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 contribution 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. The red (blue) parts correspond to the hot (cold) spots (Copyright 2011 by Daichi Kashino)...... 21 3.2 Interaction between several components in the Universe...... 28 3.3 Geometry of Thomson scattering. Blue (Red) solid and two dashed arrows denote the incident (scattered) wave number vector and its orthogonal unit vectors, respectively. We set that xˆ0 = xˆ ...... 30 3.4 Geometry for the line-of-sight direction...... 40 3.5 The CMB spectra of the II (left top panel), IE (right top one), EE (left bottom one) and BB (right bottom one) modes. Here we consider a power-law flat ΛCDM model and fix the tensor-to-scalar ratio as r = 0.1. The other cosmological parameters are fixed to the mean values reported in Ref. [27]...... 49 3.6 CMB II power spectra sourced from the scalar-mode perturbations in a power-law flat ΛCDM model. The red solid line is plotted with ωc = 0.112, ΩΛ = 0.728, ωb = 0.02249, ns = 0.967, κ = 0.088 [27]. The green dashed, blue dotted and magenta dot-dashed lines are calculated if ωc decreases to 0.1, ΩΛ increases to 0.8, and ωb increases to 0.028, respectively...... 50 4.1 Representations of triangles forming the bispectrum. This figure is adopted from Ref. [19]...... 56 4.2 Two dimensional color contour for the shapes of the primordial bispectra. Each 2 panel describes the normalized amplitude of (k1k2k3) S(k1, k2, k3) as a function of k2/k1 and k3/k1 for k3 k2 k1. The amplitude is normalized by the maxi- 2 ≤ ≤ mum value of (k1k2k3) S(k1, k2, k3). The top-left panel is the local form given by Eq. (4.7), which diverges at the squeezed configuration. The bottom-left panel is the equilateral form given by Eq. (4.10), which is amplified at the equilateral configuration. The top-right panel is the orthogonal form given by Eq. (4.11), which has a positive peak at the equilateral configuration, and a negative valley along the elongated configurations. This is quoted in Ref. [11]...... 58 5.1 The CMB III (left top panel), IIE (right top one), IEE (left bottom one), and EEE (right bottom one) bispectra induced from the local-type (red solid line), equilateral-type (green dashed one), and orthogonal-type (blue dotted one) non- Gaussianities of curvature perturbations. The three multipoles are fixed as `1 = `2 = `3 `. Here we consider a power-law flat ΛCDM model and fix the nonlinear ≡ local equil orthog parameters as fNL = fNL = fNL = 100. The other cosmological parameters are fixed to the mean values reported in Ref. [9]...... 66 5.2 The CMB III (left top panel), IIE (right top one), IEE (left bottom one), and EEE (right bottom one) bispectra induced from the local-type (red solid line), equilateral-type (green dashed one), and orthogonal-type (blue dotted one) non- local equil orthog Gaussianities of curvature perturbations with fNL = fNL = fNL = 100. Here, we fix the two multipoles as `1 = `2 = 200, and plot each curve as the function in terms of `3. The cosmological parameters are identical to those in Fig. 5.1...... 67

6.1 Shape of I/k1. For the symmetric property and the triangle condition, we limit the plot range as k1 k2 k3 and k1 k2 k3 k1 + k2...... 71 ≤ ≤ | − | ≤ ≤ 8 LIST OF FIGURES

6.2 Absolute values of the CMB reduced bispectra of temperature ﬂuctuation for `1 = `2 = `3 `. The lines correspond to the spectra generated from tensor-scalar-scalar ≡ correlation given by Eq. (6.22) with gtss = 5 (red solid line) and the primordial local non-Gaussianity in the scalar curvature perturbations with fNL = 5 (green dashed line). The other cosmological parameters are ﬁxed to the mean values limited from WMAP-7yr data reported in Ref. [10]...... 74

6.3 Absolute values of the CMB reduced bispectra of temperature fluctuation generated from tensor-scalar-scalar correlation given by Eq. (6.22) (TSS + STS + SST ) and the primordial non-Gaussianity in the scalar curvature perturbations (SSS) as a function of `3 with `1 and `2 fixed to some values as indicated. The parameters are fixed to the same values defined in Fig. 6.2...... 75

6.4 Signal-to-noise ratio normalized by gtss as a function of the maximum value between `1, `2 and `3, namely, `. Each parameter is ﬁxed to the same values deﬁned in Fig. 6.2...... 76

7.1 Absolute values of the CMB statistically anisotropic bispectrum of the intensity mode given by Eq. (7.29) with C = 1 (red solid line) and the statistically isotropic one given by Eq. (7.31) with fNL = 5 (green dashed line) for `1 = `2 = `3. The left and right figures are plotted in the configurations (m1, m2, m3) = (0, 0, 0), (10, 20, 30), respectively. The parameters are fixed to the mean values limited from the WMAP-− 7yr data as reported in Ref. [38]...... 85

7.2 Absolute values of the CMB statistically anisotropic bispectra of the intensity mode given by Eq. (7.29) for (m1, m2, m3) = (0, 0, 0) (left panel) and (10, 20, 30) (right − one) as the function with respect to `3. The lines correspond to the spectra for (`1, `2) = (102 + `3, 100) (red solid line), ( 100 `3 2, 100) (green dashed line) | − | − and (100 + `3, 100) (blue dotted line). The parameters are identical to the values deﬁned in Fig. 7.1...... 86

2 2 2 8.1 Shape of k k k SA for A = 1/2 (top left panel), 0 (top right one), 1/2 (bottom 1 2 3 − left one), and 1 (bottom right one) as the function of k2/k1 and k3/k1...... 100

8.2 Absolute values of the CMB III,IIB,IBB, and BBB spectra induced by W 3 and WWf 2 for A = 1/2, 0, 1/2, and 1. We set that three multipoles have identical values − P3 as `1 2 = `2 1 = `3. The left figures show the spectra not vanishing for n=1 `n = − − P3 even (parity-even mode) and the right ones present the spectra for n=1 `n = odd (parity-odd mode). Here, we fix the parameters as Λ = 3 106GeV, r = 0.1, and −1 × τ∗ = k∗ = 14Gpc, and other cosmological parameters are fixed as the mean values− limited from− the WMAP 7-yr data [4]...... 102

8.3 Absolute value of the CMB III spectra generated from W 3 for A = 1/2 (red solid line), 0 (green dashed one) and 1/2 (blue dotted one), and generated− from the equil equilateral-type non-Gaussianity given by Eq. (8.58) with fNL = 300 (magenta dot- dashed one). We set that three multipoles have identical values as `1 = `2 = `3 `. Here, we ﬁx the parameters as the same values mentioned in Fig. 8.2...... ≡ . . . 104

9.1 Interaction between several components in the Universe if the PMF exists...... 109 LIST OF FIGURES 9

9.2 Power spectra of the CMB intensity fluctuations. The red solid, green dashed and blue dotted lines correspond to the spectra generated from the tensor, vector and scalar components of the PMF anisotropic stress for nB = 2.9, respectively. The − upper (lower) line of the red solid and blue dotted ones are calculated when τν/τB = 1017(106). The magenta dot-dashed line expresses the spectrum sourced from the primordial curvature perturbations. The strength of PMFs is fixed to B1Mpc = 4.7nG and the other cosmological parameters are fixed to the mean values limited from WMAP-7yr data reported in Ref. [29]...... 115 9.3 CMB power spectra of the temperature fluctuations. The lines correspond to the spectra generated from vector anisotropic stress of PMFs as Eq. (9.43) (red solid line) and primordial curvature perturbations (blue dotted line). The green dashed line expresses the asymptotic power of the red solid one. The PMF parameters are fixed to nB = 2.9 and B1Mpc = 4.7nG, and the other cosmological parameters are fixed to the mean− values limited from WMAP-7yr data reported in Ref. [29]. . . . 120 9.4 Diagrams with respect to multipoles [34]. The left panel corresponds to Eq. (9.60). Due to the Wigner-6j symbol originated with the sextuplicate dependence on the Gaussian PMFs, the true multipoles `1, `2 and `3 are linked with the dummy ones L, L0, and L00 and the 1-loop structure is realized. The right panel represents the tree-structure diagram, which arises from the CMB bispectrum induced by the four- point function of the Gaussian fields as mentioned in the previous sections...... 126 9.5 The ratio of the left-hand side (exact solution) to the right-hand side (approximate solution) in Eq. (9.67). The lines correspond to the case for L1 = `1 + 2 (red solid line), for L1 = `1 (green dashed one), and for L1 = `1 2 (blue dotted one). . . . . 129 − 9.6 Absolute values of the normalized reduced bispectra of temperature fluctuations for a configuration `1 = `2 = `3 `. The red solid, green dashed, and blue dotted lines correspond to the spectra≡ generated from the auto-correlations of the PMF tensor, vector, and scalar anisotropic stresses for nB = 2.9, respectively. The upper (lower) lines of the red solid and blue dotted lines are− calculated when 17 6 τν/τB = 10 (10 ). The magenta dot-dashed line expresses the spectrum sourced local from the primordial non-Gaussianity with fNL = 5. The strength of PMFs is fixed to B1Mpc = 4.7nG and the other cosmological parameters are fixed to the mean values limited from WMAP-7yr data reported in Ref. [29]...... 135 9.7 Absolute values of the normalized reduced temperature-temperature-temperature spectra arising from the auto-correlation between the PMF vector anisotropic stresses for a configuration `1 = `2 = `3 `. The lines correspond to the spectra gener- ≡ ated from vector anisotropic stress for nB = 2.9 (red solid line) and 2.8 (green − local − dashed line), and primordial non-Gaussianity with fNL = 5 (blue dotted line). The strength of PMFs is fixed to B1Mpc = 4.7nG and the other cosmological parameters are identical to the values used in Fig. 9.6...... 136 9.8 Absolute values of the normalized reduced temperature-temperature-temperature bispectra induced by the auto-correlation between the PMF vector anisotropic stresses and generated by primordial non-Gaussianity given by Eq. (9.91) as a function of `3 with `1 and `2 fixed to some value as indicated. Each parameter is fixed to the same value defined in Fig. 9.6...... 137 10 LIST OF FIGURES

9.9 Absolute values of the normalized reduced temperature-temperature-temperature bispectra induced by the auto-correlation between the PMF tensor anisotropic stresses and generated from primordial non-Gaussianity in curvature perturbations given by Eq. (9.91) as a function of `3 with `1 = `2. Each parameter is identical to the values defined in Fig. 9.6...... 138 9.10 Absolute values of the normalized reduced bispectra of temperature fluctuation for a configuration `1 = `2 = `3 `. The red solid and green dashed lines represent the exact and approximate spectra≡ arising from the tensor-tensor-tensor correlation 17 of the PMF anisotropic stresses for B1Mpc = 4.7nG, nB = 2.9 and τν/τB = 10 , respectively. The cosmological parameters are identical to− the values defined in Fig. 9.6...... 139 LIST OF TABLES 11

List of Tables

3.1 FLRW solutions dominated by radiation, matter, curvature, or a cosmological constant...... 24 3.2 Summary of the cosmological parameters of ΛCDM with ﬁnite r model from the WMAP 7-year data [27], and the data set combined with the results of the galaxy survey [32] and Hubble constant measurement [33], respectively. Here zreion denotes the redshift at the reionization epoch, t0 is the present time of the Universe, and ωm ωb + ωc...... 51 A.1 Dipole≡ (l = 1) harmonics for spin-0 and 1...... 145 A.2 Quadrupole (l = 2) harmonics for spin-0 and 2...... 145 12 1 INTRODUCTION

1 Introduction

1.1 History of the Universe

Several observational and theoretical studies on the cosmological phenomena such as the cosmic microwave background (CMB) radiation and matter clustering established the standard cosmological scenario that our Universe starts from microscopic scale and has been cooling down via the spatial expansion. Here, we summarize this scenario on the basis of Ref. [1]. In the primeval stage, the Universe may experience the accelerated spatial expansion, so-called inflation. In this stage, physics is determined in the quantum fluctuation. Via unknown reheating process, the energy of inflation is transformed into particles. Just after reheating, (strong), weak and electromagnetic interactions are unified and almost all of particles are relativistic. However, we believe that via cooling of the Universe and some symmetry breakings, particles become massive and decouple each other. Below 100GeV, the electroweak symmetry breaking occurs and the weak interaction weakens as the temperature drops. At around 1MeV, neutrinos decouple from electrons. Below 0.5Mev, electrons become massive and e+-e− annihilation frequently occurs. If the temperature reaches 0.1MeV, the nucleus of light elements are produced from protons and neutrons. Observational abundance of these elements matches the theoretical estimation based on the Big Bang scenario. If the temperature becomes less than 1eV (1011sec), the energy density of matters dominates over that of radiations. At around 0.1eV (380000yrs), protons (and helium nucleus) and electrons combine into hydrogen (and helium) atoms. This process is called recombination. The CMB radiation is photon which decouples at that time and comes to us now. This is the black body radiation whose averaged temperature and its spatial anisotropies are, respectively, 2.725K and (10−5)K. The anisotropies of CMB intensity and polarizations reflect the density fluctuations in theO primordial Universe. Resultant contrasts of matter distributions evolve observed large-scale structures in the balance between the gravitational force and pressure of radiations. Consequently, small-scale structures are produced earlier compared with large-scale structures. First stars arise at around 108yrs. After these die, emitted photons ionize hydrogen atoms in the intergalactic medium until redshift z 6. This phenomenon is called reionization. At latter half of the age of the Universe, the second∼ accelerated expansion starts. This may be because an unknown energy with negative pressure, the so called dark energy. This expansion continues at the present epoch (13.7Gyrs).

1.2 Access to the inﬂationary epoch

At the inflationary era, the field values of physical quantities, such as metric and matters, quantum- mechanically fluctuate inside the horizon. However, the accelerated spatial expansion stretches these fluctuations beyond the horizon. Due to no causal physics, metric perturbations outside the horizon are preserved1. These constant metric perturbations re-enter the horizon just before recombination and behave as seeds of the CMB fluctuations. In this sense, detailed analyses of the patterns of the CMB anisotropies will help explain the questions about the initial condition of our Universe, e.g., what kind of field there exists, what state gravity is in, and how strong the coupling is.

1This is valid only when there are no anisotropic stress ﬂuctuations. 1.3 Concept of this thesis 13

1.3 Concept of this thesis

Conventionally, the information of the primordial density fluctuations has been extracted from the two-point functions (power spectra) of the CMB fluctuations. There is a statistical property that although a non-Gaussian variable generates both even and odd-point correlations; a Gaussian variable generates only even-point correlations. Hence, it is hard to discriminate between the Gaussian and non-Gaussian signals in the CMB power spectrum. Theoretically, whether the primordial seed fluctuations are Gaussian depends completely on the inflationary models. Therefore, the check of the non-Gaussianity of the primordial fluctuations will lead to a more precise comprehension of the early Universe. To extract the non-Gaussian signals from the CMB anisotropy, we should focus on the higher-order correlations of the CMB fluctuations such as the CMB three-point correlations (bispectra). Owing to the recent precise observation of the Universe, the CMB bispectra are be- coming detectable quantities. As a result, the CMB bispectra are good measures of the primordial non-Gaussianity. The primordial non-Gaussianities originating from the scalar components and their effects on the CMB bispectrum have been well-studied (Refs. [2, 3]). However, for some situations the vector components (vorticities) and tensor ones (gravitational waves) also act as non-Gaussian sources. This indicates that unknown signals, unlike the scalar case, may also appear in the CMB bispectra. To study these impacts in detail, we produced the general formulae for the CMB temperature and polarization bispectra from the scalar, vector and tensor non-Gaussianities [4, 5]. Next, utilizing these formulae and computing the practical CMB bispectra, we obtained new constraints on some primordial non-Gaussian sources and learned more about the nature of the early Universe [6–10]. This thesis aims to discuss the CMB bispectra induced by the primordial scalar, vector, and tensor non-Gaussianity on the basis of our recent studies [4–10]. More concrete organization of this thesis is as follows. In Secs. 2 and 3, we demonstrate how to generate the seed fluctuations in the inflationary era on the basis of some review papers and present formulae for the scalar, vector, and tensor modes of the CMB anisotropies as mentioned in Ref. [4]. We also review some observational findings obtained by the analysis of the CMB power spectra. In Secs. 4 and 5, we describe the general formulae of the CMB bispectra generated from the primordial scalar, vector, and tensor non-Gaussianities [5]. We then discuss the applications to the non-Gaussianities in two scalars and a graviton correlator [5] (Sec. 6), involving the violation of the rotational or parity invariance [9, 10] (Secs. 7 and 8), and sourced by the primordial magnetic fields [6–8] (Sec. 9). Finally, we summarize this thesis and discuss some future issues (Sec. 10). In the appendices, we describe some mathematical tools and the detailed calculations required for the conduct of our formalism.

References

[1] D. Baumann, TASI Lectures on Inﬂation, arXiv:0907.5424.

[2] N. Bartolo, E. Komatsu, S. Matarrese, and A. Riotto, Non-Gaussianity from inﬂation: Theory and observations, Phys. Rept. 402 (2004) 103–266, [astro-ph/0406398].

[3] E. Komatsu, Hunting for Primordial Non-Gaussianity in the Cosmic Microwave Background, Class. Quant. Grav. 27 (2010) 124010, [arXiv:1003.6097]. 14 REFERENCES

[4] 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].

[5] 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].

[6] 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].

[7] 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].

[8] 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].

[9] M. Shiraishi and S. Yokoyama, Violation of the Rotational Invariance in the CMB Bispectrum, Prog.Theor.Phys. 126 (2011) 923–935, [arXiv:1107.0682].

[10] 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]. 15

2 Fluctuations in inﬂation

Inflation expresses an exponential growth of the scale factor of the Universe in the early time, namely, a eHt. In Einstein gravity, this requires p ρ with p and ρ being the pressure and energy density,∼ and is often realized by the existence of∼ a − scalar field, inflaton. We believe that the small fluctuations of this field have created the curvature perturbations and the density contrasts of matters. Moreover, some vorticities and gravitational waves may also have evolved together. In this section, we briefly describe the physical treatment of these fluctuations in the inflationary era in accordance with Ref. [1].

2.1 Dynamics of inflation As the action in the inflationary era, we consider the simple one including a scalar field φ, which is called inflaton and minimally coupled with gravity as Z 4 1 2 1 µν S = d x√ g M R g ∂µφ∂νφ V (φ) , (2.1) − 2 pl − 2 −

−1/2 where R denotes the Ricci scalar, V (φ) is the potential, and Mpl (8πG) is the reduced Planck mass. The energy momentum tensor and the ﬁeld equation for ≡φ are, respectively, given by 2 δS 1 σ Tµν = ∂µφ∂νφ gµν ∂ φ∂σφ + V (φ) , (2.2) ≡ −√ g δgµν − 2 − δS 1 µ = ∂µ(√ g∂ φ) + Vφ = 0 , (2.3) δφ √ g − − where Vφ = dV/dφ. On the FLRW metric as ds2 = dt2 + a2dx2 = a2( dτ 2 + dx2) (2.4) − − with τ being the conformal time and under the assumption that φ(t, x) φ(t), the energy density and pressure of the scalar ﬁeld are written as ≡

1 2 1 2 ρφ = (∂tφ) + V (φ) , pφ = (∂tφ) V (φ) . (2.5) 2 2 − 2 Thus, if V exceeds (∂tφ) /2 and the parameter wφ pφ/ρφ becomes less than 1/3 and the accelerated expansion can be realized. The Friedmann≡ equation, the acceleration equation− and the ﬁeld equation are, respectively, given by 2 1 1 2 H = 2 (∂tφ) + V (φ) , 3Mpl 2 d2a 1 −1 2 (2.6) a 2 = 2 (ρφ + 3pφ) = H (1 H ) , dt −6Mpl − d2φ dφ + 3H + V = 0 , dt2 dt φ where H ∂ta/a is the Hubble parameter and we have introduced the so-called Hubble slow-roll parameter≡ as 2 ∂tH d ln H 3 1 ∂tφ H 2 = = (wφ + 1) = , (2.7) ≡ − H − dN 2 2 MplH 16 2 FLUCTUATIONS IN INFLATION

with N being the e-folding number. For wφ < 1/3, H < 1 is realized and the Universe experiences − d2φ an accelerated expansion. Moreover, this acceleration is kept stable if 2 3H∂tφ , Vφ . This dt | | | | corresponds to

2 1 d φ 1 dH ηH 2 = H 1 . (2.8) ≡ −H∂tφ dt − 2H dN Other slow-roll parameters are deﬁned as the function of the potential:

M 2 V 2 V (φ) pl φ , η(φ) M 2 φφ . (2.9) ≡ 2 V ≡ pl V Here, and η are called the potential slow-roll parameters, and in the slow-roll approximation, the Hubble and potential slow-roll parameters are related as

H , ηH η . (2.10) ≈ ≈ − Hence, the slow-roll inﬂation occurs also for , η 1. When these slow-roll parameters reach unity as | |

H (φend) 1 , (φend) 1 . (2.11) ≡ ≈ inﬂation stops. The e-folding number as the function of given time during inﬂation is formulated as

a Z tend Z φend H Z φ V N(φ) ln end = Hdt = dφ dφ ≡ a t φ ∂tφ ≈ φend Vφ Z φ dφ Z φ dφ = . (2.12) φend Mpl√2H ≈ φend Mpl√2

Note that N(φ) & 60 should be satisﬁed in order to solve the horizon and ﬂatness problems.

2.2 Curvature and tensor perturbations Here, we summarize the analytical solutions of curvature and tensor perturbations in the de Sitter space-time, which is derived from the action (2.1). For convenience, we adapt the comoving gauge as

2 i i δφ = 0 , gij = a [(1 + 2 )δij + hij] , ∂ih j = h i = 0 . (2.13) R Comoving curvature perturbation and the tensor perturbation hij remain constant outside horizon if there exist no extra anisotropicR stresses 1 2. The quadratic actions of Eq. (2.1) for curvature and tensor perturbations are respectively given by Z h i (2) 2 3 2 ˙ 2 2 S = M dτd xa H (∂i ) , (2.14) R pl R − R 1On superhorizon scales, this is consistent with in Refs. [2, 3], ζ in Refs. [4, 5], in Refs. [1, 6], and ζ in Ref. [7]. In a numerical code CAMBR [8, 9], the primordialR scalar-mode power spectrum−R is given by this . − 2In Sec. 9, we will show that due to the ﬁnite anisotropic stresses of the primordial magnetic ﬁeld, the curvatureR perturbations (and gravitational waves) do not remain constant even on the superhorizon 2.2 Curvature and tensor perturbations 17

M 2 Z (2) pl 3 2 h˙ ˙ i S = dτd xa hijhij ∂lhij∂lhij , (2.15) h 8 − where ˙ d/dτ. Obeying the Fourier expansion as ≡ Z d3k (x, τ) = (k, τ)eik·x , (2.16) R (2π)3 R Z 3 d k X (λ) h (x, τ) = h(λ)(k, τ)e (kˆ)eik·x , (2.17) ij (2π)3 ij λ=±2 these are rewritten as Z Z 3 h i (2) 2 d k ˙ 2 2 2 S = dτ (Mpla) H (k, τ) k (k, τ) , (2.18) R (2π)3 |R | − |R | Z 2 Z 3 (2) X Mpla d k h i S = dτ h˙ (λ)(k, τ) 2 k2 h(λ)(k, τ) 2 . (2.19) h 2 (2π)3 | | − | | λ=±2

(λ) Here, eij is the transverse-traceless polarization tensor which has two circular states λ = 2 and (λ) ˆ (λ0) ˆ ± is normalized as eij (k)eij ( k) = 2δλ,λ0 . The convention and useful properties of this tensor are − (0) ∂tφ described in Appendix D. The variable transformation as v z , z a H (for scalar mode), aM ≡ R ≡ v(±2) √ pl h(±2) (for tensor mode), and the variation principle as δS/δv(λ) = 0 lead to the ﬁeld 2 equation≡ as

2 v¨(λ) + k2 v(λ) = 0 , (2.20) k − τ 2 k where we have used a relation in the de Sitter limit:z/z ¨ =a/a ¨ = 2/τ 2. To solve these equations, we perform the quantization of the ﬁeld v(λ) as

(λ) (λ) ∗ (λ)† vk = vk(τ)ˆak + vk(τ)ˆa−k . (2.21) When we set the normalization of the mode functions as

i ∗ ∗ vk, vk (vkv˙k v˙kvk) = 1 , (2.22) h i ≡ ~ − (λ)† (λ) the canonical commutation relation between the creation (ˆak ) and annihilation (ˆak ) operators can be written as

h (λ) (λ0)†i 3 0 aˆ , aˆ 0 = (2π) δ(k k )δλ,λ0 . (2.23) k k − A vacuum state is given by

aˆ(λ) 0 = 0 . (2.24) k | i As a vacuum, one often choose the so-called Bunch-Davies Vacuum denoting the Minkowski vacuum in the far past. In this condition, i.e., τ or kτ 1, the ﬁeld equation (2.20) is reduced to → −∞ | | (λ) 2 (λ) v¨k + k vk = 0 . (2.25) 18 2 FLUCTUATIONS IN INFLATION

This is equivalent to the equation for harmonic oscillators and hence easily solved as

e−ikτ vk(τ) = . (2.26) √2k Owing to two boundary conditions (2.22) and (2.26), one can gain the solution of the mode function in the ﬁeld equation (2.20) as

e−ikτ i vk(τ) = 1 . (2.27) √2k − kτ Using this, we can express the time evolution of the primordial curvature and tensor perturbations as

2 H h (0) ∗ (0)†i (k, τ) = τ vk(τ)âk + vk(τ)â−k , (2.28) R −∂tφ (±2) H h (±2) ∗ (±2)†i h (k, τ) = √2 τ vk(τ)âk + vk(τ)â−k . (2.29) − Mpl

Finally, we summarize these power spectra on superhorizon scales ( kτ 1) as | | * 2 + 2 ! Y 3 X (kn) (2π) PR(k1)δ kn , R ≡ n=1 n=1 (2.30) 2 2 2 2 H∗ H∗ H∗ 1 H∗ 1 PR(k) = 3 = 3 3 . 2k (∂tφ)∗ Mpl 4H∗k ≈ Mpl 4∗k and

* 2 + 2 ! Y (λn) 3 Ph(k1) X h (kn) (2π) δ kn δλ ,λ , ≡ 2 1 2 n=1 n=1 (2.31) 2 H∗ 2 Ph(k) = 3 . Mpl k

Here, we have evaluated all quantities at horizon crossing, namely τ∗ = 1/k. Note that since and h(±2) are constant on superhorizon scales, these power spectra become− the initial conditionsR for the CMB power spectra of the scalar and tensor modes.

2.3 Consistency relations in the slow-roll limit As a measure of the amplitude of the primordial gravitational wave, one often use the tensor-to- scalar ratio as 2P (k) r h . (2.32) ≡ PR(k)

Comparing Eq. (2.30) with Eq. (2.31), we ﬁnd a consistency relation

r = 16H∗ 16∗ . (2.33) ≈ 2.3 Consistency relations in the slow-roll limit 19

Using Hdt = dN, we ﬁnd that r is a measure of the evolution of the inﬂaton as

8 dφ 2 r = 2 . (2.34) Mpl dN

By performing an integral over N and an approximation as r const during inﬂation, we obtain the so called Lyth bound [10]: ∼

∆φ r 1/2 . (2.35) Mpl ∼ 0.01

Therefore, if we observe r > 0.01(< 0.01), we may conclude that large-field (small-field) inflation, namely, ∆φ > Mpl(< Mpl) occurred. As measures for the shapes of the spectra, we often use the spectral indices, which are defined by

d ln PR d ln Ph ns 4 , nt 3 . (2.36) − ≡ d ln k − ≡ d ln k

From Eqs. (2.30) and (2.31), the right-hand sides are expanded as

d ln P d ln H d ln dN R = 2 ∗ H∗ 3 , d ln k dN dN d ln k − − (2.37) d ln P d ln H dN h = 2 ∗ 3 . d ln k dN d ln k −

From the deﬁnition of the Hubble slow-roll parameters (2.7) and (2.8), we obtain

d ln H∗ d ln H∗ = H∗ , = 2(H∗ ηH∗) . (2.38) dN − dN −

By using k = a∗H∗ and d ln k = dN + d ln H∗, we have

−1 dN d ln H∗ = 1 + 1 + H∗ . (2.39) d ln k dN ≈

Consequently, we can summarize the consistency relations:

ns 1 = 2ηH∗ 4H∗ 2η∗ 6∗ , − − ≈ − (2.40) nt = 2H∗ 2∗ . − ≈ −

From Eq. (2.33), we also ﬁnd the consistency relation between r and nt as

r = 8nt . (2.41) −

As shown above, r, ns and nt depend on the slow-roll parameters and hence are observables which reﬂect the nature of inﬂation. 20 REFERENCES

References

[1] D. Baumann, TASI Lectures on Inﬂation, arXiv:0907.5424.

[2] WMAP Collaboration, E. Komatsu et al., Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation, Astrophys. J. Suppl. 192 (2011) 18, [arXiv:1001.4538].

[3] L. Motta and R. R. Caldwell, Non-Gaussian features of primordial magnetic ﬁelds in power-law inﬂation, Phys.Rev. D85 (2012) 103532, [arXiv:1203.1033].

[4] J. M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013, [astro-ph/0210603].

[5] D. H. Lyth and A. R. Liddle, The primordial density perturbation: cosmology, inﬂation and the origin of structure; rev. version. Cambridge Univ. Press, Cambridge, 2009.

[6] M. Giovannini and K. E. Kunze, Generalized CMB initial conditions with pre-equality magnetic ﬁelds, Phys.Rev. D77 (2008) 123001, [arXiv:0802.1053].

[7] J. R. Shaw and A. Lewis, Massive Neutrinos and Magnetic Fields in the Early Universe, Phys. Rev. D81 (2010) 043517, [arXiv:0911.2714].

[8] A. Lewis, A. Challinor, and A. Lasenby, Eﬃcient Computation of CMB anisotropies in closed FRW models, Astrophys. J. 538 (2000) 473–476, [astro-ph/9911177].

[9] A. Lewis, CMB anisotropies from primordial inhomogeneous magnetic ﬁelds, Phys. Rev. D70 (2004) 043011, [astro-ph/0406096].

[10] D. H. Lyth, What would we learn by detecting a gravitational wave signal in the cosmic microwave background anisotropy?, Phys. Rev. Lett. 78 (1997) 1861–1863, [hep-ph/9606387]. 21

3 Fluctuations in cosmic microwave background radiation

Cosmic microwave background (CMB) radiation is composed of photons which have decoupled from electrons in the epoch of the hydrogen and helium recombination at z = 1089 and it is observed as the perfectly black body radiation whose averaged temperature is 2.725K. Historically, in 1949, Alpher and Herman predicted its existence as relics of the big bang Universe and its first detection came in 1964. More precisely, however, the CMB involves the spatial fluctuations of (10−5)K (see Fig. 3.1). We had to wait the detection of the CMB anisotropy until the data of theO COBE experiment were released in the 1990s. Theoretically, the density contrast of the CMB is computed in the system where photons, neutrinos, baryons, dark matters and dark energy exist in the gravitational potential. Compared with the observational data, the values of several key parameters have been well-determined. The WMAP experiment established the facts that the Universe is close to spatially flat and the present structure grew from the nearly scale-invariant primordial fluctuations. These consequences are almost consistent with the prediction of the standard slow-roll inflation. Furthermore, we have a compelling evidence that the Universe is dominated by dark energy and dark matter, which implies that 96% of the total energy of the Universe remains unknown. Nowadays, some bare anomalies such as the preferred direction and the parity violation are furthermore being discussed [1], and we expect to extract more detailed information from the new precise measurements [2]. In addition to the intensity of the CMB, the polarizations also lead to better understandings. The curl-free component of the polarizations, E mode, reflects the recombination history, in particular, the reionization of the Universe. The curl component of the polarizations, B mode, is generated from the primordial vector and tensor perturbations. Hence, the detection of the B- mode polarization will provide clues as to inflation and the physics beyond the standard model of the particle physics. 22 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION

In this section, we describe the original formalism of CMB ﬂuctuations including intensity and polarization anisotropies from the scalar, vector, and tensor modes partially on the basis of our paper [3], some publications [4–7] and some academic dissertations [8–10], and summarize current outputs from the analysis of the CMB power spectra.

3.1 Einstein equations Here, we derive the zeroth and ﬁrst-order Einstein equations. Let us consider the ﬂat (K = 0) FLRW metric and small perturbations in the synchronous gauge (for open and closed cases, see [11–13]):

2 2 2 i j ds = a [ dτ + (δij + hij)dx dx ] . (3.1) − We have the inverse metric to ﬁrst order in perturbations as 1 1 g00 = , g0i = 0 , gij = (δij hij) . (3.2) −a2 a2 − The Einstein equation with the cosmological constant Λ can be written as

µ µ 1 µ µ µ G ν = R ν δ νR = 8πGT ν Λδ ν , (3.3) − 2 − where the left-hand side denotes the curvature of space-time and the right-hand one is the energy momentum tensor. The Ricci tensor Rµν and Ricci scalar, namely a contracted form of the Ricci tensor, R, are expressed with the Christoﬀel symbols as

α α α α β α β Rµν = R µαν = Γ µν,α Γ µα,ν + Γ βαΓ µν Γ βνΓ µα , (3.4) − − where ,α ∂α. The Christoffel symbols in a metric space without torsion are given by ≡ λ 1 λκ Γ µν = g (gµκ,ν + gνκ,µ gµν,κ) . (3.5) 2 − Up to first order, we can express as 0 Γ 00 = , i H0 Γ 00 = Γ i0 = 0 , 0 1 ˙ Γ ij = (δij + hij) + hij , H 2 (3.6) i i 1 ˙ i Γ j0 = δ j + h j , H 2 i 1 i i i Γ jk = (∂kh j + ∂jh k ∂ hjk) , 2 − therefore each component of the Ricci tensor is calculated as 2 0 a¨ 2 1 ï ˙ i a R 0 = 3 + h i + h i , a − H 2 H 2 i 1 i ˙ j j ˙ i a R 0 = ∂ h j ∂ h j , −2 − (3.7) 2 i a¨ 2 i 1ï ˙ i 1 ˙ k i a R j = + δ j + h j + h j + h kδ j a H 2 H 2H 1 i k 2 i i k k i ∂ ∂jh k + h j ∂ ∂kh j ∂ ∂jh k . − 2 ∇ − − 3.1 Einstein equations 23

Here, a/a˙ = aH is the Hubble parameter in terms of conformal time with H being the observableH ≡ Hubble parameter. Then the Ricci scalar is also given by µ 1 a¨ ¨i ˙ i 2 i j i R = R µ = 6 + h i + 3 h i h i + ∂i∂ h j . (3.8) a2 a H − ∇

Contracting the Einstein equation (3.3) allows one to eliminate the Ricci scalar and reduce the Einstein equation to µ µ 1 µ σ µ R ν = 8πG T ν δ νT σ + Λδ ν . (3.9) − 2

µ Hence, in vacuum, we have R ν = 0.

3.1.1 Homogeneous contribution At zeroth order, the 00 and ii components of Eq. (3.3) lead to the Friedmann constraint equation and the Raychaudhuri evolution equation, respectively. Substituting Eq. (3.7) into Eq. (3.9), these are obtained as

2 2 8πG 2 ¯0 a = a T 0 + Λ , H − 3 3 (3.10) a¨ 2 8πG 2 ¯i 2 2 = a T i + a Λ . a − H − 3 The physical meaning of these equations can be illustrated with the perfect ﬂuid form as follows. The energy momentum tensor of the perfect ﬂuid is given by

µ µ µ T ν = (ρ + p)u uν + pδ ν , (3.11) hence the above equations change to

8πG Λ 2 = a2 ρ¯ + , H 3 8πG (3.12) a¨ Λ 2 2 = 8πGa2 p¯ . a − H − − 8πG Note that we may identify the cosmological constant as a component of the perfect ﬂuid as

¯µ Λ T ν = diag(1, 1, 1, 1) . (3.13) Λ 8πG − − −

To use a diﬀerent phrase, an unperturbed perfect ﬂuid of density and pressure are given by ρΛ = Λ/(8πG), pΛ = ρΛ. If we use w = p/ρ, then wΛ = 1. For convenience,− we change the Friedmann equation− to

8πG 8πG X 8πG 1 = a2ρ¯ = ρ¯ = ρ¯ , (3.14) 3 2 3H2 3H2 i H i where in third equality, we decompose the total energy density in the Universe into individual speciesρ ¯i. Introducing a quantity which means the ratio between the energy density of each 24 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION

w ρ(a) a(t) a(τ) τi rad dom 1/3 a−4 t1/2 τ 0 mat dom 0 a−3 t2/3 τ 2 0 −2 H Ω1/2τ curv dom - a t e 0 k Λ dom 1 a0 eHt τ −1 −∞ − − −∞

Table 3.1: FLRW solutions dominated by radiation, matter, curvature, or a cosmological constant.

species and the critical density in the Universe at the present time, Ωi, and which is expressed as 2 ni Ωi 8πGρ¯i0/(3H ), and using the scaling relation asρ ¯i =ρ ¯i0/a , this equation is rewritten as ≡ 0 2 X Ωi H = . (3.15) ani H i 0

For radiations, matters and cosmological constant, we have ni = 4, 3, 0, respectively. In this notation, we can also include a curvature term as a component of ni = 2. In Table 3.1, we summarize the solutions of Eq. (3.15) if the cosmological ﬂuid consists of a single component.

