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W. Buchmüller: Cosmological B-L Breaking: Baryon Asymmetry, Dark Matter and Gravitational Waves

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W. Buchmüller: Cosmological B-L Breaking: Baryon Asymmetry, Dark Matter and Gravitational Waves

  1. 1. Cosmological B−L Breaking: (Dark) Matter & Gravitational Waves Wilfried Buchm¨uller DESY, Hamburg with Valerie Domcke, Kai Schmitz & Kohei Kamada 1202.6679; 1203.0285, 1210.4105, 1304.XXX BW2013, Vrnjaˇcka Banja, Serbia, April 2013
  2. 2. I. B−L breaking, inflation & dark matter • Light neutrino masses can be explained by mixing with Majorana neutrinos with GUT scale masses from B−L breaking (seesaw mechanism) • Decays of heavy Majorana neutrinos natural source of baryon asymmetry (leptogenesis; thermal (Fukugita, Yanagida ’86) or nonthermal (Lazarides, Shafi ’91) ) • In supersymmetric models with spontaneous B−L breaking, natural connection with inflation (Copeland et al ’94; Dvali, Shafi, Schaefer ’94; ...) • LSP (gravitino, higgsino,...) natural candidate for dark matter • Consistent picture of inflation, baryogenesis and dark matter? • Possible direct test: gravitational waves 1
  3. 3. Leptogenesis and gravitinos: for thermal leptogenesis and typical superparticle masses, thermal production yields observed amount of DM, Ω ˜Gh2 = C TR 1010 GeV 100 GeV m ˜G m˜g 1 TeV 2 , C ∼ 0.5 ; ΩDMh2 ∼ 0.1 is natural value; but why TR ∼ TL ? Starting point simple observation: heavy neutrino decay width Γ0 N1 = ˜m1 8π M1 vEW 2 ∼ 103 GeV , m1 ∼ 0.01 eV , M1 ∼ 1010 GeV . yields reheating temperature (for decaying gas of heavy neutrinos) TR ∼ 0.2 · Γ0 N1 MP ∼ 1010 GeV , wanted for gravitino DM. Intriguing hint or misleading coincidence? 2
  4. 4. II. Spontaneous B−L breaking and false vacuum decay Supersymmetric SM with right-handed neutrinos, WM = hu ij10i10jHu + hd ij5∗ i 10jHd + hν ij5∗ i nc jHu + hn i nc inc iS1 , in SU(5) notation: 10 = (q, uc , ec ), 5∗ = (dc , l); electroweak symmetry breaking, Hu,d ∝ vEW , and B−L breaking, WB−L = √ λ 2 Φ v2 B−L − 2S1S2 , S1,2 = vB−L/ √ 2 yields heavy neutrino masses. Lagrangian is determined by low energy physics: quark, lepton, neutrino masses etc, but it contains all ingredients wanted in cosmology: inflation, leptogenesis, dark matter,..., all related! 3
  5. 5. Parameters of B−L breaking sector: mν = √ m2m3 = 3 × 10−2 eV, M1 M2,3 mS, m1 = (m† DmD)11/M1, vB−L. Spontaneous symmetry breaking: consider Abelian Higgs model in unitary gauge (→ massive vector multiplet, no Wess-Zumino gauge!), S1,2 = 1 √ 2 S exp(±iT) , V = Z + i 2g (T − T∗ ) . Inflaton field Φ: slow motion (quantum corrections), changes mass of ‘waterfall’ field S, rapid change after critical point where mS = 0; basic mechanism of hybrid inflation. Shift around time-dependent background, s = 1√ 2 (σ +iτ), σ → √ 2v(t)+σ with v(t) = 1√ 2 σ 2 (t, x) 1/2 x ; masses of fluctuations: 4
  6. 6. 0 1 2 3 1 0 1 0 0.5 Φ MP H MP V MP 4 S / vB-L / / m2 σ = 1 2 λ(3v2 (t) − v2 B−L) , m2 τ = 1 2 λ(v2 B−L + v2 (t)) , m2 φ = λv2 (t) , m2 ψ = λv2 (t) , m2 Z = 8g2 v2 (t) , M2 i = (hn i )2 v2 (t) ; time-dependent masses of B−L Higgs, inflaton, vector boson, heavy neutrinos, all supermultiplets! 5
