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Evolution of Scalar Fields in the Early Universe

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Evolution of Scalar Fields in the Early Universe The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Evolution of Scalar Fields in the Early Universe Louis Yang Department of Physics and Astronomy University of California, Los Angeles PACIFIC 2015 September 17th, 2015 Advisor: Alexander Kusenko Collaborator: Lauren Pearce Evolution of Scalar Fields in the Early Universe (slide 1) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs The Motivation The recent discovery of the Higgs boson with mass Mh = 125:7 0:4 GeV ± [Particle Data Group 2014] V (φ) 1 λ (φ) φ4 for φ 100 GeV ≈ 4 eff Very small or negative λeff at high scale from RGE a meta-stable electroweak vacuum ) a shallow potential at high scale ) During inflation, the scalar field with a shallow [Dario Buttazzo et al. JHEP 1312 potential can obtain a large vacuum (2013) 089] expectation value (VEV). Post-inflationary Higgs field relaxation possibility for Leptogenesis ) Evolution of Scalar Fields in the Early Universe (slide 2) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Outline 1 Quantum Fluctuations in the Inflationary Universe 2 Classical Motion of Scalar Fields 3 Possible New Physics 4 Issue with Isocurvature Perturbations Evolution of Scalar Fields in the Early Universe (slide 3) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Quantum Fluctuations in the Inflationary Universe Evolution of Scalar Fields in the Early Universe (slide 4) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Quantum fluctuations in the inflationary universe During inflation, scalar fields can obtain a large VEV through quantum fluctuations. In de Sitter space, the quantum fluctuations of scalar fields are constantly pulled to above the horizon size. Long-wave quantum fluctuations are characterized by 1 long correlation length l 2 large occupation number nk for low k => behave like (quasi) classical field. ϕ l ∆ 0 x [Figure from A. Linde - arXiv: 0503203] Evolution of Scalar Fields in the Early Universe (slide 5) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Quantum fluctuations in the inflationary universe The VEV of the field can be computed through the dispersion of p 2 the fluctuation φ0 = ∆ = φ h i In a pure de Sitter spacetime, a scalar field with mass m can obtain a large VEV 3H4 φ2 = for m2 H2: 8π2m2 [T. Bunch and P. Davies, Proc. Roy. Soc. Lond. A360, 117 (1978)] In the inflationary universe, the exponential expansion period exists for a finite time t 2 Z H 3 3 2 2 H d k H H φ 3 3 = 2 t 2 N ≈ 2 (2π) He−Ht k 4π ' 4π for m2 = 0 or m2 H2 with t 3H=m2:N Ht is the number . ' of e-folds. [A. Linde, Phys. Lett. B116, 335 (1982)] Evolution of Scalar Fields in the Early Universe (slide 6) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Hawking-Moss tunneling Hawking & Moss (1982) One can also understand the fluctuation as both the scalar field φ(x) and the metric gµν (x) experience quantum jumps. The Hawking-Moss instanton 3m4 Γ(φi ! φf ) SE (φi)−SE (φf ) pl = Ae ; where SE (φ) = − V 8V (φ) is the Euclidean action and A is some O(m4) prefactor. The entire process can then be viewed as the fields are underdoing Brownian motion and can be described by diffusion equation. V(ϕ) Quantum Jump ϕf ϕ ϕi Evolution of Scalar Fields in the Early Universe (slide 7) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Hawking-Moss tunneling Hawking & Moss (1982) One can also understand the fluctuation as both the scalar field φ(x) and the metric gµν (x) experience quantum jumps. The Hawking-Moss instanton 3m4 Γ(φi ! φf ) SE (φi)−SE (φf ) pl = Ae ; where SE (φ) = − V 8V (φ) is the Euclidean action and A is some O(m4) prefactor. The entire process can then be viewed as the fields are underdoing Brownian motion and can be described by diffusion equation. V(ϕ) Brownian motion ϕf ϕ ϕi Evolution of Scalar Fields in the Early Universe (slide 7) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Stochastic approach & Hawking-Moss tunneling Pc (φ, t): the probability distribution of finding φ at time t Diffussion equation 3 @Pc @jc @ H Pc Pc dV = − where − jc = + @t @φ @φ 8π2 3H dφ [A. A. Starobinsky (1982); A. Vilenkin (1982)] In equilibrium @Pc=@t = 0, jc = 0. One obtain the distribution SE (φmin)−SE (φ) Pc (φ) = e " 4 # −3mpl ∆V (φ) ≈ exp 2 8 V (φmin) for ∆V = V (φ) − V (φ ) V (φ ). min min The variance of the The fluctuation is not suppressed if fluctuation is R 2 2 2 φ Pc(φ)dφ 8V (φmin) φ = R ∆V (φ) < 4 Pc(φ)dφ 3mpl Evolution of Scalar Fields in the Early Universe (slide 8) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Quantum fluctuation of the Higgs field Example: the Higgs field φ on the inflationary background (inflaton I). 