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

Juan García-Bellido Galileo Galilei Institute Física Teórica UAM Florence, 2006 6th September 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 & & H & φ + φ + φ = φ = Φ 2 ( ) ( ) sin 3 m 0 t t mt ( ) 1 1 ρ = Φ + = Φ 2 2 2 2 2 2 ( ) cos sin ( ) m t mt mt m t 2 2 ( ) 1 = Φ − = 2 2 2 2 ( ) cos sin p m t mt mt 0 2 like matter − ρ 3 ( ) ~ ( ) t a t φ − = Φ 2 3 ( ) ~ ( ) n t m a t φ − Φ 1 ( ) ~ t t

Inflaton coupled to rest of the universe 1 1 1 1 1 = ∂ φ − φ + ∂ χ − χ + ξχ 2 2 2 2 2 2 2 ( ) ( ) L m m R χ 2 2 2 2 2 1 + ψ γ µ ∂ + ψ − φ ψ ψ − χ φ − φχ 2 2 2 2 2 ( ) i m h g g v µ ψ 2 & & H & φ + φ + + Π φ = 2 ( ( )) 3 0 m w Π = Γ Im ( m ) optical theorem m φ & & & & φ + φ + Γ φ + φ = 2 ( ) phenom. 3 H t m 0 φ ( ) Φ 1 − Γ d t φ φ = ⇒ ρ = − Γ ρ 3 3 0 ( ) sin 2 t e mt a a φ φ φ t dt

Perturbative decay of inflaton ψ χ φ φ g 2 v h χ ψ ∑ ∑ Γ = Γ φ → χ χ + Γ φ → ψ ψ ( ) ( ) φ i i i i i i 4 2 2 m g i v h i Γ φ → χ χ = Γ φ → ψ ψ = ( ) ( ) π π i i i i 8 m 8 ⎛ ⎞ 2 4 2 h m g v ⎜ ⎟ − Γ ≡ < < = ∑ + < < 2 2 6 eff i , m h h 10 ⎜ ⎟ φ π eff i 2 ⎝ ⎠ 8 m

Perturbative reheating 2 Γ < < = < < initially 3 H m φ t − − τ = Γ < < = 1 1 inflaton lifetime age universe t U H φ φ Γ π 2 2 2 3 M φ = Γ ⇒ ρ = ≡ P 4 ( ) ( ) H t g T T φ π reh reh reh 8 30 ≅ Γ = × ≤ 14 11 . GeV GeV 0 1 2 10 10 T M h φ eff reh P 3 m Γ ⇒ 9 ~ ~ GeV T 10 grav reh 2 M P

Non-perturbative decay of inflaton 1 1 1 = Π + ∇ χ + χ 2 2 2 2 ( ) ( ) H m t χ χ 2 2 2 = + φ 2 2 2 2 ( ) ( ) m t m g t χ χ [ ] free = ∫ 3 r r d k r χ + i k x ˆ ˆ ( , ) ( ) . . quantum field t x f t a e h c r 2 π k / 3 2 k ( ) [ ] [ ] r r r r r r ′ ′ + ′ χ Π ˆ = δ − ⇒ = δ − 3 3 ˆ ˆ ˆ h ( , ), ( , ) ( ) , ( ) t x t x i x x a a k k r r χ ′ k k & & + ω = ω = + 2 2 2 2 ( ) , ( ) ( ) time dep f t f 0 t k m t χ k k k k 1 & ∗ = = , Re( ) Wronskian g i f f g k k k k 2

Particle production (Schrödinger) ψ n Time-dependent approximation in sudden potential ψ 0

χ Occupation number of field ∫ 3 1 d k = = ( ) | | ( ) n t 0 N 0 n t π k 3 ( ) V 2 ω 1 1 = + − k 2 2 ( ) | | | | n t f g ω k k k 2 2 2 k ′ ′ + − = ( cos ) Mathieu equation f A 2 q 2 z f 0 k k k + Φ 2 2 2 2 k m ( ) g t χ = + = , A 2 q q k 2 2 m 4 m µ µ = ⇒ > > mt 2 mt solution ( ) ( ) ( ) ~ f t e p z n t e 1 k k k k

k Band structure µ k

resonance Narrow

resonance Broad

Broad resonance φ ( t )

Expanding universe

Lattice simulations n Large occupation numbers k classical fields equipartition T eff ~ n k k k

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.

