Evolution of Scalar Fields in the Early Universe

The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs

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 ⇒ |φ| < 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 √ 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 |φ| 3λ HI and |φ| 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 Φ t Φ0

1.0 16 H L LI = 10 GeV I 3 GI = 10 GeV 12 0.8 Tmax = 6.4 ´ 10 GeV

Λeff = 0.003 Φ = ´ 13 Φ 0 3.7 10 GeV log Ρ 0.6 13 HI = 2.4 ´ 10 GeV H L ΡI

0.4 Tmax Radiation- Dominated

TRH 0.2 T ΡR Coherent Oscillations (matter like) log t 4 Λ Φ0t 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

Τ 100 , 0 2 = È k 0 H = T 0 k n È W 0 log Φ 1 -100

-200 0

-300 -1 0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500

Φ0 Τ Φ0 Τ

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 + ... 2 Out of thermal equilibrium Roll down of the Higgs ﬁeld ← 3 Lepton/Baryon Next talk by Lauren Pearce number violations ← 2 One possibility is to have the lepton asymmetry L ∂0 φ ∝ A. Kusenko, L. Pearce, L. Yang, Phys. Rev. Lett. 114 (2015) 6, 061302 L. Pearce, L. Yang, A. Kusenko, M. Peloso, Phys. Rev. D 92 (2015) 2, 023509 L. Yang, L. Pearce, A. Kusenko, Phys. Rev. D 92 (2015) 043506 Similar idea for axion A. Kusenko, K. Schmitz, and T. T. Yanagida, Phys. Rev. Lett. 115 (2015) 011302

Evolution of Scalar Fields in the Early Universe (slide 16) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs

Issue with Isocurvature Perturbations

Evolution of Scalar Fields in the Early Universe (slide 17) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Isocurvature perturbations

One issue for applying to Leptogenesis p 2 φ0 = φ is the average over several h i Hubble volumes.

Each Hubble volume has different initial φ0 value. When inﬂation end, each patch of the observable universe began with different value of φ0. 2 If L ∂0 φ Different asymmetry in ∝ ⇒ each Hubble volume Large isocurvature perturbations, ⇒ which are constrainted by current CMB [Figure from Lauren Pearce] observation.

Evolution of Scalar Fields in the Early Universe (slide 18) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Isocurvature perturbations

One issue for applying to Leptogenesis p 2 φ0 = φ is the average over several h i Hubble volumes.

Evolution of Scalar Fields in the Early Universe (slide 18) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Isocurvature perturbations

One issue for applying to Leptogenesis p 2 φ0 = φ is the average over several h i Hubble volumes.

Each Hubble volume has different initial φ0 value. When inﬂation end, each patch of the observable universe began with different value of φ0. φ0′′ 2 φ0 φ0′ If L ∂0 φ Different asymmetry in ∝ ⇒ each Hubble volume Large isocurvature perturbations, ⇒ which are constrainted by current CMB [Figure from Lauren Pearce] observation.

Evolution of Scalar Fields in the Early Universe (slide 18) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Isocurvature perturbations

One issue for applying to Leptogenesis p 2 φ0 = φ is the average over several h i Hubble volumes.

Evolution of Scalar Fields in the Early Universe (slide 18) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Solutions to the isocurvature perturbation issue

Solutions:

1 IC-1: Second Minimum at Large VEVs V(ϕ) (φ vEW ) E.g. φ10 = lift 6 Second Min. L Λlift ϕ

2 IC-2: Inﬂaton-Higgs coupling V(ϕ) E.g.

2n 1 I 2 ΦI = φ Very Steep Potential L −2 M 2n−2 due to Inflaton

Evolution of Scalar Fields in the Early Universe (slide 19) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-1: Second minimum at large VEV

Motivations: 1 At large VEVs, Higgs potential is sensitive to higher-dimensional operators.

φ10 lift = 6 L Λlift 2 There seems to be a planckian minimum below our electroweak (EW) vacuum. Our EW vacuum is not stable. 3 A higher-dimensional operator can lift the possible planckian minimum and stablize our EW vacuum. The second minimum becomes metastable and higher than the EW vacuum.

Evolution of Scalar Fields in the Early Universe (slide 20) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-1: Second minimum at large VEV

The scenario: V(ϕ) Early stage of inflation 1 Large VEV at early stage of inflation 퐻4 2 The initial Higgs VEV is trapped in this second minimum (quasi-stable vacuum) at the end of inflation. Second Min. 3 Reheating destablize the quasi-stable vacuum. ϕ 4 Higgs field rolls down from the second minimum.

Evolution of Scalar Fields in the Early Universe (slide 21) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-1: Second minimum at large VEV

The scenario: V(ϕ) 1 Large VEV at early stage of inflation Trapped 2 The initial Higgs VEV is trapped in this second minimum (quasi-stable vacuum) at the end of inflation. 퐻4 Second Min. 3 Reheating destablize the quasi-stable vacuum. ϕ 4 Higgs field rolls down from the second minimum.

