1 S olution to the IR divergence problem of interacting inflaton field Y uko U rakawa (Waseda univ.)...
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Transcript of 1 S olution to the IR divergence problem of interacting inflaton field Y uko U rakawa (Waseda univ.)...
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
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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
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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
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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
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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
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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) )
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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
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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)
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Talk Plan
How to discuss the observable n-point functions
1. Introduction
2. Observable quantities
3. Proof of IR regularity
4. Summary
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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
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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
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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
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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
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4. Summary
We showed IR regularity of obeserved N-point function
for the general non-linear interaction.
Observable N-point function
Not Correlated
α