WIMP Dark Matter and the QCD Equation of State

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1 Doktoratskolleg Graz Karl-Franzens-Universität, May 2006 WIMP Dark Matter and the QCD Equation of State Owe Philipsen Universität Münster Motivation + overview Cosmology I: expansion and contents of the universe Cosmology II: thermodynamics in the universe Calculating the QCD pressure Results for WIMP relic densities M. Hindmarsh + O.P., Phys. Rev. D71 (2005)

2 Overview 25% of the mass of the Universe is made of unknown Dark Matter Leading candidate: the WIMP (Weakly Interacting Massive Particle) Supersymmetric extensions of the Standard Model of particle physics predict WIMPs in mass range GeV. WIMP density today depends on masses, couplings, and the thermodynamics of the Universe at T GeV when the major component of the Universe was a quark-gluon plasma. WIMP density known to 10% accuracy now, and 1% within 3 5 years. QCD important for precision cosmology

3 Energy Epoque Quantum Gravity GeV The Early Universe: Physics of Non-Abelian Plasmas GeV 10 3 GeV Grand Unified Theories Supersymmetry Standard Model Electroweak Symmetry Breaking 100 MeV 10 MeV 1 ev Quark Hadron Transition Nucleosynthesis Radiation Matter Decoupling

4 Cosmology I: Friedmann equation Robertson-Walker metric: ds 2 = g µν dx µ dx ν = dt 2 + a 2 (t)dx 2 Spatial sections: dx 2 = (1 Kr 2 ) 1 dr 2 + r 2 dω 2 Describes expanding space of constant curvature K. General relativity: R µν 1 2 g µνr = 8πGT µν Λg µν T µν = [p(t) + ρ(t)]u µ u ν p(t)g µν r = a(t)x, ṙ = v = (ȧ/a)r Hr defines Hubble parameter H. Reduced Planck mass m P : m 2 P = 1/8πG, Planck mass M P : M 2 P = 1/G 00-component: H 2 + K a 2 = 1 3m 2 P ρ+ 1 3 Λ

5 (Friedmann)/H 2 : K a 2 H 2 = ρ 3m 2 P H2 1= ρ ρ c 1 = Ω 1 Define critical density ρ c = 3m 2 P H2. Define density parameters Ω i = ρ i /ρ c Total density parameter Ω = i Ω i Ω > 1 closed Ω = 1 flat Ω < 1 open Species i Ω i ρ i scales as Photons γ a 4 Neutrinos ν < a 4, a 3 Baryons b ± 0.04 a 3 Total matter m 0.27 ± 0.04 a 3 Curvature K < 0.02 a 2 Dark energy Λ 0.73 ± 0.04 constant

6 WMAP: Cosmic Microwave Background temperature -200 < T < 200 µk

7 Measuring dark matter component Ω c h 2 : fitting to a model Inflation motivates an 8-parameter model for the Universe (Ω = 1) Fitting to the current data gives h = 0.75, Ω c h 2 : WMAP WMAP+ Pub ± ± Peiris et al Tegmark et al 2003 Cosmic Microwave background: Ω b h 2 = ± Big Bang Nucleosynthesis (abundances of He 4, D, He 3, Li 7 ): Ω b h 2 = ± a a Kirkman et al astro-ph/ Ω b Ω c

8 Cosmology II: thermodynamics Particle reaction rates large compared with expansion rate H 1/t equilibrium n σv H σ n v Relative speed... Thermal average Scattering cross-section Number density of scatterers Early Universe very close to thermal equilibrium: expansion isentropic. S = sa 3 = const. Entropy density s. Thermodynamic relations: s = dp dt, st = ρ + p Need to calculate pressure only ( ρ = T 2 d dt ( p T ) )

9 Pressure from non-interacting particles with g degrees of freedom: Relativistic Boson, m T (Fermion) Non-relativistic, m T p r = g π2 90 T ( 4 7 ) 8 p nr = gt ( )3 mt 2 2π exp( m/t) ρ r = g π2 30 T ( 4 7 ) 8 ρ nr = m T p nr p nr Equations of state: p r = ρ r /3, p nr 0 Pressure from interacting particles: Define effective numbers of d.o.f. for energy & entropy densities: ρ = π2 30 g eff(t)t 4, s = 2π2 45 h eff(t)t 3, Calculate from Quantum Field Theory