Evolution of Scalar Fields in the Early Universe (slide 21) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-1: Second minimum at large VEV

The scenario: V(ϕ) Reheating 1 Large VEV at early stage of inflation 2 The initial Higgs VEV is trapped in this second minimum (quasi-stable vacuum) at the end of inflation. 3 Reheating destablize the quasi-stable Thermal correction vacuum. ϕ 4 Higgs field rolls down from the second minimum.

Evolution of Scalar Fields in the Early Universe (slide 21) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-1: Second minimum at large VEV

The scenario: V(ϕ) Reheating 1 Large VEV at early stage of inflation 2 The initial Higgs VEV is trapped in this second minimum (quasi-stable Higgs VEV vacuum) at the end of inflation. Rolls Down 3 Reheating destablize the quasi-stable vacuum. ϕ 4 Higgs field rolls down from the second minimum.

Evolution of Scalar Fields in the Early Universe (slide 21) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-1: Second minimum at large VEV

1.0 < IC1 > 15 ΛI = 10 GeV 9 ΓI = 10 GeV 0.5 15 ϕ0 = 10 GeV

0 T(t) ϕ/ 0.0

-0.5

0 1000 2000 3000 4000

ϕ0 t

Evolution of Scalar Fields in the Early Universe (slide 22) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-2: Inﬂaton-Higgs coupling

Introduce coupling between the Higgs and inﬂaton ﬁeld. E.g. 2n 1 I 2 ΦI = φ . L −2 M 2n−2 Motivations: This could be obtained by integrating out heavy states in loops. Induces an large effective mass

m ( I ) = I n /M n−1 eff,φ h i h i for the Higgs field when I is large. h i If m ( I ) H in the early stage of inflation, the slow eff,φ h i roll condition is not satisfied.

Evolution of Scalar Fields in the Early Universe (slide 23) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-2: Inﬂaton-Higgs coupling

1 In the early stage of inflation, I is h i large. Higgs potential is steep. Slow-roll condition is not satisfied. The Higgs VEV stay at φ = 0 . V(ϕ) Early stage of inflation 2 At the last Nlast e-folds of inflation, 퐻4 I , m ( I ) < HI , Higgs VEV h i ↓ eff,φ h i starts to develop. Quantum jumps 3 At the end of inflation, the Higgs field Rolls down has obtained a VEV classically ϕ 휙2 ~0 p 2 HI p φ0 = φ = N . h i 2π last

4 The Higgs VEV then rolls down from φ0.

Evolution of Scalar Fields in the Early Universe (slide 24) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-2: Inﬂaton-Higgs coupling

1 In the early stage of inflation, I is h i large. Higgs potential is steep. Slow-roll condition is not satisfied. The Higgs VEV stay at φ = 0 . V(ϕ) Last N e-folds of inflation 2 At the last Nlast e-folds of inflation, 4 I , m ( I ) < HI , Higgs VEV 퐻 h i ↓ eff,φ h i starts to develop. 3 At the end of inflation, the Higgs field has obtained a VEV ϕ 휙2 starts to grow p 2 HI p φ0 = φ = N . h i 2π last

4 The Higgs VEV then rolls down from φ0.

Evolution of Scalar Fields in the Early Universe (slide 24) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-2: Inﬂaton-Higgs coupling

1 In the early stage of inflation, I is h i large. Higgs potential is steep. Slow-roll condition is not satisfied. The Higgs VEV stay at φ = 0 . V(ϕ) End of inflation 2 At the last Nlast e-folds of inflation, I , m ( I ) < HI , Higgs VEV h i ↓ eff,φ h i starts to develop. 3 At the end of inflation, the Higgs field has obtained a VEV ϕ 2 2 2 휙 = 퐻퐼 푁/4휋 p 2 HI p φ0 = φ = N . h i 2π last

4 The Higgs VEV then rolls down from φ0.

Evolution of Scalar Fields in the Early Universe (slide 24) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-2: Inﬂaton-Higgs coupling

1 In the early stage of inflation, I is h i large. Higgs potential is steep. Slow-roll condition is not satisfied. The Higgs VEV stay at φ = 0 . V(ϕ) After inflation 2 At the last Nlast e-folds of inflation, I , m ( I ) < HI , Higgs VEV h i ↓ eff,φ h i starts to develop. 3 At the end of inflation, the Higgs field Rolls down has obtained a VEV classically ϕ

p 2 HI p φ0 = φ = N . h i 2π last

4 The Higgs VEV then rolls down from φ0.

Evolution of Scalar Fields in the Early Universe (slide 24) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs IC-2: Inﬂaton-Higgs coupling

For N = 5 8, the isocurvature perturbation only last − develops on the small angular scales which are not yet constrained.

