Juan Garca-Bellido Galileo Galilei Institute Fsica Terica UAM - - PowerPoint PPT Presentation

Juan García-Bellido Física Teórica UAM 6th September 2006 Galileo Galilei Institute Florence, 2006

Outline

Reheating: Standard perturbative decay

• Oscillating inflaton field
• Perturbative decay rates
• Reheating temperature

Preheating: Very rich phenomenology

• Parametric resonance and tachyonic inst.
• Production massive part. + top. defects
• EW baryogenesis & leptogenesis
• Stochastic background gravitational waves
• Primordial magnetic fields

Reheating

Inflaton oscillating at end of inflation 3

2

= + + φ φ φ m H & & &

mt t t sin ) ( ) ( Φ = φ

( ) ( )

2 1 2 1 2 1

2 2 2 2 2 2 2 2 2 2

= − Φ = Φ = + Φ = mt mt t m p t m mt mt t m sin cos ) ( ) ( sin cos ) ( ρ

like matter

1 3 2 3 − − −

Φ Φ = t t t a m t n t a t ~ ) ( ) ( ~ ) ( ) ( ~ ) (

φ φ

ρ

Inflaton coupled to rest of the universe

2 2 2 2 2 2 2 2 2 2 2 2

2 1 2 1 2 1 2 1 2 1 2 1 φχ φ χ ψ ψ φ ψ γ ψ ξχ χ χ φ φ

ψ µ µ χ

v g g h m i R m m L − − − + ∂ + + − ∂ + − ∂ = ) ( ) ( ) (

3

2

= Π + + + φ φ φ )) ( ( w m H & & &

theorem

• ptical

φ

Γ = Π m m) ( Im phenom. 3

2

= + Γ + + φ φ φ φ

φ

m t H & & & & ) (

( )

3 3 2 1

a a dt d mt e t t

t φ φ φ

ρ ρ φ

φ

Γ − = ⇒ Φ =

Γ −

sin ) (

Perturbative decay of inflaton

∑ ∑

→ Γ + → Γ = Γ

i i i i i i

) ( ) ( ψ ψ φ χ χ φ

φ

m v gi

i i

π χ χ φ 8

2 4

= → Γ ) ( π ψ ψ φ 8

2m

hi

i i

= → Γ ) (

6 2 2 4 2 2 2

10 8

−

< < ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∑ = < < ≡ Γ m v g h h m m h

i i eff eff

, π

φ

φ χ χ

v g2

φ ψ ψ

h

Perturbative reheating

initially m H t < < = < < Γ 3 2

φ

universe age lifetime inflaton

1 1 − −

= < < Γ = H tU

φ φ

τ

4 2 2 2

30 8 3

reh reh P reh

T T g M t H ) ( ) ( π π ρ

φ φ

≡ Γ = ⇒ Γ = GeV GeV

eff 11 14

10 10 2 1 ≤ × = Γ ≅ h M T

P reh φ

. GeV

9 2 3

10 ~ ~

reh P grav

T M m ⇒ Γ

Non-perturbative decay of inflaton

2 2 2 2

2 1 2 1 2 1 χ χ

χ χ

) ( ) ( t m H + ∇ + Π = ) ( ) ( t g m t m

2 2 2 2

φ

χ χ

+ =

[ ]

field quantum . . ˆ ) ( ) ( ) , ( ˆ

/

c h e a t f k d x t

x k i k k

+ =∫

r r r

r

2 3 3

2π χ

free

[ ]

[ ]

) ( ˆ , ˆ ) ( ) , ( ˆ ), , ( ˆ k k a a x x i x t x t

k k

′ − = ⇒ ′ − = ′ Π

+ ′

r r r r h r r

r r 3 3

δ δ χ

χ

dep time ) ( ) ( , ) ( t m k t f t f

k k k k 2 2 2 2 χ

ω ω + = = + & & Wronskian 2 1 = =

∗

) Re( ,

k k k k

g f f i g &

Particle production (Schrödinger)

ψ

n

ψ

Time-dependent potential in sudden approximation

Occupation number of field

∫

= = ) ( ) ( | | ) ( t n k d N V t n

k 3 3

2 1 π 2 1 2 1 2

2 2

− + = | | | | ) (

k k k k k

g f t n ω ω equation Mathieu 2 2 = − + ′ ′

k k k

f z q A f ) cos (

χ

2 2 2 2 2 2

4 2 m t g q q m m k A

k

) ( , Φ = + + =

χ

1

2

> > ⇒ =

mt k mt k

k k

e t n z p e t f

µ µ

~ ) ( ) ( ) ( solution

k

µ k

Band structure

Narrow resonance

Broad resonance

) (t φ

Broad resonance

Expanding universe

Lattice simulations

k

n k

Large occupation numbers classical fields equipartition

k T nk

eff

~

Preheating

Very rich phenomenology after inflation

• Non-thermal production of particles (CDM)
• Production of topological defects
• EW baryogenesis & leptogenesis
• Production of gravitational waves
• Production of primordial magnetic fields
• etc.