  7. 7. Constraints from cosmic strings and inflation: upper bound on string tension (Planck Collaboration ’13) Gµ < 3.2 × 10−7 , µ = 2πB(β)v2 B−L , with β = λ/(8 g2 ) and B(β) = 2.4 [ln(2/β)] −1 for β < 10−2 ; further constraint from CMB (cf. Nakayama et al ’10), yields 3 × 1015 GeV vB−L 7 × 1015 GeV , 10−4 √ λ 10−1 . Final choice for range of parameters (analysis within FN flavour model): vB−L = 5 × 1015 GeV , 10−5 eV ≤ m1 ≤ 1 eV , 109 GeV ≤ M1 ≤ 3 × 1012 GeV . (range of m1: uncertainty of O(1) parameters) 6
  8. 8. Tachyonic Preheating Hybrid inflation ends at critical value Φc of inflaton field Φ by rapid growth of fluctuations of B−L Higgs field S (‘spinodal decomposition’): 0.001 0.01 0.1 1 5 10 15 20 25 30 <φ2(t)>1/2/v,nB(t) time: mt φ(t) (Tanh) <φ2 (t)>1/2 /v (Lattice) nB(x100) (Tanh) nB(x100) (Lattice) 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 0.1 1 Occupationnumber:nk k/m Bosons: Lattice Tanh Fermions: Lattice Tanh in addition, particles which couple to S are produced by rapid increase of ‘waterfall field’ (Garcia-Bellido, Morales ’02); no coherent oscillations! 7
  9. 9. Decay of false vacuum produces long wave-length σ-modes, true vacuum reached at time tPH (even faster decay with inflaton dynamics), σ 2 t=tPH = 2v2 B−L , tPH 1 2mσ ln 32π2 λ . Initial state: nonrelativistic gas of σ-bosons, N2,3, ˜N2,3, A, ˜A, C (contained in superfield Z), ... ; energy fractions (α = mX/mS, ρ0 = λv4 B−L/4): ρB/ρ0 2 × 10−3 gs λ f(α, 1.3) , ρF /ρ0 1.5 × 10−3 gs λ f(α, 0.8) . Time evolution: rapid N2,3, ˜N2,3, A, ˜A, C decays, yields initial radiation, thermal N1’s and gravitinos; σ decays produce nonthermal N1’s; N1 decays produce most of radiation and baryon asymmetry; details of evolution described by Boltzmann equations. 8
  10. 10. Reheating Process Major work: solve network of Boltzmann equations for all (super)particles; treat nonthermal and thermal contributions differently, varying equation of state; result: detailed time resolved description of reheating process, prediction of baryon asymmetry and gravitino density (possibly dark matter). Illustrative example for parameter choice M1 = 5.4 × 1010 GeV , m1 = 4.0 × 10−2 eV , meG = 100 GeV , m˜g = 1 TeV ; Gµ = 2.0 × 10−7 fixes (within FN flavour model) all other masses, CP asymmetries etc. Note: emergence of temperature plateau at intermediate times; final result: ηB 3.7 × 10−9 ηnt B , ΩeGh2 0.11 , i.e., dynamical realization of original conjecture. 9
  11. 11. Thermal and nonthermal number densities aRH i aRH aRH f Σ Ψ Φ N2,3 N2,3 N1 nt N1 nt N1 th N1 th 2N1 eq R B L G 100 101 102 103 104 105 106 107 108 1025 1030 1035 1040 1045 1050 10 1 100 101 102 103 Scale factor a absNa Inverse temperature M1 T Comoving number densites of thermal and nonthermla N1s,..., B−L, gravitinos and radiation as functions of scale factor a. 10
  12. 12. Time evolution of temperature: intermediate plateau aRH i aRH aRH f 100 101 102 103 104 105 106 107 108 107 108 109 1010 1011 1012 10 1 100 101 102 103 Scale factor a TaGeV Inverse temperature M1 T Gravitino abundance can be understood from ‘standard formula’ and effective ‘reheating temperature’ (determined by neutrino masses). 11
