Production of Dark Matter in the Early Universe

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1 Production of Dark Matter in the Early Universe Ulrich Feindt May 7, / 27

2 Outline Introduction Boltzmann equation Cold Dark Matter Hot Dark Matter Conclusion 2 / 27

3 Introduction Equilibrium in the Early Universe To a good approximation the early Universe was in thermal equilibrium Particles have a phase space distribution f ( x, p, t) In kinetic and chemical equilibrium f is either a Fermi-Dirac or a Bose-Einstein distribution: f ( p) = [exp ((E µ) ± 1)] 1 (1) 3 / 27

4 Introduction Freeze Out Chemical equilibrium between particle species is governed by annihilation/creation reactions Rate depends on cross-section and density Rule of thumb for freeze out Γ H thermal equilibrium Γ H decoupled evolution 4 / 27

5 Boltzmann equation Boltzmann equation Evolution of f ( x, p, t) is described by the Boltzmann equation For non-interacting, non-relativistic particles: df dt + d x dt x f + d p dt p f = 0 For the Friedmann-Robertson-Walker (FRW) model this can be rewritten to f (E, t) t Ṙ p 2 f (E, t) = 0 R E E 5 / 27

6 Boltzmann equation Boltzmann equation (cont.) Define number density: n(t) = g : g (2π) 3 d 3 p f (E, t) internal degrees of freedom Integrating the Boltzmann eq. then gives dn dt + 3Ṙ R n = 0 6 / 27

7 Boltzmann equation Collision term Evolution from equilibrium to decoupled particles depends on interaction Introduce collision term: dn dt + 3Ṙ R n = g (2π) 3 C[f ] d3 p E E.g. for reaction ψ + a i + j dn dt + 3Ṙ R n = dπ ψ dπ a dπ i dπ j (2π) 4 δ 4 (p ψ + p a p i p j ) dπ = [ M 2 ψ+a i+j f ψf a (1 ± f i )(1 ± f j ) M 2 i+j ψ+a f if j (1 ± f ψ )(1 ± f a )] g d 3 p (2π) 3 2E 7 / 27

8 Boltzmann equation Collision term (cont.) Assume stability of the Dark Matter candidate up to the age of the Universe Assume CP invariance M 2 i+j ψ+a = M 2 ψ+a i+j := M 2 Use Maxwell-Boltzmann statistics 1 ± f 1, ṅ ψ + 3Hn ψ = f i (E i ) = exp [ E ] i µ i T dπ ψ dπ a dπ i dπ j (2π) 4 δ 4 (p ψ + p a p i p j ) M 2 [f ψ f a f i f j ] 8 / 27

9 Boltzmann equation Comoving Volume Useful to scale out the effect of expansion by using comoving number density Y = n ψ /s (sr 3 = const.) sẏ = n ψ + 3Hn ψ Collision term usually depends on temperature T introduce x = m/t dy = x dπ ψ dπ a dπ i dπ j (2π) 4 δ 4 (p ψ + p a p i p j ) dx H(m)s M 2 [f ψ f a f i f j ] H(m) = 1.67g 1/2 m 2 m Pl 9 / 27

10 Freeze Out Dark Matter Particle More explicitly assume annihilation and inverse annihilation process ψ ψ X X (X = all possible annihilation products) No asymmetry between ψ and ψ X remain in equilibrium after freeze out of ψ 10 / 27

11 Freeze Out Equilibrium distribution Consider the factor [f ψ f ψ f X f X ] X, X in equilibrium (assuming µ = 0) f X = exp( E X /T ), f X = exp( E X /T ) The Maxwell-Boltzmann-distribution still valid as the X are in the high energy tail Then energy conservation (E ψ + E ψ E X E X = 0) gives: f X f X = exp( (E X + E X )/T ) = exp( (E ψ + E ψ )/T ) = f EQ ψ f ψ EQ [f ψ f ψ f X f X ] = [f ψf ψ f EQ ψ f EQ ψ ] 11 / 27

12 Freeze Out Reaction cross section Now rewrite collision term: dy dx = x σ ψ H(m) ψ X X v s ( Y 2 Y 2 EQ) with thermally averaged cross-section times velocity: σ v = (n EQ ψ ) 2 dπ ψ dπ ψdπ X dπ X (2π) 4 δ 4 (p ψ + p ψ p X p X ) M 2 exp( (E ψ + E ψ )/T ) 12 / 27

13 Freeze Out Limiting forms Look at extreme cases: ultra-relativistic Y EQ = 45ζ(3) g eff 2π 4 = g eff g s g s non-relativistic Y EQ = 45 ( π ) 1/2 g 2π 4 x 3/2 e x = g x 3/2 e x 8 g s g s 13 / 27

14 Freeze Out Final form of the Boltzmann equation Note that for radiation dominated Universe H x 2 ( H(T ) = x 2 H(m)) cast Boltzmann eq. in suggestive way: x dy Y EQ dx = Γ A H [ ( Y Y EQ Γ A n EQ σ A v ) 2 1] Note that if Γ H 1 xdy Y EQ dx 1 Annihilation freezes out 14 / 27

