Cosmological B−L Breaking:(Dark) Matter & Gravitational WavesWilfried Buchm¨ullerDESY, Hamburgwith Valerie Domcke, Kai Sch...
I. B−L breaking, inflation & dark matter• Light neutrino masses can be explained by mixing with Majorana neutrinoswith GUT ...
Leptogenesis and gravitinos: for thermal leptogenesis and typicalsuperparticle masses, thermal production yields observed ...
II. Spontaneous B−L breaking and false vacuum decaySupersymmetric SM with right-handed neutrinos,WM = huij10i10jHu + hdij5...
Parameters of B−L breaking sector: mν =√m2m3 = 3 × 10−2eV,M1 M2,3 mS, m1 = (m†DmD)11/M1, vB−L.Spontaneous symmetry breakin...
012310100.5Φ MPH MPV MP4S / vB-L//m2σ =12λ(3v2(t) − v2B−L) , m2τ =12λ(v2B−L + v2(t)) , m2φ = λv2(t) ,m2ψ = λv2(t) , m2Z = ...
Constraints from cosmic strings and inflation: upper bound on string tension(Planck Collaboration ’13)Gµ < 3.2 × 10−7, µ = ...
Tachyonic PreheatingHybrid inflation ends at critical value Φc of inflaton field Φ by rapid growthof fluctuations of B−L Higgs...
Decay of false vacuum produces long wave-length σ-modes, true vacuumreached at time tPH (even faster decay with inflaton dy...
Reheating ProcessMajor work: solve network of Boltzmann equations for all (super)particles;treat nonthermal and thermal co...
Thermal and nonthermal number densitiesaRHi aRH aRHfΣ Ψ ΦN2,3 N2,3N1ntN1ntN1thN1th2N1eqRB LG100101102103104105106107108102...
Time evolution of temperature: intermediate plateauaRHi aRH aRHf10010110210310410510610710810710810910101011101210 1100101...
III. Gravitinos & Dark MatterThermal production of gravitinos is origin of DM; depending on patternof SUSY breaking, gravi...
(10 TeV/meG)3sec. Total higgsino/wino abundanceΩew,ehh2= ΩeGew,ehh2+ Ωthew,ehh2,ΩeGLSPh2=mLSPmeGΩeGh22.7 × 10−2 mLSP100 Ge...
4HeDLSP DMobs10 410 2100m1 eV10110210310910101011mGTeVTRHGeVh DMobsw DMobswhG10 510 410 310 210 11005001000150020002500300...
IV. Gravitational WavesRelic gravitational waves are window to very early universe; contributionsfrom inflation, preheating...
∞−∞d ln k∂ρGW (k, τ)∂ ln k=132πG˙hij (x, τ) ˙hij(x, τ) ;use initial conditions for super-horizon modes from inflation, calc...
with boundary frequences (f = k/(2πa0)),f0 = 3.58 × 10−19Hzh0.70,feq = 1.57 × 10−17HzΩmh20.14,fRH = 4.25 × 10−1HzTRH107 Ge...
with characteristic scalar and vector scalesR(s)PH−1= (λ vB−L | ˙φc|)1/3, R(v)PH−1∼ mZ = 2√2 g vB−L .Contribution from cos...
inflationcosmic stringsf0feq fRH fPHfPHsfPHvpreheating10 20 10 15 10 10 10 5 100 105 101010 2510 2010 1510 1010 5 100 105 ...
Are macroscopically long cosmic strings Nambu-Goto strings? Energy loss ofstring network by ‘massive radiation’ or gravita...
Abelian-Higgs Strings vs. Nambu-Goto StringsΑ 10 6Α 10 12Nambu GotoAbelian Higgs10 20 10 15 10 10 10 5 100 105 101010 1510...
Observational Prospects783456129inflationAH cosmic stringsNG cosmic stringspreheating10 20 10 15 10 10 10 5 100 105 101010...
Summary and Outlook• Decay of false vacuum of unbroken B-L symmetry leads toconsistent picture of inflation, baryogenesis a...
(Non)thermal leptogenesis in M1 − ˜m1 plane10 1110 1010 910 810 710 610 510 410 310 210 1100108.5109109.510101010.51011101...
Gravitino Dark Matter vs. leptogenesis2 10103 10105 10107 10107 10101 10112 101110 510 410 310 210 11005102050100200500m1 ...
