1. Influence on observation from IR / UV divergence during inflation Yuko Urakawa (Waseda univ.) Y.U. and Takahiro Tanaka 0902.3209 [hep-th] Y.U. and Takahiro Tanaka 0904.4415[hep-th] Alexei Starobinsky and Y.U. in preparation

2. 1 Contents ・ Influence on observables from IR divergence ・ Influence on observables from UV divergence Y.U. and Takahiro Tanaka 0902.3209 Y.U. and Takahiro Tanaka 0904.4415 Alexei Starobinsky and Y.U. 090*.**** - Single field case - - Multi field case - Primordial fluctuation generated during inflation

3. 2 1. Introduction 3. IR divergence problem - Single field - ► Outline 2. Cosmological perturbation during inflation 4. IR divergence problem - Multi field - 5. UV divergence problem 6. Summary and Discussions

4. 3 ► Cosmic Microwave Background WMAP 1yr/3yr/5yr… 1. Introduction Almost homogeneous and isotropic universe with small inhomogeneities

5. Small scale → ← Large scale 4 ► CMB angular spectrum ΩΛ ΩΛ ΩΛ ΩΛ 1. Introduction Harmonic expansion Ωm Ωm Ωm Ωm Ωb Ωb Ωb Ωb ΩK ΩK ΩK ΩK P P Primordial spectrum

6. 5 ► CMB physics 1. Introduction K<0 K>0 K=0

7. 6 ► Sachs-Wolfe (SW) effect Flat plateau SW effect : Dominant effect ◆ Last Scattering surface z~1091 Inhomogeneity gravitational potential → red shift → temperature 1. Introduction

8. inflation ► Evolution of fluctuation Physical scale k : comoving wave number Horizon scale Horizon cross Horizon reenter

9. ► Adiabatic fluctuation inflation  ~ constant LS For at LSS

10. 9 ► WMAP 5yr date Almost scale invariant, Almost Gaussian … Consistent to the prediction from “Standard” inflation ( Single-field, Slow-roll) 95 ％ C.L. Pivot point * No running

11. 10 ► Beyond linear analysis Within linear analysis Observational date → Not exclude other models More information from Non-linear effects ･ Non-Gaussianity ･ Loop corrections 1. Introduction WMAP 5yr 95 ％ C.L. → PLANCK (2009.5)

12. 11 ► IR / UV divergences ◆ During inflation Quantum fluctuation of inflaton Quantum fluctuation of gravitational field Classical stochastic fluctuation Observation → Clarify inflation model ?? Classicalization Ultraviolet (UV) & Inflared (IR) divergence Regularization is necessary

13. 12 1. Introduction 3. IR divergence problem - Single field - ► Outline 2. Cosmological perturbation during inflation 4. IR divergence problem - Multi field - 5. UV divergence problem 6. Summary and Discussions

14. 13 Liner analysis

15. 14 ► Comoving curvature perturbation ◆ Gauge invariant quantity Spatial curvature Fluctuation of scalar field “Gauge invariant variable”

16. 15 ► Gauge invariant perturbation Gauge invariant perturbation Completely Gauge fixing Equivalent Flat gauge Gauge invariant Comoving gauge “Completely gauge fixing”

17. 16 ► Liner perturbation ◆ Single field inflation model Comoving gauge GW Non-decaying mode as k/aH → 0

18. 17 ► Adiabatic vacuum Positive frequency mode f.n. → Vacuum ( Fock space ) ◆ Initial condition In the distant past |η| → ∞, Adiabatic solution ⇔ k>>1 Much smaller than curvature scale ~ Free field at flat space-time

19. 18 ► Scalar perturbation Almost scale invariant  ~ constant hoc

20. 19 ► Chaotic inflation Reheating Inflation goes on Larger scale mode → Exit horizon earlier → Larger amplitude Red tilt n s < 1

