S olution to the IR divergence problem of interacting inflaton field

1. Solution to the IR divergence problem of interacting inflaton field Yuko Urakawa (Waseda univ.) in collaboration with Takahiro Tanaka ( Kyoto univ.)

2. IR divergence problem q 1. Introduction During inflation (Quesi-) Massless fields Scale-invariant power spectrum on large scale P (k) ∝ 1 / k3 (Ex.) inflaton φ, curvature perturbation ζ →(δ T / T )CMB Bunch-Davies vacuum u k ∝k－3/2 for k / a H << 1 → P (k) ∝ 1 / k3 [ One loop corrections ] Quadratic interaction~ ζ4 ∫d3q P (q) =　∫ d3q /q3 + ( UV contributions ) IR contributions “ Logarithmic divergence”

3. The Limit of Observations Q ⅹ ( Q - <Q> )2 << ( Q -<Q>)2 <Q> :Averaged value in observable region <Q> : Averaged value in whole universe Large fluctuation we cannot observe 1. Introduction Scale invariance --- Assured only within observable universe If ＝　∫d3k P (k) ～ ∫d3k / k3 → Include assumption on unobservable universe. → Over-estimation of fluctuations . （Ex.） Chaotic inflation Large scale fluctuation → Large amplitude

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

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

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

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

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

9. Stochastic inflation ζ3, ζ4 … Classical fluc. Quantum fluc. Stochastic evolution Coarse graining → Decohere enough → Focus on one possibility about 2. Observable quantities 2.2 Projection A.Starobinsky (1985) @ Non-linear interacting system Logarithmic divergence ← Quantum fluctuation of IR modes To discuss IR problem We should not neglect quantum fluctuation of IR modes

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

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

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

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

14. Action IR divergence from BD vacuum : Time independent Suppressed by ∂0 or　 ∂i ・ z = aφ/ H 3. Proof of IR regularity Power – low interaction without derivative All terms in S3[ζ] , S4[ζ] ∂0 or　 ∂i ζ @ Heisenberg picture ← Expand by ζ0 @ Interaction picture IR regularity for ζ0

15. IR regularity for ζ0 pk uk {vk } {uk } BD {vk } uk , k < 1/L → ζ(τ) v0 v0 → ζ(τ) vk → ζ(τ) v0 vk = vk v0 3. Proof of IR regularity uk : Mode f.n. for B-D vacuum Highly squeezed IR mode <ζk ζk > ～ uk* uk ∝ 1/ k3 LargeDispersion [ Bogoliubov transformation ×2 ] Squeezed k=0

16. How IR divergence are regulated? α Finite (β, γ) ～ Eigenstate for ζ(τi) 3. Proof of IR regularity Coherent state for ∫d β | β > < β | = 1 ∫d γ | γ > < γ | = 1 N-point function for each (β, γ) ： Finite Observed N-point f.n. Feynman rule Finite P(α) → N point f.t. ≠ 0@ Finite region {β} ※ LocalizationP(α) is essential Infinite (β, γ)

17. IR regularity for ζ0 How IR divergence are regulated? Squeezing : IR mode → ζ(τ) Finite wave packet β＝ ζ(τi) ～ Eigenstate for ζ(τi) Coherent state for IR regular function ×Πk 3. Proof of IR regularity ∫d β | β > < β | = 1 N-point function for each | β > ： Finite P(α) → Finite region {β} , N point f.t. ≠ 0 ObservedN-point f.n. Finite LocalizationP(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