2009 2 26 1 3 1.1.................................................. 3 1.2..................................................... 3 1.3...................................................... 3 1.4..................................................... 3 1.5..................................................... 5 1.6................................................... 7 1.7................................................. 8 1.8................................................. 9 1.9................................................. 10 1.10................................................. 10 1.11..................................................... 11 1.12................................................... 12 1.13................................................... 12 1.14.............................................. 13 2 14 2.1....................................................... 14 2.2................................................. 14 2.3..................................................... 14 2.4.............................................. 15 2.5................................................... 16 ( 2.6 02 ).......................................... 17 2.7.............................. 18 ( 2.8 MN ).................................................. 20 2.9............................................... 20 2.10................................................. 21 3 22 3.1....................................................... 22 3.2........................................... 22 3.3.............................................. 23 3.4.......................................... 24 1

3.5..................................... 25 3.6................................................ 25 3.7........................................... 27 3.8.................................................. 27 3.9....................................... 27 3.10....................................... 28 3.11............................................. 30 3.12............................................. 30 3.13........................................ 31 4 33 4.1....................................................... 33 4.2................................................. 33 4.3.................................................. 33 4.4................................................. 35 4.5.................................................. 36 4.6 Friedman................................................ 37 4.7................................................ 39 4.8................................................ 42 4.9.................................................. 44 4.10.............................................. 47 4.11 WIMP...................................................... 48 4.12................................................ 48 4.13................................................... 49 4.14.................................................. 51 5 51 6 51 2

1 1.1 19 GPS 1.2 30 (km/s) 1.3 1.4 3

1 1: c c x 45 O O v Ō O Ō v v = x t = tan θ 2: O Ō Ō a x = 0 x = 0 c = 1 ±1 3 Ō a 3: 4

4: Ō 4 O Ō O 45 4 a a Ō x O Ō 1.5 s 2 = t 2 + x 2 + y 2 + z 2 (1) = t 2 + r 2 (2) t O c = r t = 1 s 2 = 0 (3) r Ō c = t = 1 s 2 = 0 (4) s 2 = s 2 = 0 (5) x x s 2 = Φ(v) s 2 (6) Φ(v) = 1 ( ) 5 y P L 5 5

5: O s 2 = t 2 + r 2 (7) = P L 2 (8) = y 2 (9) O x v Ō O t = 0 A B 5 A P L B Ō P, L (6) s 2 = r 2 (10) (Ō ) Φ(v)(O ) Φ(v) = Φ( v ) (11) Ō v Ō O s 2 = Φ(v) s 2 (12) s 2 = Φ(v) s 2 (13) s 2 = Φ(v) 2 s 2 (14) Ō O s 2 = s 2 (15) Φ(v) 2 = 1 (16) Φ(v) = 1 (17) 6

O t x = 0 t = 1 Ō t x = 0 t = 1 6 O Ō 6: 1.6 Ō r = 0 s 2 = t 2 = τ 2 (18) τ O s 2 = t 2 + x 2 + y 2 + z 2 = s 2 = τ 2 τ 2 = t 2 (1 v 2 ) τ = t 1 v 2 t = τ (19) 1 v 2 O Ō 1 (19) Ō O Ō Ō 1 v 2 O Ō GPS 7

1.7 Ō Ō O x t = 0 t = 0 x = l s 2 = l 2 (20) s 2 = s 2 v = x t l 2 = t 2 + x 2 x = = x 2 (1 v 2 ) l 1 v 2 (21) 7: Ō l 2 = x 2 (1 v 2 ) (22) x = l 1 v 2 (23) t t = vl 1 v 2 (24) BA x t = v x t = x A x B t A t B = v (25) t B = 0 x B = x A vt A (26) = l v2 l 1 v 2 1 v 2 (27) (28) = l 1 v 2 (28) 8

1.8 8 8: O Ō t x Ō O x v t = αt + βx x = γt + σx ȳ = y z = z t x v vt x = 0 vx t = 0 t = α(t vx) x = σ(x vt) ȳ = y z = z t x α σ (1) t = α(t vx) x = α(x vt) t 2 + x 2 = t 2 + x 2 (29) 1 α = ± 1 v 2 (30) v = 0 Ō O 9

t = x = t vx 1 v 2 1 v 2 vt + x 1 v 2 1 v 2 (31) (32) ȳ = y (33) z = z (34) v v 1.9 x Ō W = t O W = x t Ō O v x t = t = 0 W = 1 1 v ( x + v t) 2 1 1 v 2 ( t + v x) = W + v 1 + W v (35) W = v W W = v W = W v W v 1 1.10 x ( t, x, y, z) (36) O O O x O { x α } (37) α 0 3 x 0 = t x 1 = x x 2 = y x 3 = z 10

