1 S olution to the IR divergence problem of interacting inflaton field Y uko U rakawa (Waseda univ.) in collaboration with T akahiro T anaka ( Kyoto univ.)

2 [ One loop corrections ] IR divergence problem Quadratic interaction ~ ζ 4 1. Introduction ∫d 3 q P (q) = ∫ d 3 q /q 3 + ( UV contributions ) Scale-invariant power spectrum on large scale P (k) ∝ 1 / k 3 (Ex.) inflaton φ, curvature perturbation ζ → (δ T / T ) CMB During inflation (Quesi-) Massless fields Bunch-Davies vacuum IR contributions “ Logarithmic divergence ” q u k ∝ k － 3/2 for k / a H << 1 → P (k) ∝ 1 / k 3

3 The Limit of Observations 1. Introduction If ＝ ∫ d 3 k P (k) ～ ∫ d 3 k / k 3 Scale invariance --- Assured only within observable universe → Include assumption on unobservable universe. （ Ex. ） Chaotic inflation Large scale fluctuation → Large amplitude → Over-estimation of fluctuations. :Averaged value in observable region : Averaged value in whole universe ( Q - ) 2 ) 2 Q ⅹ Large fluctuation we cannot observe

4 Topics in this Talk 1. Introduction Avoid assumptions on the region we cannot observe until today Important to clarify the early universe IR divergence Non-linear quantum effects To compute non-linear quantum effects → Need to solve the IR problem [ Our Philosophy ] (Ex.) Loop corrections, Non-Gaussianity “ The observable quantity does not include IR divergence. ” We show...

5 Talk Plan How to define the observable n-point functions 1. Introduction 2. Observable quantities 3. Proof of IR regularity 4. Summary

6 2. Observable quantities W L (x) : Window Momentum space Averaged value in observable region 2.1 Local curvature perturbation ζ obs [ Observable fluctuation ] ~ L ζ(τ)ζ(τ) ζ ⅹ → 0 ( as k or k’ → 0 ) in IR limit suppress

7 2. Observable quantities with k < 1/L is suppressed Long wavelength mode k < 1/L → Local averaged value 2.1 Local curvature perturbation ζ obs F IR suppression of [ Loop corrections ] q can regulate only external momenta k, k’ Logarithmic divergence from internal momentum q D.Lyth (2007) IR Cut off on q L ~ 1/ H 0 Log kL Local curvature perturbation Not include IR cut off for internal momentum q

8 2. Observable quantities with k < 1/L After Horizon crossing time Our local universe selects one value 2.2 Projection Superposition about Fluctuate through Non-linear interaction with short wavelength mode | Ψ > L = ∫d ζ (τ) | ζ (τ) ＞ ＜ ζ (τ) | Ψ > L State of Our universe Superposition of the eigenstate for ~ L ζ(τ)ζ(τ) Without this selection effect, is evaluated for all possibilities Over - estimation of Quantum fluctuations

9 2. Observable quantities 2.2 Projection Stochastic inflation Quantum fluc. Stochastic evolution ζ 3, ζ 4 … Classical fluc. Coarse graining → Decohere enough → Focus on one possibility about A.Starobinsky (1985) Logarithmic divergence ← Quantum fluctuation of IR Non-linear interacting system To discuss IR problem We should not neglect quantum fluctuation of IR modes

10 2. Observable quantities 2.2 Projection Localization of wave packet Observation time Decoherence Cosmic expansion Statistical Ensemble τ = τ f Various interactions Early stage of Inflation ψ ( ζ (τ) ) Superposition of Correlated Not Correlated Each wave packet Parallel Our local universe One wave packet is selected ψ ( ζ (τ) )

11 2. Observable quantities 2.2 Projection Localization of wave packet Observation time Decoherence Cosmic expansion τ = τ f Various interactions Early stage of Inflation Correlated Not Correlated Localization operator Selection Dispersion σ Not to destroy decohered wave packet σ > ( Coherent scale δ c ) α σ

12 2. Observable quantities 2.2 Projection Localization Operator N-point function with Projection Selection | 0 > a Bunch – Davies vacuum Observable N-point function IR regularity ~ L ζ(τ)ζ(τ)

13 Talk Plan How to discuss the observable n-point functions 1. Introduction 2. Observable quantities 3. Proof of IR regularity 4. Summary

14 3. Proof of IR regularity Action All terms in S 3 [ζ], S 4 [ζ] ∂ 0 or ∂ i Power – low interaction without derivative ・ z = aφ/ H IR divergence from BD vacuum : Time independent Suppressed by ∂ 0 or ∂ i Heisenberg picture← Expand by ζ Interaction picture IR regularity for ζ 0

15 3. Proof of IR regularity IR regularity for ζ 0 u k : Mode f.n. for B-D vacuum ～ u k * u k ∝ 1/ k 3 u k p k Large Dispersion Highly squeezed IR mode [ Bogoliubov transformation ×2 ] v 0 {u k } BD → ζ(τ) u k, k < 1/L v 0 → ζ(τ) v k → ζ(τ) {v k } v 0 Squeezed k=0 v k = v k {v k } v 0

16 3. Proof of IR regularity How IR divergence are regulated? Coherent state for ∫d β | β > < β | = 1 N-point function for each (β, γ) ： Finite P(α) → N point f.t. ≠ Finite region {β} ～ Eigenstate for ζ(τ i ) Observed N-point f.n. ※ Localization P(α) is essential ∫d γ | γ > < γ | = 1 Feynman rule α (β, γ) Finite (β, γ) Infinite

17 3. Proof of IR regularity IR regularity for ζ 0 IR regular function ×Πk How IR divergence are regulated? Coherent state for ∫d β | β > < β | = 1 N-point function for each | β > ： Finite P(α) → Finite region {β}, N point f.t. ≠ 0 ～ Eigenstate for ζ(τ i ) β＝ ζ(τi)β＝ ζ(τi) Observed N-point f.n. Finite Squeezing : IR mode → ζ(τ) Finite wave packet Localization P(α) is essential

18 4. Summary We showed IR regularity of obeserved N-point function for the general non-linear interaction. Observable N-point function Not Correlated α