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W. Buchmüller: Cosmological B-L Breaking: Baryon Asymmetry, Dark Matter and Gravitational Waves

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Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia

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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 WavesWilfried Buchm¨ullerDESY, Hamburgwith Valerie Domcke, Kai Schmitz & Kohei Kamada1202.6679; 1203.0285, 1210.4105, 1304.XXXBW2013, Vrnjaˇcka Banja, Serbia, April 2013
  2. 2. I. B−L breaking, inflation & dark matter• Light neutrino masses can be explained by mixing with Majorana neutrinoswith 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, naturalconnection 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 waves1
  3. 3. Leptogenesis and gravitinos: for thermal leptogenesis and typicalsuperparticle masses, thermal production yields observed amount of DM,Ω ˜Gh2= CTR1010 GeV100 GeVm ˜Gm˜g1 TeV2, C ∼ 0.5 ;ΩDMh2∼ 0.1 is natural value; but why TR ∼ TL ?Starting point simple observation: heavy neutrino decay widthΓ0N1=˜m18πM1vEW2∼ 103GeV , m1 ∼ 0.01 eV , M1 ∼ 1010GeV .yields reheating temperature (for decaying gas of heavy neutrinos)TR ∼ 0.2 · Γ0N1MP ∼ 1010GeV ,wanted for gravitino DM. Intriguing hint or misleading coincidence?2
  4. 4. II. Spontaneous B−L breaking and false vacuum decaySupersymmetric SM with right-handed neutrinos,WM = huij10i10jHu + hdij5∗i 10jHd + hνij5∗i ncjHu + hni ncinciS1 ,in SU(5) notation: 10 = (q, uc, ec), 5∗= (dc, l); electroweak symmetrybreaking, Hu,d ∝ vEW , and B−L breaking,WB−L =√λ2Φ v2B−L − 2S1S2 ,S1,2 = vB−L/√2 yields heavy neutrino masses.Lagrangian is determined by low energy physics: quark, lepton, neutrinomasses 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 × 10−2eV,M1 M2,3 mS, m1 = (m†DmD)11/M1, vB−L.Spontaneous symmetry breaking: consider Abelian Higgs model in unitarygauge (→ massive vector multiplet, no Wess-Zumino gauge!),S1,2 =1√2S exp(±iT) , V = Z +i2g(T − T∗) .Inflaton field Φ: slow motion (quantum corrections), changes mass of‘waterfall’ field S, rapid change after critical point where mS = 0; basicmechanism of hybrid inflation.Shift around time-dependent background, s = 1√2(σ +iτ), σ →√2v(t)+σwith v(t) = 1√2σ 2(t, x)1/2x ; masses of fluctuations:4
  6. 6. 012310100.5Φ MPH MPV MP4S / vB-L//m2σ =12λ(3v2(t) − v2B−L) , m2τ =12λ(v2B−L + v2(t)) , m2φ = λv2(t) ,m2ψ = λv2(t) , m2Z = 8g2v2(t) , M2i = (hni )2v2(t) ;time-dependent masses of B−L Higgs, inflaton, vector boson, heavyneutrinos, 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)v2B−L ,with β = λ/(8 g2) and B(β) = 2.4 [ln(2/β)]−1for β < 10−2; furtherconstraint from CMB (cf. Nakayama et al ’10), yields3 × 1015GeV vB−L 7 × 1015GeV ,10−4√λ 10−1.Final choice for range of parameters (analysis within FN flavour model):vB−L = 5 × 1015GeV , 10−5eV ≤ m1 ≤ 1 eV ,109GeV ≤ M1 ≤ 3 × 1012GeV .(range of m1: uncertainty of O(1) parameters)6
  8. 8. Tachyonic PreheatingHybrid inflation ends at critical value Φc of inflaton field Φ by rapid growthof fluctuations of B−L Higgs field S (‘spinodal decomposition’):0.0010.010.115 10 15 20 25 30<φ2(t)>1/2/v,nB(t)time: mtφ(t) (Tanh)<φ2(t)>1/2/v (Lattice)nB(x100) (Tanh)nB(x100) (Lattice)1e-071e-061e-050.00010.0010.010.11100.1 1Occupationnumber:nkk/mBosons: LatticeTanhFermions: LatticeTanhin 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 vacuumreached at time tPH (even faster decay with inflaton dynamics),σ2t=tPH= 2v2B−L , tPH12mσln32π2λ.Initial state: nonrelativistic gas of σ-bosons, N2,3, ˜N2,3, A, ˜A, C (containedin superfield Z), ... ; energy fractions (α = mX/mS, ρ0 = λv4B−L/4):ρB/ρ0 2 × 10−3gs λ f(α, 1.3) , ρF /ρ0 1.5 × 10−3gs 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 decaysproduce most of radiation and baryon asymmetry; details of evolutiondescribed by Boltzmann equations.8
