Brane Bianchi I, Bianchi III

Kantowski-Sachs

Brane Cosmology on Bianchi I, Bianchi III

and Kantowski-Sachs Spaces

(Ming-Fun Chun)

(Prof. Win-Fun Kao)

Brane Bianchi I, Bianchi III Kantowski-Sachs

Brane Cosmology on Bianchi I, Bianchi III and Kantowski-Sachs

Spaces

StudentMing-Fun Chun

AdvisorWin-Fun Kao

A Thesis

Submitted to Institute of Physics

College of Science

National Chiao Tung University

in partial Fulfillment of the Requirements

For the Degree of

Master

in

Physics

July 2004

Hsinchu, Taiwan, Republic of China

() 92

2

Brane Bianchi I, Bianchi III Kantowski-Sachs

()

()

()

()

:______________

()

94 7 20

94 7 20

9127508

________________

93 7 20

() 92

Brane Bianchi I, Bianchi III Kantowski-Sachs

()

9127508

________________

93 7 20

_

l>B, B, `>B; yV>B;e, B}V1,

J?D]-AdZ2-7A; .vH,

Bh Osamu Seto .w`BDBSdj; A

db2, nDu67B', F

B;d6.}0A7, ?DvDAn

u, >g',.c7nDjj, 6

7tMb4, u'N% ; rzT5I7rv

DBbnDN, '>; dY4uO j5\Bbv

vJN; 4.,, B_Av46?4

nD}-2 s-V, g)| u>7.=

: DEw.c,, 25,, -n, u.)

n; pD+n7., B7jy:7,

Fb yuBbT7v.), J_hGVz, Fb

u`7Bbr; DdQ7rA%}, BR

_Vz, BR k7r; Dkn{2B7d

b:b4 '', 'A, .>

8YH, ;BbHWu}IA[ 0vm

b>U-H7F4, , !_, db, *2 ;HN

7C

b

d2, Bb* Bianchi I,Bianchi III Kantowski-Sachs _|

; _G (Homogeneity) /4 (anisotropy) _; 7)

=gjj, (n =gjjFf#Bbk

m7 5, Bbd2d7sb cq: uBbec

_?D}0_; (perfect fluid) _, _ucqB

bF0u_JFw&vy|_&&v, 7

&v (brane spaces) 4u|2\nO

}&__=g jj(, BbJ__}w

AG (Homogeneity) /4 (Isotropy) 4, 7/w

2, b (cosmology constant) A3Obb; o

2, uBbF5?&v2&b (brane constant) A3O

bb; , Bb6)Bb5?b6}A3O

bbv`, 6u ov, Iwk&b3v5(

2

Contents

1 4

1.1 FRW_ . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 &v 10

2.1 _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Brane Bianchi type I,Bianchi type III D Kantowski-

Sachs vS . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 }&Dn 19

3.1 . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Nj5) . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.2 NjDbMj5 . . . . . . . . . . . . . . . . . . . 24

3.2 4b . . . . . . . . . . . . . . . . . . . . . . . 31

4 ! 34

5 36

5.1 Vacuum Einstein equation . . . . . . . . . . . . . . . . . . . . 36

5.2 Einstein equation with energy momentum tensor . . . . . . . . 47

3

Chapter 1

(Cosmology) w'o%7, r4}

A jD, ukh,., 6rk, /D

7|s_0J V, kDhxX, U

kA_, bW, 7/r 6\

1920, =gT|72, 7! 1929

, +N rm$Ph, 72wFrTkn

7G, 7 3z GGyA

bJ, rAIp7 nD, Bb6uw

1965,PenaiasDWilson7*5 (Cosmology Microwave

Background), 5A2.7 K, /G}0k5

7, y7Bb8, b

eV CMB hBbD? }0uG/

4 s_!4, UBb Friedmann-Roberson-Worker

_ _u5?ABuG/4G, OOh

, Bb CMB } 0wx

8%H, Dwv| jb ,ks=

n~j Bb&v3-brane , u&v\9A_&

_q, 7&&v (Bb bulk) q 7, 7.

!, &@v b ( Planck scale ) h.

7(V, :t Klein uz h9k Kaluza

&2, Fh7j, J_bHV_, *7 p

vjjJ\-yD~6-5H

.j(V. u\wW.

