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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.)


S olution to the ir divergence problem of interacting inflaton field

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”


S olution to the ir divergence problem of interacting inflaton field

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


S olution to the ir divergence problem of interacting inflaton field

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.”


S olution to the ir divergence problem of interacting inflaton field

Talk Plan

1. Introduction

2. Observable quantities

How to define the observable n-point functions

3. Proof of IR regularity

4. Summary


S olution to the ir divergence problem of interacting inflaton field

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 )


S olution to the ir divergence problem of interacting inflaton field

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


S olution to the ir divergence problem of interacting inflaton field

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


S olution to the ir divergence problem of interacting inflaton field

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


S olution to the ir divergence problem of interacting inflaton field

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


S olution to the ir divergence problem of interacting inflaton field

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


S olution to the ir divergence problem of interacting inflaton field

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


S olution to the ir divergence problem of interacting inflaton field

Talk Plan

1. Introduction

2. Observable quantities

How to discuss the observable n-point functions

3. Proof of IR regularity

4. Summary


S olution to the ir divergence problem of interacting inflaton field

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


S olution to the ir divergence problem of interacting inflaton field

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


S olution to the ir divergence problem of interacting inflaton field

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)


S olution to the ir divergence problem of interacting inflaton field

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


S olution to the ir divergence problem of interacting inflaton field

4. Summary

We showed IR regularity of obeserved N-point function

for the general non-linear interaction.

Observable N-point function

Not Correlated

α