) , ( χ φ V χ φ

Hybrid Inflation

) (1 U ∈ χ

String production @ end inflation

JGB, Linde

PRD57, 6075 (1998) PRL87, 011601 (2001) PRD64, 123517 (2001)

Felder, JGB, Kofman, Linde, Tkachev JGB, Garcia-Perez, Gonzalez-Arroyo

PRD67, 103501 (2003)

Tachyonic Preheating

Spinodal growth of long wave Higgs modes

• At the end of Hybrid Inflation
• Higgs couples to gauge fields
• Strong production of fermions

The Higgs Evolution

τ χ χ

φ 2 3 3 2 2 2 2

2 1 M t t M t t Vm m m

c c c

− = − − = − − ≈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ) ( ) (

[ ]

∫

+ +

− + = ) ( ) ( ) ( ) ( ) ( τ τ τ τ τ

k k k k

y y k p p k d H

2 3

2 1

[ ]

) ( ) ( ), ( k k i p y

k k

′ − =

′ 3

δ τ τ h

Higgs Quantum Field

) ( ) ( ) ( ) ( ) ( τ τ τ τ τ

+ − ∗

+ =

k k k k k

a f a f y

[ ]

) ( ) ( ) ( ) ( ) ( τ τ τ τ τ

+ − ∗

− − =

k k k k k

a g a g i p

k k

f i g ′ =

2

= − + ′ ′

k k

f k f ) ( τ

2

2 2 1 | ) ( | ) ( ) ( ) ( ) ( τ τ τ τ τ

k k k k k

f iF f g − = = Ω

∗ ∗

) Im( ) (

k k k

g f F

∗

= τ

Quantum Initial Conditions

2 | |

) ( , ) (

k

y k k

e N a k

−

= Ψ ⇒ = ∀ τ τ τ

Unitary Evolution

2

1

| | ) (

| | ) ( , ,

k k

y k

e f U

τ

π τ τ τ

Ω −

= Ψ ⇒ =

Occupation number of mode k

2 1 2 2 1

2 2

− + = = | | | | , ) ( , ) (

k k k k

f k g k N n τ τ τ τ

Quantum to Classical Transition

gaussian

p y G p y G ) , ( , ) ˆ , ˆ ( , ≈ τ τ 1 > > ⇒ | ) ( | τ

k

F

Quantum to Classical Transition

For longwave modes

τ < k

2 2 2 3 k B k app

e k A f k k P

) (

) ( | ) ( | ) , (

τ

τ τ τ

−

= =

2 3 3 4

2

/

) ( ) (

τ

τ π τ τ e A Bi A A ≈ = τ τ 2 = ) ( B

Power spectrum of longwave modes

Lattice Simulations

Quantum averages = Ensemble averages Initial conditions: Highly occupied modes

2

1

| | ) (

| | ) ( , ,

k k

y k

e f U

τ

π τ τ τ

Ω −

= Ψ ⇒ = π θ φ θ φ φ

φ

2

2 2

2 2

k k k f k k k

d f d e d d P

k k

| | | | | | |) (|

| | | | − Ψ

=

Histograms of Higgs field and Inflaton field

Hybrid evolution

High peaks of Higgs field

High peaks and mean of Higgs field

GGI 2006, Florence 6th September, 2006

• J. G.-B.

Dmitri Grigoriev Alex Kusenko Misha Shaposhnikov

PRD60,123504(1999)

Sakharov conditions

• B violation
• C and CP violation
• Out of equilibrium

inflation matter radiation now ?