Ou1980H, k$ Kaluza-Klein hh\T|V,

1.1 FRW_

=g&2T|, zp7vDu.}/,

H7*HvH! vDwy1920, =gT|7.&2

(T|72, un _hj, 7/6ABb

'x Bb=gj (Einstein Equation) *T

(Action Principle) ), 77T2$J[:

S =

d4x

g(R

16G+ Lmatter) (1.1)

R F[u-0" (Riemann curvature tensor),G u

b (Gravitational constant), 7Lmatteru Lagrangian. Bbgd

}(, Bb)=gj:

R 12gR = 8GT (1.2)

Tu?" (Energy Momentum Tensor),gu Metric Ten-

sor, Metric Tensor dzVz, O, 7

Ricci scalar(R) Ricci " RsBb FJ=gj

UiuD?, 7D ?SBbU

i[|V

;WBb- ( 100Mpc) h*5 (Cosmic MicrowaveBackground (CMB)) !, BbBbuG (Homogeneity) /

4 (Isotropy) , G4sBb, BbFh 4DwF

P0; 74sBb, Fhj,w4Q B

b s_!4 *=g22, p7

x| 4, 7b,, x|4&

0kb&7 &7JT& 7T

Friedmann-Roberson-Worker(F-R-W) _uJ,HF|,

6

7 _Bb)_H F-R-W _(j:

ds2 = dt2 + R2(t)( dr2

1 kr2 + r2d2 + r2sin2()d2) (1.3)

,2R(t) (Scale Factor), uOv,

6zp7 k 0b, k = 1, 0,1,k = 0v, Bbu_; k = 1v, zBbFdu

_7[; k = 1v, zBbFdu_ r, , 7,t v , (jds22!4, 2 (Lorentz

Transformation) wM\M. sBb, BbdChv, BbJ

_|_ Vdn, 7!u. y6 Metric Tensor g(j

2- trace ,M:

g = gii = (1, R2(t)

1 kr2 , R2(t)r2, R2(t)r2sin2) (1.4)

k F-R-W _4BbJ) Christofol symbol(ijk)

}:

ijk =1

2gil(

gljxk

+glkxj

gjkxl

) (1.5)

0ij =R

Rgij

i0j =R

Rij

1.4 Christofol symbol 2 Christofol symbol xM

|, sBbu4 Ricci tensor 2R = gR ,

7R uF-0, -02OBb (Bb

)0" Ricci Tensor }:

R00 = 3RR

7

Rij = (RR

+ 2(R

R)2 +

2k

R2)

(1.6)

J" Ricci Scalar:

R = 6(RR

+R2

R2+

k

R2) (1.7)

7H R(t) jJ*j=gj):

R 12gR = 8GT (1.8)

TBbu;$, w} T00 = , T 11 = T 22 = T 33 = p.

?,p 9 jjvF)00}j, OVzB

bd Fridmann Equation:

R2

R2+

k

R2=

8G

3 (1.9)

(, BbJ Fridmann Equation Z:

k

H2R2=

3H2

8G

1 1 (1.10)

u? critical density cM

c

, c 3H2

8G

,2,H RRuF+b (Hubble constant). ;Wh *

5e2Bb 1, 6

, h2Bb, =gj.hz=g7

Uj K7pb (O(VwuG7

Chapter 2

&v

Bb&v;o1919AT|,Oo1.e,1980

Hw, k$ Kaluza-Klein hh\T|V,

Su?"T.,E bulk ,^@,

E = Ciajbnanb (2.3)

Ciajbnanb bulk 2&Weyl tensor , Eq2.12?

"S(1&2?") JT[:

S =1

12TT 1

4T T +

1

24g(3T

T T 2) (2.4)

,TBbFu_x4/.$, FJu;

$, ;?"}:

T 00 = T 11 = T 22 = T 33 = p (2.5)

?";$sBb, Bbe5

>T (f... ) ,T = T u?" trace T =

T = T00 + T

11 + T

22 + T

33 ?9 p ,

[:

p = ( 1) (2.6)

_b/w 1 2 .

;Wj Eq2.1, BbJ, E = 0/k5 0v, BbO brane 2=gj, c5?&v_&v

_y2 Bb d2, .5?E bulk ,^@,

IE = 0VdBbn&v_!c q

, Bb67j brane ,6?0, 722?

