COSMIC INFLATION AND THE REHEATING OF THE UNIVERSE Francisco - PowerPoint PPT Presentation

COSMIC INFLATION AND THE REHEATING OF THE UNIVERSE Francisco Torrentí - IFT/UAM Valencia Students Seminars - December 2014

Contents 1. The Friedmann equations 2. Inflation 
 2.1. The problems of hot Big Bang Theory 
 2.2. The inflationary idea 
 2.3. Slow-roll inflation 3. Reheating theory 
 3.1. Introduction to reheating 
 3.2. Perturbative reheating 
 3.3. Preheating 4. Reheating phenomenology 
 4.1. Primordial gravitational waves 
 4.2. Reheating the universe from the SM

1. THE FRIEDMANN EQUATIONS

1. The Friedmann equations G µ ν = 8 π GT µ ν ENERGY GEOMETRY CONTENT The Einstein field equations relate the geometry of a spacetime with its energy content. Friedmann equations Application to the universe as a whole Assumption (RHS): Content of the universe is a PERFECT FLUID   0 0 0 ρ ENERGY CONTENT OF 0   T µ ν = THE UNIVERSE   0 g ij p   0 4

1. The Friedmann equations GEOMETRY OF Assumption (LHS): THE UNIVERSE (Large scales) Homogeneous Isotropic (at one point) (Assumption: (confirmed by no preferred points experiments) in the universe) Homogeneous and isotropic AT ALL POINTS K = 1 FLRW metric: K = − 1 dr 2  � ds 2 = dt 2 − a 2 ( t ) 1 − Kr 2 + r 2 d Ω 2 K = 0 a(t): Scale factor 5

1. The Friedmann equations The FRIEDMANN equations: ✓ ˙ ◆ 2 a = 8 π G ρ − k a = − 4 π G ¨ a ( ρ + 3 p ) a 2 3 a 3 Content of the Universe modelized by: ρ i = a − 3(1+ ω i ) ω i = p i / ρ i ω R = 1 / 3 ω Λ = − 1 ω M = 0 Cosm. Const. Radiation Matter Curvature Ω j = 8 π G k j = M, R, Λ 3 H 2 ρ j Ω c = − H 2 a 2 X Ω i = Ω M + Ω Λ + Ω R + Ω c = 1 Now i 6

1. The Friedmann equations The Universe goes through di ff erent epochs. In each one, a specific kind of fluid dominated the expansion. ρ i = a − 3(1+ ω i ) log[ a ( t )] a ( t ) = e Ht Λ D a ( t ) ∼ t 2 / 3 MD a ( t ) ∼ t 1 / 2 RD log[ t ]

2. INFLATION

2.1. The problems of hBB theory p = 1 2 , 2 The three problems of hot Big Bang Theory: a ( t ) ∼ t p 3 1. The horizon problem CMB is incredibly homogeneous. Not enough time for light to propagate and get 
 thermal equilibrium. Z t Comoving horizon distance: 
 dt 0 comoving distance travelled by 
 d hor ( t ) = a ( t 0 ) light since t i to t f t i 2 1 RD/MD and t i =0 d c hor ( t dec ) ∼ ( a dec H dec ) − 1 d c hor ( t 0 ) ∼ ( H 0 ) − 1 At decoupling time Now d c hor ( t dec ) a 0 H 0 ⌧ 1 hor ( t 0 ) ⇠ d c a dec H dec T γ (1) ≈ T γ (2) 9

2.1. The problems of hBB 2. The flatness problem Current observations give . Unstable point in Friedmann equations! Ω c ≈ 0 ✓ ρ 0 ,m ◆ 3 k a ( t ) + ρ 0 ,R Ω c ( t ) ↑ a ( t ) ↑ = Ω c a 2 ( t ) 8 π G We need incredible fine-tuning! Ω c ( t pl ) ≈ 10 − 60 3. The primordial monopole problem Not observed cosmic relics predicted by GUT models 10

2.2. The inflationary idea An early phase of exponential expansion can solve a ( t ) = e Ht the three problems at once Alan Guth The idea (but model failed) Andrei Linde First successful inflationary model 11