  13. 13. III. Gravitinos & Dark Matter Thermal production of gravitinos is origin of DM; depending on pattern of SUSY breaking, gravitino DM or higgsino/wino DM. Mass spectrum of superparticles motivated ‘large’ Higgs mass measured at the LHC, mLSP msquark,slepton meG . LSP is typically ‘pure’ wino or higgsino (bino disfavoured, overproduction in thermal freeze-out), almost mass degenerate with chargino. Thermal abundance of wino (w) or higgsino (h) LSP significant for masses above 1 TeV, well approximated by (Arkani-Hamed et al ’06; Hisano et al ’07, Cirelli et al ’07) Ωth ew,eh h2 = cew,eh mew,eh 1 TeV 2 , cew = 0.014 , ceh = 0.10 , Heavy gravitinos (10 TeV . . . 103 TeV) consistent with BBN, τeG 24 × 12
  14. 14. (10 TeV/meG)3 sec. Total higgsino/wino abundance Ωew,ehh2 = Ω eG ew,eh h2 + Ωth ew,eh h2 , Ω eG LSPh2 = mLSP meG ΩeGh2 2.7 × 10−2 mLSP 100 GeV TRH(M1, m1) 1010 GeV , with ‘reheating temperature’ determined by neutino masses (takes reheating process into account), TRH 1.3 × 1010 GeV m1 0.04 eV 1/4 M1 1011 GeV 5/4 . Requirement of LSP dark matter, i.e. ΩLSPh2 = ΩDMh2 0.11, yields upper bound on the reheating temperature, TRH < 4.2 × 1010 GeV; lower bound on TRH from successfull leptogenesis (depends on m1). 13
  15. 15. 4 He D LSP DM obs 10 4 10 2 100 m1 eV 101 102 103 109 1010 1011 mG TeV TRHGeV h DM obs w DM obs w h G 10 5 10 4 10 3 10 2 10 1 100 500 1000 1500 2000 2500 3000 m1 eV mLSPGeV 100mG TeV For each ‘reheating temperature’, i.e. pair (M1, m1), lower bound on gravitino mass (taken from Kawasaki et al ’08) (left panel). Requirement of higgsino/wino dark matter puts upper bound on LSP mass, dependent on m1, ‘reheating temperature’ (right panel); more stringent for higgsino mass, since freeze-out contribution larger. E.g., m1 = 0.05 eV implies meh <∼ 900 GeV, meG 10 TeV. 14
  16. 16. IV. Gravitational Waves Relic gravitational waves are window to very early universe; contributions from inflation, preheating and cosmic strings (Rubakov et al ’82; Garcia-Bellido and Figueroa ’07; Vilenkin ’81; Hindmarsh et al ’12); cosmological B−L breaking: prediction of GW spectrum with all contributions! Perturbations in flat FRW background, ds2 = a2 (τ)(ηµν + hµν)dxµ dxν , ¯hµν = hµν − 1 2 ηµνhρ ρ , determined by linearized Einstein equations, ¯hµν(x, τ) + 2 a a ¯hµν(x, τ) − 2 x ¯hµν(x, τ) = 16πGTµν(x, τ) . Spectrum of GW background, ΩGW (k, τ) = 1 ρc ∂ρGW (k, τ) ∂ ln k , 15
  17. 17. ∞ −∞ d ln k ∂ρGW (k, τ) ∂ ln k = 1 32πG ˙hij (x, τ) ˙hij (x, τ) ; use initial conditions for super-horizon modes from inflation, calculate correlation function of stress energy tensor. Contribution from inflation (primordial spectrum, transfer function; cf. Nakayama et al ’08): ΩGW(k, τ) = ∆2 t 12 k2 a2 0H2 0 T2 k (τ) = ∆2 t 12 Ωr gk ∗ g0 ∗ g0 ∗,s gk ∗,s 4/3    1 2 (keq/k) 2 , k0 k keq 1 , keq k kRH 1 2 C3 RH (kRH/k) 2 kRH k kPH 16