[:

T = 0 (2.7)

11

u} (Covariant Differentiation), Fu&v2}, T"? "5$:

T T ; = T , + T + T (2.8)

5(Bb},j7=gjD?0jjdj(,B

b)BbF; , j

12

2.1 _

,, Oh, Bb CMB }0wx

Eq2.10, Bb5(3b}un

Eq2.13 4b, BbJ*4bv

Vn__$

14

2.2 Brane Bianchi type I,Bianchi type

III D Kantowski-Sachs vS

Bbbn_G/4_ Bianchi type I,Bianchi

type III D Kantowski-Sachs(5(Bb BI,BII,KS) &-v!Z

Bbl*2JBbF_(jds2$,7(j(B

7_d"

g = (1, a1(t)2, a2(t)2, a2(t)2f()2) (2.14)

z (Metric Tensor) |, Bb-b) Christofol

symbol(ijk) sBbu", (y)[-

0R , y-0) Ricci"RD Ricci 0bR(, Bb|(y

z

,2k = 0, 1,1v}H[O BI,KS D BIII _,v?, 7 1 2b Bb*_j2BbF)uD?[ Ou?OvZU)Bb

U7j, FJBbb5?2? 0:

T = 0 (2.19)

;Wu}T?",%Jl(BbJ)Ov?D O[:

+ (a1a1

+ 2a2a2

) = 0 (2.20)

FJBb?Ov(V = a1a22):

= 0V , 0 = constant > 0 (2.21)

?), BbJ7jBbF)=jj

-uOvZ, FJBbZJn7 B

b7;)j, Bbj 2.16,2.17,2.18ZA:

d

dt(V H1) = V +

1

2k240V

1(2 ) + 112

k4520V

(1 2)(1 ) (2.22)d

dt(V H2) = V +

1

2k240V

1(2 ) + 112

k4520V

(1 2)(1 ) ka1(2.23)

3H + H21 + 2H22 = V +

1

2k240V

(2 3) + 112

k4520V

( 2)(1 3)(2.24)

J,j2, hb2:

H1 =a1a1

, H2 =a2a2

, V =1

3(H1 + 2H2) =

V

3V

16

-V, Bby,? Eq2.21, %JHb

(BbJ) }j:

V = 3V +3

2k240V

1(2 ) + 14k45

20V

12(1 ) + 2ka1(2.25)

Bb, k=0,k=-1,k=1Bb}J) Bianchi type I, a Bianchi type

III,and a Kantowski-Sachs j (;W}}l(, BbJ)

}j:

V 2 = 3V 2 + 3k240V2 +

1

4k45

20V

22 + 4k

a1dV + c

(2.26)

,2,c }b }jBb5(n3bj, Bb

Jp)|V_j u(4, 7/%rtl-, Bb

u)wj&j, , 5(}&Dn 2BbtwNj ,

BbJ_2j:

t t0 =

L(V )12 dV (2.27)

t v,t0v,V 7b L(V) 2:

L(V ) = 3V 2 + 3k240V2 +

1

4k45

20V

22 + 4k

a1dV + c (2.28)

2$, BbJQ)4b A Dai2j:

A = 3K2L(V )1 (2.29)

ai = a0V13 e

Ki

dv

vL(V )12 (2.30)

17

,2, bK2, KiBbJ;Wj Eq2.22,Eq2.23):

Hi = H +KiV

(2.31)

K1 =2

kaidt

3K2 =

1 kaidt3

(2.32)

K2 = 3i=1K2i (2.33)

j Eq2.21,Eq2.28,Eq2.32p Eq2.22,Eq2.23BbJ) K D}

b c 5O[:

K2 =2c

3(2.34)

18

Chapter 3

}&Dn

A,__R(, Bbub, FJBb

n Ov-, }AS$? Dw2

S[? OBbQ) jQj, ]Bb

nSvq, J)Nj 7_Nj

$ u, BbUKU7, Mathematica D Matlab

V)wbMj, (yDBbNjT_ BbNjJ

Wv Bbbj:

V 2 = 3V 2 + 3k240V2 +

1

4k45

20V

22 4k

a1dV (3.1)

BbJJ,j2, K-)NjV , (

BbZbzjp-j 2K-)a1

d

dt(V H1) = V +

1

2k240V

1(2 ) + 112

k4520V

12(1 )(3.2)

Bbu)a1, (BbJ4k

a1dV

, %4k

a1dVbMj(BbZJ__54 ,

19

a2l27):

b(t) =

v(t)a(t)

(3.3)

BbTbM}&vFUKlN|, Bbcqa(t0) = a0 ,

b(t0) = b0 , b(t0) = b0 , v(t0) = v0 t0 v, 7b)_,

v Bbb#8cqM: = 32,3 = p = 1 , 3k240 = m = 1 ,

14k45

20 = n = 1kKBb#:t0 = 0, a0 = 0.1, b0 = 0.1, b0 = 1. B

bhVz, Bb.UBbUKS, ]

u_b,cq 7Bb*cq|n 2, Bb

4b (b) Vn___44

74b:

