Dark Matter early kinetic decoupling
and finite temperature corrections tolong-range force systems
Tobias Binder(ITP, Göttingen)based on[1706.07433] in collaboration with T.Bringmann, M.Gustafsson, A.Hryczuk,[1808.06472] in collaboration with L. Covi, K. Mukaida.

Beyond the BSM workshop, Ikaho, Japan

October 3, 20181

Motivation
Standard description of DM chemical decoupling process:2ṅ + 3Hn = −〈σv 〉TSM n2 − neq(TSM )
Lee&Weinberg 77’, Gondolo&Gelmini 91’

É

implemented in DarkSUSY and MicrOMEGAs, efficiently andaccurately solving for DM relic density.
Captures special cases like co-annihilation, threshold and resonances.

É

relies on certain assumptions and has limitations.

Édeveloped refined description accounting for i) early kinetic
decoupling ii) self-consistent treatment of thermalcorrections.
Planck precision era: ΩDM h2 = 0.1198 ± 0.0015 (!) , refinement of DM overlosure bound.

2

Early kinetic decoupling of dark matter: when the standard
way of calculating the thermal relic density fails Binder+ 17’É

Early kd if Γsc |cd ∼ Γan |cd . Not only a theoretical possibility!

ÉDM annihilation ↔ temperatureÉdeveloped two methods to accurately compute the DM relicabundance for such case3

Two methods to deal with early kd

1

Full phase-space density solution (full BE)
Numerical solution of the non-linear integro PDE:
E (∂t − 2Hp ∂p )f (t , p ) = Can [f , feq ] + Csc [f , fSM ]É accounting for phase-space density distortion caused byvelocity-dependent annihilation.

É

2Solving coupled system of momentum moments (cBE)
ÉCoupled system of ordinary differential equations:
R 3ṅ + 3Hn = g (2dπ)p3 E Can [f , feq ]R d 3 p p2
gṪ + nṅ T + 2HTw (T ) = 3n(2π)3 E E Can [f , feq ] + Csc [f , fSM ]E

É

Closer of Boltzmann hierarchy by form Ansatz f = n n(T ) e − T .
eq4

Simplest example: Scalar singlet dark matterÉ

Two parameter model: mS , λS .

ÉBreit-Wigner enhanced annihilation for mS ∼ mH /2.

ÉHiggs couples preferably to heavy SM particles→ Boltzmann suppressed scattering5

coupled BEs, QCD=A
coupled BEs, QCD=B
coupled BEs, non-rel(Ωh2 )KD / (Ωh2 )G & G

10full BE

510.5455055606570mS [GeV]

ÉAlready in simplest scalar singlet DM model, the relicabundance can be affected by more than one order ofmagnitude!

ÉPublic phase space density solver soon available
T.Binder, T.Bringmann, M.Gustafsson, A.Hryczuk in prep. 2018, provide many more examples: Higgsand Z-portals, forbidden DM, Sommerfeld enhanced annihilation,...

6Dark matter Sommerfeld-enhanced annihilation andBound-state decay at finite temperature Binder+ 18’?

L ⊃ gχ χ̄γµ χ Aµ + gψ ψ̄γµ ψAµ?

?

?

ÉDM (χ ) longe-range force enhanced annihilation in thepresence of hot and dense plasma background.

ÉVacuum computation well understood.

ÉFinite temp. effects: at first place a conceptional question.
Charge screening? Dissociation ? No (correct) formal description available, claims about strong finitetemperature corrections based on linear response theoryÉcomprehensive ab initio derivation from non-eq. QFT Ømostly new research in quantum statistical mechanics

Émain result: new number density equation Ø7

Master equation for s-wave annihilation:

++−−ṅη + 3Hnη = −2(σv )0 Gηξ(x , x , x , x )|eq e 2βµ − 1 ,Z∞ 3Z∞d P −β P2 /4M
dE −β E ρ++−−Gηξ|eq = e −2β MeeGηξ (0, 0; E )|l =0 .
3(2π)
(2π)
−∞
−∞
É General equation, accounting for non-ideal and finite temperaturecorrections.
É Chemical potential µ[nη ] sets functional dependence of r.h.s on nη
É Four point correlator G ++−− (x , x , x , x )|eq in chemical equilibrium includes
ηξSommerfeld enhanced annihilation and bound state decay simultaneouslyρ
via two-particle spectral function Gηξ . Form invariant under interpretationof bound or scattering state at finite temperature.

É Self-consistent solution of µ[nη ] and G ++−− (x , x , x , x )|eq is important!
ηξÉ Checked various limits and compared to existing literature:

vacuum limit Ø (Lee-Weinberg eq.), linear regime close to chemicalequilibrium Ø (linear response theory)8

Vacuum spectral function
ρ

Vacuum relations between s-wave two-particle spectral function Gηξ ,Sommerfeld enhancement factor S, and bound-state decay width Γ :1Tr[12×2 ]M (ME )1/2 (σvrel )0 S (E ),2πXπρ(σvrel )0 Gηξδ(E − EBn )Γn .
(0, 0; E )|E <0,l =0 = Tr[12×2 ]2nρ(σvrel )0 Gηξ(0, 0; E )|E >0,l =0 =Coulomb potential, M=5TeV , α χ =0.1, T=M/30

Yukawa potential, M=5TeV , α χ =0.1 , T=M/30

0.20
bound states

scattering states

0.10
Sommerfeld-enhanced

0.15
ρ
Gηξ
(0,0;E)/M 2ρGηξ(0,0;E)/M 2

0.15

0.20

0.05
0.00
-0.1

mV =135GeV

on-resonancemV =75GeV

0.10

oﬀ-resonance0.05

0 0.01

-0.05
E/T

0.1

1

0.00
-0.1

0 0.01

-0.05

0.1

1

E/T

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Spectral function at finite temperature
Effective in-medium potential (in HTL resummed effective theory):
Veff (r) = −αχ mD −αχ

re −mD r − i αχ T φ (mD r ),2
where φ (0) = 0 and φ (∞) = 1, and Debye mD= gψ2 T 2 /3.
Coulomb, M=5TeV , α χ =0.1, T=M/30
mD /T=0.140.25

mD /T=0.029

0.20

mD /T=0.006
mD /T=0.001

0.15
0.10
0.05
0.00
-0.10mD /T=0.65

0.30
ρ
Gηξ
(0,0;E)/M 2ρGηξ(0,0;E)/M 2

Yukawa , M=5TeV , α χ =0.1, mV =11GeV , T=M/30

mD /T=0.65

0.30

mD /T=0.14

0.25

mD /T=0.029

0.20

mD /T=0.006
mD /T=0.001

0.15
0.10
0.05

-0.08

-0.06

-0.04
E/T-0.02

0.00

0.00
-0.10

-0.08

-0.06

-0.04

-0.02

0.00

E/T

Notation of spectral function more general than using S or Γ .
Shift towards lower energies affects total rate exponentially.
10Summary, Conclusion, Outlook
É

Questioning the assumptions entering the standard descriptionof the DM freeze-out can lead to surprising new phenomena.

É

Early kinetic decoupling can have huge impact on DM
overclosure bound.

É

Public phase-space density code with many more examples
soon available.

ÉDeveloped rigorous theoretical understanding of long-range
force systems at finite temperature.

ÉIf bound state solution exist (mV ® αχ M) they are important
to include in the relic density computation.

É

New number density equation allows to include finitetemperature effects self-consistently.

É

Impact of finite temperature effects on relic density needs
careful investigation, work in progress.
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