(37) i j k 1 3 x Ō x Ō { xᾱ} (38) 3 xᾱ = Λᾱβ x β (39) β=0 Λᾱβ O Ō 3 β=0 xᾱ = Λᾱβ x β (40) 3 β=0 1.11 9. A Ō U 9: U = eō (41) ( MCR ) ( 10) 9 10, A MCR eō U O U α = Λ ᾱ β ( e ō) = Λ ᾱ o (42) U 0 = (1 v 2 ) 1 2 U 1 = v(1 v 2 ) 1 2 (43) U 2 = 0 U 3 = 0 11

10: 1.12, ( ),. p = m U (44). p 0 = m(1 v 2 ) 1 2 p 1 = mv(1 v 2 ) 1 2 p 2 = 0 p 3 = 0 (45) O p (E, p 1, p 2, p 3 ), O.,. ( ). n p a k (46) k=1,1 i. 1.13. A 2 = (A 0 ) 2 + (A 1 ) 2 + (A 2 ) 2 + (A 3 ) 2 (47) = (A 0 ) 2 + (A 1 ) 2 + (A 2 ) 2 + (A 3 ) 2,.,.,,,,.,.,.,.,,. A B = A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3 (48) 12

,.., (48),.,( ).. O e α e β = 0 (α β) (49), Ō. eᾱ e β = 0 (ᾱ β) (50). 11: t x, Ō.,,. 1.14.,. p p = m 2 U U = m 2 (51), p p = E 2 + (p 1 ) 2 + (p 2 ) 2 + (p 3 ) 2 3 = E 2 + (p i ) 2 (52) i=1. 3 E 2 = m 2 + (p i ) 2 (53) i=1 0,E = m, c = 1 c, E = mc 2,,,. 13

2 2.1 2.2 T A B φ T( A, B) = φ (54) ( 0) 2 x y 12: ( ) ( ) ( ) ( 0 1 ) 2.3 s 2 = t 2 + x 2 + y 2 + z 2 (55) 14

s 2 = x x (56) s 2 = g( x, d) (57) = g ij x i x j (58) g g ij x i x j g ij = η ij η ij i = j = 0 1 i = j = 1, 2, 3 +1 g ij 2.4. p p α p α = p( e α ) e α,. A, e β p p( A) = p(a α e α ) = A α p( e α ) (59) = A α p α = A 0 p 0 + A 1 p 1 + A 2 p 2 + A 3 p 3 (60) p β p( e β) = p(λ ᾱ β e α) = Λ ᾱ β p( e α) = Λ ᾱ β p α, O Ō.. p( A) = Aᾱpᾱ = Λ µ ᾱλᾱβ Aβ p µ = δ µ β Aβ p µ = A β p β (61) δ µ β = { 1 µ = β 0 µ β Ō O p( A),.,,. { ω α }. p = p α ω α (62) 15

, p( A) = p α A α (63), p( A) = p α A α = p α ω α ( A) = p α ω α (A β e β ) = p α A β ω α ( e β ). ω α ( e β ) = δβ α (64). α e β., O ω 0 (1, 0, 0, 0) ω 1 (0, 1, 0, 0) ω 2 (0, 0, 1, 0) ω 3 (0, 0, 0, 1).,. (61) p = p α ω α = p β ω β p α ω α = Λ α β p β ω β p α ω α Λ α β p α ω β = 0 p α ( ω α Λ α β p α ω β ) = 0 (65) p α p α. ω α Λ α β ω β = 0 ω α = Λ α β ω β (66),,. 2.5 φ x φ,x φ x α φ,α (67) 16

