インフレーション期の量子トンネリングと 非線形量子ゆらぎ

杉村 和幸 (京大基研 D3) Y TPYUKAWA INSTITUTE FOR THEORETICAL PHYSICS

K. S., D. Yamauchi and M. Sasaki, EPL 100, 29004 (2012) K. S., Phys. Rev. D 88, 025037 (2013)

K. S. and E. Komatsu, JCAP 1311, 065 (2013)

共同研究者：　山内大介（RESCEU）、佐々木節（YITP） 　　　　　　　 小松英一郎（MPA）

（ゆらぎの非ガウス性）

Contents •  Introduction

•  量子トンネリング中の非線形発展

•  Slow-roll inflation中のバブル生成

•  Conclusion

K. S., D. Yamauchi and M. Sasaki, EPL 100, 29004 (2012)

•  Open inflation K. S. and E. Komatsu, JCAP 1311, 065 (2013)

K. S., Phys. Rev. D 88, 025037 (2013)

Contents •  Introduction

•  量子トンネリング中の非線形発展

•  Slow-roll inflation中のバブル生成

•  Conclusion

K. S., D. Yamauchi and M. Sasaki, EPL 100, 29004 (2012)

•  Open inflation K. S. and E. Komatsu, JCAP 1311, 065 (2013)

K. S., Phys. Rev. D 88, 025037 (2013)

インフレーションの背後に潜む物理

p String landscape (Susskind, 2003)

potential for scalar fields

p  Inflation

-  複数スカラー場＆ 複雑なポテンシャル

-  観測とよく一致

-  インフレーションの背後に潜む物理は?

(flatness, homogeneity, power spectrum)

p String landscapeにいかに迫るか？

I.  第一原理からポテンシャルの導出を目指す (string theory側)

-  余剰次元の形状の自由度がスカラー場に対応

II.  観測から証拠を探す (cosmology側)

2

RESEARCH PLAN

My research interests are focused on deepening our understanding of the history of our universe

from its beginning. Inflation, accelerated expansion of the universe in its very early phase, is now

part of the standard cosmology, because it reproduces the current observational data with amazing

precision. However, we have only very little knowledge of the physics behind inflation, which lead to

the following question: what is the physical origin of inflation? All existing inflation models remain to

be just phenomenological, because we don’t have theory at such high energy scales. The energy scale

of inflation is far beyond that of the standard model of particle physics and there is no hope to realize

accelerator experiments at such high energy. However, string theory, the most studied candidate of

the ultimate theory, has the possibility to give the answer to the physics behind inflation. Although

string theory is not completed yet, remarkable progress in it recently offered a new framework on the

cosmology: string theory landscape.

String theory landscape is very attractive framework, which potentially gives the physical origin

of inflation as well as a resolution to the cosmological constant problem. I’m very much interested

in testing it by observation, since the confirmation/exclusion of it has an extremely large impact on

both cosmology and string theory. String theory landscape is a huge landscape of potential for scalar

fields (FIG. 1), which are originated from the degrees of freedom of the extra dimension geometry

in string theory. Although details of landscape are not clear, string theory suggests the existence of

a large number of scalar fields and local potential minima. Scalar fields jump between these local

minima by quantum tunneling, which differentiates inflation models in string theory landscape from

conventional inflation models, generating distinctive observational signatures.

Having contributed to establishing a theoretical foundation on testing string theory landscape

by observation, I set my next goal as making observational predictions from string theory landscape

which enhance the detectability by fully making use of high-precision observational data in the coming

era. For example, while higher order correlation of the fluctuations has started to be observed, future

galaxy and 21cm-line surveys will give us very large volume three dimensional maps of the large scale

structures. For future research, I plan to pay special attention to the following two possible sources

of observable signatures from string theory landscape.

FIG. 1. Image of string theory landscape. The horizontal directions correspond to scalar field values, say φ1

and φ2, and the vertical direction corresponds to potential V (φ1,φ2).

2

RESEARCH PLAN

My research interests are focused on deepening our understanding of the history of our universe

from its beginning. Inflation, accelerated expansion of the universe in its very early phase, is now

part of the standard cosmology, because it reproduces the current observational data with amazing

precision. However, we have only very little knowledge of the physics behind inflation, which lead to

the following question: what is the physical origin of inflation? All existing inflation models remain to

be just phenomenological, because we don’t have theory at such high energy scales. The energy scale

of inflation is far beyond that of the standard model of particle physics and there is no hope to realize

accelerator experiments at such high energy. However, string theory, the most studied candidate of

the ultimate theory, has the possibility to give the answer to the physics behind inflation. Although

string theory is not completed yet, remarkable progress in it recently offered a new framework on the

cosmology: string theory landscape.

