Physics of the Big Bang 3

Home , Topological defect

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 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

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

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

−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

(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)

(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

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.

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)

¯ 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.

High temperature effective potential VT T>Tc

1 2 1 2 ¯2 1 ¯4 T=Tc VT = 2 (−|µ| + 24 λT )φ + 4! λφ

Equilibrium at T

2 2 I Critical temperature Tc = 24|µ |/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)

Effective potential with multiple ﬁelds: cubic term important

VT T=T γ 1 4 1 2 2 ¯ 2 ¯ 3 ¯ T>Tc ∆VT ' (T − T2 )|φ| − AT |φ| + λ|φ| 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|

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

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. by integrating out singlet 2 4 6 I VT (φ) ' c0 + c1(T )h + c2h + c3h + ··· I c2 < 0 gives 1st order transition at tree level.

I etc. etc. etc.

(12)Dorsch, Huber, No (2015) (13)Ashoorioon, Konstandin (2009) (14)Grojean, Servant, Wells (2005) (15)Huber et al (2006)

V T T

I Lowest energy path to equilibrium state via critical bubble R c w I Energy of critical bubble Eb I Nucleation rate per unit volume r 4 (high T ): Γ/V ' T exp(−Eb/T ) Critical bubble ﬁeld proﬁle

I Thermal ﬂuctuations produce bubbles at 4 Symmetric phase rate/volume Γ/V ' T exp(−Eb/T )

I Bubbles growth speed vb set by interaction with medium

I Bubble merger completes phase transition Higgs phase

I Thermal ﬂuctuations produce bubbles at 4 rate/volume Γ/V ' T exp(−Eb/T )

I Bubbles growth speed vb set by interaction with medium

I Bubble merger completes phase transition

Higgs phase

I Thermal ﬂuctuations produce bubbles at 4 rate/volume Γ/V ' T exp(−Eb/T )

I Bubbles growth speed vb set by interaction with medium

I Bubble merger completes phase transition

Higgs phase

V Z T T

γ 2 2 2 1 3 1 4 ∆V ' (T − T )φ − AT φ + λφ V ( φ m ) T 2 2 3 4 T V ( 0 ) T

δE[φ] φm φ |φ| Critical bubble φ (r) solves = 0 + I b δφ(x) 0

I Radius Rc

I Phase boundary surface energy Rc 3 w σ ' φ+/λ Free energy difference I r ∆V = VT (φ+) − VT (0)

V Z T T

γ 1 1 V ( φ ) 2 2 2 3 4 T m ∆VT ' (T − T2 )φ − AT φ + λφ 2 3 4 V ( 0 ) T φ φ I Estimate: m + |φ| 0 4πR3 E ' − c ∆V + 4πR2σ b 3 c R c w

I Critical bubble radius Rc ' σ/∆V 3 2 r I Critical bubble energy Eb ' σ/ ∆V

Sakharov conditions for baryogenesis:

I B violation:

I C and CP violation: Antimatter excess violates C and CP

I non-equilibrium: /B processes reduce B asymmetry in equilibrium

Sakharov conditions for baryogenesis:

I B violation: Electroweak theory has unstable topological defects – sphalerons (S)(16) Formation and decay ofS results in change in B + L of LH fermions(17)

I C and CP violation: C violation automatic in SM. CP violation needs more than CKM at high T

I non-equilibrium: Supercooling at 1st order phase transition?

(16)Klinkhamer, Manton (1984) (17)Kuzmin, Rubakov, Shaposhnikov (1985)

Mechanism:(18)

I CP-violation in bubble wall ﬁeld proﬁle

I CP-asymmetry in reﬂection of fermions

I Chiral asymmetry → (Sphalerons) → baryon asymmetry

v v L Lc R Rc v v

Signal: gravitational waves when bubbles collide(19)

(18)Cohen, Kaplan, Nelson 1991 (19)Witten 1984, Kosowsky, Turner, Watkins 1986

I Relevant approximation for GWs: classical scalar field, classical relativistic fluid µν µ ν µν 1 α I Tφ = ∂ φ∂ φ − g 2 ∂αφ∂ φ µν µ ν µν I Tfluid = [ + p] U U + g p (20) I Equations: ¨ 2 ∂V ˙ i I −φ + ∇ φ − ∂φ = ηW (φ + V ∂i φ) ˙ i ˙ i ∂V ˙ i 2 ˙ i 2 I E + ∂i (EV ) + P[W + ∂i (WV )] − ∂φ W (φ + V ∂i φ) = ηW (φ + V ∂i φ) . ˙ j ∂V ˙ j I Zi + ∂j (Zi V ) + ∂i P + ∂φ ∂i φ = −ηW (φ + V ∂j φ)∂i φ. i i I W – relativistic γ-factor; V – fluid 3-velocity, E – fluid energy density; Z – fluid momentum density. I η(φ) – coupling

(20)Enqvist et al 1992; Kurki-Suonio, Laine 1996, Hindmarsh, Rummukainen, Weir (2013)

