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Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum

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Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum Maresuke Shiraishi November 1, 2018 arXiv:1210.2518v2 [astro-ph.CO] 23 May 2013 Abstract Although cosmological observations suggest that the fluctuations of seed fields are almost Gaus- sian, the possibility of a small deviation of their fields from Gaussianity is widely discussed. The- oretically, there exist numerous inflationary scenarios which predict large and characteristic non- Gaussianities in the primordial perturbations. These model-dependent non-Gaussianities act as sources of the Cosmic Microwave Background (CMB) bispectrum; therefore, the analysis of the CMB bispectrum is very important and attractive in order to clarify the nature of the early Uni- verse. Currently, the impacts of the primordial non-Gaussianities in the scalar perturbations, where the rotational and parity invariances are kept, on the CMB bispectrum have been well-studied. However, for a complex treatment, the CMB bispectra generated from the non-Gaussianities, which originate from the vector- and tensor-mode perturbations and include the violation of the rotational or parity invariance, have never been considered in spite of the importance of this information. On the basis of our current studies [1{7], this thesis provides the general formalism for the CMB bispectrum sourced by the non-Gaussianities not only in the scalar-mode perturbations but also in the vector- and tensor-mode perturbations. Applying this formalism, we calculate the CMB bispectrum from two scalars and a graviton correlation and that from primordial magnetic fields, and we then outline new constraints on these amplitudes. Furthermore, this formalism can be easily extended to the cases where the rotational or parity invariance is broken. We also compute the CMB bispectra from the non-Gaussianities of the curvature perturbations with a preferred direction and the graviton non-Gaussianities induced by the parity-violating Weyl cubic terms. We also present some unique impacts to the violation of these invariances on the CMB bispectrum. 2 REFERENCES References [1] M. Shiraishi, S. Yokoyama, D. Nitta, K. Ichiki, and K. Takahashi, Analytic formulae of the CMB bispectra generated from non- Gaussianity in the tensor and vector perturbations, Phys. Rev. D82 (2010) 103505, [arXiv:1003.2096]. [2] M. Shiraishi, D. Nitta, S. Yokoyama, K. Ichiki, and K. Takahashi, CMB Bispectrum from Primordial Scalar, Vector and Tensor non-Gaussianities, Prog. Theor. Phys. 125 (2011) 795{813, [arXiv:1012.1079]. [3] M. Shiraishi, D. Nitta, S. Yokoyama, K. Ichiki, and K. Takahashi, Cosmic microwave background bispectrum of vector modes induced from primordial magnetic fields, Phys. Rev. D82 (2010) 121302, [arXiv:1009.3632]. [4] M. Shiraishi, D. Nitta, S. Yokoyama, K. Ichiki, and K. Takahashi, Computation approach for CMB bispectrum from primordial magnetic fields, Phys. Rev. D83 (2011) 123523, [arXiv:1101.5287]. [5] M. Shiraishi, D. Nitta, S. Yokoyama, K. Ichiki, and K. Takahashi, Cosmic microwave background bispectrum of tensor passive modes induced from primordial magnetic fields, Phys. Rev. D83 (2011) 123003, [arXiv:1103.4103]. [6] M. Shiraishi and S. Yokoyama, Violation of the Rotational Invariance in the CMB Bispectrum, Prog.Theor.Phys. 126 (2011) 923{935, [arXiv:1107.0682]. [7] M. Shiraishi, D. Nitta, and S. Yokoyama, Parity Violation of Gravitons in the CMB Bispectrum, Prog.Theor.Phys. 126 (2011) 937{959, [arXiv:1108.0175]. REFERENCES 3 Acknowledgments First, I acknowledge the support of my supervisor, Prof. Naoshi Sugiyama. He backed my studies with much well-directed advice, enhancement of computing resources, and recruitment of high- caliber staff. In addition, owing to his powerful letters of recommendation, I could spend a fulfilling life in academia as a JSPS research fellow with ample opportunities to make presentations about our studies and communicate with top-level researchers in a variety of fields. The work presented in this thesis is based on seven papers written in collaboration with Shuichiro Yokoyama, Kiyotomo Ichiki, Daisuke Nitta, and Keitaro Takahashi. Shuichiro Yokoyama supported and encouraged me in all seven papers. He mainly provided advice on nature in the inflationary era and on the cosmological perturbation theory. Kiyotomo Ichiki's comments focused mainly on the impacts of the primordial magnetic fields on CMB and on the technical side of numerical calculations. A preface to these works arose from a chat with Shuichiro Yokoyama and Kiyotomo Ichiki. They also corrected my drafts in a tactful way. Daisuke Nitta's main contribu- tion was in telling me about several mathematical skills and fresh ideas necessary for calculating the primordial non-Gaussianities and the CMB bispectrum. Keitaro Takahashi gave me a polite lecture about the CMB scalar-mode bispectrum and discussed an application of our formalism to the