1 V (φ, I) = V (φ) + V (I) + ::: ≈ λ φ4 + Λ4 + ::: H I 4 eff I The quantum transition of the Higgs field from 0 to φ is not suppressed if 2 4 1 4 8 ΛI 4 −1=4 λeffφ < ∼ HI ) jφj < 0:62λeff HI 4 3 mpl Even though hφi = 0 due to the even potential, the variance of the fluctuation of φ is not zero. p 2 ∼ −1=4 φ0 = hφ i = 0:36λeff HI Generally, during inflation, we expect the scalar field to obtain a large VEV φ0 such that 4 VH (φ0) H ∼ I Evolution of Scalar Fields in the Early Universe (slide 9) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Classical Motion of Scalar Fields Evolution of Scalar Fields in the Early Universe (slide 10) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Slow rolling during inflation Scalar field in an expanding universe @V φ¨ + 3Hφ_ + Γ φ_ + = 0 φ @φ During inflation, the scalar field can be in slow-roll. @V φ¨ and φ_2 V @φ The slow-roll conditions are 2 p 2 @ V (φ, I) 2 V (φ, I) @V (φ, I) 9H 2 = meff (φ) and 48π : @φ mpl @φ The first condition can be understood as the time scale for rolling down s !−1 @2V τ ∼ m−1 = H−1: eff @φ2 As long as meff (φ) H, there is insufficient time for the scalar field to roll down. Evolution of Scalar Fields in the Early Universe (slide 11) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Slow rolling of the Higgs field 1 4 For 4 λφ or the Higgs potential, the slow-roll conditions are 1=6 −1=2 27 −1=3 21=3 jφj 3λ HI and jφj λ mplH : eff 4π eff I The conditions for all the quantum fluctuations to be unable to roll are: 2 5 mpl λeff 4800 and λeff 3 × 10 ; ΛI which are easily satisfied when ΛI < mpl. In other words, during inflation, the Higgs field can jump quantum mechanically but cannot roll down classically. ) a large Higgs VEV is developed. V(ϕ) Quantum Jump Roll Down Classically ϕ ϕmin Evolution of Scalar Fields in the Early Universe (slide 12) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Brief summary Quantum fluctuation Brings the field to a VEV φ0 such that 4 Vφ (φ0) H ∼ Slow rolling The field won’t roll down if m2 H2 eff V(ϕ) Quantum Jump Roll Down Classically ϕ ϕmin Evolution of Scalar Fields in the Early Universe (slide 13) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Relaxation of the Higgs field after inflation As inflation ends, the inflaton enters the Inflaton coherent oscillations regime, H < meff (φ0). V(I) The Higgs field is no longer in slow-roll. Slow-Roll The Higgs then rolls down and oscillates around φ = 0 with decreasing amplitude −1 Coherent within τroll H . Oscillations ∼ ΛI f t f0 1.0 16 H L LI = 10 GeV I 3 GI = 10 GeV 12 0.8 Tmax = 6.4 ´ 10 GeV leff = 0.003 f = ´ 13 f 0 3.7 10 GeV log r 0.6 13 HI = 2.4 ´ 10 GeV H L rI 0.4 Tmax Radiation- Dominated TRH 0.2 T rR Coherent Oscillations (matter like) log t 4 l f0t 1 10 100 1000 10 t = 1 G End of Inflation t ' I H L End of Inflation at t = 0 Evolution of Scalar Fields in the Early Universe (slide 14) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Relaxation of the Higgs field after inflation During the oscillation of the Higgs field, the Higgs condensate can decay into several product particles: Non-perturbative decay: W and Z bonsons. 4 300 200 3 L t 100 , 0 2 = È k 0 H = T 0 k n È W 0 log f 1 -100 -200 0 -300 -1 0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500 f0 t f0 t 15 9 ΛI = 10 GeV and ΓI = 10 GeV for IC-1 Perturbative decay (thermalization): top quark. Those decay channels do affect the oscillation of the Higgs field but they becomes important only after several oscillations. Evolution of Scalar Fields in the Early Universe (slide 15) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Possible New Physics The relaxation from such large VEV opens a great channel for many interesting physics including matter-antimatter asymmetry (Leptogenesis or Baryogenesis). Sakharov conditions: 1 C and CP violations Time-dependent background Higgs field + ..
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