ρ log a log

Evolution of Universe

Λ GUT EW QGP

] ) [( Φ Φ + − =

+ µ µ µν µν

D D Tr F F L

a a

4 1

c b abc w a a a a a w

A A g A A F A g i D

ν µ µ ν ν µ µν µ µ µ

ε τ + ∂ − ∂ = − ∂ = 2

2 2 2 2 2 2 2 2 2

2 1 2 4 2 1 2 1 χ χ φ φ λ χ φ φ φ φ φ m g v V Tr

a a

+ + − = ≡ + = Φ Φ+ ) ( ) , ( ) ( ] [

) , ( ) ( χ χ

µ

Φ − ∂ + V

2

2 1

The SU(2) Higgs-Inflaton model

Chern-Simons Number

] ~ [

µν µν

π F F Tr x d dt g N

t t w CS

i ∫

∫

= ∆

3 2 2

16 ) , ( t x Q x d dt

t ti ∫

∫

≡

3 2

16 1 π ) ( ) ( t N dt d Vm t

CS 2 4

1 ∆ ≡ Γ ) ( ) ( t mdt mt I

t ti

Γ =∫

5

10 64 07 12 45

−

⋅ ± = ) . . ( ) ( I

Chern-Simons Charge

) , ( t x Q

Chern-Simons Charge

) , ( t x Q

Peak in Chern-Simons Charge

Sphaleron Production

2 CS

N ∆

Sphaleron Production

2 CS

N ∆

CS

N ∆

sph

E 1 1 −

Cold EW Baryogenesis (I)

Baryonic current

L L

j ψ γ ψ

µ µ =

) , ( t x Q j j

L B 2

16 3 π

µ µ µ µ

≡ ∂ = ∂

Chiral anomaly

CS

N L B ∆ = ∆ = ∆ ⇒ 3

µν µν

π δ F F g M

w CP CP

~

2 2 2

16 3

new

Φ Φ =

+

L

CP violation

CP violation

CS

N ∆

sph

E 1 1 −

eff

µ induces a bias

CP violation

CS

N ∆

sph

E 1 1 −

eff

µ induces a bias

Cold EW Baryogenesis (II)

Φ Φ =

+

dt d M

CP 2 new eff

δ µ

Effective potential

B B B

n T n dt d Γ − Γ =

eff sph eff

µ

Boltzman equation

CP CP B

M v s n δ δ

8 2 2 6

10 10 2

− −

= × = ⇒

new

EW Symmetry Breaking can lead to the production of baryons via sphaleron production at tachyonic preheating after hybrid inflation The right amount of baryons depends on CP violation param.

hep-lat/0509094

• J. G.-B.

Andres Diaz-Gil Margarita Garcia-Perez Antonio Gonzalez-Arroyo

GGI 2006, Florence 6th September, 2006

EW Tachyonic Preheating

Spinodal growth of long wave Higgs modes

• At the end of EW Hybrid Inflation
• Inflaton couples to Higgs
• Higgs couples to SM fields
• Strong production of fermions

and gauge fields

The SU(2)xU(1) Higgs-Inflaton model

] ) [( Φ Φ + − − =

+ µ µ µν µν µν µν

D D Tr G G F F L

a a

4 1 4 1

µ ν ν µ µν ν µ µ ν ν µ µν µ µ µ µ

ε τ τ B B F A A g A A G B g i A g i D

c b abc w a a a Y a a w

∂ − ∂ = + ∂ − ∂ = − − ∂ =

3

2 2

) , ( ) ( χ χ

µ

Φ − ∂ + V

2

2 1

2 2 2 2 2 2 2 2 2

2 1 2 4 2 1 2 1 χ χ φ φ λ χ φ φ φ φ φ m g v V Tr

a a

+ + − = ≡ + = Φ Φ+ ) ( ) , ( ) ( ] [

) , ( χ φ V χ φ

Hybrid Inflation

High peaks and mean of Higgs field

Evolution after EWSB

Kinetic Turbulence & Scaling

1 2 1 5 3 − = = =

− −

m p p q kt n t t k n

p q

. ) ( ) , ( 1 2 2

2 2 2

− = ∝ − = ∆

−

m t ν φ φ φ

ν

The amplitude of magnetic fields

4 2 2 4 4

10 10 10 ) ( GeV v m V

H mag

= ≈ ≤

− −

ρ

4 42 2

10 39 1 8 1 GeV Gauss

−

× = . π

Conversion factor

2 4

3 ) . (

) (

G a a

mag rh mag

µ ρ ρ ≈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

The coherence scale of magnetic fields

) (t a t ∝ ∝ ξ ξ

During kinetic turbulence After e+e- annihilation

kpc today 100 10 − ≈ ) ( ξ

EW Symmetry Breaking can lead to the production of primordial magnetic fields at tachyonic preheating after hybrid inflation The right amplitude and scale

• f magnetic fields depends on

the extent of kinetic turbulence Initial conditions for magneto- Hydrodynamic simulations