A 3

i=1

(Hi H)23H2

(3.4)

n4b6uU7_jV48$

, Bb = 1, = 32, = 2, VT__}&

20

3.1

BbI = 32, /UM0.001VTbM}&M, 7w

FbqlF2,BbndNjbkwF10,100,1000

B10000IVdBbK, Bb63k240d

(, BbJwNS _-Bb

Nj8u/v(v

2. 14k45

20V

223,

3. 3k240V23, Njj:

V 2 ' 3k240V 2 = mV 2 (3.13)

Bb)Nj$:

v(t) = (

m

2(t t)) 2

6)a1(t),a2(t):

b(t) = c3(t t)23 , a(t) =

1

c23(

m

2)

2 (t t) 23 (3.14)

b:t = t0 2mv20 , c3 = b0(

2

mv

20 )

23 = 3

2v, Bb

)

v(t) =3

43 (m)

23 (t t) 432

83

b(t) = c4(t t) 49 , a(t) = 3103 (m)

23 (t t) 49

283 c23

Bbn

f(t) | 3V2 + 1

4k45

20V

22 + 4k

a1dV

3k240V2 | 1 (3.15)

(BbwvS Bb_-, Bb

Njvu v'oq, /ks_Nj

v5

23

3.1.2 NjDbMj5

Bbn__, b}u Kantowski-Sachs moddle,Bianchi Type

I D Bianchi Type III _. QOBbnFb3V 2,14k45

20V

22J

k240V2'bv, bF) Nj, 1DbM}

&Fd|VjWn, ;W k}530QVdBb}&

nYW. QOBbZ J.3vVWn.

BbI = 3/2, /UM0.001VTbM}& M, 7B

bbq = 1/3, k240 = 1000/3 J112

k4520 = 1/3. Bb

nbk240Vv Bbk240 > 1000/3v8,

7Bbk240 = 1000/3Vd yzpBbFn

[:

V (t)2 = 3V 2 + 3k240V (t)2 +

1

4k45

20V (t)

22 + 4k

a1dV (3.16)

QOBbbM}&)(DNjd, ;WBb

}j23vn, Bb|10%JqA3D

bMj$,Bb Error2 Error = (NumericalaproximationNumerical

)100%, J-Bb}._VT}&:

(1)KS: 3.1 KS _bMj(BbJcA

}Avq 7j; 3.2b

3bMjD_j, BbJ vdDbM

_(N; 3.3 brane 3vN$, Bb*$2

J)|VvVQvbM_DBbNjV

, Jc)v Qv brane Tub; J3.4

b3v`NjDbMj$

24

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10x 10

5

Time

Vol

ume

Sca

le F

acto

r (V

)

Numerical Plotting

Numerical

Figure 3.1: KSv(: = 1/3, k240 = 1000/3 J 1

12k45

20 = 1/3, = 3/2

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100

1

2

3

4

5

6

7

8

9

10x 10

5

Vo

lum

e S

cale

Fac

tor

(V)

Dominance of Cosmological Constant

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100

0.005

0.01

0.015

0.02

0.025

0.03

Time

Err

or

(V)

data1

NumericanceApproxivation

Figure 3.2: KS_3V 23vNjD(: = 1/3,k240 = 1000/3 J

112

k4520 = 1/3, = 3/2

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 105

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4x 10

3

Vo

lum

e S

cale

Fac

tor

(V)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 105

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time

Err

or

(%)

Dominance of 5D Gravitational Constant

Error

NumericanceApproxivation

Figure 3.3: 14k45

20V

223vNjvqD KS (: = 1/3, k240 = 1000/3 J

112

k4520 = 1/3, = 3/2

0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.060.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4Dominance of 4D Gravitational Constant

Vo

lum

e S

cale

Fac

tor

(V)

0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.060

2

4

6

8

10

12

14

16

Time

Err

or

(%)

Error

NumericanceApproxivation

Figure 3.4: 3k240V23vNjvqD KS

(: = 1/3, k240 = 1000/3 J112

k4520 = 1/3, = 3/2

26

(2)BI: 3.5 BI _bMj(; 3.6b

3bMjDNj, *_BbJ3vvu

v((Vv3Nj}}_(; 3.7

brane 3v$, BbJc brane vQv

; 3.8b3v`NjDbMj$, Bb

bbvBb J)yN8Jy

0 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10x 10

5 Numercal Plotting of Bianchi Type I

Time

Vo

lum

e S

cale

Fac

tor

Numericance

Figure 3.5: BIv(: = 1/3, k240 = 1000/3 J 1

12k45

20 = 1/3, = 3/2

3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10x 10

5 Dominance of Cosmological Constant

time

Vo

lum

e S

cale

Fac

tor

(V)

