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, 1305.3392 GGI, Florence, June 2013
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
Leptogenesis and gravitinos: for thermal leptogenesis and typical superparticle masses, thermal production yields observed amount of DM, ✓ ◆ ✓ 100 GeV ◆ ⇣ m ˜ ⌘ 2 T R G h 2 = C g , C ⇠ 0 . 5 ; Ω ˜ 10 10 GeV m ˜ 1 TeV G Ω DM h 2 ⇠ 0 . 1 is natural value; but why T R ⇠ T L ? Starting point simple observation: heavy neutrino decay width ✓ M 1 ◆ 2 N 1 = ˜ m 1 ⇠ 10 3 GeV , M 1 ⇠ 10 10 GeV . Γ 0 m 1 ⇠ 0 . 01 eV , e 8 ⇡ v EW yields reheating temperature (for decaying gas of heavy neutrinos) q N 1 M P ⇠ 10 10 GeV , Γ 0 T R ⇠ 0 . 2 · wanted for gravitino DM. Intriguing hint or misleading coincidence? 2
II. Spontaneous B � L breaking and false vacuum decay Supersymmetric SM with right-handed neutrinos, W M = h u ij 10 i 10 j H u + h d i n c j H u + h n i n c i n c i 10 j H d + h ⌫ ij 5 ⇤ ij 5 ⇤ i S 1 , in SU (5) notation: 10 = ( q, u c , e c ) , 5 ⇤ = ( d c , l ) ; electroweak symmetry breaking, h H u,d i / v EW , and B � L breaking, p � � � v 2 W B � L = B � L � 2 S 1 S 2 , 2 Φ p h S 1 , 2 i = v B � 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
m ⌫ = p m 2 m 3 = 3 ⇥ 10 � 2 eV, Parameters of B � L breaking sector: m 1 = ( m † M 1 ⌧ M 2 , 3 ' m S , e D m D ) 11 /M 1 , v B � L . Spontaneous symmetry breaking: consider Abelian Higgs model in unitary gauge ( ! massive vector multiplet, no Wess-Zumino gauge!), S 1 , 2 = 1 V = Z + i 2 S 0 exp( ± iT ) , p 2 g ( T � T ⇤ ) . Inflaton field Φ : slow motion (quantum corrections), changes mass of ‘waterfall’ field S , rapid change after critical point where m S = 0 ; basic mechanism of hybrid inflation. p Shift around time-dependent background, s 0 = 2 ( � 0 + i ⌧ ) , � 0 ! 1 2 v ( t )+ � p x ) i 1 / 2 1 2 h � 0 2 ( t, ~ with v ( t ) = x ; masses of fluctuations: p ~ 4
� = 1 ⌧ = 1 m 2 2 � (3 v 2 ( t ) � v 2 m 2 2 � ( v 2 B � L + v 2 ( t )) , m 2 � = � v 2 ( t ) , B � L ) , m 2 = � v 2 ( t ) , m 2 Z = 8 g 2 v 2 ( t ) , M 2 i = ( h n i ) 2 v 2 ( t ) ; time-dependent masses of B � L Higgs, inflaton, vector boson, heavy neutrinos, all supermultiplets! 5
Constraints from cosmic strings and inflation: upper bound on string tension (Planck Collaboration ’13) Gµ < 3 . 2 ⇥ 10 � 7 , µ = 2 ⇡ B ( � ) v 2 B � L , with � = � / (8 g 2 ) and B ( � ) = 2 . 4 [ln(2 / � )] � 1 for � < 10 � 2 ; further constraint from CMB (cf. Nakayama et al ’10) , yields 3 ⇥ 10 15 GeV . v B � L . 7 ⇥ 10 15 GeV , p 10 � 4 . � . 10 � 1 . Final choice for range of parameters (analysis within FN flavour model): v B � L = 5 ⇥ 10 15 GeV , 10 � 5 eV e m 1 1 eV , 10 9 GeV M 1 3 ⇥ 10 12 GeV . (range of e m 1 : uncertainty of O (1) parameters) 6
Tachyonic Preheating Hybrid inflation ends at critical value Φ c of inflaton field Φ by rapid growth of fluctuations of B � L Higgs field S 0 (‘spinodal decomposition’): 10 φ (t) (Tanh) Bosons: Lattice < φ 2 (t)> 1/2 /v (Lattice) Tanh n B (x100) (Tanh) Fermions: Lattice 1 n B (x100) (Lattice) Tanh 1 0.1 Occupation number: n k < φ 2 (t)> 1/2 /v, n B (t) 0.01 0.1 0.001 0.0001 0.01 1e-05 1e-06 0.001 1e-07 5 10 15 20 25 30 0.1 1 time: mt k/m in addition, particles which couple to S 0 are produced by rapid increase of ‘waterfall field’ (Garcia-Bellido, Morales ’02) ; no coherent oscillations! 7