2.2. The inflationary idea ✓ a ◆ a ( t ) = e Ht Number of e-folds: N ≡ log = Ht a i 1. Horizon problem: Due to the inflationary epoch, all points in the CMB were causally connected in the past. 2. Flatness problem: 
 During inflation, Ω k =0 is an attractor point. 3 k Ω c ρ 0 , Λ a 2 ( t ) = a ( t ) ↑ Ω c ( t ) ↓ 8 π G 3. Primordial monopole problem: Inflation washes out any cosmic relics. THE THREE PROBLEMS ARE SOLVED WITH N ≈ 60 12

2.3. Slow-roll inflation d 2 a HOW TO IMPLEMENT IT? Definition of INFLATION: dt 2 = 0 Action of the inflaton: Inflationary potential ✓ 1 ◆ Z p d 4 x S = | g | 2 ∂ µ φ∂ µ φ − V ( φ ) φ = φ ( t ) Field and Friedmann equations: Energetic content: H ( t ) ≡ ˙ a a ρ φ = 1 φ 2 + V ( φ ) φ + ∂ V ( φ ) ˙ φ + 3 H ( t ) ˙ ¨ = 0 2 ∂φ E.o.m: p φ = 1 ˙ φ 2 − V ( φ ) 1 ✓ 1 ◆ H 2 = φ 2 + V ( φ ) ˙ 2 3 m 2 2 p 13

2.3. Slow-roll inflation • First requirement: 2 ˙ 1 � 2 V ( φ ) >> 1 ✏ = 3 V ( � ) << 1 ˙ φ 2 2 Potential energy First SLOW-ROLL dominates over kinetic parameter d 2 a H 2 ≈ V ( φ ) 1 dt 2 ≈ + V ( φ ) Fr. eqns: 3 m 2 3 m 2 a p p INFLATION! R t H ( φ ) dt 0 a ( t ) ≈ a i e (Quasi) de Sitter 2 ˙ 1 � 2 − V ( � ) ≈ − 1 + 2 ! ≡ p φ Energy: = 3 ✏ 2 ˙ � 2 + V ( � ) 1 ⇢ φ 14

2.3. Slow-roll inflation • Second requirement: We must ensure that ε <<1 is sustained for 
 at least 60 e-folds or more. (¨ � ↑↑→ ˙ The field must not accelerate � ↑↑→ ✏ ↑↑ ) φ + ∂ V ( φ ) φ + 3 H ( t ) ˙ ¨ = 0 ∂φ ¨ φ Second η ≡ − << 1 We need: | ¨ φ | << 3 H ˙ SLOW-ROLL φ , V 0 ( φ ) H ˙ φ parameter SLOW-ROLL CONDITIONS IN TERMS OF POTENTIAL: ✏ , ⌘ << 1 ◆ 2 ✏ V ≡ m 2 ✓ V 00 ◆ ✓ V 0 p η V ≡ m 2 p V 2 V ✏ V , ⌘ V << 1 ( ⌘ ≈ ⌘ V − ✏ V ) ( ✏ ≈ ✏ V ) 15

2.3. Slow-roll inflation V ( φ ) = 1 Working example: 2 m 2 φ 2 END of inflation: φ end = M P 2 √ π ≈ M P 2 2 ✏ = ⌘ = p � 2 = ✏ 3 . 5 m 2 p � 2 m 2 the field starts to oscillate around the minimum of its potential 16

3. REHEATING (getting the “bang” from the Big Bang)

3.1. Introduction to reheating What is the origin of all matter and radiation present in our universe today? During inflation, the universe is But now… empty and cold M ≈ 10 23 M � S ≈ 10 89 S ≈ 0 M ≈ 0 T ≈ 0 (Inflation dilutes any relic species left from (and T >> 0 in the early universe ) a hypohetical earlier period of the universe) We need to “reheat” the universe after inflation 18

3.1. Introduction to reheating Hot Big Bang Inflation Reheating theory the universe V ( φ ) T r energy transfer to created particles starts to get cold again.. final reheating (dominant energy) (the universe gets hot) temperature 19

