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

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1 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 + 3p) + (4) 3c2 3. ρ. K = 0, Λ = 0 (Einstein-de Sitter universe), eq. (3), ρ c 3c2 H 2 8πG (5)., eq. (3) Ω ρ = 8πG ρ c 3c 2 ρ, H2 (6) Ω k c2 K H 2 a 2, (7) Ω Λ c2 Λ 3H 2, (8) q äa ȧ 2 (9) Ω Ω k + Ω Λ = 1 (10) 1

2 ,, q = 1 2 ( p ) Ω Ω Λ (11) ρ. recombination, p = 0. eq (10), H 2 H 2 0 a 3, a 2, a 0 *1. = Ω 0 a 3 Ω k,0 a 2 + Ω Λ,0 (12) *1 ρ = ρ m + ρ r,, ρ r a 4, H2 H = Ω m,0 a 3 + Ω r,0 a 4 Ω k,0 a 2 + Ω Λ,0.

3 2 Newton. ρ ( /c 2 )., *2., Euler Lagrange. comoving, ρ + ρv = 0, t (13) v p + (v )v = Ψ t ρ, (14) Ψ = 4πGρ. (15) dρ = ρ v, dt (16) dv dt = 1 p Ψ ρ (17) (18) r(t) = a(t)x, (19) v(t, r) = ȧx + aẋ Hr + u, (20) r 1 a x, (21) t r t ȧ x a x x (22) ( x ), x/ t = 0, (23) ρ t + 3ȧ a ρ + 1 (ρu) = 0 (24) a dv dt = t (ȧx + u) + u (ȧx + u) a = äx + u t + ȧ a u + u a u = p aρ Ψ a (25) *2 Λ Poisson Ψ = 4πGρ Λc 2. 3

4 . comoving Φ Ψ a + ä 2 x2 (26), u t + Hu + 1 a u u = 1 p Φ (27) aρ., Poisson,. Ψ = 4πGρa 2 (28) 3. ρ = ρ + ρ 1 (29) v = ȧx + u (30) Ψ = Ψ 0 + φ (31) p = p 0 + p 1. (32) 3.1 Background eq. (24), (25), and (28),, ρ t + 3H ρ = 0, (33) äx = 1 a Ψ 0, (34) Ψ 0 = 4πG ρa 2 (35)., 1 a 3 t ( ρa3 ) = 0 (36), ( ).,, p = 0, Λ = 0 eq. (4). 3ä = 1 a Ψ 0 = 4πG ρa (37) 4

5 3.2 ( ),. δ(x, t) ρ 1 ρ (ρ = ρ(1 + δ)), δ 1. ρ(1 + δ) +3H ρ(1+δ)+ 1 ( ) ρ t a { ρ(1+δ)u} = (1+δ) + 3H ρ + ρ δ t t + ρ a u+ ρ (δu) = 0, a (33),, ρ δ t + ρ a u = 0, δ t + 1 a u = 0 (38).,,, u t + Hu = 1 a ρ p 1 1 a φ = 1 a c2 sρ 1 ρ 1 a φ = c2 s a δ 1 φ (39) a., c 2 s p ρ, S (40) p 1 = c 2 sρ 1 + p S δs ρ (41) ( δs = 0). Poisson φ = 4πG ρδa 2. (42) eq. (39), eq. (38), eq. (42) δ δ = 2H δ + 4πG ρδ + c2 s a 2 δ (43). δ.,,. 5

6 3.2.1 Jeans δ, exp(ωt + ik x),, (k, λ comoving). eq. (43), ω 2 + 2Hω 4πG ρ + k 2 c2 s a 2 = 0,. ω > 0,, k < k J = 4πG ρ a2 c 2 s = 3 2 a2 ΩH2 c 2, (44) s, λ > λ J = 2π πc = 2 s k J G ρa 2 (45). Einstein-de Sitter H 2 = 8πG ρ/3, λ > λ J c s ah (46). aλ > c s c l H. l H c/h. c s c/ 3. equal time recombination CDM p = 0, CDM., baryon, c 2 s a 1, λ J a 1/2 /ȧ const.. recommbination, baryon p ρ 1+γ, c s ρ γ/2 a 1, λ J a 1/ equal time. eq. (43) c s = 0. Einstein-de Sitter universe., Einstein-de Sitter universe a(t). eq. (3) ρ = ρ c = ρ c,0 a 3 = 3H0 2 /(8πGa 3 ), K = Λ = 0, ȧ 2 = H0 2 a 1 (47). a = At α, α = 2/3 A 3 = 9/4H 2 0., a = ( ) H2 2 0 t 3, (48) H = 2 3 t 1 (49). eq. (43), δ δt δt 2 = 0 (50) 6

