 39 (1977) 165.8
Insert relation of time and temperature (c = heat capacity):
dT
dt= −3Hs
c, ∂t =
dT
dt∂T .
Defining a “yield parameter” through Y ≡ n/s we get
∂TY =c〈σv rel〉3H
(Y2 − Y
2eq) .
Inserting c ∼ T 2, 〈σv rel〉 ∼ α2/M2, H ∼ T 2/mPl, where M
and α denote the WIMP mass and coupling, we get
∂TY ∼ mPlα2
M2(Y
2 − Y2eq) .
9
The equilibrium number density is a known function of T,M :
n eq ∝∫
d3p
(2π)31
e√
p2+M2/T ± 1≈
(MT
2π
)3/2
e−M/T
.
The entropy density is dominated by massless modes: s ∝ T 3.
Linearize around equilibrium (Y = Y eq + δY , δY = Y − Y eq):
z ≡ M
T⇒ Y
′(z) ∼ − mPlα
2
M
Y eq(z)[Y (z) − Y eq(z)]
z2.
The differential equation has a “thermal fixed point” at Y (z) =
Y eq(z) but cannot keep close to it for Y eq(z) ≪ 1. 10
Numerical solution shows a “freeze-out”:
0 5 10 15 20 25 30z = M / T
1e-12
1e-08
1e-04
1e+00
Y
Yeq
M = 7 TeV
M = 5 TeV
M = 3 TeV
M = 1 TeV
α = 0.01, mPl
= 1.2 ∗ 1016
TeV
11
Final energy density (e ≡ Mn) compared to radiation ∼ T 4:
1 2 3 4 5 6 7 8
M / TeV
0e+00
1e-05
2e-05
3e-05
e /
T4 @
T =
1 G
eV
α = 0.01, mPl
= 1.2 ∗ 1016
TeV
WIMP miracle
overclosure
LHC pushes up M , so there is a danger “overclosure”!12
Could increased 〈σvrel〉 help?
13
Much discussed:5 “Sommerfeld effect”6:
〈σv rel〉 −→ 〈σ tree v rel S(v rel)〉 .
For attractive Coulomb-like interaction,
S(v rel) ∼ α
v rel
for v rel<∼α .
5e.g. J. Hisano, S. Matsumoto, M. Nagai, O. Saito and M. Senami, Non-perturbative
effect on thermal relic abundance of dark matter, hep-ph/0610249; J.L. Feng, M. Kaplinghatand H.-B. Yu, Sommerfeld Enhancements for Thermal Relic Dark Matter, 1005.4678.
6L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, Third
Edition, §136; V. Fadin, V. Khoze and T. Sjostrand, On the threshold behavior of heavy top
production, Z. Phys. C 48 (1990) 613.14
More recent:7 bound states (“WIMPonium”):
Mbound = 2M − ∆E ⇒ e−Mbound/T > e
−2M/T.
This is quantum mechanics in a statistical background.
7e.g. B. von Harling and K. Petraki, Bound-state formation for thermal relic dark
matter and unitarity, 1407.7874.15
Some quantum statistical physics
16
Particles in the initial state: most energy is carried by mass.
E rest ∼ 2M , Ekin ∼k2
2M∼ T .
Particles in the final state: all energy is carried by momentum.
Ekin ∼ 2k ∼ 2M ⇒ ∆x ∼ 1
k∼ 1
M≪ 1
T.
DM
DM
Therefore the “hard” annihilation process is local.8
8e.g. G.T. Bodwin, E. Braaten and G.P. Lepage, Rigorous QCD analysis of inclusive
annihilation and production of heavy quarkonium, hep-ph/9407339; L.S. Brown andR.F. Sawyer, Nuclear reaction rates in a plasma, astro-ph/9610256.
17
But before the annihilation there are “soft” initial-state effects:
. . .
soft hard
18
A “linear response” analysis shows that
〈σv rel〉 ∼ α2
M2n2eq
1
Z
∑
m,n
e−Em/T〈m|φ†
φ†
⇒1
︷ ︸︸ ︷|n〉〈n|φφ|m〉
︸ ︷︷ ︸
=:〈O†(0)O(0)〉T
.
Here |m〉 are eigenstates containing a DM-DM pair,
and φφ annihilates the DM-DM pair.
19
The DM-DM problem can be reduced to a 1-body problem:
Em =: E′+
[
2M +k2
4M
]
︸ ︷︷ ︸
center-of-mass energy
.
