Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook Physics of the Big Bang 3 Mark Hindmarsh1;2 1Department of Physics & Astronomy University of Sussex 2Department of Physics and Helsinki Institute of Physics Helsinki University Spåtind Conference 5. tammikuuta 2016 Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook Outline Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook Departures from equilbrium in the early Universe I “Freeze-out" (loss of chemical equilibrium) - dark matter, neutrinos I “Decoupling" (loss of kinetic equilibrium) - photons/CMB I Phase transitions: I 1st order: metastable states I 2nd order: critical slowing down I Cross-over: negligible departure from equilibrium Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook Phase transitions & cosmology Phase transitions happened in real time in early Universe: Thermal Changing T (t) Vacuum Changing field σ(t) I QCD phase transition I Thermal (First order: strangelets, axion balls, magnetic fields) I Electroweak phase transition (1) I Thermal (First order: electroweak baryogenesis ) (2) I Vacuum: cold electroweak baryogenesis I Grand Unified Theory & other high-scale phase transitions (3) I Thermal: topological defects (4) I Vacuum: hybrid inflation, topological defects, ... (1)Kuzmin, Rubakov, Shaposhnikov 1988 (2)Smit and Tranberg 2002-6; Smit, Tranberg & Hindmarsh 2007 (3)Kibble 1976; Zurek 1985, 1996; Hindmarsh & Rajantie 2000 (4)Copeland et al 1994; Kofman, Linde, Starobinsky 1996 Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook Degrees of freedom of SM: mostly coloured Mass g Mass g γ 0 2 g 0 16 νe ∼< 1 eV 2 u 3 MeV 12 νµ ∼< 1 eV 2 d 7MeV 12 ντ ∼< 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 GeV4 t 174 GeV 12 W 80 GeV6 Z 91 GeV3 h 125 GeV3 7 7 40 GeV: 8 18 + 2 8 60 + 16 68.5/84.25 7 7 0.4 GeV: 8 14 + 2 8 36 + 16 47.5/61.75 QCD interactions important Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook QCD phase diagram −10 (5) I ηB = nB=nγ = (6:15 ± 0:15) × 10 (WMAP7 + H0 + BAO) I Cross-over at low chemical potential 200 175 Quark-gluon plasma D 150 MeV @ 125 100 75 Hadron phase 2SC Temperature 50 25 CFL NQ 250 500 750 1000 1250 1500 1750 2000 Baryon chemical potential @MeVD Ruester et al hep-ph/0503184 (5)Komatsu et al 2010 Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook QCD equation of state (6) I Budapest-Marseille-Wuppertal lattice (physical quark masses) I Shown: pressure and trace anomaly I(T ) = ρ(T ) − 3p(T ) (with fit) I Can model with hadronic resonance gas (low T ) and dimensional reduction (high T ) (6)Borsányi et al. (2010) Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook QCD and cosmology (7) I WIMP density depends on eqn. of state at T ' 4(mX =100GeV) GeV 2 (8) I Planck: ∆ΩX h = 0:1198 ± 0:0015 - QCD effects can be a few % (9) I Production of sterile neutrinos . I Sterile neutrinos density depends on geff(T ) at T ∼ TQCD (10) I Production of gravitinos I Gravitinos produced by bremsstrahlung during scattering I Most scattering is by strongly coupled states (7)Hindmarsh, Philipsen 2005 (8)Ade et al et al 2015 (9)Dodelson, Widrow 1994; Asaka, Laine, Shaposhnikov 2006; Laine & Shröder 2012 (10)Weinberg 1982, Nanopoulos, Olive Srednicki 1983; Ellis, Kim, Nanopoulos 1984; Bolz, Brandenburg, Buchmuller 2002; Rychkov, Strumia 2007 Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook Free energy of an ideal gas I Free energy density f = ρ − Ts (also f = −p) I To find equilibrium state we minimise free energy 4 2 I Dimensions: f = T φ(m=T ) with φ(0) = −gπ =90. Pressure due to particles of mass m in equilibrium (zero chemical potential) η = ±1 (FD/BE)): Z 2 1 3 1 k 2 2 p = d k ; E = (k + m ) 2 eE=T + η 3E Free energy density (f = −kT ln Z=V ): Z 3 f = −ηT d k ln(1 + ηe−E=T ) Note f = −p by partial integration. Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook Free energy: exact formulae in high T expansion Bosons: 2 2 2 2 3 4 2 π m T (m ) 2 T m m = − 4 + − − fB T 2 ln 2 90 24 12π 64π abT 4 2 ` m X ` ζ(2` + 1) m − (−1) 5 (` + 1)! 4π2T 2 16π 2 ` Fermions: π2 7 m2T 2 m4 m2 = − 4 + + fF T 2 ln 2 90 8 48 64π af T 4 ` m X ` ζ(2` + 1) − `− 2 + (−1) (1 − 2 2 1)Γ(` + 1 ) m 5 (` + 1)! 2 4π2T 2 16π 2 ` 2 3 ab = 16π ln( 2 − 2γE ), af = ab=16, γE = 0:5772 ::: (Euler’s constant) Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook Effective potential for scalar field with gauge fields and fermions ¯ I scalars (MS(φ)), Let scalar field give masses to I vectors (MV (φ¯)) I (Dirac) fermions (MF (φ¯)) ¯ 1 2 ¯2 1 ¯4 VT (φ) = VT (0) + 2 µ φ + 4! λφ ! T 2 X X X + M2(φ¯) + 3 M2 (φ¯) + 2 M2 (φ¯) 24 S V F S V F ! T X 2 3 X 2 3 − (M (φ¯)) 2 + 3 (M (φ¯)) 2 + ··· 12π S V S V Neglect higher order terms where M2(φ)=T 2 1. Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook Symmetry restoration at high T High temperature effective potential VT T>Tc 1 2 1 2 ¯2 1 ¯4 T=Tc VT = 2 (−|µj + 24 λT )φ + 4! λφ Equilibrium at T<Tc 1 φ¯2 = 6(jµ2j − λT 2)/λ 24 2 2 2 = v (1 − T =Tc ) -v +v φ 2 2 I Critical temperature Tc = 24jµ j/λ I Above Tc , equilibrium state is φ¯ = 0 I φ ! −φ symmetry is restored (11) I Second-order phase transition discontinuity in specific heat, correlation length diverges ξ = 1=m(T ) (11)Kirzhnitz & Linde (1974), Dolan & Jackiw (1974) Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook First order phase transition Effective potential with multiple fields: cubic term important VT T=T γ 1 4 1 2 2 ¯ 2 ¯ 3 ¯ T>Tc ∆VT ' (T − T2 )jφj − AT jφj + λjφj 2 4! T=Tc T=T2 I Second minimum develops at T1 T=0 I Critical temperature Tc : free energies are equal. I System can supercool below Tc . First order transition I +v discontinuity in free energy |φ| Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook Phase transition in the Standard Model Standard Model phase diagram p Supercritical Crossover T(GeV) Symmetric phase No transition 2nd order 110 2nd order transition Liquid !st order transition Vapour Higgs phase 1st order 75 m (GeV) h T Kajantie et al 1996 Water phase diagram (sketch) SM is cross-over Mark Hindmarsh Physics of the Big Bang 3 Introduction: phase transitions in the early universe QCD phase transition Phase transitions in weakly coupled gauge theories Other phase transitions: topological defect formation Summary and outlook 1st order phase transitions in SM extensions I 2HDM (2 Higgs doublet model) 0 0 ± I Extra scalars (A , H , H ) increase strength of cubic term. (12) I Strong phase transition when mA0 & 400 GeV I Extra singlet scalars I Tree level first order phase transition (13) I Strong phase transition with SM-like phenomenology allowed 6 (14) I Effective field theory with h operator (15) I e.g.