2.6 ( 0 2 ) ( 0 2 ) p( A) q( B) p q( A, B) (68). ( 0 2 ) ( 02 ) f αβ f( e α, e β ) (69) 0 3 16 A, B f( A, B) = f(a α e α, B β e β ) = A α B β f( e α, e β ) (70) = A α B β f αβ f = f αβ ω αβ (71) (69) f µν = f( e µ, e ν ) (72) (71) f µν = f αβ ω αβ ( e µ, e ν ) (73) ω αβ ( e µ, e ν ) = δ α µδ β ν (74) (74) δ α µ e µ ω α δ β ν e ν ω β ω αβ ω αβ = ω α ω β (75) ( 0 2 ) ( 02 ) f = f αβ ω α ω β (76). f( A, B) = f( B, A) (77) A = e α, B = e β f αβ = f βα (78) (77) ( 0 2 ) h(s) h (s) ( A, B) = 1 2 h( A, B) + 1 2 h( B, A) (79) A = e α, B = e β h (s)αβ = 1 2 (h αβ + h βα ) (80) 17

h (αβ) = 1 2 (h αβ + h βα ) (81) ( 0 2 ) A = e α, B = e β f( A, B) = f( B, A) f αβ = f βα (82) h (A) ( A, B) = 1 2 h( A, B) 1 2 h( B, A) (83) h (A)αβ = 1 2 (h αβ h βα ) (84) h [αβ] = 1 2 (h αβ h βα ) (85) ( 0 2 ) h αβ = 1 2 (h αβ + h βα ) + 1 2 (h αβ h βα ) = h (αβ) + h [αβ] (86) ( 0 2 ) g( A, B) = A B = A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3 = B A = g( B, A) (87) 2.7 V g g( V, ) Ṽ g( V, ) Ṽ ( ) (88) A Ṽ ( A) g( V, A) = V A (89) g g(, V ) Ṽ ( ) (90) Ṽ e α V α Ṽ ( e α) = V e α = e α V = e α (V β e β ) = ( e α e β )V β. (91) 18

V α = η αβ V β (92) 1 µ = β = 0 η αβ = 1 µ = β = 1, 2, 3 0 µ β α = 0 V 0 = η 0β V β = ( 1) V 0 + 0 V 1 + 0 V 2 + 0 V 3 (93) = V 0 α = 1 V 0 = 0 + V 1 (1) + 0 + 0 = V 1 (94) V (a, b, c, d) (95) Ṽ ( a, b, c, d) (96) η αβ ( 0) 2 ηαβ (η αβ 0) η αβ Ã A (190 A α η αβ A β (97) A β = η βα A α (98) g A α A β 1 1 dφ φ V 0 V φ = dφ = dφ V = 0 (99) η αβ η 00 = 1,η 0i = 0,η ij = δ ij η αβ η αβ p 2 = p 2 = η αβ p α p β A α = η αβ A β p 2 = η αβ (η αµ p µ )(η βν p ν ) η αβ η βν η αβ η βν = δ ν α 19

p 2 = (p 0 ) 2 + (p 1 ) 2 + (p 2 ) 2 + (p 3 ) 2 (100) p q = 1 2 [( p + q)2 p 2 q 2 ] = p 0 q 0 + p 1 q 1 + p 2 q 2 + p 3 q 3 (101) 2.8 ( M) N V p V ( p) p( V ) p α V α < p, V > (102) ( M) 0 ( M) ( 0 M 20 ) V W V ( p) W ( p) p( V ) q( W ) V α p α W β q β (103) V W V α W β ( 2) 0 eα e β ( M) 0 ω α ( M) 0 ( M) ( N MN ) M N R ( 1) R( p; A) R( ω α ; e β ) = Rβ α (104) Ō Rᾱ β = R( ωᾱ; e β) = R(Λᾱµ ω µ ; Λ ν β e ν ) = ΛᾱµΛ ν βr µ ν (105) 2.9 ( M) ( N M 1 ) ( N + 1 M + 1 N 1) ( 2) 1 T αβ γ Tβγ α Tαγ β η βµ T αµ γ η αµ T µβ γ (106) α β ( 1) 2 T α βγ T γ ( 2) ( 1 12 ) 0 1, 2, 3 η αµ η µβ,. η αµ η µβ = η α β (107) η µβ ηβ α = η αµ 20

2.10 ( 0 0 ) ( 00 ) ( 0 1 ) ( 11 ) T T = T α β ω β ẽ α (108) T dt dτ = lim T(τ + τ) T(τ) τ 0 τ (109) ω α (τ + τ) = ω α (τ) (110) T dt ( dt α ) dτ = β ω β e α (111) dτ dt α β dτ = T α β,γu γ (112) (111),(112) ( 1 1 ) dt dτ = (T α β,γ ω β e α )U γ (113) T T α β,γ ω β ω γ e α (114) ( ) 21