3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Err

or

(%)

NumericanceApproxivation

Error

Figure 3.6: 3V 23vNjvqD BI (: = 1/3, k240 = 1000/3 J

112

k4520 = 1/3, = 3/2

27

0 0.2 0.4 0.6 0.8 1 1.2

x 105

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4x 10

3

Vo

lum

e S

cale

Fac

tor

(V)

0 0.2 0.4 0.6 0.8 1 1.2

x 105

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5Dominance of 5D Gravitaional Constant

Time

Err

or

(%)

Error

NumericanceApproxivation

Figure 3.7: 14k45

20V

223vNjvqD BI (: = 1/3, k240 = 1000/3 J

112

k4520 = 1/3, = 3/2

0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.060.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Time

Vo

lum

e S

cale

Fac

tor

(V)

0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.060

2

4

6

8

10

12

14

16Dominance of 4D Gravitational Constant

Err

or

(%)

Error

NumericanceApproxivation

Figure 3.8: 3k240V23vNjvqD BI

(: = 1/3, k240 = 1000/3 J112

k4520 = 1/3, = 3/2

28

(3)BIII: 3.9_bMj(; 3.10b3bMjD_

j; 3.11 brane 3v$J3.12b3

v`NjDbMj$

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10x 10

5

Time

Vol

ume

Sca

le F

acto

r (V

)

Numerical Plotting

Numerical

Figure 3.9: BIIIv(: = 1/3, k240 = 1000/3 J 1

12k45

20 = 1/3, = 3/2

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100

1

2

3

4

5

6

7

8

9

10x 10

5

Vo

lum

e S

cale

Fac

tor

(V)

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045Dominance of Cosmological Constant

Time

Err

or

(%)

NumericanceApproxivation

Error

Figure 3.10: 3V 23vNjvqD BIII (: = 1/3, k240 = 1000/3 J

112

k4520 = 1/3, = 3/2

29

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 105

1

1.5

2

2.5x 10

3

Vo

lum

e S

cale

Fac

tor

(V)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 105

0

2

4

6

8

10

12

Time

Err

or

(%)

Dominance of 5D Gravitational Constant

Error

NumericanceApproxivation

Figure 3.11: 14k45

20V

223vNjvqD BIII (: = 1/3, k240 = 1000/3 J

112

k4520 = 1/3, = 3/2

0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.0650.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Vo

lum

e S

cale

Fac

tor

(V)

0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.0650

2

4

6

8

10

12

14

16

Time

Err

or

(%)

Dominance of 4D Gravitational Constant

Error

NumericanceApproxivation

Figure 3.12: 3k240V23vNjvqD BIII

(: = 1/3, k240 = 1000/3 J112

k4520 = 1/3, = 3/2

Bb, DMv BbNjJbMj'

N, uIAE, [Bb uW ,VzBbJ

__ KS,BI,BIII 3k240d_|c(, xA.v

Nj, 7__3b.

4k

a1dV TuU__wNjv KS wk BI

wk BIII 7o BIII ok BI ok KS *

}j26J|4k

a1dVT.}T, .}3v`

30

3.2 4b

42:

A 3

i=1

(Hi H)23H2

*l2BbJ4bBbF_-2$:

A = 3K2L(V )1

7w2K2DL(V )

L(V ) = 3V 2 + 3k240V2 +

1

4k45

20V

22 + 4k

a1dV + c

K2 =2c

3

n4b6uU Mathematica 2bM7_j

V48$ , Bb = 1, = 32, = 2, VT__

}& Bbb u: = 1/3, k240 = 1/3,112

k4520 = 1/3, = 3/2

c = 1 %7QbM}&(, BbJ)__4b

(, 3.13 KS (; 3.14 BI (; 3.15 BIII

(F

31

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Ani

sotr

opic

Par

amet

er

KantowskiSachs

gamma=1gamma=2gamma=3/2

Figure 3.13: Kantowski-Sachs__4: = 1/3, k240 =1/3, 1

12k45

20 = 1/3, = 3/2, c = 1

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Ani

sotr

opy

Anisotropy of Bianchi Type I on Brane Cosmology

23/21

Figure 3.14: Bianchi type I __4: = 1/3, k240 = 1/3,112

k4520 = 1/3, = 3/2, c = 1

32

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Ani

sotr

opic

Par

amet

er

Bianchi Type III

gamma=1gamma=2gamma=3/2

Figure 3.15: Bianchi type III __4: = 1/3, k240 =1/3, 1

12k45

20 = 1/3, = 3/2, c = 1

;WJ,

Chapter 4

!

