C10-075 26 2 12 1

1 4 1.1............................. 4 1.2............................ 5 1.3................................... 5 2 6 2.1............................ 6 2.2......................... 6 2.3................... 7 2.4............................. 7 3 8 3.1............................. 8 3.2........................ 8 3.2.1............................. 9 3.2.2........................... 9 3.2.3................ 11 3.2.4................... 11 3.2.5.................... 11 4 13 4.1 Runge-Kutta............................. 13 4.2.......................... 14 4.3 γ................... 15 4.4................ 18 4.4.1......................... 18 4.4.2................. 19 5 21 5.1............................. 21 5.2............................ 21 5.3............................. 22 5.4................ 22 6 23 6.1................................. 23 6.2............................. 23 6.3................................. 24 2

6.4.............................. 30 6.4.1 I.......................... 30 6.4.2 II.......................... 31 6.4.3 III......................... 32 7 33 3

1 1.1 137 138 38 4 90 1: ([6] ) 4

1.2 1000 100 138 1.3 ( ) 5

2 2.1 (Albert Einstein) 1915 1922 (Aleksandr Friedmann) (Georges Lemaitre) 1927 2.2 (Edwin Hubble) 1929 6

2.3 (George Gamow) 1948 1965 (Arno Penzias) (Robert Wilson) 2.4 (Seul Perlmutter) (Brian Schmidt) WMAP 73 23 4 7

3 3.1 (1) G µν + Λg µν = 8πG c 4 T µν (1) G µν R µν R G µν = R µν 1 2 Rg µν Λ g µν G c T µν Λ T µν 3.2 8

3.2.1 2 100 0 3.2.2 K 2 3 3 3 9

2: ds 2 (r, θ, ϕ) (2) ds 2 = dr2 1 Kr 2 + r2 ( dθ 2 + sin 2 θdϕ 2) (2) K r r = 2πr 4πr 2 r 2 ( dθ 2 + sin 2 θdϕ 2) (2 ) θ z ϕ x 3 3: ([8] ) 10

3.2.3 3 4 (2) ds 2 (3) ds 2 = c 2 dt 2 + a 2 dr 2 (t)[ 1 Kr + ( 2 r2 dθ 2 + sin 2 θdϕ 2) ] (3) (Robertson-Walker) t c a(t) 3.2.4 T µν T µν T µν = ρc 2 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p ρc 2 p (4) 3.2.5 (5) (6) (5) 00 (6) 11

2ä a + { (ä ) 2 + k a a 2 c2 = 8πG 3 ρ + Λ 3 c2 (5) (ȧ a ) 2 + k a 2 c2} = 8πG p c 2 + Λc2 (6) (5) (5) Λ = 0 (7) (8) µ T µν = 0 (7) ρ = 3ȧ (ρ + p ) a c 2 p ρ (9) (10) (8) p = p (ρ) (9) p = (γ 1) ρ (10) γ (5) (8) (10) a(t) 12

4 k k k 4.1 Runge-Kutta a Runge-Kutta Runge-Kutta 4 4 Runge-Kutta Runge-Kutta (x n, y n ) x n+1 = x + h (x n+1, y n+1 ) 4 Runge-Kutta (11) (12) y n+1 = y n + h 6 (k 1 + 2k 2 + 2k 3 + k 4 ) (11) k 1 = f (x n, y n ) k 2 = f ( x n + h, y 2 n + 1k ) 2 1 k 3 = f ( x n + h 2, y n + 1k ) 2 2 k 4 = f (x n + h, y n + k 3 ) (12) (11) y n+1 y n x x h 4 (12) 4 k 1 x n k 2 k 1 x n + h x n + h 2 k 3 k 2 k 2 y k 4 k 3 y 13

4 4: ( )([7] ) 4.2 Runge-Kutta 3 (5) k +1 0 1 (5) (8) (10) (10) γ γ = 4/3 Λ = 0 G = c = 1 a(t) 5 14

5: t = 0 k a(t) 3 2 k = 1 k = 0 k = 1 4.3 γ (10) γ 5 γ = 4/3 γ = 1 p = 0 k = 1, 0, +1 γ = 4/3 6 8 15

6: k = 1 (γ = 1, γ = 4/3 ) 7: k = 0 (γ = 1, γ = 4/3 ) 16

8: k = +1 (γ = 1, γ = 4/3 ) γ = 1 k 9 9: γ = 1 γ = 4/3 k 17

4.4 t = 0 t = 0 4.4.1 (Vesto Melvin Slipher) 1910 z λ s λ 0 (13) z = λ 0 λ s λ s (13) 18 v c v c z (14) z = v c (14) (14) z d d v (15) v = H 0 d (15) 18

H 0 t H(t) H 0 = H (0) 4.4.2 t v d a v = ȧ d = a H 0 (16) H 0 = ȧ (16) a H 0 a H 0 t = 0 H 0 H 0 = 0.045 t = 0 10 11 10: 19

11: H 0 (15) 18 H 70 11 20

5 5.1 k = 0 Λ = 0 ρ c = 3H2 8πG N (17) Ω tot ρ 0 ρ c Ω tot = ρ 0 ρ c (18) (18) (16) (5) k a 2 H 2 0 = Ω tot 1 + Λ 3H 2 0 (19) 5.2 Λ Ω Λ (19) Ω Λ 21

Ω Λ = Λ 3H 2 0 (19) Ω tot + Ω Λ = 1 + k a 2 H0 2 (20) (21) 5.3 3 k Ω k (21) k Ω k Ω k = k a 2 H0 2 (22) 5.4 Ω tot Ω Λ Ω k (5) Ω tot + Ω Λ + Ω k = 1 (23) a 22

6 6.1 Ω tot + Ω Λ + Ω k = 1 3 Java GUI Swing Java Java JFreeChart [9] JFreeChart Swing 6.2 12 Gakushu1 Gakushu2 Gakushu3 prefclear 23

12: 6.3 13 24

13: 1. 25

14: 2. 15: 3. 26

I Ω Λ = 0 Ω k II Ω k = 0 Ω tot 0.1 1.0 0.1 10 III 10 III 1 16: 1 Ω tot Ω Λ 17: Ω tot 27

18: Ω Λ 4. t 0 0 19: 5. I III 6. III 7. Ω tot + Ω Λ + Ω k = 1 28

8. 0 9. t a t H 29

6.4 6.4.1 I I k I 20 20: I Ω k Ω k 0 30

6.4.2 II Ω k = 0.0 II Ω k = 0.0 10 Ω tot 1.0 0.1 10 II Ω k = 0.0 21 21: II 31

6.4.3 III III Planck Ω tot = 0.25 Ω Λ = 0.75 Ω k = 0.0 138 III 3 III t = 0 22 22: III(Ω tot = 0.25, Ω Λ = 0.75, Ω k = 0.0) 138 32

7 33

[1] p165-p.194 ( 1996 ) [2] p.200-p.201 ( 2010 ) [3] ( 2008 ) [4] I ( 2008 ) [5] II ( 2007 ) [6] Timeline of the Universe http://map.gsfc.nasa.gov/media/060915/index.html [7] http://www.me.tokushima-u.ac.jp/ miw/wadai/ip3/06/10/10.html [8] http://homepage2.nifty.com/eman/relativity/flrw.html [9] JFreeChart http://www.jfree.org/jfreechart/ 34