  18. 18. with boundary frequences (f = k/(2πa0)), f0 = 3.58 × 10−19 Hz h 0.70 , feq = 1.57 × 10−17 Hz Ωmh2 0.14 , fRH = 4.25 × 10−1 Hz TRH 107 GeV , fPH = 1.99 × 104 Hz λ 10−4 1/6 vB−L 5 × 1015 GeV 2/3 TRH 107 GeV 1/3 . Contribution from preheating (cf. Dufaux et al ’07): ΩGW (kPH) h2 cPH (RPHHPH)2 aPH aRH Ωrh2 gRH ∗ g0 ∗ g0 ∗,s gRH ∗,s 4/3 , 17
  19. 19. with characteristic scalar and vector scales R (s) PH −1 = (λ vB−L | ˙φc|)1/3 , R (v) PH −1 ∼ mZ = 2 √ 2 g vB−L . Contribution from cosmic strings (Abelian Higgs): ΩGW (k) 1 6π2 Fr vB−L MPl 4 Ωrh2    (keq/k)2 , k0 k keq 1, keq k kRH (kRH/k)2 , kRH k constant Fr recently determined in numerical simulation (cf. Figueroa, Hindmarsh, Urrestilla ’12). Result similar to contribution from inflation, but very different normalization! 18
  20. 20. inflation cosmic strings f0 feq fRH fPH fPH s fPH v preheating 10 20 10 15 10 10 10 5 100 105 1010 10 25 10 20 10 15 10 10 10 5 100 105 1010 1015 1020 1025 f Hz GWh2 k Mpc 1 19
  21. 21. Are macroscopically long cosmic strings Nambu-Goto strings? Energy loss of string network by ‘massive radiation’ or gravitational waves? GW radiation from NG strings, radiated by loops of length l(t, ti) = αti − ΓGµ(t − ti) ; rate for amplitude h and frequency f (cf. Kuroyanagi et al ’12), d2 R dzdh (f, h, z) 3 4 θ2 m (1 + z)(α + ΓGµ)h 1 αγ2t4(z) dV (z) dz Θ(1 − θm) ; can be approximately integrated analytically over z and h; results differs qualitatively from Abelian-Higgs prediction; five orders of magnitude difference in normalization! Truth somewhere inbetween? 20
  22. 22. Abelian-Higgs Strings vs. Nambu-Goto Strings Α 10 6 Α 10 12 Nambu Goto Abelian Higgs 10 20 10 15 10 10 10 5 100 105 1010 10 15 10 10 10 5 10 5 100 105 1010 1015 1020 1025 f Hz GWh2 k Mpc 1 21
  23. 23. Observational Prospects 78 3 4 5 6 1 2 9 inflation AH cosmic strings NG cosmic strings preheating 10 20 10 15 10 10 10 5 100 105 1010 10 25 10 20 10 15 10 10 10 5 100 10 5 100 105 1010 1015 1020 1025 f Hz GWh2 k Mpc 1 22
  24. 24. Summary and Outlook • Decay of false vacuum of unbroken B-L symmetry leads to consistent picture of inflation, baryogenesis and dark matter (everything from heavy neutrino decays) • Prediction: relations between neutrino and superparticle masses for gravitino or higgsino/wino dark matter • Possible direct test: detection of relic gravitational wave background, may provide determination of reheating temperature 23
  25. 25. (Non)thermal leptogenesis in M1 − ˜m1 plane 10 11 10 10 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 100 108.5 109 109.5 1010 1010.5 1011 1011.5 1012 1012.5 1013 m1 eV M1GeV ΗB m1,M1 vBL5.01015 GeV Inflation & strings ΗB nt ΗB th ΗB th ΗB obs ΗB nt ΗB obs ΗB ΗB obs Upper bound on M1 from inflation; lower bound from baryogenesis 24
  26. 26. Gravitino Dark Matter vs. leptogenesis 2 1010 3 1010 5 1010 7 1010 7 10101 1011 2 1011 10 5 10 4 10 3 10 2 10 1 100 5 10 20 50 100 200 500 m1 eV mG GeV M1 GeV such that G h2 0.11 vBL5.01015 GeVmg1TeV M1 GeV ΗB nt ΗB obs ΗB ΗB obs Gravitino mass range: 10 GeV <∼ meG <∼ 700 GeV; heavy neutrino mass range: 2 × 1010 GeV <∼ M1 <∼ 2 × 1011 GeV (more stringent than inflation) 25

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