1.0 < IC2> 17 0.8 ΛI = 10 GeV 8 ΓI = 10 GeV 0.6 N last =8 15 0 0.4 ϕ0 = 10 GeV ϕ/

0.2

0.0

-0.2

0 200 400 600 800 1000 1200

ϕ0 t

Evolution of Scalar Fields in the Early Universe (slide 25) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Summary

During inflation, the Higgs field can obtain a large VEV through quantum fluctuation, but the field cannot roll down due to inflationary background. As the inflation end, the Higgs field rolls down within around Hubble time scale and oscillates around its minimum. Through the relaxation of the Higgs or other scalar fields, Letpogenesis and Baryongenesis are possible. Possible issue with isocurvature perturbation can be solved by introducing higher dimensional operators.

Thank you for your listening!

Evolution of Scalar Fields in the Early Universe (slide 26) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Inﬂation

The Universe appears to be almost homogeneous and isotropic today ⇒ Inflation In the early universe, the energy density was dominated by vacuum energy. Inflation from a real scalar field: Inflaton I (x) 1 L = gµν ∂ I∂ I − V (I) I 2 µ ν I

The equation of motion is 2 ¨ ˙ ˙ dVI (I) 2 a˙ 8π I+3HI+ΓI I+ = 0, with H ≡ = 2 (ρI + ρother) dI a 3mpl

where we assume a uniform ﬁeld conﬁguration and a FRW spacetime ds2 = dt2 − a (t)2 dr2 + r2dΩ2.

Evolution of Scalar Fields in the Early Universe (slide 27) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs The Brief History of the Early Universe

¨ dV ˙2 1 Slow-roll (inﬂation) regime: I dI and I V .

ΓI is not active.

˙ ∼ dV 2 ∼ 8π V(I) 3HI = − , and H = 2 VI (I) dI 3mpl Slow-Roll

Inﬂaton acts like vacuum energy. a(t) ∝ eHt Coherent Oscillations ΛI 2/3 2 Coherent oscillations regime: a (t) ∝ (t − ti) I Inﬂaton acts like non-relativistic particle. The log Ρ

Universe is matter-dominated. H L ΡI Inﬂaton then decays into relativistic particles ρR. Tmax Radiation- Dominated

4 TRH ΛI −Γ t ΡR ρ˙ + 3Hρ + Γ ρ = 0 ⇒ ρ (t) = e I Coherent I I I I I 3 Oscillations a (t) (matter like) log t t = 1 G End of Inflation t ' I H L 1/2 3 Radiation-dominated regime: a (t) ∝ (t − ti)

At t = 1/ΓI , most of the inﬂatons decay into ρR, Evolution of Scalarand Fields the in the reheating Early Universe is complete. (slide 28) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs The Hawking-Moss Tunneling

If V (φf ) V (φi) V (φi) , we have | − | 4 4 3mpl 1 1 3mpl V (φf ) − V (φi) SE (φi)−SE (φf ) = − − ≈ − 2 8 V (φi) V (φf ) 8 V (φi)

The transition rate is then

4 ! Γ 3mpl V (φf ) − V (φi) ∝ exp − 2 V 8 V (φi)

Thus, the transition is not suppressed as long as

8 2 V (φf ) − V (φi) < 4 V (φi) 3mpl

Evolution of Scalar Fields in the Early Universe (slide 29) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Reheating

As inflation ends, the inflatons enter the coherent oscillations regime, the Higgs field is no longer in slow-roll. In this case, we have to consider the full equation of motion ∂V (φ) φ¨ + 3Hφ˙ + Γ φ˙ = − H . φ ∂φ

The Hubble parameter and the temperature of the plasma are determined by

ρ˙r + 4Hρr = ΓI ρI , 8πG H2 = (ρ + ρ ) , 3 I r π2 ρ = g T 4. r 30 ∗

While the decay of Higgs may produce some non-zero lepton number by itself, most of the plasma are generated by the decay of inﬂaton.

Evolution of Scalar Fields in the Early Universe (slide 30) PACIFIC 2015 The Motivation Quantum Fluctuations Classical Motion Leptogenesis ICs Perturbative decay (thermalization) to top quark

Thermalization rate is comparable to the Hubble parameter only after the maximum reheating has been reached.

3.0 ´ 1011

H(t) 11 2.5 ´ 10 GH(t)

2.0 ´ 1011 D 1.5 ´ 1011 GeV @

1.0 ´ 1011

5.0 ´ 1010

0 0 1. ´ 10-12 2. ´ 10-12 3. ´ 10-12 4. ´ 10-12 5. ´ 10-12 6. ´ 10-12 7. ´ 10-12 t GeV-1

@ D 15 H(t) vs ΓH (t) through top quark for IC-1, with the parameters ΛI = 10 9 GeV and ΓI = 10 GeV. The vertical lines: the ﬁrst time the Higgs VEV crosses zero, and the time of maximum reheating, from left to right.

Evolution of Scalar Fields in the Early Universe (slide 31) PACIFIC 2015