3 3.1 3.2 13: F G = m G g F I = m I a m g m I m G = m I ( ) 22

3.3 ABCD 14: 14 V x β V x β = V α x β e α + V α e α x β (115) 0 e α x β = Γµ αβ e µ (116) Γ α β µ V x β = V α x β e α + V α Γ µ αβ e µ (117) 23

V V α + V µ Γ α x β x β µβ V V α x β = ( V x β + V µ Γ α µβ) e α (118) V α ;β ( V ) α β (119) ( β V α ) (120) V α,β + V µ Γ α µβ (121) V x β = V α ;β e α (122) V α ;β = V α,β 3.4 φ dφ = φ (123) φ = p α V α (124) β ( ) p β φ = φ,β = p α x β V α + p α V α x β (125) = p α x β V α + p α V α ;β p α V µ Γ α µβ (126) = ( p α x β p µγ µ αβ )V α + p α V;β α ( β p) α ( p) αβ p α;β = p α,β p µ Γ µ αβ (127) β φ = β (p α V α ) = p α;β V α + p α V α ;β Γ Γ 24

3.5 φ φ φ,β φ φ,β;α ( 0 2 ) φ,β;α x α x β φ (128) α β φ,β,α = φ,α,β φ,β;α = φ,α;β (129) φ,β;α φ,µ Γ µ αβ = φ,α;β φ,µ Γ µ βα (130) Γ µ αβ φ,µ = Γ µ βα φ,µ (131) Γ µ αβ = Γµ βα (132) g Γ 2 g αγ g αβ,µ = Γ ν αµg νβ + Γ ν βµg αν (133) g αµ,β = Γ ν αβg νµ + Γ ν µβg αν (134) g βµ,α = Γ ν βαg νµ Γ ν µαg βν (135) g αβ,µ + g αµ,β g βµ,α (136) = (Γ ν αµ Γ ν µα)g νβ + (Γ ν αβ Γ ν βα)g νµ + (Γ ν βµ + Γ ν µβ)g αν (137) = 2g αν Γ ν βµ (138) Γ γ βµ = 1 2 gαγ (g αβ,µ + g αµ,β g βµ,α ) (139) Γ ( ) 3.6 ABCD R δ 1 δ 2 A i δa i : Āi (R 2 ) Āi (R 1 ) = RjklA i j δ 1 x k δ 2 x l (140) 25

15: Rjkl i ( 1) 3 δ P Q A Ā i (Q s ) = A i (P ) Γ i jk(p )A j (P )δ s x k (141) s 1, 2 Āi (R 1 ) Āi (R 2 ) Ā i (R 1 ) = A i (Q 1 ) Γ i jk(q 1 )A j (Q 1 )(δ 1 x k + δ 2 x k ) (Γ i jl,k(q 1 ) Γ i nl(q 1 )Γ n jk(q 1 ))A j (Q 1 )δ 1 x k δ 2 x l Ā i (R 2 ) = A i (Q 2 ) Γ i jk(q 2 )A j (Q 2 )(δ 1 x k + δ 2 x k ) (Γ i jk,l(q 2 ) Γ i nk(q 2 )Γ n jl(q 2 ))A j (Q 2 )δ 1 x k δ 2 x l (142) Rjkl i δa i = Āi (R 2 ) Āi (R 1 ) = (Γ i jl,k Γ i jk,l + Γ i nkγ n jl Γ i nlγ n jk)a j δ 1 x k δ 2 x l Rjkl i = Γ i jl,k Γ i jk,l + Γ i nkγ n jl Γ i nlγ n jk (143),R jkl i R ijkl = R klij = R [ij][kl] (144) R i[jkl] = 0 (145) R ij := Rilj l (146) R := g ij R ij (147) G ij G ij := R ij 1 2 gij R (148) 26

3.7 ρ ρc 2 p p ρ 0 0 0 T ij = 0 p 0 0 0 0 p 0 (149) 0 0 0 p = ( ρ + p ) U i U j + pη ij (150) c = 1 3.8 U U λ U α = dxα dλ U β = dxβ dλ U U = 0 (151) U β U α ;β = U β U α,β + Γ α µβu µ U β = 0 (152) d (dx α ) + Γ α dx µ dλ dλ µβ dλ dx β dλ = 0 (153). 3.9 (1) (2) ( 2 φ = 4π F = m φ 27

