ENFPC 2013

191 views

Published on

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
191
On SlideShare
0
From Embeds
0
Number of Embeds
3
Actions
Shares
0
Downloads
0
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

ENFPC 2013

  1. 1. Outline Warm inflation Starobinsky I Observations Starobinsky II Non-isentropic stochastic inflation, single field potentials and Planck data Leandro A. da Silva, Rudnei O. Ramos1 XXXIV Encontro Nacional de F´ ısica de Part´ ıculas e Campos 27/08/2013 1 Universidade do Estado do Rio de Janeiro Final remarks
  2. 2. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Two main questions: How the introduction of dissipative and temperature effects impacts the compatibility between theoretical predictions and observational data? Can temperature and dissipation stop eternal self-reproduction of the universe?
  3. 3. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Warm inflation: Same basic ideas of standard inflation. Inflaton interacts with its environment → radiation production during inflation. No reheating mechanism is necessary. Smooth transition to radiation domination era.
  4. 4. Outline Warm inflation Starobinsky I Observations Starobinsky II Warm inflation Microscopic motivation: L[φ, χ, σ] = L[φ] + L[χ] + L[σ] + Lint [φ, χ] + Lint [χ, σ] Procedure: functional integration over χ e σ. Non-equilibrium dynamics → Real time formalism Markovian approximation (local dissipation) → system characteristic time scale relaxation time scale Effective equation of motion: ∂ 1 ∂2 + (3H + Υ) − 2 2 ∂t ∂t a 2 Φ+ ∂Veff (Φ) = ξT ∂Φ Final remarks
  5. 5. Outline Warm inflation Starobinsky I Observations Starobinsky II Warm inflation: basic equations 1 ∂2 ∂ + (3H + Υ) − 2 ∂t2 ∂t a 2 Φ+ ∂V (Φ) = ξT ∂Φ ξT (x, t)ξT (x , t ) = 2ΥT a−3 δ(x − x )δ(t − t ) a=− ¨ 8π ˙ ρr + Φ2 − V (Φ) a 3m2 pl a˙ ˙ ˙ ˙ ρΦ = −3 Φ2 − ΥΦ2 + ξT Φ , ˙ a = m2 pl 16π < 1 + Q, V V 2 , η= η <1+Q m2 pl 8π e a ˙ ˙ ˙ ρr = −4 ρr + ΥΦ2 − ξT Φ ˙ a V V , β= β <1+Q, m2 pl 8π ΥV ΥV Q≡ Υ 3H Final remarks
  6. 6. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Contributions to the power spectrum Important characteristic of inflation: Natural mechanism to generation of nearly scale invariant density perturbations. Cold inflation: quantum fluctuations contributions Warm inflation: thermal fluctuations contributions
  7. 7. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Contributions to the power spectrum Important characteristic of inflation: Natural mechanism to generation of nearly scale invariant density perturbations. Cold inflation: quantum fluctuations contributions Warm inflation: thermal fluctuations contributions ⇓ extreme cases... Non-isentropic stochastic inflation: Quantum and thermal fluctuations taken in account explicitly and in a transparent way. Recovers standard results both from cold and warm inflation.
  8. 8. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky I: perturbative approach Central idea: Φ(x, t) → Φ> (x, t) + Φ< (x, t) Φ> (x, t) → ϕ(t) + δϕ(x, t) Mode separation implemented through an “Window function”: W (k, t) ≡ θ(k − aH) Goal: Effective dynamics for δϕ. Φ< (x, t) ≡ φq (x, t) = d3 k W (k, t) φk (t)e−ik·x ak + φ∗ (t)eik·x a† ˆ ˆk k 3/2 (2π) √ H π (1) φk (τ ) = (|τ |)3/2 Hµ (k|τ |) , 2 where µ = 9/4 − 3η.