Fluid energy density

−1 t = 400Tc

−1 t = 1200Tc

T N η=0.1 c, b=37 0.1

0.01 Metric perturbations hij from )

6 0.001 transverse-traceless part of EM c GT ( 0.0001 tensor Πij : k ln d / 1e-05 t /T =500 c 2 GW t /T ¨ ρ =1000 c hij − ∇ hij = 16πGΠij d 1e-06 t /T =1500 c t=2000/T 1e-07 c TT t=2500/T Πij = [( + p)Vi Vj + ∂i φ∂j φ] c 1e-08 0.01 0.1 k T ( c)

dρ (k) k 3 Z Gravitational wave power spectrum: GW = dΩ h˙ (t, k)h˙ ∗(t, k) d ln k 32πG ij ij

Space-based GW detector eLISA I e.g. strong EW transition with (launch 2034): eLISA conﬁgurations: best (r), worst (g) (Caprini et al. 2015)

I Total (k) sound (g) turbulence (r)

10-8

10-10 L f H GW

W -12

2 10 h

10-14

10-16 10-5 10-4 0.001 0.01 0.1 LISA Pathﬁnder launch 3rd Dec 2015 f@HzD

Abelian Higgs model: complex scalar ﬁeld φ(x), vector ﬁeld Aµ(x). 1 L = − F F µν + |Dφ|2 − V (φ), eff 4 µν eff 1 V (φ) ' V + m2 |φ2| + λ|φ|4 eff 0 eff 4 1 2 2 2 2 12 (λ + 3e )T − |µ | Thermal where meff(T ) = 1 2 2 . 2 κσ(t) − |µ | Vacuum VT T>Tc

T=Tc Potential energy function VT (φ) changes shape at T

Z 1 S = − d 4x p−g gµν D φ∗D φ + V (φ) + gµρgνσF F , µ ν 4e2 µν ρσ

Complex scalar ﬁeld φ(x, t), vector ﬁeld Aµ(x, t) Covariant derivative Dµ = ∂µ − iAµ. 1 2 2 2 Potential V (φ) = 2 λ(|φ| − v ) . Metric ds2 = a2(τ)(−dτ 2 + dx2) 1 τ: conformal time, ∝ t, t 2

Temporal gauge (A0 = 0) ﬁeld equations (index raised with Minkowski metric). a˙ φ¨ + 2 φ˙ − D2φ + λa2(|φ|2 − v 2)φ = 0, a 1 ∂µ F − ia2(φ∗D φ − D φ∗φ) = 0, e2 µν ν ν

(21) I Numerical solution of classical ﬁeld equations

I Initial conditions: φ(x) Gaussian random ﬁeld, correlation length small

I Initial cooling, then expansion damping

I φ(x) currents generates magnetic ﬁelds B

I B conﬁned to ﬂux tubes (Nielsen-Olesen strings) as φ(x) relaxes 2 I Strings have mass per unit length µ ' 2πφ0

(21)Parallel simulation framework LATﬁeld2 http://www.latﬁeld.org/

500 C C 20 11 vv ⇥ I Tµν (string) source for gravitational 400 perturbations 300 200 I Measure correlation functions of 100 energy-momentum tensor 0 C C 20 300 22 tt ⇥ φ4 U (k, τ, τ 0) = √ 0 C (kτ, kτ 0) 200 ab 0 ab ττ 100

[split into scalar-scalar (2x2); 0 101 102 103 C12 vector-vector; tensor-tensor] 300 k⌧

I Self-similar (“scaling”) correlators 200 → 100

Scale-free perturbations 0 101 102 103 k⌧

Strings normalised to ` = 10 (Bevis, Hindmarsh, Kunz, Urrestilla 2006)

I Multipole moments: R ∗ alm = dΩ∆T (n)Ylm(n)

I Angular power spectrum: l X 2 Cl = |alm| m=−l

I Anisotropy power: l(l + 1)Cl /(2π) Limits from Planck: f10 < 3.0% Gµ < 3.3 × 10−7 15 φ0 < 2.8 × 10 GeV

(22) (23) I Gravitational waves. Generic features:

I scale-invariant spectrum −3 4 I amplitude Ωgw(ω) ∼ 10 (v/MP) , (24) I Cosmic rays

I GeV-scale γ-rays (EGRET, FERMI/LAT) I UHECRs (Auger) I Neutrinos (Ice Cube)

I Decaying defects are sources of (25) I baryon number (26) I dark matter

(22)Krauss 1992, Fenu et al 2009 (23) 2 Figueroa, Hindmarsh, Urrestilla 2012. Cosmic string loops ∼ (v/MP) (Vilenkin 1981) (24)Bhattacharjee, Sigl 1999 (25)Bhattacharjee, Kibble, Turok 1984 (26)Jeannerot, Zhang, Brandenberger 1999

I There were phase transitions in the early Universe

I QCD phase transition affects production of weakly-interacting particles

I Electroweak transition can make baryon number and gravitational waves in SM extensions

I Phase transitions in extensions of the standard model can produce cosmic strings −7 I if formed, have Gµ < 3.3 × 10 (constrains on hybrid inﬂation models)

I Gravitational wave production from 1st order phase transition

I Promising sensitivity from eLISA I Higgs structure ↔ EW phase transition (27) I Collider signals and phase transition strength

I CMB polarisation, GW and particle production from cosmic strings

I Search for high scale broken gauge symmetries

(27)Dorsch, Huber, No (2015); Ashoorioon Konstandin (2009)

Mark Hindmarsh Physics of the Big Bang 3