21. 20 ► Tensor perturbation ◆ Initial condition In the distant past |η| → ∞, Adiabatic solution ◆ Power spectrum Almost scale invariant, Red tilt

22. 21 Quantum correlation

23. 22 ► Linear theory x yz 0 x y (i) Two point fn.(ii) Three point fn. Transition from y to x

24. 23 ► Non-linear theory ← Expansion by free field x y λ (i) Two point fn. x y x y x y x y O(λ 0 )O(λ 1 )O(λ 2 ) etc

25. 24 ► Non-linear theory λ (ii) Three point fn. O(λ 1 )O(λ 3 ) etc x yz x yz λ x yz

26. 25 ► Summary of Interaction picture Propagator ↑↑ Vertex 1. Write down all possible connected graphs 2. Compute the amplitude of each graph Feynman rule x y z kk q Fourier trans. Loop integral

27. 26 Non-linear perturbation

28. 27 ► Interests on Non-linear corrections Primordial perturbation ζ x yz xy x yz w x y z x y and so on… More information on inflation

29. 28 Comoving gauge ► ADM formalism S = S EH + S φ = S [ N, N i, ζ ] Hamiltonian constraint ∂ L / ∂ N = 0 N = N[ζ] Momentum constraint ∂ L / ∂ N i = 0 N i = N i [ζ] → e ρ : scale factor S [ N, N i, ζ ] = S [ ζ ] ◆ Lagrange multiplier N / N i Maldacena (2002)

30. 29 ► Non-linear action 1 st order constraints 2 nd order constraints (ex) 1 st order constraints

31. 30 ► NGs / Loop corrections 2002 J.Maldacena “Quantum origin” ( Mainly until Horizon crossing) Single field with canonical kinetic term NG → Suppressed by slow-roll parameters 2005 Seery &Lidsey 2005, 2006 S.Weinberg Loop correction amplified at most logarithmic order Single & Multi field(s) with non-canonical kinetic term NG → Dependence on the evolution of sound speed IR divergence in Loop corrections → Logarithmic 2007 M.Sloth 2007 D.Seery 2008 Y.U. & K.Maeda 2004 D.Boyanovsky and so on

32. 31 ► IR divergence problem ∫d 3 q |ζ q | 2 = ∫ d 3 q /q 3 q kk' Momentum ( Loop )integral Scale-invariant ◆ One Loop correction to power spectrum Mass-less field ζ Next to leading order Log. divergence

33. 32 1. Introduction 3. IR divergence problem - Single field - ► Outline 2. Cosmological perturbation during inflation 4. IR divergence problem - Multi field - 5. UV divergence problem 6. Summary and Discussions

34. 33 Primordial perturbation ► Our purpose Loop integral To extract information from loop corrections, we need to discuss … diverge “ Physically reasonable regularization scheme ” Increasing IR corrections Spectrum : Large Dependence on IR cut off ( Note )

35. 34 Fluctuations computed by Conventional perturbation Fluctuations we actually observe ex. CMB ・ Prove “Regularity of observables” ・ Propose “How to compute observables” Strategy ► IR divergence problem Vertex integral ◆ Loop corrections Diverge Finite

36. 35 Violation of Causality

37. 36 ► Non local system ◆ Constraint eqs. : Solutions of Elliptic type eqs. Hamiltonian constraint ∂ L / ∂ N = 0 N = N[ζ] N i = N i [ζ] → Momentum constraint ∂ L / ∂ N i = 0 (ex) 1 st order Hamiltonian constraint Non local term

38. 37 ► Causality A portion of Whole universe η x Observation Initial ζ (x) p. We can observe fluctuations within “Causal past J - (p) ”

39. 38 δQ (x) = Q(x) ‐ Q Q : Average value ◆ Definition of fluctuation t x Observation Initial ζ (x) p. ζ (x) x ∈ J - (p) affected by { J - (p) } c Conventional perturbation theory Q : Average value in whole universe ► Violation of Causality