3.1.2 Perturbed contribution

At ﬁrst order, 00 and ij components of Eq. (3.9) generate the evolution equations as

¨i ˙ i 2 0 i h i + h i = 8πGa δT 0 δT i , H − h¨i + 2 h˙ i + h˙ k δi (∂i∂ hk + 2hi ∂k∂ hi ∂ ∂ihk ) j j k j j k j j k k j (3.16) H H − ∇ − − 2 i 1 i µ = 16πGa δT j δ jδT µ , − 2 and 00 and i0 components of Eq. (3.3) generate the constraint equations as

˙ i j i 2 i 2 0 2 h i + ∂ ∂ih j h i = 16πGa δT 0 , H − ∇ − (3.17) j ˙ i i ˙ j 2 i ∂ h j ∂ h j = 16πGa δT 0 . − From here, let us express these equations with the variables in the helicity states. To do it, we decompose all kind of vectors and tensors, such as metric, velocities and energy momentum tensors, into each helicity part in accordance with the formulae:

Z 3 ! d k (0) X (λ) ω (x, τ) = ω(0)O + ω(λ)O eik·x , i (2π)3 i i λ=±1 (3.18) Z 3 ! d k 1 (0) (0) X (λ) (λ) X (λ) (λ) ik·x χij(x, τ) = χisoδij + χ O + χ O + χ O e , (2π)3 −3 ij ij ij λ=±1 λ=±2 3.1 Einstein equations 25 where we deﬁne the projection vectors and tensors as (0) ˆ ˆ O (k) ika , a ≡ O(±1)(kˆ) i(±1)(kˆ) , a ≡ − a (0) ˆ ˆ ˆ 1 O (k) kakb + δa,b , (3.19) ab ≡ − 3 (±1) ˆ ˆ (±1) ˆ ˆ (±1) ˆ O (k) ka (k) + kb (k) , ab ≡ b a O(±2)(kˆ) e(±2)(kˆ) . ab ≡ ab (±1) (±2) The polarization vector and tensor, i , eij , satisfy the divergenceless and transverse-traceless conditions as ˆ (±1) ˆ ˆ (±2) ˆ (±2) ˆ kii (k) = kieij (k) = eii (k) = 0 . (3.20) The prescription for the scalar-vector-tensor decomposition and explicit forms of the polarization vector and tensor are presented in Appendix D. Then, from Eqs. (3.16) and (3.17), we can rewrite the evolution equations as ¨ ˙ 2 0 iso hiso + hiso = 8πGa δT 0 + δT , H − t ¨(0) ˙ (0) 1 2 (0) 2 (0) h + 2 h + k (hiso h ) = 16πGa δTt , H 3 − (3.21) h¨(±1) + 2 h˙ (±1) = 16πGa2δT (±1) , H t h¨(±2) + 2 h˙ (±2) + k2h(±2) = 16πGa2δT (±2) , H t and the constraint equations as

˙ 1 2 (0) 2 0 hiso + k (hiso h ) = 8πGa δT 0 , H 3 − ˙ ˙ (0) 2 (0) (3.22) k hiso h = 24πGa δT , − v kh˙ (±1) = 16πGa2δT (±1) . − v Here, we have obeyed the convention as

Z 3 ! d k (0) X (λ) δT i (x, τ) = δT (0)O + δT (λ)O eik·x , 0 (2π)3 v i v i λ=±1 (3.23) Z 3 ! i d k 1 iso (0) (0) X (λ) (λ) X (λ) (λ) ik·x δT j(x, τ) = δT δij + δT O + δT O + δT O e , (2π)3 −3 t t ij t ij t ij λ=±1 λ=±2 Our perturbation quantities are related to the variables for the scalar mode in the synchronous gauge of Ref. [14], namely h and η, as (0) hiso = h, h = (h + 6η) . (3.24) − − Hence, we understand the correspondence to the gauge-invariant variables by Bardeen (ΦA, ΦH ) [15] and Kodama-Sasaki (Ψ, Φ) [16]:

1 ¨(0) ˙ (0) ΦA = Ψ = 2 h + h , −2k H (3.25) 1 (0) 1 ˙ (0) ΦH = Φ = h hiso h . 6 − − 2k2 H 26 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION

3.2 Boltzmann equations The distribution function of several species evolves in accordance with the Boltzmann equation as df ∂f ∂f ∂xi ∂f ∂pµ ∂f = + i + µ = , (3.26) dτ ∂τ ∂x ∂τ ∂p ∂τ ∂τ C where τ is the conformal time, pµ is the proper momentum of species, and a subscript C denotes the collision term. In this Boltzmann equation, there exist two contributions: the gravitational redshift and the effect of scattering, which correspond to the third term of the first equality and the term of the second equality, respectively. For convenience, we introduce the comoving momentum and energy as qi api, app2 + m2. Setting a unit vector parallel to the fluid momentum as q = qnˆ and expanding≡ the≡ distribution function up to first order:

f(x, q, nˆ, τ) = f (0)(q) 1 + f (1)(x, q, nˆ, τ) , (3.27) the above Boltzmann equation is rewritten as

df ∂f (1) ∂f (1) dxi = f (0) + dτ ∂τ ∂xi dτ (0) (1) (0) (1) i ∂f dq (0) ∂f dq (1) ∂f dq (0) ∂f dnˆ ∂f + + f + f + f i = . (3.28) ∂q dτ ∂q dτ ∂q dτ ∂nˆ dτ ∂τ C To estimate dq/dτ, dnˆi/dτ, we consider the geodesic equation as dP µ P 0 + Γµ P αP β = 0 , (3.29) dτ αβ where P µ is the canonical momentum as µ 1 ij 1 ij j 1 P = , qj δ h ,Pµ = , q δij + hij . (3.30) a2 − 2 − 2

µ 2 2 2 The contraction is given by P Pµ = p (/a) = m . From µ = 0 component, we obtain − − dq 1 ∂h = qnˆinˆj ij . (3.31) dτ −2 ∂τ Similarly, from the spatial components, we have

i mn ij dnˆ i ∂h ∂h 2q m in q i mn 2 =n ˆ nˆmnˆn nˆj nˆmnˆn∂ h + nˆmnˆn∂ h (h) . (3.32) dτ ∂τ − ∂τ − ≈ O Furthermore, since we have the zeroth order expression as dxi/dτ = (q/)ˆni, the Boltzmann equations up to ﬁrst order are expressed as

(1) (1) (0) ∂f q i ∂f 1 i j ∂hij ∂ ln f 1 ∂f + nˆ i nˆ nˆ = (0) . (3.33) ∂τ ∂x − 2 ∂τ ∂ ln q f ∂τ C The general expression for the energy momentum tensor is given by Z µ µ −1/2 dP1dP2dP3 P Pν T ν = gdeg ( g) f , (3.34) − (2π)3 P 0 3.3 Stokes parameters 27

where gdeg denotes the degree of freedom. Substituting the relations: 1 i 2 −1/2 −4 1 i dP1dP2dP3 = 1 + h i q dqdΩn , ( g) = a 1 h i , (3.35) 2 − − 2 and noting that Z 4π Z Z nînˆ dΩ = δi , nˆ dΩ = nˆ nˆ nˆ dΩ = 0 , (3.36) j n 3 j i n i j k n the homogeneous and linearized components of the energy momentum tensor are obtained as ZZ 0 1 (0) (1) 2 T 0 = ρ = f (1 + f )q dqdΩn − −(2π)3a4 Z ZZ 1 (0) 2 1 (0) (1) 2 ¯0 0 = f q dq f f q dqdΩn T 0 + δT 0 , −2π2a4 − (2π)3a4 ≡ ZZ i 1 i 1 ij (0) (1) 2 T 0 = q nˆ nˆjh f (1 + f )q dqdΩn −(2π)3a4 − 2 ZZ (3.37) 1 i (0) (1) 2 i = qnˆ f f q dqdΩn δT 0 , −(2π)3a4 ≡ ZZ 2 i 1 q i 1 ia 1 b (0) (1) 2 T j = nˆ nâh nˆj + nˆ hjb f (1 + f )q dqdΩn (2π)3a4 − 2 2 Z 2 ZZ 2 1 i q (0) 2 1 q i (0) (1) 2 ¯i i = δ j f q dq + nˆ nˆjf f q dqdΩn T j + δT j . 6π2a4 (2π)3a4 ≡ i i These components correspond to the density contrast δ, velocity v and anisotropic stress Π j of fluid as 1 0 i 0 i ¯0 δT 0 δT 0 δT i i ¯i i δT j i T 0 = ρ¯ , = δ , = = v , T j =pδ ¯ j , = Π j . (3.38) − ρ¯ − ρ¯ +p ¯ −ρ¯ +p ¯ − p¯

Therefore, equating the integral of Eq. (3.33) over q1, q2, q3 with Eqs. (3.37) and (3.38), we can i i see that the Boltzmann equation (3.33) becomes the differential equations with respect to δ, v , Π j for each species. These equations correspond to the Euler and continuity equations. Generally, as the species of the cosmological fluid which mainly generate the inhomogeneity of the cosmological structure, there exist baryon, photon, neutrino and cold dark matter (CDM), hence we can trace the evolution of their fluctuations due to solving these Boltzmann equations coupled with the Einstein equations (3.21) and (3.22). Between baryons and photons, Thomson scattering is effective, so that their Boltzmann equations have the collision term. On the other hand, for neutrinos and CDMs, since there are no short-length interactions, the right-hand side of the Boltzmann equations vanishes. All these species couple with the metric via gravity. This relation is illustrated in Fig. 3.2.

3.3 Stokes parameters Here, we introduce the Stokes parameters to characterize the polarization states of the radiation field. For simplicity, at first, we consider a plane electromagnetic wave propagating along the z axis. The Fourier decomposition of the radiation field is expressed as Z ∞ iφx iφy i(kz−ωt) E(z, t) = dk xÊxe + yÊye e , (3.39) −∞ 1The anisotropic stress of the magnetic field is often normalized by photon’s energy density as Eq. (9.2). 28 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION

photon

Thomson

electron neutrino

gravity metric

Coulomb

proton dark matter

Figure 3.2: Interaction between several components in the Universe.

where Ex,Ey and φx, φy are the real quantities describing the amplitudes and phases in the xˆ yˆ plane, respectively, and ω = kc denotes the frequency of the wave. − The Stokes parameters are given by

2 2 I Ex + Ey , ≡ | |2 | |2 Q Ex Ey , ≡ | | − |∗ | (3.40) U 2Re[ExEy] , ≡ − ∗ V 2Im[E Ey] , ≡ − x where these are all real quantities. For the monochromatic wave, I2 = Q2 + U 2 + V 2 is satisﬁed. I measures the intensity of radiation and is always positive. The other parameters represent the polarization states and can take ether positive or negative values. Q and U quantify the magnitude of the linear polarization, and V parametrizes the circular polarization. While I and Q are parity- even quantities, U and V are parity-odd ones. In order to see the transformation rule of Q and U under rotation of axes, let us introduce the new coordinate (x0, y0), which is related to the original coordinate (x, y) by the rotation around the z axis as

x0 cos ψ sin ψ x = . (3.41) y0 sin ψ cos ψ y − 3.4 Boltzmann equations for photons 29

Then, the radiation ﬁeld is converted into

Z ∞ ˆ0 0 iφx ˆ0 0 iφy i(kz−ωt) E(z, t) = dk x Exe + y Eye e , −∞ (3.42) 0 0 ±iψ E iE = e (Ex iEy); x ± y ± hence we have

Q0 iU 0 = e∓2iψ(Q iU) . (3.43) ± ± This implies that the linear combination, Q iU, are the spin- 2 quantities. Therefore, the anisotropy of the linear polarization should be± expanded with the± spin-2 spherical harmonics.

3.4 Boltzmann equations for photons Here, for quantifying the CMB anisotropy, let us focus on the linearized Boltzmann equation for photons. The distribution function of photons is given by

p −1 f = exp 1 , (3.44) T [1 + Θ(x, nˆ, t)] − hence we have [14]

1 q ∂f (0) ∂ ln f (0) f (0) = , f (1)(x, q, nˆ, τ) = Θ = Θ , (3.45) ep/T 1 −f (0) ∂q − ∂ ln q − where Θ ∆T/T . Substituting Eq. (3.45) into Eq. (3.33), the Boltzmann equation in terms of Θ is expressed≡ as

(0) ∂f ∂Θ i ∂Θ 1 i j ∂hij ∂f q +n ˆ i + nˆ nˆ = . (3.46) − ∂q ∂τ ∂x 2 ∂τ ∂τ C The first two terms in the bracket denote the free-streaming of the photon, whereas the remaining third term in the bracket account for the gravitational redshift. As for the CMB polarization, there exist no gravitational effects because Q, U and V themselves are first-order quantities and the third term in the left-hand side of Eq. (3.33) does not appear. Hence, we summarize the Boltzmann equations of Θ, Q, U and V :

∂Θ ∂Θ 1 ∂h +n î + nînˆj ij = Θ˙ , ∂τ ∂xi 2 ∂τ T ∂Q i ∂Q ˙ +n ˆ i = QT , ∂τ ∂x (3.47) ∂U ∂U +n î = U˙ , ∂τ ∂xi T ∂V ∂V +n î = V˙ , ∂τ ∂xi T where ΘT ,QT ,UT and VT denote the collision terms of Thomson scattering and ˙ ∂/∂τ is the derivative with respect to the conformal time. Next we consider the contribution of≡ these terms. 30 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION

Figure 3.3: Geometry of Thomson scattering. Blue (Red) solid and two dashed arrows denote the incident (scattered) wave number vector and its orthogonal unit vectors, respectively. We set that xˆ0 = xˆ

3.5 Thomson scattering The process of scattering off a photon by a charged particle without the energy exchange of photons is called the Rayleigh scattering. In particular, when the charged particle is an electron, the process is known as Thomson scattering. During the epoch of recombination, electrons scattered off photons by Thomson scattering. Here, we consider an incoming plane wave of the radiation with a wave number vector kI parallel to the z axis and an outgoing radiation scattered off by an electron with a wave number vector kS. We take the plane spanned by kI and kS as the scattering plane as shown in Fig. 3.3. The differential cross section of Thomson scattering is given by

dσ 3σT ˆ ˆ 2 = kI kS , (3.48) dΩ 8π | · | where dΩ = d(cos θ)dφ and σT is the cross section of Thomson scattering. This equation quantiﬁes the change of the intensity by the scattering. For simplicity, we consider the case of the x0 axis parallel to the x axis. Here, we suppose the incident radiation with the polarization states I0 = 0 0 0 0 0 0 0 0 0 0 (Iy0 ,Ix0 ,U ,V ), where I = Ix0 + Iy0 and Q = Iy0 Ix0 . When there is no dependence on the azimuthal angle, namely φ0 = 0, from the notation− of Stokes parameters (3.40) and (3.48), we obtain

3σT 2 0 3σT 0 3σT 0 I = cos θI 0 ,I = I 0 ,U = cos θU , (3.49) y 16π y x 16π x 16π where we have normalized these equations so that the number of photons is conserved during a single scattering. For a general case with a non-vanishing azimuthal angle φ, Q0 and U 0 are replaced 3.5 Thomson scattering 31 as Q0 iU 0 e∓2iφ(Q0 iU 0) . (3.50) ± ⇒ ± Then, the changes of the Stokes parameters between the incident radiation from nˆ0 = zˆ (θ0 = 0, φ0 = 0), and the scattered radiation with nˆ = (θ, φ) are given by [17] " # 1 3 X 3 s ∆Θ(nˆ0 = zˆ, nˆ) = (1 + cos2 θ)Θ0 sin2 θe−siφ Q0 + iU 0 , 4π 4 8 2 − s=±2 " # (3.51) 1 3 X 3 s 2 s ∆(Q iU)(nˆ0 = zˆ, nˆ) = sin2 θΘ0 + 1 cos θ e−siφ Q0 + iU 0 , 4π 4 8 2 2 ± − s=±2 ± where we use Θ = ∆I/I/4. Using the explicit formulae of the spin-0 and spin-2 spherical harmonics described in Table A.2, we can extend this expression to the form corresponding to an arbitrary direction of nˆ0: X 1 ∆Θ(nˆ0, nˆ) = Y (nˆ)Y ∗ (nˆ0) + Y (nˆ)Y ∗ (nˆ0) Θ0 10 2m 2m 0m 0m m r # X 3 2 s Y (nˆ) Y ∗ (nˆ0) Q0 + iU 0 , 20 3 2m s 2m 2 (3.52) − s=±2 " r # X 3 2 X s (∆Q i∆U)(nˆ0, nˆ) = Y (nˆ) Y ∗ (nˆ0)Θ0 + Y ∗ (nˆ0) Q0 + iU 0 . 10 ±2 2m 3 2m s 2m 2 ± m − s=±2 We will use Eq. (3.52) in the frame satisfying k zˆ, where k is the wave number vector. Integrating these equations over all directions nˆ0, we expressk the scattered fields as 2 Z ˙ 0 0 ΘT (nˆ) = κ˙ Θ(nˆ) dΩ ∆Θ(nˆ , nˆ) vb nˆ , − − − · (3.53) Z Q˙ iU˙ (nˆ) = κ˙ (Q iU)(nˆ) dΩ0(∆Q i∆U)(nˆ0, nˆ) , ± T − ± − ± where we define the differential optical depth asκ ˙ aσT nexe with nexe being the density of ionized electrons, and its total value at time τ is given by≡

Z τ0 κ(τ) κ˙ (τ 0)dτ 0 , (3.54) ≡ τ with τ0 being the present conformal time. From here, we discuss the polarization property in more detail. One of the key points in Eq. (3.52) is that the temperature anisotropy generates the polarization of the CMB photons. Then, what mode of the temperature anisotropy is related to the generation of the polarization? For simplicity, we suppose that the incident radiation ﬁeld is unpolarized, Q0 = U 0 = V 0 = 0 and consider the case for nˆ = zˆ. Integrating Eq. (3.52) over all incident radiation, we gain r Z 3σT 2π 0 0 0 0 0 0 Q iU(zˆ) = dΩ Y22(θ , φ )Θ (θ , φ ) . (3.55) ± 4π 15 2The Stokes parameter, V , which means the circular polarization of photon, can be ignored because it cannot be generated through Thomson scattering if this is initially absent. 32 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION

When the incident temperature (intensity) anisotropy is expanded with the spherical harmonics 0 0 0 P 0 0 0 as Θ (θ , φ ) = `m a`mY`m(θ , φ ), Eq. (3.55) is replaced with 3σ r2π Q iU(zˆ) = T a0 . (3.56) ± 4π 15 22 Thus, if there exists no quadrupole moment (` = 2) in the unpolarized radiation ﬁeld, the total scattered radiation along the z direction would be never polarized. Long before recombination, in the thermal equilibrium, the polarization states of photons are equally populated and the incident radiation should not have any polarization. Therefore, there are only the unpolarized radiations before recombination. Allowing the polarization at the last scatters just before the photons begin to stream freely, the polarized emission can lead to the multipole anisotropy and one has polarized quadrupole and octupole and so on. These eﬀects are automatically involved in the Boltzmann equation.

3.6 Transfer functions Here, we derive the CMB anisotropy sourced from scalar-, vector- and tensor-mode perturbations. We obey a Fourier transformation as Z d3k X(nˆ, τ) = ∆ (τ, k, nˆ) , (3.57) (2π)3 X

ik·nˆ where X = Θ,Q iU and ∆X is called the transfer function. Note that we include the factor e ± in ∆X . At ﬁrst, for convenience, we derive the transfer functions of photons when k zˆ. For k zˆ, the scalar-vector-tensor decomposition of the gravitational redshift and Doppler termk in the Boltzmannk equation of photons (3.47) and (3.53) are given by 1 i j ˙ 1 1 ˙ ˙ (0) 2 1 nˆ nˆ hij(k zˆ, τ) = hiso(k zˆ, τ) + h (k zˆ, τ) cos θk,n + 2 k 2 −3 k k − 3 1 X λiφk,n ˙ (λ) + sin θk,n cos θk,ne h (k zˆ, τ) √ k λ=±1 2 1 X 2 λiφk,n ˙ (λ) + sin θk,ne h (k zˆ, τ) √ k λ=±2 2 2 (0) 1 ˙ (S) ˙ (S) 2 1 ξ (k zˆ) h (k, τ) + h (k, τ) cos θk,n ≡ k 3 iso − 3

X λiφk,n (λ) ˙ (V ) + sin θk,n cos θk,ne ξ (k zˆ)h (k, τ) (3.58) k λ=±1

X 2 λiφk,n (λ) ˙ (T ) + sin θk,ne ξ (k zˆ)h (k, τ) , k λ=±2 i (0) X λiφk,n (λ) vb(k zˆ, τ) nˆ = i cos θk,nv (k zˆ, τ) + − sin θk,ne v (k zˆ, τ) k · b k √ b k λ=±1 2 (0) (S) i cos θk,nξ (k zˆ)v (k, τ) ≡ k b X λiφk,n (λ) (V ) + i sin θk,ne ξ (k zˆ)v (k, τ) , − k b λ=±1 3.6 Transfer functions 33 where ξ(0), ξ(±1) and ξ(±2) are the initial stochastic variables of scalar, vector and tensor modes, and we use the calculation results from Appendix D as  1  (±1) 1 j (zˆ) =  i  , √2 ±0  0 0 1  (±1) 1 Oij (zˆ) =  0 0 i  , (3.59) √2 1 i ±0 ±  1 i 0  (±2) (±2) 1 ± Oij (zˆ) = eij (zˆ) =  i 1 0  . √2 ±0− 0 0 According to Refs. [14, 18, 19], if we equate ξ(0) with the comoving curvature perturbation on superhorizon scales , the initial conditions of the metric perturbations and the baryon velocity are 3 R

(S) 1 2 h (k, τini) = (kτini) , iso −4 (S) 5 2 h (k, τini) = 3 (kτini) , (3.61) − − 2(15 + 4Rν) 1 v(S)(k, τ ) = (kτ )3 . b ini 36 ini In the tensor mode, equating ξ(±2) with the primordial gravitational wave on superhorizon scales h(±2), it is satisfied that (T ) 1 5 2 h (k, τini) = 1 (kτini) . (3.62) 2√2 − 2(15 + 4Rν) Note that the Doppler effect does not affect only the tensor-mode perturbation. We introduce the transfer function in the Fourier space as [4, 20] (S) (0) ˜ (S) ∆ (τ, k zˆ, nˆ) = ξ (k zˆ)∆ (τ, k, µk,n) , I k k I (S) (S) (0) ˜ (S) (∆ i∆ )(τ, k zˆ, nˆ) = ξ (k zˆ)∆ (τ, k, µk,n) , Q ± U k k P (V ) q (V ) X 2 λiφk,n (λ) ˜ ∆ (τ, k zˆ, nˆ) = i 1 µ e ξ (k zˆ)∆ (τ, k, µk,n) , I k − − k,n k I λ=±1 (V ) (V ) q (V ) X 2 λiφk,n (λ) ˜ (∆ i∆ )(τ, k zˆ, nˆ) = λ(1 λµk,n) 1 µ e ξ (k zˆ)∆ (τ, k, µk,n) ,(3.63) Q ± U k ∓ ∓ − k,n k P λ=±1 (T ) (T ) 2 X λiφk,n (λ) ˜ ∆ (τ, k zˆ, nˆ) = (1 µ ) e ξ (k zˆ)∆ (τ, k, µk,n) , I k − k,n k I λ=±2 2 (T ) (T ) λ (T ) X λiφk,n (λ) ˜ (∆ i∆ )(τ, k zˆ, nˆ) = 1 µk,n e ξ (k zˆ)∆ (τ, k, µk,n) , Q ± U k ∓ 2 k P λ=±2

3Then, the parameters in Ref. [14] are given by 5 + 4R η(k, τ ) = 1 ν (kτ )2 , ini 12(15 + 4R ) ini −R − ν (3.60) 1 h(k, τ ) = (kτ )2 . ini −2R ini 34 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION

ˆ where µk,n k nˆ. Then, from Eqs. (3.52) and (3.53), we can write the collision term of Thomson scattering for≡ the· scalar mode:

Z Z d3k Z X 1 dΩ0∆Θ(S)(nˆ0, nˆ) = dΩ0 ξ(0)(k zˆ) (2π)3 10 m k hn o ∗ 0 ∗ 0 ˜ (S) 0 Y2m(nˆ)Y (nˆ ) + 10Y0m(nˆ)Y (nˆ ) ∆ (τ, k, µ ) × 2m 0m I r ( ) # 3 X (S) Y (nˆ) Y ∗ (nˆ0) ∆˜ (τ, k, µ0) , 2 2m s 2m P − s=±2 (3.64) Z Z d3k Z X √6 dΩ0(∆Q(S) i∆U (S))(nˆ0, nˆ) = dΩ0 Y (nˆ)ξ(0)(k zˆ) (2π)3 10 ±2 2m ± − m k h Y ∗ (nˆ0)∆˜ (S)(τ, k, µ0) × 2m I r ( ) # 3 X (S) Y ∗ (nˆ0) ∆˜ (τ, k, µ0) , 2 s 2m P − s=±2

for the vector mode:

r Z Z d3k Z X X 1 2π dΩ0∆Θ(V )(nˆ0, nˆ) = dΩ0 λξ(λ)(k zˆ) (2π)3 5 3 k m λ=±1 h n o ∗ 0 ∗ 0 i Y2m(nˆ)Y (nˆ ) + 10Y0m(nˆ)Y (nˆ ) × 2m 0m ˆ0 ˜ (V ) 0 Y1λ(n )∆I (τ, k, µ ) × ( ) # 3 X (V ) Y (nˆ) Y ∗ (nˆ0) Y (nˆ0) ∆˜ (τ, k, µ0) , √ 2m s 2m s 2λ P − 5 s=±2 (3.65) Z Z 3 Z 0 (V ) (V ) 0 d k 0 X 2√π X (λ) dΩ (∆Q i∆U )(nˆ , nˆ) = dΩ ±2Y2m(nˆ) λξ (k zˆ) ± − (2π)3 5 k m λ=±1 h ∗ 0 0 ˜ (V ) 0 iY (nˆ )Y1λ(nˆ )∆ (τ, k, µ ) × 2m I ( ) # 3 X (V ) Y ∗ (nˆ0) Y (nˆ0) ∆˜ (τ, k, µ0) , √ s 2m s 2λ P − 5 s=±2 3.6 Transfer functions 35 for the tensor mode: r Z Z d3k Z X 2 2π X dΩ0∆Θ(T )(nˆ0, nˆ) = dΩ0 ξ(λ)(k zˆ) (2π)3 5 15 k m λ=±2 hn o ∗ 0 ∗ 0 Y2m(nˆ)Y (nˆ ) + 10Y0m(nˆ)Y (nˆ ) × 2m 0m ˆ0 ˜ (T ) 0 Y2λ(n )∆I (τ, k, µ ) × ( ) # X ∗ ˆ0 ˆ0 ˜ (T ) 0 3Y2m(nˆ) sY2m(n )sY2λ(n ) ∆P (τ, k, µ ) , − s=±2 (3.66) Z Z 3 Z r 0 (T ) (T ) 0 d k 0 X 4 π X (λ) dΩ (∆Q i∆U )(nˆ , nˆ) = dΩ ±2Y2m(nˆ) ξ (k zˆ) ± − (2π)3 5 5 k m λ=±2 h ∗ 0 0 ˜ (T ) 0 Y (nˆ )Y2λ(nˆ )∆ (τ, k, µ ) × 2m I ( ) # X ∗ ˆ0 ˆ0 ˜ (T ) 0 3 sY2m(n )sY2λ(n ) ∆P (τ, k, µ ) . − s=±2 Here, we use r r p 2 λiφ 8π p 2 λiφ 16π 1 µ e = λ Y1λ , 1 µ (1 λµ)e = ±2Y2λ (for λ = 1) , − − 3 − ∓ ∓ 5 ± (3.67) r 2 r 2 λiφ 2π λ λiφ π (1 µ )e = 4 Y2λ , 1 µ e = 8 ±2Y2λ (for λ = 2) . − 15 ∓ 2 5 ± Using the multipole expansion as

˜ (S/V/T ) 0 X lp 0 ˜ (S/V/T ) ∆ (τ, k, µ ) = ( i) 4π(2l + 1)Yl0(nˆ )∆ (τ, k) (3.68) I/P − I/P,l l and the Ω0-integrals for λ = 0: Z 0 ∗ dΩ Y2mYl0 = δl,2δm,0 , Z 0 ∗ dΩ Y0mYl0 = δl,0δm,0 , (3.69) Z r 0 ∗ 5 1 dΩ ±2Y Yl0 = δm,0 δl,0 δl,2 , 2m 6 − √5 for λ = 1: ± Z r r ! 0 ∗ 3 3 dΩ Y Y1λYl0 = δm,λ δl,1 δl,3 , 2m 20π − 7 Z 0 ∗ dΩ Y0mY2λYl0 = 0 , (3.70) Z ! 0 ∗ 1 √3 √5 √7 2 dΩ ±2Y ±2Y2λYl0 = δm,λ δl,0 λδl,1 δl,2 λδl,3 δl,4 , 2m √4π ∓ 3 − 7 ± 7 − 21 36 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION and for λ = 2: ± Z ! 0 ∗ 1 2√5 1 dΩ Y Y2λYl0 = δm,λ δl,0 δl,2 + δl,4 , 2m √4π − 7 7 Z 0 ∗ dΩ Y0mY2λYl0 = 0 , (3.71) Z ! 0 ∗ 1 √3 2√5 √7 1 dΩ ±2Y ±2Y2λYl0 = δm,λ δl,0 λδl,1 + δl,2 λδl,3 + δl,4 , 2m √4π ∓ 3 7 ∓ 28 42 we can obtain the anisotropies generated via Thomson scattering for the scalar mode:

Z Z d3k rπ 0 (S) 0 (0) ˜ (S) (S) dΩ ∆Θ (nˆ , nˆ) = ξ (k zˆ) ∆ Y20(nˆ)ψ (k, τ) , (2π)3 k I,0 − 5 Z Z 3 "r # 0 (S) (S) 0 d k (0) 6π (S) (3.72) dΩ (∆Q i∆U )(nˆ , nˆ) = ξ (k zˆ) ±2Y20(nˆ)ψ (k, τ) , ± (2π)3 k 5 ψ(S)(k, τ) ∆˜ (S) + ∆˜ (S) + ∆˜ (S) , ≡ I,2 P,0 P,2 for the vector mode:

Z Z 3 " r # 0 (V ) 0 d k X 8π (λ) (V ) dΩ ∆Θ (nˆ , nˆ) = λ Y2λ(nˆ)ξ (k zˆ) ψ (k, τ) , (2π)3 15 k λ=±1 Z Z 3 " r # 0 (V ) (V ) 0 d k X 16π (λ) (V ) (3.73) dΩ (∆Q i∆U )(nˆ , nˆ) = λ ±2Y2λ(nˆ)ξ (k zˆ) ψ (k, τ), ± − (2π)3 5 k λ=±1 3 3 3 3 6 ψ(V )(k, τ) ∆˜ (V ) + ∆˜ (V ) ∆˜ (V ) ∆˜ (V ) + ∆˜ (V ) , ≡ 10 I,1 10 I,3 − 5 P,0 − 7 P,2 35 P,4 and for the tensor mode:

Z Z 3 " r # 0 (T ) 0 d k X 32π (λ) (T ) dΩ ∆Θ (nˆ , nˆ) = Y2λ(nˆ)ξ (k zˆ) ψ (k, τ) , (2π)3 15 k λ=±2 Z Z 3 " r # 0 (T ) (T ) 0 d k X 64π (λ) (T ) (3.74) dΩ (∆Q i∆U )(nˆ , nˆ) = ±2Y2λ(nˆ)ξ (k zˆ) ψ (k, τ) , ± − (2π)3 5 k λ=±2 1 1 3 3 6 3 ψ(T )(k, τ) ∆˜ (T ) + ∆˜ (T ) + ∆˜ (T ) ∆˜ (T ) + ∆˜ (T ) ∆˜ (T ) . ≡ 10 I,0 7 I,2 70 I,4 − 5 P,0 7 P,2 − 70 P,4 Thus, from the Boltzmann equation (3.47), the gravitational redshift and the Doppler terms (3.58), and the collision term of Thomson scattering (3.53) and (3.72) - (3.74), we derive the Boltzmann equation for the scalar mode: ˜˙ (S) ˜ (S) 1 ˙ (S) 2 1 ˙ (S) ∆ + ikµk,n∆ = h + µ h I I − 3 iso k,n − 3 ˜ (S) ˜ (S) 3 2 1 (S) (S) κ˙ ∆ ∆ + µ ψ iµk,nv , (3.75) − I − I,0 4 k,n − 3 − b ˜˙ (S) ˜ (S) ˜ (S) 3 2 (S) ∆ + ikµk,n∆ = κ˙ ∆ (1 µ )ψ , P P − P − 4 − k,n 3.6 Transfer functions 37 for the vector mode: h i ˜˙ (V ) ˜ (V ) ˙ (V ) ˜ (V ) (V ) (V ) ∆ + ikµk,n∆ = iµk,nh κ˙ ∆ + iµk,nψ v , I I − − I − b h i (3.76) ˜˙ (V ) ˜ (V ) ˜ (V ) (V ) ∆ + ikµk,n∆ = κ˙ ∆ + ψ , P P − P and for the tensor mode: h i ˜˙ (T ) ˜ (T ) ˙ (T ) ˜ (T ) (T ) ∆ + ikµk,n∆ = h κ˙ ∆ ψ , I I − − I − h i (3.77) ˜˙ (T ) ˜ (T ) ˜ (T ) (T ) ∆ + ikµk,n∆ = κ˙ ∆ + ψ . P P − P The multipole expansion of these equations gives

k h i ∆˜˙ (S/V/T ) + (l + 1)∆˜ (S/V/T ) l∆˜ (S/V/T ) = V (S/V/T ) , (3.78) I/P,l 2l + 1 I/P,l − I/P,l−1 I/P,l where (S) 1 ˙ (S) 2 ˙ (S) ˜ (S) 1 (S) 1 (S) V = h δl,0 h δl,2 κ˙ ∆ (1 δl,0) ψ δl,2 + v δl,1 , I,l − 3 iso − 15 − I,l − − 10 3 b (S) ˜ (S) 1 (S) V = κ˙ ∆ ψ (δl,2 + 5δl,0) , P,l − P,l − 10 (V ) 1 ˙ (V ) ˜ (V ) 1 (V ) (V ) V = h δl,1 κ˙ ∆ ψ δl,1 v δl,0 , I,l 3 − I,l − 3 − b (3.79) h i (V ) ˜ (V ) (V ) V = κ˙ ∆ + ψ δl,0 , P,l − P,l h i (T ) ˙ (T ) ˜ (T ) (T ) V = h δl,0 κ˙ ∆ ψ δl,0 , I,l − − I,l − h i (T ) ˜ (T ) (T ) V = κ˙ ∆ + ψ δl,0 . P,l − P,l Following the line of sight integration [21], we can give the explicit solution of this Boltzmann equation as follows. At ﬁrst, let us present the derivation for tensor mode. Using de−κ/dτ =κe ˙ −κ and multiplying eikµk,nτ−κ in both sides of Eq. (3.77), we have

d (T ) ˜ ikµk,nτ−κ ikµk,n ˙ (T ) −κ (T ) ∆I e = e [ h e + gψ ] , dτ − (3.80) d (T ) ∆˜ eikµk,nτ−κ = eikµk,n gψ(T ) . dτ P − where g(τ) κe˙ −κ is the visibility function which describes the probability that a given CMB photon last scattered≡ at a given time. Through the integral over conformal time and some treatments, we obtain each from of the transfer function at τ = τ0 as

Z τ0 (T ) h i ∆˜ = dτe−iµk,nx h˙ (T )e−κ + gψ(T ) , I − 0 (3.81) Z τ0 (T ) ˜ −iµk,nx (T ) ∆P = dτe gψ , − 0 38 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION

Here we use κ(τ0) = 0, κ(τ = 0) and x k(τ0 τ). In the same manner, we also obtain the scalar-mode function: → ∞ ≡ −

Z τ0 (S) 1 (S) 1 ˜ −iµk,nx ˙ 2 ˙ (S) −κ ∆I = dτe hiso + µk,n h e 0 − 3 − 3 ˜ (S) 3 2 1 (S) (S) +g ∆ µ ψ + iµk,nv , I,0 − 4 k,n − 3 b " ... ! Z τ0 1 h (S) 1 −iµk,nx ˙ (S) ˙ (S) −κ = dτe hiso + 2 + h e 0 −3 k 3 ! (3.82) 2h¨(S) 3 ψ¨(S) 1 v˙ (S) +g + ∆˜ (S) + + ψ(S) b k2 I,0 4 k2 4 − k ! # h˙ (S) 3 ψ˙ (S) v(S) 3 gψ¨ (S) +g ˙ + b + , k2 2 k2 − k 4 k2 Z τ0 (S) 3 ˜ −iµk,nx 2 (S) ∆P = dτe (1 µk,n) gψ , 0 − 4 and the vector-mode one:

Z τ0 (V ) h (V )i ˜ −iµk,nx ˙ (V ) −κ (V ) ∆I = dτe iµk,n h e + gψ + gvb 0 − " ! # Z τ0 h¨(V ) h˙ + ψ˙ (V ) ψ(V ) −iµk,nx −κ (V ) = dτe e + g + vb +g ˙ , (3.83) 0 k k k Z τ0 (V ) ˜ −iµk,nx (V ) ∆P = dτe gψ . − 0 where in the second equality of each equation, we neglect the topological terms. Consequently, we can summarize the transfer functions when k zˆ for the scalar mode: k

Z τ0 (S) (0) −iµk,nx (S) ∆I (τ0, k zˆ, nˆ) = ξ (k zˆ) dτe SI (k, τ) , k k 0 r 4 6π Z τ0 (S) (S) (0) −iµk,nx (S) (∆Q i∆U )(τ0, k zˆ, nˆ) = ±2Y20(nˆ)ξ (k zˆ) dτe SP (k, τ) , ± k 3 5 k 0 ... ! 1 h (S) 1 S(S)(k, τ) h˙ (S) + + h˙ (S) e−κ I ≡ −3 iso k2 3 ! (3.84) 2h¨(S) 3 ψ¨(S) 1 v˙ (S) + g + ∆˜ (S) + + ψ(S) b k2 I,0 4 k2 4 − k ! h˙ (S) 3 ψ˙ (S) v(S) 3 gψ¨ (S) +g ˙ + b + , k2 2 k2 − k 4 k2 3 S(S)(k, τ) gψ(S)(k, τ) , P ≡ 4 3.7 All-sky formulae for the CMB scalar-, vector- and tensor-mode anisotropies 39 for the vector mode: " r # 8π Z τ0 (V ) X (λ) −iµk,nx (V ) ∆ (τ0, k zˆ, nˆ) = iλ Y1λ(nˆ)ξ (k zˆ) dτe S (k, τ) , I k 3 k I λ=±1 0 " r # 16π Z τ0 (V ) (V ) X (λ) −iµk,nx (V ) (∆ i∆ )(τ0, k zˆ, nˆ) = λ ±2Y2λ(nˆ)ξ (k zˆ) dτe S (k, τ) , Q ± U k 5 k P λ=±1 0 (3.85) ! h¨(V ) h˙ + ψ˙ (V ) ψ(V ) S(V )(k, τ) e−κ + g + v(V ) +g ˙ , I ≡ k k b k S(V )(k, τ) gψ(V ) , P ≡ − and for the tensor mode: " r # 32π Z τ0 (T ) X (λ) −iµk,nx (T ) ∆ (τ0, k zˆ, nˆ) = Y2λ(nˆ)ξ (k zˆ) dτe S (k, τ) , I k 15 k I λ=±2 0 " r # 64π Z τ0 (T ) (T ) X (λ) −iµk,nx (T ) (∆ i∆ )(τ0, k zˆ, nˆ) = ±2Y2λ(nˆ)ξ (k zˆ) dτe S (k, τ) , (3.86) Q ± U k 5 k P λ=±2 0 S(T )(k, τ) h˙ (T )e−κ + gψ(T ) , I ≡ − S(T )(k, τ) gψ(T ) . P ≡ − (S/V/T ) Here, we have introduced the source function, SI/P (k, τ).

3.7 All-sky formulae for the CMB scalar-, vector- and tensor-mode anisotropies

In this subsection, let us formulate the all-mode CMB coeﬃcients a`m in the all-sky analysis on the basis of the derivation in Refs. [3, 7]. Since the CMB anisotropy is described in the spherical coordinate system, its intensity (I) and two polarization (Q and U) ﬁelds should be expanded by the spin-0 and spin-2 spherical harmonics, respectively, as

(Z) X (Z) Θ (nˆ) = aI,`mY`m(nˆ) , `,m (3.87) (Z) (Z) X (Z) (Q iU )(nˆ) = a ±2Y`m(nˆ) . ± ±2,`m `,m

Here, the index Z denotes the mode of perturbations: Z = S (scalar), = V (vector) or = T (tensor). The main difficulty when computing the spectrum of polarization arises from the variance under rotations in the plane perpendicular to nˆ. While Q and U are easily calculated in a coordinate system where k zˆ, the superposition of the different modes is complicated by the behavior of Q k and U under rotations. However, using the spin raising and lowering operators ∂0 , ∂0 defined in Appendix A, we can obtain spin-0 quantities. This leads to the rotational invariant fields like the intensity one and there are no ambiguities connected with the rotation of coordinate system arise. 40 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION

z n : line-of sight direction (photon’s traveling direction)

Ωk,n Ωn

x Ωk k : wave number vector y

x y

Figure 3.4: Geometry for the line-of-sight direction.

