Á§Ää(Decelerating Parameter) ín

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Time

Decelerating Factor

Decelerating Factor of Bianchi Type I on General Relativity

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Figure 13: 2óú-Bianchi type I _íÁ§ÄäÆ“Ç : ™-õ( (γ = 2) , ™(

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0 0.5 1 1.5 2 2.5 3 3.5

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Time

Decelerating Factor

Decelerating Factor of Bianchi Type I on Brane Cosmology

2 4/3 3/2 1

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−0.9

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−0.8

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Time

Decelerating Factor

Decelerating Factor of Bianchi Type V on General Relativity

2 4/3 3/2 1

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£ C=1.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−1 0 1 2 3 4 5

Time

Decelerating Factor

Decelerating Factor of Bianchi Type V on Brane Cosmology

2 4/3 3/2 1

Figure 16: Brane ‡‹-- Bianchi type V _íÁ§ÄäÆ“Ç : ™-õ( (γ = 2) , ™(

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A Binachi Type I _  í R û

A.1  2 ó ú  í c q

Here, we want to complete the procedure of deriving Einstein equation by Christoffel connection symbol Γⁱ_jk, Riemann curvature tensor R_αβγ^σ , Ricci curvature tensor Rµν and Ricci scalar tensor R. Starting from the line element of Bianchi type I model

ds² = gµνdx^µdx^ν = −dt²+ a1(t)²dx²+ a2(t)²dy²+ a3(t)²dz² (97) The metric tensor g_µν can be show as below forms, without no-diagonal terms.

g₀₀ = −1 ; g₁₁= a₁(t)² ; g₂₂ = a₂(t)² ; g₃₃= a₃(t)² (98) g⁰⁰ = −1 ; g¹¹= 1

a²₁ ; g²²= 1

a₂(t)² ; g³³ = 1

a₃(t)² (99) The Christoffel connection symbol Γⁱ_jk can be expressed as

Γⁱ_jk = 1

2g^ih[∂_jg_hk+ ∂_kg_hj − ∂_hg_jk] (100) By ways of simple calculation, we will acquire the non-zero Christoffel curvature symbol Γⁱ_jk as below terms.

Γ⁰_ii = −a_i˙(t)

ai(t)g⁰⁰g_ii ; i = 1 ∼ 3 (101) Γⁱ_0i = a_i˙(t)

a_i(t)gⁱⁱg_ii ; i = 1 ∼ 3 (102) Hence, by relation of Ricci tensor Rµν and Riemann tensor R^α_µνβ, we can obtain the non-zero Ricci tensor from the non-non-zero Christoffel symbol(101)(102). And from the character of geometry we only get the diagonal elements.

R_µν = R^σ_µσν = Γ^σ_µν,σ− Γ^σ_µσ,ν + Γ^β_µνΓ^σ_βσ− Γ^β_µσΓ^σ_βν (103)

The Einstein field equation could be represented as differential forms via Ricci tensor(104)(105), scalar tensor(106). But, we must figure out the right part of below equation in the G.R.

G_µν = −Λg_µν + k²₄T_µν (107) Now, we must discuss this issue as two parts with different energy-density assumptions, (a) the perfect cosmological fluid, p = (γ − 1)ρ, γ = constant and 1 ≤ γ ≤ 2 ; (b) the scalar field, ρ_φ= ˙φ²/2 + U(φ) andp_φ = ˙φ²/2 − U(φ).

J The Perfect Cosmological Fluid

By the assumption of perfect cosmological fluid model obeying the barotropic form, p = (γ − 1)ρ, we could show the complete Einstein field Eq(107) as much simpler form by substituting Eq(18)(19) in Eq (107) neglecting the second order term of γ. First, the non-zero energy-momentum terms could be showed as

T₀₀= −ρ ; T_ii= p = ρ(γ − 1) , i = 1 ∼ 3 (108) Then we could substitute above equation into the Einstein equations as below forms.