primordial magnetic fields and cosmological defects. Owing to these four collaborators, I could gain a better understanding of the primordial non-Gaussianities and the CMB bispectrum. Financially, these individual works that comprise this thesis were supported in part by a Grant- in-Aid for JSPS Research under Grant No. 22-7477, a Grant-in-Aid for Scientific Research on Priority Areas No. 467 \Probing the Dark Energy through an Extremely Wide and Deep Survey with the Subaru Telescope", and a Grant-in-Aid for the Nagoya University Global COE Program \Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos" from the Ministry of Education, Culture, Sports, Science and Technology of Japan. I also acknowledge the Kobayashi-Maskawa Institute for the Origin of Particles and the Uni- verse, Nagoya University, for providing computing resources useful in conducting the research reported in this thesis. I was helped by many useful comments from Takahiko Matsubara, Chiaki Hikage, Kenji Kadota, Toyokazu Sekiguchi, and several researchers in other laboratories and institutions. My friends Masanori Sato and Shogo Masaki, my junior fellows Takahiro Inagaki and Yoshitaka Takeuchi, and other students provided advice on not only science and computers but also life. I owed my energy to them. Daichi Kashino provided a beautiful picture of the cosmic microwave background sky. Owing to Shohei Saga's careful reading, I could fix many typos in this thesis. Finally, I would like to thank my family for their concerns and hopes. 4 CONTENTS Contents 1 Introduction 12 1.1 History of the Universe . 12 1.2 Access to the inflationary epoch . 12 1.3 Concept of this thesis . 13 2 Fluctuations in inflation 15 2.1 Dynamics of inflation . 15 2.2 Curvature and tensor perturbations . 16 2.3 Consistency relations in the slow-roll limit . 18 3 Fluctuations in cosmic microwave background radiation 21 3.1 Einstein equations . 22 3.1.1 Homogeneous contribution . 23 3.1.2 Perturbed contribution . 24 3.2 Boltzmann equations . 26 3.3 Stokes parameters . 27 3.4 Boltzmann equations for photons . 29 3.5 Thomson scattering . 30 3.6 Transfer functions . 32 3.7 All-sky formulae for the CMB scalar-, vector- and tensor-mode anisotropies . 39 3.8 Flat-sky formulae for the CMB scalar-, vector- and tensor-mode anisotropies . 44 3.9 CMB power spectrum . 47 4 Primordial non-Gaussianities 55 4.1 Bispectrum of the primordial fluctuations . 55 4.2 Shape of the non-Gaussianities . 56 4.2.1 Local type . 56 4.2.2 Equilateral type . 57 4.2.3 Orthogonal type . 57 4.3 Observational limits . 58 4.4 Beyond the standard scalar-mode non-Gaussianities . 59 5 General formalism for the CMB bispectrum from primordial scalar, vector and tensor non-Gaussianities 65 6 CMB bispectrum induced by the two scalars and a graviton correlator 69 6.1 Two scalars and a graviton interaction during inflation . 69 6.2 Calculation of the initial bispectrum . 70 6.3 Formulation of the CMB bispectrum . 71 6.4 Estimation of the signal-to-noise ratio . 74 6.5 Summary and discussion . 75 7 Violation of the rotational invariance in the CMB bispectrum 78 7.1 Statistically-anisotropic non-Gaussianity in curvature perturbations . 78 7.2 CMB statistically-anisotropic bispectrum . 81 7.2.1 Formulation . 82 CONTENTS 5 7.2.2 Behavior of the CMB statistically-anisotropic bispectrum . 84 7.3 Summary and discussion . 86 8 Parity violation of gravitons in the CMB bispectrum 90 8.1 Parity-even and -odd non-Gaussianity of gravitons . 91 8.1.1 Calculation of the primordial bispectrum . 91 8.1.2 Running coupling constant . 94 8.2 CMB parity-even and -odd bispectrum . 95 8.2.1 Formulation . 95 (r) (r) 8.2.2 Evaluation of f 3 and f ........................... 99 W WWf 2 8.2.3 Results . 101 8.3 Summary and discussion . 104 9 CMB bispectrum generated from primordial magnetic fields 108 9.1 CMB fluctuation induced from PMFs . 108 9.1.1 Setting for the PMFs . 109 9.1.2 Scalar and tensor modes . 111 9.1.3 Vector mode . 112 9.1.4 Expression of a`m's . 114 9.1.5 CMB power spectrum from PMFs . 114 9.2 Formulation for the CMB bispectrum induced from PMFs . 120 9.2.1 Bispectrum of the anisotropic stress fluctuations . 121 9.2.2 CMB all-mode bispectrum . 121 9.3 Treatment for numerical computation . 126 9.3.1 Thin LSS approximation . 128 9.3.2 Selection rules of the Wigner-3j symbol . 130 9.4 Shape of the non-Gaussianity . 130 9.5 Analysis . 133 9.6 Summary and discussion . 137 10 Conclusion 143 A Spin-weighted spherical harmonic function 144 B Wigner D-matrix 146 C Wigner symbols 147 C.1 Wigner-3j symbol . 147 C.2 Wigner-6j symbol . 148 C.3 Wigner-9j symbol . 149 C.4 Analytic expressions of the Wigner symbols . 150 D Polarization vector and tensor 152 (a) (a) E Calculation of f 3 and f 155 W WWf 2 6 CONTENTS F Graviton non-Gaussianity from the Weyl cubic terms 157 F.1 W 3 .............................................157 F.2 WWf 2 ...........................................159 LIST OF FIGURES 7 List of Figures 3.1 CMB anisotropy on the last scattering surface.
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