10 Decoupling or freeze-out...when interaction rates become less than expansion rate Weakly interacting, cross-section σ G 2 F E2. Estimate decoupling temperature: n σv H = ( g eff /3)T 2 /m P Neutrinos: νe νe Cross-section: σ G 2 F T 2 Gives T f,ν 1MeV Hot dark matter (T f,ν m ν ) WIMPs: XX q q Cross-section: σv G 2 F m2 q Gives T f,x m X / GeV Cold dark matter (T f,x m X )

11 WIMP relic density calculation Mass m, number density n, annihilations XX... with total cross-section σ. Boltzmann equation (non-rel. particles): a ṅ + 3ȧ a n = σv Møl (n 2 n 2 eq) Let x = m/t, Y = n/s: where: g 1/2 (T) = h eff g 1/2 eff dy dx = ( π 45 ( 1 + T 3 d lnh eff dt ) 1 2 ). g 1 2 (T)(mM P ) σv Møl (Y 2 Y 2 eq) 1 x 2 Approximate solution: σv Møl = a (0) + a (1) /x + Ω X h GeV 2 a (0) x f g 1 2 (T) a Møller velocity: v Møl = `(v 1 v 2 ) 2 (v 1 v 2 ) 2 12 (Gondolo & Gelmini 1991)

12 Degrees of freedom for T = GeV: mostly coloured species m g species m g T=40 GeV: T=0.4 GeV: γ 0 2 g 0 16 ν e < 1 ev 2 u 3 MeV 12 ν µ < 1 ev 2 d 7MeV 12 ν τ < 1 ev 2 s 76 MeV 12 e 0.5 MeV 4 c 1.2 GeV 12 µ 106 MeV 4 b 4.2 GeV 12 τ 1.7 GeV 4 t 174 GeV 12 W 80 GeV 6 Z 91 GeV / /61.75 Cannot ignore strong QCD interactions!

13 p QCD, theory of strong interactions L QCD = 1 4g 2 N 2 c 1 a=1 F a µνf a µν + N f i=1 ψ i [γ µ D µ + m i ]ψ i. N c =3; N f =2,..., 6; m u, m d, m s, m c, m b, m t ; g 1. Confinement, non-perturbative physics High temperature/density: phase transition (schematically) asymptotic freedom: α s (q ) 0 T, µ B Hadrongas Quark-Gluon-Plasma

14 The QCD pressure...is minus the grand canonical free energy density p(t, µ) f(t, µ) = lim V For early universe: µ = 0 T V lnz = lim V T V ln QFT at finite T: two problems of perturbation theory a) strong coupling g(t) { Tr [ exp ( ĤQCD µ ˆB T )]} b) infrared problem at T 0: g 2 e E/T 1 E,p T g 2 T m divergent for m = 0 current dark matter calculations use simple potential models of QCD interactions: a here: lattice around T c, hadronic gas model T < T c, effective theory T T c in principle this can be done better! a Olive 1981

15 How experiment probes the phase transition & QGP...

16 T c and equation of state from the lattice Karsch, Laermann, Peikert 2000 pure gauge : T c = (271 ± 2) MeV continuum chiral limit, N f = 2 : T c = (173 ± 8) MeV on coarse lattice N f = 3 : T c = (154 ± 8) MeV Ideal gas: Stefan-Boltzmann p SB T 4 = {3 π2 90 ( N f) π2 90, T < T c, T > T c p/t 4 p SB /T 4 3 flavour 2+1 flavour 2 flavour pure gauge 0 T [MeV] T > T c : more degrees of freedom, but significant interaction! sqgp? (not yet right, m π 300 MeV, coarse lattices etc.) limited to T < 5T c