Acting these operators on Q iU in Eq. (3.87), we have ±

1/2 X (` + 2)! (Z) ∂0 2(Q(Z) + iU (Z))(nˆ) = a Y (nˆ) , (` 2)! 2,`m `m `m − (3.88) 1/2 0 2 (Z) (Z) X (` + 2)! (Z) ∂ (Q iU )(nˆ) = a Y`m(nˆ) , − (` 2)! −2,`m `m −

(Z) Instead of a±2,`m, it is convenient to introduce their linear combinations as [22]

(Z) 1 (Z) (Z) aE,`m a2,`m + a−2,`m , ≡ −2 (3.89) i a(Z) a(Z) a(Z) . B,`m ≡ 2 2,`m − −2,`m

These ﬁelds E and B have parity-even and odd properties, respectively, in analogy with the electric 3.7 All-sky formulae for the CMB scalar-, vector- and tensor-mode anisotropies 41 and magnetic ﬁelds. Then, from Eqs. (3.88) and (3.89), we can express

Z Z d3k a(Z) = dΩ Y ∗ (Ω ) ∆(Z)(τ , k, nˆ) , X,`m n `m n (2π)3 X 0 1/2 (Z) 1 (` 2)! h 0 2 (Z) (Z) 0 2 (Z) (Z) i ∆ (τ0, k, nˆ) − ∂ (∆ + i∆ ) + ∂ (∆ i∆ ) (τ0, k, nˆ) , (3.90) E ≡ −2 (` + 2)! Q U Q − U 1/2 (Z) i (` 2)! h 0 2 (Z) (Z) 0 2 (Z) (Z) i ∆ (τ0, k, nˆ) − ∂ (∆ + i∆ ) ∂ (∆ i∆ ) (τ0, k, nˆ) . B ≡ 2 (` + 2)! Q U − Q − U

Here, X discriminates between intensity and two polarization (electric and magnetic) modes, respectively, as X = I,E,B. When k zˆ, using Eqs. (3.84) - (3.86) and the operations derived from Eq. (A.2) as k

!2 λ 0 2 −iµk,nx 2 −iµk,nx ∂ 2Y2λ(nˆ)e = ∂µk,n + 2 (1 µk,n)2Y2λ(Ωk,n)e , − 1 µk,n − − (3.91) !2 λ 0 2 −iµk,nx 2 −iµk,nx ∂ −2Y2λ(nˆ)e = ∂µ (1 µ )−2Y2λ(Ωk,n)e , − k,n − 1 µ2 − k,n − k,n we obtain more explicit expressions as

1/2 Z τ0 (S) (` 2)! (S) (0) ˆ(S) −iµk,nx ∆E (τ0, k zˆ, nˆ) = − ξ (k zˆ) dτSP (k, τ) (x)e , k (` + 2)! k 0 E 1/2 " r # (V ) (` 2)! X 8π (λ) ∆ (τ0, k zˆ, nˆ) = − λi Y1λ(nˆ)ξ (k zˆ) E k (` + 2)! − 3 k λ=±1 Z τ0 (V ) ˆ(V ) −iµk,nx dτSP (k, τ) (x)e , × 0 E 1/2 " r # (V ) (` 2)! X 8π (λ) ∆ (τ0, k zˆ, nˆ) = − i Y1λ(nˆ)ξ (k zˆ) B k (` + 2)! − 3 k λ=±1 Z τ0 (V ) ˆ(V ) −iµk,nx (3.92) dτSP (k, τ) (x)e , × 0 B 1/2 " r # (T ) (` 2)! X 32π (λ) ∆ (τ0, k zˆ, nˆ) = − Y2λ(nˆ)ξ (k zˆ) E k (` + 2)! 15 k λ=±2 Z τ0 (T ) ˆ(T ) −iµk,nx dτSP (k, τ) (x)e , × 0 E 1/2 " r # (T ) (` 2)! X λ 32π (λ) ∆ (τ0, k zˆ, nˆ) = − Y2λ(nˆ)ξ (k zˆ) B k (` + 2)! − 2 15 k λ=±2 Z τ0 (T ) ˆ(T ) −iµk,nx dτSP (k, τ) (x)e × 0 B 42 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION with the operators ˆ, ˆ deﬁned as E B ˆ(S)(x) (1 + ∂2)2x2 , E ≡ x ˆ(V ) 2 2 2 3 (x) 4x + (12 + x )∂x + 8x∂ + x ∂ , E ≡ x x ˆ(V ) 2 2 2 (x) x + 4x∂x + x ∂ , B ≡ x (3.93) ˆ(T ) 2 2 (x) 12 + x (1 ∂ ) 8x∂x , E ≡ − − x − ˆ(T ) 2 (x) 8x + 2x ∂x . B ≡

From here, we want to show analytical expressions of a`m’s. In the above discussion, we have analytical formulae of the transfer functions when k zˆ. This implies that we consider the physics in the blue basis of Fig. 3.4 and their transfer functionsk are completely determined by not the angle between zˆ and nˆ, namely Ωn, but the angle between k and nˆ, namely Ωk,n. However, as shown in Eq. (3.90), to obtain a`m’s, we have to consider the physics in the red basis Fig. 3.4 and perform the Ωn-integral. Here, instead of the transformation of the transfer functions, it is a better way to transform the integration variable in the a`m as Ωn Ωk,n. This can be done by using the Wigner (`) → D-matrix Dmm0 , which is the unitary irreducible matrix of rank 2` + 1 that forms a representation of the rotational group. The property of this matrix and the relation with spin-weighted spherical harmonics are explained in Appendix B. If we consider the rotational matrix   cos θk cos φk sin φk sin θk cos φk − S(Ωk)  cos θk sin φk cos φk sin θk sin φk  (3.94) ≡ sin θk 0 cos θk − corresponding to the conﬁguration (α = φk, β = θk, γ = 0) of Eq. (B.3) and satisfying

Ωn = S(Ωk)Ωk,n , (3.95) the transformation equation (B.1) can be equated with

∗ X (`) ∗ Y`m(Ωn) = Dmm0 (S(Ωk)) Y`m0 (Ωk,n) . (3.96) m0

Using this equation and the relation of the coordinate transformation as dΩn = dΩk,n, the a`m of arbitrary mode is written as

Z 3 " Z # (Z) d k X (`) ∗ (Z) a = D 0 (S(Ωk)) dΩk,nY 0 (Ωk,n)∆ (τ0, k zˆ, nˆ) X,`m (2π)3 mm `m X k m0 1/2 Z 3 " Z # 4π d k X m0 ∗ ∗ (Z) = ( 1) −m0 Y (Ωk) dΩk,nY 0 (Ωk,n)∆ (τ0, k zˆ, nˆ) . 2` + 1 (2π)3 − `m `m X k m0 (3.97)

In the second equality, we have obeyed the relation of Eq. (B.4) in this case as

1/2 (`) 4π m0 ∗ D 0 (S(Ωk)) = ( 1) −m0 Y (Ωk) . (3.98) mm 2` + 1 − `m 3.7 All-sky formulae for the CMB scalar-, vector- and tensor-mode anisotropies 43

Using the mathematical results of the Ωk,n-integrals r 2L + 1 −iµk,nx X L e = 4π( i) jL(x) YL0(Ωk,n) , − 4π L Z ∗ −iµk,nx ` p dΩk,nY 0 e = ( i) δm0,0 4π(2` + 1)j`(x) , `m − s (3.99) Z 3 (` + 1)! j (x) ∗ −iµk,nx `−1 ` dΩk,nY 0 Y1±1e = ( i) δm0,±1 (2` + 1) , `m − 2 (` 1)! x − s Z 15 (` + 2)! j (x) ∗ −iµk,nx `−2 ` dΩk,nY 0 Y2±2e = ( i) δm0,±2 (2` + 1) , `m − 8 (` 2)! x2 − we can ﬁnd the general formulae of the a`m for all-mode perturbations:

Z 3 (Z) ` d k X λ+x ∗ (λ) (Z) a = 4π( i) [sgn(λ)] −λY (Ωk)ξ (k) (k) , (3.100) X,`m − (2π)3 `m TX,` λ where the helicity of the perturbations is expressed by λ: λ = 0 for (Z = S), = 1 for (Z = V ) or = 2 for (Z = T ), the index x discriminates between the two parity states: x ±= 0 for X = I,E, ± x = 1 for X = B, and the time-integrated transfer functions (Z)(k) are expressed as 4 TX,` Z τ0 (S) (S) I,` (k) = dτSI (k, τ)j`(x) , T 0 1/2 Z τ0 (S) (` 2)! (S) ˆ(S) E,` (k) = − dτSP (x)j`(x) , T (` + 2)! 0 E (` + 1)!1/2 Z τ0 j (x) (V )(k) = dτS(V )(k, τ) ` , TI,` (` 1)! I x − 0 1/2 Z τ0 (` + 1)! (` 2)! j`(x) (V )(k) = − dτS(V )(k, τ) ˆ(V )(x) , TE,` − (` 1)! (` + 2)! P E x − 0 (3.101) 1/2 Z τ0 (` + 1)! (` 2)! j`(x) (V )(k) = − dτS(V )(k, τ) ˆ(V )(x) , TB,` − (` 1)! (` + 2)! P B x − 0 1/2 Z τ0 (T ) (` + 2)! (T ) j`(x) I,` (k) = dτSI (k, τ) 2 , T − (` 2)! 0 x Z τ0− (T ) (T ) ˆ(T ) j`(x) E,` (k) = dτSP (k, τ) (x) 2 , T − 0 E x Z τ0 (T ) (T ) ˆ(T ) j`(x) B,` (k) = dτSP (k, τ) (x) 2 . T 0 B x

Note that in the all-sky analysis, due to the dependence of transfer functions on φk,n, a`m’s depend on the helicity state through the spin spherical harmonics. In the above discussion, we take the synchronous gauge and derive the a`m. However, in the same manner, we can obtain the identical form of Eq. (3.100) even in another gauge. In this case, the diﬀerent points can be conﬁned only

4In Ref. [3], there are three typos: right-hand sides of Eqs. (B21), (B22) and (B23) must be multiplied by a factor 1, respectively. − 44 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION in the transfer function (3.101). In a numerical code CAMB [13, 19], these transfer functions of polarization modes are expanded as, e.g.,

" # Z τ0 (T ) ˙ (T ) ¨(T ) (T ) ˙ (T ) (T ) gψ¨ + 2g ˙ψ + gψ 4 gψ˙ + gψ (T ) 6 E,` (k) = dτ 2 + gψ 1 2 j`(x) , T − 0 k k x − − x " ! # (3.102) Z τ0 (T ) ˙ (T ) (T ) (T ) 2ψ ψ gψ˙ B,` (k) = 2 dτ g + + j`(x) . T − 0 x k k

3.8 Flat-sky formulae for the CMB scalar-, vector- and tensor-mode anisotropies

The flat-sky approximation uses the (2D) plane wave expansion of the CMB fluctuation instead of the spherical harmonics one, and it is valid if we restrict observed direction (parallel to nˆ) only close to the z axis [4, 23]. As confirmed in Ref. [4], the flat-sky power spectra of E- and B-mode polarizations sourced from the primordial scalar and tensor perturbations are in good agreement with the all-sky ones for ` & 40. On the basis of these studies, we have also compared the all-sky power spectra with the flat-sky ones for the I,E, and B modes from scalar, vector, and tensor perturbations and found their consistencies at ` & 40.

As mentioned in the previous subsection, in order to estimate the a`m, one must construct the transfer functions for arbitrary k. In other words, we want to obtain the transfer functions expressed by arbitrary k (whose direction is denoted by Ωk) and nˆ (denoted by Ωn) instead of Ωk,n in Fig. 3.4. In the I modes, only by changing Ωk,n to Ωk and Ωn with the relation (3.95), the transfer functions for arbitrary k can be obtained. In the E and B modes, in addition to this treatment, one must consider the mixing between ∆Q and ∆U under the transformation S(Ωk) as described in Ref. [4]. This eﬀect is expressed as

0 0 (Z) (Z) ∓2iψ (Z) (Z) (∆ i∆ )(τ0, k, nˆ) = e (∆ i∆ )(τ0, k zˆ, nˆ) . (3.103) Q ± U Q ± U k

ˆ ˆ with the mixing angle ψ. The angle ψ represents the rotation angle between θk,n and θn, where ˆ ˆ θk,n and θn are the unit vectors orthogonal to nˆ in a particular basis in which k zˆ and a general basis, respectively (see Fig. 3.4). k

In the ﬂat-sky analysis, i.e., θn 0, by using Eqs. (3.84) - (3.86) and by using the limit of ψ → 3.8 Flat-sky formulae for the CMB scalar-, vector- and tensor-mode anisotropies 45

as ψ φn φk + π, the transfer functions for arbitrary k are derived as → − Z τ0 (S) (S) −ik·nˆD ∆I (τ0, k, nˆ) ξ(k) dτSI (k, τ)e , → 0 Z τ0 (S) (S) ∓2i(φn−φk) 2 (S) −ik·nˆD (∆Q i∆U )(τ0, k, nˆ) e sin θkξ(k) dτSP (k, τ)e , ± → 0 ! Z τ0 (V ) X (λ) (V ) −ik·nˆD ∆ (τ0, k, nˆ) i sin θk ξ (k) dτS (k, τ)e , I → I λ=±1 0

(V ) (V ) ∓2i(φn−φk) (∆Q i∆U )(τ0, k, nˆ) e sin θk ± → " ! !# X (λ) X (λ) cos θk ξ (k) λξ (k) × − ± λ=±1 λ=±1 (3.104) Z τ0 (V ) −ik·nˆD dτSP (k, τ)e , × 0 ! Z τ0 (T ) 2 X (λ) (T ) −ik·nˆD ∆ (τ0, k, nˆ) sin θk ξ (k) dτS (k, τ)e , I → I λ=±2 0 " ! (T ) (T ) ∓2i(φn−φk) 2 X (λ) (∆ i∆ )(τ0, k, nˆ) e (1 + cos θk) ξ (k) Q ± U → λ=±2 !# Z τ0 X λ (λ) (T ) −ik·nˆD 2 cos θk ξ (k) dτS (k, τ)e . ∓ 2 P λ=±2 0

It is important to note that the φk dependence which are inherent in the vector and tensor perturbations vanishes in the flat-sky approximation, besides a trivial φk dependence due to a spin-2 nature of the Stokes Q and U parameters. One may explicitly see that φk,n dependence vanishes in the transfer functions when taking θn 0 because the S(Ωk) matrix rotates the basis with the new z axis always being on the x z plane→ in a particular basis in which k zˆ (see Fig. 3.4). This − k approximation means that for θn 1, it is valid to calculate the CMB fluctuation on the basis of vector and tensor perturbations fixed as θn = 0, namely, φk,n = π. In the flat-sky limit, a`m in the all-sky analysis described as Eq. (3.90) is modified by using the plane wave as

Z Z 3 (Z) 2 −i`·Θ d k (Z) (Z) a d Θe ∆ (τ0, k, nˆ) a (`) , X,`m → (2π)3 X ≡ X " # (Z) 1 X (Z) s (Z) ∆ (τ , k, nˆ) ∆ + i∆ (τ , k, nˆ)e−si(φ`−φn) , E 0 2 Q 2 U 0 (3.105) → s=±2 " # (Z) i X s (Z) s (Z) ∆ (τ , k, nˆ) ∆ + i∆ (τ , k, nˆ)e−si(φ`−φn) , B 0 2 2 Q 2 U 0 → − s=±2

5 where Θ is the 2D vector projecting nˆ to the ﬂat-sky plane expressed as Θ = (Θcosφn, Θsinφn) . (T ) For example, in order to obtain aI,`m, we substitute Eq. (3.104) into Eq. (3.105) and calculate

5Not confuse Θ with the CMB temperature ﬂuctuation. 46 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION as follows:

! Z 3 Z τ0 Z (T ) d k X k (T ) a (`) = sin2 θ ξ(λ)(k) dτ d2Θe−i(k D+`)·ΘS (k, τ)e−ikzD I (2π)3 k I λ=±2 0 ! Z 3 Z τ0 d k X (T ) = sin2 θ ξ(λ)(k) dτ(2π)2δ(kkD + `)S (k, τ)e−ikzD (2π)3 k I λ=±2 0 ! Z τ0 Z ∞ dkz X (λ) k = dτ ξ (k = `/D, kz) 2π − 0 −∞ λ=±2 `2 1 (T ) p 2 2 −ikzD 2 2 SI (k = kz + (`/D) , τ) 2 e , (3.106) ×(kzD) + ` D

where D = τ0 τ is the conformal distance and we have decomposed k into two-dimensional vector − k parallel to the ﬂat sky and that orthogonal to it, k = (k , kz). In order to obtain the last equation, we use following relations which are satisﬁed under kk = `/D as −

s ` 2 k = k2 + , z D ` ` sin θk = = , p 2 2 kD (kzD) + ` (3.107) s ` 2 cos θk = sgn(kz) 1 , − kD

φk = φ` + π .

The other-mode a`m’s can be calculated in the same manner. Thus, we summarize the all-mode 3.9 CMB power spectrum 47 a(`)’s:

Z τ0 Z ∞ (Z) dkz X x (λ) k 1 −ikzD (Z) a (`) = dτ [sgn(λ)] ξ (k = `/D, kz) e (kz, τ) X 2π − D2 SX,` 0 −∞ λ (S) (S) p 2 2 (kz, τ) = S (k = k + (`/D) , τ) , SI,` I z 2 (S) ` (S) p 2 2 E,` (kz, τ) = 2 2 SP (k = kz + (`/D) , τ) , S (kzD) + `

(V ) ` (V ) p 2 2 (kz, τ) = i S (k = k + (`/D) , τ) , I,` p 2 2 I z S (kzD) + ` s 2 (V ) ` ` (V ) p 2 2 (kz, τ) = sgn(kz) 1 S (k = k + (`/D) , τ) , E,` p 2 2 2 2 P z S − (kzD) + ` − (kzD) + ` (3.108)

(V ) ` (V ) p 2 2 (kz, τ) = i S (k = k + (`/D) , τ) , B,` p 2 2 P z S − (kzD) + ` 2 (T ) ` (T ) p 2 2 I,` (kz, τ) = 2 2 SI (k = kz + (`/D) , τ) , S (kzD) + ` 2 (T ) ` (T ) p 2 2 E,` (kz, τ) = 2 2 2 SP (k = kz + (`/D) , τ) , S − (kzD) + ` s 2 (T ) ` (T ) p 2 2 B,` (kz, τ) = 2i sgn(kz) 1 2 2 SP (k = kz + (`/D) , τ) , S − (kzD) + `

(Z) where we label X,` as the modiﬁed source function. One can formulate the ﬂat-sky CMB power spectrum and bispectrumS by using these formulae [3, 24–26].

3.9 CMB power spectrum To extract several information about the Universe from the observational data, the two-point correlation function of the CMB ﬂuctuations (called CMB power spectrum) is often-used. Here, we formulate the CMB power spectrum and summarize the constraints on several model parameters from the current observational data. If we assume the Gaussianity and the symmetry under the parity and rotational transformations in the initial stochastic variables, their power spectrum can be expressed as ( 1 (Z = S) (λ1) (λ2)∗ 3 ξ (k1)ξ (k2) = (2π) PZ (k1)δ(k1 k2)δλ ,λ (3.109) − 1 2 × 1/2 (Z = V,T )

This implies that the couplings between the diﬀerent modes of the perturbation vanish in the power spectrum. Then, from the formula of the all-sky a`m (3.100), the CMB power spectra of all modes are derived as

* 2 + Y (Zn) (Z1) m1 a C ( 1) δ` ,` δm ,−m δZ ,Z δx ,x , Xn,`nmn ≡ X1X2,`1 − 1 2 1 2 1 2 1 2 n=1 (3.110) Z (Z1) 2 2 (Z1) (Z1) C = k dk1PZ (k1) (k1) (k1) X1X2,`1 π 1 1 TX1,`1 TX2,`1 48 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION where we use the relation derived by the reality condition of the metric perturbation:

ξ(λ)(k) = ( 1)λξ(λ)∗( k) . (3.111) − − In Fig. 3.5, we plot the CMB intensity and polarization power spectra of the scalar and tensor modes. Here we think that the scalar and tensor perturbations are sourced from the primordial curvature perturbations (ξ(0) = ) and primordial gravitational waves (ξ(±2) = h(±2)), respectively. The ratio between these powerR spectra, called the tensor-to-scalar ratio r, is defined as 6 2P (k) r T . (3.112) ≡ PS(k) At first, let us focus on the II spectra in the left top panel. In the scalar spectrum, the dominant signal is generated from the acoustic oscillation of the photon and baryon fluid. The first acoustic peak is located at `1 220. This scale is corresponding to the angle of the sound horizon at ∼ the recombination epoch as `1 2πτ0/rs(z∗). At small scales corresponding to ` `1, due to the difference between the photon∼ and baryon speeds, the coupling between photons and baryons is ineffective and the acoustic oscillation is highly damped. This effect is well-known as the Silk damping [28]. On the other hand, the gravitational blue shift due to the potential decay in the late time affects the fluctuations for ` `1. This is called the integrate Sachs Wolfe (ISW) effect [29] −κ (Z) and arises from the terms proportional to e of the source function SI in Eqs. (3.84) and (3.86). In the tensor spectrum, the ISW effect leads to the dominant signals and the scattering effect (the (T ) second term of the source function SI in Eq. (3.86)) is subdominant [6]. As shown in Eqs. (3.84) and (3.86), the EE and BB spectra (the bottom two panels) have no gravitational redshift term and there is no ISW effect. Instead of it, the scattering term generates the CMB fluctuation. The most interesting signature in the polarization power spectra is the enhancement at ` . 10. This can be caused by Thomson scattering at reionization of hydrogen which may have occurred at z 10 [4]. Therefore, these signals can be important to determine the optical depth of the Universe,∼ κ. The IE spectrum described in the right top panel seems to include the features of both the II and EE spectra and has both positive and negative values. Unlike the above three cases, the BB spectrum never arises from the scalar mode because the scalar mode has only one helicity, namely λ = 0 7. Hence, we believe that the BB spectrum directly tells us the amplitude of the primordial gravitational waves depending on the energy scale of inflation. Theoretically, the CMB power spectrum depends on the parameters which determine the dynamics of the Universe as the energy density of the cosmic fluids, curvature, and the Hub- (S) ble constant H0. Figure 3.6 shows the dependence of CII,` on the density parameters of cold 2 dark matters, the cosmological constant and baryons as ωc Ωc(H0/100sec Mpc/km) , ΩΛ and 2 ≡ · ωb Ωb(H0/100sec Mpc/km) , respectively. From this figure, one can observe that as ωc de- ≡ · (S) creases, the overall amplitude of CII,` enlarges. This behavior is understood as follows. If ωc decreases, since the radiation dominated era is lengthened, the gravitational potential for smaller (S) k enters the horizon and decays. Thus, CII,` at corresponding multipoles as ` kτ is boosted due to the gravitational blue shift. This is the so called early ISW effect [29]. Next,∼ focusing on the (S) blue dotted line, one can find that if ΩΛ becomes large, CII,` is boosted for ` . 10. This is due to the late ISW effect, that is, Λ dominates the Universe earlier and the potential at larger scales is

6This deﬁnition is consistent with Eq. (2.32), and the notation of Ref. [27] and CAMB [13] 7The vector mode generates the BB spectrum due to its two helicities. However, due to the decay of the vector potential via the Einstein equation, this becomes the subdominant signal. To avoid this, the sources such as cosmic strings and magnetic ﬁelds need to exist and support. 3.9 CMB power spectrum 49

10000 1000

1000 100

] 10 ] 2 2

100 K K µ µ [ [ 2 1 2 0 0

10 )T )T π π 0.1 |/(2 /(2

II,l 1 IE,l 0.01

l(l+1)C 0.1 l(l+1)|C 0.001

0.01 0.0001 scalar scalar tensor tensor 0.001 1e-05 10 100 1000 10 100 1000 l l 100 0.01 scalar tensor 10 ] ] 2 1 2 K K

µ µ 0.001 [ [ 2 2 0 0.1 0 )T )T π π /(2 /(2 0.01 EE,l BB,l 0.0001 0.001 l(l+1)C l(l+1)C

0.0001

tensor 1e-05 1e-05 10 100 1000 10 100 1000 l l Figure 3.5: The CMB spectra of the II (left top panel), IE (right top one), EE (left bottom one) and BB (right bottom one) modes. Here we consider a power-law flat ΛCDM model and fix the tensor-to-scalar ratio as r = 0.1. The other cosmological parameters are fixed to the mean values reported in Ref. [27].

(S) destroyed, hence CII,` at corresponding `’s is amplified. We also notice that when ωb enlarges, the ratio of the amplitude between the first and second peaks of the magenta dot-dashed line increases. Solving the coupled Boltzmann equations, the acoustic oscillation of the baryon-photon fluid in the matter dominated era is roughly given by

1 Θ + R Φ cos(krs) RΦ , (3.113) ∼ 3 − where Φ is the potential of the conformal Newtonian gauge, rs denotes the sound horizon and R 3ρb/(4ργ). Then if ωb increases and R becomes large, this equation experiences increase in ≡ (S) 2 amplitude and suppression of the intercept. Hence, the diﬀerence of CII,` Θ between the odd- and even-number peaks increases. These parameters are limited with the others∝ from the current observational data set as Table 3.2. Other than these parameters, massive neutrinos and some relativistic components also make impacts on the CMB ﬂuctuation (e.g. Refs. [30, 31]). The CMB power spectrum also depends on the primordial curvature perturbations and primordial gravitational waves. Conventionally, these spectra are parametrized as

k3P (k) k ns−1 S = A , (3.114) 2π2 S 0.002Mpc−1 50 3 FLUCTUATIONS IN COSMIC MICROWAVE BACKGROUND RADIATION

7000 reference ω 6000 c decrease ΩΛ increase ω ] b increase 2 5000 K µ [ 2

0 4000 )T π

/(2 3000 (S) II,l 2000 l(l+1)C 1000

0 10 100 1000 l

Figure 3.6: CMB II power spectra sourced from the scalar-mode perturbations in a power-law flat ΛCDM model. The red solid line is plotted with ωc = 0.112, ΩΛ = 0.728, ωb = 0.02249, ns = 0.967, κ = 0.088 [27]. The green dashed, blue dotted and magenta dot-dashed lines are calculated if ωc decreases to 0.1, ΩΛ increases to 0.8, and ωb increases to 0.028, respectively. and Eq. (3.112). As shown in Eq. (3.100), the magnitudes of the primordial curvature perturbations A and gravitational waves rA simply determine the overall amplitude of the C(S) and S S X1X2,`1 C(T ) , respectively. The spectral index of the power spectrum of the curvature perturbations n X1X2,`1 s changes in slope of C(Z1) . From the observational data, the constraints on A , n , r are given X1X2,`1 S s as Table 3.2. Here, we want to note that ns = 1 is excluded at about 3-sigma level. This implies the deviation from the exact de Sitter expansion in inflation. As shown in the bound on r, unlike the primordial curvature perturbation, the primordial gravitational wave has not been detected yet. However, some experimental groups aim to discover the BB spectrum through remove of some noisy foreground emission and improvement of the instruments [2, 34–36]. If these projects achieve, it will be possible to judge the existence of the primordial gravitational waves of r < 0.01. So far, we discussed under the assumption that the parity and rotational invariances are kept. On the other hand, there are a lot of studies which probe the somewhat exotic scenarios where these invariances violate. As a theoretical prediction, if parity-violating action such as the Chern- Simon term exists in the early Universe, ξ(+2)(k)ξ(+2)∗(k0) = ξ(−2)(k)ξ(−2)∗(k0) and the IB and EB spectra appear [37–40]. Using the parametrization as 6 obs C CIE,` sin(2∆α) , (3.115) IB,` ≡ the parity violation is limited as 5.0◦ < ∆α < 2.8◦ (95% CL) [27]. The rotational invariance is broken if the Universe has the preferred− direction and this situation is realized by the anisotropic inflation [41–43]. This leads to the direction-depending power spectrum as iso ˆ 2 PS(k) = P (k)[1 + g(k nˆ) ] , (3.116) S · and produces the off-diagonal components in the CMB power spectrum as `1 = `2. From the CMB observational data, the magnitude of the statistical anisotropy has been limited6 as g = 0.15 0.039 [44] ± REFERENCES 51

Parameter WMAP 7-yr WMAP + BAO + H0 +0.056 100ωb 2.249 2.255 0.054 −0.057 ± ωc 0.1120 0.0056 0.1126 0.0036 ±+0.030 ± ΩΛ 0.727 0.725 0.016 −0.029 ± ns 0.967 0.014 0.968 0.012 κ 0.088 ± 0.015 0.088 ± 0.014 ± −9 ± −9 AS (2.43 0.11) 10 (2.430 0.091) 10 r± < 0.36× ±< 0.24 × H0 70.4 2.5 km/s/Mpc 70.2 1.4 km/s/Mpc ± ± Ωb 0.0455 0.0028 0.0458 0.0016 ± ± Ωc 0.228 0.027 0.229 0.015 ±+0.0056 ± ωm 0.1345 0.1352 0.0036 −0.0055 ± zreion 10.6 1.2 10.6 1.2 ± ± t0 13.77 0.13 Gyr 13.76 0.11 Gyr ± ± Table 3.2: Summary of the cosmological parameters of ΛCDM with ﬁnite r model from the WMAP 7-year data [27], and the data set combined with the results of the galaxy survey [32] and Hubble constant measurement [33], respectively. Here zreion denotes the redshift at the reionization epoch, t0 is the present time of the Universe, and ωm ωb + ωc. ≡

Furthermore, owing to the progress of the observational accuracy, the deviation of the Gaus- sianity can be measured. Beyond the power spectrum, this is achieved by using the three-point function (bispectrum). In the next section, we discuss about how to extract the information on the early Universe from the CMB scalar, vector and tensor bispectrum. In addition, we can add other components of fluids in the analysis of the CMB spectrum. From Sec. 9, we focus on the effect of the primordial magnetic fields on the CMB.

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4 Primordial non-Gaussianities

The study of non-Gaussian impacts in the cosmological fluctuations provides an important information of the early Universe [1]. The primordial non-Gaussianities are measures of the interactions in inflation, hence constraining this will lead to a great deal about the inflationary dynamics. It may also puts strong constraints on alternatives to the inflationary paradigm (e.g., Refs. [2–8]). In the previous part, we expanded the inflationary action to second order in the comoving curvature perturbation and the gravitational waves h(±2). These actions allowed us to compute R (±2) the power spectra PR(k) and Ph(k). If the fluctuations and h obey the exact Gaussian statistics, the power spectrum (or two-point correlation function)R contains all the information 1. However, for the non-Gaussian fluctuations, higher-order correlation functions beyond the two- point function contain additional information about inflation. Estimating the leading non-Gaussian effects requires the expansion of the action to third order since we must take into account the leading non-trivial interaction terms. In this section, we review recent studies about the primordial non- Gaussianity based on e.g., Refs. [9–11].

4.1 Bispectrum of the primordial ﬂuctuations

At ﬁrst, we give the deﬁnition of the bispectrum of the initial perturbations ξ(λ) of the scalar (λ = 0), vector (λ = 1), and tensor (λ = 2) modes. The Fourier transformation of the two-point function is the± power spectrum ±

* 2 + 3 ! ( 1 (Z = S) Y (λn) 3 X λ1 ξ (kn) = (2π) PZ (k1)δ kn δλ1,λ2 ( 1) . (4.1) n=1 n=1 − × 1/2 (Z = V,T )

Similarly, the Fourier mode of the three-point function is so called the bispectrum

* 3 + 3 ! Y (λn) 3 λ1λ2λ3 X ξ (kn) = (2π) F (k1, k2, k3)δ kn . (4.2) n=1 n=1

Here, the translation invariance of the background results in the delta function denoting the momentum conservation. If the scale invariance is kept, we have

λ1λ2λ3 −6 λ1λ2λ3 F (bk1, bk2, bk3) = b F (k1, k2, k3) . (4.3)

Moreover, due to the rotational invariance, the independent variables are reduced to k2/k1 and k3/k1. In order to compute the primordial bispectrum, it is necessary to deal with the vacuum evolution under the ﬁnite interactions carefully. This is not the leading order eﬀect in calculating the power spectrum. The in-in formalism is a powerful method to compute the primordial higher- order cosmological correlation [12–16]. In Sec. 8, using this formalism, we actually discuss the computation for the bispectrum of the gravitational waves.

1Odd-point correlation functions of Gaussian ﬂuctuations vanish while their even-point functions can be expanded by two-point functions due to the Wick’s theorem. 56 4 PRIMORDIAL NON-GAUSSIANITIES

Figure 4.1: Representations of triangles forming the bispectrum. This ﬁgure is adopted from Ref. [19].

4.2 Shape of the non-Gaussianities The delta function in Eq. (4.2) results in a closed triangle in Fourier space. The triangle configuration at which the primordial bispectrum is amplified is dependent on the inflationary models; therefore the shape of the non-Gaussianity is a powerful clue to inflation [17, 18]. 2 λ1λ2λ3 We can study the bispectrum shape by plotting the magnitude of (k1k2k3) F (k1, k2, k3) as a function of k2/k1 and k3/k1 for k3 k2 k1. For identification of each triangle, one often use ≤ ≤ the following names: squeezed (k1 k2 k3), elongated (k1 = k2 + k3), folded (k1 = 2k2 = 2k3), ≈ isosceles (k2 = k3), and equilateral (k1 = k2 = k3). In Fig. 4.1, the visual representations of these triangles are depicted. From here, we concentrate on three representative shapes of the primordial bispectrum: “local”, “equilateral”, and “orthogonal”. Then, it may be convenient to decompose the non-Gaussianity of the curvature perturbations into the magnitude-depending and shape-depending parts: 3 F 000(k , k , k ) = f (2π2A )2S(k , k , k ) , (4.4) 1 2 3 5 NL S 1 2 3 where AS is the magnitude of curvature perturbations defined in Eq. (3.114).

4.2.1 Local type The simplest way to parametrize the non-Gaussianity of curvature perturbation is to expand by Gaussian ﬂuctuation g [20] as R 3 local 2 2 (x) = g(x) + f g(x) g(x) . (4.5) R R 5 NL R − R 4.2 Shape of the non-Gaussianities 57

From this equation, we can see that the non-Gaussianity is localized at a given point in the real local space. Therefore, we call this the local-type non-Gaussianity, and fNL is called the local-type nonlinearity parameter. Then, the bispectrum of the local-type non-Gaussianity of the curvature perturbations is derived as 6 F 000 (k , k , k ) = f local [P (k )P (k ) + 2 perms.] . (4.6) local 1 2 3 5 NL R 1 R 2

Fixing the spectral index as ns = 1 and equating this equation with Eq. (4.4), we can write local 1 S (k1, k2, k3) = 2 3 + 2 perms. . (4.7) (k1k2)

This is boosted in the squeezed limit: k3 k1 k2 as shown in the top-left panel of Fig. 4.2. In this limit, the bispectrum reaches ≈

000 12 local F (k1, k2, k3 0) = f PR(k1)PR(k3) . (4.8) local → 5 NL In Refs. [12, 21–24], the authors found that the local-type non-linearity parameter is tiny in the single ﬁeld slow-roll inﬂation as

local 5 5 5 f = (1 ns) = (2H ηH ) = (3 η) , (4.9) NL 12 − 6 − 6 − local local which gives fNL = 0.015 for ns = 0.963. In contrast, large fNL can be realized in the models with multiple light ﬁelds during inﬂation [25–35], the curvaton scenario [36–38], and inhomogeneous reheating [39, 40].

4.2.2 Equilateral type The equilateral bispectrum is parametrized as [41] equil 1 2 1 S (k1, k2, k3) = 6 3 + 2 perms. 2 + 2 3 + 5 perms. . (4.10) −(k1k2) − (k1k2k3) k1k2k3 This bispectrum is obtained in the inflationary models with non-canonical kinetic terms for the equil scalar field. For example, the so-called Dirac-Born-Infeld (DBI) inflation [42, 43] predicts fNL 2 ∝ 1/cs for cs 1 with cs being the effective sound speed of the scalar field fluctuation. We can also − equil find a lot of other large fNL models [44–48]. This bispectrum has a peak at the equilateral limit, namely, k1 = k2 = k3 as described in the bottom-left panel of Fig. 4.2. Due to the orthogonality between the local- and equilateral-type bispectra, these can be measured almost independently. In Sec. 8, we confirm that the graviton non-Gaussianity originated from the Weyl cubic action is also categorized as the equilateral type.

4.2.3 Orthogonal type In the orthogonal type, we use the following parametrization: orthog 3 8 3 S (k1, k2, k3) = 6 3 + 2 perms. 2 + 2 3 + 5 perms. . (4.11) −(k1k2) − (k1k2k3) k1k2k3 This is nearly orthogonal to both the local-type and equilateral-type non-Gaussianities [49]. This bispectrum can arise from a linear combination of higher-derivative scalar-field interaction terms which produce the equilateral bispectra. This function has a positive peak at the equilateral configuration and negative valley along the elongated configurations as seen in the top-right panel of Fig. 4.2. 58 4 PRIMORDIAL NON-GAUSSIANITIES

Figure 4.2: Two dimensional color contour for the shapes of the primordial bispectra. Each panel 2 describes the normalized amplitude of (k1k2k3) S(k1, k2, k3) as a function of k2/k1 and k3/k1 for 2 k3 k2 k1. The amplitude is normalized by the maximum value of (k1k2k3) S(k1, k2, k3). The top-left≤ panel≤ is the local form given by Eq. (4.7), which diverges at the squeezed configuration. The bottom-left panel is the equilateral form given by Eq. (4.10), which is amplified at the equilateral configuration. The top-right panel is the orthogonal form given by Eq. (4.11), which has a positive peak at the equilateral configuration, and a negative valley along the elongated configurations. This is quoted in Ref. [11].

4.3 Observational limits Using the optimal estimators [11, 49–52], the constraints on the nonlinearity parameters from the CMB data (WMAP 7-yr data) are obtained as [53]

10 < f local < 74 , 214 < f equil < 266 , 410 < f orthog < 6 (95% CL) . (4.12) − NL − NL − NL As another approach for extracting the non-Gausssinity from the CMB data, the methods with the Minkowski functionals have been developed [54–56]. The PLANCK satelite [57] and the proposed CMBPol mission [58] will give tighter bounds as local local σ(fNL ) 5 and 2. At the level of fNL = (1), we need to be concerned about the contamination of the signals∼ from late-time secondary eﬀects.O Studies on the gravitational non-linear evolution at late times can be seen in Refs. e.g., [59–62]. In addition, the primordial non-Gaussianity also imprints its signatures on the large scale structure in the Universe. Estimating the primordial non-Gaussianity from the data of the matter distribution is hard due to large contamination of late-time gravitational nonlinear evolution. Regardless of it, the scale-dependence of the bias parameter between biased objects and linearized matter density ﬁelds is a good indicator for the primordial local-type non-Gaussianity [63, 64]) . From the luminous red galaxies (LRGs) sample of SDSS, Ref. [64] obtained a bound as

29 < f local < 70 (95% CL) . (4.13) − NL This is comparable to bounds from the CMB data. The way to extract the information on the 4.4 Beyond the standard scalar-mode non-Gaussianities 59 primordial non-Gaussianity from the matter distribution has continuously been studied (see, e.g., Refs. [65–67]).

4.4 Beyond the standard scalar-mode non-Gaussianities Historically, as described above, only the non-Gaussianity of curvature perturbations has been well-known studied. However, the non-Gaussianity of vector- or tensor-mode perturbation can be generated from the cosmological defects [68, 69], the magnetic ﬁelds [70], the nonlinear gravitational waves [12, 71–74], and so on. Furthermore, somewhat exotic non-Gaussianities including the violation of the rotatoional or parity invariance in the early Universe have recently been discussed (see, e.g., Refs. [72, 75–77]). Hence, for precise comprehension of the early Universe, detailed analyses of these signals are necessary. This is the main motivation of our studies: construction of the general formulae for the CMB bispectrum with not only scalar- but also vector- and tensor-mode contributions, and computation and analysis of the CMB bispectrum sources from these novel non-Gaussianities. In the following sections, we focus on our studies.

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5 General formalism for the CMB bispectrum from primordial scalar, vector and tensor non-Gaussianities

In this section, on the basis of the formulation of the CMB anisotropy in Sec. 3, we develop the formulae for the CMB bispectrum sourced from scalar-, vector-, and tensor-mode non-Gaussianity. These results have been published in our paper [1]. At first, we should remember an expression of CMB fluctuation discussed in Sec. 3. In the all-sky analysis, the CMB fluctuations of intensity, and E and B-mode polarizations are expanded with the spin-0 spherical harmonics, respectively. Then, the coefficients of CMB fluctuations, called a`m’s, are described as

Z 3 (Z) ` d k X λ+x ∗ ˆ (λ) (Z) a = 4π( i) [sgn(λ)] −λY (k)ξ (k) (k) , (5.1) X,`m − (2π)3 `m TX,` λ where the index Z denotes the mode of perturbations: Z = S (scalar), = V (vector) or = T (tensor) and its helicity is expressed by λ; λ = 0 for Z = S, = 1 for Z = V or = 2 for Z = T , X discriminates between intensity and two polarization (electric± and magnetic) modes,± respectively, as X = I,E,B and x is determined by it: x = 0 for X = I,E or = 1 for X = B, (λ) (Z) ξ is the initial perturbation decomposed on each helicity state and X,` is the time-integrated transfer function in each sector given by Eq. (3.101)1. T Next, we expand ξ(λ) with spin-( λ) spherical harmonics as − (λ) X (λ) ˆ ξ (k) ξ (k)−λY`m(k) , (5.2) ≡ `m `m and eliminate the angular dependence in Eq. (5.1) by performing kˆ-integral:

Z ∞ 2 (Z) k dk X (λ) (Z) a = 4π( i)` [sgn(λ)]λ+xξ (k) (k) . (5.3) X,`m − (2π)3 `m TX,` 0 λ Here, we use the orthogonality relation of the spin-λ spherical harmonics described in Appendix A as Z 2 ∗ d nˆλY`0m0 (nˆ)λY`m(nˆ) = δ`,`0 δm,m0 . (5.4)

Then the CMB bispectrum generated from the primordial non-Gaussianity of the scalar, vector and tensor perturbations is written down as

* 3 + " 3 Z ∞ 2 # Y (Z ) Y k dkn (Z ) X a n = 4π( i)`n n n (k ) [sgn(λ )]λn+xn Xn,`nmn 3 Xn,`n n n − 0 (2π) T n=1 n=1 λn * 3 + Y (λ ) ξ n (k ) , (5.5) `nmn n × n=1 * 3 + " 3 Z #* 3 + Y (λ ) Y Y ξ n (k ) = d2kˆ Y ∗ (kˆ ) ξ(λn)(k ) . `nmn n n−λn `nmn n n n=1 n=1 n=1

1Here, we set 00 = 1. 5 GENERAL FORMALISM FOR THE CMB BISPECTRUM FROM PRIMORDIAL 66 SCALAR, VECTOR AND TENSOR NON-GAUSSIANITIES

3.5 0.1

3 local equilateral 0.01 2.5

orthogonal 16 16 10 10

2 × 0.001 × 2 ) 2 ) π π 1.5 |/(2 /(2 0.0001 III,lll 1 IIE,lll b |b 2 2 0.5 1e-05 (l+1) (l+1) 2 l 2 l 0 local 1e-06 -0.5 equilateral orthogonal -1 1e-07 10 100 1000 10 100 1000 l l 0.01 0.01

local 0.001 local 0.001 equilateral equilateral 0.0001

orthogonal 16 orthogonal 16 10 10 × × 0.0001 1e-05 2 2 ) ) π π

|/(2 1e-06 |/(2 1e-05 IEE,lll

EEE,lll 1e-07 |b |b 2 2 1e-06 1e-08 (l+1) (l+1) 2 2 l l 1e-09 1e-07 1e-10

1e-08 1e-11 10 100 1000 10 100 1000 l l Figure 5.1: The CMB III (left top panel), IIE (right top one), IEE (left bottom one), and EEE (right bottom one) bispectra induced from the local-type (red solid line), equilateral-type (green dashed one), and orthogonal-type (blue dotted one) non-Gaussianities of curvature perturbations. The three multipoles are fixed as `1 = `2 = `3 `. Here we consider a power-law flat ΛCDM local ≡ equil orthog model and fix the nonlinear parameters as fNL = fNL = fNL = 100. The other cosmological parameters are fixed to the mean values reported in Ref. [9].

