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1 Lagrange Introduction CfA survey 2dF SDSS high-z Euler Lagrange Lagrange CDM Lagrange Lagrange 2 Newton Friedmann Newton Newton Newton [1,2,3,4,5] ρ + r ρu = 0, 1 t r u +u r u = 1 t r ρ rp + g, 2 g = r Φ, g = 4πGρ, 3 x = δ r,at : scale factor 4 at u = ṙ =ȧx + vx,t, v aẋ, 5 x = a r, 6 fx = r/a, t f = ȧ t r t x a x xf, 7 ρ = ρ b t{1+δx,t}, ρ b a [5] 1

2 Poisson δ t + 1 a x {v1 + δ} = x φ =4πGρ ba 2 δ, φ Φ 2 3 πga2 x 2 ρ b. 10 Euler v t + 1 ȧ v v + a a v = 1 a φ 1 P. 11 aρ 9, 10, 11 3 Euler v,δ v,δ δ δ Euler 9 11 v δ t + 1 v =0, a 12 v t + ȧ a v + 1 a φ + 1 dp δ =0. a dρ 13 2 δ t 2 +2ȧ δ a t =4πGρ bδ + 1 dp a 2 dρ 2 δ, 14 δ 2 δ t 2 +2ȧ δ a t =4πGρ bδ, 15 E-dS 2 δ t δ 3t t = 2δ 3t 2, 16 2/3 1 t t δ + t =δ + t i at, δ t =δ t i. 17 t i t i δ 1 δ Euler 1 4 Lagrange 4.1 Lagrange Zel dovich [6] Lagrange Eular Lagrange Euler x Lagrange q Lagrange Euler [7,8,9] Lagrange 2

3 Lagrange 9, Euler 11 d dt t + 1 a v x. 18 dρ dt +3ȧ a ρ + ρ a x v = 0, 19 dv dt + ȧ a v = g 1 ρa xp. 20 Lagrange Lagrange St, q x = q + St, q. 21 x q Jacobian J [ ] J = det xi = det [δ ij + S i,j]. 22 q j J 19 ρxjd 3 q = ρqd 3 q δ = 1 J. 23 J Euler 20 Lagrange S v = aṡ. 24 Lagrange g = a S +2ȧ aṡ 1 dp a 2 dρ J 1 x J P =0 25 rot div x S +2ȧ = 0, 26 aṡ x S +2ȧ = 4πGρ b J aṡ x q q i = x j q i x j = x i = q i S j q i = q i S j q i = q i S j q i x i + S j q i x j S k q j q j x j, 28 x k q j + S j q i S k q j x k + 29 [10, 11, 12, 13, 14, 15] q 3

4 Lagrange Lagrange longitudinal mode transverse mode S = S 1 + S 2 + S 3 + oε S n = ψ n + s T n. 31 transverse mode 26 s T ȧ = aṡt transverse mode E-dS longitudinal mode Lagrange ψ s T 1 = s a q+t 1/3 s b q. 33 ψ = Dtψ 1 q+etψ 2 q+f tψ 3 q+ 34 μ 1 ψ = ψ,ii, 35 μ 2 ψ m,ψ n = 1 2 [ψm,ii ψ n,jj ψm,ij ψn,ji ], 36 μ 2 ψ m = μ 2 ψ m,ψ m, 37 μ 3 ψ = det[ψ,ij ]. 38 Jacobian J =1+μ 1 ψ+μ 2 ψ+μ 3 ψ, 39 longitudinal mode 27 1st order: D +2ȧ aḋ 4πGρ bd ψ 1,ii = nd order: 3rd order: Ë +2ȧ aė 4πGρ be ψ 2,ii = 2πGρ b D 2 μ 2 ψ 1 ψ 3,ii +2 F +2ȧ F a DE + DË +2ȧ +2ȧ aḋe ] = 4πGρ b [Fμ 1 ψ 3 +2DEμ 2 ψ 1,ψ 2 +D 3 μ 3 ψ 1 [ = 2πGρ b D 2 ψ 1,ii ψ1,ij ψ1,ji a DĖ μ 2 ψ 1,ψ 2 +3D 2 ]. 41 D +2ȧ a D μ 3 ψ 1 F +2ȧ F a ] 4πGρ b F ψ 3,ii = 8πGρ b [DE D 2 μ 2 ψ 1,ψ 2 +D 3 μ 3 ψ

5 Lagrange 40 Zel dovich Euler Doroshkevich et al. [16] τ flat τ = t 1/3 flat τ = 1 Ω 1/2 = ȧ a flat 2. dτ a 2 dt. 43, E-dS 40 D +2ȧ aḋ 3 2 ȧ 2 D =0. 45 a D t 2/3,t 1 a, a 3/2. 46 ψ 1 WMAP flat Λ 0 Ω+ Λ =1, 47 3H2 [13] x 3h 2 1 d2 dh 2 +2h d x = 2δ, dh h = Ht H, 48 d dh =, 49 h Λ=0 Lagrange 3h 2 1 D +2hḊ =2D, 50 D + h = dh 2 h h h 2 h 2 1 =2hB 1/3 1/h 2 3, 5, 6 51 D h = h, 52 q 2 [14] 5

