LCS 理論的穩定性分析和 WKK 理論

Abstract

在這篇論文，我們將會討論兩種修改重力模型。第一種模型是 LCS 重力，我們會分析 LCS 重力在 FRW、Bianchi I 和 Bianchi II 空間的解。在 FRW 空間這部分，我們會試著和宇宙標準模型所得的解互相比較，作為評斷 LCS 重力是不是一個合適的宇宙論模型。在 Bianchi I 空間這部分，我們會顯示所得到的解確實是宇宙無毛定理的完美例子，並分析這個解的穩定性。在 Bianchi II 空間這部分，我們會重現 Kaloper 所得到的解，再做解的穩定性分析，用來評斷 Kaloper 所得到的解，是不是確實是宇宙無毛定理的反例。

第二種模型，我們將會討論 WKK 理論。就字面上，WKK 理論正是結合 Weyl 重力和 KK 理論所得到的新的修改重力理論。我們除了會一步一步的展示怎麼建立 WKK 理論，也會顯示 WTEGR，而 WTEGR 也會簡化成 TEGR。最後，我們會分析 WGR 理論的宇宙學，這裡主要是針對在 de Sitter 空間和 Bianchi I 空間的解。在 de Sitter 空間這部分，我們仍會試著和宇宙標準模型所得的解互相比較。在 Bianchi I 空間這部分，我們會展示所得到的非均向解，並分析這組解的穩定性。

In this thesis, we will discuss two models of modified gravity. The first one is the Lorentz Chern-Simons gravity. We will analyze the LCS gravity in FRW space, Bianchi I space and Bianchi II space. In the sector of FRW space, we will try to compare our solutions with the solutions of Standard-model Cosmology to see whether LCS gravity could be a good candidate of modified gravity. In the sector of Bianchi I space, we will show that the solutions we find are indeed the perfect examples of cosmic no-hair theorem. We will also analyze the stabilities of our solutions. In the sector of Bianchi II space, we reproduce the solutions found by Kaloper. We will also analyze the stabilities to verify whether Kaloper's solutions are the counterexamples of no-hair theorem.

The second model we will discuss is the Weyl-invariant Kaluza-Klein theory. Literally, we integrate both Weyl gravity and Kaluza-Klein theory to form a new modified gravity. We will first demonstrate how to construct the Weyl-invariant Kaluza-Klein theory step by step. We then show the Teleparallel equivalence of Weyl-invariant general relativity is also true and will reduce to the Teleparallel equivalence of general relativity. Finally, we analyze the Cosmology of Weyl General Relativity in de Sitter space and Bianchi I space. In the sector of de Sitter space, we will compare the solutions of de Sitter space with the solutions of Standard-model Cosmology again. In the sector of Bianchi I space, we demonstrate the anisotropic solutions we find and analyze the stabilities of these solutions.