  10. 10. Reheating ProcessMajor work: solve network of Boltzmann equations for all (super)particles;treat nonthermal and thermal contributions differently, varying equationof state; result: detailed time resolved description of reheating process,prediction of baryon asymmetry and gravitino density (possibly dark matter).Illustrative example for parameter choiceM1 = 5.4 × 1010GeV , m1 = 4.0 × 10−2eV ,meG = 100 GeV , m˜g = 1 TeV ; Gµ = 2.0 × 10−7fixes (within FN flavour model) all other masses, CP asymmetries etc.Note: emergence of temperature plateau at intermediate times; final result:ηB 3.7 × 10−9ηntB , ΩeGh20.11 ,i.e., dynamical realization of original conjecture.9
  11. 11. Thermal and nonthermal number densitiesaRHi aRH aRHfΣ Ψ ΦN2,3 N2,3N1ntN1ntN1thN1th2N1eqRB LG10010110210310410510610710810251030103510401045105010 1100101102103Scale factor aabsNaInverse temperature M1 TComoving 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 plateauaRHi aRH aRHf10010110210310410510610710810710810910101011101210 1100101102103Scale factor aTaGeVInverse temperature M1 TGravitino abundance can be understood from ‘standard formula’ andeffective ‘reheating temperature’ (determined by neutrino masses).11
  13. 13. III. Gravitinos & Dark MatterThermal production of gravitinos is origin of DM; depending on patternof SUSY breaking, gravitino DM or higgsino/wino DM. Mass spectrum ofsuperparticles motivated ‘large’ Higgs mass measured at the LHC,mLSP msquark,slepton meG .LSP is typically ‘pure’ wino or higgsino (bino disfavoured, overproductionin thermal freeze-out), almost mass degenerate with chargino. Thermalabundance of wino (w) or higgsino (h) LSP significant for masses above1 TeV, well approximated by (Arkani-Hamed et al ’06; Hisano et al ’07, Cirelli et al ’07)Ωthew,ehh2= cew,ehmew,eh1 TeV2, cew = 0.014 , ceh = 0.10 ,Heavy gravitinos (10 TeV . . . 103TeV) consistent with BBN, τeG 24 ×12
  14. 14. (10 TeV/meG)3sec. Total higgsino/wino abundanceΩew,ehh2= ΩeGew,ehh2+ Ωthew,ehh2,ΩeGLSPh2=mLSPmeGΩeGh22.7 × 10−2 mLSP100 GeVTRH(M1, m1)1010 GeV,with ‘reheating temperature’ determined by neutino masses (takes reheatingprocess into account),TRH 1.3 × 1010GeVm10.04 eV1/4M11011 GeV5/4.Requirement of LSP dark matter, i.e. ΩLSPh2= ΩDMh20.11, yieldsupper bound on the reheating temperature, TRH < 4.2 × 1010GeV; lowerbound on TRH from successfull leptogenesis (depends on m1).13
  15. 15. 4HeDLSP DMobs10 410 2100m1 eV10110210310910101011mGTeVTRHGeVh DMobsw DMobswhG10 510 410 310 210 110050010001500200025003000m1 eVmLSPGeV100mGTeVFor each ‘reheating temperature’, i.e. pair (M1, m1), lower bound ongravitino mass (taken from Kawasaki et al ’08) (left panel). Requirement ofhiggsino/wino dark matter puts upper bound on LSP mass, dependenton m1, ‘reheating temperature’ (right panel); more stringent for higgsinomass, since freeze-out contribution larger. E.g., m1 = 0.05 eV impliesmeh<∼ 900 GeV, meG 10 TeV.14
  16. 16. IV. Gravitational WavesRelic gravitational waves are window to very early universe; contributionsfrom inflation, preheating and cosmic strings (Rubakov et al ’82; Garcia-Bellido andFigueroa ’07; Vilenkin ’81; Hindmarsh et al ’12); cosmological B−L breaking: predictionof GW spectrum with all contributions!Perturbations in flat FRW background,ds2= a2(τ)(ηµν + hµν)dxµdxν, ¯hµν = hµν −12ηµνhρρ ,determined by linearized Einstein equations,¯hµν(x, τ) + 2aa¯hµν(x, τ) − 2x¯hµν(x, τ) = 16πGTµν(x, τ) .Spectrum of GW background,ΩGW (k, τ) =1ρc∂ρGW (k, τ)∂ ln k,15