Carrying out the integral over k we are left with
〈O†(0)O(0)〉T = e
−2M/T
(MT
π
)3/2 ∫ ∞
−Λ
dE′
πe−E′/T
ρ(E′) .
20
The “spectral function” ρ(E′) represents the solution of a
Schrodinger equation for a Green’s function.
[HT − iΓT (r) − E
′]G(E
′; r, r
′) = δ
(3)(r − r
′) ,
limr,r′→0
ImG(E′; r, r
′) = ρ(E
′) .
Here the Hamiltonian has a standard from
HT = −∇2r
M+ VT (r) , r = |r| ,
whereas −iΓT (r) accounts for real scatterings with the plasma.
21
Procedure in a nutshell
• Compute thermal self-energy of the exchanged particle
• Determine the corresponding time-ordered propagator
• Fourier-transform for potential VT (r) and width ΓT (r)
• Solve for ρ(E′) = ImG(E′; 0, 0)
• Laplace-transform with weight e−E′/T for 〈O†(0)O(0)〉T
• Integrate Lee-Weinberg equation numerically
22
Results for simple cases
23
Spectral function for typical WIMPs
-1×10-2 0 1×10
-22×10
-23×10
-2
E’ / M
0
2×10-3
4×10-3
ρ 1 /
(ω
2 Ν1)
free
free ∗ S1
M = 4 TeV
T = 200 GeV
24
It becomes interesting if M is large and T low
-4×10-4
-2×10-4 0 2×10
-4
E’ / M
0
2×10-3
4×10-3
6×10-3
ρ 1 /
(ω
2 Ν1)
T = 600 GeV
T = 100 GeV
T = 10 GeV
M = 12 TeV
25
Even more interesting if we decrease T further
-4e-06 -2e-06 0e+00 2e-06E’ / M
0e+00
2e-03
4e-03
6e-03ρ 1
/ (
ω2 Ν
1)
T = 600 GeV
T = 100 GeV
T = 10 GeV
T = 1 GeV
M = 12 TeV
26
A nice relation to heavy ion collision experiments
S. Chatrchyan et al. [CMS Collaboration], Suppression of excited Υ states in PbPb collisions
at√sNN = 2.76 TeV, Phys. Rev. Lett. 107 (2011) 052302 [1105.4894].
This follows a general pattern previously predicted theoretically.9
9e.g. F. Karsch, D. Kharzeev and H. Satz, Sequential charmonium dissociation, Phys.
Lett. B 637 (2006) 75 [hep-ph/0512239].27
Dark matter density evolution in the Early Universe
101
102
103
z = M / T
10-16
10-14
10-12
10-10
10-8
Y
λ3,4,5
= 0
λ3,4,5
= 1
λ3,4,5
= π
M = 12.0 TeV, ∆M = 10-3
M
Yeq
28
Examples of overclosure bounds
0 2 4 6 8 10 12 14M / TeV
0.0
0.1
0.2
0.3
Ωdm
h2
λ3,4,5
= 0
λ3,4,5
= 1
λ3,4,5
= π
overclosure
viable
29
Relation to indirect non-detection
30
Why is Early Universe not much constrained by present day?
Long ago: t ∼ 10−12 s, T ∼ 100 GeV.
DM annihilation:
DM
DM
DM’ annihilation:
DM′
DM′
DM ↔ DM’ is in thermal equilibrium ⇒ annihilation can proceed
through the heavier DM’ channel if this is more efficient.
31
Today: t ∼ 1017 s, T ≪ eV.
DM annihilation is active:
DM
DM
DM’ decayed long ago.
Therefore present day indirect detection constraints cannot
directly fix 〈σv rel〉 relevant for Early Universe cosmology.
32
If DM’ interacts strongly, there are large effects.10
-4e-02 -2e-02 0e+00 2e-02E’ / M
0e+00
2e-02
4e-02
6e-02
8e-02
ρ / ω
2
free ∗ S1
free
T = 100 GeV
T = 150 GeV
T = 200 GeV
T = 300 GeV
T = 500 GeV
Gluon exchange, M = 3 TeV
10e.g. J. Ellis, F. Luo and K.A. Olive, Gluino Coannihilation Revisited, 1503.07142.
33
Summary
• Apart from model uncertainties, generic dark matter studies
also contain theoretical uncertainties.
• In general, both quantum-mechanical effects (bound states,
multiple interactions) and statistical physics phenomena (Debye
screening, frequent scatterings on plasma particles) play a role.
• Strong interactions may enhance the annihilation rate because
of bound states, and this may help to avoid overclosure.
• Model-specific studies are needed for definite conclusions.
34
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