(1) 1 2. ( ) ( ), 1. 2. (2) ds 2 = (1 + 2φ)dt 2 + (1 2φ)(dx 2 + dy 2 + dz 2 ) (154). φ 0 1 3.10 U = d x dτ (155) p = m U (156) U U = 0 (157) U α U µ ;α = 0 = U α U µ,α + Γ µ αβ U α U β (158) d (dx α ) + Γ α dx µ dλ dλ µβ dλ dx β dλ = 0 (159) λ, m τ m U p p = p α p µ,α + Γ µ αβ pα p β (160) = 0 (161) p α p 0,α + Γ 0 αβp α p β = 0 (162) 28

2 p 0 >> p i p α α = mu α α = m d dτ ( α = (162) x α ), m d dτ p0 + Γ 0 00(p 0 ) 2 = 0 (163) Γ µ βγ = 1 2 gµα (g αβ,γ + g αγ,β g γβ,α ) (164) Γ 0 00 = 1 2 g0α (g α0,0 + g α0,0 g 00,α ) (165) g αβ g αβ g 0α α = 0 0 Γ 0 00 = 1 2 g00 g 00,0 = 1 ( 1 2 1 + 2φ = 1 ( 1 2 1 + 2φ = (φ),0 1 + 2φ ) ( (1 + 2φ)),0 ) ( 2φ),0 = φ,0 + O(φ 2 ) (166) O(φ 2 ) (163) p 0 = m(1 v 2 ) 1 2 m (167) m d dτ p0 + Γ 0 00(p 0 ) 2 = 0 m d dτ p0 = m 2 (φ,0 + O(φ 2 )) d dτ p0 mφ,0 = m φ τ φ τ p 0 p 0 p 0 1,2 (168) p α p i,α + Γ i αβp α p β = 0 (169) m d dτ pi + Γ i 00(p 0 ) 2 = 0 d dτ pi = mγ i 00 (170) Γ i 00 = 1 2 giα (g α0,0 + g α0,0 g 00,α ) (171) g iα g iα = 1 1 2φ δiα (172) 29

δ i0 = 0 g j0 = 0 Γ i 00 = 1 1 2 1 2φ δiα (2g j0,0 g 00,j ) (173) Γ i 00 = 1 2 g 00,jδ ij 1 O(φ 2 ) = 1 2 g 00,jδ ij + O(φ 2 ) (174) g 00,j = (1 + 2φ),j = ( 2φ),j (170) Γ i 00 = 1 2 ( 2φ),jδ ij (175) d dτ pi = mφ,j δ ij (176) F = m φ F = ma 1,2 1,2 (1) 3.11 (1) (2) (1) 0 (2) 3.12 p p α p β;α = p α p β,α Γ γ βα pα p γ = 0 (177) m dp β dτ = Γγ βα pα p γ (178) Γ γ βα pα p γ = 1 2 gγν (g να,β + g νβ,α g αβ,ν )p α p γ = 1 2 (g να,β + g νβ,α g αβ,ν )p α p ν (179) 30

p α p ν ν α ( ) ν α Γ γ βα pα p γ = 1 2 gγν 2g να,β p α p ν (180) m dp β dτ = 1 2 gγν 2g να,β p α p ν (181) g να x β g να p β p p p = m 2 U U = m 2 (182) = g αβ p α p β φ << 1, p << m = (1 + 2φ)(p 0 ) 2 + (1 2φ)((p 1 ) 2 + (p 2 ) 2 + (p 3 ) 2 ) (183) (p 0 ) 2 = [ m 2 + (1 2φ)p 2] (1 + 2φ) 1. (184) (p 0 ) 2 m 2 (1 2φ + p2 m 2 ) (185) p 0 m(1 φ + p2 2m 2 ) (186) p 0 = g 0α p α = g 00 p 0 = (1 + 2φ)p 0 (187) p 0 m(1 + φ + p2 2m 2 ) = m + mφ + p2 2m m mφ p2 2m p 0 (188) 3.13 (2) G φ 2 φ = 4πGρ (189) T 00 T 31

O(g) = kt (190) O g ( 0 2 ) gµν,λσ, g µν,λ, g µν, µ Λ 2 (190) O αβ = R αβ + µg αβ R + Λg αβ (191) T αβ,β = 0 (192) T αβ ;β = 0 (193) O αβ ;β = 0 (194) (191) (R αβ + µg αβ R) ;β = 0 (195) 0 G αβ = R αβ 1 2 gαβ R = G βα (196) G αβ ;β = 0 (197) µ = 1 2 (198) G αβ + Λg αβ = kt αβ (199) G + Λg = kt (200) (1) 2 φ = 4πG (2) (3) 32