40. 39 Large scale fluctuation → Large amplitude Q ⅹ Large fluctuation we cannot observe Q on whole universe ( Q - Q ) 2 < < ( Q - Q ) 2 Q on observable region - Chaotic inflation - δ 2 ζ ∝ H 2 / ɛ Amplitude of ζ

41. 40 ◆ Gauge fixing ζ (x) x ∈ J - (p) affected by { J - (p) } c Gauge invariant Completely gauge fixing at whole universe☠ Impossible - We can fix our gauge only within J - (p). - Change the gauge at { J - (p) } c → Influence on ζ (x) x ∈ J - (p) ► Violation of Causality 2

42. 41 Gauge degree of freedom

43. 42 ► Gauge choice NGs / Loop corrections Computed in Comoving gauge Flat gauge Maldacena (2002), Seery & Lidsey (2004) etc.. - Gauge degree of freedom DOF in Boundary condition : Solutions of Elliptic type eqs.

44. 43 ► Boundary condition Solution 2 Solution 1 Arbitrary integral region

45. 44 ► Scale transformation keeping Gauge condition Scale transformation x i → x i = e - f(t) x i ～

46. 45 Solution 2 Solution 1 ► Scale transformation

47. 46 ► Gauge condition Additional gauge condition Change homogeneous mode ＝０ (1) Observable fluctuation (2) Solution of Poisson eq. ∂ -2 “Causal evolution” : Not affected by { J - (p) } c Averaged value at J - (p)

48. 47 ► Gauge invariant perturbation ∂ L / ∂ N = 0 ◆ Naïve understanding Local gauge condition No Influence from { J - (p) } c Fix Gauge within J - (p) → Determine ζ (x) x ∈ J - (p) Recovery of Gauge invariance

49. 48 ► Quantization Adiabatic vacuum ◆ Initial condition : Curvature at local comoving gauge : Curvature at ordinal comoving gauge P (k) ∝ 1 / k 3 Divergent IR mode Gauge transformation We prove IR corrections of are regular.

50. 49 ► Regularization scheme “Cancel” IR divergence Extremely long inflation Higher order corrections might dominate lower ones. Validity of Perturbation ?? Effective cut off by k~ 1/L t Exceptional case L t : Scale of causally connected region

51. 50 ► Regularization scheme Dotted line : horizon scale Thick line : “Suppression scale” → UV cut off (i) Well suppressed (ii) For l-th loop corrections s.t. Dominated by contributions from the early time

52. 51 1. Introduction 3. IR divergence problem - Single field - ► Outline 2. Cosmological perturbation during inflation 4. IR divergence problem - Multi field - 5. UV divergence problem 6. Summary and Discussions

53. 52 ► Multi-field generalization Background trajectory （2） ♯ ≧ 2（2） ♯ ≧ 2（2） ♯ ≧ 2（2） ♯ ≧ 2 Gauge invariant → Still diverges δσ (x) = δσ (x) ‐ δσ Local average = 0 ～ （ 1 ） ♯ = 1 δs δσ IR regular ◆ Local flat gauge ♯ : Number of IR divergent fields ✔

54. 53 πk πk δs k IR mode ∝ 1 / k 3 Highly squeezed phase space ＜ O(x) O(y) O(z)… ＞ O = δσ, δ s ～ ～ ► IR divergence in Multi-field model ☠ Origin of IR divergence ◆ Squeezed wave packet

55. 54 πk πk δs k IR mode ∝ 1 / k 3 Highly squeezed phase space ＜ O(x) O(y) O(z)… ＞ O = δσ, δ s ～ ～ A portion of wave packet ► IR divergence in Multi-field model Observable fluctuation Prove IR regularity of observables ◆ Squeezed wave packet

56. 55 Decoherence | δs > One of Wave packet → Realized Wave packet of Wave packet of | δs > ► Wave packet of universe Observation time t = t f Early stage of Inflation Superposition of CorrelatedUncorrelated Statistical Ensemble Cosmic expansion Various interactions

57. 56 t = t f ► Parallel world t = t i Pick up Causally disconnected universe Our universe In, another wave packet may be picked up. However, we cannot know what happens there.

