Structure formation in Void Universes

Osaka City University (OCU)Ryusuke Nishikawa

collaboratorKen-ichi Nakao (OCU) ,Chul-Moon Yoo (YITP)

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Dark Energy & Copernican PrincipleStandard cosmological model

General Relativity 　＋　 Copernican Principle 　＋　Observations

Dark Energy(homogeneous and isotropic spacetime)

Inhomogeneous cosmological model Tomita (2000) , Celerier (2000)

We live close to the center in spherically symmetric spacetime.

General Relativity 　＋　 Copernican Principle 　＋　Observations

Dark Energy(inhomogeneous and isotropic spacetime)

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Void cosmological modelsdust, spherically symmetric

Lemaitre-Tolman-Bondi (LTB) solutions

Homogeneous Big Bang time

only growing mode

two functional degree (growing mode and decaying mode)

We consider homogeneous Big Bang Void models. large void

Clarkson, Regis (2010) 3/15

Observational Tests

•CMB acoustic peak positions

•Radial BAO

•redshift drift

•kSZ effect

etc.

consistency

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Tests using the large-scale structure evolution have not been performed.

Clarkson, Regis (2010), Yoo, Nakao, Sasaki (2010) ・・・Zibin, Moss, Scott (2008), Garcia-Bellido, Haugbolle (2008)

Yoo, Kai, Nakao (2008)

Yoo, Nakao, Sasaki (2011)

The symmetry of the background LTB is less than FLRW.

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Void structure

Clarkson, Regis model (2010)

nonlinear

density contrast :

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density contrast on past light-cone

This was first pointed out by Enqvist, Mattsson, Rigopoulos (2009).

We can use perturbative analysis for void structure inside the past light-cone.

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Linear approximation for the void universe

background FLRW

The relative error is within 20%.

linear perturbation

linear growing factor

density

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Hubble parameter

blue line : linear approximationblack line : exact LTB

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Perturbation in the approximated void universe

Second order perturbations in homogeneous and isotropic spacetime

We can solve.

Spherically symmetric ：

synchronous comoving gauge

We assume and neglect terms.

Tomita (1967), ・・・

(we consider only scalar-scalar coupling)

Non-spherically symmetric ：

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Non-spherically symmetric density perturbation

sub-horizon scale :

Fourier transform

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Angular power spectrum & Effective growth rate

3D power spectrum in FLRW.effective growth rate

We assume

In linear approximation,11/15

Effective growth rate

If we observe the growth rate of , we can test the void model.

summary

Void model (CR model)

ΛCDM

Open FLRW

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Future work

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redshift space distortions

Guzzo et al. (2005)

この図にヴォイドモデルを書き入れたい．

線形摂動でヴォイドの効果が入る．

Kaiser (1987)Matsubara, Suto (1996)

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redshift space real space

2-parameter の摂動の場合：

redshift space distortions

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視線方向redshift space

real space

void の効果

視線方向の相関を強める．>0

参考

redshift space distortions

空間曲率無視

redshift space distortions

redshift space distortions

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FLRW

FLRW + void effect

LTB solution球対称 、ダスト時空は LTB (Lemaitre-Tolman-Bondi) 解で記述される .

known function

任意関数は

・　　　　　　　　　　を仮定．

・　　　　　は座標を選ぶ自由度．

（宇宙初期は一様等方時空）

second-order perturbation

linear perturbation equations

second-order perturbation

second-order perturbation equation

density contrast on past light-cone

Garcia-Bellido & Haugbolle model (2008)

（遠方は Einstein de-Sitter universe に近づく void model ）

近似して LTB摂動方程式を解いた例Zibin (2008)

silent approximation

neglecting the coupling between density perturbations and gravitational waves

Dunsby et al. (2010)

メモRedshift は FLRW の redshift 用いて書く．

２次摂動まで入れると，１次の効果まで取り入れたredshift を考える必要があるか．

球対称ゆらぎのみ存在するときに distortions はどうみえるか？

固有速度は（一様等方時空に比べて）外に行くほど小さくなる．-> 　 redshift space では集まるようにみえる．