1. Structure formation in Void Universes Osaka City University (OCU) Ryusuke Nishikawa collaborator Ken-ichi Nakao (OCU),Chul-Moon Yoo (YITP) ? 1/15

2. Dark Energy & Copernican Principle Standard 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) 2/15

3. Void cosmological models dust, 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

4. Observational Tests CMB acoustic peak positions Radial BAO redshift drift kSZ effect etc. consistency ○ △ ？ ×? 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. 4/15

5. Void structure Clarkson, Regis model (2010) nonlinear density contrast : 5/15

6. 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. 6/15

7. Linear approximation for the void universe background FLRW The relative error is within 20%. linear perturbation linear growing factor density 7/15

8. Hubble parameter blue line : linear approximation black line : exact LTB 8/15

9. 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 ： 9/15

10. Non-spherically symmetric density perturbation sub-horizon scale : Fourier transform 10/15

11. Angular power spectrum & Effective growth rate 3D power spectrum in FLRW. effective growth rate We assume In linear approximation, 11/15

12. Effective growth rate If we observe the growth rate of, we can test the void model. summary Void model (CR model) ΛCDM Open FLRW 12/15

13. Future work 13/15

14. 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) 14/15 redshift spacereal space 2-parameter の摂動の場 合：

15. redshift space distortions 15/15 redshift space real space void の効果 視線方向の相関を強める． >0>0

16. 参考

17. redshift space distortions 空間曲率無視

18. redshift space distortions

19. 19/15 FLRW FLRW + void effect

20. 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）

21. second-order perturbation linear perturbation equations

22. second-order perturbation second-order perturbation equation

23. density contrast on past light-cone Garcia-Bellido & Haugbolle model (2008) （遠方は Einstein de-Sitter universe に近づく void model ）

24. 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)

25. 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．