7 . δ t α, α = 2 3 and 1 (51) 2,, δ(t) = C 1 t C2 t 1 (52) 2. grwoing mode, decaying mode. growing mode D(t), Einstein-de Sitter ( ), D(t) a(t) *3. Einstein-de Sitter eq. (43), growing mode, decaying mode a. equal time D + H da 0 a 3 H 3, (53) D H (54). H = H 0 Ω0 a 3 + Ω Λ,0 + 1 Ω 0 Ω Λ,0 a 2 (55) 3.3 eq. (38), Poisson eq. (42),, δ = Ḋ D δ, u = Ḋ D φ 4πG ρa (56). Growth factor, f Ḋ a D ȧ Ω0.6 (57) u = fh 4πG ρa φ (58).. φ u = Ḋ + ω. (59) D 4πG ρa *3, super horizon scale D(t) a 2,. 7

8 u T ω. eq. (39) rotation, u, t u T + ȧ a u T (60)., u T (t, x) = U T(x) a(t) ; U T = 0 (61), ω decaying mode,., grwoing mode, *4. u = Ḋ D φ 4πG ρa = fh φ (62) 4πG ρa Zel dovich approximation δ 1,,. Euler x, Lagrange (δ < 3 4). First order Zel dovich approximation (Zel dovich 1970)., Lagrange q, r = a(t)[q + b(t)s(q)], (63) i.e. x(t, q) = q + b(t)s(q) (64). q grid. b(t)s(q) displacement, s. u u(t, q) = a(t)ẋ(t, q) = a(t)ḃ(t)s(q) (65) *5. Deformation tensor,., D ij = x i q j = δ ij + b(t) s i q i (66) ρd 3 x = ρd 3 q. (67) *4 (redshift distortion) u δ, f Ω 0.6 /b (b linear bias). *5 u. s = ψ,. 8

9 , ρ = d 3 x d 3 q 1 ρ D 1 ρ ρ = δ ij b(t) s i (68) q j. Jacobina, D 3 D = (1 bλ i ) (69) i=1. λ i D. b(t),,.,,, *6., b(t)s(q) 1, ρ = ρ[1 b q s] (70)., δ = b s, (71) b(t) = D(t). u = aḋs Eq. (38), s = δ/d *7. *8. * 9, s = 0, * 10. δ(q) = s(q) = ˆδ(k)e ik q d 3 k, (72) ŝ(k)e ik q d 3 k, (73) ŝ(k) = i k k 2 1 D ˆδ(k) (74) *6 orbit crossing ( q ),. *7 s = ψ, ψ = δ/d Poisson. *8. *9 (2π) 1. *10 powerspectrum, public code... 9

10 3.4 (collapse),. r, M.,, E < 0 cycloid * 11,, E > 0,. GM(< r) r = r 2. (75) 2ṙ2 1 GM(< r) = E = const. (76) r r = A 2 (1 cos θ), (77) t = A3 GM (θ sin θ). (78) r = A 2 (cosh θ 1), (79) t = A3 GM (sinh θ θ) (80) collapse (E < 0). ρ = 3M/(4πr 3 ). Einstein-de Sitter eq. (49) ρ = 1/(6πGt 2 )., eq. (78), δ(θ) = ρ ρ ρ = 9 2 (θ sin θ) 2 (1 cos θ) 3 1 (81). θ 1, θ, t L = A3 6 GM θ3, (82). 2, δ L (t) = 3 20 δ L (θ) = 3 20 θ2, (83) ( 6 GMt L A 3 ) 2 3. (84) *11. Bernoulli Newton Bernoulli, Bernoulli Newton cycroid.. 10

11 ,. θ = π, r (maximum expansion) turnaround. δ δ ta = 9π2 16., δ L,ta = 3 20 ( 6 GMt ta A 3 * 12. redshift z 0 δ 0 1, (85) ) 2/ (86) 1 + z ta = δ 0 δ L,ta (1 + z 0 ) (87) z ta, maximum expansion., r 0 (ρ = and θ = 2π) collapse,, δ L,c = 3 20 (12π) (88).., ρ =, violent relaxation * 13, virial. r vir. turnaround 0., virial 2T + W = 0, E = GM r ta (89), E = T + W = 1 2 W = 1 2 GM r vir. (90) r vir = 1 2 r ta (91). turnaround 2 3, ρ = 1/(6πGt 2 ), t c = 2t ta, turnaround 1/4., vir ρ ρ = 32 ρ ρ = 32(δ ta + 1) = 18π (92) vir ta Einstein-de Sitter universe * 14 virial overdensity. *12 t ta, t L (π), t(π) *13. *14 Ω 0 1 and Ω Λ > 0. Nakamura & Suto (1997). fitting formula Somerville & Primack (1999). Appendix. 11