String theory landscape is very attractive framework, which potentially gives the physical origin

of inflation as well as a resolution to the cosmological constant problem. I’m very much interested

in testing it by observation, since the confirmation/exclusion of it has an extremely large impact on

both cosmology and string theory. String theory landscape is a huge landscape of potential for scalar

fields (FIG. 1), which are originated from the degrees of freedom of the extra dimension geometry

in string theory. Although details of landscape are not clear, string theory suggests the existence of

a large number of scalar fields and local potential minima. Scalar fields jump between these local

minima by quantum tunneling, which differentiates inflation models in string theory landscape from

conventional inflation models, generating distinctive observational signatures.

Having contributed to establishing a theoretical foundation on testing string theory landscape

by observation, I set my next goal as making observational predictions from string theory landscape

which enhance the detectability by fully making use of high-precision observational data in the coming

era. For example, while higher order correlation of the fluctuations has started to be observed, future

galaxy and 21cm-line surveys will give us very large volume three dimensional maps of the large scale

structures. For future research, I plan to pay special attention to the following two possible sources

of observable signatures from string theory landscape.

FIG. 1. Image of string theory landscape. The horizontal directions correspond to scalar field values, say φ1

and φ2, and the vertical direction corresponds to potential V (φ1,φ2).

弦理論的ランドスケープは見えるか

バブル生成

-  量子トンネリングは一次相転移なのでバブル生成を伴う

p スカラー場の量子トンネリング／バブル生成

V(φ)

量子トンネリング

φ

量子トンネリング／バブル生成に特有な観測シグナルを用いて、弦理論的ランドスケープの観測的検証をめざす

-  インフレーション中にポテンシャル極小点間を遷移

本研究では、特に非線形量子ゆらぎに注目！！

Contents •  Introduction

•  量子トンネリング中の非線形発展

•  Slow-roll inflation中のバブル生成

•  Conclusion

K. S., D. Yamauchi and M. Sasaki, EPL 100, 29004 (2012)

•  Open inflation K. S. and E. Komatsu, JCAP 1311, 065 (2013)

K. S., Phys. Rev. D 88, 025037 (2013)

量子トンネリングと時間発展

V(φ)

量子トンネリング

φ

•  φFからφTへの遷移は時間発展？

No： ある瞬間に突然遷移するのが量子トンネリング

虚時間を用いた方法で状態の発展が記述可能

•  φFからφTへの遷移の過程は記述できる？

Yes:

インスタントンの方法 (Coleman and De Luccia 1980)

φT φF

In-in formalismの拡張

(Maldacena 2003) p N-点関数の表式（量子トンネリングなしの場合）

L = L0 + Lint

P: path ordering operator

-  Cに沿った積分で非線形時間発展の寄与を評価

Re t Im t ��(1� i�)

��(1 + i�) t0

量子トンネリング後の宇宙におけるN-点関数が計算可能に

C

量子状態の非線形発展を記述する手法 ラグランジアン:

D'(t0,x1)'(t0,x2) · · ·'(t0,xN )

E=D0���P '(t0,x1)'(t0,x2) · · ·'(t0,xN )ei

RC dt

Rd3

xLint(t,x)��� 0

E

p 量子トンネリングありの場合

-  トンネリング中の状態の非線形発展の寄与も取り入れる

-  Cを虚時間領域を含むように拡張

(K. S., Phys. Rev. D 88, 025037 (2013))

K. S., Phys. Rev. D 88, 025037 (2013)

Contents •  Introduction

•  量子トンネリング中の非線形発展

•  Slow-roll inflation中のバブル生成

•  Conclusion

K. S., D. Yamauchi and M. Sasaki, EPL 100, 29004 (2012)

•  Open inflation K. S. and E. Komatsu, JCAP 1311, 065 (2013)

K. S., Phys. Rev. D 88, 025037 (2013)

量子トンネリングのシナリオ1: Open inflation

3

tunnelingslow-roll

V(φ)

φφF φN

FIG. 1: Potential for single field open inflation. Slow-rollinflation is triggered by quantum tunneling from the falsevacuum φF to the nucleation point φN .

