1

olution to the divergence problem of interacting inflaton field

uko rakawa (Waseda univ.)

in collaboration with akahiro anaka ( Kyoto univ.)

2

[ One loop corrections ]

IR divergence problem

Quadratic interaction 　 ~ ζ4

1. Introduction

∫d3q (q) = 　∫ d3q /q3 + ( UV contributions )

Scale-invariant power spectrum on large scale (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 → (k) 1 / ∝k3

3

The Limit of Observations 1. Introduction

If 　 ∫ d3k (k) ～ ∫ d3k / k3 　 Scale invariance --- Assured only within observable universe

→ Include assumption on unobservable universe.

（ Ex. ） Chaotic inflation

Large scale fluctuation → Large amplitude

→ Over-estimation of fluctuations .

<Q> :Averaged value in observable region

<Q> : Averaged value in whole universe

( Q - <Q> )2 　　 < < ( Q - <Q> )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

WL(x) : Window function

@ Momentum space

Averaged value in observable region

2.1 Local curvature perturbation ζobs

[ Observable fluctuation ]

~ L

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 ζobs

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)

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

ζ3, ζ4 …

Classical fluc.

Coarse graining → Decohere enough→ Focus on one possibility about

A.Starobinsky (1985)

Logarithmic divergence ← Quantum fluctuation of IR modes

@ 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 = τf

Various interactions

Early stage of Inflation

ψ ( ζ (τ) )

Superposition of

Correlated Not Correlated

Each wave packet 　　 Parallel World

@ 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 = τf

Various interactions

Early stage of Inflation

Correlated Not Correlated

Localization operator Selection

Dispersion σNot to destroy decohered wave packetσ > ( Coherent scale δc )

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)

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 S3[ζ] , S4[ζ] ∂0 or 　 ∂ i

Power – low interaction without derivative

・z = aφ/ H

IR divergence from BD vacuum : Time independent Suppressed by ∂0 or 　 ∂ i

ζ @ Heisenberg picture ← Expand by ζ0 @ Interaction picture

IR regularity for ζ0

15

3. Proof of IR regularity IR regularity for ζ0

uk : Mode f.n. for B-D vacuum

<ζk ζk > ～ uk* uk 　∝ 1/ k3

uk

　

pk

　 Large Dispersion

Highly squeezed IR mode

[ Bogoliubov transformation ×2 ]

v0

　

{uk } BD

→ ζ(τ)

uk , k < 1/L v0 → ζ(τ)

vk → ζ(τ)

{vk }

v0

　

Squeezed k=0

vk = vk

{vk }

v0

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. ≠ 0 @ Finite region {β}

～ Eigenstate for ζ(τi)

Observed N-point f.n.

※ Localization P(α) is essential

∫d γ | γ > < γ | = 1

Feynman rule

α

(β, γ)Finite

Finite

(β, γ)Infinite

17

3. Proof of IR regularity IR regularity for ζ0

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(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