This formalism will be applicable to diverse sources of the scalar, vector and tensor non-Gaussianities, such as, a scalar-graviton coupling [1] (Sec. 6), cosmic strings [2, 3], primordial magnetic ﬁelds [4–6] (Sec. 9), and statistically-anisotropic and parity-violating interactions [7, 8] (Secs. 7 and 8). To compute the CMB bispectrum composed of arbitrary perturbation modes, we have to reduce D E the expanded primordial bispectrum, Q3 ξ(λn) (k ) , involving the contractions of the wave n=1 `nmn n number vector and polarization vector and tensor, and the integrals over the angles of the wave number vectors. As shown later, this is elegantly completed by utilizing the Wigner symbols and spin-weighted spherical harmonics. If the initial bispectrum satisﬁes the rotational invariance, the CMB bispectrum is divided into the Wigner-3j symbol depending on the azimuthal quantum numbers and the angle-averaged function as * 3 + Y (Zn) `1 `2 `3 (Z1Z2Z3) a` m = BX X X ,` ` ` . (5.6) n n m1 m2 m3 1 2 3 1 2 3 n=1 Let us focus on the CMB bispectrum from the curvature perturbations. Then, from Eqs. (4.2), (4.4), (4.7), (4.10), (4.11), (5.5), (5.6) and the knowledge of Appendix C, we derive the reduced 67

10 1

local 16 16 0.1 equilateral 1 10 10 × × orthogonal 2 2 ) ) π π 0.01 |/(2 |/(2 3 3 l l

0.1 2 2 l l 1 1 III,l IIE,l 0.001 +1)|b

0.01 +1)|b 3 3 (l (l 3

3 0.0001 local +1)l +1)l 1

equilateral 1 (l

0.001 (l 1 1 l orthogonal l 1e-05

0.0001 1e-06 10 100 10 100

l3 l3 0.01 1e-05

local 16 16 equilateral 1e-06 10

10 0.001 × × 2

2 orthogonal ) ) π π 1e-07 |/(2 |/(2 3 3 l l 2 2

0.0001 l l 1 1

IEE,l 1e-08 EEE,l

1e-05 +1)|b +1)|b 3 3 (l (l 1e-09 3 3 +1)l +1)l 1 1e-06 1 local (l (l 1

1 1e-10 l l equilateral orthogonal 1e-07 1e-11 10 100 10 100

l3 l3 Figure 5.2: The CMB III (left top panel), IIE (right top one), IEE (left bottom one), and EEE (right bottom one) bispectra induced from the local-type (red solid line), equilateral-type (green dashed one), and orthogonal-type (blue dotted one) non-Gaussianities of curvature perturbations local equil orthog with fNL = fNL = fNL = 100. Here, we ﬁx the two multipoles as `1 = `2 = 200, and plot each curve as the function in terms of `3. The cosmological parameters are identical to those in Fig. 5.1. bispectra as

b(SSS) = I0 0 0 −1 B(SSS) X1X2X3,`1`2`3 `1`2`3 X1X2X3,`1`2`3 Z ∞ " 3 Z ∞ # 2 Y 2 2 (S) 000 = y dy k dkn (kn)j` (kny) F (k1, k2, k3) , (5.7) π n Xn,`n n 0 n=1 0 T where we have introduced the I symbol as r s1s2s3 (2l1 + 1)(2l2 + 1)(2l3 + 1) l1 l2 l3 Il l l . (5.8) 1 2 3 ≡ 4π s1 s2 s3 In Figs. 5.1 and 5.2, we plot these CMB bispectra for each multipole conﬁguration. From these, one can see that depending on the shape of the primordial non-Gaussianity, the overall magnitudes of the bispectra do not change, but the shapes in ` space change. Comparing these with the CMB data, the bounds on the nonlinearity parameters (4.12) are obtained. In this simple CMB bispectrum, there exists no dependence of the initial bispectrum on the polarization vector and tensor, hence we can derive the above formulae easily. Considering the 68 REFERENCES vector- and tensor-mode contributions in the CMB bispectrum, however, the computation becomes so cumbersome due to the complicated angular dependence of the polarization vector and tensor. This diﬃculty is also true for the bispectrum where the rotational or parity invariance is broken. From the next section, we deal with these complicated bispectra depending on the several scenarios by applying the wonderful mathematical tools such as the Wigner symbols and the spin-weighted spherical harmonics.

References

[1] 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].

[2] K. Takahashi et al., Non-Gaussianity in Cosmic Microwave Background Temperature Fluctuations from Cosmic (Super-)Strings, JCAP 0910 (2009) 003, [arXiv:0811.4698].

[3] M. Hindmarsh, C. Ringeval, and T. Suyama, The CMB temperature bispectrum induced by cosmic strings, Phys. Rev. D80 (2009) 083501, [arXiv:0908.0432].

[4] 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].

[5] 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].

[6] 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].

[7] M. Shiraishi and S. Yokoyama, Violation of the Rotational Invariance in the CMB Bispectrum, Prog.Theor.Phys. 126 (2011) 923–935, [arXiv:1107.0682].

[8] 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].

[9] WMAP Collaboration, E. Komatsu et al., Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation, Astrophys. J. Suppl. 192 (2011) 18, [arXiv:1001.4538]. 69

6 CMB bispectrum induced by the two scalars and a graviton correlator

In this section, adapting Eq. (5.5) to the primordial non-Gaussianity in two scalars and a graviton correlation [1], we compute the CMB scalar-scalar-tensor bispectrum. This discussion is based on our paper [2].

6.1 Two scalars and a graviton interaction during inflation We consider a general single-field inflation model with Einstein-Hilbelt action [3] : Z M 2 S = d4x√ g pl R + p(φ, X) , (6.1) − 2

2 where g is the determinant of the metric, R is the Ricci scalar, Mpl 1/(8πG), φ is a scalar ﬁeld, µν ≡ and X g ∂µφ∂νφ/2. Using the background equations, the slow-roll parameter and the sound speed for≡ −perturbations are given by ˙ H Xp,X 2 p,X 2 = 2 2 , cs , (6.2) ≡ −H H Mpl ≡ 2Xp,XX + p,X where H is the Hubble parameter, the dot means a derivative with respect to the physical time t and p,X denotes partial derivative of p with respect to X. We write a metric by ADM formalism

ds2 = N 2dt2 + a2eγab (dxa + N adt)(dxb + N bdt) , (6.3) − a where N and N are respectively the lapse function and shift vector, γab is a transverse and γab traceless tensor as γaa = ∂aγab = 0, and e δab + γab + γacγcb/2 + . On the flat hypersurface, the gauge-invariant curvature perturbation ζ≡is related to the first-order··· fluctuation of the scalar field ϕ as ζ = Hϕ/φ˙ 1. Following the conversion equations (D.15) and (D.22), we decompose ζ − and γab into the helicity states as 1 ξ(0)(k) = ζ(k) , ξ(±2)(k) = e(∓2)(kˆ)γ (k) . (6.4) 2 ab ab (±2) Here, eab is a transverse and traceless polarization tensor explained in Appendix D. The interaction parts of this action have been derived by Maldacena [1] up to the third-order terms. In particular, we will focus on an interaction between two scalars and a graviton. This is because the correlation between a small wave number of the tensor mode and large wave numbers of the scalar modes will remain despite the tensor mode decays after the mode reenters the cosmic horizon. We find a leading term of the two scalars and a graviton interaction in the action coming from the matter part of the Lagrangian through X as

−2 p,X X 3rd−order a γab∂aϕ∂bϕ , (6.5) | ⊃ 2 therefore, the interaction part is given by Z 4 Sint d x agtssγab∂aζ∂bζ . (6.6) ⊃ 1Here, ζ and γ are equivalent to and h in Eq. (2.13), respectively. ab R ab 6 CMB BISPECTRUM INDUCED BY THE TWO SCALARS AND A GRAVITON 70 CORRELATOR

Here, we introduce a coupling constant gtss. From the definition of ζ, γab and the slow-roll parameter, gtss = . For a general consideration, let us deal with gtss as a free parameter. In this sense, constraining this parameter may offer a probe of the nature of inflation and gravity in the early Universe. The primordial bispectrum is then computed using in-in formalism in the next subsection.

6.2 Calculation of the initial bispectrum In the same manner as discussed in Ref. [1], we calculate the primordial bispectrum generated from two scalars and a graviton in the lowest order of the slow-roll parameter:

3 ! (±2) (0) (0) 3 ±200 X ξ (k1)ξ (k2)ξ (k3) = (2π) F (k1, k2, k3)δ kn , n=1 4g I(k , k , k )k k H4 F ±200(k , k , k ) tss 1 2 3 2 3 ∗ e(∓2)(kˆ )kˆ kˆ , (6.7) 1 2 3 Q 3 2 2 4 ab 1 2a 3b ≡ i(2ki ) 2cs∗∗Mpl k1k2 + k2k3 + k3k1 k1k2k3 I(k1, k2, k3) kt + + 2 , ≡ − kt kt where kt k1 + k2 + k3, and means that it is evaluated at the time of horizon crossing, i.e., ≡ ∗ (∓2) ˆ ˆ ˆ a∗H∗ = k. Here, we keep the angular and polarization dependences, eab (k1)k2ak3b, which have sometimes been omitted in the literature for simplicity [4–6]. We show, however, that expanding this term with spin-weighted spherical harmonics enables us to formulate the rotational-invariant bispectrum in an explicit way. The statistically isotropic power spectra of ξ(0) and ξ(±2) are respectively given by

(0) (0)∗ 0 3 0 ξ (k)ξ (k ) (2π) PS(k)δ(k k ) , 3 ≡ 2 − k PS(k) H∗ 2 = 2 2 AS , 2π 8π cs∗∗Mpl ≡ (6.8) D (λ) (λ0)∗ 0 E 3 PT (k) 0 0 ξ (k)ξ (k ) (2π) δ(k k )δλ,λ0 (for λ, λ = 2) , ≡ 2 − ± 3 2 k PT (k) H∗ r 2 = 2 2 = 8cs∗∗AS AS , 2π π Mpl ≡ 2 where r is the tensor-to-scalar ratio and AS is the amplitude of primordial curvature perturbations 2. Note that the power spectra satisfy the scale invariance because we consider them in the lowest order of the slow-roll parameter. Using these equations, we parametrize the initial bispectrum in this case from Eq. (6.7) as

±200 (TSS) (∓2) ˆ ˆ ˆ F (k1, k2, k3) = f (k1, k2, k3)eab (k1)k2ak3b , (6.9) 4 2 (TSS) 16π ASgtss I(k1, k2, k3) kt f (k1, k2, k3) 2 2 2 . (6.10) ≡ k1k2k3 kt k1 Note that f (TSS) seems not to depend on the tensor-to-scalar ratio. In Fig. 6.1 , we show the shape (TSS) of I/k1. From this, we conﬁrm that the initial bispectrum f (6.10) dominates in the squeezed limit as k1 k2 k3 like the local-type bispectrum of scalar modes given by Eq. (4.7). ' 2 For cs∗ = 1, these results are identical to Eqs. (2.30) and (2.31). 6.3 Formulation of the CMB bispectrum 71

1.0

k2k3 0.8

0.6

ÈIk1È 10

0 0.0 0.5

k1k3 1.0

Figure 6.1: Shape of I/k1. For the symmetric property and the triangle condition, we limit the plot range as k1 k2 k3 and k1 k2 k3 k1 + k2. ≤ ≤ | − | ≤ ≤

In the squeezed limit, the ratio of f (TSS) to the scalar-scalar-scalar counterpart f (SSS) = 6 5 fNLPS(k1)PS(k2), which has been considered frequently, reads

(TSS) f 10gtss I ktk2 20gtss I (SSS) = 2 . (6.11) f 3fNL kt k3 → 3fNL kt

In the standard slow-roll inflation model, this ratio becomes (1) and does not depend on the O tensor-to-scalar ratio because gtss and fNL are proportional to the slow-roll parameter , and I/kt has a nearly flat shape. The average of amplitude is evaluated as I/kt 0.6537. Therefore, it manifests the comparable importance of the higher order correlations of tensor≈ − modes to the scalar ones in the standard inflation scenario.

6.3 Formulation of the CMB bispectrum Here, using Eqs. (5.5), (6.7) and (6.9), we explicitly calculate the CMB tensor-scalar-scalar bispectrum as

" 3 # D E Z ∞ 2 (T ) (S) (S) Y `n kndkn (Zn) a a a = 4π( i) (kn) X1,`1m1 X2,`2m2 X3,`3m3 (2π)3 Xn,`n n=1 − 0 T x1 " 3 Z # X λ1 Y 2 ˆ ∗ ˆ ∗ ˆ ∗ ˆ d kn −λ Y (k1)Y (k2)Y (k3) × 2 1 `1m1 `2m2 `3m3 λ1=±2 n=1 3 ! 3 (TSS) (∓2) ˆ ˆ ˆ Y (2π) f (k1, k2, k3)eab (k1)k2ak3bδ kn . (6.12) × n=1

At ﬁrst, we express all parts containing the angular dependence with the spin spherical har- 6 CMB BISPECTRUM INDUCED BY THE TWO SCALARS AND A GRAVITON 72 CORRELATOR monics3: 3/2 (∓2) ˆ ˆ ˆ (8π) X ∗ ˆ ∗ ˆ ∗ ˆ 2 1 1 eab (k1)k2ak3b = ±2Y2M (k1)Y1ma (k2)Y1m (k3) , (6.13) 6 b M ma mb Mmamb 3 ! " 3 # X Z ∞ Y X δ k = 8 y2dy ( 1)Li/2j (k y)Y ∗ (kˆ ) i Li i LiMi i 0 − i=1 i=1 LiMi 0 0 0 L1 L2 L3 IL1L2L3 , (6.14) × M1 M2 M3 where we used the relations listed in Appendices C and D and r s1s2s3 (2l1 + 1)(2l2 + 1)(2l3 + 1) l1 l2 l3 Il l l . (6.15) 1 2 3 ≡ 4π s1 s2 s3 Secondly, using Eq. (C.8), we replace all the integrals of spin spherical harmonics with the Wigner symbols: Z 2 ˆ ∗ ˆ ∗ ˆ ∗ ˆ ±20∓2 `1 L1 2 d k1 ∓2Y`1m1 (k1)YL1M1 (k1)±2Y2M (k1) = I` L 2 , 1 1 m1 M1 M Z 2 ˆ ∗ ˆ ∗ ˆ ∗ ˆ 0 0 0 `2 L2 1 d k2 Y`2m2 (k2)YL2M2 (k2)Y1ma (k2) = I`2L21 , (6.16) m2 M2 ma Z 2 ˆ ∗ ˆ ∗ ˆ ∗ ˆ 0 0 0 `3 L3 1 d k3 Y`3m3 (k3)YL3M3 (k3)Y1m (k3) = I`3L31 . b m3 M3 mb Thirdly, using the summation formula of ﬁve Wigner-3j symbols as Eq. (C.20), we sum up the Wigner-3j symbols with respect to azimuthal quantum numbers in the above equations and express with the Wigner-9j symbol as X L1 L2 L3 2 1 1 M1 M2 M3 M ma mb M1M2M3 Mmamb `1 L1 2 `2 L2 1 `3 L3 1 × m1 M1 M m2 M2 ma m3 M3 mb  ` ` `  ` ` `  1 2 3  = 1 2 3 L L L . (6.17) m m m 1 2 3 1 2 3  2 1 1 

After these treatments and the summation over λ1 = 2 as ± x1 ( 20−2 X λ1 2I (for x1 + L1 + `1 = even) , Iλ10−λ1 = `1L12 (6.18) `1L12 2 0 (for x1 + L1 + `1 = odd) , λ1=±2 we can obtain the CMB angle-averaged bispectrum induced from the nonlinear coupling between two scalars and a graviton as   3/2 `1 `2 `3 (TSS) (8π) X L1+L2+L3 0 0 0 20−2 0 0 0 0 0 0   B = ( 1) 2 I I I I L1 L2 L3 X1X2X3,`1`2`3 3 − L1L2L3 `1L12 `2L21 `3L31 L1L2L3  2 1 1 

3Equations (3.14) and (3.21) in Ref. [2] include typos. 6.3 Formulation of the CMB bispectrum 73

Z ∞ " 3 Z ∞ # 2 Y 2 `n 2 (Zn) (TSS) y dy ( i) k dkn jL (kny) f (k1, k2, k3)(6.19). π n Xn,`n n × 0 n=1 − 0 T

Note that the absence of the summation over m1, m2 and m3 in this equation means that the tensor-scalar-scalar bispectrum maintains the rotational invariance. As described above, this con- sequence is derived from the angular dependence in the polarization tensor. Also in vector modes, if their power spectra obey the statistical isotropy like Eq. (6.8), one can obtain the rotational invariant bispectrum by considering the angular dependence in the polarization vector as Eq. (D.11). Considering Eq. (6.18) and the selection rules of the Wigner symbols explained in Appendix C, we can see that the values of L1,L2 and L3 are limited as ( `1 2 , `1 (for X1 = I,E) L1 = | ± | ,L2 = `2 1 ,L3 = `3 1 , `1 1 (for X1 = B) | ± | | ± | | ± | (6.20) 3 X L1 L2 L3 L1 + L2 , Li = even , | − | ≤ ≤ i=1 and the bispectrum (6.19) has nonzero value under the conditions:

3 ( X even (for X1 = I,E) , `1 `2 `3 `1 + `2 , `i = (6.21) | − | ≤ ≤ i=1 odd (for X1 = B) . In Figs. 6.2 and 6.3, we describe the reduced CMB bispectra of intensity mode sourced from two scalars and a graviton coupling: b(TSS) + b(STS) + b(SST ) = I0 0 0 −1 B(TSS) + B(STS) + B(SST ) , (6.22) III,`1`2`3 III,`1`2`3 III,`1`2`3 `1`2`3 III,`1`2`3 III,`1`2`3 III,`1`2`3 and primordial curvature perturbations (5.7):

b(SSS) = I0 0 0 −1 B(SSS) . (6.23) III,`1`2`3 `1`2`3 III,`1`2`3 For the numerical computation, we modify the Boltzmann Code for Anisotropies in the Microwave Background (CAMB) [7, 8] 4. In the calculation of the Wigner-3j and 9j symbols, we use the Com- mon Mathematical Library SLATEC [9] and the summation formula of three Wigner-6j symbols (C.21). As the radiation transfer functions of scalar and tensor modes, namely, (S) and (T ) , we TXi,ì TXi,ì use Eq. (3.101). From the behavior of each line shown in Fig. 6.3 at small `3 that the reduced CMB bispectrum is roughly proportional to `−2, we can confirm that the tensor-scalar-scalar bispectrum has a nearly squeezed-type configuration corresponding to the shape of the initial bispectrum as discussed above. From Fig. 6.2, by comparing the green dashed line with the red solid line roughly estimated as

4The CMB bispectra generated from the two scalars and a graviton correlator in Figs. 6.2 and 6.3 become slightly smaller than those in Ref. [2] due to the accuracy enhancement of the numerical calculation. 6 CMB BISPECTRUM INDUCED BY THE TWO SCALARS AND A GRAVITON 74 CORRELATOR

0.1 16 10

× 0.01 2 ) π

|/(2 0.001 lll |b 2 0.0001 (l+1) 2 l 1e-05 TSS + STS + SST SSS 1e-06 10 100 1000 l

Figure 6.2: Absolute values of the CMB reduced bispectra of temperature ﬂuctuation for `1 = `2 = `3 `. The lines correspond to the spectra generated from tensor-scalar-scalar correlation ≡ given by Eq. (6.22) with gtss = 5 (red solid line) and the primordial non-Gaussianity in the scalar local curvature perturbations with fNL = 5 (green dashed line). The other cosmological parameters are ﬁxed to the mean values limited from WMAP-7yr data reported in Ref. [10].

6.4 Estimation of the signal-to-noise ratio Here, we compute the signal-to-noise ratio by comparing the intensity bispectrum of Eq. (6.19) with the zero-noise (ideal) data and examine the bound on the absolute value of gtss. The formulation of (the square of) the signal-to-noise ratio (S/N) is reported in Refs. [11] and [12]. In our case, it can be expressed as

2 (TSS) (STS) (SST ) 2 B + B + B S X III`1`2`3 III`1`2`3 III`1`2`3 = 2 , (6.25) N σ` ` ` 2≤`1≤`2≤`3≤` 1 2 3

where σ`1`2`3 denotes the variance of the bispectrum. Assuming the weakly non-Gaussianity, the variance can be estimated as [13, 14]

2 σ C` C` C` ∆` ` ` , (6.26) `1`2`3 ≈ 1 2 3 1 2 3 where ∆` ` ` takes 1, 6 or 2 for `1 = `2 = `3, `1 = `2 = `3, or the case that two `’s are the same, 1 2 3 6 6 respectively. C` denotes that the CMB angular power spectrum included the noise spectrum, which is neglected in our case. In Fig. 6.4, the numerical result of Eq. (6.25) is presented. We ﬁnd that (S/N) is a monotonically increasing function roughly proportional to ` for ` < 2000. It is compared with the order estimation of Eq. (6.25) as Ref. [12]

r 3 r 3 3 (TSS) (STS) (SST ) S ` (2`) ` ` ` ` bIII``` + bIII``` + bIII``` | 2 3/2 | N ∼ 24 × 4π 0 0 0 (` C`) 6.5 Summary and discussion 75

16 0.1 10 × 2 ) π 0.01 |/(2 3 l 2 l 1 l 0.001 +1)|b 3 (l

3 0.0001

TSS + STS + SST: l1 = l2 = 220 +1)l 1 1004

(l 1e-05 1 l SSS: l1 = l2 = 220 1004 1e-06 10 100 1000

Figure 6.3: Absolute values of the CMB reduced bispectra of temperature fluctuation generated from tensor-scalar-scalar correlation given by Eq. (6.22) (TSS + STS + SST ) and the primordial non-Gaussianity in the scalar curvature perturbations (SSS) as a function of `3 with `1 and `2 fixed to some values as indicated. The parameters are fixed to the same values defined in Fig. 6.2.

−5 ` 5.4 10 gtss . (6.27) ∼ × × | | ` ` ` 2 Here, we use Eq. (6.24) and the approximations as P `3/24, `3 0.36 `, and ∼ 0 0 0 ∼ × 2 −10 ` C` 6 10 . We confirm that this is consistent with Fig. 6.4, which justifies our numerical calculation∼ × in some sense. This figure shows that from the WMAP and PLANCK experimental data [10, 15], which are roughly noise-free at ` . 500 and 1000, respectively, expected (S/N)/gtss values are 0.05 and 0.11. Hence, to obtain (S/N) > 1, we need gtss > 20 and 9. The latter value | | is close to a naive estimate gtss 5, which was discussed at the end of the previous subsection. | | . 6.5 Summary and discussion In this section, we present a full-sky formalism of the CMB bispectrum sourced from the primordial non-Gaussianity not only in the scalar but also in the vector and tensor perturbations. As an extension of the previous formalism discussed in Ref. [5], the new formalism contains the contribution of the polarization vector and tensor in the initial bispectrum. In Ref. [5], we have shown that in the all-sky analysis, the CMB bispectrum of vector or tensor mode cannot be formed as a simple angle-averaged bispectrum in the same way as that of scalar mode. This is because the angular integrals over the wave number vectors have complexities for the non-orthogonality of spin spherical harmonics whose spin values differ from each other if one neglects the angular dependence of the polarization vector or tensor. In this study, however, we find that this difficulty vanishes if we maintain the angular dependence in the initial bispectrum. To present how to use our formalism, we compute the CMB bispectrum induced by the nonlinear mode-coupling between the two scalars and a graviton [1]. The typical value of the reduced bispectrum in temperature fluctuations is calculated as a function of the coupling constant between 76 REFERENCES

0.1

0.01 tss

S/N/g 0.001

0.0001

1e-05 10 100 1000 l

Figure 6.4: Signal-to-noise ratio normalized by gtss as a function of the maximum value between `1, `2 and `3, namely, `. Each parameter is ﬁxed to the same values deﬁned in Fig. 6.2.

(TSS) (STS) (SST ) −4 −18 scalars and gravitons gtss: bIII,```+bIII,```+bIII,``` ` 8 10 gtss . Through the computation of the signal-to-noise ratio,| we find that the two| scalars ∼ × and× a graviton| | coupling can be detected by the WMAP and PLANCK experiment if gtss (10). Although we do not include the effect | | ∼ O of the polarization modes in the estimation of gtss in this study, they will provide more beneficial information of the nonlinear nature of the early Universe.

References

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[15] PLANCK Collaboration, The Scientiﬁc programme of planck, astro-ph/0604069. 78 7 VIOLATION OF THE ROTATIONAL INVARIANCE IN THE CMB BISPECTRUM

7 Violation of the rotational invariance in the CMB bispectrum

The current cosmological observations, particularly Cosmic Microwave Background (CMB), tell us that the Universe is almost isotropic, and primordial density fluctuations are almost Gaussian random fields. However, in keeping with the progress of the experiments, there have been many works that verify the possibility of the small deviation of the statistical isotropy, e.g., the so-called “Axis of Evil”. The analyses of the power spectrum by employing the current CMB data suggest that the deviation of the statistical isotropy is about 10% at most (e.g. [1–6]). Toward more precise measurements in future experiments, there are a lot of theoretical discussions about the effects of the statistical anisotropy on the CMB power spectrum, [7–11], e.g., the presence of the off-diagonal configuration of the multipoles in the CMB power spectrum, which vanishes in the isotropic spectrum. As is well known, it might be difficult to explain such statistical anisotropy in the standard inflationary scenario. However, recently, there have been several works about the possibility of generating the statistically anisotropic primordial density fluctuations in order to introduce non- trivial dynamics of the vector field. [12–24]. In Ref. [14], the authors considered a modified hybrid inflation model where a waterfall field couples not only with an inflaton field but also with a mass- less vector field. They have shown that, owing to the effect of fluctuations of the vector field, the primordial density fluctuations may have a small deviation from the statistical isotropy and also the deviation from the Gaussian statistics. If the primordial density fluctuations deviate from the Gaussian statistics, they produces the non-zero higher order spectra (corresponding to higher order correlation functions), e.g., the bispectrum (3-point function), the trispectrum (4-point function) and so on. Hence, in the model presented in Ref. [14], we can expect that there are characteristic signals not only in the CMB power spectrum but also in the CMB bispectrum. With these motivations, in this work, we calculate the CMB statistically anisotropic bispectrum sourced from the curvature perturbations generated in the modified hybrid inflation scenario proposed in Ref. [14], on the basis of the useful formula presented in Ref. [25]. Then, we find the peculiar configurations of the multipoles which never appear in the isotropic bispectrum, like off-diagonal components in the CMB power spectrum. These discussions are based on Ref. [26]. This section is organized as follows. In the next subsection, we briefly review the inflation model where the scalar waterfall field couples with the vector field and calculate the bispectrum of curvature perturbations based on Ref. [14]. In Sec. 7.2, we give an exact form of the CMB statistically anisotropic bispectrum and analyze its behavior by numerical computation. Finally, we devote the final subsection to the summary and discussion. Throughout this section, we obey the definition of the Fourier transformation as

Z d3k f(x) f˜(k)eik·x , (7.1) ≡ (2π)3

−1/2 and a normalization as Mpl (8πG) = 1. ≡ 7.1 Statistically-anisotropic non-Gaussianity in curvature perturbations In this subsection, we briefly review the mechanism of generating the statistically anisotropic bispectrum induced by primordial curvature perturbations proposed in Ref. [14], where the authors 7.1 Statistically-anisotropic non-Gaussianity in curvature perturbations 79 set the system like the hybrid inflation wherein there are two scalar fields: inflaton φ and waterfall field χ, and a vector field Aµ coupled with a waterfall field. The action is given by Z 4 1 1 µν 1 µν ρσ 2 S = dx √ g R g (∂µφ∂νφ + ∂µχ∂νχ) V (φ, χ, Aν) g g f (φ)FµρFνσ . (7.2) − 2 − 2 − − 4

Here, Fµν ∂µAν ∂νAµ is the field strength of the vector field Aµ, V (φ, χ, Aµ) is the potential of fields and ≡f(φ) denotes− a gauge coupling. To guarantee the isotropy of the background Universe, we need the condition that the energy density of the vector field is negligible in the total energy of the Universe and we assume a small expectation value of the vector field. Therefore, we neglect the effect of the vector field on the background dynamics and also the evolution of the fluctuations of the inflaton. In the standard hybrid inflation (only with the inflaton and the waterfall field), inflation suddenly ends owing to the tachyonic instability of the waterfall field, which is triggered when the inflaton reaches a critical value φe. In the system described using Eq. (7.2), however, φe may fluctuate owing to the fluctuation of the vector field and it generates additional curvature perturbations. Using the δN formalism [27–32], the total curvature perturbation on the uniform-energy-density hypersurface at the end of inflation t = te can be estimated in terms of the perturbation of the e-folding number as

ζ(te) = δN(te, t∗) 2 ∂N 1 ∂ N 2 = δφ∗ + 2 δφ∗ ∂φ∗ 2 ∂φ∗ 2 2 ∂N dφe(A) µ 1 ∂N d φe(A) ∂ N dφe(A) dφe(A) µ ν + µ δAe + µ ν + 2 µ ν δAe δAe . (7.3) ∂φe dA 2 ∂φe dA dA ∂φe dA dA

Here, t∗ is the time when the scale of interest crosses the horizon during the slow-roll inflation. As- suming the sudden decay of all fields into radiations just after inflation, the curvature perturbations on the uniform-energy-density hypersurface become constant after inflation ends 1. Hence, at the leading order, the power spectrum and the bispectrum of curvature perturbations are respectively derived as * 2 + 2 ! Y 3 2 X ζ(kn) = (2π) N∗ Pφ(k1)δ kn n=1 n=1

2 dφe(A) dφe(A) µ ν +N δA (k1)δA (k2) , (7.4) e dAµ dAν h e e i * 3 + 3 ! Y 3 2 X ζ(kn) = (2π) N∗ N∗∗[Pφ(k1)Pφ(k2) + 2 perms.]δ kn n=1 n=1

3 dφe(A) dφe(A) dφe(A) µ ν ρ +N δA (k1)δA (k2)δA (k3) e dAµ dAν dAρ h e e e i 2 4 dφe(A) dφe(A) 1 d φe(A) Nee dφe(A) dφe(A) +Ne µ ν ρ σ + 2 ρ σ dA dA Ne dA dA Ne dA dA µ ν ρ σ [ δA (k1)δA (k2)(δA ? δA )e(k3) + 2 perms.] , (7.5) × h e e i 2 3 where Pφ(k) = H∗ /(2k ) is the power spectrum of the fluctuations of the inflaton, N∗ ∂N/∂φ∗, 2 2 2 2 ≡ N∗∗ ∂ N/∂φ∗, Ne ∂N/∂φe, Nee ∂ N/∂φe, and ? denotes the convolution. Here, we ≡ ≡ ≡ µ assume that δφ∗ is a Gaussian random field and δφA = 0. h i 1This ζ is consistent with in Eq. (2.13). R 80 7 VIOLATION OF THE ROTATIONAL INVARIANCE IN THE CMB BISPECTRUM

For simplicity, we estimate the fluctuation of the vector fields in the Coulomb gauge: δA0 = 0 i and kiA = 0. Then, the evolution equation of the fluctuations of the vector field is given by ¨ ¨ f 2 j i i a ∂j∂ i = 0 , (7.6) A − f A − A where i fδAi, ˙ denotes the derivative with respect to the conformal time, and we neglect the contributionA ≡ from the potential term. When f a, a−2 with appropriate quantization of the fluctuations of the vector field, we have the scale-invariant∝ power spectrum of δAi on superhorizon scale as [14, 18, 33]

2 ! i j 3 −2 ij ˆ X δAe(k1)δAe(k2) = (2π) Pφ(k)fe P (k1)δ kn , (7.7) n=1

ij ˆ ij ˆiˆj where a is the scale factor, P (k) = δ k k , ˆ denotes the unit vector, and fe f(te). Therefore, substituting this expression into Eq. (7.4),− we can rewrite the power spectrum≡ of the primordial curvature perturbations, ζ, as

* 2 + 2 ! Y 3 X ζ(kn) (2π) Pζ (k1)δ kn , ≡ n=1 n=1 (7.8) " 2 # 2 Ne i j ˆ Pζ (k) = Pφ(k) N∗ + q q Pij(k) , fe

i 2 i j where qi dφe/dA , qij d φe/(dA dA ). We can divide this expression into the isotropic part and the anisotropic≡ part≡ as [7]

2 iso ˆ Pζ (k) P (k) 1 + gβ qˆ k , (7.9) ≡ ζ · with

iso 2 β P (k) = N Pφ(k)(1 + β) , gβ = , (7.10) ζ ∗ −1 + β

2 2 where β = (Ne/N∗/fe) q . The bispectrum of the primordial curvature perturbation given by Eq. (7.5) can be written| as|

* 3 + 3 ! Y 3 X ζ(kn) (2π) Fζ (k1, k2, k3)δ kn , (7.11) n=1 ≡ n=1 g 2 F (k , k , k ) = β P iso(k )P iso(k ) ζ 1 2 3 β ζ 1 ζ 2 N∗∗ 2 a b 1 cd Nee c d ˆ ˆ 2 + β qˆ qˆ qˆ + 2 qˆ qˆ Pac(k1)Pbd(k2) × N∗ Ne Ne +2 perms.. (7.12)

Hereq ˆcd qcd/ q 2 and we have assumed that the ﬂuctuation of the vector ﬁeld δAi almost obeys ≡ | | µ ν ρ Gaussian statistics; hence δA (k1)δA (k2)δA (k3) = 0. h e e e i 7.2 CMB statistically-anisotropic bispectrum 81

Hereinafter, for calculating the CMB bispectrum explicitly, we adopt a simple model whose potential looks like an Abelian Higgs model in the unitary gauge as [14]

i λ 2 2 2 1 2 2 2 1 2 2 1 2 µ 2 V (φ, χ, A ) = (χ v ) + g φ χ + m φ + h A Aµχ , (7.13) 4 − 2 2 2 where λ, g, and h are the coupling constants, m is the inflaton mass, and v is the vacuum expectation value of χ. Since the effective mass squared of the waterfall field is given by

2 2 ∂ V 2 2 2 2 i m = λv + g φ + h A Ai = 0 , (7.14) χ ≡ ∂χ2 − e and the critical value of the inﬂaton φe can be obtained as

2 2 2 2 i g φ = λv h A Ai , (7.15) e − we can express β, qi and qij in Eq. (7.12) in terms of the model parameters as

" 2 # 2 2 i ˆi ij 1 gφe ij ˆi ˆj 1 h A qˆ = A , qˆ = δ + A A , β 2 2 , (7.16) − −φe hA ' fe g φe

2 where we have used N∗ Ne 1/√2 with (∂V/∂φ/V ) /2 being a slow-roll parameter and A A. Substituting' these − quantities' into Eq.≡ (7.12), the bispectrum of primordial curvature perturbations| | ≡ is obtained as

iso iso â ˆb cd ˆ ˆ Fζ (k1, k2, k3) = CPζ (k1)Pζ (k2)A A δ Pac(k1)Pbd(k2) + 2 perms., 2 (7.17) 2 φe g C gβ . ≡ − Ne hA Note that in the above expression, we have neglected the effect of the longitudinal polarization in the vector field for simplicity 2 and the terms that are suppressed by a slow-roll parameter η 2 2 2 2 −1 ≡ ∂ V/∂φ /V because N∗∗/N∗ Nee/Ne (Neφe) η. Since the current CMB observations − ' ' −−1 ' suggest gβ < (0.1) (e.g., Refs. [1, 2]) and N √2, the overall amplitude of the bispectrum O e ' − in this model, C, does not seem to be sufficiently large to be detected. However, even if gβ 1 2 and 1, C can become greater than unity for (g/h/A) φe 1. Thus, we expect meaningful signals also in the CMB bispectrum. Then, in the next subsection, we closely investigate the CMB bispectrum generated from the primordial bispectrum given by Eq. (7.17) and discuss a new characteristic feature of the CMB bispectrum induced by the statistical anisotropy of the primordial bispectrum.

7.2 CMB statistically-anisotropic bispectrum In this subsection, we give a formula of the CMB bispectrum generated from the primordial bispectrum, which has statistical anisotropy owing to the ﬂuctuations of the vector ﬁeld, given by Eq. (7.17). We also discuss the special signals of this CMB bispectrum, which vanish in the statistically isotropic bispectrum.