3.1. Introduction to reheating Model for other L = 1 2( ∂ µ φ ) 2 − V ( φ ) + 1 2( ∂ µ χ ) 2 − 1 interaction χ χ 2 − 1 + 2 m 2 2 g 2 φ 2 χ 2 reheating: fields inflaton-fields Interaction term Inflaton Scalar field V ( φ ) = 1 2 m 2 φ 2 (Parabolic potential) φ ( t ) Inflation equation (neglecting interaction): φ + 3 H ( t ) ˙ ¨ φ + m 2 φ = 0 (+ Friedmann eqn.) t Φ ( t ) = Φ 0 φ ( t ) ∼ Φ ( t ) sin mt t a ( t ) ∼ t 2 / 3 OSCILLATIONS AROUND THE MINIMUM OF THE POTENTIAL Matter-dominated 20

3.2. Perturbative reheating L = 1 2( ∂ µ φ ) 2 − V ( φ ) + 1 2( ∂ µ χ ) 2 − 1 − 1 − g φχ 2 + ¯ 2 g 2 φ 2 χ 2 2 m 2 χ χ 2 − h φ ¯ ψ ( i γ µ ∂ µ − m ψ ) ψ ψψ Fermion Inflaton Interaction terms Scalar Inflaton couples (weakly) to other particles, and then it decays: g 2 Γ ( φ → χχ ) = h 2 eff m 8 π m i + g 2 ✓ ◆ Γ φ = Γ ( φ + χ i χ i ) + Γ ( φ → ¯ X i ψψ ) = h 2 h 2 eff = m ψψ ) = h 2 m 8 π i Γ ( φ → ¯ 8 π new friction term φ + 3 H ( t ) ˙ ¨ φ + Γ φ ˙ φ + m 2 φ = 0 Inflaton e ff ective e.o.m: Solution d φ ( t ) ∼ Φ ( t ) sin mt dt ( ρ φ a 3 ) = − Γ φ ρ φ a 3 comoving inflaton energy density and particle d Φ ( t ) = Φ 0 e − 1 2 (3 H + Γ ) t dt ( n φ a 3 ) = − Γ φ n φ a 3 number decays into particles 21

3.2. Perturbative reheating Γ φ << H = 2 total comoving energy and number of inflaton (1) particles is conserved 3 t Inflaton decays suddenly H − 1 ≈ Γ − 1 (2) Γ φ ≈ H φ into particles Inflaton lifetime Age of the universe • It may take many many inflation oscillations to get to (2). 
 (we need to wait until the Universe is old enough!) 
 • When (2) arrives, reheating is instantaneous. It realises all ρ φ into , and in an exponential burst of energy. ψ χ THERMALIZATION (3) to a reheating Created particles interact among themselves temperature T r 22

3.2. Perturbative reheating Estimation of the reheating temperature: t r ∼ 2 3 Γ − 1 φ p T rh ≈ 0 . 1 Γ φ M p Reheating time For our chaotic model: m ≈ 10 13 GeV h 2 eff m Reheating T rh ≤ 10 11 GeV Γ φ = h eff ≤ 10 − 3 temperature 8 π Problems: 1) Low temperature. 
 2) In some models, always. 
 H > Γ φ BUT: Inflaton is not composed of individual inflaton quanta, it is 
 a coherently oscillating field with large amplitude. We need new formalism! 23

3.3. Preheating New formalism: Particle production in the presence of strong background fields. L = 1 2( ∂ µ φ ) 2 − V ( φ ) + 1 2( ∂ µ χ ) 2 − 1 χ χ 2 − 1 Minkowski 2 m 2 2 g 2 φ 2 χ 2 spacetime (for the moment) Inflaton Interaction term Scalar field φ ( t ) = Φ sin mt A free field with χ + g 2 Φ 2 sin 2 mt m 2 χ ( t ) = m 2 φ � r 2 φ + m 2 m 2 ¨ χ ( t ) = 0 χ ( t ) time-dependent mass! f k : (field mode) In momentum space d 2 f k k ( t ) = k 2 + m 2 ω 2 χ ( t ) dt 2 + ω 2 k ( t ) f k = 0 Kofman, Linde & Starobinsky, “Towards Time-dependent the theory of reheating frequency Field-mode equation after inflation” (1997) 24