Decay of false vacuum produces long wave-length � -modes, true vacuum reached at time t PH (even faster decay with inflaton dynamics), ✓ 32 ⇡ 2 ◆ � 1 � h � 0 2 i = 2 v 2 B � L , t PH ' ln . � 2 m � � t = t PH Initial state: nonrelativistic gas of � -bosons, N 2 , 3 , ˜ N 2 , 3 , A , ˜ A , C (contained in superfield Z ), ... ; energy fractions ( ↵ = m X /m S , ⇢ 0 = � v 4 B � L / 4 ): ⇢ B / ⇢ 0 ' 2 ⇥ 10 � 3 g s � f ( ↵ , 1 . 3) , ⇢ F / ⇢ 0 ' 1 . 5 ⇥ 10 � 3 g s � f ( ↵ , 0 . 8) . Time evolution: rapid N 2 , 3 , ˜ N 2 , 3 , A , ˜ A , C decays, yields initial radiation, thermal N 1 ’s and gravitinos; � decays produce nonthermal N 1 ’s; N 1 decays produce most of radiation and baryon asymmetry; details of evolution described by Boltzmann equations. 8
Reheating Process Major work: solve network of Boltzmann equations for all (super)particles; treat nonthermal and thermal contributions di ff erently, 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 M 1 = 5 . 4 ⇥ 10 10 GeV , m 1 = 4 . 0 ⇥ 10 � 2 eV , e Gµ = 2 . 0 ⇥ 10 � 7 m e G = 100 GeV , m ˜ g = 1 TeV ; fixes (within FN flavour model) all other masses, CP asymmetries etc. Note: emergence of temperature plateau at intermediate times; final result: ⌘ B ' 3 . 7 ⇥ 10 � 9 ' ⌘ nt G h 2 ' 0 . 11 , B , Ω e i.e., dynamical realization of original conjecture. 9
Thermal and nonthermal number densities Inverse temperature M 1 ê T 10 - 1 10 0 10 1 10 2 10 3 10 50 R 10 45 s+y+f é nt nt + N N 1 1 10 40 B - L abs N H a L eq 2 N 1 é G 10 35 é th th + N N 1 1 10 30 é N 2,3 + N 2,3 f 10 25 i a RH a RH a RH 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Scale factor a Comoving number densites of thermal and nonthermla N 0 1 s ,..., B � L , gravitinos and radiation as functions of scale factor a . 10
Time evolution of temperature: intermediate plateau Inverse temperature M 1 ê T 10 - 1 10 0 10 1 10 2 10 3 10 12 10 11 T H a L @ GeV D 10 10 10 9 10 8 f i a RH a RH a RH 10 7 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Scale factor a Gravitino abundance can be understood from ‘standard formula’ and e ff ective ‘reheating temperature’ (determined by neutrino masses). 11
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, m LSP ⌧ m squark , slepton ⌧ m e G . LSP is typically ‘pure’ wino or higgsino (bino disfavoured, overproduction in thermal freeze-out), almost mass degenerate with chargino. Thermal w ) or higgsino ( e abundance of wino ( e h ) LSP significant for masses above 1 TeV , well approximated by (Arkani-Hamed et al ’06; Hisano et al ’07, Cirelli et al ’07) ✓ m e ◆ 2 w, e h h 2 = c e h Ω th , c e w = 0 . 014 , c e h = 0 . 10 , w, e w, e h e 1 TeV Heavy gravitinos ( 10 TeV . . . 10 3 TeV ) consistent with BBN, ⌧ e G ' 24 ⇥ 12
G ) 3 sec . Total higgsino/wino abundance (10 TeV /m e h h 2 = Ω h h 2 + Ω th h h 2 , e G Ω e w, e w, e w, e e e ⌘ ✓ T RH ( M 1 , e ◆ G h 2 ' 2 . 7 ⇥ 10 � 2 ⇣ m LSP LSP h 2 = m LSP m 1 ) e G Ω Ω e , 10 10 GeV m e 100 GeV G with ‘reheating temperature’ determined by neutino masses (takes reheating process into account), ✓ ◆ 1 / 4 ✓ ◆ 5 / 4 m 1 e M 1 T RH ' 1 . 3 ⇥ 10 10 GeV . 10 11 GeV 0 . 04 eV Requirement of LSP dark matter, i.e. Ω LSP h 2 = Ω DM h 2 ' 0 . 11 , yields upper bound on the reheating temperature, T RH < 4 . 2 ⇥ 10 10 GeV ; lower bound on T RH from successfull leptogenesis (depends on e m 1 ). 13
3000 obs é > W DM W w 10 11 obs W LSP > W DM D é 2500 w é @ TeV D m LSP @ GeV D T RH @ GeV D 2000 10 10 m é 1 @ eV D 100 ¥ m G 1500 10 0 obs é > W DM W h 10 9 10 - 2 1000 é 4 He h é G 10 - 4 500 10 - 5 10 - 4 10 - 3 10 - 2 10 - 1 10 0 10 1 10 2 10 3 é é @ TeV D m G m 1 @ eV D For each ‘reheating temperature’, i.e. pair ( M 1 , e m 1 ) , 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 e m 1 , ‘reheating temperature’ (right panel); more stringent for higgsino mass, since freeze-out contribution larger. E.g., m 1 = 0 . 05 eV implies h < m e ⇠ 900 GeV, m e G & 10 TeV. 14
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