H₁H₂+ H₂H₃+ H₁H₃ = Λ + k²₄ρ (109) H˙₂+ H₂²+ ˙H₃ + H₃²+ H₂H₃ = Λ + k²₄ρ(1 − γ) (110) H˙1+ H₁²+ ˙H3 + H₃²+ H1H3 = Λ + k²₄ρ(1 − γ) (111) H˙₂+ H₂²+ ˙H₁ + H₁²+ H₂H₁ = Λ + k²₄ρ(1 − γ) (112) In order to let the analysis much simpler, we could sum equation (110)(111)(112) then minus(109) to obtain (26) and sum equation (109)(111)(112) then minus (110) to get (313) as i = 1. Finally, we had derived the complete procedure of Einstein field equation in the G.R so that we could use them to analyze the dynamical evolution of our Universe.

We could show the equations representing the dynamical evolution as below forms.

3 ˙H +

X3 i=1

H_i² = Λ − 3γ − 2

2 k₄²ρ (113)

1 V

d

dt(V H_i) = Λ − γ − 2

2 k₄²ρ , i = 1 ∼ 3 (114)

˙ρ + 3γHρ = 0 (115)

A.2 Brane ^ @ í h õ

In fact, we can connect the G.R and Brane world scenario by adding the additional brane effect term, k⁴₅S_µν and E_µν, into the gravitational field equation (G_µν) then the Einstein field equation with brane effect can be show as

Gµν = −Λ + k₄²Tµν+ k₅⁴Sµν− Eµν (116) The terms of G_µν, representing the geometrical structure, are still the same with the left parts of equation (109)(110)(111)(112). In my analysis, I don’t consider the bulk Weyl tensor, E_µν, in the course of concise mathematical form.

(♣). The Perfect Cosmological Fluid

We still make the same assumption of perfect cosmological fluid. So that we can get the Einstein field equations easily through equation (18)(19). First, the non-zero energy-momentum (Tµν) and nero-zero quadratic forms (Sµν) could be expressed as

T₀₀ = −ρ ; T_ii = p = ρ(γ − 1) , i = 1 ∼ 3 (117) S00 = 1

12ρ² ; Sii= 1

12ρ²(1 − 2γ) , i = 1 ∼ 3 (118) From above equations, we could get the Einstein’s equation as below forms.

H₁H₂+ H₂H₃+ H₁H₃ = Λ + k₄²ρ + 1 We still can sum equation (120)(121)(122) then minus (119) to acquire equation (47);

sum (119)(120)(121) then minus (122) to gain (48) when i = 3. They could be showed as below forms.

Equation (125) is the differential form of energy-momentum conservation law.

B Bianchi Type V _  í R û

B.1  2 ó ú  í h õ

The line element of Bianchi type V is more complicate that Bianchi type I. Then we still obtain the non-zero Christoffel connection symbol Γ^k_ij, Ricci tensor R_µν and scalar tensor R via the line element of Bianchi type I in order to derived the gravitational field equation. First, the metric tensor was expressed as the forms.

g₀₀ = −1 ; g₁₁= a²₁(t) ; g₂₂ = a²₂(t)e^−2x ; g₃₃= a²₃(t)e^−2x (126) g⁰⁰ = −1 ; g¹¹= 1

a²₁(t) ; g²² = 1

a²₂(t)e^−2x ; g³³ = 1

a²₃(t)e^−2x (127) The non-zero Christoffel connection symbol Γ^k_ij (100) take the form

Γ⁰_ii = −g⁰⁰g_iia˙_i(t)

a_i(t) , Γⁱ_0i = a˙_i(t)

a_i(t) ; i = 1 ∼ 3 (128) Γ¹_jj = g¹¹g_jj , Γ^j_1j = g₀₀ ; j = 2 , 3 (129) Comparing with equation (128)(129) and (101)(102) we understand that geometrical mode of Bianchi type V is more complicate with Bianchi type I. Replacing equation (128)(129) for (103), the non-zero Ricci scalar tensor R_µν and scalar tensor R take the forms, the diagonal terms.

Using variables (18,19) Substituting equation (130∼134) for (107) with the supposition of perfect fluid and neglecting the Weyl tensor, flat Universe E_µν = 0. We still discuss this issue for two different models, as the explaining in the Bianchi type I model.

