臺灣博碩士論文加值系統

The main task of this Ph.D. thesis is seeking Bianchi type I metrics, which are homogeneous but anisotropic space, and studying their stability in some interesting cosmological models/theories to see whether they respect the well-known cosmic no-hair conjecture proposed by Hawking and his colleagues.

In particular, chapters 2 and 3 of this thesis include results in some extended scenarios of a supergravity motivated model proposed by Kanno, Soda, and Watanabe recently, which includes a coupling between the scalar field $
hi$ and the $U(1)$ field $A_\mu$ such as $f^2(

hi) F_{\mu\nu}F^{\mu\nu}$. As a result, this coupling causes stable spatial anisotropies in all the studied scenarios, in which the scalar field $
hi$ can be either canonical or non-canonical forms like the Dirac-Born-Infeld (DBI) or supersymmetric Dirac-Born-Infeld (SDBI) form. In other word, the existence of this coupling always leads to counterexamples to the cosmic no-hair conjecture. In order to make this conjecture alive, we introduce a phantom field $
si$, whose kinetic energy is negative definite, to these models. As a result, the inclusion of the phantom field $
si$ makes the following spatial hairs unstable during the inflationary phase, no matter the form of the scalar field $
hi$.

In chapter 4, we study the cosmological implications of a non-linear massive gravity theory proposed by de Rham, Gabadadze, and Tolley (dRGT) recently, which has been shown to be free of the Boulware-Deser ghost. In particular, we are able to find a simple stable anisotropic cosmological solution to the dRGT theory. More interestingly, we are also able to show the cosmological constant-like behavior of massive graviton terms in the dRGT theory. This result might give us a hint in order to investigate the nature of cosmological constant $\Lambda$. Similar to the previous chapters, we introduce the phantom field into the system and see that this extra field does lead the anisotropic cosmological solution to unstable state in general.

According to the study presented in this Ph.D. thesis, we might come to a conclusion that the phantom field is closely associated with the validity of the cosmic no-hair conjecture by causing, at least, one unstable mode to anisotropic metric(s).

The main task of this Ph.D. thesis is seeking Bianchi type I metrics, which are homogeneous but anisotropic space, and studying their stability in some interesting cosmological models/theories to see whether they respect the well-known cosmic no-hair conjecture proposed by Hawking and his colleagues.

In particular, chapters 2 and 3 of this thesis include results in some extended scenarios of a supergravity motivated model proposed by Kanno, Soda, and Watanabe recently, which includes a coupling between the scalar field $
hi$ and the $U(1)$ field $A_\mu$ such as $f^2(

hi) F_{\mu\nu}F^{\mu\nu}$. As a result, this coupling causes stable spatial anisotropies in all the studied scenarios, in which the scalar field $
hi$ can be either canonical or non-canonical forms like the Dirac-Born-Infeld (DBI) or supersymmetric Dirac-Born-Infeld (SDBI) form. In other word, the existence of this coupling always leads to counterexamples to the cosmic no-hair conjecture. In order to make this conjecture alive, we introduce a phantom field $
si$, whose kinetic energy is negative definite, to these models. As a result, the inclusion of the phantom field $
si$ makes the following spatial hairs unstable during the inflationary phase, no matter the form of the scalar field $
hi$.

In chapter 4, we study the cosmological implications of a non-linear massive gravity theory proposed by de Rham, Gabadadze, and Tolley (dRGT) recently, which has been shown to be free of the Boulware-Deser ghost. In particular, we are able to find a simple stable anisotropic cosmological solution to the dRGT theory. More interestingly, we are also able to show the cosmological constant-like behavior of massive graviton terms in the dRGT theory. This result might give us a hint in order to investigate the nature of cosmological constant $\Lambda$. Similar to the previous chapters, we introduce the phantom field into the system and see that this extra field does lead the anisotropic cosmological solution to unstable state in general.

According to the study presented in this Ph.D. thesis, we might come to a conclusion that the phantom field is closely associated with the validity of the cosmic no-hair conjecture by causing, at least, one unstable mode to anisotropic metric(s).

1 Introduction 1
2 Two scalar fields model 16
2.1 Basic setup . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Anisotropic power-law solutions . . . . . . . . . . . . . . . . . . . . . 17
2.3 Stability analysis of the expanding solutions . . . . . . . . . . . . . . . . . . . 20
3 Dirac-Born-Infeld models 26
3.1 Basic setup . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Anisotropic power-law solutions . . . . . . . . . . . . . . . . . . . . . 27
3.3 Stability analysis of the expanding solutions . . . . . . . . . . . . . . . . . . . 33
3.3.1 Dynamical equations . . . . . . . . . . . . . . . . . . . . . 33
3.3.2 Power-law perturbations . . . . . . . . . . . . . . . . . . . 36
3.4 The two-scalar-fields DBI model . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.1 The model . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.2 Anisotropic power-law solutions . . . . . . . . . . . . . . . . . . . . . 41
3.4.3 Stability analysis of the expanding solutions . . . . . . . . . . . . . . . 44
3.5 Supersymmetric extension of DBI model . . . . . . . . . . . . . . . . . . . . . 48
4 Non-linear massive gravity theory 50
4.1 Introduction . . . . . . . . . . . . . . . . . 50
4.2 The Stuckelberg formulation . . . . . . . . . . . . . . . . . . . 51
4.2.1 Effective cosmological constant . . . . . . . . . . . . . . . . . . . . . . 56
4.2.2 Alternative proof . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Some additional properties of the fiducial field equations . . . . . . . . . . . . 58
4.3.1 The recurrence relation of the massive Lagrangian . . . . . . . . . . . . 58
4.3.2 The traceless quartic terms . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Bianchi type I physical metric and fiducial metric . . . . . . . . . . . . . . . . 60
4.5 The constraint equations . . . . . . . . . . . . . . . . . . . . . 62
4.5.1 A = B solutions . . . . . . . . . . . . . . . . . . . . . 63
4.5.2 A $\neq$ B solutions . . . . . . . . . . . . . . . . . . . . . 64
4.6 Anisotropic cosmological solution and its stability . . . . . . . . . . . . . . . . 66
4.6.1 Stability analysis . . . . . . . . . . . . . . . . . . . . . 68
4.6.2 Effect of scalar
fields . . . . . . . . . . . . . . . . . . . . . . 71
4.6.3 Isotropic fiducial metric and global stability analysis . . . . . . . . . . 72
5 Conclusions 75
6 Appendix 78
6.1 Supersymmetric basic of KSW model . . . . . . . . . . . . . . . . . . . . . . . 78
6.2 Derivations for field equations of KSW model . . . . . . . . . . . . . . . . . . 81
6.2.1 Einstein equations . . . . . . . . . . . . . . . . . . . . . 82
6.2.2 Vector field equation . . . . . . . . . . . . . . . . . . . . . 84
6.2.3 Scalar field equations . . . . . . . . . . . . . . . . . . . . 86
6.2.4 Component equations of Einstein equations . . . . . . . . . . . . . . . 87
6.3 Additional derivations for the massive gravity theory . . . . . . . . . . . . . . 88
6.3.1 Massive graviton terms in the modified Einstein equation . . . . . . . . 88
6.3.2 Another way to obtain the vanishing $Y_{\mu\nu}$ . . . . . . . . . . . . . . . . . 89
6.3.3 Additional calculations for global stability analysis . . . . . . . . . . . 91

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