臺灣博碩士論文加值系統

　　我們建立了一個兼具尺度對稱和有質量重力的模型、研究它的場方程式並且找到了幾個存在於第一型比昂其時空中可解析的解。在這個模型中，有質量重力場提供了一個等效的宇宙常數，而尺度對稱性提供了冷暗物質。在我們把Weyl規範向量的四次位能項加到拉格朗日密度之中以後，雖然尺度對稱的物質在今天看起來是冷的，它的初始狀態卻會是極端地熱，即它表現的像是冷和熱物質的混合。除此之外，多加一個自由的純量場來控制運動方程式中的非均向參數也使得我們的解能夠兼容均向背景。

We construct a model with Weyl-invariance and massive gravity, study its eld equations, and nd some analytic solutions in the Bianchi type I space. In this model, massive gravity provides an eective cosmological constant, while Weyl-invariance provides a cold dark matter. After we add a quartic potential of the Weyl gauge vector into the Lagrangian density, the Weyl-invariant matter will be extreme hot initially although cold today, which acts like a mixture of cold matter and hot matter. Furthermore, a free
in aton is added to control an anisotropy in equations of motion so that our solution can contain an isotropic background.

1 Introduction 1
2 ΛCDM model 2
2.1 Equation of state . . . . . . . . . . . . . . . . . . 3
3 Weyl-invariance 4
3.1 Conformal frame and unitary gauge . . . . . . . . . . 6
3.2 Equations of motion . . . . . . . . . . . . . . . . . 7
4 Massive gravity 8
4.1 Variations . . . . . . . . . . . . . . . . . . . . . 10
4.2 Relations between the determinant and eigenvalues . 11
4.3 Energy-momentum tensor . . . . . . . . . . . . . . . 12
4.4 Weyl-invariant massive gravity . . . . . . . . . . . 14
5 Weyl-invariant massive bi-gravity 15
5.1 Field equations . . . . . . . . . . . . . . . . . . 15
5.2 Conservation laws . . . . . . . . . . . . . . . . . 15
6 Bianchi type I expanding universe 17
6.1 Energy-momentum tensor and field equations . . . . . 17
6.2 Weyl-invariant Bianchi type I solution of an isotropic
energy-momentum tensor . . . . . . . . . . . . . . . 18
6.3 Analysis of the solution . . . . . . . . . . . . . . 20
7 Quartic potential and inflaton 21
7.1 The solution of the isotropic energy-momentum tensor 22
7.2 Conformal time . . . . . . . . . . . . . . . . . . . 25
7.3 Hubble diagram . . . . . . . . . . . . . . . . . . . 26
7.4 Initial behavior and final state . . . . . . . . . . 28
7.5 Equation of state ω and ratios of energy densities Ω 28
7.6 e-folding number . . . . . . . . . . . . . . . . . . 30
7.7 Perturbations . . . . . . . . . . . . . . . . . . . 30
8 Summary 35
9 References 36
A Curvature tensor in diagonal Bianchi spaces 38
B Total derivative combinations in the decoupling limit 38
C Constant solution 39

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