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Loop corrections to the primordial perturbations

Yuko 　 Urakawa (Waseda university)

Kei-ichi Maeda (Waseda university)

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　 Motivation

Non-linear perturbations

Quantum fluc. of inflaton

Transition from Quantum fluctuation to Classical perturbation

Observable quantity

[Inflation model]

Minimally coupled single scalar field

+ Einstein – Hilbert action

Loop corrections from “Stochastic gravity ”.

More information about the inflation model Non-linear perturbations

3

Closed Time Path formalism

B.L.Hu and E.Verdaguer (1999) Stochastic gravity

h h

φ

φ

h

h hh

Interacting system : Scalar field φ & Gravitational field Fluc. h

Evolution of the in-in expectation value. < in 　 | ** | in > 　

Effective action in the CTP formalism

Stochastic gravity

[ Effective action in CTP]

Sub-Planck region Quantum fluc. of scalar φ ＞＞ Quantum fluc. h

Integrate out only φ

time

“Coarse–graining ”

h External line, h ∈ ∈ 　 Internal line h ～ Classical external field

@ Path integral of ΓCTP

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A.A.Starobinsky (1987)

Evolution of Gravitational field ← Quantum φ @ Sub-Planck region

Evolution of Long-wave mode, φsp ← Quantum fluc. of Short-wave mode, φsb

Imaginary part in ΓCTP [g]

Quantum Fluc. of φ

　

Stochastic inflation

Stochastic gravity

Quantum Fluc. of φsb

integrated outφsp φsb

Self-interaction of φ

g ab φintegrated out

Interaction between φ and g

Imaginary part in ΓCTP [φsp]

→ 　 Stochastic variable ξab

Transition from Quantum fluc. to Classical perturbations

ΓCTP with “Coarse–graining ”

“ Loop corrections “

Langevin type equation

→ 　 Stochastic variable ξ

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Application to the inflationary universe

ξab → Fluc. of Tab for φ on 　 g

Memory term

Habcd (x , y) ← 　 Re[ΓCTP ]

Quantum effect of φ

Background g : Slow-roll inflation

Fluctuations (h , φ ) 　→　 ΓCTP

h h

φ

φ

h hφ h h

φ φ

φ

etcδΓCTP / δhab = 0

Nabcd (x , y) ← 　 Im[ΓCTP]

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Perturbation

Non-linear effect of φ 　→ Couples these tree modes

Coupling

1. Stochastic variable ξab has also Vector and Tensor part.

2. Memory term

scalar + vector + tensor δgab

One loop corrections to Scalar & Tensor perturbations

Flat slicing

h h

φ

φ

Metric ansatz

scalar tensor scalar

Coupling among the three modes: scalar ,vector, and tensor

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　 UV divergence

Renormalization 　 　 IR divergence

[ Initial condition ] for － k τi > 1

IR divergenceh h

φ

φ q

k - q

kk

Neglection

Mode eq. for φI in Interacting picture

UV divergent part ・・・ Decaying mode in superhorizon

D.Podolsky and A.A.Starobinsky (1996)

Unphysical initial condition

Beginning of Inflation τisubhorizon superhorizon

Quantum effect ： Like in Minkowski sp.

Cut off

Need not care about UV divergence in “Observable quantity”

→ Quantum fluc. ～ Classical stochastic fluc.(Observable)

∝ k-3

q

Hi ０

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Scalar perturbations

superhorizon limit

Gauge invariant ζ ∝ 　 δT / T

( Leading part of Loop corrections ) / (Linear perturbation) ～ (H/mpl)2.

[ Results ]

Nk < exp[1/2(ε － ηV)]If 2 (ε － ηV) log(k/Hi) < 1

Amplified by the Nk

Similar ampfilication @ S.Weinberg (2005) & M.S.Sloth(2006).

ττi -

1/k

e-foldings Nk

for ηV 　 log k|τ| ＜＜　 1

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Tensor perturbations

c.f. Linear perturbation

( LHS ) Evolution eq. for HT(t) in Linear perturbation

( RHS ) Amplification from Quantum φ (Due to Non-linear interactions)

[ Results ]

No amplification in terms of the e-foldings.

No IR divergence.

( Leading part of the loop corrections ) / (Linear perturbation) ～ (H/mpl)2.

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　 Summary

Stochastic gravity

Stochastic gravity

Amplitude (H/m∝ pl)4

Amplified by Nk

・ Non-linear quantum effect

・ Transition from Quantum fluc. to Classical perturbations

One Loop corrections

Both the scalar perturbations and the tensor perturbations

Scalar perturbations

Tensor perturbations

No Amplification by Nk

No IR divergence.

h h

φ

φ q

k - q

kk