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Loop corrections to the primordial perturbations

Yuko 　 Urakawa (Waseda university)

Keiichi Maeda

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　 Motivation

Global dependence on the potential of inflaton among loop corrections

Two point correlation function of curvature perturbation can be determined from the behavior on the horizon crossing time

We can pick up only local information about the inflation model.

Linear perturbations

Loop effect to two point correlation function of the curvature perturbations and tensor perturbations

from Stochastic gravity

(ex) V(φ) ，　 V’(φ) ， V’’(φ) ．．．

How about non-linear perturbations ?

h h

φ

φ

vertex 　 hφφ vertex 　 φ4

etc

To Search Global time dependence among loop effects

1. Dependence on Vertex operators

2. Dependence on the Background field evolution

3

Compton wavelengthPlanck scale

10-33cm 10-13cm

great difference

Basic idea of Stochastic gravity

There is some region where we can approximate as,

gravitational field 　→　 partially quantized

matter field 　→　 fully quantized

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

Effective action in stochastic gravityFeynman-Vernon’s influence function (or IN-IN formalism)

Effective action SIF , which describes the quantum effect of the matter field

[Total effective action]

Stochastic gravity

h h

φ

φ

h

h hh

Gravitational field is treated as a external field.

It does not contribute as the propagator of the internal line.

Causal equation

Gauge invariant equation

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Einstein-Langevin equation

Stochastic gravity

Due to the direct change of the gravitational field

Due to the back reaction throughφ

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

These three modes are not independent each other due to the non-linear effect of the scalar field.

Coupling

１． Stochastic variable ξab 　（← Quantum fluctuations of the scalar field ） Tensor type and vector type equation also have an-isotropic pressure of ξab.

２． Memory termscalar + vector + tensor δgab

These quantum effects are described by the propagator of the scalar field.

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　 Effective action〔 Quantum correction of 　 φ 〕

Global feature of the inflation model

quantum fluctuation

１． Vertex operator

αm = O ((εSR)m/2) ← 　 We can prove by the mathematical deduction.

In case, the slow-roll condition are satisfied. 　　 εSR ≡ε ， ηV, ηH, η

２． Propagator

~ H2

ηV ≡ V’’/ κ2 V

As its coefficient, the vertex operators include the information of the potential.

In principle, the higher loop correction include the more global information of the potential.

The propagator depends on the evolution of the background spacetime.

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　 Effective action

Loop corrections 　

αm = O ((εSR)m/2)

２． Propagator

~ H2

１． Vertex

h h

ψ

ψ

◆ 　 (κH)2 [tree graph]

coupling between g and φ

α1

h hψ

　 α1

(κH)2 V’(φ), H

vertex 　 h ・ ψ 　←　 This interaction is included in linear analysis.

◆ 　 (κH)4 [loop graph]

◆ 　 (κH)6 [loop graph]

α2 α2 (κH)2

+ V’’(φ)

+ V(3)(φ) + V(4)(φ)

(κH)2

(κH)2h h

ψ

ψ

ψ

α3α3

(κH)2

(κH)2

　 α1

h

　　 α4

α1

h (κH)2 (κH)2

(κH)2

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Perturbations of Einstein-Langevin equation

Scalar perturbations

Metric ansatz

・ Gauge condition for scalar perturbations

・ We have neglected vector perturbations.

superhorizon limit

Memory term includes the metric perturbations.

two-component Einstein equation

But they are suppressed by slow-roll parameters.

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Perturbations of Einstein-Langevin equation

Tensor perturbations

Metric ansatz

・ Gauge condition for scalar perturbations

・ We have neglected vector perturbations.

c.f. Linear perturbation → source free

+ Slow-roll parameter constant

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Global time dependence among Loop correctionsUp to second order perturbations

S. Weinberg (2005), (2006)

Comparison with preceding researches

【 Interaction 】

massless scalar field 　（→ σ ） , fermions, etc

In most of standard inflation models, although regularization problem has been left, there are no global time dependence among loop corrections. ( Loop corrections for fixed internal momentum)

【 Results 】　

scalar perturbations & inflaton → curvature perturbation in comoving slicing 　 ζtensor perturbations → γ

ζ ζ

σ

σ

h h

φ

φ

ζ

ζ

ζ

Stochastic gravity

Comparison Number of σ field

Constant number

c.f.

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　 Ultraviolet divergence

Renormalization 　 　 Infrared divergence

ηV ≡ V’’/ κ2 V

[ Initial condition ]

This infrared divergence is due to the unphysical initial condition.

in superhorizon region – k τ<< 1

h h

ψ

ψ q

k - q

kk

We cannot impose this initial condition to the mode which was outside horizon on the beginning of the inflation.

To avoid this unphysical divergence, we have introduced the cutoff Hi = H (τi) . (i.e. q H≧ i )

Based on the discussion with A.A.Starobinsky

h h

ψ

ψ q

k - q

kk

If m > 0, this ultraviolet divergence part shall decay in superhorizon region.

We have neglected this decaying part in superhorizon region.

c.f. Physical meaning of the neglection of the decaying mode A.A.Starobinsky C.Q.G. 13 (1996) 377

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　Summary

・ Vertex operator

◆ 　 (κH)2 [tree graph] V’(φ), H 　　　←　線形摂動

◆ 　 (κH)4 [loop graph]

◆ 　 (κH)6 [loop graph]

+ V’’(φ)

+ V(3)(φ) 　＆　 V(4)(φ)

Scalar perturbation も Tensor perturbation も、 super horizon でほぼ一定。

Global time dependence on the potential of inflaton among loop corrections

・ Evolution of the background field

◆ 　 (κH)4 [loop graph]

Tensor perturbation の方が slow-roll parameter に対する依存性が弱い。