Conservation and evolution of the curvature perturbation

Transcription

1 Conservation and evolution of the curvature perturbation University of Wisconsin-Madison 1150 University Avenue, Madison WI USA Cosmo 08 University of Wisconsin-Madison, USA 28th August, 2008 Based on D. J. H. Chung and JG, in preparation K.-Y. Choi, JG and D. Jeong, in preparation

2 Outline 1 Introduction Curvature perturbation? 2 Single field inflation Equation of motion Subtleties 3 Multi field inflation Single and multi field inflation Evolution after multi field inflation 4 Conclusions

3 Curvature perturbation Err? Gauge invariant curvature perturbation ζ ζ = ψ + H δρ ρ

4 Curvature perturbation Err? Gauge invariant curvature perturbation ζ ζ = ψ + H δρ ρ Generic hypersurface the one with uniform energy density 1 Wiggles in the spatial curvature 2 Strongly constrained by observations P ζ 10 5, n 0.96, dn d logk 0.01, (The only?) Window to the early universe

5 Curvature perturbation Err? Gauge invariant curvature perturbation ζ ζ = ψ + H δρ ρ Generic hypersurface the one with uniform energy density 1 Wiggles in the spatial curvature 2 Strongly constrained by observations P ζ 10 5, n 0.96, dn d logk 0.01, (The only?) Window to the early universe Q: What do we know about ζ?

6 Equation of motion of ζ From perturbed Klein-Gordon equation / Einstein equation / variation principle we can obtain ζ + No potential dependence / exact ( 2 φ φ 2Ḣ ) ζ H + 3H 2 a 2 ζ = 0

7 Equation of motion of ζ From perturbed Klein-Gordon equation / Einstein equation / variation principle we can obtain ( 2 φ ζ + φ 2Ḣ ) ζ H + 3H 2 a 2 ζ = 0 No potential dependence / exact 1 On super-horizon scales, ζ = 0 is always a solution

8 Equation of motion of ζ From perturbed Klein-Gordon equation / Einstein equation / variation principle we can obtain ( 2 φ ζ + φ 2Ḣ ) ζ H + 3H 2 a 2 ζ = 0 No potential dependence / exact 1 On super-horizon scales, ζ = 0 is always a solution 2 Negative coefficient: (exponentially) growing solution

9 Equation of motion of ζ From perturbed Klein-Gordon equation / Einstein equation / variation principle we can obtain ( 2 φ ζ + φ 2Ḣ ) ζ H + 3H 2 a 2 ζ = 0 No potential dependence / exact 1 On super-horizon scales, ζ = 0 is always a solution 2 Negative coefficient: (exponentially) growing solution Conservation of ζ: Slow-roll is necessary even for single field case ζ+o(h) ζ = 0

10 Case of particle production (1/2) Conservation of ζ conservation of energy L = 1 2 g2 φ 2 χ 2 At φ = m χ /g, χ particles are resonantly produced

11 Case of particle production (1/2) Conservation of ζ conservation of energy L = 1 2 g2 φ 2 χ 2 At φ = m χ /g, χ particles are resonantly produced χ = χ 0 + δχ χ field is quantum: How to treat it?

12 Case of particle production (2/2) With Q = δφ + ( φ/h)ψ = ( φ/h)ζ, { k Q k = 3H Q 2 k a 2 + V + g 2 N χ [( m 2 Pl H 3H + Ḣ ) φ + 2V gn (m χ gφ) χ 2 ]} Q H k Change in the background HUGE change in P ζ

13 Case of particle production (2/2) With Q = δφ + ( φ/h)ψ = ( φ/h)ζ, { k Q k = 3H Q 2 k a 2 + V + g 2 N χ [( m 2 Pl H 3H + Ḣ ) φ + 2V gn (m χ gφ) χ 2 ]} Q H k Change in the background HUGE change in P ζ P 1/2-5 R 10 N = 2 N = 1 N = 0.5 N = k/a0h0

14 Case of particle production (2/2) With Q = δφ + ( φ/h)ψ = ( φ/h)ζ, { k Q k = 3H Q 2 k a 2 + V + g 2 N χ [( m 2 Pl H 3H + Ḣ ) φ + 2V gn (m χ gφ) χ 2 ]} Q H k Change in the background HUGE change in P ζ ρ χ ρ second order effect ρ2 χ H From conservation equation: P ζ = 6(ρ + p) 2 P χ Many subtleties How to turn on χ? What is the range of wavenumber and time? small change in P ζ

15 Case of particle production (2/2) With Q = δφ + ( φ/h)ψ = ( φ/h)ζ, { k Q k = 3H Q 2 k a 2 + V + g 2 N χ [( m 2 Pl H 3H + Ḣ ) φ + 2V gn (m χ gφ) χ 2 ]} Q H k Change in the background HUGE change in P ζ ρ χ ρ second order effect ρ2 χ H From conservation equation: P ζ = 6(ρ + p) 2 P χ Many subtleties How to turn on χ? What is the range of wavenumber and time? small change in P ζ We are trying to find a new perspective on ζ regarding its conservation

16 What is different from single field inflation? There are more than one orthogonal directions into which the field can be kicked

17 What is different from single field inflation? There are more than one orthogonal directions into which the field can be kicked

18 What is different from single field inflation? There are more than one orthogonal directions into which the field can be kicked Different histories

19 What is different from single field inflation? There are more than one orthogonal directions into which the field can be kicked Different histories Inflation Oscillation / Reheating RD MD

20 What is different from single field inflation? There are more than one orthogonal directions into which the field can be kicked Different histories All possible trajectories coalesce here Inflation Oscillation / Reheating RD MD

21 Multi field inflation and afterwards For multi field inflation, inflationary estimates are not enough and the evolution after inflation should be taken into account e.g. curvaton σ should satisfy 1 flat potential: m σ H 2 non-zero amplitude: σ 10 8 m Pl 3 small energy fraction: V σ V tot Easily satisfied by individual inflaton field after multi field inflation inflaton = curvaton

22 Evolution of ζ in multi field inflation Multiple chaotic inflation: V = 1 2 i m 2 i φ2 i with φ i Γ (i) γ, Γ(i) m

23 Evolution of ζ in multi field inflation Multiple chaotic inflation: V = 1 2 i m 2 i φ2 i with φ i Γ (i) γ, Γ(i) m

24 Evolution of ζ in multi field inflation Multiple chaotic inflation: V = 1 2 i m 2 i φ2 i with φ i Γ (i) γ, Γ(i) m

25 Matter-radiation isocurvature perturbation S αβ = 3 ( ) ζ α ζ β Smγ = δρ m 3 δρ γ ρ m 4 ρ γ 1 S mγ = 0 in single field inflation 2 Observationally 10% contribution 3 One of the signatures of multi field inflation?

26 Matter-radiation isocurvature perturbation S αβ = 3 ( ) ζ α ζ β Smγ = δρ m 3 δρ γ ρ m 4 ρ γ 1 S mγ = 0 in single field inflation 2 Observationally 10% contribution 3 One of the signatures of multi field inflation? We can find 1 S mγ 0 with large e-folds: ζ i = adiabatic + non-adiabatic 2 ζ does change after inflation: non-zero δp nad DO exist 3 P ζ and P S have slightly different scale dependence

27 Conclusions Conservation of the curvature perturbation is not as simple as a piece of cake 1 Single field inflation Slow-roll is required to ensure the conservation of ζ But many subtleties regarding the conservation of ζ 2 Multi field inflation ζ varies throughout the whole evolution of the universe Inflationary estimates may not work Possibly non-zero, detectable S mγ