臺灣博碩士論文加值系統

這篇論文主要研究,在Einstein-Hilbert以及Induce gravity模型,Bianchi第二型度規下場方程式以及解的行為,我們將用
變分方式得到場方程式,用愛因斯坦方程式來檢驗我們變分的結果是否正確,在此過程中,我們主要討論Bianchi typeII,並
且詳盡地推導每一個步驟,在試著將情況推廣成所有Bianchi模型通式。接下來對場方程式微擾,檢驗哈伯常數$H_i(t)$的
穩定性,且驗證Bianchi模型會從不均向性趨向於均向。在此論文中,我們亦用數值模擬的方式,探討在不同時期,尺度因子
的膨脹情況。另外我們也針對早期宇宙的暴脹情形做了計算,利用純量場的誘發重力理論,探討宇宙在大霹靂後,10^{-44}秒
到10^{-36}秒,模擬尺度因子的暴脹情形。

In this thesis we study one of the Bianchi type spaces, type II, which is a homogeneous and anisotropic space.
We follow the procedure to calculate the Chrystoffel symbols and curvature tensors and obtain the field equations with
the variational method. We first consider the Einstein-Hilbert model with perfect fluid and the induced gravity model at early universe.
. We gain the numerical solution and test that the scale factor indeed expands. And we also check the induced
gravity model that will indeed inflate the space within the $10^{-44}$ to $10^{-36}$ seconds that can solve
the flatness problem. The anisotropy to isotropy is also checked in these models.

1 導論
1.1 廣義相對論簡介....................1
1.2 標準大霹靂模型以及暴脹理論........1
1.3 Friedmann-Robertson-Walker 度規與標準宇宙模型..2
2 Bianchi 第二型度規下的E-H模型.........5
2.1 基本計算............................5
2.1.1度規...............................5
2.1.2Chritoffel符號與曲率張量...........6
2.1.3Ricci張量與Ricci純量...............9
2.1.4高階修正項的計算...................10
2.2 場方程式............................12
2.2.1Friedmann場方程式..................12
2.2.2g11場方程式通式....................13
2.2.3帶入g11場方程式通式所得到的場方程式14
2.2.4與愛因斯坦場方程式做比較...........15
2.2.5直接對a(t)做辯分...................16
2.3Bianchi恆等式所給出的場方程式連結....17
2.4場方程式的解的討論...................18
2.4.1趨向均向性(isotropy)的討論與模擬...18
2.4.2穩定性的分析.......................19
2.4.3數值模擬...........................20
2.5考慮理想流體下的場方程式.............21
2.5.1數值模擬...........................23
3 Induced Gravity 模型..................25
3.1原新暴脹與渾沌暴脹...................26
3.2場方程式及近似解.....................26
3.3數值模擬.............................30
Appendix
A
所使用的單位 .................32
B
Bianchi Type Space 簡介..............33
B.1黎曼流形與偽黎曼流形..............33
B.2運動群............................33
B.3Killing方程式.....................34
B.4群的種類..........................35
B.4.1可有G1群的空間..................35
B.4.2九種運動類型....................37
C 較廣義化的度規.....................39

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