2Owing to this treatment, we can use the quantities estimated in the Coulomb gauge as Eq. (7.12). In a more precise discussion, we should take into account the contribution of the longitudinal mode in the unitary gauge. 82 7 VIOLATION OF THE ROTATIONAL INVARIANCE IN THE CMB BISPECTRUM

7.2.1 Formulation The CMB ﬂuctuation can be expanded in terms of the spherical harmonic function as

∆X(nˆ) X = a Y (nˆ) , (7.18) X X,`m `m `m where nˆ is a unit vector pointing toward a line-of-sight direction, and X denotes the intensity ( I) and polarizations ( E,B). According to Eq. (5.3), the coeﬃcient, a`m, generated from primordial≡ curvature perturbations,≡ ζ, is expressed as

Z ∞ 2 ` k dk aX,`m = 4π( i) 3 ζ`m(k) X,`(k) (for X = I,E) , (7.19) − 0 (2π) T Z 2 ˆ ∗ ˆ ζ`m(k) d kζ(k)Y (k) , (7.20) ≡ `m where X,` is the time-integrated transfer function of scalar modes as described in Eq. (3.101). Using theseT equations, the CMB bispectrum generated from the bispectrum of the primordial curvature perturbations is given by

* 3 + " 3 Z ∞ 2 #* 3 + Y Y k dkn Y a = 4π( i)`n n (k ) ζ (k ) , (7.21) Xn,`nmn (2π)3 Xn,`n n `nmn n n=1 n=1 − 0 T n=1 with * 3 + " 3 Z # 3 ! Y Y 2 ˆ ∗ ˆ 3 X ζ`nmn (kn) = d knY`nmn (kn) (2π) δ kn Fζ (k1, k2, k3) . (7.22) n=1 n=1 n=1

We expand the angular dependencies which appear in the Dirac delta function, δ(k1 + k2 + k3), and the function, Fζ (k1, k2, k3), given by Eq. (7.17) with respect to the spin spherical harmonics as

3 ! Z ∞ " 3 # X 2 Y X Ln/2 ∗ ˆ δ kn = 8 y dy ( 1) jLn (kny)YLnMn (kn) 0 − n=1 n=1 LnMn 0 0 0 L1 L2 L3 IL1L2L3 , (7.23) × M1 M2 M3 3 0 ˆa ˆb cd ˆ ˆ 4π X 01−1 01−1 000 LL LA A A δ Pac(k1)Pbd(k2) = 4 I I 0 I L11 L 11 11LA 1 1 1 − 3 0 L,L ,LA=0,2 0 X ∗ ˆ ∗ ˆ ∗ ˆ LL LA Y (k1)Y 0 0 (k2)Y (A) ,(7.24) LM L M LAMA MM 0 M × 0 A MM MA where the 2 3 matrices of a bracket and a curly bracket denote the Wigner-3j and 6j symbols, respectively,× and

r s1s2s3 (2l1 + 1)(2l2 + 1)(2l3 + 1) l1 l2 l3 Il l l . (7.25) 1 2 3 ≡ 4π s1 s2 s3 7.2 CMB statistically-anisotropic bispectrum 83

Here, we have used the expressions of an arbitrary unit vector and a projection tensor as Appendices ∗ ˆ C and D. Note that for Y00(A) = 1/√4π, the contribution of LA = 0 in Eq. (7.24) is independent of the direction of the vector ﬁeld. Therefore, the statistical anisotropy is generated from the signals of LA = 2. By integrating these spherical harmonics over each unit vector, the angular dependences on k1, k2, k3 can be reduced to the Wigner-3j symbols as Z 2 ˆ ∗ ∗ ∗ 0 0 0 `1 L1 L d k1Y`1m1 YL1M1 YLM = I`1L1L , m1 M1 M Z 0 2 ∗ ∗ ∗ 0 0 0 `2 L2 L ˆ 0 0 0 (7.26) d k2Y`2m2 YL2M2 YL M = I`2L2L 0 , m2 M2 M Z 2 ˆ ∗ ∗ m3 d k3Y Y = ( 1) δL ,` δM ,−m . `3m3 L3M3 − 3 3 3 3

From these equations, we obtain an alternative explicit form of the bispectrum of ζ`m as

* 3 + Z ∞ Y X L1+L2+`3 3 2 2 0 0 0 ζ`nmn (kn) = (2π) 8 y dy ( 1) IL1L2`3 − 0 − n=1 L1L2 iso iso P (k1)jL (k1y)P (k2)jL (k2y)Cj` (k3y) × ζ 1 ζ 2 3 4π 3 m3 X 01−1 01−1 4 ( 1) IL11 IL011 × 3 − 0 L,L ,LA=0,2 0 0 0 0 0 0 0 000 LL LA I I 0 I × `1L1L `2L2L 11LA 1 1 1 X L L ` Y ∗ (Aˆ ) 1 2 3 LAMA M M m × 0 1 2 3 M1M2MM MA − 0 0 `1 L1 L `2 L2 L LL LA 0 0 × m1 M1 M m2 M2 M MM MA +2 perms.. (7.27)

This equation implies that, owing to the vector ﬁeld A, the CMB bispectrum has a direction dependence, and hence, the dependence on m1, m2, m3 cannot be conﬁned only to a Wigner-3j symbol, namely, * 3 + Y 3 `1 `2 `3 ζ`nmn (kn) = (2π) `1`2`3 (k1, k2, k3) . (7.28) m1 m2 m3 n=1 6 F This fact truly indicates the violation of the rotational invariance in the bispectrum of the primordial curvature perturbations and leads to the statistical anisotropy on the CMB bispectrum. Let us consider the explicit form of the CMB bispectrum. Here, we set the coordinate as ˆ ∗ A = zˆ. Then, by substituting Eq. (7.27) into Eq. (7.21) and using the relation YL M (zˆ) = p A A (2LA + 1)/(4π)δMA,0, the CMB bispectrum is expressed as

* 3 + " 3 # Y Z ∞ Y 2 Z ∞ a = y2dy k2dk (k ) Xn,`nmn π n n Xn,`n n n=1 − 0 n=1 0 T X `1+`2+L1+L2 +` 0 0 0 ( 1) 2 3 I × − L1L2`3 L1L2 84 7 VIOLATION OF THE ROTATIONAL INVARIANCE IN THE CMB BISPECTRUM

iso iso P (k1)jL (k1y)P (k2)jL (k2y)Cj` (k3y) × ζ 1 ζ 2 3 4π 3 m3 X 01−1 01−1 4 ( 1) IL11 IL011 × 3 − 0 L,L ,LA=0,2 0 0 0 0 0 0 0 000 LL LA I I 0 I × `1L1L `2L2L 11LA 1 1 1 r 2 2LA + 1 X L1 L2 `3 × 4π m1 M m2 + M m3 M=−2 − − − − 0 `1 L1 L `2 L2 L × m1 m1 MM m2 m2 + M M − − − − LL0 L A + 2 perms.. (7.29) × M M 0 −

By the selection rules of the Wigner symbols described in Appendix C, the ranges of `1, `2, `3, m1, m2 and m3, and the summation ranges in terms of L1 and L2 are limited as

3 3 X X `n = even , mn = 0 , n=1 n=1 (7.30) L1 = `1 2 , `1, `1 + 2 ,L2 = `2 2 , `2, `2 + 2 , | − | | − | L2 `3 L1 L2 + `3 . | − | ≤ ≤ 7.2.2 Behavior of the CMB statistically-anisotropic bispectrum On the basis of Eq. (7.29), we compute the CMB bispectra for the several `’s and m’s. Then, we modify the Boltzmann Code for Anisotropies in the Microwave Background (CAMB) [34, 35] and use the Common Mathematical Library SLATEC [36]. In Fig. 7.1, the red solid lines are the CMB statistically anisotropic bispectra of the intensity mode given by Eq. (7.29) with C = 1, and the green dashed lines are the statistically isotropic one sourced from the local-type non-Gaussianity of curvature perturbations given by Eq. (4.7)

* 3 + Z ∞ " 3 Z ∞ # Y 0 0 0 `1 `2 `3 2 Y 2 2 aXn,`nmn = I`1`2`3 y dy kndkn Xn,`n (kn)j`n (kny) m1 m2 m3 π n=1 0 n=1 0 T iso iso 6 P (k1)P (k2) fNL + 2 perms. , (7.31) × ζ ζ 5 with fNL = 5 for `1 = `2 = `3 and two sets of m1, m2, m3. From this figure, we can see that the red solid lines are in good agreement with the green dashed line in the dependence on ` for both configurations of m1, m2, m3. This seems to be because the bispectrum of primordial curvature perturbations affected by the fluctuations of vector field given by Eq. (7.17) has not only the anisotropic part but also the isotropic part and both parts have the same amplitude. In this sense, it is expected that the angular dependence on the vector field Aˆ does not contribute much to a change in the shape of the CMB bispectrum. We also find that the anisotropic bispectrum for C 0.3 is comparable in magnitude to the case with fNL = 5 for the standard local type, which corresponds∼ to the upper bound on the local-type non-Gaussianity expected from the PLANCK experiment [37]. 7.2 CMB statistically-anisotropic bispectrum 85

16 0.01 16 0.01 10 10 × × 2 2 ) ) π π 0.001 0.001 >|/(2 >|/(2 3 3 m m 3 3 l l a a

2 0.0001 2 0.0001 m m 2 2 l l a a 1 1 m m 1 1 l 1e-05 l 1e-05 +1)|

(l 1e-06 (l 1e-06 3 3 +1)l +1)l 1 1 (l (l

1 1e-07 1 1e-07 l 10 100 1000 l 100 1000

l3 l3 Figure 7.1: Absolute values of the CMB statistically anisotropic bispectrum of the intensity mode given by Eq. (7.29) with C = 1 (red solid line) and the statistically isotropic one given by Eq. (7.31) with fNL = 5 (green dashed line) for `1 = `2 = `3. The left and right figures are plotted in the configurations (m1, m2, m3) = (0, 0, 0), (10, 20, 30), respectively. The parameters are fixed to the mean values limited from the WMAP-7yr data− as reported in Ref. [38].

In the discussion of the CMB power spectrum, if the rotational invariance is violated in the primordial power spectrum given by Eq. (7.9), the signals in the off-diagonal configurations of ` also have nonzero values [7, 8, 10]. Likewise, there are special configurations in the CMB bispectrum induced from the statistical anisotropy on the primordial bispectrum as Eq. (7.17). The selection rule (7.30) suggests that the statistically anisotropic bispectrum (7.29) could be nonzero in the multipole configurations given by

`1 = `2 `3 4, `2 `3 2, `2 + `3 + 2, `2 + `3 + 4 , (7.32) | − | − | − | − and two permutations of `1, `2, `3. In contrast, in these conﬁgurations, the isotropic bispectrum ` ` ` (e.g., Eq. (7.31)) vanishes owing to the triangle condition of the Wigner-3j symbol 1 2 3 m1 m2 m3 and the nonzero components arise only from

`2 `3 `1 `2 + `3 . (7.33) | − | ≤ ≤ Therefore, the signals of the configurations (7.32) have the pure information of the statistical anisotropy on the CMB bispectrum. Figure 7.2 shows the CMB anisotropic bispectra of the intensity mode given by Eq. (7.29) with C = 1 for the several configurations of `’s and m’s as a function of `3. The red solid line and green dashed line satisfy the special relation (7.32), namely, `1 = `2 + `3 + 2, `2 `3 2, | − | − and the blue dotted line obeys a configuration of Eq. (7.33), namely, `1 = `2 + `3. From this figure, we confirm that the signals in the special configuration (7.32) are comparable in magnitude to those for `1 = `2 + `3. Therefore, if the rotational invariance is violated on the primordial bispectrum of curvature perturbations, the signals for `1 = `2 + `3 + 2, `2 `3 2 can also become beneficial observables. Here, note that the anisotropic bispectra in the| other− special| − configurations: `1 = `2 + `3 + 4, `2 `3 4 are zero. It is because these signals arise from only the contribution of 0 | − | − L = L = LA = 2,L1 = `1 2,L2 = `2 2 in Eq. (7.29) owing to the selection rules of the Wigner symbols, and the summation± of the four± Wigner-3j symbols over M vanishes for all `’s and m’s. Hence, in this anisotropic bispectrum, the additional signals arise from only two configurations `1 = `2 + `3 + 2, `2 `3 2 and these two permutations. | − | − 86 7 VIOLATION OF THE ROTATIONAL INVARIANCE IN THE CMB BISPECTRUM

16 1 16 1 10 10 × × 2 2 ) )

π π 0.1 0.1 >|/(2 >|/(2

3 3 0.01 m m 3 3 l l a a

2 0.01 2

m m 0.001 2 2 l l a a 1 1 m m 1 1 l 0.001 l 0.0001 1e-05 +1)|

(l 0.0001 (l 3 3 1e-06 +1)l +1)l 1 1 (l (l

1 1e-05 1 1e-07 l 10 100 1000 l 100 1000

l3 l3 Figure 7.2: Absolute values of the CMB statistically anisotropic bispectra of the intensity mode given by Eq. (7.29) for (m1, m2, m3) = (0, 0, 0) (left panel) and (10, 20, 30) (right one) as the − function with respect to `3. The lines correspond to the spectra for (`1, `2) = (102 + `3, 100) (red solid line), ( 100 `3 2, 100) (green dashed line) and (100 + `3, 100) (blue dotted line). The parameters are| identical− | − to the values deﬁned in Fig. 7.1.

7.3 Summary and discussion

In this section, we investigated the statistical anisotropy in the CMB bispectrum by considering the modified hybrid inflation model where the waterfall field also couples with the vector field [14]. We calculated the CMB bispectrum sourced from the non-Gaussianity of curvature perturbations affected by the vector field. In this inflation model, owing to the dependence on the direction of the vector field, the correlations of the curvature perturbations violate the rotational invariance. Then, interestingly, even if the magnitude of the parameter gβ characterizing the statistical anisotropy of the CMB power spectrum is too small, the amplitude of the non-Gaussianity can become large depending on several coupling constants of the fields. Following the procedure of Sec. 5 [25], we formulated the statistically anisotropic CMB bispectrum and confirm that three azimuthal quantum numbers m1, m2 and m3 are not confined only to ` ` ` the Wigner symbol 1 2 3 . This is evidence that the rotational invariance is violated m1 m2 m3 in the CMB bispectrum and implies the existence of the signals not obeying the triangle condition of the above Wigner symbol as `2 `3 `1 `2 + `3. We demonstrated that the signals of the | − | ≤ ≤ CMB bispectrum for `1 = `2 +`3 +2, `2 `3 2 and these two permutations do not vanish. In fact, the statistically isotropic bispectra are| − exactly|− zero for these configurations; hence, these signals have the pure information of the statistical anisotropy. Because the amplitudes of these intensity bispectra are comparable to those for `1 = `2 +`3, it might be possible to detect these contributions of the statistical anisotropy in future experiments, which would give us novel information about the physics of the early Universe. Of course, also for the E-mode polarization, we can give the same discussions and results. Although we assume a specific potential of inflation to show the statistical anisotropy on the CMB bispectrum explicitly, the above calculation and discussion will be applicable to other inflation models where the rotational invariance violates. REFERENCES 87

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8 Parity violation of gravitons in the CMB bispectrum

Non-Gaussian features in the cosmological perturbations include detailed information on the nature of the early Universe, and there have been many works that attempt to extract them from the bispectrum (three-point function) of the cosmic microwave background (CMB) anisotropies (e.g., Refs. [1–4]). However, most of these discussions are limited in the cases that the scalar-mode contribution dominates in the non-Gaussianity and also are based on the assumption of rotational invariance and parity conservation. In contrast, there are several studies on the non-Gaussianities of not only the scalar-mode perturbations but also the vector- and tensor-mode perturbations [5–7]. These sources produce the additional signals on the CMB bispectrum [8] and can give a dominant contribution by considering such highly non-Gaussian sources as the stochastic magnetic fields [9]. Furthermore, even in the CMB bispectrum induced from the scalar-mode non-Gaussianity, if the rotational invariance is violated in the non-Gaussianity, the characteristic signals appear [10]. Thus, it is very important to clarify these less-noted signals to understand the precise picture of the early Universe. Recently, the parity violation in the graviton non-Gaussianities has been discussed in Refs. [11, 12]. Maldacena and Pimentel first calculated the primordial bispectrum of the gravitons sourced from parity-even (parity-conserving) and parity-odd (parity-violating) Weyl cubic terms, namely, W 3 and WWf 2, respectively, by making use of the spinor helicity formalism. [11] Soda et al. proved that the parity-violating non-Gaussianity of the primordial gravitational waves induced from WWf 2 emerges not in the exact de-Sitter space-time but in the quasi de-Sitter space-time, and hence, its amplitude is proportional to a slow-roll parameter. [12] In these studies, the authors assume that the coupling constant of the Weyl cubic terms is independent of time. In this section, we estimate the primordial non-Gaussianities of gravitons generated from W 3 and WWf 2 with the time-dependent coupling parameter [13]. We consider the case where the coupling is given by a power of the conformal time. We show that in such a model, the parity violation in the non-Gaussianity of the primordial gravitational waves would not vanish even in the exact de-Sitter space-time. The effects of the parity violation on the CMB power spectrum have been well-studied, where an attractive result is that the cross-correlation between the intensity and B-mode polarization is generated [14–17]. On the other hand, in the CMB bispectrum, owing to the mathematical property of the spherical harmonic function, the parity-even and parity-odd signals should arise from just the opposite configurations of multipoles [18, 19]. Then, we formulate and numerically calculate the CMB bispectra induced by these non-Gaussianities that contain all the correlations between the intensity (I) and polarizations (E,B) and show that the signals from W 3 (parity-conserving) appear in the configuration of the multipoles where those from WWf 2 (parity-violating) vanish and vice versa. These discussions are based on Ref. [20]. This section is organized as follows. In the next subsection, we derive the primordial bispectrum of gravitons induced by W 3 and WWf 2 with the coupling constant proportional to the power of the conformal time. In Sec. 8.2, we calculate the CMB bispectra sourced from these non- Gaussianities, analyze their behavior and find some peculiar signatures of the parity violation. The final subsection is devoted to summary and discussion. In Appendices E and F, we describe the detailed calculations of the contractions of the polarization tensors and unit vectors, and of the intial bispectra by the in-in formalism.

Throughout this section, we use Mpl 1/√8πG, where G is the Newton constant and the rule that all the Greek characters and alphabets≡ run from 0 to 3 and from 1 to 3, respectively. 8.1 Parity-even and -odd non-Gaussianity of gravitons 91

8.1 Parity-even and -odd non-Gaussianity of gravitons In this subsection, we formulate the primordial non-Gaussianity of gravitons generated from the Weyl cubic terms with the running coupling constant as a function of a conformal time, f(τ), whose action is given by

Z f(τ) S = dτd3x √ gW 3 + WWf 2 , (8.1) Λ2 − with

3 αβ γδ σρ W W γδW σρW αβ , ≡ (8.2) 2 αβµν γδ σρ WWf WµνγδW σρW αβ , ≡ αβ αβµν 0123 where W γδ denotes the Weyl tensor, is a 4D Levi-Civita tensor normalized as = 1, and Λ is a scale that sets the value of the higher derivative corrections [11]. Note that W 3 and WWf 2 have the even and odd parities, respectively. In the following discussion, we assume that the coupling constant is given by

τ A f(τ) = , (8.3) τ∗ where τ is a conformal time. Here, we have set f(τ∗) = 1. Such a coupling can be readily realized by considering a dilaton-like coupling in the slow-roll inﬂation as discussed in Sec. 8.1.2.

8.1.1 Calculation of the primordial bispectrum

Here, let us focus on the calculation of the primordial bispectrum induced by W 3 and WWf 2 of Eq. (8.1) on the exact de-Sitter space-time in a more straightforward manner than those of Refs. [11, 12]. At ﬁrst, we consider the tensor perturbations on the Friedmann-Lemaitre-Robertson-Walker metric as

ds2 = a2( dτ 2 + eγij dxidxj) , (8.4) − j where a denotes the scale factor and γij obeys the transverse traceless conditions; γii = ∂γij/∂x = 0 1. Up to the second order, even if the action includes the Weyl cubic terms given by Eq. (8.1), the gravitational wave obeys the action as [11, 12]

2 Z Mpl 3 2 S = dτdx a (γ ˙ ijγ˙ ij γij,kγij,k) , (8.5) 8 −

i where ˙ ∂/∂τ and ,i ∂/∂x . We expand the gravitational wave with a transverse and traceless ≡ (λ)≡ (λ)† (λ) polarization tensor eij and the creation and annihilation operators a , a as

Z 3 d k X (λ) (λ) γ (x, τ) = γ (k, τ)a e (kˆ)eik·x + h.c. ij (2π)3 dS k ij λ=±2

1 γij is identical to hij in Sec. 2. 92 8 PARITY VIOLATION OF GRAVITONS IN THE CMB BISPECTRUM

Z 3 d k X (λ) = γ(λ)(k, τ)e (kˆ)eik·x , (8.6) (2π)3 ij λ=±2 with

(λ) (λ) ∗ (λ)† γ (k, τ) γdS(k, τ)a + γ (k, τ)a . (8.7) ≡ k dS −k Here, λ 2 denotes the helicity of the gravitational wave and we use the polarization tensor satisfying≡ the ± relations as Eq. (D.13). The creation and annihilation operators a(λ)†, a(λ) obey the relations as (λ) ak 0 = 0 , | i (8.8) h (λ) (λ0)†i 3 0 a , a 0 = (2π) δ(k k )δλ,λ0 , k k − where 0 denotes a vacuum eigenstate. Then, the mode function of gravitons on the de Sitter | i space-time γdS satisﬁes the ﬁeld equation as

2 2 γ¨dS γ˙ dS + k γdS = 0 , (8.9) − τ and a solution is given by H e−ikτ γdS = i 3/2 (1 + ikτ) , (8.10) Mpl k where H = (aτ)−1 is the Hubble parameter and has a constant value in the exact de Sitter space-time. − On the basis of the in-in formalism (see, e.g., Refs. [5, 21]) and the above results, we calculate the tree-level bispectrum of gravitons on the late-time limit. According to this formalism, the expectation value of an operator depending on time in the interaction picture, O(t), is written as

D R 0 0 R 0 0 E O(t) = 0 T¯ ei Hint(t )dt O(t)T e−i Hint(t )dt 0 , (8.11) h i ¯ where T and T are respectively time-ordering and anti-time-ordering operators and Hint(t) is the interaction Hamiltonian. Applying this equation, the primordial bispectrum of gravitons at the tree level can be expressed as

* 3 + * " 3 # + Y Z τ Y (λn) 0 0 (λn) γ (kn, τ) = i dτ 0 : Hint(τ ):, γ (kn, τ) 0 , (8.12) n=1 −∞ n=1 where : : denotes normal product. Up to the ﬁrst order with respect to γij, the nonzero components of the Weyl tensor are written as 1 W 0i = (Hτ)2γ , 0j 4 ij,αα ij 1 2 W 0k = (Hτ) (γ ˙ ki,j γ˙ kj,i) , 2 − (8.13) 0i 1 2 W jk = (Hτ) (γ ˙ ik,j γ˙ ij,k) , 2 − ij 1 2 W kl = (Hτ) ( δikγjl,αα + δilγjk,αα + δjkγil,αα δjlγik,αα) , 4 − − 8.1 Parity-even and -odd non-Gaussianity of gravitons 93

2 3 2 where γij,αα γ¨ij + γij. Then W and WWf respectively reduce to ≡ ∇ 3 ij kl mn 0i jk lm W = W klW mnW ij + 6W jkW lmW 0i 0i 0j kl 0i 0j 0k + 12W 0jW klW 0i + 8W 0jW 0kW 0i , (8.14) 2 ijk pq lm pq 0m WWf = 4η Wjkpq W lmW 0i + 2W 0mW 0i 0p lm 0p 0m +2Wjk0p W lmW 0i + 2W 0mW 0i ,

ijk 0ijk R where η . Using the above expressions and dτHint = Sint, up to the third order, the ≡ − interaction Hamiltonians of W 3 and WWf 2 are respectively given by Z A 3 −2 2 τ HW 3 = d xΛ (Hτ) − τ∗ 1 γij,αα [γjk,ββγki,σσ + 6γ ˙ kl,iγ˙ kl,j + 6γ ˙ ik,lγ˙ jl,k 12γ ˙ ik,lγ˙ kl,j] , × 4 − (8.15) Z τ A H = d3xΛ−2(Hτ)2 WWf 2 − τ∗ ijk η [γkq,αα( 3γjm,ββγ˙ iq,m + γmi,ββγ˙ mq,j) + 4γ ˙ pj,kγ˙ pm,l(γ ˙ il,m γ˙ im,l)] . × − − Substituting the above expressions into Eq. (8.12), using the solution given by Eq. (8.10), and considering the late-time limit as τ 0, we can obtain an explicit form of the primordial bispectra: → * 3 + 3 ! Y (λn) 3 X (r) (a) ˆ ˆ ˆ γ (kn) = (2π) δ kn fint (k1, k2, k3)fint (k1, k2, k3) , (8.16) n=1 int n=1 with2

6 2 0 Z 0 (r) H H −A 0 05+A −iktτ fW 3 = 8 Re τ∗ dτ τ e , Mpl Λ −∞ 1 3 f (a) = e(−λ1) e(−λ2)e(−λ3) + e(−λ2)e(−λ3)kˆ kˆ W 3 ij 2 jk ki 4 kl kl 2i 3j 3 3 (−λ2) (−λ3) ˆ ˆ (−λ2) (−λ3) ˆ ˆ + e e k2 k3 e e k2 k3 + 5 perms, 4 ki jl l k − 2 ik kl l j (8.17) 6 2 0 H H Z 0 f (r) = 8 Im τ −A dτ 0τ 05+Ae−iktτ , WW 2 ∗ f Mpl Λ −∞ h n o (a) ijk (−λ1) (−λ2) (−λ3) ˆ (−λ2) (−λ3) ˆ f = iη e 3e e k3m + e emq k3j WWf 2 kq − jm iq mi n oi (−λ1) (−λ2) ˆ ˆ (−λ3) ˆ (−λ3) ˆ +e e k1 k2 e k3 e k3 + 5 perms . pj pm k l il m − im l

P3 3 2 Here, kt n=1 kn, int = W and WWf , “5 perms” denotes the five symmetric terms under ≡ ˆ ˆ ˆ the permutations of (k1, λ1), (k2, λ2), and (k3, λ3). From the above expressions, we find that the bispectra of the primordial gravitational wave induced from W 3 and WWf 2 are proportional to 0 −A R 0 0 05+A −iktτ the real and imaginary parts of τ∗ −∞ dτ τ e , respectively. This difference comes from the number of γij,αα andγ ˙ ij,k. HW 3 consists of the products of an odd number of the former

2 Here, we set that τ∗ < 0. 94 8 PARITY VIOLATION OF GRAVITONS IN THE CMB BISPECTRUM terms and an even number of the latter terms. On the other hand, in H , the situation is WWf 2 2 0 3/2 −ikτ 0 the opposite. Since the former and latter terms containγ ¨dS k γdS = (2Hτ /Mpl)k e and 0 1/2 −ikτ 0 − γ˙ dS = i(Hτ /Mpl)k e , respectively, the total numbers of i are diﬀerent in each time integral. (r) (r) Hence, the contributions of the real and imaginary parts roll upside down in f 3 and f . Since W WWf 2 the time integral in the bispectra can be analytically evaluated as

0 Z 0 h i −A 0 05+A −iktτ π π −6 −A τ∗ dτ τ e = cos A + i sin A Γ(6 + A)kt ( ktτ∗) , (8.18) −∞ 2 2 −

(r) (r) f 3 and f reduce to W WWf 2 6 2 (r) H H π −6 −A fW 3 = 8 cos A Γ(6 + A)kt ( ktτ∗) , (8.19) Mpl Λ 2 − 6 2 (r) H H π −6 −A f = 8 sin A Γ(6 + A)k ( ktτ∗) , (8.20) WW 2 t f Mpl Λ 2 − where Γ(x) is the Gamma function. For more detailed derivation of the graviton bispectrum, see Appendix F. From this equation, we can see that in the case of the time-independent coupling, which corresponds to the A = 0 case, the bispectrum from WWf 2 vanishes. This is consistent with a claim in Ref. [12] 3. On the other hand, interestingly, if A deviates from 0, it is possible to realize the nonzero bispectrum induced from WWf 2 even in the exact de Sitter limit. Thus, we expect the signals from WWf 2 without the slow-roll suppression, which can be comparable to those from W 3 and become suﬃciently large to observe in the CMB.

8.1.2 Running coupling constant Here, we discuss how to realize f τ A within the framework of the standard slow-roll inflation. During the standard slow-roll inflation,∝ the equation of motion of the scalar field φ, which has a potential V , is expressed as

˙ p −1 φ 2φMplτ , (8.21) ' ± 2 2 where φ [∂V/∂φ/(3MplH )] /2 is a slow-roll parameter for φ, + and signs are taken to be for ∂V/∂φ >≡0 and ∂V/∂φ < 0, respectively, and we have assumed that aH−= 1/τ. The solution of the above equation is given by − p τ φ = φ∗ 2φMpl ln . (8.22) ± τ∗

Hence, if we assume a dilaton-like coupling as f e(φ−φ∗)/M , we have ≡ A τ p Mpl f(τ) = ,A = 2φ , (8.23) τ∗ ± M where M is an arbitrary energy scale. Let us take τ∗ to be a time when the scale of the present −1 horizon of the Universe exits the horizon during inﬂation, namely, τ∗ = k 14Gpc. Then, the | | ∗ ∼ 3 In Ref. [12], the authors have shown that for A = 0, the bispectrum from WWf 2 has a nonzero value upward in the ﬁrst order of the slow-roll parameter. 8.2 CMB parity-even and -odd bispectrum 95 coupling f, which determines the amplitude of the bispectrum of the primordial gravitational wave induced from the Weyl cubic terms, is on the order of unity for the current cosmological scales. p From Eq. (8.23), we have A = 1/2 with M = 8φMpl. As seen in Eqs. (8.19) and (8.20), ± this leads to an interesting situation that the bispectra from W 3 and WWf 2 have a comparable (r) (r) magnitude as f 3 = f . Hence, we can expect that in the CMB bispectrum, the signals from W WWf 2 these terms are almost± the same. In the next subsection, we demonstrate these through the explicit calculation of the CMB bispectra.

8.2 CMB parity-even and -odd bispectrum In this subsection, following the calculation approach discussed in Sec. 6, we formulate the CMB bispectrum induced from the non-Gaussianities of gravitons sourced by W 3 and WWf 2 terms discussed in the previous subsection.

8.2.1 Formulation Conventionally, the CMB ﬂuctuation is expanded with the spherical harmonics as

∆X(nˆ) X = a Y (nˆ) , (8.24) X X,`m `m `m where nˆ is a unit vector pointing toward a line-of-sight direction, and X means the intensity ( I) and the electric and magnetic polarization modes ( E,B). By performing the line-of-sight ≡ ≡ (±2) integration, the coeﬃcient, a`m, generated from the primordial ﬂuctuation of gravitons, γ , is given by [corresponding to Eq. (5.3)]

Z ∞ 2 x ` k dk X λ (λ) aX,`m = 4π( i) X,`(k) γ (k) , (8.25) − (2π)3 T 2 `m 0 λ=±2 Z (λ) 2 ˆ (λ) ∗ ˆ γ (k) d kγ (k)−λY (k) , (8.26) `m ≡ `m where x discriminates the parity of three modes: x = 0 for X = I,E and x = 1 for X = B, and X,` is the time-integrated transfer function of tensor modes (3.101). Like Eq. (5.5), we can obtain theT CMB bispectrum generated from the primordial bispectrum of gravitons as

* 3 + " 3 Z 2 xn # Y Y `n kndkn X λn aX ,` m = 4π( i) X ,` (kn) n n n − (2π)3 T n n 2 n=1 n=1 λn=±2 * 3 + Y (λ ) γ n (k ) . (8.27) `nmn n × n=1 In order to derive an explicit form of this CMB bispectrum, at ﬁrst, we need to express all the functions containing the angular dependence on the wave number vectors with the spin spherical (a) (a) harmonics. Using the results of Appendix E, f 3 and f can be calculated as W WWf 2 0 00 (a) 3/2 X X 2 L L f = (8π) W 3 MM 0 M 00 L0,L00=2,3 M,M 0,M 00 96 8 PARITY VIOLATION OF GRAVITONS IN THE CMB BISPECTRUM

∗ ˆ ∗ ˆ ∗ ˆ λ1 Y2M (k1)λ2 YL0M 0 (k2)λ3 YL00M 00 (k3) × " r 0 1 7 L λ20−λ2 λ30−λ3 δL0,2δL00,2 + ( 1) I 0 I 00 × −20 3 − L 12 L 12   2 L0 L00  π 2 L0 L00   π 1 1 2  5 2 1 1 × − −  1 2 1  2 1 L0 2 L0 L00 +2π + 5 perms , (8.28) 2 1 1 2 1 1 0 00 (a) 3/2 X X 2 L L f = (8π) 0 00 WWf 2 MM M L0,L00=2,3 M,M 0,M 00 00 ∗ ˆ ∗ ˆ ∗ ˆ L λ30−λ3 λ Y (k1)λ Y 0 0 (k2)λ Y 00 00 (k3)( 1) I 00 × 1 2M 2 L M 3 L M − L 12   r  2 2 L00  2π 2 2 L00   δ 0 3 2√2π 1 1 1  L ,2  5 1 2 1  × −  1 1 2    0 00    2 L L  r 0 00 λ1 λ20−λ2 4π 2π 7 2 L L + I 0 1 2 1 + 2 L 12  3 15 3 1 2 2  −  1 1 2  +5 perms , (8.29) where the 2 3 matrix of a bracket, and the 2 3 and 3 3 matrices of a curly bracket denote the Wigner-3×j, 6j and 9j symbols, respectively, and× × r s1s2s3 (2l1 + 1)(2l2 + 1)(2l3 + 1) l1 l2 l3 Il l l . (8.30) 1 2 3 ≡ 4π s1 s2 s3 The delta function is also expanded as 3 ! Z ∞ " 3 # X 2 Y X Ln/2 ∗ ˆ δ kn = 8 y dy ( 1) jLn (kny)YLnMn (kn) 0 − n=1 n=1 LnMn 0 0 0 L1 L2 L3 IL1L2L3 . (8.31) × M1 M2 M3 ˆ ˆ ˆ Next, we integrate all the spin spherical harmonics over k1, k2, k3 as Z ` L 2 2 ˆ ∗ ∗ ∗ λ10−λ1 1 1 d k1−λ1 Y`1m1 YL1M1 λ1 Y2M = I` L 2 , 1 1 m1 M1 M Z 0 2 ∗ ∗ ∗ λ20−λ2 `2 L2 L ˆ 0 0 d k2−λ2 Y`2m2 YL2M2 λ2 YL M = I` L L0 0 , (8.32) 2 2 m2 M2 M Z 00 2 ∗ ∗ ∗ λ30−λ3 `3 L3 L ˆ 00 00 d k3−λ3 Y`3m3 YL3M3 λ3 YL M = I` L L00 00 . 3 3 m3 M3 M Through the summation over the azimuthal quantum numbers, the product of the above ﬁve Wigner-3j symbols is expressed with the Wigner-9j symbols as 0 00 X L1 L2 L3 2 L L 0 00 M1 M2 M3 MM M M1M2M3 MM 0M 00 8.2 CMB parity-even and -odd bispectrum 97

0 00 `1 L1 2 `2 L2 L `3 L3 L 0 00 × m1 M1 M m2 M2 M m3 M3 M  ` ` `  ` ` `  1 2 3  = 1 2 3 L L L . (8.33) m m m 1 2 3 1 2 3  2 L0 L00 

Finally, performing the summation over the helicities, namely λ1, λ2 and λ3, as x ( X λ 2I20−2 (` + L + x = even) Iλ0−λ = `L2 , 2 `L2 0 (` + L + x = odd) λ=±2 x ( 20−2 20−2 X λ λ0−λ λ0−λ 2I`LL0 IL012 (` + L + x = odd) I 0 I 0 = , (8.34) 2 `LL L 12 0 (` + L + x = even) λ=±2 x+1 ( X λ 2I20−2 (` + L + x = odd) Iλ0−λ = `L2 , 2 `L2 0 (` + L + x = even) λ=±2 and considering the selection rules of the Wigner symbols as described in Appendix C, we derive the CMB bispectrum generated from the non-Gaussianity of gravitons induced by W 3 as * 3 + Z ∞ Y `1 `2 `3 X L1+L2+L3 2 2 0 0 0 aXn,`nmn = y dy ( 1) IL1L2L3 m1 m2 m3 0 − n=1 W 3 L1L2L3 " 3 Z # Y 2 (r) ( i)`n k2dk (k )j (k y) f (k , k , k ) π n n Xn,`n n Ln n W 3 1 2 3 × n=1 − T   `1 `2 `3 3/2 X 20−2 20−2 20−2   (8π) 8I I 0 I 00 L L L `1L12 `2L2L `3L3L 1 2 3 × 0 00 L0,L00=2,3  2 L L  " r 3 ! 1 7 Y (e) δL0,2δL00,2 20 3 Ln,`n,xn × − n=1 D L0 20−2 20−2 (e) (o) (o) +( 1) I 0 I 00 − L 12 L 12 DL1,`1,x1 DL2,`2,x2 DL3,`3,x3   2 L0 L00  π 2 L0 L00   π 1 1 2  5 2 1 1 × − −  1 2 1  2 1 L0 2 L0 L00 +2π + 5 perms , (8.35) 2 1 1 2 1 1 and WWf 2 as * 3 + Z ∞ Y `1 `2 `3 X L1+L2+L3 2 2 0 0 0 aXn,`nmn = y dy ( 1) IL1L2L3 m1 m2 m3 0 − n=1 WWf 2 L1L2L3 " 3 Z # Y 2 `n 2 (r) ( i) kndkn Xn,`n (kn)jLn (kny) f (k1, k2, k3) π WWf 2 × n=1 − T   `1 `2 `3 3/2 X 20−2 20−2 20−2   L00 20−2 (8π) 8I I 0 I 00 L L L ( 1) I 00 `1L12 `2L2L `3L3L 1 2 3 L 12 × 0 00 − L0,L00=2,3  2 L L  98 8 PARITY VIOLATION OF GRAVITONS IN THE CMB BISPECTRUM

h (e) (e) (o) δL0,2 × DL1,`1,x1 DL2,`2,x2 DL3,`3,x3  r  2 2 L00  2π 2 2 L00   3 2√2π 1 1 1  5 1 2 1  × −  1 1 2  3 ! 20−2 Y (o) +I 0 L 12 Ln,`n,xn n=1 D   2 L0 L00  r  4π   2π 7 2 L0 L00 1 2 1 + + 5 perms . (8.36)  3 15 3 1 2 2  × −  1 1 2 

Here, “5 perms” denotes the ﬁve symmetric terms under the permutations of (`1, m1, x1), (`2, m2, x2), and (`3, m3, x3), and we introduce the ﬁlter functions as

(e) (δL,`−2 + δL,` + δL,`+2)δx,0 + (δL,`−3 + δL,`−1 + δL,`+1 + δL,`+3)δx,1 , DL,`,x ≡ (8.37) (o) (δL,`−2 + δL,` + δL,`+2)δx,1 + (δL,`−3 + δL,`−1 + δL,`+1 + δL,`+3)δx,0 , DL,`,x ≡ where the superscripts (e) and (o) denote L+`+x = even and = odd, respectively. From Eqs. (8.35) and (8.36), we can see that the azimuthal quantum numbers m1, m2, and m3 are conﬁned only ` ` ` in a Wigner-3j symbol as 1 2 3 . This guarantees the rotational invariance of the m1 m2 m3 CMB bispectrum. Therefore, this bispectrum survives if the triangle inequality is satisﬁed as `1 `2 `3 `1 + `2. | −Considering| ≤ ≤ the products between the functions in Eq. (8.35) and the selection rules as P3 D 3 n=1 Ln = even, we can notice that the CMB bispectrum from W does not vanish only for

3 X (`n + xn) = even . (8.38) n=1

3 P3 Therefore, W contributes the III,IIE,IEE,IBB,EEE, and EBB spectra for n=1 `n = even P3 and the IIB,IEB,EEB, and BBB spectra for n=1 `n = odd. This property can arise from any sources keeping the parity invariance such as W 3. On the other hand, in the same manner, we understand that the CMB bispectrum from WWf 2 survives only for

3 X (`n + xn) = odd . (8.39) n=1

By these constraints, we ﬁnd that in reverse, WWf 2 generates the IIB,IEB,EEB, and BBB P3 P3 spectra for n=1 `n = even and the III,IIE,IEE,IBB,EEE, and EBB spectra for n=1 `n = odd. This is a characteristic signature of the parity violation as mentioned in Refs. [18, 19]. Hence, P3 if we analyze the information of the CMB bispectrum not only for n=1 `n = even but also for P3 n=1 `n = odd, it may be possible to check the parity violation at the level of the three-point correlation. The above discussion about the multipole conﬁgurations of the CMB bispectra can be easily understood only if one consider the parity transformation of the CMB intensity and polarization 8.2 CMB parity-even and -odd bispectrum 99

ﬁelds in the real space (8.24). The III, IIE, IEE, IBB, EEE and EBB spectra from W 3, and the IIB,IEB,EEB, and BBB spectra from WWf 2 have even parity, namely,

* 3 + * 3 + Y ∆Xi(nî) Y ∆Xi( nî) = − . (8.40) X X i=1 i i=1 i Then, from the multipole expansion (8.24) and its parity flip version as

∆X( nˆ) X X ` − = aX,`mY`m( nˆ) = ( 1) aX,`mY`m(nˆ) , (8.41) X − − `m `m P3 one can notice that n=1 `n = even must be satisﬁed. On the other hand, since the IIB,IEB,EEB, and BBB spectra from W 3, and the III, IIE, IEE, IBB, EEE, and EBB spectra from WWf 2 have odd parity, namely,

* 3 + * 3 + Y ∆Xi(nˆi) Y ∆Xi( nˆi) = − , (8.42) X X i=1 i − i=1 i P3 one can obtain n=1 `n = odd. In Sec. 8.2.3, we compute the CMB bispectra (8.35) and (8.36) when A = 1/2, 0, 1, that is, ± the signals from W 3 become as large as those from WWf 2 and either signals vanish.