6 B 1/h 2 B z p, q = z 0 t p 1 1 t q 1 dt, 53 Λ=0 E-dS Lagrange deformation tensor X αβ X αβ = x α q β X αβ X αβ = diag1 S 1 11 δ = J 1 1=[1 S 1 11 = δ αβ + S α q β, 54 qdt, 1 S1 qdt, 1 S1 qdt, qdt1 S1 22 qdt1 S1 33 qdt] 1 1, 56 S 1 11 q q S 1 11 q maxdt =1, 57 S 1 11 q, S1 22 q, S1 33 q Zel dovich 4.4 growing mode growing mode growing mode D Dft = 0, 58 Dgt = Qt, E-dS [10, 11] ψ 2,ii = 3 14 a2 μ 2 ψ 1 = 3 [ ] 14 a2 ψ 1,ij ψ1,ji ψ1,ii ψ1,jj 42 E-dS [12, 13, 14] ψ 3,ii = a { μ 2ψ 1,ψ 2 1 } 3 μ 3ψ { = a 3 5 [ ] ψ 2,ii 9 ψ1,jj ψ2,ij ψ1,ji 1 } 3 detψ1,ij. 61 6

7 Λ 0 [13] Lagrange ψ 2 = Etψ 2 t 0, 62 ψ 3 = F a tψ a 3 + F b tψ 3 b E-dS a b 3h 2 1Ë +2hĖ = 2E 2D2 +, 64 3h 2 1 F a +2hF a = 2F a 2D+ 3, 65 3h 2 1 F b +2hF b = 2F b +2D+ 3 1 E D Λ= P = κρ γ κ = constant, ρ = ρ b J 1 dp dρ J 1 x J = κγ a 2 J γ x J, Longitudinal mode [17, 18] S +2ȧ aṡ 4πGρ bs κγργ 1 b a 2 2 S =0. 69 Transverse mode Longitudinal mode 69 S Fourier Lagrange ȧ Ŝ +2 Ŝ 4πGρ b Ŝ + κγργ 1 b a a 2 K 2 Ŝ =0. 70 E-dS γ 4/3 ŜK,a a 1/4 2C2 K J ±5/8 6γ C 1 4 3γ a4 3γ/2, 71 J ν ν Bessel γ =4/3 ŜK,a a 1/4± 25/16 C 2 K 2 /2C 1, 72 C 1 4πGρ b a in ain 3/3 C 2 κγρ b a in γ 1 a 3γ 1 in a in E-dS [19] [18, 19, 20, 21] 7

8 1: Lagrange γ Bouchet et al. [10] Morita and Tatekawa [18] Buchert and Ehlers [11] Tatekawa et al. [19] Buchert [12] Tatekawa [20] Bouchet et al. [13] Tatekawa [21] Catelan [14] Sasaki and Kasai [15] 5 Lagrange Lagrange N Lagrange [22, 23] [5,7,8,9] d a 2 dx [ = 2a2 x x0 ] 3 1 dt dt 9t 2, 73 x x x 0 = xt 0 a 0 δ = a 73 2 dr 1 = a da R 3, 74 5 Rθ =atx/x 0 δ <0 74 Rθ = 3 cosh θ 1, aθ = 3 [ ] 2/3 3 sinh θ θ δx = = x0 3 1 x 9θ sinh θ2 2cosh θ E-dS ] n Rt =R 0 [1 1 k C k a k, 78 k=1 C k Lagrange Munshi, Sahni, and Starobinsky [7] Sahni and Shandarin [8] δ>0 δ <0 8

9 2: Lagrange C k 1 /C k k C k 1 /C k k C k 1 /C k [5] 78 a C k a c lim. 79 k C k+1 a >a c 2 Padé fx M k=0 a kx k [9, 24] 1+ N k=1 b kx k, 80 6 Lagrange N [25] WMAP 73 % p = wρ, 81 1 w< 0.79 w LCDM Lagrange w = 0.8 w = 1 8h 1 Mpc z =2 1% 9