Bbd7rn, ]iII7, czpZ

1. Bb__ KS,BI,BIII 3k240d_|c(, xA.

v Nj wv, Bb)

ju

V = V0e

3t

*j2Jc)bw27b o,

Bb)j:

v(t) = (

n(t t)) 1

,2n = 14k45

20_v, u brane ,bO3b

7s_v5, BbJv3k240d_|c(,

j:

v(t) = (

m

2(t t)) 2

,2m = 3k240, vubrObiH

34

2. 7__3b.4k

a1dVTuU__w N

jv KS wk BI wk BIII 7o BIII ok BI

ok KS *BbbM_ 26J|4k

a1dVT.}

T, .}3v`OFz, }v A

Chapter 5

5.1 Vacuum Einstein equation

_2#(j:

ds2 = gdxdx (5.1)

= d(t)2dt2 + a(t)2dr2 + b(t)2(d2 + f()2d2)

d":

g =

d(t)2 0 0 00 a(t)2 0 00 0 b(t)2 00 0 0 b(t)2f()2

d"5}:g11 = d(t)2, g22 = a(t)2, g33 = b(t)2, g44 = b(t)2f()2

:

g =

1d(t)2 0 0 0

0 1a(t)2

0 0

0 0 1b(t)2

0

0 0 0 1b(t)2f()2

36

d"5}:

g11 = 1d(t)2 , g22 = 1

a(t)2, g33 = 1

b(t)2, g44 = 1

b(t)2f()2

_2:

f() =

, Bianchi type Isin , Kantowski-Sachssinh , Bianchi type III

l:

1. ) Christofol symble j:

2. ) Ricci tensor R

Curvature tensor 2: R = mm ( )R R

3. 7) Ricci scalar :

R gR

4. |(J) Einstein equation :

R 12gR = kT g

1. I

Christofol symbol 2:

=1

2g(g + g g)

=1

2g(g, + g, g,)

Bbt) Christofol symbol } :

(a)

000 =1

2g0(g0,0 + g0,0 g00,) = 1

2g0(g0,0)

=1

2g0g00,0 =

1

2

1

d2(t)d(d2(t))

dt

=d(t)

d(t)

(b)

00i =1

2g0(g0,i + gi,0 g0i,) = 1

2g0(g0,i + gi,0)

=1

2g00g00,i +

1

2g0kgki,0

= 0

(c)

i00 =1

2gi(g0,0 + g0,0 g00,) = 1

2gi(2g0,0 g00,)

= gi0g00,0 12gi0g00,0

= 0

38

(d)

0ij =1

2g0(gi,j + gj,i gij,)

=1

2g00(g0i,j + g0j,i) 1

2g00gij,0

= 12g00gij,0

0rr =aa

d2

0 =bb

d2

0 =bbf 2()

d2

(e)

i0j =1

2gi(g0,j + gj,0 g0j,) = 1

2gi(g0,j + gj,0)

=1

2gi0g00,j +

1

2gigj,0

=1

2gikgkj,0

r0r =1

2grrgrr,0 =

1

2a2 2aa = a

a

0 =1

2gg,0 =

b

b

0 =1

2gg,0 =

1

2

1

b2f 2() 2bbf 2() = b

b

(f)

ijk =1

2gi(gj,k + gk,j gjk,) = 1

2gil(glj,k + glk,j gjk,l)

rrr =1

2grr(grr,r + grr,r grr,r) = 1

2grrgrr,r = 0

39

rr =1

2grr(grr, + gr,r gr,r) = 0

rr = 0 r =

1

2grr(gr, + gr, g,r) = 0

r = 0 r =

1

2grr(gr, + gr, g,r) = 0

= 0 r = 0 =

1

2gl(gl, + gl, g,l) = 0

rr = 0 r = 0 =

1

2gl(gl, + gl, g,l) = 1

2gg, = ff,

= 0 r = 0 =

1

2gl(gl, + gl, g,l) = 1

2gg, =

f,f

rr = 0 r = 0 = 0

FJBb) }:

000 =dd

0rr =aad2

0 =bbd2

0 =bbf2

d2r0r =

aa

0 =bb

0 =bb

=f,f

= ff,

40

2. II

Curvature tensor 2:

R = , , + mm mm

Bbl Ricci tensor }:

R R

(a)