(r) (r) 8.2.2 Evaluation of f 3 and f W WWf 2 Here, to compute the CMB bispectra (8.35) and (8.36) in ﬁnite time, we express the radial func- (r) (r) tions, f 3 and f , with some terms of the power of k1, k2, and k3. Let us focus on the dependence W WWf 2 on k1, k2, and k3 in Eqs. (8.19) and (8.20) as

(r) (r) −6 −A SA(k1, k2, k3) f 3 f k ( ktτ∗) = , (8.43) W WW 2 t A/3 A ∝ f ∝ − (k1k2k3) ( τ∗) − −6 where we deﬁne SA to satisfy SA k as ∝ A/3 (k1k2k3) SA(k1, k2, k3) 6+A . (8.44) ≡ kt

In Fig. 8.1, we plot SA for A = 1/2, 0, 1/2, and 1. From this, we notice that the shapes of SA are similar to the equilateral-type− conﬁguration as Eq. (4.10) 1 1 1 2 Sequil(k1, k2, k3) = 6 3 3 3 3 3 3 2 2 2 −k1k2 − k2k3 − k3k1 − k1k2k3 1 1 1 1 1 1 + 2 3 + 2 3 + 2 3 + 2 3 + 2 3 + 2 3 . (8.45) k1k2k3 k1k3k2 k2k3k1 k2k1k3 k3k1k2 k3k2k1 To evaluate how a function S is similar in shape to a function S0, we introduce a correlation function as [3, 22] S S0 cos(S S0) · , (8.46) · ≡ (S S)1/2(S0 S0)1/2 · · 100 8 PARITY VIOLATION OF GRAVITONS IN THE CMB BISPECTRUM

0.0000 0.0000 1.0 1.0 0.5 0.5 0.0 0.0 k2k1 k2k1 1.0 1.0

0.8 0.8 k3k1 k3k1

0.6 0.6

0.0008 0.0004 0.0006 0.0003 0.0004 0.0002

0.0002 0.0001

0.0000 0.0000 1.0 1.0 0.5 0.5 0.0 0.0 k2k1 k2k1

2 2 2 Figure 8.1: Shape of k k k SA for A = 1/2 (top left panel), 0 (top right one), 1/2 (bottom left 1 2 3 − one), and 1 (bottom right one) as the function of k2/k1 and k3/k1. with 0 X S(k1, k2, k3)S (k1, k2, k3) S S0 · ≡ P (k1)P (k2)P (k3) ki Z 1 Z 1 4 4 0 dx2 dx3x2x3S(1, x2, x3)S (1, x2, x3) , (8.47) ∝ 0 1−x2 −3 where the summation is performed over all ki, which form a triangle and P (k) k denotes the power spectrum. This correlation function gets to 1 when S = S0. In our case,∝ this is calculated as 0.968 , (A = 1/2)  − 0.970 , (A = 0) cos(SA Sequil) (8.48) · ' 0.971 , (A = 1/2)  0.972 , (A = 1) that is, an approximation that SA is proportional to Sequil seems to be valid. Here, we also calculate the correlation functions with the local- and orthogonal-type non-Gaussianities [4] and conclude that these contributions are negligible. Thus, we determine the proportionality coeﬃcient as

 −4 4.40 10 Sequil , (A = 1/2)  × − S S 2.50 10−4S , (A = 0) S A equil S = equil (8.49) A · equil × −4 ' Sequil Sequil 1.42 10 Sequil , (A = 1/2) ·  ×  −5 8.09 10 Sequil . (A = 1) × 8.2 CMB parity-even and -odd bispectrum 101

Substituting this into Eqs. (8.19) and (8.20), we obtain reasonable formulae of the radial functions for A = 1/2 as

2 4 2 r −4 (r) (r) π Mpl 10395 π 1.42 10 Sequil f 3 = f rAS , (8.50) W WW 2 1×/2 1/6 f ' 2 Λ 8 2 ( τ∗) (k1k2k3) − and for A = 1/2 as − 2 4 2 r (r) (r) π Mpl 945 π −4 1/2 1/6 f 3 = f rAS 4.40 10 ( τ∗) (k1k2k3) Sequil . (8.51) W − WWf 2 ' 2 Λ 4 2 × −

Here, we also use

H 2 π2 = rAS , (8.52) Mpl 2 where AS is the amplitude of primordial curvature perturbations and r is the tensor-to-scalar (r) ratio [4, 8]. For A = 0, the signals from WWf 2 disappear as f = 0 and the ﬁnite radial WWf 2 function of W 3 is given by

2 4 2 (r) π Mpl −4 f 3 rAS 960 2.50 10 Sequil . (8.53) W ' 2 Λ × ×

(r) 2 In contrast, for A = 1, since fW 3 = 0, we have only the parity-violating contribution from WWf as

2 4 2 −5 (r) π Mpl 8.09 10 Sequil f rAS 5760 . (8.54) WW 2 × 1/3 f ' 2 Λ × ( τ∗)(k1k2k3) −

8.2.3 Results On the basis of the analytical formulae (8.35), (8.36), (8.50), (8.51), (8.53) and (8.54), we compute the CMB bispectra from W 3 and WWf 2 for A = 1/2, 0, 1/2, and 1. Then, we modify the Boltzmann Code for Anisotropies in the Microwave Background− (CAMB) [23, 24]. In calculating the Wigner symbols, we use the Common Mathematical Library SLATEC [25] and some analytic (r) formulae described in Appendices C and D. From the dependence of the radial functions fW 3 and (r) f on the wave numbers, we can see that the shapes of the CMB bispectra from W 3 and WWf 2 WWf 2 are similar to the equilateral-type configuration. Then, the significant signals arise from multipoles satisfying `1 `2 `3. We confirm this by calculating the CMB bispectrum for several `’s. Hence, ' ' in the following discussion, we give the discussion with the spectra for `1 `2 `3. However, we P3 ' ' do not focus on the spectra from n=1 `n = odd for `1 = `2 = `3 because these vanish due to the asymmetric nature. In Fig. 8.2, we present the reduced CMB III,IIB,IBB, and BBB spectra given by

* 3 + −1 X `1 `2 `3 Y bX1X2X3,`1`2`3 = (G`1`2`3 ) aXn,`nmn , (8.55) m1 m2 m3 m1m2m3 n=1 102 8 PARITY VIOLATION OF GRAVITONS IN THE CMB BISPECTRUM

1 + 2 + 3 = even 1 + 2 + 3 = odd WW~ 2 III 1 W3 IIB WW~ 2 IBB 0.01 W3 BBB 0.0001 1e-06 1e-08 1e-10 1e-12 W3 III 16 WW~ 2 IIB 1e-14 W3 IBB A = -1/2 A = -1/2 ~ 2 10 WW BBB W3 IIB

× 1 W3 BBB

2 0.01 ) 0.0001 π 1e-06 (2

/ 1e-08 |

3 1e-10 ℓ

2 1e-12 ℓ

1 1e-14 W3 III A = 0 A = 0 ℓ W3 IBB b | WW~ 2 III 1 W3 IIB WW~ 2 IBB 0.01 W3 BBB

+ 1) 0.0001

2 1e-06 ℓ

( 1e-08 2

ℓ 1e-10 1e-12 W3 III WW~ 2 IIB 1e-14 W3 IBB A = 1/2 A = 1/2 + 1) WW~ 2 BBB

1 WW~ 2 III ℓ 1 WW~ 2 IBB (

1 0.01 ℓ 0.0001 1e-06 1e-08 1e-10 1e-12

1e-14 WW~ 2 IIB A = 1 A = 1 WW~ 2 BBB 10 100 1000 10 100 1000

Figure 8.2: Absolute values of the CMB III,IIB,IBB, and BBB spectra induced by W 3 and 2 WWf for A = 1/2, 0, 1/2, and 1. We set that three multipoles have identical values as `1 2 = − P3 − `2 1 = `3. The left figures show the spectra not vanishing for n=1 `n = even (parity-even − P3 mode) and the right ones present the spectra for n=1 `n = odd (parity-odd mode). Here, we fix 6 −1 the parameters as Λ = 3 10 GeV, r = 0.1, and τ∗ = k∗ = 14Gpc, and other cosmological parameters are fixed as the× mean values limited from the− WMAP− 7-yr data [4]. 8.2 CMB parity-even and -odd bispectrum 103

4 for `1 2 = `2 1 = `3. Here, the G symbol is defined by [19] − − s p Q3 2 `3(`3 + 1)`2(`2 + 1) n=1(2`n + 1) `1 `2 `3 G`1`2`3 . (8.57) ≡ `1(`1 + 1) `2(`2 + 1) `3(`3 + 1) 4π 0 1 1 − − − At first, from this figure, we can confirm that there are similar features of the CMB power spectrum of tensor modes [26, 27]. In the III spectra, the dominant signals are located in `3 < 100 due to the enhancement of the integrated Sachs-Wolfe effect. On the other hand, since the fluctuation of polarizations is mainly produced through Thomson scattering at around the recombination and reionization epoch, the BBB spectra have two peaks for `3 < 10 and `3 100, respectively. The cross-correlated bispectra between I and B modes seem to contain both∼ these effects. These features back up the consistency of our calculation. The curves in Fig. 8.2 denote the spectra for A = 1/2, 0, 1/2, and 1, respectively. We notice that the spectra for large A become red compared with− those for small A. The difference in tilt of ` between these spectra is just one corresponding to the difference in A. The curves of the left P3 and right figures obey n=1 `n = even and = odd, respectively. As mentioned in Sec. 8.2.1, we stress again that in the ` configuration where the bispectrum from W 3 vanishes, the bispectrum from WWf 2 survives, and vice versa for each correlation. This is because the parities of these terms are opposite each other. For example, this predicts a nonzero III spectrum not only for P3 3 P3 2 n=1 `n = even due to W but also for n=1 `n = odd due to WWf . We can also see that each bispectrum induced by W 3 has a different shape from that induced by WWf 2 corresponding to the difference in the primordial bispectra. Regardless of this, the overall amplitudes of the spectra for A = 1/2 are almost identical. However, if we consider A deviating ± from these values, the balance between the contributions of W 3 and WWf 2 breaks. For example, if 1/2 < A < 1/2, the contribution of W 3 dominates. Assuming the time-independent coupling, − namely, A = 0, since f (r) = 0, the CMB bispectra are generated only from W 3. Thus, we WWf 2 will never observe the parity violation of gravitons in the CMB bispectrum. On the other hand, when 3/2 < A < 1/2 or 1/2 < A < 3/2, the contribution of WWf 2 dominates. In an extreme − − (r) 2 case, if A = odd, since fW 3 = 0, the CMB bispectra arise only from WWf and violate the parity P3 invariance. Then, the information of the signals under n=1 `n = odd will become more important in the analysis of the III spectrum. 3 In Fig. 8.3, we focus on the III spectra from W for `1 = `2 = `3 ` to compare these with the III spectrum generated from the equilateral-type non-Gaussianity≡ of curvature perturbations given by

Z ∞ " 3 Z ∞ # (SSS) 2 Y 2 2 (S) b = y dy k dkn (kn)j` (kny) III,`1`2`3 π n I,`n n 0 n=1 0 T

3 equil 2 2 f (2π AS) Sequil(k1, k2, k3) , (8.58) ×5 NL 4The conventional expression of the CMB-reduced bispectrum as

* 3 + 0 0 0 −1 X `1 `2 `3 Y bX1X2X3,`1`2`3 (I` ` ` ) aXn,`nmn (8.56) ≡ 1 2 3 m1 m2 m3 m1m2m3 n=1 breaks down for P3 ` = odd due to the divergence behavior of (I0 0 0 )−1. Here, replacing the I symbol with n=1 n `1`2`3 the G symbol, this problem is avoided. Of course, for P3 ` = even, G is identical to I0 0 0 . n=1 n `1`2`3 `1`2`3 104 8 PARITY VIOLATION OF GRAVITONS IN THE CMB BISPECTRUM

100

16 0.01 10 × |

2 0.0001 ) π

/(2 1e-06 lll |b 2 1e-08 W3 (A = -1/2) (l+1)

2 3 l 1e-10 W (A = 0) 3 1e-12 W (A = 1/2) SSS 1e-14 10 100 1000 l

Figure 8.3: Absolute value of the CMB III spectra generated from W 3 for A = 1/2 (red solid line), 0 (green dashed one) and 1/2 (blue dotted one), and generated from the equilateral-type− equil non-Gaussianity given by Eq. (8.58) with fNL = 300 (magenta dot-dashed one). We set that three multipoles have identical values as `1 = `2 = `3 `. Here, we ﬁx the parameters as the same values mentioned in Fig. 8.2. ≡

equil (S) where fNL is the nonlinearity parameter of the equilateral non-Gaussianity and I,` is the transfer T P3 function of scalar mode described in Eq. (3.101). Note that these three spectra vanish for n=1 `n = odd. From this ﬁgure, we can estimate the typical amplitude of the III spectra from W 3 at large scale as 2 4 −4 −2 GeV r b``` ` 3.2 10 . (8.59) | | ∼ × × Λ 0.1 This equation also seems to be applicable to the III spectra from WWf 2. On the other hand, the CMB bispectrum generated from the equilateral-type non-Gaussianity on a large scale is evaluated equil with fNL as

f equil −4 −15 NL b``` ` 4 10 . (8.60) | | ∼ × × 300 equil From these estimations and ideal upper bounds on fNL estimated only from the cosmic variance equil 6 for ` < 100 [28–30], namely fNL . 300 and r 0.1, we ﬁnd a rough limit: Λ & 3 10 GeV. P3 ∼ × Here, we use only the signals for n=1 `n = even due to the comparison with the parity-conserving bispectrum from scalar-mode non-Gaussianity. Of course, to estimate more precisely, we will have P3 to calculate the signal-to-noise ratio with the information of n=1 `n = odd [19].

8.3 Summary and discussion In this section, we have studied the CMB bispectrum generated from the graviton non-Gaussianity induced by the parity-even and parity-odd Weyl cubic terms, namely, W 3 and WWf 2, which have a REFERENCES 105 dilaton-like coupling depending on the conformal time as f τ A. Through the calculation based ∝ on the in-in formalism, we have found that the primordial non-Gaussianities from WWf 2 can have a magnitude comparable to that from W 3 even in the exact de Sitter space-time. Using the explicit formulae of the primordial bispectrum, we have derived the CMB bispectra of the intensity (I) and polarization (E,B) modes. Then, we have confirmed that, owing to the difference in the transformation under parity, the spectra from W 3 vanish in the ` space where those from WWf 2 survive and vice versa. For example, owing to the parity-violating WWf 2 term, P3 P3 the III spectrum can be produced not only for n=1 `n = even but also for n=1 `n = odd, and P3 the IIB spectrum can also be produced for n=1 `n = even. These signals are powerful lines of evidence the parity violation in the non-Gaussian level; hence, to reanalyze the observational data P3 for n=1 `n = odd is meaningful work. When A = 1/2, 0, 1/2, and 1, we have obtained reasonable numerical results of the CMB − bispectra from the parity-conserving W 3 and the parity-violating WWf 2. For A = 1/2, we have ± found that the spectra from W 3 and WWf 2 have almost the same magnitudes even though these have a small difference in the shapes. In contrast, if A = 0 and 1, we have confirmed that the signals from WWf 2 and W 3 vanish, respectively. In the latter case, we will observe only the parity-violating signals in the CMB bispectra generated from the Weyl cubic terms. We have also found that the shape of the non-Gaussianity from such Weyl cubic terms is quite similar to the equilateral-type non-Gaussianity of curvature perturbations. In comparison with the III spectrum generated from 6 the equilateral-type non-Gaussianity, we have found that if r = 0.1, Λ & 3 10 GeV corresponds equil × approximately to fNL . 300. Strictly speaking, to obtain the bound on the scale Λ, we need to calculate the signal-to-noise P3 P3 ratio with the information of not only n=1 `n = even but also n=1 `n = odd for each A by the application of Ref. [19]. This will be discussed in the future.

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9 CMB bispectrum generated from primordial magnetic ﬁelds

Recent observational consequences have shown the existence of (10−6)G magnetic fields in galaxies and clusters of galaxies at redshift z 0.7 2.0 [1–3]. OneO of the scenarios to realize this is an amplification of the magnetic fields by∼ the− galactic dynamo mechanism (e.g. [4]), which requires (10−20)G seed fields prior to the galaxy formation. A variety of studies have suggested the possibilityO of generating the seed fields at the inflationary epoch [5, 6], the cosmic phase tran- sitions [7, 8], and cosmological recombination [9–11] and also there have been many studies about constraints on the strength of primordial magnetic fields (PMFs) through the impact on the cosmic microwave background (CMB) anisotropies, in particular, the CMB power spectrum sourced from the PMFs [12–17]. The PMFs excite not only the scalar fluctuation but also the vector and tensor fluctuations in the CMB fields. For example, the gravitational waves and curvature perturbations, which come from the tensor and scalar components of the PMF anisotropic stresses, produce additional CMB fluctuations at large and intermediate scales [15, 17]. In addition, it has been known that the magnetic vector mode may dominate the CMB small-scale fluctuations by the Doppler effect (e.g. [14, 15]). The PMF anisotropic stresses depend quadratically on the magnetic seed fields. Thus, assuming the gaussianity of the PMF, the anisotropic stress and CMB fluctuation obey the highly non- Gaussian statistics [18, 19]. Owing to the Wick’s theorem, the CMB bispectrum contains the pure non-Gaussian information. Hence, to extract the information of the PMF from the CMB fields, the analysis of the CMB bispectrum is of great utility. Recently, in Refs. [20–23], the authors investigated the contribution of the scalar-mode anisotropic stresses of PMFs to the bispectrum of the CMB temperature fluctuations. From current CMB experimental data, some authors obtained rough limits on the PMF strength smoothed on 1Mpc scale as B1Mpc < (1)nG. However, since in all these studies, the complicated angular dependence on the wave numberO vectors are neglected, there may exist any uncertainties. In addition, the authors have never considered the dependence on the vector- and tensor-mode contributions and hence more precise discussion including these concerns should be realized. With these motivations, we have studied the CMB scalar, vector and tensor bispectra induced from PMFs and firstly succeeded in the exact computation of them with the full-angular dependence [24–26] by applying the all-sky formulae for the CMB bispectrum [27] 1. In our studies, we also updated constraints on the PMF strength. In this section, after reviewing the impact of PMFs on the CMB anisotropies, we present the derivation of the CMB bispectra induced from PMFs and discuss their behaviors. In addition, we put limits on the PMFs by considering the WMAP data and the expected PLANCK data [29, 30]. Finally, we mention our future works. These discussions are based on our studies [24–26],

9.1 CMB fluctuation induced from PMFs The PMFs drive the Lorentz force and the anisotropic stress, and change the motion of baryons (protons and electrons) and the growth of the gravitational potential via the Euler and Einstein equations. Consequently, the photon’s anisotropy is also affected. We illustrate this in Fig. 9.1. In the following discussion, we summarize the impacts of PMFs on the CMB fluctuations in detail

1In Ref. [28], after us, the authors presented an analytic formula for the CMB temperature bispectrum generated from vector anisotropic stresses of the PMF. 9.1 CMB ﬂuctuation induced from PMFs 109

photon

Thomson

electron neutrino

gravity metric Lorentz Coulomb

PMF

proton dark matter

Figure 9.1: Interaction between several components in the Universe if the PMF exists. and current constraints on the PMFs obtained from the CMB power spectrum.

9.1.1 Setting for the PMFs

Let us consider the stochastic PMFs Bb(x, τ) on the homogeneous background Universe which is characterized by the Friedmann-Robertson-Walker metric,

2 2 2 b c ds = a(τ) dτ + δbcdx dx , (9.1) − where τ is a conformal time and a(τ) is a scale factor. The expansion of the Universe makes the amplitude of the magnetic ﬁelds decay as 1/a2 and hence we can draw oﬀ the time dependence as Bb(x, τ) = Bb(x)/a2. Each component of the energy momentum tensor (EMT) of PMFs is given by

0 µ 1 2 µ T (x ) = ρB = B (x) ργ(τ)∆B(x ) , 0 − −8πa4 ≡ − 0 µ b µ T c(x ) = T 0(x ) = 0 , (9.2) 2 b µ 1 B (x) b b µ b b µ T (x ) = δ B (x)Bc(x) ργ(τ) ∆B(x )δ + Π (x ) . c 4πa4 2 c − ≡ c Bc 110 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS

The Fourier components of the spatial parts are described as

b b b T (k, τ) ργ(τ) δ ∆B(k) + Π (k) , c ≡ c Bc Z 3 0 1 d k b 0 0 ∆B(k) = 3 B (k )Bb(k k ) , 8πργ,0 (2π) − (9.3) Z 3 0 b 1 d k b 0 0 ΠBc(k) = 3 B (k )Bc(k k ) , −4πργ,0 (2π) − where we have introduced the photon energy density ργ in order to include the time dependence −4 of a and ργ,0 denotes the present energy density of photons. In the following discussion, for simplicity of calculation, we ignore the trivial time-dependence. Hence, the index is lowered by δbc and the summation is implied for repeated indices. Assuming that Ba(x) is a Gaussian ﬁeld, the statistically isotropic power spectrum of PMFs 2 PB(k) is deﬁned by

3 PB(k) ˆ Ba(k)Bb(p) = (2π) Pab(k)δ(k + p) , (9.4) h i 2 with a projection tensor

ˆ X (σ) ˆ (−σ) ˆ ˆ ˆ Pab(k) a (k)b (k) = δab kakb , (9.5) ≡ σ=±1 −

(±1) which comes from the divergence free nature of PMFs. Here kˆ denotes a unit vector and a is a a (±1) normalized divergenceless polarization vector which satisﬁes the orthogonal condition; kˆ a = 0. The details of the relations and conventions of the polarization vector are described in Appendix D. Although the form of the power spectrum PB(k) is strongly dependent on the production mechanism, we assume a simple power law shape given by

nB PB(k) = ABk , (9.6) where AB and nB denote the amplitude and the spectral index of the power spectrum of magnetic fields, respectively. In order to parametrize the strength of PMFs, we smooth the magnetic fields with a conventional Gaussian filter on a comoving scale r:

Z ∞ 2 2 k dk −k2r2 Br 2 e PB(k) , (9.7) ≡ 0 2π then, AB is calculated as

nB +5 2 (2π) Br AB = , (9.8) nB +3 nB +3 Γ( 2 )kr where Γ(x) is the Gamma function and kr 2π/r. We focus on the scalar, vector and tensor≡ contributions induced from the PMFs, which come from the anisotropic stress of the EMT, i.e., ΠBab. Following the deﬁnition of the projection

2Here we neglect the helical component. This effect will be considered in Ref. [31]. 9.1 CMB fluctuation induced from PMFs 111 operators in Appendix D, the PMF anisotropic stress fluctuation is decomposed into 3 Π(0)(k) = O(0)(kˆ)Π (k) , Bs 2 ij Bij 1 Π(±1)(k) = O(∓1)(kˆ)Π (k) , (9.9) Bv 2 ij Bij 1 Π(±2)(k) = O(∓2)(kˆ)Π (k) . Bt 2 ij Bij These act as sources of the CMB scalar-, vector- and tensor-mode fluctuations as follow.

9.1.2 Scalar and tensor modes If the seed magnetic fields exist in the early Universe, the scalar and tensor components of the PMF anisotropic stress are not compensated prior to neutrino decoupling [15, 17], and the scalar and tensor metric perturbations generated from them survive passively. These residual metric perturbations generate the CMB anisotropies of the scalar and tensor modes. These kind of CMB anisotropies are so called “passive mode” and may dominate at intermediate and large scales depending on the PMF strength [17]. To estimate curvature perturbation and gravitational wave driven by PMFs on superhorizon scales, we shall focus on the Einstein equation at the radiation dominated era. Before neutrino decoupling, the Universe is dominated by the radiative fluid. The fluid is tightly coupled to baryons and can not create any anisotropic stress. Hence, in this period, total anisotropic stress comes (0/±2) from only PMFs, namely, constant ΠBs/t . Until neutrino decoupling, this survives and it will be a source of metric perturbations via the Einstein equation. Then, at the superhorizon limit, the Einstein equation for scalar and tensor modes on the synchronous gauge (3.21) reduces to same form as 2 6 h¨(0/±2)(k, τ) + h˙ (0/±2)(k, τ) = R Π(0/±2)(k) , (9.10) τ τ 2 γ Bs/t 2 2 where we have used =a/a ˙ = 1/τ, = 8πGρa /3 and Rγ ργ/ρ 0.6. This is analytically solved as H H ≡ ≈ (0/±2) C2 (0/±2) τ h (k, τ) = C1 + + 6RγΠBs/t (k) ln , (9.11) τ τB where τB is the conformal time at the generation of the PMF. On the other hand, after neu- −1 trino decoupling (τ > τν = 1MeV ), resultant neutrino anisotropic stress compensates the PMF anisotropic stress. Hence, right-hand side of the above Einstein equation becomes zero and the growth of metric perturbations ceases. Accordingly, for τ τν, superhorizon-scale comoving curvature and tensor perturbations are evaluated as (0) τν (k) = (k, τB) + RγΠBs(k) ln , (9.12) R R τB (±2) (±2) (±2) τν h (k) = h (k, τB) + 6RγΠBt (k) ln , (9.13) τB where we have used a relation of the scalar perturbations on superhorizon scales: h(0) = 6 3. This logarithmic growth and saturation of metric perturbations even on superhorizon scalesR are

3 and h(±2) are equal to ζ and √3H of Refs. [15, 17], respectively. R − − T 112 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS caused by only the property of sources as ρ a−4, hence the above discussion is applicable to the case of general radiation fluid other than PMFs∝ [32]. These metric perturbations act as a source of the CMB fluctuations of scalar and tensor modes. (S) (T ) Note that the effects of PMFs on the transfer functions X,` and X,` are inconsiderable at larger scales [17], hence it is safe to use the non-magneticT transfer functionsT (3.101) in the computation of the CMB spectra induced from scalar and tensor modes of the PMF anisotropic stress.

9.1.3 Vector mode The vector mode has no equivalent passive mode as the gravitational potential of the vector mode decays away via the Einstein equation posterior to neutrino decoupling. Thus, in the vector mode, we need to consider the impact of the PMF on the transfer function. In Refs. [13, 14, 25, 28], it is discussed that the temperature ﬂuctuations are generated via Doppler and integrated Sachs-Wolfe eﬀects on the CMB vector modes. On the basis of them, we derive the transfer function of the vector magnetic mode as follows. When we decompose the metric perturbations into vector components as

2 δg0c = δgc0 = a Ac , 2 (V ) (V ) (9.14) δgcd = a ∂chd + ∂dhc , we can construct two gauge-invariant variables, namely a vector perturbation of the extrinsic curvature and a vorticity, as

V A h˙ , ≡ − (9.15) Ω v A , ≡ − where v is the spatial part of the four-velocity perturbation of a stationary ﬂuid element and a dash denotes a partial derivative of the conformal time τ. Here, choosing a gauge as h˙ = 0, we can express the Einstein equation

16πGρ (Π(V) + Π(V) + Π(V)) V˙ + 2 V = γ,0 γ ν B , (9.16) H − a2k and the Euler equations for photons and baryons ˙ Ωγ +κ ˙ (vγ vb) = 0 , (9.17) − (V) ˙ κ˙ L Ωb + Ωb (vγ vb) = 4 . (9.18) H − R − a (ρb + pb)

(V) (V) (V ) ˆ Here L kργ,0ΠB is the Lorentz force of vector mode and Πa = ikbPacΠbc, p is the isotropic pressure, the≡ indices γ, ν and b denote the photon, neutrino and baryon,− κ is the optical depth, and R (ρb + pb)/(ργ + pγ) 3ρb/(4ργ). In the tight-coupling limit as vγ vb, the photon vorticity ≡ ' ' is comparable to the baryon one: Ωγ Ωb Ω. Then, the Euler equations (9.17) and (9.18) are combined into ' ≡

(V) ˙ L (1 + R)Ω + R Ω = 4 , (9.19) H a (ργ + pγ) 9.1 CMB ﬂuctuation induced from PMFs 113 and this solution is given by τL(V)(k) Ω(k, τ) , (9.20) ' (1 + R)(ργ,0 + pγ,0) Note that Eq. (9.19) and the above solution are valid for perturbation wavelengths larger than the comoving Silk damping scale LS 2π/kS, namely, k < kS, where photon viscosity can be ≡ neglected compared to the Lorentz force. For k > kS, due to the eﬀect of the photon vorticity, the Euler equation (9.19) is changed as [12]

2 (V) ˙ k χ L (1 + R)Ω + R + Ω = 4 , (9.21) H aρb a (ργ + pγ)

−1 where χ = (4/15)ργLγa is the photon shear viscosity coeﬃcient and Lγ =κ ˙ is the photon comoving mean-free path. We can obtain this analytical solution: L(V)(k) Ω(k, τ) 2 . (9.22) ' (k Lγ/5)(ργ,0 + pγ,0) Hence, we can summarize the vorticity of the baryon and photon ﬂuids as

(V) Ω(k, τ) β(k, τ)ΠB (k) , ' ( ρ kτ/(1 + R) for k < k (9.23) β(k, τ) = γ,0 S . ργ,0 + pγ,0 × 5κ/k ˙ for k > kS As mentioned above, the CMB temperature anisotropies of vector modes are produced through the Doppler and integrated Sachs-Wolfe effects as ∆I(nˆ) Z τ0 τ0 ˙ = vγ nˆ τ∗ + dτV nˆ , (9.24) I − · | τ∗ · ˆ where τ0 is today and τ∗ is the recombination epoch in conformal time, µk,n k nˆ, x k(τ0 τ), and nˆ is a unit vector along the line-of-sight direction. Because of compensation≡ · of the≡ anisotropic− stresses, a solution of the Einstein equation (9.16) expresses the decaying signature as V a−2 after neutrino decoupling. Therefore, in an integrated Sachs-Wolfe effect term, the contribution∝ around the recombination epoch is dominant. Furthermore, neglecting dipole contribution due to v today, we can form the coefficient of anisotropies as Z ∆I(nˆ) a(V ) d2nˆ Y ∗ (nˆ) I,`m ≡ I `m Z d3k Z 2 (V) ∗ −iµk,nx∗ d nˆ[Π (k) nˆ]Y (nˆ)β(k, τ∗)e . (9.25) ' (2π)3 B · `m

In the transformation nˆ (µk,n, φk,n), the functions are rewritten as → s 2 (V) 1 µk,n X (λ) Π (k) nˆ i − Π (k)eiλφk,n , B · → − 2 Bv λ=±1 (9.26) ∗ X (`) ˆ ∗ Y (nˆ) D 0 S(k) Y 0 (Ωk,n) , `m → mm `m m0 2 d nˆ dΩk,n , → 114 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS

(V ) P (λ) (λ) where we use the relation: Πa = iΠ a and the Wigner D matrix under the rotational λ=±1 − Bv transformation of a unit vector parallel to z axis into kˆ corresponding to Eq. (3.96). Therefore, performing the integration over Ωk,n in the same manner as Sec. 3, we can obtain the explicit form (V ) of aI,`m and express the radiation transfer function introduced in Eq. (3.100) as Z 3 (V ) ` d k X ∗ ˆ (λ) (V ) a = 4π( i) λ−λY (k)Π (k) (k) , I,`m − (2π)3 `m Bv TI,` λ=±1 (9.27) 1/2 (V ) (` + 1)! β(k, τ∗) j`(x∗) I,` (k) . T ' (` 1)! √2 x∗ − This is consistent with the results presented in Refs. [15, 33]. In the same manner, the vector-mode transfer functions of polarizations are derived [14, 19]. Then, we can also express as

Z 3 (V ) ` d k X x+1 ∗ ˆ (λ) (V ) a = 4π( i) λ −λY (k)Π (k) (k) . (9.28) X,`m − (2π)3 `m Bv TX,` λ=±1

9.1.4 Expression of a`m’s From the above results and Eq. (5.3), the CMB intensity and polarization ﬂuctuations induced from PMFs are summarized as Z 2 (Z) k dk X (λ) (Z) a = 4π( i)` [sgn(λ)]λ+xξ (k) (k) , X,`m − (2π)3 `m TX,` λ Z (0) 2 ˆ ∗ ˆ τν (0) ξ`m(k) d kY`m(k) Rγ ln ΠBs(k) , ≈ τB (9.29) Z (±1) 2 ˆ ∗ ˆ (±1) ξ (k) d k∓1Y (k)Π (k) , `m ≈ `m Bv Z (±2) 2 ˆ ∗ ˆ τν (±2) ξ`m (k) d k∓2Y`m(k) 6Rγ ln ΠBt (k) , ≈ τB

(±2) (0) (±2) (±2) where we take (k, τB) = h (k, τB) = 0, and regard ξ and ξ as and h , respectively 4. R R

9.1.5 CMB power spectrum from PMFs From the formulae (9.29), the CMB power spectra from PMFs are written as

* 2 + " 2 Z 2 #* 2 + Y (Zn) Y `n kndkn (Zn) X λn+xn Y (λn) a = 4π( i) (kn) [sgn(λn)] ξ (kn) . (9.30) Xn,`nmn − (2π)3 TXn,`n `nmn n=1 n=1 λn n=1 D E To compute the initial power spectrum Q2 ξ(λn) (k ) , we need to deal with the power spectrum n=1 `nmn n of the anisotropic stresses as

" 2 Z 3 0 # −2 Y d k Π (k )Π (k ) = ( 4πρ ) n Bab 1 Bcd 2 γ,0 (2π)3 h i − n=1

4In Refs. [31, 34], we equate ξ(0) to . −R 9.1 CMB ﬂuctuation induced from PMFs 115

10000

1000 ] 2 K µ [

2 100 0 )T π /(2 l 10 l(l+1)C

1 PMF tensor PMF vector PMF scalar curvature 0.1 10 100 1000 l

Figure 9.2: Power spectra of the CMB intensity fluctuations. The red solid, green dashed and blue dotted lines correspond to the spectra generated from the tensor, vector and scalar components of the PMF anisotropic stress for nB = 2.9, respectively. The upper (lower) line of the red solid and − 17 6 blue dotted ones are calculated when τν/τB = 10 (10 ). The magenta dot-dashed line expresses the spectrum sourced from the primordial curvature perturbations. The strength of PMFs is fixed to B1Mpc = 4.7nG and the other cosmological parameters are fixed to the mean values limited from WMAP-7yr data reported in Ref. [29].

0 0 0 0 Ba(k )Bb(k1 k )Bc(k )Bd(k2 k ) × h 1 − 1 2 − 2 i 2 ! Z Y −2 3 0 0 0 = δ kn ( 4πργ,0) d k1PB(k1)PB( k1 k1 ) n=1 − | − | 1 ˆ0 0 ˆ0 0 [Pad(k )Pbc(k\1 k ) + Pac(k )Pbd(k\1 k )] . (9.31) ×4 1 − 1 1 − 1 Note that this equation includes the convolution integral and the complicated angular dependence. In Refs. [16, 35–37], the authors performed the numerical and analytical computation of this 0 convolution integral over k1 and provided the fitting formulae with respect to the magnitude of the wave numbers k1 for each value of the magnetic spectral index nB. In Fig. 9.2, we plot the power spectra of the intensity anisotropies (9.30) for the scalar, vector and tensor modes when magnetic spectrum is nearly scale invariant as nB = 2.9. Here, we assume that the PMFs generate from the epoch of the grand unification to that of− the electroweak phase 17 6 transition, i.e., τν/τB = 10 10 . Firstly, we will see that the shapes of the tensor and scalar power spectra are similar to those− of the non-magnetic case coming from the scale-invariant primordial spectra shown in Fig. 3.5. This is because PMFs impact on only the primordial gravitational waves and primordial curvature perturbations, and do not change the transfer functions of the tensor and scalar modes. For ` . 100, the tensor mode dominates over the intensity signal. The scalar mode seems to dominate in the intermediate scale as 100 . ` . 2000. The vector-mode spectrum monotonically increases for larger than Silk damping scale, namely, ` kSτ0 2000, and decreases . ∼ for ` & 2000. These features seem to trace the scaling relation of the transfer function in terms of 116 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS the wave number (9.23). Hence, we can understand that the latter damping effect arises from the viscosity of photons. The vector mode seems to show up for very small scale, namely, ` & 2000. In this figure, for comparison, we also plot the CMB intensity power spectrum from the primordial curvature perturbations not depending on the PMF. In principle, comparing this spectrum with that sourced from PMFs leads to bounds on the PMF parameters. Actually, the researchers perform the parameter estimation by the Malkov Chain Monte Carlo approach [35, 36, 38–42]. So far, the most stringent limit on the PMF strength from the CMB two-point function data of the intensity and polarizations are B1Mpc < 5nG and nB < 0.12 [40]. In Refs. [35, 42], combining the CMB data with the information of the matter power− spectrum, tighter bounds are gained. As discussed above, conventionally, the CMB power spectra from PMFs are computed by using the fitting formulae for the power spectra of the magnetic anisotropic stresses. However, without these formulae, we can obtain the CMB power spectra by applying the mathematical tools such as the Wigner symbols [25]. In the remaining part, focusing on the vector mode, we present this new approach and show the consistency with the conventional result. Then, we use some colors in the equations for readers to follow the complex equations more easily. From Eq. (9.30), the CMB power spectrum of the intensity mode induced from the magnetic- vector-mode anisotropic stress is formulated as

" 2 # D E Z ∞ k2dk (V ) (V )∗ Y n n (V ) `1 `2 a a = 4π (kn) ( i) i I,`1m1 I,`2m2 (2π)3 I,`n n=1 0 T −

X D (λ1) (λ2)∗ E λ1λ2 Π (k1)Π (k2) × Bv,`1m1 Bv,`2m2 λ1,λ2=±1 (V ) C δ` ,` δm ,m , (9.32) ≡ II,`1 1 2 1 2 where the initial power spectrum, which is expanded by the spin spherical harmonics, is

D E Z Z (λ1) (λ2)∗ −2 2 ˆ 2 ˆ ∗ ˆ ˆ Π (k1)Π (k2) = ( 4πργ,0) d k1 d k2−λ Y (k1)−λ Y` m (k2) Bv,`1m1 Bv,`2m2 − 1 `1m1 2 2 2 Z kD Z kD 02 0 0 02 0 0 k1 dk PB(k1) k2 dk2PB(k2) × 0 0 Z Z 2 ˆ0 2 ˆ0 0 0 0 0 d k d k δ(k1 k k )δ(k2 k k ) × 1 2 − 1 − 2 − 2 − 1 1 h i ˆ (−λ1) ˆ ˆ (λ2) ˆ ˆ0 ˆ0 ˆ0 ˆ0 k1 (k1)k2 (k2) Pad(k )Pbc(k ) + Pac(k )Pbd(k ) , ×4 a b c d 1 2 1 2 (9.33)

(±1) ˆ where we use Eq. (9.9) and the definition of the vector projection operator, Oab (k), in Ap- pendix D. Note that we rewrote the power spectrum (9.31) as more symmetric form in terms of k1, k2 and k3. Then, we should simplify this initial power spectrum. For the first part in two permutations, we calculate δ-functions and the summations with 9.1 CMB fluctuation induced from PMFs 117 respect to a, b, c and d:

Z ∞ X L1+3L2+3L3 0 0 2 2 0 0 0 0 0 δ(k1 k1 k2) = 8 A dA ( 1) IL1L2L3 jL1 (k1A)jL2 (k1A)jL3 (k2A) − − 0 − L1L2L3 M1M2M3 L L L ∗ ˆ ˆ0 ∗ ˆ0 M2 1 2 3 YL1M1 (k1)YL2M2 (k1)YL3−M3 (k2)( 1) , × − M1 M2 M3 − − Z ∞ L0 +3L0 +3L0 0 0 2 X 1 2 3 0 0 0 0 0 δ(k2 k k ) = 8 B dB ( 1) 2 I 0 0 0 jL0 (k2B)jL0 (k B)jL0 (k B) 2 1 L1L2L3 1 2 2 3 1 − − 0 0 0 0 − L1L2L3 0 0 0 M1M2M3 0 0 0 0 L L L ∗ ˆ ˆ0 ∗ ˆ0 M2 1 2 3 (9.34) YL0 M 0 (k2)YL0 M 0 (k2)YL0 −M 0 (k1)( 1) 0 0 0 , × 1 1 2 2 3 3 − M M M 1 − 2 − 3 2 (λ ) X X 4π kˆ 2 (kˆ )P (kˆ0 ) = λ 1a d 2 ad 1 3 2 σ=±1 ma,md=±1,0 ˆ ˆ ∗ ˆ0 ∗ ˆ0 Y1m (k1)λ Y1m (k2)−σY (k )σY (k ) , × a 2 d 1ma 1 1md 1 4π 2 ˆ (−λ1) ˆ ˆ0 X X k2cb (k1)Pbc(k2) = ( λ1) 0 3 − σ =±1 mc,mb=±1,0 ˆ ˆ ∗ ˆ0 ∗ ˆ0 Y1m (k2)−λ Y1m (k1)−σ0 Y (k )σ0 Y (k ) , × c 1 b 1mc 2 1mb 2 perform the angular integrals of the spin spherical harmonics:

Z 2 ˆ0 ∗ ∗ ∗ d k −σY YL M σY Y 0 0 1 1ma 2 2 1md L3−M3 0 X σ+ma 0−σ−S 0−σ−S L3 1 L L2 1 L = ( 1) IL0 1L IL 1L 0 , − 3 2 M md M M2 ma M LMS − 3 − Z 2 0 ∗ ∗ ∗ d kˆ 0 Y Y 0 0 0 Y Y 2−σ 1mc L2M2 σ 1mb L3−M3 0 0 0 0 0 0 0 0 X σ +mc 0−σ −S 0−σ −S L3 1 L L2 1 L = ( 1) I 0 I 0 , L31L L 1L0 0 0 0 − 2 M3 mb M M2 mc M L0M 0S0 − − Z (9.35) 2 ˆ ∗ ∗ d k1−λ1 Y1mb Y1ma −λ1 Y`1m1 YL1M1 X 0λ1−Sk 0λ1−Sk L1 `1 Lk 1 1 Lk = IL ` L I11L , 1 1 k k M1 m1 Mk ma mb Mk LkMkSk Z 2 ˆ ∗ d k2λ Y1m Y1m −λ Y` m Y 0 0 2 d c 2 2 2 L1M1 0 X m2+λ2 0λ2−Sp 0λ2−Sp L1 `2 Lp 1 1 Lp = ( 1) I 0 I , L `2Lp 11Lp 0 − 1 M1 m2 Mp mc md Mp LpMpSp − − − 118 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS sum up the Wigner-3j symbols over the azimuthal quantum numbers: X 1 1 L L L L ( 1)M2+ma k 1 2 3 − ma mb Mk M1 M2 M3 M1M2M3 − − Mkmamb 0 L3 1 L L2 1 L L1 `1 Lk 0 × M3 mb M M2 ma M M1 m1 Mk − −  0  0  L L `1  M+`1+L3+L+1 L L `1 = ( 1) 0 L3 L2 L1 , − M M m1 −  1 1 Lk  0 0 0 (9.36) X M 0 +m 1 1 Lp L L L ( 1) 2 c 1 2 3 m m M M 0 M 0 M 0 0 0 0 − c d p 1 2 3 M1M2M3 − − − − Mpmcmd 0 0 0 0 L2 1 L L3 1 L L1 `2 Lp 0 0 0 0 M mc M M md M M m2 Mp 2 − − 3 − 1  0  0 L L `2 0 0 L L `   M +`2+L2+L+1+Lp 2 0 0 0 = ( 1) 0 L2 L3 L1 , − M M m2 −  1 1 Lp  and sum up the Wigner-3j symbols over M and M 0:

0 0 m2 X M+M 0 L L `1 L L `2 ( 1) ( 1) 0 0 = − δ`1,`2 δm1,m2 . (9.37) − M M m1 M M m2 2`1 + 1 MM 0 − −

Following the same procedures in the other permutation and calculating the summation over Lp as  L0 L `   L0 L `  Lp  2   2  X 0λ2−λ2 0λ2−λ2 1 + ( 1) 0 0 0 3 0λ2−λ2 0 0 0 I 0 I L L L = I 0 L L L , (9.38) L ` L 11Lp − 2 3 1 L ` 2 3 2 1 1 2 p 2 −2√2π 1 2 Lp  1 1 Lp   1 1 2  we can obtain the exact solution of Eq. (9.33) as

