Primordial Non-Gaussianity in Multi-Scalar
arXiv:0705.3178
Slow-Roll Inflation
In collaboration with S.Yokoyama& T.Tanaka
Teruaki Suyama
（Institute for Cosmic Ray Research, University of Tokyo, Japan）
Introduction
Very appealing idea
Inflation in the early universe
•Flatness problem
solves
•Horizon problem
•Monopole problem
•generate primordial perturbation
But,
the mechanism of inflation itself is still unkown.
Inflaton is scalar field?
What kind of scalar field?
How many field?
What kind of potential energy?
If these are resolved, it would be a great progress for physics
and cosmology.
Observational cosmology is entering a precision era.
COBE (1992)
WMAP (2003)
http://map.gsfc.nasa.gov/
discovered temperature
anisotropy
Excluded some inflation
models
PLANCK (200X)
http://www.rssd.esa.int/index.php?project=Planck
Constraints will be stronger
•One characteristic of PLANCK
Useful information of second
order perturbations will be
obtained
Consistent with inflation
（Non-Gaussianity）
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
PLANCK will detect the non-Gaussianity if
(Komatsu et al. 2003)
(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 ・・・・・
(Maldacena 2003)
Detection is hopeless
•Curvaton model ・・・・・
：ratio of energy density of the curvaton
field to total energy density at curvaton
decay
(Moroi&Takahashi 2001, Lyth&Wands 2002)
・
・
・
etc.
If r is small,
There are models that generate large non-Gaussianity.
Then in what situation is detectable large non-Gaussianity generated?
It is important to have theoretical understanding about the generative
mechanism of non-Gaussianity.
We have calculated
generated in the multi-scalar slow-roll inflation
for arbitrary number of scalar fields and arbitrary form of the potential.
situation
Length scale
Super horizon scale
Matter dominated
Radiation
dominated
inflation
reheating
totay
CMB
obseved
= primordial
+
＋
generated in post-inflationary era
generated after horizon-reentry by secondary effect
Evolution of the curvature perturbation on super-horizon scale
(e.g., Sasaki&Stewart, 96)
Separate universe approach
e-folding number between
Friedmann universe
Friedmann universe
and
Local expansion = expansion of the unperturbed universe
δN formalisim
Curvature perturbation
If we know correlation functions of
functions of
.
, we can calculate correlation
Curvature perturbation at
If slow-roll conditions are satisfied,
are Gaussian to a good approximation.
(Seery&Lidsey, 2005)
To leading order
(Lyth&Rodriguez, 2005)
Slow-roll conditions
One problem
We choose
as a time after the complete convergence of
background trajectories in field space occurred.
Field space
Violation of the slow-roll
condition
After
, the curvature perturbation
remains constant as long as the
relevant scale is super-horizon scale.
(Lyth, Malik&Sasaki, 2005)
Background trajectories
Hubble=const. surface
We assume that the complete convergence occurs during
slow-roll conditions are satisfied.
Result
where
What we have found
can be written by two D-component vectors
and
.
( D is a number of scalar fields.)
Only 2D informations are enough.
Useful for numerical calculation !!
Equations for two vectors
Solve until
under initial condition
Solve until
under initial condition
.
.
•Order estimate of
Order counting
We multiply
the potential.
whenever field derivative appears in
Rough order estimate gives
Possible loophole
1)
may become large.
2) Violate the condition
Summary
We studied the generation of non-Gaussianity in multi-scalar slow-roll inflation.
Final expression of
shows that
• 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.
（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）
おまけ
Inflation models with specific form of the potential
Two-field inflation model
(Vernizzi&Wands 2006, Choi et al. 2007)
(Choi et al. 2007)
N-flation model
(Kim&Liddle, 2006)
(Battefeld&Easther, 2006)
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
多成分の場によるインフレーションのダイナミクス
以下、D個のスカラー場の場合を考える
Slow-roll 近似
•Slow-roll 条件
を仮定
Field space
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
trajectory
このとき
は、保存する
(Lyth et al., 2005)
以降を考えなくてもよい
Hubble=一定面
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)
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
Field space
今回は、そこまでの解析は無理
Slow-roll 条件が破れる
trajectory
Hubble=一定面
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。
Analytic formula for the non-linear parameter
と
と展開
を求める
• 一次の解
• 二次の解
•上の式から分かること
は、二つのベクトル
と
で決まる。 ( 2D個の情報で十分 )
Naïveな予想
なので、
要かな
個の情報が必
•二つのベクトルの従う式
を初期条件に
を初期条件に
まで解く
まで解く
と
の表式
Field space
V=一定
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
これを δN について解くと
最終的な表式
ここで
•オーダー評価
仮定
Fieldの微分が一発掛かるたびに
大雑把に見積もると
だけオーダーが下がる
Possible loophole
1)
もしかすると
は、大きくなるかも
2)
三階微分が小さく抑えられている必要はない
Slow-roll inflationの枠組みで
というlarge non-Gaussianityが生成される可能性は残っている
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、
• ただし、
（観測は絶望的）
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）