4 4.1 Galilei X 4.2 1929 Hubble Hubble ( ) Hubble 4.3 4.3.1 Hubble point Hubble Hubble z λ 0 λ e λ e (201) ( λ e λ 0 33

v 1 + v/c z = 1 v/c 1 (202) ( v c v ( d1 ) m 1 m 2 = 5 log 10 d 2 ( m i d i (202) (203) (203) 16: Hubble Hubble v = H 0 d (204) ( H 0 Hubble (s 1 )) (204) 17: 34

A, B A, B v OA = H 0 doa (205) v OB = H 0 dob (206) v AB = H 0 ( d OB d OA ) (207) = H 0 dab (208) Hubble Hubble Hubble H 0 = 72±8(km s 1 Mpc 1 ) (209) = 2.3 10 18 (s 1 ) (210) pc 1pc 3.09 10 18 m t = d(t 0) v(t 0 ) = H 0 1 = 1.37 10 10 (yr) yr 9.461 10 15 (m) 137 9.461 10 15 (m/yr) 1.37 10 10 (yr) = 1.29 10 26 (m) = 1.29 10 23 (km) = 4.2(Gpc) Hubble Hubble 4.3.2 1:pc 1AU 1pc 1AU 1 1AU = 1.496 10 11 (m) (211) 4.4 point 1/4 35

18: pc 4.5 point Hubble 36

4.6 Friedman point 4.6.1 100Mpc 4.6.2 Robertson-Walker ( ) ( ) ( ) 19: dl 2 = dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 (212) dl 2 = dl 2 = dr 2 1 r 2 /R 2 + r2 dθ 2 + r 2 sin 2 θdφ 2 (213) dr 2 1 + r 2 /R 2 + r2 dθ 2 + r 2 sin 2 θdφ 2 (214) 37

R dl 2 = dr2 1 Kr 2 + r2 dθ 2 + r 2 sin 2 θdφ 2 (215) K = 0 K = 1 R 2 K = 1 R 2 K θ = π 2 dr = dθ = 0 l dl 2 = r 2 dφ 2 (216) l = 2π dθ = dφ = 0 0 dl (217) = 2πr (218) dl 2 = 0 r D D = = dr2 1 Kr 2 (219) r 0 r 0 dl (220) dr 1 Kr 2. (221) K 180 180 180 (215) a(t) 20: dtdr dtdθ dl 2 = dt 2 + a(t) 2( dr 2 1 Kr 2 + r2 dθ 2 + r 2 sin 2 θdφ 2) (222) Robertson-Walker 38

4.6.3 Friedman Robertson-Walker Einstein Friedman Einstein G µν = 8πGT µν + Λg µν (223) Λ Einstein T µν = (ρ + p)u µ U ν + pg µν (224) 0 U µ = U ν = ( 1, 0, 0, 0) (225) Robertson-Walker Friedman G 00 = a(t),00 = 4 a(t) 3 πg(ρ + 3p) + Λ 3 ( a(t),0 ) 2 8πG G ii = = a(t) 3 ρ K a(t) 2 + Λ 3 (251) Einstein (254) (254) t (251) (226) (227) ρ,0 + 3(ρ + p) a(t),0 a(t) p<ρ ( 1 ) ρ = ρ 0 a(t) 3 = 0 (228) p r = ρ r 3 ( 1 ) ρ = ρ 0 a(t) 4 (251) (254) (259) Friedman T µν ;ν (251) ρ p Λ ρ + 3p Λ (254) (229) (230) (230) (229) (230) 4.7 point Hubble Hubble 39

4.7.1 Hubble Hubble Hubble d 0 d(t) = a(t 0) a(t) (231) d 0 d(t) a(t 0 ) ( ) a(t) ( ) v(t) (231) v(t) = d dt d(t) = d 0 da(t) a(t 0 ) dt = 1 a(t) d(t)da(t) dt = a(t),0 d(t) (232) a(t) H(t) = a(t),0 a(t) (233) v(t) = H(t)d(t) (234) Hubble H(t) Hubble Hubble Hubble Hubble Hubble 72±8(km s 1 Mpc 1 ) 4.7.2 a(t) a(t) 0 a(t) 2 Hubble 1 2 a(t)2,0 = 1 a(t) G4 3 πa(t)3 ρ (235) H(t) 2 = 8 πgρ (236) 3 ρ = 3H(t)2 8πG (237) 40