D E √2π 8(2π)1/2 2 Π(λ1) (k )Π(λ2)∗ (k ) = /(2` + 1)δ δ Bv,`1m1 1 Bv,`2m2 2 1 `1,`2 m1,m2 − 3 3ργ,0 L +L0 X X P3 i i 0 0 0 0 0 0 ( 1) i=1 2 I I 0 0 0 L1L2L3 L1L2L3 × 0 − LL L1L2L3 0 0 0 L1L2L3  0   0  L L `1 L L `2 0     X L +L3 0 0 0 ( 1) 2 L3 L2 L1 L L L × − 2 3 1 Lk  1 1 Lk   1 1 2  Z ∞ Z ∞ 2 2 A dAj (k A) B dBj 0 (k B) L1 1 L1 2 × 0 0 Z kD 02 0 0 0 0 k dk P (k )j (k A)j 0 (k B) 1 1 B 1 L2 1 L3 1 × 0 Z kD 02 0 0 0 0 k dk P (k )j 0 (k B)j (k A) 2 2 B 2 L2 2 L3 2 × 0 9.1 CMB ﬂuctuation induced from PMFs 119

0 0 0 0 0 0 X L2+L2+L3+L3 0S−S 0S−S 0S −S 0S −S ( 1) IL0 1L IL 1L IL 1L0 IL0 1L0 × − 3 2 3 2 S,S0=±1

0λ1−λ1 0λ1−λ1 0λ2−λ2 λ1λ2IL ` L I11L IL0 ` 2 . (9.39) × 1 1 k k 1 2

Note that in this equation, the dependence on the azimuthal quantum number is included only in

δm1,m2 . In the similar discussion of the CMB bispectrum, this implies that the CMB vector-mode power spectrum generated from the magnetized anisotropic stresses is rotationally-invariant if the PMFs satisfy the statistical isotropy as Eq. (9.4). Furthermore, using such evaluations as

0 0 0 0 0 0 X L2+L2+L3+L3 0S−S 0S−S 0S −S 0S −S ( 1) IL0 1L IL 1L IL 1L0 IL0 1L0 − 3 2 3 2 S,S0=±1 ( 01−1 01−1 01−1 01−1 0 0 4IL0 1LIL 1LIL 1L0 IL0 1L0 (L3 + L2,L3 + L2 = even) = 3 2 3 2 , (9.40) 0 (otherwise)

( 01−1 01−1 01−1 0 4I I I 0 (L1 + `1,L1 + `2 = even) X 0λ1−λ1 0λ1−λ1 0λ2−λ2 L1`1Lk 11Lk L1`22 IL ` L I11L IL0 ` 2 = , (9.41) 1 1 k k 1 2 0 (otherwise) λ1,λ2=±1 " 2 Z ∞ 2 # Z ∞ Z ∞ Y kndkn (V ) 2 2 4π (kn) A dAjL (k1A) B dBjL0 (k2B) (2π)3 I,`n 1 1 n=1 0 T 0 0 Z kD Z kD 02 0 0 0 0 02 0 0 0 0 k dk P (k )j (k A)j 0 (k B) k dk P (k )j 0 (k B)j (k A) 1 1 B 1 L2 1 L3 1 2 2 B 2 L2 2 L3 2 × 0 0 " 2 Z ∞ 2 # Y kndkn (V ) 4π (kn)j` (kn(τ0 τ∗)) (2π)3 I,`n n ' n=1 0 T − 2 2 4 τ∗ −(nB +1) −(nB +1) AB(τ0 τ∗) L L0 (τ0 τ∗) L0 L (τ0 τ∗) , (9.42) × − 5 K 2 3 − K 2 3 − the CMB angle-averaged power spectrum is formulated as

1/2 2 Z ∞ 2 2 (V ) √2π 32(2π) k dk (V ) CII,` /(2` + 1) 4π 3 I,` (k)j`(k(τ0 τ∗)) ' − 3 3ργ,0 0 (2π) T − 2 X X 01−1 01−1 01−1 X X 2 4 τ∗ I I I 0 AB(τ0 τ∗) L1`Lk 11Lk L1`2 × 0 0 0 − 5 L1L1 Lk LL L2L2 0 L3L3

−(nB +1) −(nB +1) L L0 (τ0 τ∗) L0 L (τ0 τ∗) ×K 2 3 − K 2 3 − 0 P3 Li+Li 0 i=1 +L2+L3 0 0 0 0 0 0 01−1 01−1 01−1 01−1 ( 1) 2 I I 0 0 0 I 0 I I 0 I 0 L1L2L3 L L L L 1L L 1L L 1L L 1L0 × − 1 2 3 3 2 3 2  L0 L `   L0 L `     0 0 0  L3 L2 L1 L L L . (9.43) × 2 3 1  1 1 Lk   1 1 2 

Here, we use the thin LSS approximation described in Sec. 9.3.1. This has nonzero value in the 120 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS

10000

1000 ] 2 K µ [ 2

0 100 )T π / (2

II,l 10 l(l+1)C 1 PMF vector ∝ l2.1 primary scalar 0.1 100 1000 l

Figure 9.3: CMB power spectra of the temperature fluctuations. The lines correspond to the spectra generated from vector anisotropic stress of PMFs as Eq. (9.43) (red solid line) and primordial curvature perturbations (blue dotted line). The green dashed line expresses the asymptotic power of the red solid one. The PMF parameters are fixed to nB = 2.9 and B1Mpc = 4.7nG, and the other cosmological parameters are fixed to the mean values limited− from WMAP-7yr data reported in Ref. [29]. configurations:

0 (Lk,L1) = (2, ` 2 ), (2, `), (1, `) ,L = ` 2 , ` , | ± | 1 | ± | ` L L0 ` + L, | − 0| ≤ ≤ (L2,L3) = ( L 1 , L 1 ), (L, L), (L + 1, L 1 ) , 0 | 0− | | ±0 | 0 0 0 | ± 0 | (9.44) (L2,L3) = ( L 1 , L 1 ), (L ,L ), (L + 1, L 1 ) , | − | | ± 0 | 0 0 | ± | L1 + L2 + L3 = even ,L1 + L2 + L3 = even , 0 0 0 0 0 L1 L2 L3 L1 + L2 , L L L L + L . | − | ≤ ≤ | 1 − 2| ≤ 3 ≤ 1 2 This shape is described in Fig. 9.3. From this figure, we confirm that the amplitude and the overall behavior of the red solid line are in broad agreement with the green dashed line of Fig. 9.2 and the previous studies (e.g. [15–17, 43]). For ` . 2000, using the scaling relations of the Wigner symbols at the dominant configuration L `, L0 1 as discussed in Sec. 9.5, we analytically find (V ) nB +3 ∼ ∼ that CI,` ` . This traces our numerical results as shown by the green dashed line. This computation∝ approach is of great utility in the higher-order correlations. In the next subsection, in accordance with this approach, we compute the CMB bispectra sourced from PMFs.

9.2 Formulation for the CMB bispectrum induced from PMFs In this subsection, we derive the explicit form of the CMB bispectra induced from PMFs by calculating the full-angular dependence which has never been considered in the previous studies [20–23, 28]. The following procedures are based on the calculation rules discussed in Ref. [25]. Note 9.2 Formulation for the CMB bispectrum induced from PMFs 121 that we use some colors in the following equations for readers to follow the complex equations more easily.

9.2.1 Bispectrum of the anisotropic stress ﬂuctuations

µ According to Eq. (9.2), EMT of PMF at an arbitrary point, T ν(x), depends quadratically on the Gaussian magnetic ﬁelds at that point. This non-Gaussian structure is identical to the local-type non-Gaussianity of the curvature perturbations as mentioned in Sec 4.2, hence it is expected that the statistical properties of the magnetic ﬁelds obey those of the local-type non-Gaussianity. This will be automatically shown in Sec. 9.4. Using Eq. (9.4) and the Wick’s theorem, the bispectrum of the anisotropic stresses is calculated as

" 3 Z 3 0 # −3 Y d k Π (k )Π (k )Π (k ) = ( 4πρ ) n Bab 1 Bcd 2 Bef 3 γ,0 (2π)3 h i − n=1 0 0 0 0 0 0 Ba(k1)Bb(k1 k1)Bc(k2)Bd(k2 k2)Be(k3)Bf (k3 k3) × h " 3 − − # − i Z kD Z −3 Y 02 0 0 2 ˆ0 = ( 4πργ,0) kn dknPB(kn) d kn − n=1 0 0 0 0 0 0 0 δ(k1 k + k )δ(k2 k + k )δ(k3 k + k ) × − 1 3 − 2 1 − 3 2 1 ˆ0 ˆ0 ˆ0 [Pad(k )Pbe(k )Pcf (k ) + a b or c d or e f ] , ×8 1 3 2 { ↔ ↔ ↔ } (9.45)

−1 where kD is the Alfv´en-wave damping length scale [44, 45] as kD (0.1)Mpc and the curly bracket denotes the symmetric 7 terms under the permutations of indices:∼ O a b, c d, or e f. Note that we express in a more symmetric form than that of Ref. [18] to↔ perform↔ the angular↔ integrals which is described in Sec. 9.2. To avoid the divergence of ΠBab(k1)ΠBcd(k2)ΠBef (k3) h i in the IR limit, the value range of the spectral index is limited as nB > 3. We note that this bispectrum depends on the Gaussian PMFs to six, hence this is highly non-Gaussian− compared with the bispectrum of primordial curvature perturbations proportional to the Gaussian variable to four as shown in Sec. 4.2.

9.2.2 CMB all-mode bispectrum Following the general formula (5.5) and using Eq. (9.29), the CMB bispectra induced from PMF are written as

* 3 + " 3 Z ∞ 2 #* 3 + Y (Z ) Y k dkn (Z ) X Y (λ ) a n = 4π( i)`n n n (k ) [sgn(λ )]λn+xn ξ n (k ) . Xn,`nmn 3 Xn,`n n n `nmn n − 0 (2π) T n=1 n=1 λn n=1 (9.46) Remember that the index Z denotes the mode of perturbations: Z = S (scalar), = V (vector) or = T (tensor) and its helicity is expressed by λ; λ = 0 for (Z = S), = 1 for (Z = V ) or = 2 for (Z = T ), X discriminates between intensity and two polarization (electric± and magnetic) modes,± respectively, as X = I,E,B and x is determined by it: x = 0 for X = I,E or = 1 for X = B. In D E the following discussion, we calculate Q3 ξ(λn) (k ) and ﬁnd an explicit formulae of the CMB n=1 `nmn n bispectra corresponding to an arbitrary Z. 122 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS

Using Eqs. (9.29) and (9.45), we can write

* 3 + " 3 # Z Z kD Z Y (λ ) Y ξ n (k ) = ( 4πρ )−3 d2kˆ Y ∗ (kˆ ) k02dk0 P (k0 ) d2kˆ0 `nmn n γ,0 n−λn `nmn n n n B n n n=1 − n=1 0 0 0 0 0 0 0 δ(k1 k + k )δ(k2 k + k )δ(k3 k + k ) × − 1 3 − 2 1 − 3 2 0 (−λ1) ˆ 0 (−λ2) ˆ 0 (−λ3) ˆ C O (k1)C O (k2)C O (k3) × −λ1 ab −λ2 cd −λ3 ef ˆ0 ˆ0 ˆ0 Pad(k )Pbe(k )Pcf (k ) , (9.47) × 1 3 2 with  3 τν Rγ ln (λ = 0)  2 τB 0  1 Cλ (λ = 1) . (9.48) ≡ 2 ±  τν 3Rγ ln (λ = 2) τB ± Let us consider this exact expression by expanding all the angular dependencies with the spin- weighted spherical harmonics and rewriting the angular integrals with the summations in terms of the multipoles and azimuthal quantum numbers. In the ﬁrst step, in order to perform all angular integrals, we expand each function of the wave number vectors with the spin-weighted spherical harmonics. By this concept, three delta functions are rewritten as Z ∞ X L1+3L2+L3 0 0 2 2 0 0 0 0 0 δ(k1 k1 + k3) = 8 A dA ( 1) IL1L2L3 jL1 (k1A)jL2 (k1A)jL3 (k3A) − 0 − L1L2L3 M1M2M3 L L L ∗ ˆ ˆ0 ∗ ˆ0 M2 1 2 3 YL1M1 (k1)YL2M2 (k1)YL3M3 (k3)( 1) , × − M1 M2 M3 − Z ∞ L0 +3L0 +L0 0 0 2 X 1 2 3 0 0 0 0 0 δ(k2 k + k ) = 8 B dB ( 1) 2 I 0 0 0 jL0 (k2B)jL0 (k B)jL0 (k B) 2 1 L1L2L3 1 2 2 3 1 − 0 0 0 0 − L1L2L3 0 0 0 M1M2M3 (9.49) 0 0 0 0 L L L ∗ ˆ ˆ0 ∗ ˆ0 M2 1 2 3 YL0 M 0 (k2)YL0 M 0 (k2)YL0 M 0 (k1)( 1) 0 0 0 , × 1 1 2 2 3 3 − M M M 1 − 2 3 Z ∞ L00+3L00+L00 0 0 2 X 1 2 3 0 0 0 0 0 δ(k3 k + k ) = 8 C dC ( 1) 2 I 00 00 00 jL00 (k3C)jL00 (k C)jL00 (k C) 3 2 L1 L2 L3 1 2 3 3 2 − 0 00 00 00 − L1 L2 L3 00 00 00 M1 M2 M3 00 00 00 00 L L L ∗ ˆ ˆ0 ∗ ˆ0 M2 1 2 3 YL00M 00 (k3)YL00M 00 (k3)YL00M 00 (k2)( 1) 00 00 00 , × 1 1 2 2 3 3 − M M M 1 − 2 3 where r s1s2s3 (2`1 + 1)(2`2 + 1)(2`3 + 1) `1 `2 `3 I` ` ` . (9.50) 1 2 3 ≡ 4π s1 s2 s3 The other functions in Eq. (9.47), which depend on the angles of the wave number vectors, can be also expanded in terms of the spin-weighted spherical harmonics as

(−λ1) ˆ (−λ2) ˆ (−λ3) ˆ ˆ0 ˆ0 ˆ0 Oab (k1)Ocd (k2)Oef (k3)Pad(k1)Pbe(k3)Pcf (k2) 9.2 Formulation for the CMB bispectrum induced from PMFs 123

r 2 X 3 X 4π = C−λ − 1 8π 3 σ1,σ2,σ3=±1 µ1mamb ∗ ˆ 2 1 1 ∗ ˆ0 ∗ ˆ0 λ1 Y2µ1 (k1) −σ1 Y1ma (k1)σ3 Y1m (k3) × µ1 ma mb b r 3 4π 2 X ∗ ˆ 2 1 1 ∗ ˆ0 ∗ ˆ0 C−λ2 λ2 Y2µ2 (k2) −σ2 Y1mc (k2)σ1 Y1m (k1) × 8π 3 µ2 mc md d µ2mcmd r 3 4π 2 X ∗ ˆ 2 1 1 ∗ ˆ0 ∗ ˆ0 C−λ3 λ3 Y2µ3 (k3) −σ3 Y1me (k3)σ2 Y1m (k2) , × 8π 3 µ3 me mf f µ3memf (9.51) where we have used Eqs. (9.5), some relations in Appendices C and D, and the deﬁnition (D.23) as r 3 2 1 1 (λ) ˆ X ∗ ˆ ma mb Oab (k) = Cλ −λY2M (k)αa αb , 8π M ma mb Mmamb  2 (λ = 0) (9.52) − Cλ = 2√3λ (λ = 1) . ± 2√3 (λ = 2) ± In the second step, let us consider performing all angular integrals and replacing them with the ˆ0 ˆ0 ˆ0 Wigner-3j symbols. Three angular integrals with respect to k1, k2 and k3 are given as Z 2 ˆ0 ∗ ∗ ∗ d k −σ Y YL M σ Y Y 0 0 1 1 1ma 2 2 1 1md L3M3 0 X X ma 0−σ1−S 0−σ1−S L3 1 L L2 1 L = ( 1) IL0 1L IL 1L 0 , − − 3 2 M md M M2 ma M LM S=±1 3 − Z 2 ˆ0 ∗ ∗ ∗ d k −σ Y YL0 M 0 σ Y Y 00 00 2 2 1mc 2 2 2 1mf L3 M3 00 0 0 0 (9.53) 0 0 X X mc 0−σ2−S 0−σ2−S L3 1 L L2 1 L = ( 1) IL001L0 IL0 1L0 00 0 0 0 , − − 3 2 M3 mf M M2 mc M L0M 0 S0=±1 − Z 2 0 ∗ ∗ ∗ d kˆ Y Y 00 00 Y Y 3−σ3 1me L2 M2 σ3 1mb L3M3

00 00 00 00 00 X X me 0−σ3−S 0−σ3−S L3 1 L L2 1 L = ( 1) I 00 I 00 , L31L L 1L00 00 00 00 − − 2 M3 mb M M2 me M L00M 00 S00=±1 − where we have used a property of spin-weighted spherical harmonics given by Eq. (C.7). We can ˆ ˆ ˆ also perform the angular integrals with respect to k1, k2 and k3 as Z L ` 2 2 ˆ ∗ ∗ ∗ 0λ1−λ1 1 1 d k1YL1M1 −λ1 Y`1m1 λ1 Y2µ1 = IL ` 2 , 1 1 M1 m1 µ1 Z L0 ` 2 2 ˆ ∗ ∗ ∗ 0λ2−λ2 1 2 d k Y 0 0 Y Y = I 0 , (9.54) 2 L M −λ2 `2m2 λ2 2µ2 L `22 0 1 1 1 M1 m2 µ2 Z L00 ` 2 2 ˆ ∗ ∗ ∗ 0λ3−λ3 1 3 d k Y 00 00 Y Y = I 00 . 3 L M −λ3 `3m3 λ3 2µ3 L `32 00 1 1 1 M1 m3 µ3 124 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS

At this point, all the angular integrals in Eq. (9.47) have been reduced into the Wigner-3j symbols. As the third step, we consider summing up the Wigner-3j symbols in terms of the azimuthal quantum numbers and replacing them with the Wigner-6j and 9j symbols, which denote Clebsch- Gordan coeﬃcients between two other eigenstates coupled to three and four individual momenta [27, 46–48]. Using these properties, we can express the summation of ﬁve Wigner-3j symbols with a Wigner-9j symbol:

X L L L 2 1 1 ( 1)M2+ma 1 2 3 − M1 M2 M3 µ1 ma mb M1M2M3 − µ1mamb 00 L3 1 L L2 1 L L1 `1 2 00 × M3 mb M M2 ma M M1 m1 µ1 −  00  00  L L `1  M+`1+L3+L L L `1 = ( 1) 00 L3 L2 L1 , − − M M m1 −  1 1 2  0 0 0 X M 0 +m L L L 2 1 1 ( 1) 2 c 1 2 3 M 0 M 0 M 0 µ m m 0 0 0 − 1 2 3 2 c d M1M2M3 − µ2mcmd 0 0 0 0 L3 1 L L2 1 L L1 `2 2 0 0 0 0 (9.55) × M md M M mc M M m2 µ2 3 2 − 1  0  0 LL `2 0 0 0 LL `   M +`2+L3+L 2 0 0 0 = ( 1) 0 L3 L2 L1 , − − M M m2 −  1 1 2  00 00 00 X M 00+m L L L 2 1 1 ( 1) 2 e 1 2 3 M 00 M 00 M 00 µ m m 00 00 00 − 1 2 3 3 e f M1 M2 M3 − µ3memf 00 0 00 00 00 L3 1 L L2 1 L L1 `3 2 00 0 00 00 00 × M mf M M me M M m3 µ3 3 2 − 1  0 00  0 00 L L `3 00 00 00 L L `   M +`3+L3 +L 3 00 00 00 = ( 1) 0 00 L3 L2 L1 . − − M M m3 −  1 1 2 

Furthermore, we can also sum up the renewed Wigner-3j symbols arising in the above equations over M,M 0 and M 00 with the Wigner-6j symbol as [49]

00 0 0 00 X M+M 0+M 00 L L `1 LL `2 L L `3 ( 1) 00 0 0 00 − M M m1 M M m2 M M m3 MM 0M 00 − − − L+L0+L00 `1 `2 `3 `1 `2 `3 = ( 1) 0 00 . (9.56) − m1 m2 m3 L L L

With this prescription, one can ﬁnd that the three azimuthal numbers are conﬁned only in the D E `1 `2 `3 Q3 (λn) Wigner-3j symbol as . This 3j symbol arises from n=1 ξ` m (kn) and exactly m1 m2 m3 n n ensures the rotational invariance of the CMB bispectrum as pointed out above. 9.2 Formulation for the CMB bispectrum induced from PMFs 125

Consequently, we can obtain an exact form of the primordial angular bispectrum given by

* 3 + " 3 # Z kD Y (λn) `1 `2 `3 −3 Y 02 0 0 ξ` m (kn) = ( 4πργ,0) kn dknPB(kn) n n m1 m2 m3 n=1 − n=1 0 X X `1 `2 `3 × L0 L00 L LL0L00 S,S0,S00=±1 00 0 0 00 S Sλ1 0 0 SS λ2 0 0 S S λ3 0 0 f 00 (k , k , k1)f 0 (k , k , k2)f 0 00 (k , k , k3), (9.57) × L L`1 3 1 LL `2 1 2 L L `3 2 3 where Z ∞ S00Sλ X 2 fL00L` (r3, r2, r1) = y dyjL3 (r3y)jL2 (r2y)jL1 (r1y) 0 L1L2L3  L00 L `  L1+L2+L3 00 00   `+L2+L3 0 0 0 0S −S 0S−S 0λ−λ ( 1) ( 1) 2 I I 00 I I L L L L1L2L3 L31L L21L L1`2 3 2 1 × − −  1 1 2   √2 (8π)3/2R ln (τ /τ )(λ = 0)  3 γ ν B 2 3/2 . (9.58) 3 (8π) λ (λ = 1) × 3/2 ±  4(8π) Rγ ln (τν/τB)(λ = 2) − ± Here, the coeﬃcients have been calculated as

 2 3/2 √ (8π) Rγ ln (τν/τB)(λ = 0) r 3 4π 2 − 3 C C0 8 = 2 (8π)3/2λ (λ = 1) . (9.59) −λ −λ 8π 3 3 − 3/2 ± 4(8π) Rγ ln (τν/τB)(λ = 2) ± Substituting Eq. (9.57) into Eq. (9.46), we can formulate the CMB bispectra generated from arbitrary three modes such as the scalar-scalar-vector and tensor-tensor-tensor correlations with the f function as

* 3 + Y (Zn) `1 `2 `3 −3 aX ,` m = ( 4πργ,0) n n n m1 m2 m3 n=1 − " 3 # Z 2 Z kD Y k dkn (Z ) X ( i)`n n n (k ) [sgn(λ )]λn+xn k02dk0 P (k0 ) 2 Xn,`n n n n n B n × − 2π T 0 n=1 λn X X `1 `2 `3 × L0 L00 L LL0L00 S,S0,S00=±1 00 0 0 00 S Sλ1 0 0 SS λ2 0 0 S S λ3 0 0 f 00 (k , k , k1)f 0 (k , k , k2)f 0 00 (k , k , k3) . (9.60) × L L`1 3 1 LL `2 1 2 L L `3 2 3 From this form, it can be easily seen that due to the sextuplicate dependence on the Gaussian PMFs, the Wigner-6j symbol connects the true multipoles (`1, `2 and `3) and the dummy ones (L, L0 and L00), and the 1-loop calculation with respect to these mulipoles is realized as illustrated in the left panel of Fig. 9.4. Due to the extra summations over L, L0 and L00, it takes a lot of time to compute this compared with the tree-level calculation presented in the previous sections. 126 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS

l1 l1

L L’’

L’ l2 l3 l2 l3

Figure 9.4: Diagrams with respect to multipoles [34]. The left panel corresponds to Eq. (9.60). Due to the Wigner-6j symbol originated with the sextuplicate dependence on the Gaussian PMFs, 0 00 the true multipoles `1, `2 and `3 are linked with the dummy ones L, L , and L and the 1-loop structure is realized. The right panel represents the tree-structure diagram, which arises from the CMB bispectrum induced by the four-point function of the Gaussian ﬁelds as mentioned in the previous sections.

9.3 Treatment for numerical computation

In order to perform the numerical computation of the CMB bispectra, we give the explicit angle- averaged form of Eq. (9.60) as

(Z1Z2Z3) −3 X `1 `2 `3 B = CZ CZ CZ ( 4πργ,0) X1X2X3,`1`2`3 1 2 3 − L0 L00 L LL0L00 0 00 X P3 Ln+Ln+Ln+2`n n=1 2 0 0 0 0 0 0 0 0 0 ( 1) IL L L IL0 L0 L0 IL00L00L00 × − 1 2 3 1 2 3 1 2 3 L1L2L3 0 0 0 L1L2L3 00 00 00 L1 L2 L3  00   0   0 00  L L `1 LL `2 L L `3    0 0 0   00 00 00  L3 L2 L1 L3 L2 L1 L3 L2 L1 ×  1 1 2   1 1 2   1 1 2  " 3 Z ∞ 2 # Y `n kndkn (Zn) ( i) (kn) 2π2 Xn,`n × n=1 − 0 T Z ∞ Z ∞ Z ∞ 2 2 2 A dAj (k A) B dBj 0 (k B) C dCj 00 (k C) L1 1 L1 2 L1 3 × 0 0 0 Z kD Z kD 02 0 0 0 0 02 0 0 0 0 k dk P (k )j (k A)j 0 (k B) k dk P (k )j 0 (k B)j 00 (k C) 1 1 B 1 L2 1 L3 1 2 2 B 2 L2 2 L3 2 × 0 0 Z kD 02 0 0 0 0 k dk P (k )j 00 (k C)j (k A) 3 3 B 3 L2 3 L3 3 × 0 0 00 0 00 0 0 0 0 00 00 00 00 X L2+L2+L2 +L3+L3+L3 0S−S 0S−S 0S −S 0S −S 0S −S 0S −S ( 1) IL0 1L IL 1L IL001L0 IL0 1L0 IL 1L00 IL001L00 × − 3 2 3 2 3 2 S,S0,S00=±1 9.3 Treatment for numerical computation 127

X x1 0λ1−λ1 x2 0λ2−λ2 x3 0λ3−λ3 [sgn(λ1)] IL ` 2 [sgn(λ2)] IL0 ` 2 [sgn(λ3)] IL00` 2 , (9.61) × 1 1 1 2 1 3 λ1λ2λ3 with

 √2 (8π)3/2R ln (τ /τ )(Z = S)  3 γ ν B C 2 3/2 . (9.62) Z 3 (8π) (Z = V ) ≡ 3/2  4(8π) Rγ ln (τν/τB)(Z = T ) − We consider performing the summations with respect to the helicities. By considering the selection rules of the Wigner-3j symbol, the summations over S,S0 and S00 are performed as

0 00 0 00 0 0 0 0 00 00 00 00 X L2+L2+L2 +L3+L3+L3 0S−S 0S−S 0S −S 0S −S 0S −S 0S −S ( 1) IL0 1L IL 1L IL001L0 IL0 1L0 IL 1L00 IL001L00 − 3 2 3 2 3 2 S,S0,S00=±1 01−1 01−1 01−1 01−1 01−1 01−1 = I 0 I I 00 0 I 0 0 I 00 I 00 00 L31L L21L L3 1L L21L L31L L2 1L 8 (L0 + L ,L00 + L0 ,L + L00 = even) 3 2 3 2 3 2 . (9.63) × 0 (otherwise)

By the same token, the summations over λ1, λ2 and λ3 are given by

X x1 0λ1−λ1 x2 0λ2−λ2 x3 0λ3−λ3 [sgn(λ1)] I [sgn(λ2)] I 0 [sgn(λ3)] I 00 L1`12 L1`22 L1 `32 λ1λ2λ3

0|λ1|−|λ1| 0|λ2|−|λ2| 0|λ3|−|λ3| = I I 0 I 00 L1 `1 2 L1 `2 2 L1 `3 2 23−NS (L + ` + x ,L0 + ` + x ,L00 + ` + x = even) 1 1 1 1 2 2 1 3 3 , (9.64) × 0 (otherwise)

5 where NS is the number of the scalar modes constituting the CMB bispectrum . Thus, we rewrite the bispectrum as (Z1Z2Z3) −3 X `1 `2 `3 B = CZ CZ CZ ( 4πργ,0) X1X2X3,`1`2`3 1 2 3 − L0 L00 L LL0L00 0 00 X P3 Ln+Ln+Ln+2`n n=1 2 0 0 0 0 0 0 0 0 0 ( 1) IL L L IL0 L0 L0 IL00L00L00 × − 1 2 3 1 2 3 1 2 3 L1L2L3 0 0 0 L1L2L3 00 00 00 L1 L2 L3  00   0   0 00  L L `1 LL `2 L L `3    0 0 0   00 00 00  L3 L2 L1 L3 L2 L1 L3 L2 L1 ×  1 1 2   1 1 2   1 1 2  " 3 Z ∞ 2 # Y `n kndkn (Zn) ( i) (kn) 2π2 Xn,`n × n=1 − 0 T Z ∞ Z ∞ Z ∞ 2 2 2 A dAj (k A) B dBj 0 (k B) C dCj 00 (k C) L1 1 L1 2 L1 3 × 0 0 0 Z kD Z kD 02 0 0 0 0 02 0 0 0 0 k dk P (k )j (k A)j 0 (k B) k dk P (k )j 0 (k B)j 00 (k C) 1 1 B 1 L2 1 L3 1 2 2 B 2 L2 2 L3 2 × 0 0

5Caution about a fact that λ is determined by Z, namely, λ = 0, 1, 2 for Z = S, V, T , respectively. | | | | 128 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS

Z kD 02 0 0 0 0 k dk P (k )j 00 (k C)j (k A) 3 3 B 3 L2 3 L3 3 × 0 01−1 01−1 01−1 01−1 01−1 01−1 0 00 0 0 00 00 8IL0 1LIL 1LIL001L0 IL0 1L0 IL 1L00 IL001L00 L ,L2,L L ,L ,L L3,L ,L × 3 2 3 2 3 2 Q 3 Q 3 2 Q 2 3−N 0|λ1|−|λ1| 0|λ2|−|λ2| 0|λ3|−|λ3| S 0 00 2 IL ` 2 IL0 ` 2 IL00 ` 2 L1,`1,x1 L ,`2,x2 L ,`3,x3 , (9.65) × 1 1 1 2 1 3 U U 1 U 1 where we introduce the ﬁlter functions as

L0 ,L ,L (δL0 ,L+1 + δL0 ,|L−1|)(δL ,L+1 + δL ,|L−1|) + δL0 ,LδL ,L Q 3 2 ≡ 3 3 2 2 3 2 (9.66) L ,` ,x (δL ,` −2 + δL ,` + δL ,` +2)δx ,0 + (δL ,` −1 + δL ,` +1)δx ,1 . U 1 1 1 ≡ 1 1 1 1 1 1 1 1 1 1 1 1 The above analytic expression seems to be quite useful to calculate the CMB bispectrum induced from PMFs with the full-angular dependence. However, it is still too hard to calculate numerically, because the full expression of the bispectrum has six integrals. In addition, when we calculate the spectra for large `’s, this situation becomes worse since we spend a lot of time calculating the Wigner symbols for large `’s. The CMB signals of the vector mode appear at ` > 2000, hence we need the reasonable approximation in calculation of the CMB bispectra composed of the vector modes. In what follows, we introduce an approximation, the so-called thin last scattering surface (LSS) approximation to reduce the integrals.

9.3.1 Thin LSS approximation Let us consider the parts of the integrals with respect to A, B, C, k0, p0 and q0 in the full expression of the bispectrum (9.65) of B(VVV ) . In the computation of the CMB bispectrum, the integral in III,`1`2`3 R 2 (V ) terms of k,(p and q) appears in the form as k dk (k)jL (kA). We find that this integral is TI,`1 1 sharply-peaked at A τ0 τ∗, where τ0 is the present conformal time and τ∗ is the conformal time of the recombination' epoch.− According to Sec. 9.1.3, the vorticity of subhorizon scale sourced by magnetic fields around the recombination epoch mostly contributes to generate the CMB vector perturbation. On the other hand, since the vector mode in the metric decays after neutrino decoupling, the integrated Sachs-Wolfe effect after recombination is not observable. Such a behavior of the transfer function would be understood on the basis of the calculation in Sec. 9.1 and we expect (V )(k) j (k(τ τ )), and the k-integral behaves like δ(A (τ τ )). By the numerical I,`1 `1 0 ∗ 0 ∗ computation,T we∝ found that− − − Z ∞ Z ∞ τ Z A2dA k2dk (V )(k )j (k A) (τ τ )2 ∗ k2dk (V )(k )j (k (τ τ )), (9.67) 1 1 I,`1 1 L1 1 0 ∗ 1 1 I,`1 1 `1 1 0 ∗ 0 0 T ' − 5 T − is a good approximation for L1 = `1 2, `1 as described in Fig. 9.5. Note that only the cases ± L1 = `1 2, `1 should be considered due to the selection rules for Wigner-3j symbols as we shall see later. From± this figure, we can find that the approximation (the right-handed term of Eq. (9.67)) has less than 20% uncertainty for `1 L1 & 100, and therefore this approximation leads to only less than 10% uncertainty in the bound' on the strength of PMFs if we place the constraint from 6 the bispectrum data at `1, `2, `3 & 100 . Using this approximation, namely A = B = C τ0 τ∗ R R R 0 0 0 → − and dA = dB = dC τ∗/5, the integrals with respect to A, B, C, k , p and q are estimated as → " 3 Z ∞ 2 # Z ∞ Z ∞ Z ∞ Y `n kndkn (V ) 2 2 2 4π( i) (kn) A dAjL (k1A) B dBjL0 (k2B) C dCjL00 (k3C) (2π)3 I,`n 1 1 1 n=1 − 0 T 0 0 0

6Of course, if we calculate the bispectrum at smaller multipoles and the CMB bispectra are produced by other modes than the vector one, we may perform the full integration without this approximation. 9.3 Treatment for numerical computation 129

2 L1 = l1 + 2 L1 = l1 L1 = l1 - 2 1.5

1 exact / approximate 0.5

0 10 100 1000

Figure 9.5: The ratio of the left-hand side (exact solution) to the right-hand side (approximate solution) in Eq. (9.67). The lines correspond to the case for L1 = `1 +2 (red solid line), for L1 = `1 (green dashed one), and for L1 = `1 2 (blue dotted one). −

Z kD Z kD 02 0 0 0 0 02 0 0 0 0 k dk P (k )j (k A)j 0 (k B) k dk P (k )j 0 (k B)j 00 (k C) 1 1 B 1 L2 1 L3 1 2 2 B 2 L2 2 L3 2 × 0 0 Z kD 02 0 0 0 0 k dk P (k )j 00 (k C)j (k A) 3 3 B 3 L2 3 L3 3 × 0 " 3 # Z ∞ 2 3 Y `n kndkn (V ) 3 6 τ∗ 4π( i) (kn)j` (kn(τ0 τ∗)) A (τ0 τ∗) (2π)3 I,`n n B 5 ' n=1 − 0 T − −

−(nB +1) −(nB +1) −(nB +1) L L0 (τ0 τ∗) L0 L00 (τ0 τ∗) L00L (τ0 τ∗) . (9.68) ×K 2 3 − K 2 3 − K 2 3 −

N Here the function 0 is deﬁned as Kll

Z ∞ N 1−N ll0 (y) dkk jl(ky)jl0 (ky) K ≡ 0 0 π yN−1 Γ(N)Γ( l+l +2−N ) = 2 (for y, N, l + l0 + 2 N > 0) , 2y 2N l−l0+1+N −l+l0+1+N l+l0+2+N Γ( 2 )Γ( 2 )Γ( 2 ) − (9.69)

N −N 0 0 which behaves asymptotically as 0 (y) l for l l 1. Here we have evaluated the k Kll ∝ ∼ integrals by setting kD . This is also a good approximation because the integrands are 0→0 ∞0 −1 suppressed enough for k , p , q < kD (10)Mpc . ∼ O 130 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS

9.3.2 Selection rules of the Wigner-3j symbol From the selection rules of the Wigner symbols as described in Appendix C, we can further limit the summation range of the multipoles as

0 0 00 0 L `2 L L + `2 , Max[ L `1 , L `3 ] L Min[L + `1,L + `3] , | − | ≤ ≤ 0 |0 − 0 | | − | ≤00 ≤00 00 L1 + L2 + L3 = even ,L1 + L2 + L3 = even ,L1 + L2 + L3 = even , (9.70) 0 0 0 0 0 00 00 00 00 00 L1 L2 L3 L1 + L2 , L L L L + L , L L L L + L , | − | ≤ ≤ | 1 − 2| ≤ 3 ≤ 1 2 | 1 − 2| ≤ 3 ≤ 1 2 and from the above restrictions the multipoles in the bispectrum, `1, `2 and `3, are also limited as

`1 `2 `3 `1 + `2 . (9.71) | − | ≤ ≤ Therefore, these selection rules signiﬁcantly reduce the number of calculation. In these ranges, while L0 and L00 are limited by L, only L has no upper bound. However, we can show that the summation of L is suppressed at `1 `2 `3 L as follows. When the summations with respect 0 00 ∼ ∼ 0 00 0 00 0 to L, L and L are evaluated at large L, L and L , namely `1, `2, `3 L L L ,L2 L3 0 00 0 00 00 ∼ ∼ ∼ ∼ L, L L L and L L3 L , we get 2 ∼ 3 ∼ 2 ∼ ∼ Z kD X `1 `2 `3 X 02 0 0 0 0 k dk PB(k )jL (k A)jL0 (k B) L0 L00 L 1 1 1 2 1 3 1 0 00 0 00 0 LL L L2L2L2 0 00 L3L3 L3 Z kD Z kD 02 0 0 0 0 02 0 0 0 0 k dk P (k )j 0 (k B)j 00 (k C) k dk P (k )j 00 (k C)j (k A) 2 2 B 2 L2 2 L3 2 3 3 B 3 L2 3 L3 3 × 0 0 0 00 P3 Li+Li+Li i=1 0 0 0 0 0 0 0 0 0 01−1 01−1 01−1 01−1 01−1 01−1 ( 1) 2 I I 0 0 0 I 00 00 00 I 0 I I 00 I 0 I 00 I 00 L1L2L3 L L L L L L L 1L L 1L L 1L0 L 1L0 L 1L L 1L00 × − 1 2 3 1 2 3 3 2 3 2 3 2  00   0   0 00  L L `1 LL `2 L L `3    0 0 0   00 00 00  L3 L2 L1 L3 L2 L1 L3 L2 L1 ×  1 1 2   1 1 2   1 1 2  X (LL0L00)nB +4/3 . (9.72) ∝ LL0L00 Therefore, we may obtain a stable result with the summations over a limited number of L when 0 we consider the magnetic power spectrum is as red as nB 2.9, because the summations of L and L00 are limited by L. Here, we use the analytic formulas∼ of− the I symbols which are given by

 00   L L `1  `1 `2 `3 0 00 −1/6 −(nB +1) nB +1 00 −1/2 0 00 (LL L ) , L L0 L , L3 L2 L1 (L L) , (9.73) L L L 2 3 ∝ K ∝  1 1 2  ∝ as described in detail in Appendix C. Using the thin LSS approximation and the summation rules described above, we can perform the computation of the CMB bispectrum containing full-angular dependence in a reasonable time.