L RCwall

reheating

nucleation

E1 or E2

FIG. 2: Penrose-like diagram of the universe where open in-flation occurs. The top half corresponds to the universe afterbubble nucleation where time is real. The bottom half corre-sponds to either one of two hemispheres of the Euclidean sphereE1 or E2 where time is imaginary, see also Fig. 3. In an openinflation scenario, slow-roll inflation of an open FLRW universe(homogeneous spatial hypersurfaces are 3-hyperboloids) occursin the R-region, and the universe evolves to the present universethrough reheating.

For simplicity, we ignore multiple-field effects or the possibility of a rapid-roll era soon after the quantum tunneling,though these effects are known to be potentially significant[14, 15]. We leave these issues for future study.

While there are many papers on the power spectrum for these types of model[16, 17, 18], the bispectrum has notyet been studied. The main contribution to the power spectrum can be calculated using free Quantum Field Theory(QFT) theory, however, QFT with interactions is needed to calculate the bispectrum. In this paper, we calculatethe bispectrum by applying the in-in formalism to a universe where the background evolution is given by a CDLinstanton as was done in [19]. It should be noted that though this is a natural extension of the usual in-in formalismin a flat universe[20], as far as we know, there has been no stringent proof for the in-in formalism on a CDL instantonbackground. The proof of this formalism will appear elsewhere[21].

It has recently been shown that a modified initial state (i.e. non Bunch-Davies vacuum state) gives rise to anobservationally interesting bispectrum[22, 23]. In open inflation models, such a modified initial state1 is naturallyrealized as a result of the evolution from the Bunch-Davies vacuum state before the quantum tunneling[16, 17, 18]. Byassuming a modified initial state at some initial time, it has been shown that the bispectrum behaves as B(k, k, k3) ∝k−43 k2 in the squeezed configuration k3 ≪ k1 ∼ k2(≡ k), which is enhanced by the factor k/k3 compared to the usual

local type bispectrum, which behaves as B(k, k, k3) ∝ k−33 k3.

In open inflation models, there are two possible sources of non-Gaussianity. Firstly, as pointed out in [19], a certainamount of non-Gaussianity may be generated in the presence of a large self-interaction near the potential barrier, or

1 In this paper we use “non Bunch-Davies vacuum” state to refer to a state whose positive frequency functions differ from those of the“adiabatic vacuum” state[24]. With this terminology, the state associated with the de-Sitter invariant Wightman function, often referredto as the “Bunch-Davies vacuum” state, is also a “non Bunch-Davies vacuum” state.

-  インフラトンの量子トンネリング

-  インフラトン一定面が3次元双曲面に（青線）

bubble wall

reheating

nucleation time

-  バブル内側でのみスローロール進行

Penrose diagram

-  スローロールの長さで現在の曲率ΩK決まる

-  「我々の宇宙」＝バブル内側

Open inflation

時間

空間 -  Bubble wall（赤線）と交差せず

バブルは見えない

Open inflationからのBispectrum

3

tunnelingslow-roll

V(φ)

φφF φN

FIG. 1: Potential for single field open inflation. Slow-rollinflation is triggered by quantum tunneling from the falsevacuum φF to the nucleation point φN .

L RCwall

reheating

nucleation

E1 or E2

FIG. 2: Penrose-like diagram of the universe where open in-flation occurs. The top half corresponds to the universe afterbubble nucleation where time is real. The bottom half corre-sponds to either one of two hemispheres of the Euclidean sphereE1 or E2 where time is imaginary, see also Fig. 3. In an openinflation scenario, slow-roll inflation of an open FLRW universe(homogeneous spatial hypersurfaces are 3-hyperboloids) occursin the R-region, and the universe evolves to the present universethrough reheating.

For simplicity, we ignore multiple-field effects or the possibility of a rapid-roll era soon after the quantum tunneling,though these effects are known to be potentially significant[14, 15]. We leave these issues for future study.

While there are many papers on the power spectrum for these types of model[16, 17, 18], the bispectrum has notyet been studied. The main contribution to the power spectrum can be calculated using free Quantum Field Theory(QFT) theory, however, QFT with interactions is needed to calculate the bispectrum. In this paper, we calculatethe bispectrum by applying the in-in formalism to a universe where the background evolution is given by a CDLinstanton as was done in [19]. It should be noted that though this is a natural extension of the usual in-in formalismin a flat universe[20], as far as we know, there has been no stringent proof for the in-in formalism on a CDL instantonbackground. The proof of this formalism will appear elsewhere[21].