R00 = R00

= R0000 + Ri0i0

= {000,0 000,0 + m0000m m0000m}+{i00,i i0i,0 + m00iim m0ii0m}

= i00,i i0i,0 + m00iim m0ii0m= i00,i i0i,0 + 000ii0 + j00iij 00ii00 j0ii0j= i0i,0 + 000ii0 j0ii0j= [r0r,0 + 0,0 + 0,0] + 000[rr0 + 0 + 0]

[(r0r)2 + (0)2 + (0)2]

= [( aa)0 + 2(

b

b)0] +

d

d[a

a+ 2

b

b] [( a

a)2 + 2(

b

b)2]

Ha aa

,Hb bb

R00 = [ ddHa 2 d

dHb + Ha + H

2a + 2Hb + 2H

2b ]

41

(b)

Rij = Rij

= R0i0j + Rkikj

= {0ji,0 00i,j + mij 00m mi00jm}+{kji,k kki,j + mij kkm mikkjm}

= [0ji,0 00i,j + 0ij000 + lij00l 0i00j0 li00jl]= +[kji,k kki,j + 0ijkk0 + lijkkl 0ikkj0 likkjl]= [0ji,0 +

0ij

000 li00jl]

= +[kji,k kki,j + 0ijkk0 + lijkkl 0ikkj0 likkjl] R0i0j = 0ji,0 + 0ij000 li00jl Rkikj = kji,k kki,j + 0ijkk0 + lijkkl 0ikkj0 likkjl

i.

R0i0j = 0ji,0 +

0ij

000 li00jl

R0r0r = 0rr,0 + 0rr000 rr00rr= (

aa

d2),0 + (

aa

d2)(

d

d) ( a

a)(

aa

d2)

= (aa

d2),0 + (

aad

d3) ( a

2

d2)

R00 = 0,0 + 0000 00= (

bb

d2),0 + (

bb

d2)(

d

d) ( b

b)(

bb

d2)

= (bb

d2),0 + (

bbd

d3) ( b

2

d2)

R00 = 0,0 + 0000 00= (

bbf 2

d2),0 + (

bbf 2

d2)(

d

d) ( b

b)(

bbf 2

d2)

= f 2[R00]

42

ii.

Rkikj = kij,k kki,j + 0ijkk0 + lijkkl 0ikkj0 likkjl

Rkrkr = Rrrrr + Rrr + Rrr= rrr,r rrr,r + 0rrrr0 + lrrrrl 0rrrr0 lrrrrl

+rr, r,r + 0rr0 + lrrl 0rr0 lrrl+rr, r,r + 0rr0 + lrrl 0rr0 lrrl

= 0rr0 +

0rr

0

= (aa

d2)(

b

b+

b

b)

= (2aab

d2b)

Rkk = Rrr + R + R= Rrr + R

= r,r rr, + 0rr0 + lrrl 0rr0 lrrl+, , + 00 + ll 00 ll

= 0rr0 , + 00 ll

= (bb

d2)(

a

a) (f,

f), + (

bb

d2)(

b

b) (f,

f)2

= (bb

d2)(

a

a) + (

b

d)2 f,,

f

Rkk = Rrr + R + R= Rrr + R

= r,r rr, + 0rr0 + lrrl 0rr0 lrrl+, , + 00 + ll 00 ll

= 0rr0 +

, +

0

0 ll

= (f 2bb

d2)(

a

a) (ff,), + (f 2 bb

d2)(

b

b) (f,

f)(ff,)

= f 2[(bb

d2)(

a

a+

b

b) f,,

f]

43

)Rij }: :

Rrr = R0r0r + R

krkr

= (a

d)2[ d

dHa + Ha + 2HaHb + H

2a ]

R = R00 + R

krr

= (b

d)2[ d

dHb + Hb + HaHb + 2H

2b ]

f,,f

R = R00 + R

krr

= (bf

d)2[ d

dHb + Hb + HaHb + 2H

2b ] ff,,

44

3. III )0b:

R = gR

= g00R00 + grrRrr + g

R + gR

= 2 1d2{ dd(Ha + 2Hb) Ha 2Hb 2HaHb H2a 3H2b }+

2k

b2

hence k = f,, f

4. IV , The vacuum Einstein equation :

R 12gR = g

=gj} (component):

(a) 00 component

R00 12g00R = g00

= 1d2

[H2b + 2HaHb] +k

d2b2(5.2)

(b) rr component

Rrr 12grrR = grr

= 1d2

[ dd(2Hb) + 2Hb + 3H

2b ] +

k

b2

(5.3)

(c) component

R 12gR = g

45

= 1d2

[ dd(Ha + Hb) + Ha + Hb + HaHb + H

2a + H

2b ]

(5.4)

(d) component

R 12gR = g

= 1d2

[ dd(Ha + Hb) + Ha + Hb + HaHb + H

2a + H

2b ]

(5.5)

d = 1, BbJ):

=

H2b + 2HaHb +kb2

, 00 component

2Hb + 3H2b +

kb2

, rr component

Ha + Hb + HaHb + H2a + H

2b , or component

(5.6)

00 component uF Friedmann equation .