10 7 Euler Lagrange Lagrange [7, 8, 9] Lagrange shell crossing shell crossing Appendix shell crossing Lagrange [26] [1]P.J.E.PeeblesThe Large Scale Structure of the Universe, Princeton University Press, Princeton, [2] P. Coles and F. Lucchin, Cosmology: The Origin and Evolution of Cosmic Structure John Wiley & Sons, Chichester, [3] V. Sahni and P. Coles, Phys. Rep. 262, [4] B. J. T. Jones, V. J. Martinez, E. Saar, and V. Trimble, Rev. Mod. Phys. 76, [5] T. Tatekawa, Recent Research Development in Astrophysics Vol. 2, 1-26 Research Signpost: Kerala, India, 2005 astro-ph/ [6] Ya. B. Zeldovich, Astron. Astrophys. 5, [7] D. Munshi, V. Sahni, and A. A. Starobinsky, Astrophys. J. 436, [8] V. Sahni and S. F. Shandarin, Mon. Not. R. Astron. Soc. 282, [9] A. Yoshisato, T. Matsubara, and M. Morikawa, Astrophys. J., 498, [10] F. R. Bouchet, R. Juszkiewicz, S. Colombi, and R. Pellat, Astrophys. J. 394, L [11] T. Buchert and J. Ehlers, Mon. Not. R. Astron. Soc. 264, [12] T. Buchert, Mon. Not. R. Astron. Soc. 267, [13] F. R. Bouchet, S. Colombi, E. Hivon, and R. Juszkiewicz, Astron. Astrophys., 296, [14] P. Catelan, Mon. Not. R. Astron. Soc. 276, [15] M. Sasaki and M. Kasai, Prog. Theor. Phys. 99, [16] A. G. Doroshkevich, V. S. Ryabenkin, and S. F. Shandarin, Astrofizica 9, [17] S. Adler and T. Buchert, Astron. Astrophys. 343, [18] M. Morita and T. Tatekawa, Mon. Not. R. Astron. Soc. 328,

11 [19] T. Tatekawa, M. Suda, K. Maeda, M. Morita, and H. Anzai, Phys. Rev. D66, [20] T. Tatekawa Phys. Rev. D71, [21] T. Tatekawa Phys. Rev. D72, [22] T. Buchert, astro-ph/ Stars to the Universe, Shanghai PR China 1998, Annals of Shanghai Observatory. [23] T. Tatekawa, Phys. Rev. D69, [24] T. Matsubara, A. Yoshisato, and M. Morikawa, Astrophys. J. 504, [25] T. Tatekawa and S. Mizuno, JCAP 0602, [26] T. Buchert, M. Kerscher and C. Sicka, Phys. Rev. D62, [27] L. Kofman, D. Pogosyan, S. F. Shandarin, and A. L. Melott, Astrophys. J. 393, [28] P. Coles, A. L. Melott, and S. F. Shandarin, Mon. Not. R. Astron. Soc. 260, [29] S. N. Gurbatov, A. I. Saichev, and S. F. Shandarin, Mon. Not. R. Astron. Soc. 236, [30] T. Buchert and A. Domínguez, Astron. Astrophys. 335, [31] T. Buchert, A. Domínguez, and J. Perez-Mercader, Astron. Astrophys. 349, [32] T. Tatekawa, Phys. Rev. D70, [33] H. Sotani and T. Tatekawa, Phys. Rev. D73, [34] A. Domínguez, Mon. Not. R. Astron. Soc. 334, A Lagrange A.1 Truncated Zel dovich Truncated Zel dovich TZA [27, 28] shell crossing P ini k knl D 2 t P ini k =1, 82 k 0 t k NL k NL P ini k P ini exp k 2 /knl 2, 83 Gauss Zel dovich N Zel dovich CDM dump [23] 11

12 A.2 adhesion adhesion [29] Burgers shell crossing E-dS adhesion Euler Bergers Euler ũ a +ũ xũ = 0, 84 ũ x a = ẋ ȧ, 85 u t + uu x = ν 2 u xx, 86 ũ a +ũ xũ = ν 2 xũ. 87 adhesion Hopf-Cole ũ = ν log θ, 88 θ D = ν 2 θ, 89 Hopf-Cole Burgers ũx,t= x qα j α exp I α / a 2ν α α q α I α Lagrange j α exp I α, 90 2ν I α Ix,a; q α =S 0 q α + x q α 2 =min., 91 2a [ ] 1/2 j α det δ ij + 2 S 0 = q α, 92 q i q j q S 0 = Sq,t 0, 93 S 0 Lagrange u Lagrange Adhesion TZA [30, 31] Jeans adhesion [32, 33] [34] N 12

1 ( ) Einstein, Robertson-Walker metric, R µν R 2 g µν + Λg µν = 8πG c 4 T µν, (1) ( ds 2 = c 2 dt 2 + a(t) 2 dr 2 ) + 1 Kr 2 r2 dω 2, (2) (ȧ ) 2 H 2

1 ( ) Einstein, Robertson-Walker metric, R µν R 2 g µν + Λg µν = 8πG c 4 T µν, (1) ( ds 2 = c 2 dt 2 + a(t) 2 dr 2 ) + 1 Kr 2 r2 dω 2, (2) (ȧ ) 2 H 2 = 8πG a 3c 2 ρ Kc2 a 2 + Λc2 3 (3), ä a = 4πG Λc2 (ρ