  17. 17. ∞−∞d ln k∂ρGW (k, τ)∂ ln k=132πG˙hij (x, τ) ˙hij(x, τ) ;use initial conditions for super-horizon modes from inflation, calculatecorrelation function of stress energy tensor.Contribution from inflation (primordial spectrum, transfer function; cf.Nakayama et al ’08):ΩGW(k, τ) =∆2t12k2a20H20T2k (τ)=∆2t12Ωrgk∗g0∗g0∗,sgk∗,s4/312 (keq/k)2, k0 k keq1 , keq k kRH12 C3RH (kRH/k)2kRH k kPH16
  18. 18. with boundary frequences (f = k/(2πa0)),f0 = 3.58 × 10−19Hzh0.70,feq = 1.57 × 10−17HzΩmh20.14,fRH = 4.25 × 10−1HzTRH107 GeV,fPH = 1.99 × 104Hzλ10−41/6vB−L5 × 1015 GeV2/3TRH107 GeV1/3.Contribution from preheating (cf. Dufaux et al ’07):ΩGW (kPH) h2cPH (RPHHPH)2 aPHaRHΩrh2 gRH∗g0∗g0∗,sgRH∗,s4/3,17
  19. 19. with characteristic scalar and vector scalesR(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)16π2Fr vB−LMPl4Ωrh2(keq/k)2, k0 k keq1, keq k kRH(kRH/k)2, kRH kconstant Frrecently determined in numerical simulation (cf. Figueroa, Hindmarsh,Urrestilla ’12). Result similar to contribution from inflation, but very differentnormalization!18
  20. 20. inflationcosmic stringsf0feq fRH fPHfPHsfPHvpreheating10 20 10 15 10 10 10 5 100 105 101010 2510 2010 1510 1010 5 100 105 1010 1015 1020 1025f HzGWh2k Mpc 119
  21. 21. Are macroscopically long cosmic strings Nambu-Goto strings? Energy loss ofstring network by ‘massive radiation’ or gravitational waves? GW radiationfrom NG strings, radiated by loops of lengthl(t, ti) = αti − ΓGµ(t − ti) ;rate for amplitude h and frequency f (cf. Kuroyanagi et al ’12),d2Rdzdh(f, h, z)34θ2m(1 + z)(α + ΓGµ)h1αγ2t4(z)dV (z)dzΘ(1 − θm) ;can be approximately integrated analytically over z and h; resultsdiffers qualitatively from Abelian-Higgs prediction; five orders of magnitudedifference in normalization! Truth somewhere inbetween?20
  22. 22. Abelian-Higgs Strings vs. Nambu-Goto StringsΑ 10 6Α 10 12Nambu GotoAbelian Higgs10 20 10 15 10 10 10 5 100 105 101010 1510 1010 510 5 100 105 1010 1015 1020 1025f HzGWh2k Mpc 121
  23. 23. Observational Prospects783456129inflationAH cosmic stringsNG cosmic stringspreheating10 20 10 15 10 10 10 5 100 105 101010 2510 2010 1510 1010 510010 5 100 105 1010 1015 1020 1025f HzGWh2k Mpc 122
  24. 24. Summary and Outlook• Decay of false vacuum of unbroken B-L symmetry leads toconsistent picture of inflation, baryogenesis and dark matter(everything from heavy neutrino decays)• Prediction: relations between neutrino and superparticlemasses for gravitino or higgsino/wino dark matter• Possible direct test: detection of relic gravitationalwave background, may provide determination of reheatingtemperature23
  25. 25. (Non)thermal leptogenesis in M1 − ˜m1 plane10 1110 1010 910 810 710 610 510 410 310 210 1100108.5109109.510101010.510111011.510121012.51013m1 eVM1GeVΗB m1,M1vBL5.01015GeVInflation& stringsΗBntΗBthΗBthΗBobsΗBntΗBobsΗB ΗBobsUpper bound on M1 from inflation; lower bound from baryogenesis24
  26. 26. Gravitino Dark Matter vs. leptogenesis2 10103 10105 10107 10107 10101 10112 101110 510 410 310 210 11005102050100200500m1 eVmGGeVM1 GeV such that G h20.11vBL5.01015GeVmg1TeVM1 GeVΗBntΗBobsΗB ΗBobsGravitino mass range: 10 GeV <∼ meG<∼ 700 GeV; heavy neutrino massrange: 2 × 1010GeV <∼ M1 <∼ 2 × 1011GeV (more stringent than inflation)25

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