1 Cosmology and Dark Matter V.A. Rubakov Lecture 2 2 Outline of Lecture 2 WIMP search Dark matter axions Theory Cosmology Search Fuzzy dark matter Warm dark matter Sterile neutrino Gravitino Dark matter summary 3 WIMP search: direct Difficulties in direct search for WIMPs Recoil energy of nucleus A 1 2 2 10 100 Erec . MAvX 2 keV (1 + MA/MX ) ∼ ÷ vX 200 km/s = WIMP velocity in our Galaxy. ≃ Rate in detector of mass Mdet (spin-independent) 23 ρloc 2 6 10 Mdet Γ vX (σNA ) · ≃ MX A g 100 σ 0 2events GeV A Mdet N . 100 10 45 2 ≃ yr · MX tonn − cm 3 ρloc 0.4 GeV/cm = local DM mass density. ≃ 4 Direct searches 5 SUSY neutralino strongly constrained NB: gray points require a lot of fine-tuning. 6 The LHC is sensitive to WIMPs spin-independent spin-dependent µ 5 µ 5 X¯ γµ X q¯γ q X¯ γµ γ X q¯γ γ q · · 2 σAX ∝ A σAX ∝ JA(JA + 1) 7 Indirect searches DM annihilation in centers of Sun, Earth ¯ X + X π± , K± + ... ν , ν¯ + ... → → High Baksan Underground Scintillation Telescope energy = ⇒ Super-K neutrinos IceCube Baikal GVD DM annihilation in space e+, p¯ in cosmic rays (PAMELA, AMS), annihilation γ’s (Fermi-LAT, MAGIC, HESS, CTA ...). 8 Limits from annihilation γ’s -24 10 χχ bb (CMB) -25 H.E.S.S. 10 Planck (Einasto) /s) 3 (NFW) thermal DM CTA (cm 〉 v (dSphs) -26 (Albert+, 2016) (Einasto) 〈σ 10 CTA Fermi-LAT Fermi-LAT+MAGIC (NFWc) Fermi-LAT(45dSphs,15y) -27 10 Fermi-LAT 10 10 2 10 3 10 4 mχ (GeV) Current limits, solid Projected limits, dashed NFW, Einasto: dark matter profiles in galaxies Thermal DM: WIMP annihilation cross section, assuming domination of X bb¯ → 9 TeV SCALE PHYSICS MAY WELL BE RESPONSIBLE FOR GENERATION OF DARK MATTER Is this guaranteed? By no means. Other good DM candidates: axion, sterile neutrino, gravitino Plus a lot of exotica... 10 Axions Motivation: solution of strong CP problem What’s the problem? Point No. 1: global symmetries of QCD in chiral limit mu = md = ms = 0 1 a aµν ¯ γ µ LQCD,m=0 = 4Gµν G +∑qii Dµ qi − i 1 a aµν µ µ = Gµν G +∑ q¯L iiγ Dµ qL i + q¯R iiγ Dµ qR i −4 , , , , i Naively: symmetry under independent SU(3) U(1) rotations of left and right quarks, × SU(3)L U(1)L SU(3)R U(1)R = SU(3)L SU(3)R U(1)B U(1)A × × × × × × iα iβγ5 U(1)B: qi e qi; U(1)A: qi e qi → → 11 Symmetry partially broken due to quark condensate in QCD vacuum, ¯ 1 3 u¯LuR = dLdR = qq¯ Λ h i h i 2h i ∼ QCD Remaining symmetry SU(3)V : rotates left and right quarks together; U(1)B Expect 9 Nambu–Goldstone bosons, 8 from SU(3)L SU(3)R SU(3)V plus 1 from U(1)B U(1)A U(1)B. × → × → 0 ¯ 0 But there are only 8 in Nature: π±, π , K±, K0, K , η. 2 2 NB: mπ = mu d qq¯ / fπ , h i η′ is heavy, does not behave like Nambu–Goldstone boson. Reason: U(1)A is not a symmetry in QCD A Axial current has triangle anomaly, ∂µ Jµ = 0. 6 12 Point No. 2 Quark Yukawa interactions = quark mass matrix ⇒ (d) ¯i j (u) ¯i j (d) ¯i j (u) i j LY = y Q Hd + y Q H∗u + h.c. = Lm = m d d + m u¯ u + h.c. ij L R ij L R ⇒ ij L R ij L R (u,d) (u,d) 2 1 2 3 mij = yij v/√ complex; i, j = , , = generation label. Standard lore: diagonalize = CKM matrix, 3 angles, 1 phase. ⇒ This is not quite true 13 One more phase: common phase of all Yukawa couplings/masses, iθ CKM mij = e m or θ=(Arg det m)/3 · ij At first sight: rotate away, 5 qi eiθ/2qi , qi e iθ/2qi , i.e., qi e iγ θ/2qi L → L R → − R → − But this is not allowed: U(1)A is not a symmetry! θ is a physical parameter Yet another subtlety: can have a term in QCD Lagrangian α s θ ε µνλρ a a 16π 0 Gµν Gλρ What matters is θ0 + Arg det m In what follows: θ0 moved to quark mass matrix. 