(^J) The Perfect Cosmological Fluid

We still neglected the quadratic term of γ in the Einstein field equation and took the forms by the variables of (17∼19).

H₁H₂+ H₁H₃+ H₂H₃− 1

a²₁(t) = Λ + k²₄ρ (135) H˙2+ ˙H3+ H₂²+ H₃²+ H2H3 + 1

a²₁(t) = Λ + k²₄ρ(1 − γ) (136) H˙₁+ ˙H₃+ H₁²+ H₃²+ H₁H₃ + 1

a²₁(t) = Λ + k²₄ρ(1 − γ) (137) H˙₂+ ˙H₁+ H₂²+ H₁²+ H₂H₁− 3

a²₁(t) = Λ + k²₄ρ(1 − γ) (138) Sum equation (136 ∼ 138) and minus (135), we could obtain equation (a). Equation (b) in the situation of i = 1 still can be gotten by summing (137,138) and (135) then minus (136). They can be expressed as below forms.

3 ˙H +

X3 i=1

H_i² = Λ − 3γ − 2

2 k₄²ρ (139)

1 V

d

dt(V H_i) − 2

a²₁(t) = Λ − γ − 2

2 ; i = 1 ∼ 3 (140)

2H₁− H₂ − H₃ = 0 (141)

˙ρ + 3γHρ = 0 (142)

We could understand that the energy-momentum conservation law of Bianchi type V is the same with it of Bianchi type I cosmology.

B.2 Brane ^ @ í h õ

The Christoffel connection symbol Γ^k_ij, Ricci tensor Rµν and scalar tensor R are the same with G.R under Bianchi type V cosmology. So we still use the same quantity of Einstein field equation, G_µν = R_µν−¹₂g_µνR. We still discussed two different models of matter field assumptions for analysis.

(♠). The Perfect Cosmological fluid

According to the perfect cosmological fluid and the geometrical forms of space-time under the Bianchi type V cosmology, we could express the equations as below forms. The non-zero energy-momentum tensor (T_µν) and non-zero quadratic forms (S_µν) are the same as

the derivations in appendix A, the perfect cosmological fluid.

H₁H₂+ H₁H₃+ H₂H₃− 1

a²₁(t) = Λ + k₄²ρ + 1

12k⁴₅ρ² (143) H˙₂ + ˙H₃+ H₂²+ H₃²+ H₂H₃+ 1

a²₁(t) = Λ + k₄²ρ(1 − γ) + 1

12k₅⁴ρ²(1 − 2γ) (144) H˙₁ + ˙H₃+ H₁²+ H₃²+ H₁H₃+ 1

a²₁(t) = Λ + k₄²ρ(1 − γ) + 1

12k₅⁴ρ²(1 − 2γ) (145) H˙2+ ˙H1 + H₂²+ H₁²+ H2H1− 3

a²₁(t) = Λ + k₄²ρ(1 − γ) + 1

12k₅⁴ρ²(1 − 2γ) (146) We still could derive equations (77), (78) and (79) by using some mathematical technique.

Equation (77) could be derived by summing (144∼ 146) and subtracting (143). Equation (78) whose i = 1 could be shown when we sum (145), (146) and (143) then minus (144).We can obtain the relation of H₁ = H₂ by expanding equation (78) with i = 1 and i = 2.The same processes could be operated to get the equation (79), 2H₁ − H₂ − H₃ = 0. The general forms of dynamical evolution of Bianchi type V in our brane anisotropic universe could be showed as

3 ˙H +

X3 i=1

H_i² = Λ − 3γ − 2

2 k₄²ρ −3γ − 1

12 k₅⁴ρ² (147) 1

V d

dt(V H_i) − 2

a²₁(t) = Λ −γ − 2

2 − γ − 1

12 k₅⁴ρ² (148) 2H₁− H₂− H₃ = 0 (149) We could find the evolution of our anisotropic universe from solving equation (147) ∼ (149).

References

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×çÓÜçû˝Fî=d (2003)

在文檔中 Bianchi Type I 以及 Bianchi Type V宇宙模型在Brane時空下的演化 (頁 41-53)