Plan of the talk

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Locality and Evolution of Cosmological Perturbations

Ko Furuta (Riken)

in collaboration with

Chu Chong-Sun and Lin Feng-Li


Plan of the talk

・Ekpyrotic Scenario

・Cosmic Microwave Background

・Metric Perturbation (Scalar, Tensor, Vector Perturbation)

・Matching Condition with locality


Ekpyrotic (Cyclic) Scenario

1. Static universe 3. Big bang

4. Expansion of the brane 5. Present time 5. Go back to 1 2. Branes approaching


time evolution of the scale factor

ekpyrotic inflation

t t

x x


Typical potential for ekpyrotic scenario

1 2

3 4



理論物理学 研究室


現在の宇宙はおよそ140億年の歴史を持つと思われています.宇宙誕生当時は非常に高温,高密度の世界であったと考 えられています.近年の観測技術の進歩によって,宇宙誕生初期の様子が次第に明らかになってきました.このコーナー では,宇宙を調べることとミクロの世界の物理とがどのように関係しているのかを説明していきます.



インフレーションが終わるとともに宇宙が熱せられたと考えられています.やがて温度が下がり,ばらばらに運動していた素粒子 が結びつき,ついには原子が構成されていきます.その後,原子たちは重力によって引き合い大きな塊となり,多くの星や銀河 系などが生まれました.

下の図はWMAP衛星による観測の様子を表しています.宇宙に原子が生まれようになると,光はさえぎるものが無くなりまっすぐ 進むようになります.このときの光が長い時間をかけて現在の地球に届いています.WMAP衛星はこの光(宇宙背景放射Cosmic Micro Wave Background)を詳しく調べています.

WMAP衛星によって得られ たデータ.観測される光の 強度の揺らぎを表している.

Wilkinson Microwave Anisotropy Probe (WMAP)衛星


Scale Invariant Fluctuation in CMB from Inflation


Spatially homogeneous metric




tdec Cosmological Microwave Background



Metric Perturbation

Temperature anisotropy and metric potential

We study the time evolution of . Metric Perturbation

Full metric

Scalar fluctuation

Gauge invariant quantities

Perfect fluid


Power spectrum

Expand the einstein eq. around the b.g. .

e.o.m. for Φ


Φ from quantum fluctuations

Short wave length limit (well before the bounce)

Metric perturbation

Beyond the horizon scale (right before the bounce)


Determining S and D.

Connecting at

…Scale inv. spectrum?

After the bounce


Scale inv. fluct. in CMB is in the constant mode.

Matching cond. needs mixing of and .

We want


Matching condition

(1) Determine a hypersurface on which matching condition is given.

(2) Determine a matching condition.


before the bounce bounce

after the bounce

We consider the matching condition from

the view point of locality .



No scale invariant spectrum in . Mixing in . … Anisotropic stress

during the bounce.

Class of matching conditions is unaltered from that in general relativity.


Fix the gauge

Cf. Israel matching condition

Ekpyrotic scenario



Fix the coordinate system in spatial direction before the discontinuity.

Introduce time coordinate so that

for the surfaces of the matching.

Locality enables one to associate to by local physics.

Spatial coordinate after is fixed.


This can be the case even if goes to zero at the bounce since we are using conformal time .


Expand the matching condition around a homogeneous background.

Only the positive powers of derivatives appear in linear perturbation since all the components of the metric after the bounce is

unambiguously determined from local physics.


Choosing the surface of the matching

We consider the fluctuation of the form

: time dependent quantities.

Ex. Longitudinal gauge


Coordinate transformation


Metric and scalar field fluctuation becomes

Farther Local coordinate transformation around .

Smaller compared to the original with the factor . ( Entropy perturbation . )


Note on surface of the matching


Fluctuation of the order parameter

Surface of the phase transition is given by some order parameter

Consider a time translation

Order parameter transforms as


Surface of the matching is given by up to order.


Metric fluctuations after the bounce in the gauge

Only positive powers of derivatives enter.

Go back to the global coordinate system Apply the matching condition


Gauge invariant quantities

Perfect fluid condition determines the time dependence of and .

Finally one obtains


is solved by

Absorbed by a jump in after the bounce.

Anisotropic stress during the bounce

Depends on and .



No scale invariant spectrum in . Mixing in . … Anisotropic stress

during the bounce.

Class of matching conditions is unaltered from that in general relativity.


Other works

・ matching condition in terms of

and .

scale inv. spectrum. (Steinhardt et al)

E.o.m. determining includes spatial derivatives of scalar field and metric.

・ matching condition with modified Israel cond. introducing

“surface tension” at the bounce. (Durrer and Vernizzi)

Surface tension also should be determined by

initial conditions on metric and scalar fields.



・We have shown no scale inv. spectrum can be obtained in ekpyrotic scenario without non-local effects or non-linear matching conditions.

・Matching cond. for general initial fluctuations are obtained.

include the results in the standard matching cond.


・Consistent with the result focusing on the attractor property around the bounce. (Creminelli et al.)


・ Non-linear effects?

・ Noncommutativity around the bounce?




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