12 3.5, M (dark halos) n(m)dm Press-Schechter (Press & Schechter 1974) δ, δ 1 ( ) random Gaussian. σ 2 = δ 2, f(δ)dδ = 1 ) exp ( δ2 2πσ 2σ 2 dδ. (93) Fourier, * 15 ˆδ(k) = ˆδ(k) e iφ(k) = δ(x)e ik x d 3 x (94), power spectrum, P (k) ˆδ(k) 2, (95) ˆδ(k)ˆδ(k ) = P (k)δ D (k k ) (96). ensemble. P (k) = P (k). ξ(r) δ(x)δ(x ) Fourier P (k) = ξ(r)e ik x d 3 x (97). random Gaussian field P (k) ( ξ(r)) Smoothed density field: window function W M (x). W M (x)d 3 x = 1. window function, smoothing scale R M., M ρr 3. (98), mass scale M, δ M (x) = W M (x x)δ(x )d 3 x (99) *15 Fourier... 12

13 . Eq. (99) Fourier,., δ M (x) = = 1 (2π) 3 = 1 (2π) 3 = 1 (2π) 3 = 1 (2π) 3 δ(x) = 1 (2π) 3 ˆδ(k)e ik x d 3 k, (100) W M (x 1 x) ˆδ(k)e ik x (2π) 3 d 3 k d 3 x W M (x x)e ik x d 3 x ˆδ(k)d 3 k W M (y)e ik (y+x) d 3 y ˆδ(k)d 3 k W M (y)e ik y d 3 y ˆδ(k)e ik x d 3 k Ŵ (kr)ˆδ(k)e ik x d 3 k (101) Ŵ (kr) = W M (x)e ik x d 3 x (102) window function Fourier * 16. smoothing δ = δ c M ( ) collapse. window function Top-hat filter 2. Gaussian filter 3. Sharp k-space filter W M (r) = 3 (1 4πR 3 θ r ), R (103) Ŵ (kr) = 3 [sin(kr) kr cos(kr)]. (kr) 3 (104) 1 W M (r) = (2πR 2 ) 2 3 Ŵ (kr) = exp ( k2 R 2 2 exp ( r2 2R 2 ), (105) ). (106) W M (r) = 1 2π 2 r 3 [sin(k cr) k c r cos(k c r)], (107) Ŵ (kr) = θ(k c k), (108) where k c R 1. *16 e,. 13

14 3.5.3 Press-Schechter mass function. Mass scale M, σ(m) 2 δm 2 = Ŵ (kr) 2 P (k)d 3 k. (109) δ M δ c mass scale M collapse., 1 f(δ M δ c ) = 2πσ(M) δ c ) exp ( δ2 2σ(M) 2 dδ. (110), (M > M)., M M + dm collapse n(m)dm, mass scale M collapse, Mn(M) ρ dm = f(δ M δ c ) f(δ M+dM δ c ) f(δ M δ c ) dm (111) M ( ) δ c = exp δ2 c dm. (112) 2πσ(M) 2 σ(m) M 2σ(M) 2 δ > 0. δ < 0 2 * 17, ( ) 2 ρδ c σ(m) n(m) = π Mσ(M) 2 M exp δ2 c 2σ(M) 2 (113). Press-Schechter (PS) mass function. factor 2 cloudin-cloud problem, Peacock & Heavens (1990) Bond et al. (1991) 25. Eq. (113) sharp k-space filter., sharp k-space filter, smoothing scale R mass scale M. Sheth & Tromen (1999) mass function N. ellipsoidal collapse model (Sheth et al. 2001). *17 ΩΩΩ < 14

15 4 collapse Eulerian x-space volume V, L(t) = d 3 r ρr v = ρa 4 d 3 x (1 + δ)x u. (114) a 3 V volume V. Eulerian volume V Lagrangian volume Γ, L(t) = ρa 5 Γ V d 3 q (q + S) ds dt, (115) Eulerian coordinates x Lagrangian coordinate q mapping x(t) = q + S(q, t) (116). mapping q x(q, t) Jacobian J (where d 3 x J 1 = d 3 q). Eq.(115) exact L. Zel doich,. Eq.(115), first order 1 + δ[x(q, t), t] = J(q, t) 1 (117) S(q, t) D(t)s(q) = D(t) φ(q) (118) L(t) ρa 5 dd dt = ρa 5 dd dt Γ Γ d 3 q (q + D φ) φ d 3 q q φ (119). φ(q) 0 Tayler,,, φ φ(q) = φ(0) + q i q i + 1 q=0 2 q 2 φ i q i q j q j + (120) q=0 L i (t) ρa 5 Ḋ d 3 q ɛ ijk q j q l Γ 2 φ q k q l (121) q=0 Potential φ D kl 2 φ q k q l, (122) q=0 inertia tensor I jl ρa 3 Γ d 3 q q j q l (123) 15

16 L i (t) = a 2 Ḋɛ ijk I jl D kl. (124) L(t) a 2 Ḋ t, (125) Einstein-de Sitter high-z. turnaround tidal field, turnaround (δ = δ ta 1.05), Lagrange collapse. 16

17 5 ( masa/research/nao08/) ( taka/lectures/cosmology/) P.J.E. Peebles, The Large Scale Structure of the Universe Padmanabhan, T., Structure formation in the Universe 17

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