58. 57 If finite ← Finite ► Projection α σ ＜ P(α) O(x) O(y) … ＞ Proof of IR regularity Our “Observables” ≩ Actual observable correlation fn. ⊋

59. 58 π δs ＜ P(α) O(x) O(y) O(z)…. ＞ O = δσ, δ s ～ ～ After decoherence, a portion of wave packet contributes Momentum integrals Regular Temporal integral Logarithmic secular evolution ► Regularization scheme phase space ◆ Loop integrals

60. 59 1. Introduction 3. IR divergence problem - Single field - ► Outline 2. Cosmological perturbation during inflation 4. IR divergence problem - Multi field - 5. UV divergence problem 6. Summary and Discussions

61. 60 ► UV regularization ζ : curvature perturbation / h ij : GW Diverge in x → y limit ◆ Adiabatic regularization UV mode ~ Solution in adiabatic approximation Parker & Fulling (1974) Regular ☠ Divergent Adiabatic expansion

62. 61 ► Influence from Adiabatic reg. Parker (2007), Parker et.al. (2008/2009) ×( Slow-roll parameter ) Amplitudes of ζ / GW suppressed by subtraction terms at horizon crossing time Single field inflation

63. 62 ► No Influence from Adiabatic reg. Exact solution Solution in Adiabatic approx. Constant value Decay Super horizon limit A.Starobinsky & YU (2009) Negligible

64. 63 ► Summary - IR regularization - Comoving gauge Flat gauge + Local gauge G ： “Causality” is preserved ◆ Single field case NGs/Loop corrections are free from IR divergence ( except for models with extremely long durations ) ◆ Multi field case Gauge fixing is not enough to discuss observables To consider them, we need to consider “decoherence”. We cannot deny the existence of secular evolution.

65. 64 ► Summary - UV regularization - ◆ Adiabatic regularization Regularize UV divergence We should introduce subtraction terms for all modes Subtraction term decays during cosmic evolution → No-influence on observables, which appears in the coincidence limit However…

66. 65 - Supplement -

67. 66 τ = τ i 3. Regularization scheme ~ Multi field ~ ► Decoherence process Initial state : adiabatic vacuum | 0 > ad Correlated Superposition of | > | 0 > ad = ∫d | > ad Include the contribution from all wave packets ad ad Overestimation

68. 67 @ Causally connected region Expansion by G F, G D, G +, G － Evolution of time G R : Regular in IR limit GFGF GDGD G + or G － x ・ x’ ≠0, Finite value CTP Expansion by G R ◆ Closed Time Path ◆ Expansion by Retarded Green f.n. ► Expansion by Retarded Green fn.

69. 68 (Ex.) N = 4 R [ Detailed exp. ] ► Expansion by Retarded Green fn. = ・・・・・・ + R = R ++RR =+ R+ RR + ・・・

70. 69 ► Expansion by Retarded Green fn.2 ◆ Contraction R Contraction k k' R

71. 70 IR regular ?? 3. Proof of IR regularity [ Detailed exp. ] = ∑ ( IR regular functions G R m ) × a a ~ Eigenetate for ζ I with finite wave packet FiniteFinite region FiniteInfinite region → ∞ ・ Without P (α) ・ With P (α) ► Mode expansion

72. 71 Stochastic inflation → Decoherence Necessity to consider Local quantity ( |x| < L )Lyth (2007) Local quantity Cut off only for external momentum → Introduction of IR Cut off 1/L Bartolo et. al (2008) Riotto & Sloth (2008) Enqvist et.al. (2008) k < kc Stochastic fluctuation Neglecting a part of quantum fluctation Include the artificial cut-off scale Under-estimation of IR corrections → Doubtful ► Recent topics