46

5.2 Einstein equation with energy momen-

tum tensor

Energy momentum tensor }:

T 00 = , T 11 = T 22 = T 33 = p p ( 1)

T 2:

T = T = p3

Bb,?"(=gjA::

R 12gR = g + k24T (5.7)

k24 = 8G u 4&b.

47

1. The solution of Einstein equation : T = T g , FJ:

(a)

T00 = T00 g00 = ()(d2)

;W A.2 00 }Jh Einstein equation B.1 BbJ):

1

d2[H2b + 2HaHb] +

k

d2b2= + k24

as d = 1 H2b + 2HaHb +k

b2= + k24 (5.8)

(b)

Trr = Trr grr = pa

2 = ( 1)a2

;W A.3 rr }Jh Einstein equation B.1 BbJ):

2Hb + 3H2b +

k

b2= + k24(1 )

(5.9)

(c)

T = T g = pb

2 = ( 1)b2

;W A.2 }Jh Einstein equation B.1 BbJ):

Ha + Hb + HaHb + H2a + H

2b = + k

24(1 )

(5.10)

48

(d)

T = T g = pb

2f 2 = ( 1)b2f 2

;W,}Jh Einstein equation BbJ):

Ha + Hb + HaHb + H2a + H

2b = + k

24(1 )

(5.11)

c(Bb)_j:

H2b + 2HaHb +kb2

= + k24

2Hb + 3H2b +

kb2

= + k24(1 )Ha + Hb + HaHb + H

2a + H

2b = + k

24(1 )

49

(Zj:

(a2a2

)2 + 2(a1a1

)(a2a2

) +k

a22= + k24 (5.12)

2a2a2

+ (a2a2

)2 +k

a22= + k24(1 )

(5.13)a1a1

+a2a2

+ (a1a1

)(a2a2

) = + k24(1 )(5.14)

b2:

Ha =a1a1

, Hb =a2a2

50

2. Energy conservation Low

T = 0

Bb:

T T ; = T , + T + T

:

T = T g T 00 = , T rr = p

a21, T =

p

a22, T =

p

a22f2

FJ:

T = 0T 0 +rT r +T T = (a) + (b) + (c) + (d)

(a)

0T 0 = T 00,0 + T00 + T 00= + (T i0i0 + T

0000) + (T0000 + T

0ii0)

=

(b)

rT r = T rr,r + T rr + T rr= 0 + (T 00r0r + T

0irir) + (Tr00r + T

riir)

= T 00r0r + Triir

51

= T 00r0r + Trr0rr

= a1a1

+p

a21a1a1

= a1a1

(c)

T = T , + T + T = 0 + T 000 + T

0

= a2a2

+p

a22a2a2

= a2a2

(d)

T = T , + T + T = 0 + T 000 + T

+ T + T

0

= a2a2

+p

a22

f,f

+p

a22

ff,f 2

+p

a22

f 2a2a2f

= a2a2

+ pa2a2

= a2a2

T = 0T 0 +rT r +T T

= (a) + (b) + (c) + (d)

= + a1a1

+ a2a2

+ a2a2

= 0

52

+ (a1a1

+ 2a2a2

) = 0 (5.15)

;?Ovt: (V = a1a22)

= 0V , 0 = constant > 0

53

3. BbJZ Eq5.12 , Eq5.13 , Eq5.14 , :

d

dt(V Ha) = V +

1

2k240V

1(2 ) (5.16)d

dt(V Hb) = V +

1

2k240V

1(2 ) ka1(5.17)

3H + H2a + 2H2b = V +

1

2k240V

(2 3) (5.18)

b2:

Ha =a1a1

, Hb =a2a2

, V =1

3(Ha + 2Hb) =

V

3V

;Ws_j Eq5.16 , Eq5.17 , V jJA:

V = 3V +3

2k240V

1(2 ) 2ka1 (5.19)

and

Ha = H +2

3VK

Hb = H 13V

K

K =

ka1dt

j%}}(:

V 2 = 3V 2 + 3k240V2 4k

a1dV (5.20)

54

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55

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56