Primordial Non-Gaussianity in Multi-Scalar Slow-Roll Inflation

In collaboration with S.Yokoyama& T.Tanaka

Teruaki Suyama（ Institute for Cosmic Ray Research, University of Tokyo, Japan ）

arXiv:0705.3178

Inflation in the early universe

IntroductionIntroduction

Very appealing idea

•Flatness problem

•Horizon problem

•Monopole problem

solves

•generate primordial perturbation

But,the mechanism of inflation itself is still unkown.

Inflaton is scalar field? What kind of scalar field?

How many field?

If these are resolved, it would be a great progress for physics and cosmology.

What kind of potential energy?

Observational cosmology is entering a precision era.

http://www.rssd.esa.int/index.php?project=Planck

http://map.gsfc.nasa.gov/

COBE (1992) WMAP (2003) PLANCK (200X)

Consistent with inflation

Excluded some inflation models

Constraints will be stronger

Useful information of second order perturbations will be obtained

（ Non-Gaussianity ）

•One characteristic of PLANCK

discovered temperature anisotropy

Non-Gaussianity

Non-Gaussianity of the perturbation

・・・ perturbation that does not obey the Gaussian distribution

Form of non-Gaussianity frequently used in literatures

： primordial perturbation of the metric

： Gaussian variable

Observational limit on

(Komatsu et al. 2003)

PLANCK will detect the non-Gaussianity if

(Komatsu&Spergel, 2001)

We can expect non-Gaussianity will be an useful approach to probe the early universe.

Then how useful ?? (What situation is the large non-Gaussianity generated?)

Non-Gaussianity in some inflation models

•Single field inflation model ・・・・・

Detection is hopeless

•Curvaton model ・・・・・

： ratio of energy density of the curvaton field to total energy density at curvaton decay

If r is small,

(Moroi&Takahashi 2001, Lyth&Wands 2002)

etc.

・・・

(Maldacena 2003)

There are models that generate large non-Gaussianity.

Then in what situation is detectable large non-Gaussianity generated?

We have calculated generated in the multi-scalar slow-roll inflation for arbitrary number of scalar fields and arbitrary form of the potential.

It is important to have theoretical understanding about the generative mechanism of non-Gaussianity.

situation

inflation

Radiation dominated

Matter dominated

totayCMB

reheating

Length scale

Super horizon scale

obseved 　 = primordial ＋ generated in post-inflationary era

+ generated after horizon-reentry by secondary effect

Evolution of the curvature perturbation on super-horizon scale

Separate universe approach

Local expansion = expansion of the unperturbed universe

(e.g., Sasaki&Stewart, 96)

e-folding number between and

Friedmann universeFriedmann universe

Curvature perturbation

If we know correlation functions of , we can calculate correlation functions of .

δN formalisim

Curvature perturbation at

If slow-roll conditions are satisfied, are Gaussian to a good approximation. (Seery&Lidsey, 2005)

(Lyth&Rodriguez, 2005)

To leading order

Slow-roll conditions

One problem

Violation of the slow-roll condition

Field space

Hubble=const. surface

Background trajectories

We assume that the complete convergence occurs during slow-roll conditions are satisfied.

We choose as a time after the complete convergence of background trajectories in field space occurred.

After , the curvature perturbation remains constant as long as the relevant scale is super-horizon scale.

(Lyth, Malik&Sasaki, 2005)

Result

where

can be written by two D-component vectors and .

What we have found

Only 2D informations are enough.

Equations for two vectors

Solve until under initial condition .

Useful for numerical calculation !!

( D is a number of scalar fields.)

Solve until under initial condition .

We multiply whenever field derivative appears in the potential.

•Order estimate of Order counting

Rough order estimate gives

Possible loophole

may become large.

1)

2) Violate the condition

Summary

We studied the generation of non-Gaussianity in multi-scalar slow-roll inflation.

Final expression of shows that

（ detection of such non-Gaussianity in the near future is hopeless. ）

• There remains some possibilities to generate large non-Gaussianity. （ e.g. larger third derivatives of the potential ）

• At most 2D quantities are enough to obtain .

Quite useful for the numerical calculation.

• Rough order estimate gives small non-Gaussianity even in the models with non-separable potential.

おまけ

Inflation models with specific form of the potential

Two-field inflation model

N-flation model

(Kim&Liddle, 2006)

(Vernizzi&Wands 2006, Choi et al. 2007)

(Battefeld&Easther, 2006)

(Choi et al. 2007)

in Multi-scalar slow-roll inflation

•Multi-scalar slow-roll inflationインフレーションのダイナミクスが、 single field で記述されるとは限らない

以下、 D 個のスカラー場の場合を考える

多成分の場によるインフレーションのダイナミクス

Slow-roll 近似

•Slow-roll 条件

を仮定

Field space

Hubble= 一定面

trajectory

Slow-roll 条件が破れる

： trajectory が収束する時刻

では、断熱揺らぎのみ

以降を考えなくてもよい

での曲率揺らぎ

Slow-roll 条件のもとでは、 は非常に良い精度でGaussian

(Seery&Lidsey, 2005)

このとき　　　は、保存する(Lyth et al., 2005)

の三点相関から

(Lyth&Rodriguez, 2005)

と　　　　　を求めればよい　

• 一つの問題点

もし が、 slow-roll 条件が破れた後の時刻ならば、 slow-roll 条件が破れた後の進化も知る必要がある　　　

今回は、そこまでの解析は無理Slow-roll 条件が破れる

Field space

Hubble= 一定面

trajectory

Slow-roll の間に生成される non-Gaussianity だけを評価する。あるいは、 slow-roll 中にtrajectory が収束すると仮定する。

と　　　　 を求める　

Analytic formula for the non-linear parameter

と展開

• 一次の解

• 二次の解

は、二つのベクトル　　　　　 と　　　　　 で決まる。

•上の式から分かること

Naïve な予想

( 2D 個の情報で十分 )

なので、　　　　　　　　　　　　　　　　　 個の情報が必要かな

•二つのベクトルの従う式

を初期条件に 　　　まで解く

を初期条件に 　　　まで解く

と　　　 の表式

Slow-roll 条件を使うとHubble 一定 ≒ ポテンシャル V 一定 なので

これから δN と 　の間に関係が付く

Field space

V= 一定

これを δN について解くと

最終的な表式

•オーダー評価

仮定 Field の微分が一発掛かるたびに　　　　　だけオーダーが下がる

ここで

大雑把に見積もると

Possible loophole

もしかすると　　　 は、大きくなるかも

1)

2)

三階微分が小さく抑えられている必要はない

という large non-Gaussianity が生成される可能性は残っている

Slow-roll inflation の枠組みで

まとめ

Multi-scalar slow-roll inflation で生成される non-Gaussianity について調べた

得られた　　　　　の表式を見ると

• D×D 個もの情報はいらず、高々 2D 個 だけの情報で決まる ( 数値計算に便利 )• 大雑把なオーダー評価では、 （観測は絶望

的）

• ただし、　　　　　　　となる可能性は残されている

（ポテンシャルの三階微分を大きくするとか）