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
宇宙の進化とミクロの物理
理論物理学 研究室
宇宙の歴史
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．
上の図は宇宙がどのように進化してきたかを表しています．宇宙誕生初期のことはあまりよくわかっていません．理論的には，
弦理論と呼ばれるもので記述されるのではないかと考えられています．その後，インフレーションと呼ばれる急激な膨張が起こり，
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
t
x
t0
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.
x
before the bounce bounce
after the bounce
We consider the matching condition from
the view point of locality .
Conclusion
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
Locality
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.
and
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
with
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
Suppose
Fluctuation of the order parameter
Surface of the phase transition is given by some order parameter
Consider a time translation
Order parameter transforms as
Thus
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 .
Conclusion
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.
Conclusion
・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.
(Brandenberger,Finelli)
・Consistent with the result focusing on the attractor property around the bounce. (Creminelli et al.)