14 Vacuum energy density for non-zero θ, call it V(θ) (useful in what follows). Keep only u,d-quarks, take their masses equal for simplicity, mu = md mq 4 MeV (heavier quarks less important), ≡ ∼ iθ ¯ Lm = e mq(u¯LuR + dLdR) + h.c. Perturbation theory in mq: V(θ)= Lm . Recall quark condensate −h i ¯ 1 3 u¯LuR = dLdR = qq¯ = real Λ h i h i 2h i ∼ QCD NB: No arbitrary phase here, otherwise η′ would be pseudo-Nambu–Goldstone boson! Get 2 2 mπ fπ V(θ) = Lm = mq qq¯ cosθ= cosθ −h i − h i − 4 θ is a physical parameter! Violates CP. NB: Minimum of V(θ) is at θ = 0. No use if θ is just a parameter. NB: V(θ) more complicated for mu = md. 6 15 26 CP-violation within QCD. Neutron edm dn < 3 10 e cm = · − · ⇒ 10 θ . 10− Strong CP problem. Fine tuning? Mechanism that ensures θ = 0 Peccei–Quinn: promote θ to a field. Simple version: two Englert–Brout–Higgs fields, Y1 = 1, Y2 = 1: − (d) ¯ (u) ¯ 2 2 L = y QLH1dR + y QLH2uR + Dµ H1 + Dµ H2 V(H1,H2) | | | | − Classical level: require global U(1)PQ symmetry (PQ symmetry) 5 i iγ θ/2 i iθ iθ q e q , H1 e H1 , H2 e H2 → → → Vev’s: H1 = v1/√2, H2 = v2/√2, break PQ symmetry spontaneously.h i h i 16 Parametrize iθ(x) iθ(x) H1 = e v1/√2 , H2 = e v2√2 If not for QCD, θ(x) would be a massless Nambu–Goldstone boson, axion. Kinetic term 1 1 1 2 ∂ θ 2 2 ∂ θ 2 2 ∂ θ 2 2v1( µ ) + 2v2( µ ) = 2 fPQ( µ ) i θ(x) Quark masses md,u = yd,uv1,2e h i. Turn on QCD: shift θ θ + const is NOT a symmetry. → Consequences θ(x) is such that V(θ) is at minimum = θ = 0 automatically.h i Strong CP problem solved.⇒ θ(x) gets a mass 17 1 2 2 1 2 Lθ = f (∂µθ) V(θ) , V(θ) mq qq¯ cosθ = mq qq¯ θ 2 PQ − ≃ − h i 2 h i Axion field θ(x) = a(x)/ fPQ: 2 2 12 m qq¯ mπ fπ 10 GeV m2 q m 6 10 6 eV a h2 i 4 2 = a = − ≃ fPQ ≃ fPQ ⇒ · · fPQ Interactions: Axion-photon-photon α α a(x) Caγγ θ εµνλρFµν Fλρ =Caγγ (~E H~ ) , Caγγ 1 roughly 16π · 2π fPQ · ∼ 18 Two Englert–Brout–Higgs fields are not enough 2 2 fPQ= v1 + v2 = 246 GeV too small: q (Weinberg–Wilczek) axion is too heavy (ma 15 keV), its couplings too large, ruled out experimentally.∼ Add heavy fields, make fPQ large. Dine–Fischler–Srednicki–Zhitnitsky (DFSZ); Kim–Shifman–Vainshtein–Zakharov (KSVZ) General picture: PQ field Φ(x) = ρ(x)eiθ(x), vev Φ = fPQ h i axion field a(x) = θ(x)/ fPQ m qq¯ ∂ Φ 2 ¯ θ 2 2 ∂ θ 2 ¯ θ 2 ∂ 2 q 2 ( µ ) mq qq ( ) + = fPQ( µ ) mq qq = ( µa) h2 ia − h i ··· − h i − fPQ Light axions, interact very weakly 19 To summarize Peccei–Quinn solution to strong CP problem predicts axion with mass 1012 GeV ma = 6 µeV · fPQ and aγγ interaction α a(x) Caγγ (~E H~ ) 2π fPQ · where Caγγ 1 is model-dependent, and fPQ is the only free ∼ parameter. Larger fPQ = smaller ma, weaker interactions. ⇒ 20 Why is this interesting for cosmology? Axion is practically stable: 5 α 2 m3 eV Γ a γγ C2 a τ 1017 yrs ( ) = aγγ 8π 4π 2 = a = → fPQ ⇒ ma Interacts very weakly = dark matter candidate ⇒ May never be in thermal equilibrium = cold dark matter if momenta are negligibly small. ⇒ Q. How can one arrange for negligibly small momenta for particles with sub-eV masses? A. Condensates (not the only option) 21 Axion production: misalignment Recall V(θ) mq qq¯ cosθ ≃ − h i Early Universe, high T: qq¯ = 0 = V(θ) = 0. h i ⇒ No preferred value of θ = Initial condinion θ0 anywhere between π and⇒π. − At QCD epoch (T 200 MeV) potential V(θ) builds up. θ starts to roll down. ∼ V(θ) V(θ) • • 0θ0 2π θ θ high T lower T 22 Estimating axion mass density Rolling down starts when ma(T) H(T): before that time scale of ∼ rolling m 1 is larger than the cosmological time scale H 1. a− ∼ − After initial rolling, θ oscillates about minimum θ = 0. Homogeneous oscillating field = condensate = collection of quanta with zero spatial momentum. Just what we need for cold dark matter! Estimate for present mass density: Energy density at beginning of rolling 2 2 2 2 2 V(θ0,T) m (T)a0 = m (T) f θ0 ≃ a a PQ Number density of quanta at that time 2 2 na(T) V(θ0,T)/ma(T) = ma(T) f θ0 ≃ PQ 23 2 Recall ma(T) H(T) = T /M = number-to-entropy ∼ Pl∗ ⇒ H(T) f 2 θ 2 f 2 θ 2 na # PQ 0 # PQ 0 = 3 = s g T √g MPlT ∗ ∗ with T = TQCD 200 MeV and # 1.