  9. 9. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky I: perturbative approach ∂ϕ ∂2ϕ + [3H + Υ(ϕ)] + V,ϕ (ϕ) = 0 , 2 ∂t ∂t 1 2 ∂2 ∂ ˜ − + [3H + Υ(ϕ)] + Υ,ϕ (ϕ)ϕ + V,ϕϕ (ϕ) δϕ = ξq + ξT , ˙ ∂t2 ∂t a2 ˜ ξq = − ∂2 1 ∂ − 2 + [3H + Υ(ϕ)] 2 ∂t ∂t a 2 + Υ,ϕ (ϕ)ϕ + V,ϕϕ (ϕ) φq , ˙ ˜ ξq → generalized quantum noise term ˜ ˜ ξq (x, t), ξq (x , t ) = 0 → classical behavior preserved Equation of motion as a function of z = k/(aH): 1 η − βQ/(1 + Q) δϕ (k, z) − (3Q + 2)δϕ (k, z) + 1 + 3 δϕ(k, z) = z z2 1 ˜ ξT (k, z) + ξq (k, z) . H 2z2
  10. 10. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky I: perturbative approach Using the EoM solution, we define the inflaton power spectrum: Pδϕ = k3 2π 2 d3 k (th) (qu) δϕ(k, z)δϕ(k , z) = Pδϕ (z) + Pδϕ (z) (2π)3 2 3Q 2α 2ν−2α Γ (α) Γ (ν − 1) Γ (α − ν + 3/2) √ 2 z 1 2 π Γ ν − 2 Γ (α + ν − 1/2) ≈ HT 4π 2 + H coth T zH 2η z 2T ,
  11. 11. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky I: perturbative approach Using the EoM solution, we define the inflaton power spectrum: Pδϕ = k3 2π 2 d3 k (th) (qu) δϕ(k, z)δϕ(k , z) = Pδϕ (z) + Pδϕ (z) (2π)3 2 3Q 2α 2ν−2α Γ (α) Γ (ν − 1) Γ (α − ν + 3/2) √ 2 z 1 2 π Γ ν − 2 Γ (α + ν − 1/2) ≈ HT 4π 2 + H coth T zH 2η z 2T , As expected, nearly scale invariant. Alternative derivation of the enhancement term (Mohanty et al, Phys. Rev. Lett. 97, 251301 (2006))
  12. 12. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky I: perturbative approach Using the EoM solution, we define the inflaton power spectrum: Pδϕ = k3 2π 2 d3 k (th) (qu) δϕ(k, z)δϕ(k , z) = Pδϕ (z) + Pδϕ (z) (2π)3 2 3Q 2α 2ν−2α Γ (α) Γ (ν − 1) Γ (α − ν + 3/2) √ 2 z 1 2 π Γ ν − 2 Γ (α + ν − 1/2) ≈ HT 4π 2 + H coth T zH 2η z 2T , Recovers all results of cold and warm inflation: Q 1 and T H ⇒ Pδϕ ∝ HT (Berera and Fang, Phys. Rev. Lett. 74 (1995)) √ Q 1 and T H ⇒ Pδϕ ∝ T HΥ (Hall, Moss and Berera, Phys. Rev. D 69, 083525 (2004) ) Q 1 and T H ⇒ Pδϕ ∝ H 2 (cold inflation)
  13. 13. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky I: perturbative approach V (φ) = 4 λMpl p φ Mpl p , Υ(φ, T ) = Cφ φ2a T c , m2b X c + 2a − 2b = 1 Figure: blue lines, Υ(φ), red Υ = cte. Dashed lines p = 2, full lines p = 4, dotted lines p = 6
  14. 14. Outline Warm inflation Starobinsky I Observations Starobinsky II Cosmological parameters: Curvature perturbations: H2 P = ∆2 (k0 ) R ˙ δϕ φ2 8 H2 ∆2 = 2 h Mpl 4π 2 ∆2 = R ns −1 k k0 Spectral index (and running ns ): ns − 1 = d ln ∆2 R d ln k ns ≡ dns d ln k Tensor-to-scalar ratio: r≡ ∆2 4 H2 h 2 = (1 + Q)2 π 2 P ∆R δϕ Final remarks
  15. 15. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Cosmological parameters: Some interesting limits: spectral index: Q → 0 and T → 0 ns = 1 + 2η − 6ε r ≈ 16 1 and T H ⇒ 1 9 9 3 − ε − β + η + O(1/Q3/2 ) + O(1/(Q3/2 T 2 )) ns = 1 + Q 4 4 2 16 H r≈ √ 3πT Q5/2 (Hall, Moss and Berera, Phys. Rev. D 69, 083525 (2004) ) Q Q 1 and T H ns = 1 + 2η − 6ε + (8ε − 2η)Q + O(Q2 )
  16. 16. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Cosmological parameters: WMAP-9yr (arXiv:1212.5226) Figure: Green, eCMB, red, eCMB+BAO+H0 .Light colors → 95% CL, dark colors → 68% CL.
  17. 17. Outline Warm inflation Starobinsky I Observations Starobinsky II Cosmological parameters: Planck (arXiv:1303.5082) Final remarks
  18. 18. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Cosmological parameters: WMAP-9yr (arXiv:1212.5226)
  19. 19. Outline Warm inflation Starobinsky I Results: V ∝ φ2 68%CL Observations Starobinsky II Final remarks
  20. 20. Outline Warm inflation Starobinsky I Results: V ∝ φ4 95%CL Observations Starobinsky II Final remarks
  21. 21. Outline Warm inflation Starobinsky I Results: V ∝ φ6 95%CL Observations Starobinsky II Final remarks