17 Pressure for T < T c, hadronic resonance gas model Below T c, q, q, g confined into hadrons. Treat as ideal gas, including resonances Z = n exp( E n/t) π 0 π ± K ± K 0 η ρ ω K ± K 0 p n η φ. 135 MeV 139 MeV 494 MeV 498 MeV 547 MeV 771 MeV 782 MeV 892 MeV 896 MeV 938 MeV 940 MeV 958 MeV 1019 MeV ε/t 4 T/T c Karsch, Redlich, Tawfik 2003

18 Effective high temperature theory: dimensional reduction typical scale R Scale hierarchy 2πT, gt, g 2 T Perturbative integration of p > T (Matsubara modes 0, fermions!), expansion in g(t)/(4π) Ginsparg; Appelquist, Pisarski... 1 > R 1 T T R, effectively 3d! simulation of 3d effective theory IR modes p gt,g 2 T, bosonic; good V, a 0 limits dim.red.for pressure: a write A a µ(τ,x) = n Aa nµ(x)e i2πnτ/β 1. Integrate out A a nµ(x) (n 0, masses 2nπT ), ψ, ψ: O(g 0 ) 2. Integrate out A a 0(x) (mass gt ): O(g 3 ) 3. Integrate A a i (x) (scale g2 T, non-perturbative): O(g 6 lng, g 6 ) a Kajantie, Laine, Rummukainen & Schröder 2001

19 Example: spatial string tension computed in 3d and 4d (N f = 0): 1.2 T c / Λ MS _ = loop 1.0 T/σ s 1/ loop 0.6 4d lattice, N τ 1 = T / T c

20 Pressure for T > T c Kajantie et al 2002, Karsch et al p/p SB _ T/Λ MS g 2 g 3 g 4 g 5 g 6 (ln(1/g)+0.7) 4d lattice Pure glue (N f = 0) Λ MS = 237 MeV p SB (T) = 16 π2 90 T 4 Fit order g 6 term (not yet computed) also fits slope! N.B.: In principle no fit required, but four-loop computation... will it match?

21 Lattice QCD pressure: not quite right yet Lattice data for N f 0 not at continuum limit or physical quark masses. Dimensional Reduction method matched only for pure glue N f = 0 Define QCD correction factor f from pure glue and scale to correct T c : Definition: f(t, N f ) = p(t, N f )/p SB (T, N f ) Observe approximate scaling: a f(t, N f ) f (T/T c (N f )) a Karsch, Laermann, Peikert p/p SB 3 flavour 2 flavour 2+1 flavour T/T c

22 Modelling the QCD pressure correction factor f Scale to N f = 0, T c (0) = 269 MeV using T c listed below. T > 4.43T c Dim. Red. O(g 6 ln(g)), matched to lattice 4.43T c > T > T HG Interpolated lattice T HG > T Hadronic gas EOS model T c /MeV T HG /MeV Notes A i.e. no hadronic gas B C chosen to keep f(t) smooth. Multiply coloured degrees of freedom by f

23 Results: degrees of freedom standard a (dashed) and modified (solid) h eff 60 g 1/2 * T/GeV h eff (T) T/GeV g 1/2 (T) = h eff g 1/2 eff Weak effect because p, ρ each shrink, but p ρ/3 still a Olive 1981 ( 1 + T 3 d lnh eff dt )

24 Calculating relic densities: packages DarkSUSY a MicrOMEGAs b Both use same equation of state c, incorporated as a look-up table. New look-up tables for DarkSUSY from a Gondolo et al 2004 b Bélanger et al 2001, 2004 c Olive 1981

25 Relic densities in benchmark SUSY models Battaglia et al 2003 QCD uncertainty estimated by scaling DR f by (0.9) (1.1) and re-matching. Model m χ /GeV T f /GeV Ω c h 2 (DS) Ω c h 2(0.9f) (1.1f) A (54) (42) B (56) (31) C (83) (65) G (33) (13) H (71) (53) I (62) (40) J (90) (79) L (18) (03) (%)

26 Conclusions The Universe is filled with Dark Matter: leading candidate is a (SUSY) WIMP QCD plasma effects increase WIMP density by few % Not insignificant: Planck may be able to reach (Ω c h 2 ) = Precision QCD is necessary for precision cosmology Needed: lattice & dim. red. calculations for N f 0. Eqn. of State information from Heavy Ion Collisions?