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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.). IR divergence problem. q. 1. Introduction. During inflation. (Quesi-) Massless fields.

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S olution to the ir divergence problem of interacting inflaton field

Solution to the IR divergence problem of interacting inflaton field

Yuko Urakawa (Waseda univ.)

in collaboration with

Takahiro Tanaka ( Kyoto univ.)


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”


The Limit of Observations

Q

( 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


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.”


Talk Plan

1. Introduction

2. Observable quantities

How to define the observable n-point functions

3. Proof of IR regularity

4. Summary


ζ(τ)

ζ

~ 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 )


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


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


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


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


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


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


Talk Plan

1. Introduction

2. Observable quantities

How to discuss the observable n-point functions

3. Proof of IR regularity

4. Summary


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


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


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)


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


4. Summary

We showed IR regularity of obeserved N-point function

for the general non-linear interaction.

Observable N-point function

Not Correlated

α


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