φ χ Hybrid Inflation ) χ , φ ( V

χ ∈ ( 1 ) U String production @ end inflation

JGB, Linde PRD57, 6075 (1998) PRL87, 011601 (2001) Felder, JGB, Kofman, PRD64, 123517 (2001) Linde, Tkachev JGB, Garcia-Perez, PRD67, 103501 (2003) Gonzalez-Arroyo

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 ⎛ ⎞ χ 2 ⎜ ⎟ = − ≈ − − 2 2 3 ( ) 1 2 m m Vm t t ⎜ ⎟ φ χ c 2 ⎝ ⎠ c = − − = − τ 3 2 ( ) M t t M c [ ] ∫ 1 + + = τ τ + − τ τ τ 3 2 ( ) ( ) ( ) ( ) ( ) H d k p p k y y k k k k 2 [ ] ′ τ τ = δ − 3 h ( ), ( ) ( ) y p i k k ′ k k

Higgs Quantum Field ∗ + τ = τ τ + τ τ ( ) ( ) ( ) ( ) ( ) y f a f a − k k k 0 k k 0 [ ] ∗ + τ = − τ τ − τ τ ( ) ( ) ( ) ( ) ( ) p i g a g a − k k k 0 k k 0 ′ = ′ ′ + − τ = 2 g i f ( ) f k f 0 k k k k ∗ τ − τ ( ) ( ) 1 2 g iF Ω τ = = k k ( ) ∗ τ τ k 2 ( ) | ( ) | f 2 f k k ∗ τ = ( ) Im( ) F f g k k k

Quantum Initial Conditions 2 − ∀ τ τ = ⇒ Ψ τ = 0 | | k y ( ) , ( ) k a 0 0 N e k k 0 0 0 0 0 Unitary Evolution 1 2 − Ω τ τ = τ ⇒ Ψ τ = 0 ( ) | | y , , ( ) 0 U 0 e k k 0 0 π | | f k Occupation number of mode k 1 k 1 τ = τ τ τ = + − 2 2 ( ) , ( ) , | | | | n 0 N 0 g f 0 k k k k 2 k 2 2

1 > Quantum to Classical Transition > | ) τ ( gaussian k F | ) ⇒ p , y ( 0 G ≈ τ , 0 ) p ˆ , y ˆ ( G τ , 0

Quantum to Classical Transition < τ For longwave modes k 2 − τ τ = τ = τ ( ) 3 2 2 B k ( , ) | ( ) | ( ) P k k f A k e app k A τ / 3 2 4 τ = τ ≈ 2 0 ( ) ( ) A A Bi e 3 0 π τ τ = τ ( ) B 2

Power spectrum of longwave modes

Lattice Simulations Quantum averages = Ensemble averages Initial conditions: Highly occupied modes 1 2 − Ω τ τ = τ ⇒ Ψ τ = 0 ( ) | | y , , ( ) 0 U 0 e k k 0 0 π | | f k φ 2 | | φ θ − k 2 | | d d φ φ θ = 2 | | f k k (| |) | | P d d e k Ψ π k k k 2 | | f 2 k

Higgs field and Inflaton field Histograms of

Hybrid evolution

High peaks of Higgs field

High peaks and mean of Higgs field

PRD60,123504(1999) J. G.-B. Dmitri Grigoriev Alex Kusenko GGI 2006, Florence 6 th September, 2006 Misha Shaposhnikov

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

ρ log Evolution of Universe GUT EW QGP now ? Λ inflation radiation matter log a

The SU(2) Higgs-Inflaton model 1 µν + µ = − + Φ Φ a [( ) ] L F F Tr D D µν µ a 4 1 + ∂ χ − Φ χ 2 ( ) ( , ) V i µ = ∂ − τ a 2 D g A µ µ µ w a 2 = ∂ − ∂ + ε a a a abc b c F A A g A A µν µ ν ν µ µ ν w 1 1 Φ + Φ = φ + φ φ ≡ φ 2 a 2 [ ] ( ) Tr 0 a 2 2 λ 2 g 1 φ χ = φ − + φ χ + χ 2 2 2 2 2 2 ( , ) ( ) V v m 4 2 2

Chern-Simons Number t ∫ i ∫ 2 ~ g µν ∆ = 3 w [ ] N dt d x Tr F F µν π CS 2 16 t t ∫ t i ∫ 1 ≡ 3 ( , ) dt d x Q x t π 2 16 t = ∫ 1 d Γ Γ ≡ ∆ 2 ( ) ( ) I mt mdt t ( ) ( ) t N t CS 4 Vm dt t i − = ± ⋅ 5 ( ) ( . . ) I 45 12 07 0 64 10

Chern-Simons Charge ( , ) Q x t

Chern-Simons Charge ( , ) Q x t

Peak in Chern-Simons Charge

∆ Sphaleron Production 2 N CS

∆ Sphaleron Production 2 N CS E sph ∆ − N 1 0 1 CS

Cold EW Baryogenesis (I) µ = µ ψ γ ψ Baryonic current j L L 3 µ µ ∂ = ∂ ≡ ( , ) j j Q x t µ µ π B L 2 16 Chiral anomaly ⇒ ∆ = ∆ = ∆ B L 3 N CS + Φ Φ 2 ~ 3 g µν = δ CP violation L w F F µν π CP CP 2 2 M 16 new

CS induces a bias N ∆ 1 eff µ 0 sph E CP violation 1 −

CS induces a bias N ∆ 1 eff µ 0 sph E CP violation 1 −

Cold EW Baryogenesis (II) δ d + µ = Φ Φ Effective potential CP eff 2 M dt new Γ d = µ − Γ sph Boltzman equation n n eff B B B dt T eff 2 n v − − ⇒ = × δ = δ 6 8 B 2 10 10 CP CP 2 s M 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.