21: (critical) cr ρ cr = 3H(t)2 8πG 0. (238) Ω 0 = ρ 0 ρ cr0 (239) = 8πGρ 0 3H 0 2 (240) 4.7.3 Hubble Hubble ρ cr0 = 3H2 0 8πG (241) = 3 (2.2977 10 18 ) 2 8 π 6.67259 10 11 (kg m 3 ) (242) = 9.5 10 27 (kg m 3 ) (243) = 9.5 10 30 (g cm 3 ) (244) = 9.5 10 24 (g m 3 ) (245) 1.6736 10 24 g 1m 4.7.4 Hubble H(t) Ω 0 Friedman 41

Friedman ρ = ρ 0 ( 1 a(t) 3 ) ( a(t),0 ) 2 8πG = a(t) 3 ρ K a(t) 2 + Λ (246) 3 ( = H0 2 Ω0 a(t) 3 K a(t) 2 H0 2 + Λ ) 3H0 2 (247) ( = H0 2 Ω0 a(t) 3 k ) 0 a(t) 2 + Ω Λ0 (248) k 0 = K H0 2 Ω Λ0 = Λ 3H 2 0 (249) (250) ρ Λ,Ω Λ0 ρ Λ = Ω Λ0 = Λ 8πG = Λ 3H 2 0 (251) (252) Λ 8πGρ cr0 (253) = ρ Λ ρ cr0 (254) Ω Λ0 Hubble H(t 0 ) = H 0 (255) a(t 0 ) = 1 (256) ( a(t),0 ) 2 = H 2 a(t) 0 = 1 = Ω 0 a(t 0 ) 3 k 0 a(t 0 ) 2 + Ω Λ0 (257) = Ω 0 k 0 + Ω Λ0 (258) k 0 = K H 2 0 = Ω 0 + Ω Λ0 1 (259), K Ω 0 + Ω Λ0 1 Hubble H(t) Ω 0 + Ω Λ0 4.8 95 point 4.8.1 42

4.8.2 ( ) 4.8.3 δρ ( ) a(t), 22: 43

4.8.4 E = mc 2 25 75 4.8.5 25 75 0.5 0.5 5 20 75 4.9 20 MACHO W IMP point 4.9.1 4.9.2 0.7M pc 3 40 ( ) 4.9.3 0 30kpc 44

23: 24: ρ(r) r m M M(r) = 2π r 0 ρ(r )r dr (260) m v F = G Mm r 2 (261) F = m v2 r (262) F = F M(r) = v2 G r (263) M r r M(r) m F = G Mm r 2 (264) 45

v F = m v2 r (265) F = F G Mm r 2 = m v2 r M r = v2 G G M(r) (266) 46

4.9.4 1.. 2.. 3. 4. (CDM ) 5.. 1 2 3 4 (CDM) ( ) (HDM) 5 5 20 4.10 MACHO WIMP 4.10.1 MACHO MACHO 5 20 5 MACHO MACHO ( ) MACHO MACHO MACHO 0.16 47

( ) 4.11 WIMP WIMP 20 WIMP HDM CDM 0.5 CDM (Axion) CDM 10 14 1 (LHC) CDM CMB CDM HDM CDM 4.11.1 1. ( ) 20 2. 1. MACHO WIMP 4.12 75 point 4.12.1 48

4.13 Ia Ia Ia 1. 2. Ia 3. 4. Hubble 5. Hubble Ia 1. Ia 2. Ia 3. Ω M = 0.25 Ω Λ = 0.75 Ω Λ = 0.73 Ia 60 Ia SCP HZSS 49

4.13.1 0 0 4.13.2 Einstein 250 Einstein T µν 0 ( ) ρ p ρ = wpc 2 (267) w w w=-1 w 0 w 4.13.3 w Ia w w w 1 95 0.7 4.13.4 1. ( ) 75 2. 1. ( ) 50

4.14 1. 2. 137 3. 95 4. 20 75 5. 6. 5 X 6 [1] BERNARD F.SCHUTZ,,,1988 [2],,,1996 [3],,,1996 [4],,,1998 [5],, II-,,2007 [6],,,2001 51

[7],,,2005 [8],,2003 52