  22. 22. Outline Warm inflation Starobinsky I Observations Starobinsky II Extending Starobinsky II: large scales Similar prescription: Φ(x, t) = ϕ(x, t) + φq (x, t) Large scales ( ¨ H −1 ): ≈ homogeneous dynamics, Φ ≈ 0 Resulting equation of motion: ϕ=− ˙ V,ϕ (ϕ) H 3/2 + 3H(1 + Q) 2π 1+ 2 eH/T Two-point function: ζ(t)ζ(t ) = δ(t − t ) . −1 ζ(t) Final remarks
  23. 23. Outline Warm inflation Starobinsky I Observations Starobinsky II Extending Starobinsky II: large scales Associated Fokker-Planck equation: ∂ ∂ 1 ∂2 P (ϕ, t) = − D(1) P (ϕ, t) + D(2) P (ϕ, t) ∂t ∂ϕ 2 ∂ϕ2 ≡ LF P P (ϕ, t) Drift and difusion coefficients: V,ϕ (ϕ) ≡ −f (ϕ) , 3H(1 + Q) H3 2 = 2 1 + H/T . 4π e −1 D(1) = − D(2) Final remarks
  24. 24. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky II: large scales Eternal inflation: Inflation doesn’t end globally Qualitative condition to self-reproduction regime of H-regions: f (ϕ) H(ϕ) D(2) H(ϕ)
  25. 25. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky II: large scales Global picture: physical probability distribution function, PV : ∂ ∂ ∂PV = −D(1) (ϕ)PV + D(2) (ϕ)PV ∂t ∂ϕ ∂ϕ PV (ϕ, t) ≡ PV (ϕ, t) exp 3 dtH +3 [H(ϕ) − H ] PV
  26. 26. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky II: large scales Global picture: physical probability distribution function, PV : ∂ ∂ ∂PV = −D(1) (ϕ)PV + D(2) (ϕ)PV ∂t ∂ϕ ∂ϕ PV (ϕ, t) ≡ +3 [H(ϕ) − H ] PV PV (ϕ, t) exp 3 dtH Dimensionless version:   ∂ ∂  x2n−1 PV (x, t ) = − PV (x, t ) ∂t ∂x xn 1 + γ 3xn 2 3n ∂ λ x 2 3 n + 1 + xn /T PV (x, t ) + [x − xn ] PV (x, t ) , ∂x2 12n2 8π 2 2n e −1 V (ϕ) = 4 λMp 2n ϕ Mp 2n
  27. 27. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky II: large scales Transforming to a Schr¨dinger-like equation: o dx σ→ D(2) (x) 1 ∂ (2) (x) ∂σ D ∂ → ∂x Pn (x) → 1 1 exp (2) (x)−3/4 2 D dx D(1) (x) ψn (σ) , D(2) (x) ∂2 ψn (σ) − VS (σ)ψn (σ) = Λn ψn (σ) , ∂σ 2 where (2) (2) (2) (1) 3 (D,x )2 D,xx D,x D(1) D,x (D(1) )2 VS (σ) = − − + + . 16 D(2) 4 2 2D(2) 4D(2)
  28. 28. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky II: large scales Schr¨dinger-like equation ⇒ Sturm-Liouville problem: o L≡ d d p(x) y + q(x)y = −λw(x)y dx dx ca y(a) + da y (a) = 0 cb y(b) + db y (b) = 0 . cn φn (x) y(x) = n Lφn = −λn w(x)φn λ0 < λ1 < λ2 . . . −py λ0 = miny(x) dy dx b b a + a b a dx p dy dx dx y 2 w(x) 2 − qy 2 (Rayleigh quotient)
  29. 29. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Extending Starobinsky II: large scales Eternal inflation case: Cn ψn (σ)eΛn t Ψ(σ, t) = n dσ Λ0 = −minψ(σ) dψn dσ 2 2 + VS (σ)ψn 2 dσ ψn Λ0 > 0 ⇔ there is some interval σ1 < σ < σ2 so that VS < 0
  30. 30. Outline Warm inflation Starobinsky I Observations Starobinsky II Extending Starobinsky II: large scales Ex.: φ4 potential. Cold limit: n=2 VS =− 3 −2 5 σ − + σ 2 − 3π 16 2 3 −1 σ λ Final remarks
  31. 31. Outline Warm inflation Starobinsky I Observations Starobinsky II Extending Starobinsky II: large scales High dissipation and temperature limit: n=2 VS = Then, if λT < 288π 2 γ(γ+2) 72π 2 288π 2 − γλT (2 + γ) σ −2 (γλT )2 eternal self-reproduction is suppressed! Final remarks
  32. 32. Outline Warm inflation Starobinsky I Observations Starobinsky II Final remarks Answers to the proposed questions: Non-isentropic stochastic polynomial chaotic inflation model is viable in face of the most recent results for the cosmological parameters. (more details and results: Rudnei O. Ramos and L. A. da Silva JCAP03(2013)032) Temperature and dissipative effects can suppress the eternal self-reproduction of H-regions.

×