It has recently been shown that a modified initial state (i.e. non Bunch-Davies vacuum state) gives rise to anobservationally interesting bispectrum[22, 23]. In open inflation models, such a modified initial state1 is naturallyrealized as a result of the evolution from the Bunch-Davies vacuum state before the quantum tunneling[16, 17, 18]. Byassuming a modified initial state at some initial time, it has been shown that the bispectrum behaves as B(k, k, k3) ∝k−43 k2 in the squeezed configuration k3 ≪ k1 ∼ k2(≡ k), which is enhanced by the factor k/k3 compared to the usual

local type bispectrum, which behaves as B(k, k, k3) ∝ k−33 k3.

In open inflation models, there are two possible sources of non-Gaussianity. Firstly, as pointed out in [19], a certainamount of non-Gaussianity may be generated in the presence of a large self-interaction near the potential barrier, or

1 In this paper we use “non Bunch-Davies vacuum” state to refer to a state whose positive frequency functions differ from those of the“adiabatic vacuum” state[24]. With this terminology, the state associated with the de-Sitter invariant Wightman function, often referredto as the “Bunch-Davies vacuum” state, is also a “non Bunch-Davies vacuum” state.

p Bispectrum of ζ

Open inflationに特有な寄与はεと(kcurv/ki)の積で抑制される

kcurv: 曲率スケール

B⇣(k1, k2, k3) =

✓1� ns +O

✓✏⇥ kcurv

ki

◆◆P⇣(k)P⇣(k3)

-  量子トンネリングが状態を励起 (非Bunch-Davies真空が自然に実現する系)

K. S. and E. Komatsu, JCAP 1311, 065 (2013)

-  Power spectrumへの影響は既知 (Yamamoto, Sasaki, Tanaka 1995)

D⇣(k1)⇣(k2)⇣(k3)

E= (2⇡)3�(k1 + k2 + k3)B⇣(k1, k2, k3)

ε: スローロールパラメータ

(in squeezed limit: k3 � k1 � k2(� k)

k1

k2 k3

)

ns: スペクトルインデックス

通常と同様の寄与（Maldacena 2003)

Contents •  Introduction

•  量子トンネリング中の非線形発展

•  Slow-roll inflation中のバブル生成

•  Conclusion

K. S., D. Yamauchi and M. Sasaki, EPL 100, 29004 (2012)

•  Open inflation K. S. and E. Komatsu, JCAP 1311, 065 (2013)

K. S., Phys. Rev. D 88, 025037 (2013)

量子トンネリングのシナリオ２： スローロールインフレーション中のバブル生成

-  インフラトンΨがスローロール中

-  バブル生成の影響は摂動的

reheating

nucleation timebubble wall

Ψ

Vinfl(Ψ)

σ

Vtun(σ)

-  他のスカラー場σのバブルが生成

-  インフラトン一定面は平坦空間（青線）

バブルが見える

-  Bubble wall（赤線）と交差

通常のインフレーション

Penrose diagram

時間

空間

非一様なスキューネス

p 観測量(CMBゆらぎ）へのmapping

p skewnessの生成

-  右図のような配置を仮定

バブル痕跡 last scattering surface

high skewness low skewness

-  特定の方向のみskewnessが大きい

-  バブルとの非線形相互作用により、量子ゆらぎのskewnessがバブル近辺で生成

バブル痕跡

大 小

skewness

ポテンシャルを仮定して具体的に計算

�(n̂) ⌘ �T (n̂)

T̄-  Sが一様と仮定したときの制限 （Planck衛星2013）

NG bubbleの特徴： 円対称の構造、バブルのフチで最大、etc

バブル生成から生じる特有なシグナル！！

K. S., D. Yamauchi and M. Sasaki, EPL 100, 29004 (2012)

バブル状の構造を持った高スキューネス領域

S ⇠ (3± 6)⇥ �4

Contents •  Introduction

•  量子トンネリング中の非線形発展

•  Slow-roll inflation中のバブル生成

•  Conclusion

K. S., D. Yamauchi and M. Sasaki, EPL 100, 29004 (2012)

•  Open inflationからのbispectrum K. S. and E. Komatsu, JCAP 1311, 065 (2013)

K. S., Phys. Rev. D 88, 025037 (2013)

Conclusions

p 弦理論的ランドスケープは、インフレーション中のスカラー場の量子トンネリングを示唆する

p 弦理論的ランドスケープから生じる観測シグナルとして、量子トンネリングが生成するゆらぎの非線形性に注目した

p 実際の観測と比較することで、将来的に弦理論的ランドスケープの観測的証拠が得られる可能性がある

p 本研究において、計算手法の開発をおこない、さらにその手法をいくつかのモデルに応用し、予想されるゆらぎの非線形性を計算した

（難しいかもしれないが、インパクトは大！！）