15 Freeze Out Final form of the Boltzmann equation (cont.) No closed solutions consider qualitative behaviour In relativistic regime: n EQ T 3, Γ A T n In NR regime: n EQ (mt ) 3/2 exp( m/t ) Γ decreases exponentially freeze out But for Γ H 1 Y Y EQ 15 / 27

16 Cold Dark Matter Cold Dark Matter Consider cold dark matter (i.e. DM that was non-relativistic at freeze out) For NR species Y EQ decreases exponentially with x Parameterize temperature dependence of σ A v : σ A v v 2n (n = 0 for s-wave, n = 1 for p-wave, etc.) v T 1/2 σ A v T n σ A v σ 0 ( T m ) n = σ 0 x n (x 3) 16 / 27

17 Cold Dark Matter Solution of the Boltzmann equation The Boltzmann eq. now becomes dy dx = λx n 2 (Y 2 Y 2 EQ ) ( λ [ ] ) σa v s H(m) x=1 Define departure from equilibrium Y Y EQ = Y EQ λx n 2 (2Y EQ + ) At early times both and are small Set = 0 xn+2 2λ 17 / 27

18 Cold Dark Matter Solution of the Boltzmann equation (cont.) At late times: Y Y EQ neglect Y EQ and Y EQ = λx n 2 2 Y = = n + 1 λ x n+1 f = [ ] H(m)(n + 1) σ A v s x n+1 f x=1 Need to determine x f (time of freeze out) Define criterion: (x f ) = cy EQ with c a constant of order unity Plug in early time solution (x f ) x n+2 /λ(2 + c) and solve for x f x f ln[(2 + c)λac] a = 0.145(g/g s ) f ( n + 1 ) ln[ln[(2 + c)λac]] 2 18 / 27

19 Cold Dark Matter Numerical solution of the Boltzmann equation Figure: Freeze out of a massive particle species. Dashed line: actual abundance, solid line: equilibrium abundance [1] 19 / 27

20 Cold Dark Matter Prediction for Cosmology Choosing c(c + 2) = n + 1 gives best fit to numerical results (better than 5%) Straightforward calculation then gives: n ψ0 = s 0 Y = (n + 1)x n+1 f (g s /g 1/2 cm 3 )m Pl mσ 0 1/2 Ω ψ h 2 9 (n + 1)(g /g s )x f = m Pl σ A v Note: Ω ψ σ A v 1 20 / 27

21 Cold Dark Matter WIMPs as Dark Matter candidates Assume ψ has 2 degrees of freedom and σ A v = a cm 2 (n = 0) and g = 60 [ ( m )] x f = ln a GeV Y = a(m/gev) Ω ψ h 2 = 0.043x f a a Hence for Ω ψ of order unity the annihilation cross section needs to be characteristic of a weak process. 21 / 27

22 Cold Dark Matter Restriction of SUSY parameters from Cosmology Figure: The (m 1/2, m 0 ) plane for tan β = 10. The turquoise shaded area is preferred by cosmology. [2] 22 / 27

23 Cold Dark Matter Exceptions in the calculation There are exception to the simple T -dependence assumed above [3] Neutralino DM with an additional sparticle Annihilation of the sparticle can control the relic abundance if m T f DM particle lies near a mass threshold for annihilation to an additional species Forbidden annihilation channel can dominate the cross section Annihilation takes place near a pole in the cross section. Special care needs to be taken in the thermal averaging of the cross section. 23 / 27

24 Hot Dark Matter Hot Dark Matter Now consider a particle species that decoupled while still relativistic (e.g. neutrinos) Y EQ does not change with time Y insensitive to the details of freeze out Y is just the equilibrium value at decoupling g eff Y = Y EQ (x f ) = g s (x f ) 24 / 27

25 Hot Dark Matter Neutrinos as Dark Matter? For a particle species ψ this gives: Ω ψ h 2 = g ( eff m ) g s ev Light neutrinos decouple when T few MeV and g s = g = A two-component neutrino species has g eff = 1.5 and hence Ω ν ν h 2 = m ν 91.5eV For neutrinos to make up all the measured dark matter, the mass needs to be of order 10 ev Experiments suggest lower neutrino masses 25 / 27

26 Conclusion Conclusion Relic particle abundances can be calculated with the Boltzmann equation Abundance of cold relics depends (almost) only on annihilation cross section WIMPs are good dark matter candidates Neutrinos as hot dark matter can be excluded because of low mass Caveat: all calculation depend on the underlying model for cross section 26 / 27

27 References References [1] M.E. Turner and E.W. Kolb, The Early Universe [2] J. Ellis and K.A. Olive, Supersymmetric Dark Matter Candidates, arxiv: v1 [3] K. Griest and D. Seckel, Three exceptions in the calculation of relic abundances, Phys. Rev. D 43, (1991) 27 / 27

Topic 3. Evidence for the Big Bang

Topic 3. Evidence for the Big Bang Topic 3 Primordial nucleosynthesis Evidence for the Big Bang! Back in the 1920s it was generally thought that the Universe was infinite! However a number of experimental observations started to question