9.4 Shape of the non-Gaussianity In this subsection, in order to understand the shape of the non-Gaussianities arising from PMFs, we reduce the bispectra of the PMF anisotropic stress by the pole approximation [34]. 9.4 Shape of the non-Gaussianity 131

Let us focus on the structure of the bispectrum of the PMF anisotropic stresses (9.45). If the magnetic spectrum is enough red as nB 3, the integral over the wave number vectors is almost ∼ − 0 0 0 determined by the behavior of the integrand around at three poles as k1, k2, k3 0. Considering the eﬀects around at these poles, we can express the bispectrum of the PMF anisotropic∼ stresses approximately as

3 ! αAB 8π X Π (k )Π (k )Π (k ) ( 4πρ )−3 knB +3 δ k Bab 1 Bcd 2 Bef 3 γ,0 n + 3 ∗ 3 n h i ∼ − B n=1 1 h ˆ ˆ ˆ ˆ PB(k1)PB(k2)δadPbe(k1)Pcf (k2) + PB(k2)PB(k3)Pad(k2)Pbe(k3)δcf ×8 ˆ ˆ i +PB(k1)PB(k3)Pad(k1)δbePcf (k3) + a b or c d or e f , (9.74) { ↔ ↔ ↔ } where we evaluate as

Z Z k∗ Z αAB 8π 3 0 0 ˆ0 02 0 0 2 ˆ0 ˆ0 nB +3 d k PB(k )Pab(k ) α k dk PB(k ) d k Pab(k ) = k∗ δab . (9.75) ∼ 0 nB + 3 3 Note that α is an unknown parameter and should be determined by the comparison with the exact −1 bispectra [(9.60) or (9.65)], and we take k∗ = 10Mpc . Under this approximation, the angular bispectrum of the primordial tensor perturbations (λ1, λ2, λ3 = 2) is given by ± * 3 + " 3 Z # 3 3 ! Rγ ln(τν/τB) αAB 8π Y (λn) Y 2 ˆ ∗ ˆ nB +3 X ξ (kn) d kn−λ Y (kn) k δ kn `nmn n `nmn 4πρ n + 3 ∗ 3 n=1 ∼ n=1 γ,0 B n=1 h ˆ ˆ ˆ ˆ PB(k1)PB(k2)δadPbe(k1)Pcf (k2) + PB(k2)PB(k3)Pad(k2)Pbe(k3)δcf × ˆ ˆ i +PB(k1)PB(k3)Pad(k1)δbePcf (k3)

(−λ1) ˆ (−λ2) ˆ (−λ3) ˆ ( 27)e (k1)e (k2)e (k3) . (9.76) × − ab cd ef Using Eq. (D.23), we reduce the contraction of the subscripts in the TTT spectrum to h (−λ1) ˆ (−λ2) ˆ (−λ3) ˆ ˆ ˆ Oab (k1)Ocd (k2)Oef (k3) PB(k1)PB(k2)δadPbe(k1)Pcf (k2) ˆ ˆ ˆ ˆ i +PB(k2)PB(k3)Pad(k2)Pbe(k3)δcf + PB(k1)PB(k3)Pad(k1)δbePcf (k3)

(−λ1) ˆ (−λ3) ˆ (−λ2) ˆ = eae (k1)eef (k3)efa (k2)[PB(k1)PB(k2) + 2 perms.] 5/2 (8π) 01−1 01−1 2 2 2 = I I [PB(k1)PB(k2) + 2 perms.] − 3 211 211 1 1 1 X ∗ ˆ ∗ ˆ ∗ ˆ 2 2 2 λ Y (k1)λ Y 0 (k2)λ Y 00 (k3) . (9.77) × 1 2M 2 2M 3 2M MM 0 M 00 M,M 0,M 00

The delta function is also expanded with the spin spherical harmonics as Eq. (6.14)

3 ! " 3 # X Z ∞ Y X δ k = 8 y2dy ( 1)Li/2j (k y)Y ∗ (kˆ ) i Li i LiMi i 0 − i=1 i=1 LiMi 132 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS

0 0 0 L1 L2 L3 IL1L2L3 . (9.78) × M1 M2 M3 Then, the angular integrals are performed as Z ` L 2 2 ˆ ∗ ∗ ∗ λ10−λ1 1 1 d k1−λ1 Y`1m1 YL1M1 λ1 Y2M = I` L 2 , 1 1 m1 M1 M Z 2 ∗ ∗ ∗ λ20−λ2 `2 L2 2 ˆ 0 d k2−λ2 Y`2m2 YL2M2 λ2 Y2M = I` L 2 0 , (9.79) 2 2 m2 M2 M Z 2 ∗ ∗ ∗ λ30−λ3 `3 L3 2 ˆ 00 d k3−λ3 Y`3m3 YL3M3 λ3 Y2M = I` L 2 00 , 3 3 m3 M3 M and all the Wigner-3j symbols are summed up as X L1 L2 L3 2 2 2 0 00 M1 M2 M3 MM M M1M2M3 MM 0M 00 `1 L1 2 `2 L2 2 `3 L3 2 0 00 × m1 M1 M m2 M2 M m3 M3 M  ` ` `  ` ` `  1 2 3  = 1 2 3 L L L . (9.80) m m m 1 2 3 1 2 3  2 2 2 

Thus the initial bispectrum (9.76) is rewritten as

* 3 + R ln(τ /τ )3 αA 8π Y (λn) `1 `2 `3 γ ν B B nB +3 ξ` m (kn) k∗ n n m1 m2 m3 4πρ n + 3 3 n=1 ∼ γ,0 B Z ∞ " 3 # 2 Y X Ln/2 0 0 0 8 y dy ( 1) jLn (kny) IL1L2L3 × 0 − n=1 Ln    `1 `2 `3  2 2 2 01−1 01−1 λ10−λ1 λ20−λ2 λ30−λ3 I I I I I L1 L2 L3 1 1 1 211 211 `1L12 `2L22 `3L32 ×  2 2 2  5/2 9(8π) [PB(k1)PB(k2) + 2perms.] . (9.81) × Comparing the exact initial bispectrum of the tensor modes (9.57) with this equation, we can see that the number of the time-integrals and summations in terms of the multipoles decreases. This means that corresponding to the pole approximation, the 1-loop calculation (the left panel of Fig. 9.4) reaches the tree-level one (the right one of that ﬁgure). This approximation seems to be applicable to the non-Gaussianity generated from the chi-squared ﬁelds without the complicated angular dependence [50]. Note that the scaling behaviors of these initial bispectra with respect to k1, k2 and k3 are in agreement with that of the local-type non-Gaussianity (4.7). Thus, if the pole approximation is valid, we can expect that the PMFs generate the CMB bispectra coming from the local-type non-Gaussianity. Via the summation over λ1, λ2 and λ3 as Eq. (9.64), the approximate CMB bispectra of the tensor modes are quickly formulated: 3 app(TTT ) Rγ ln(τν/τB) αAB n +3 8π X L1+L2+L3 0 0 0 B (α) = k B ( 1) 2 I X1X2X3,`1`2`3 ∗ L1L2L3 4πργ,0 nB + 3 3 − L1L2L3 9.5 Analysis 133

 ` ` `  2 2 2  1 2 3  I01−1I01−1 L L L 1 1 1 211 211 1 2 3 ×  2 2 2  20−2 20−2 20−2 8I I I L ,` ,x L ,` ,x L ,` ,x × `1L12 `2L22 `3L32U 1 1 1 U 2 2 2 U 3 3 3 Z ∞ " 3 Z ∞ 2 # 2 Y `n kndkn (T ) 8 y dy ( i) (kn)jL (kny) 2π2 Xn,`n n × 0 n=1 − 0 T 5/2 9(8π) [PB(k1)PB(k2) + 2perms.] , (9.82) × where the multipoles are limited as

3 X Ln = even , L1 L2 L3 L1 + L2 , (9.83) n=1 | − | ≤ ≤ and the triangle inequality imposes

`1 `2 `3 `1 + `2 . (9.84) | − | ≤ ≤ In the next subsection, we compare these approximate spectra with the exact spectra given by Eq. (9.65) and evaluate the validity of the pole approximation.

9.5 Analysis In this subsection, we show the result of the CMB intensity-intensity-intensity spectra induced from the auto-correlations of the each-mode anisotropic stress. In order to compute numerically, we insert Eq. (9.65) into the Boltzmann code for anisotropies in the microwave background (CAMB) [15, 51]. We use the transfer functions shown in Sec. 9.1. In the calculation of the Wigner-3j, 6j and 9j symbols, we use a common mathematical library called SLATEC [52] and analytical expressions in Appendix C. In Fig. 9.6, we plot the CMB reduced bispectra of these modes deﬁned as [53]

b(Z1Z2Z3) I0 0 0 −1 B(Z1Z2Z3) , (9.85) III,`1`2`3 ≡ `1`2`3 III,`1`2`3 for `1 = `2 = `3. Here, for comparison, we also write the bispectrum generated from the local-type primordial non-Gaussianities of curvature perturbations given by Eq. (4.7). From the red solid lines, we can find the enhancement at ` . 100 in tensor-tensor-tensor bispectra. It is because the ISW effect gives the dominant signal like in the CMB anisotropies of tensor modes [17, 54]. From the green dashed line, one can see that the peak of the vector-vector- vector bispectrum is located at ` 2000 and the position is similar to that of the angular power (V ) ∼ spectrum CI,` induced from the vector mode as calculated in Sec. 9.1. At small scales, the vector mode contributes to the CMB power spectrum through the Doppler effect. Thus, we can easily find that the Doppler effect can also enhance the CMB bispectrum on small scale. From the blue dotted lines, we can see that the scalar-scalar-scalar bispectra are boosted around at ` 200 due to the acoustic oscillation of the fluid of photons and baryons. On the other hand, as `∼enlarges, the spectra are suppressed by the Silk damping effect. These features are also observed in the non-magnetic case (the magenta dot-dashed line), however, owing to the difference of the angular dependence on the wave number vectors in the source bispectra, the location of the nodes slightly differs. Comparing the behaviors between the three spectra arising from PMFs, we confirm that 134 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS the tensor, scalar and vector modes become effective for ` . 100, 100 . ` . 2000 and ` & 2000, respectively, like the behaviors seen in the power spectra. Thus, for ` < 1000, namely the current instrumental limit of the angular resolution such as the PLANK experiment [30], we expect that the auto- and cross-correlations between the scalar and tensor modes will be primary signals of PMFs in the CMB bispectrum. 3/2 3/2 (SSS) (TTT ) h (S) i h (T ) i The overall amplitudes of bIII,``` and bIII,``` seem to be comparable to CII,` and CII,` . (VVV ) However, we find that the amplitude of bIII,``` is smaller than the above expectation. This is because the configuration of multipoles, corresponding to the angles of wave number vectors, is limited to the conditions placed by the Wigner symbols. We can understand this by considering the scaling relation with respect to ` at high `. If the magnetic power spectrum given by Eq. (9.4) is close to the scale-invariant shape, the configuration that satisfies L L00 ` and L0 1 contributes dominantly in the summations. Furthermore, the other multipoles∼ are∼ evaluated as∼

0 00 00 0 0 00 L1 L L ` , L2 L L3 L ` , L L 1 , (9.86) ∼ 1 ∼ 1 ∼ ∼ 2 ∼ ∼ 3 ∼ 2 ∼ 3 ∼ (VVV ) from the triangle conditions described in Appendix C. Then we can find b `2nB +4 for III,``` ∝ ` . 1000, where we have also used the following relations Z 2 (V ) `1 `2 `3 −1 −(nB +1) −(nB +1) nB +1 k dk I,` (k)jì (k(τ0 τ∗)) ` , 0 00 ` , L L0 L00L ` , T i − ∝ L L L ∝ K 2 3 ∼ K 2 3 ∝  00   0   0 00  L L `1 LL `1 L L `1 (9.87)   −3/2  0 0 0   00 00 00  −1 L3 L2 L1 ` , L3 L2 L1 L3 L2 L1 ` ,  1 1 2  ∝  1 1 2  ∼  1 1 2  ∝ which, except for the first relation, are also coming from the triangle conditions of the Wigner 3-j symbols. Therefore, combining with the scaling relation of the CMB power spectrum mentioned (VVV ) (nB −1)/2 (V )3/2 in Sec. 9.1, we find that bIII,``` is suppressed by a factor ` from CII,` . (VVV ) In Fig. 9.7, we show bIII,``` for `1 = `2 = `3 for the different spectral index nB. Red solid and green dashed lines correspond to the bispectrum with the spectral index of the power spectrum of PMFs fixed as nB = 2.9 and 2.8, respectively. From this figure, we find that the CMB − − bispectrum becomes steeper if nB becomes larger, which is similar to the case of the power spectrum. These spectra trace the scaling relation in the above discussion. These will lead to another constraint on the strength of PMFs. In Figs. 9.8 and 9.9, we show the reduced bispectrum b(VVV ) and b(TTT ) with respect to III,`1`2`3 III,`1`2`3 ` with setting ` = ` , respectively. From Fig. 9.8, we can see that b(VVV ) for ` , ` , ` 100 is 3 1 2 III,`1`2`3 1 2 3 & nearly flat and given as 6 (VVV ) −19 B1Mpc `1(`1 + 1)`3(`3 + 1) b 2 10 . (9.88) | III,`1`2`3 | ∼ × 4.7nG We can understand this by the analytical evaluation as follows. As mentioned above, in the sum- 0 00 mations of Eq. (9.65), the configuration that L `1,L 1 and L `3 contributes dominantly. By using this and the approximations that ∼ ∼ ∼

0 00 0 0 00 0 00 00 L1 `1 ,L `2 ,L `3 ,L2 L L,L L L ,L L3 L , (9.89) ∼ 1 ∼ 1 ∼ ∼ 3 ∼ 2 ∼ 3 ∼ 2 ∼ ∼ which again come from the triangle conditions from the Wigner symbols, the scaling relation (VVV ) nB +1 of `3 at large scale is evaluated as b ` . From this estimation we can ﬁnd that III,`1`2`3 ∝ 3 9.5 Analysis 135

10 PMF tensor PMF vector 1 PMF scalar curvature

16 0.1 10 × 2 )

π 0.01 |/(2 lll |b

2 0.001 (l+1) 2

l 0.0001

1e-05

1e-06 10 100 1000 l

Figure 9.6: Absolute values of the normalized reduced bispectra of temperature fluctuations for a configuration `1 = `2 = `3 `. The red solid, green dashed, and blue dotted lines correspond to the spectra generated from≡ the auto-correlations of the PMF tensor, vector, and scalar anisotropic stresses for nB = 2.9, respectively. The upper (lower) lines of the red solid and blue dotted − 17 6 lines are calculated when τν/τB = 10 (10 ). The magenta dot-dashed line expresses the spectrum local sourced from the primordial non-Gaussianity with fNL = 5. The strength of PMFs is fixed to B1Mpc = 4.7nG and the other cosmological parameters are fixed to the mean values limited from WMAP-7yr data reported in Ref. [29].

` (` + 1)` (` + 1)b(VVV ) `0.1, for n = 2.9, and `0.2 for n = 2.8, respectively, which 1 1 3 3 III,`1`2`3 3 B 3 B match the behaviors of the bispectra∝ in Fig. 9.8.− − From Fig. 9.9, we can also see that if the PMF spectrum obeys the nearly scale-invariant shape, b(TTT ) for ` , ` , ` 100 is given by III,`1`2`3 1 2 3 . 6 (TTT ) −16 B1Mpc `1(`1 + 1)`3(`3 + 1) b (130 6) 10 , (9.90) | III,`1`2`3 | ∼ − × 4.7nG 17 6 where the factor 130 corresponds to the τν/τB = 10 case and 6 corresponds to 10 . In order to obtain a rough constraint on the magnitude of the PMF, we compare the bispectrum induced from the PMF with that from the local-type primordial non-Gaussianity in the curvature perturbations, which is typically estimated as [55]

−18 local `1(`1 + 1)`3(`3 + 1)b` ` ` 4 10 f . (9.91) 1 2 3 ∼ × NL By comparing this with Eq. (9.90), the relation between the magnitudes of the PMF with the local nearly scale-invariant power spectrum and fNL is derived as B 1Mpc (1.22 2.04) f local 1/6 . (9.92) 1nG ∼ − | NL | Using the above equation, we can obtain the rough bound on the PMF strength. As shown in Fig. 9.6, because the tensor bispectrum is highly damped for ` & 100, we should use an upper 136 9 CMB BISPECTRUM GENERATED FROM PRIMORDIAL MAGNETIC FIELDS

0.1 PMF: nB = -2.9 -2.8 curvature

0.01 16 10 × 2 )

π 0.001 |/(2 lll |b 2 0.0001 (l+1) 2 l

1e-05

1e-06 100 1000 l

Figure 9.7: Absolute values of the normalized reduced temperature-temperature-temperature spectra arising from the auto-correlation between the PMF vector anisotropic stresses for a conﬁgu- ration `1 = `2 = `3 `. The lines correspond to the spectra generated from vector anisotropic ≡ stress for nB = 2.9 (red solid line) and 2.8 (green dashed line), and primordial non-Gaussianity local − − with fNL = 5 (blue dotted line). The strength of PMFs is ﬁxed to B1Mpc = 4.7nG and the other cosmological parameters are identical to the values used in Fig. 9.6.

local local bound on fNL obtained by the current observational data for ` < 100, namely fNL < 100 [56]. This value is consistent with a simple prediction from the cosmic variance [53]. From this value, we derive B1Mpc < 2.6 4.4nG. From here, let us discuss− the validity and possibility of the CMB bispectra under the pole approximation (9.82). Figure 9.10 shows the shapes of the CMB tensor-tensor-tensor spectra based on the exact form (9.65) and approximate one (9.82). Both spectra seem to have a good agreement in the shape of the ` space. To discuss more precisely, using the correlation

0 X 0 b b b```b , (9.93) · ∝ ``` ` we calculate a correlation coeﬃcient as bex bapp · = 0.99373 , (9.94) p(bex bex)(bapp bapp) · · where bex and bapp are the exact and approximate reduced bispectra, respectively. This fact, which this quantity approaches unity, implies that the pole approximation can produce an almost exact copy. An unknown parameter, α, is derived from the relation as

bex bex bapp(α = 1) α = ``` · = 0.2991 . (9.95) bapp(α = 1) ≈ bapp(α = 1) bapp(α = 1) ``` · The cases other than the tensor-tensor-tensor spectrum will be presented in Ref. [34]. 9.6 Summary and discussion 137

0.01 PMF: (nB, l1) = (-2.9, 500) (-2.9, 1004) (-2.8, 500)

16 (-2.8, 1004)

10 0.001 curvature × 2 ) π |/(2 3 l

2 0.0001 l 1 l +1)|b 3

(l 1e-05 3 +1)l 1 (l 1

l 1e-06

1e-07 100 1000

Figure 9.8: Absolute values of the normalized reduced temperature-temperature-temperature bispectra induced by the auto-correlation between the PMF vector anisotropic stresses and generated by primordial non-Gaussianity given by Eq. (9.91) as a function of `3 with `1 and `2 fixed to some value as indicated. Each parameter is fixed to the same value defined in Fig. 9.6.

As shown in the previous subsections, the CMB bispectra from PMFs arise from the six-point correlation of the Gaussian magnetic ﬁelds and have one-loop structure due to the summation over the additional multipoles. Hence, it takes so long hours to estimate all `’s contribution and it is actually impossible to compute the signal-to-noise ratio. However, using the pole approximation, since the summation reaches the tree-level calculation, we will obtain more precise bound through the estimation of the signal-to-noise ratio including the contribution of the cross-correlations between scalar and tensor modes [34].

9.6 Summary and discussion In this section, on the basis of our recent works [24–26], we presented the all-sky formulae for the CMB bispectra induced by the scalar, vector, and tensor non-Gaussianities coming from the PMFs by dealing with the full-angular dependence of the bispectrum of the PMF anisotropic stresses. Then, expressing the angular dependence with the spin-weighted spherical harmonics and converting the angular integrals into the Wigner symbols were key points of the formulation. From the practical calculation, it is found that the CMB bispectra from the magnetic tensor, scalar, and vector modes dominate at large (` . 100), intermediate (100 . ` . 2000), and small (` & 2000) scales. For the discussion about the shape of the non-Gaussianity in the PMF anisotropic stresses, we performed the pole approximation, which is the evaluation of the convolutions at around the divergence points of the integrands, and found that the bispectra of the PMF anisotropic stresses are classified as the local-type configuration. Owing to this, we had some significant signals of the CMB bispectra on the squeezed limit also in the multipole space. Compared with the exact formula, the approximate one reduces the computing time, hence we expect the use for the calculation of the signal-to-noise ratio [34]. We also investigated the dependence of the CMB bispectrum on the 138 REFERENCES

100

16 10 10 × 2 ) π |/(2

3 1 l 2 l 1 l +1)|b

3 0.1 (l 3 +1)l 1 (l

1 0.01 l PMF : l1 = l2 = 10 PMF : l1 = l2 = 20 PMF : l1 = l2 = 40 curvature 0.001 10 100

Figure 9.9: Absolute values of the normalized reduced temperature-temperature-temperature bispectra induced by the auto-correlation between the PMF tensor anisotropic stresses and generated from primordial non-Gaussianity in curvature perturbations given by Eq. (9.91) as a function of `3 with `1 = `2. Each parameter is identical to the values defined in Fig. 9.6. spectral index of the PMF power spectrum and confirmed that the CMB bispectrum induced from the PMFs is sensitive to it. Since the characteristic scale varies with the value of the spectral index, it is important to consider not only the contribution from the scalar mode, but also those from the vector and tensor modes. By translating the current bound on the local-type non-Gaussianity from the CMB bispectrum into the bound on the amplitude of the magnetic fields, we obtain a new limit: B1Mpc < 2.6 4.4nG. This is a rough estimate coming from the large scale information of the tensor mode and a− precise constraint is expected if one considers the full ` contribution by using an appropriate estimator of the CMB bispectrum induced from the primordial magnetic fields. Because of the complicated discussions and mathematical manipulations, here we restrict our numerical results to the intensity bispectra of auto-correlations between scalar, vector and tensor modes despite the fact that our formula for the CMB bispectra (9.60) contains the polarizations and the cross-correlations between scalar, vector and tensor modes. However, like the non-magnetic case [57], the modes other than our numerical results, such as B(SVT ) , will bring in more IEB,`1`2`3 reasonable bounds on the PMFs [34]. Furthermore, the effect on the CMB four-point correlation (trispectrum) is just beginning to be roughly discussed [58]. Applying our studies, this should be taken into account more precisely.

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16 1 10 × 2

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10 Conclusion

The main purpose of this thesis was to present the formalism for the CMB bispectrum induced by the non-Gaussianities not only in the standard scalar-mode perturbations but also in the vector- and tensor-mode ones where the violation of the rotational or parity invariance is also involved, and some attempts to prove the nature of the early Universe by applying our formalism. To do this, we have discussed the following things. In Sec. 1, we gave the introduction of this thesis. Then, we quickly summarized the history of the Universe, the paradigm in the early Universe, and the concept of this thesis. In Sec. 2, we summarized how to generate the curvature perturbations and gravitational waves and the consistency relations in the slow-roll inflation. In Sec. 3, we showed how to construct the a`m’s generated from the primordial scalar, vector and tensor sources in order to formulate the CMB bispectrum easily. We also summarized the constraints on several key parameters, which characterize the nature of inflation and the dynamics of the Universe, obtained from the current CMB data. In Sec. 4, we focused on the topic of the primordial non-Gaussianities. In Sec. 5, we gave the general formulae for the CMB bispectrum coming from not only scalar-mode but also vector- and tensor- mode perturbations, which includes both the auto- and cross-correlations between the intensity and polarizations. Next, applying this formalism, we computed the CMB bispectra from several kinds of the non-Gaussianities. In Sec. 6, we treated the two scalars and a graviton correlator and obtained the CMB bispectrum including the tensor-mode perturbation. Here, we had a bound on the nonlinear scalar-scala-tensor coupling by the computation of the signal-to-noise ratio. In Sec. 7, we considered the non-Gaussianity which has the preferred direction. Through the analysis, we found that the finite signals arise from the multipoles except for the triangle inequality. We furthermore confirmed that these special signals are comparable in magnitude with the signals keeping the triangle inequality. In Sec. 8, we dealt with the graviton non-Gaussianity arising from the parity-conserving and parity-violating Weyl cubic terms. Calculating the CMB intensity and polarization bispectra, we clarified that the intensity-intensity-intensity spectrum from the parity- P3 violating non-Gaussianity obeys the condition as n=1 `n = odd. These configurations will be very beneficial to check the parity violation of the Universe in the non-Gaussian level observationally. In Sec. 9, we took into account the effect of the non-Gaussianities due to the primordial magnetic fields. Depending quadratically on the magnetic fields, the magnetic anisotropic stresses obey the chi-square distributions. Since these non-Gaussian anisotropic stresses become the sources of the CMB fluctuations, their bispectra have the finite values. Computing the CMB intensity-intensity- intensity spectra, we clarified that the tensor (vector) mode dominates at large (small) scales and the scalar mode shows up at intermediate scales. By the pole approximation, we also found that the bispectrum of the magnetic anisotropic stresses is similar to the local-type bispectrum. Com- paring the theoretical results with the observational limit on the local-type non-Gaussianity, we obtained a bound on the strength of the magnetic fields, B1Mpc < 2.6 4.4nG. We expect that this bound will be updated by considering the impacts of the cross-correlations− between scalar, vector and tensor modes, and the additional information from polarizations. Our formalism for the CMB bispectrum is general enough to be applicable to the non-Gaussian sources other than the above ones. Moreover, this will be easily extended to the higher-order correlations. Therefore, the studies in this thesis will be very beneficial to quest for the true picture of the origin of the Universe. 144 A SPIN-WEIGHTED SPHERICAL HARMONIC FUNCTION

A Spin-weighted spherical harmonic function

Here, we review the properties of the spin-weighted spherical harmonic function. In the past, this was mainly applied to the analysis of the gravitational wave (see e.g. Ref. [1]). This discussion is based on Refs. [2–4]. The spin-weighted spherical harmonic function on 2D sphere, sYlm(θ, φ), is more general expression than the ordinaly spherical harmonic function, Ylm(θ, φ), and has additional U(1) symmetry characterized by a spin weight s. The spin-s function such as sYlm(θ, φ) obeys the spin raising and 0 0 −i(s+1)ψ 0 0 0 −i(s−1)ψ 0 lowering rule as (∂ sf) = e ∂ sf and (∂ sf) = e ∂ sf. Here, the spin raising and lowering operators are given by 0 s −s ∂ sf(θ, φ) = sin θ [∂θ + i csc θ∂φ] sin θsf(θ, φ) , − (A.1) 0 −s s ∂ sf(θ, φ) = sin θ [∂θ i csc θ∂φ] sin θsf(θ, φ) , − − Speciﬁcally, the spin raising and lowering operators acting twice on the spin- 2 function ±2f(µ, φ) such as the CMB polarization ﬁelds can be expressed as ± 2 0 2 m 2 ∂ 2f(θ, φ) = ∂µ + (1 µ )2f(µ, φ) , − 1 µ2 − (A.2) − 2 0 2 m 2 ∂ −2f(θ, φ) = ∂µ (1 µ )−2f(µ, φ) , − − 1 µ2 − − ˜ imφ where µ cos θ and ±2f(θ, φ) = ±2f(µ)e . Utilizeing these properties, we can express sYlm(θ, φ) ≡ in terms of 0Ylm(θ, φ) = Ylm(θ, φ) as

1 2 (l s)! 0 s sYlm(θ, φ) = − ∂ Ylm(θ, φ) (0 s l) , (l + s)! ≤ ≤ 1 (A.3) 2 (l + s)! s 0 −s sYlm(θ, φ) = ( 1) ∂ Ylm(θ, φ)( l s 0) , (l s)! − − ≤ ≤ − where these equations contain

0 1 ∂ sYlm(θ, φ) = [(l s)(l + s + 1)] 2 s+1Ylm(θ, φ) , − 0 1 ∂ sYlm(θ, φ) = [(l + s)(l s + 1)] 2 s−1Ylm(θ, φ) , (A.4) − − 0 0 ∂ ∂ sYlm(θ, φ) = (l s)(l + s + 1)sYlm(θ, φ) . − − These properties reduce to an explicit expression: 1/2 imφ (l + m)!(l m)! 2l + 1 2l sYlm(θ, φ) = e − sin (θ/2) (l + s)!(l s)! 4π − X l s l + s − ( 1)l−r−s+mcot2r+s−m(θ/2) . (A.5) × r r + s m − r − This holds the orthogonality and completeness conditions as Z 2π Z 1 ∗ dφ d cos θsYl0m0 (θ, φ)sYlm(θ, φ) = δl0,lδm0,m , 0 −1 (A.6) X ∗ 0 0 0 0 sY (θ, φ)sYlm(θ , φ ) = δ(φ φ )δ(cos θ cos θ ). lm − − lm 145

m Y1m 1Y1m q q 1 m 3 sin θemiφ 1 3 (1 m cos θ)emiφ ± − 8π − 2 4π − 1 q 3 q 3 0 2 π cos θ 8π sin θ

Table A.1: Dipole (l = 1) harmonics for spin-0 and 1.

m Y2m 2Y2m q q 2 1 15 sin2 θ emiφ 1 5 1 m cos θ2 emiφ ± 4 2π 8 π − 2 q 15 miφ 1 q 5 miφ 1 m 8π sin θ cos θ e 4 π sin θ (1 m cos θ) e ± − q − q − 0 1 5 (3 cos2 θ 1) 3 5 sin2 θ 2 4π − 4 6π Table A.2: Quadrupole (l = 2) harmonics for spin-0 and 2.

The reactions to complex condugate and parity transformation are given by

∗ s+m sYlm(θ, φ) = ( 1) −sYl−m(θ, φ) , − l (A.7) sYlm(π θ, φ + π) = ( 1) Ylm(θ, φ) . − − −s Finally, we give the speciﬁc expressions for some simple cases in Tables A.1 and A.2. 146 B WIGNER D-MATRIX

B Wigner D-matrix

(l) Here, on the basis of Refs. [3, 5, 6], we introduce the properties of the Wigner D-matrix Dmm0 , which is the unitary irreducible matrix of rank 2l + 1 that forms a representation of the rotational group as SU(2) and SO(3). With this matrix, the change of the spin wighted spherical harmonic function under the rotational transformation as nˆ Rnˆ is expressed as → ∗ X (l) ∗ sYlm(Rnˆ) = Dmm0 (R)sYlm0 (nˆ) . (B.1) m0 This satisﬁes the relation as

(l)∗ m−m0 (l) (l) −1 D 0 (R) = ( 1) D 0 (R) = D 0 (R ) . (B.2) mm − −m,m m m When we express the rotational matrix with three Euler angles (α, β, γ) under the z y z convention as − −  cos α cos β cos γ sin α sin γ cos β sin γ cos α cos γ sin α cos α sin β  − − − R =  cos α sin γ + cos γ cos β sin α cos α cos γ cos β sin α sin γ sin β sin α  , (B.3) cos γ sin β sin−γ sin β cos β − we can write a general relationship between the Wigner D-matrix and the spin weighted spherical harmonics as r (l) s 4π ∗ −isγ D (α, β, γ) = ( 1) −sY (β, α)e . (B.4) ms − 2l + 1 lm Like the spin weighted spherical harmonics, there also exists the orthogonality of the Wigner D-matrix as

Z 2π Z 1 Z 2π 2 (l0)∗ (l) 8π dα d cos β dγDm0s0 (α, β, γ)Dms(α, β, γ) = δl0,lδm0,mδs0,s . (B.5) 0 −1 0 2l + 1 147

C Wigner symbols

Here, we brieﬂy review the useful properties of the Wigner-3j, 6j and 9j symbols. The following discussions are based on Refs. [5, 7–12].

C.1 Wigner-3j symbol In quantum mechanics, considering the coupling of two angular momenta as

l3 = l1 + l2 , (C.1) the scalar product of eigenstates between the right-handed term and the left-handed one, namely, a Clebsch-Gordan coeﬃcient, is related to the Wigner-3j symbol:

l1−l2+m3 l l l ( 1) l1m1l2m2 (l1l2)l3m3 1 2 3 − h | i . (C.2) m1 m2 m3 ≡ √2l + 1 − 3 This symbol vanishes unless the selection rules are satisﬁed as follows:

m1 l1 , m2 l2 , m3 l3 , m1 + m2 = m3 , | | ≤ | | ≤ | | ≤ (C.3) l1 l2 l3 l1 + l2 (the triangle condition) , l1 + l2 + l3 Z . | − | ≤ ≤ ∈ Symmetries of the Wigner-3j symbol are given by l1 l2 l3 P3 l l2 l1 l3 P3 l l1 l3 l2 = ( 1) i=1 i = ( 1) i=1 i m1 m2 m3 − m2 m1 m3 − m1 m3 m2 (odd permutation of columns) l l l l l l = 2 3 1 = 3 1 2 m2 m3 m1 m3 m1 m2 (even permutation of columns) P3 l l1 l2 l3 = ( 1) i=1 i − m1 m2 m3 − − − (sign inversion of m1, m2, m3) . (C.4)

The Wigner-3j symbols satisfy the orthogonality as X l1 l2 l3 l1 l2 l3 (2l + 1) = δ 0 δ 0 , 3 0 0 m1,m1 m2,m2 m1 m2 m3 m1 m2 m3 l3m3 (C.5) 0 X l1 l2 l3 l1 l2 l3 (2l + 1) = δ 0 δ 0 . 3 0 l3,l3 m3,m3 m1 m2 m3 m1 m2 m3 m1m2

P3 For a special case that i=1 li = even and m1 = m2 = m3 = 0, there is an analytical expression as

3 li p p p −l P l1 l2 l3 P3 i i=1 2 ! ( l1 + l2 + l3)! (l1 l2 + l3)! (l1 + l2 l3)! = ( 1) i=1 2 − − q − . (C.6) 0 0 0 − −l1+l2+l3 l1−l2+l3 l1+l2−l3 P3 2 ! 2 ! 2 ! i=1 li + 1 ! 148 C WIGNER SYMBOLS

P3 This vanishes for i=1 li = odd. The Wigner-3j symbol is related to the spin-weighted spherical harmonics as 2 Y X l l l ˆ ∗ ˆ −s1−s2−s3 1 2 3 si Ylimi (n) = s3 Yl3m3 (n)Il l l , (C.7) 1 2 3 m1 m2 m3 i=1 l3m3s3 which leads to the “extended” Gaunt integral including spin dependence: Z 2 −s1−s2−s3 l1 l2 l3 d nˆs1 Yl1m1 (nˆ)s2 Yl2m2 (nˆ)s3 Yl3m3 (nˆ) = Il l l . (C.8) 1 2 3 m1 m2 m3 q l l l Here Is1s2s3 (2l1+1)(2l2+1)(2l3+1) 1 2 3 . l1l2l3 4π ≡ s1 s2 s3

C.2 Wigner-6j symbol Considering two other ways in the coupling of three angular momenta as

l5 = l1 + l2 + l4 (C.9)

= l3 + l4 (C.10)

= l1 + l6 , (C.11) the Wigner-6j symbol is deﬁned using a Clebsch-Gordan coeﬃcient between each eigenstate of l5 corresponding to Eqs. (C.10) and (C.11) as

l1+l2+l4+l5 l1 l2 l3 ( 1) (l1l2)l3; l4; l5m5 l1;(l2l4)l6; l5m5 − ph | i . (C.12) l4 l5 l6 ≡ (2l3 + 1)(2l6 + 1) This is expressed with the summation of three Wigner-3j symbols: X P6 l −m l5 l1 l6 l6 l2 l4 l4 l3 l5 ( 1) i=4 i i − m5 m1 m6 m6 m2 m4 m4 m3 m5 m4m5m6 − − − − − − l l l l l l = 1 2 3 1 2 3 ; (C.13) m1 m2 m3 l4 l5 l6 hence, the triangle conditions are given by

l1 l2 l3 l1 + l2, l4 l5 l3 l4 + l5 , | − | ≤ ≤ | − | ≤ ≤ (C.14) l1 l5 l6 l1 + l5, l4 l2 l6 l4 + l2 . | − | ≤ ≤ | − | ≤ ≤ The Wigner-6j symbol obeys 24 symmetries such as l l l l l l l l l 1 2 3 = 2 1 3 = 2 3 1 (permutation of columns) l4 l5 l6 l5 l4 l6 l5 l6 l4 l l l l l l = 4 5 3 = 1 5 6 l1 l2 l6 l4 l2 l3 (exchange of two corresponding elements between rows). (C.15) Geometrically, the Wigner-6j symbol is expressed using the tetrahedron composed of four triangles obeying Eq. (C.14). It is known that the Wigner-6j symbol is suppressed by the square root of the volume of the tetrahedron at high multipoles. C.3 Wigner-9j symbol 149

C.3 Wigner-9j symbol Considering two other ways in the coupling of four angular momenta as

l9 = l1 + l2 + l4 + l5 (C.16)

= l3 + l6 (C.17)

= l7 + l8 , (C.18) where l3 l1 + l2, l6 l4 + l5, l7 l1 + l4, l8 l2 + l5, the Wigner 9j symbol expresses a ≡ ≡ ≡ ≡ Clebsch-Gordan coeﬃcient between each eigenstate of l9 corresponding to Eqs. (C.17) and (C.18) as   l1 l2 l3   (l1l2)l3;(l4l5)l6; l9m9 (l1l4)l7;(l2l5)l8; l9m9 l4 l5 l6 h p | i . (C.19) ≡ (2l3 + 1)(2l6 + 1)(2l7 + 1)(2l8 + 1)  l7 l8 l9  This is expressed with the summation of ﬁve Wigner-3j symbols: X l4 l5 l6 l7 l8 l9 m4 m5 m6 m7 m8 m9 m4m5m6 m7m8m9 l4 l7 l1 l5 l8 l2 l6 l9 l3 × m4 m7 m1 m5 m8 m2 m6 m9 m3   l1 l2 l3 l1 l2 l3   = l4 l5 l6 , (C.20) m1 m2 m3  l7 l8 l9  and that of three Wigner-6j symbols:   l1 l2 l3   X 2x l1 l4 l7 l2 l5 l8 l3 l6 l9 l4 l5 l6 = ( 1) (2x + 1) ; (C.21) − l8 l9 x l4 x l6 x l1 l2  l7 l8 l9  x hence, the triangle conditions are given by

l1 l2 l3 l1 + l2 , l4 l5 l6 l4 + l5 , l7 l8 l9 l7 + l8 , | − | ≤ ≤ | − | ≤ ≤ | − | ≤ ≤ (C.22) l1 l4 l7 l1 + l4 , l2 l5 l8 l2 + l5 , l3 l6 l9 l3 + l6 . | − | ≤ ≤ | − | ≤ ≤ | − | ≤ ≤ The Wigner-9j symbol obeys 72 symmetries:       l1 l2 l3 l2 l1 l3 l1 l2 l3   P9   P9   li li l4 l5 l6 = ( 1) i=1 l5 l4 l6 = ( 1) i=1 l7 l8 l9 − −  l7 l8 l9   l8 l7 l9   l4 l5 l6  (odd permutation of rows or columns)      l2 l3 l1   l4 l5 l6  = l5 l6 l4 = l7 l8 l9  l8 l9 l7   l1 l2 l3  (even permutation of rows or columns)      l1 l4 l7   l9 l6 l3  = l2 l5 l8 = l8 l5 l2  l3 l6 l9   l7 l4 l1  (reﬂection of the symbols) . (C.23) 150 C WIGNER SYMBOLS

C.4 Analytic expressions of the Wigner symbols Here, we show some analytical formulas of the Wigner symbols. The I symbols, which are deﬁned as r s1s2s3 (2l1 + 1)(2l2 + 1)(2l3 + 1) l1 l2 l3 Il l l , (C.24) 1 2 3 ≡ 4π s1 s2 s3 are expressed as s 3 Q −l 0 0 0 i=1(2li + 1) P3 i I = ( 1) i=1 2 l1l2l3 4π − P3 li p p p i=1 2 ! ( l1 + l2 + l3)! (l1 l2 + l3)! (l1 + l2 l3)! − − q − × −l1+l2+l3 l1−l2+l3 l1+l2−l3 P3 2 ! 2 ! 2 ! ( i=1 li + 1)!

(for l1 + l2 + l3 = even)

= 0 (for l1 + l2 + l3 = odd) , (C.25) r s 5 (l2 1)(l2 + 1) I0 1 −1 = ( 1)l2+1 (for l = l 2, l = 2) l1 l2 l3 − 1 2 3 8π − l2 1/2 − − s r 15 l + 1/2 l2 2 = ( 1) (for l1 = l2, l3 = 2) 16π − (l2 1/2)(l2 + 3/2) − s r 5 l (l + 2) l2 2 2 = ( 1) (for l1 = l2 + 2, l3 = 2) 8π − l2 + 3/2 r 3 l3+1p = ( 1) l3 + 1 (for l1 = l3 1, l2 = 1) 8π − − r 3 l3+1p = ( 1) l3 + 1/2 (for l1 = l3, l2 = 1) 4π − r 3 l3+1p = ( 1) l3 (for l1 = l3 + 1, l2 = 1) . (C.26) 8π − The Wigner-9j symbols are calculated as   l1 l2 l3 r   2(l3 1) + 1 l1 l4 1 l2 l5 1 l4 l5 l6 = ± (for l6 = l3 2) 5 l3 2 l3 1 l5 l3 1 l3 l1 ±  1 1 2  ± ± ± s (2l3 1)(2l3 + 2)(2l3 + 3) l l 1 l l 1 = − 1 4 2 5 30(2l )(2l + 1) l3 l3 1 l5 l3 1 l3 l1 3 3 − − s 2(2l3 1)(2l3 + 1)(2l3 + 3) l l 1 l l 1 + − 1 4 2 5 15(2l3)(2l3 + 2) l3 l3 l5 l3 l3 l1 s (2l3 1)(2l3)(2l3 + 3) l l 1 l l 1 + − 1 4 2 5 30(2l3 + 1)(2l3 + 2) l3 l3 + 1 l5 l3 + 1 l3 l1

(for l6 = l3) , (C.27) C.4 Analytic expressions of the Wigner symbols 151 where these Wigner-6j symbols are analytically given by s P P l1 l2 1 l1+l4+l6+1 l1+l4+l6+2 2 l1+l4−l6+1 2 = ( 1) (for l2 = l1 1, l5 = l4 + 1) l4 l5 l6 − 2l4+3P3 2l1+1P3 − s 2(l1 + l4 + l6 + 2)(l1 + l4 l6 + 1) = ( 1)l1+l4+l6+1 − − 2l4+3P3 s ( l1 + l4 + l6 + 1)(l1 l4 + l6) − − (for l2 = l1, l5 = l4 + 1) × 2l1+2P3 s P P l1+l4+l6+1 −l1+l4+l6+1 2 l1−l4+l6+1 2 = ( 1) (for l2 = l1 + 1, l5 = l4 + 1) − 2l4+3P3 2l1+3P3

l1+l4+l6+1 = ( 1) [l4(l4 + 1) + l1(l1 1)(l4 + 1) l6(l6 + 1) l1(l1 + 1)l4] −s − − − 2(l1 + l4 + l6 + 1)(l1 + l4 l6) − (for l2 = l1 1, l5 = l4) × ( l1 + l4 + l6 + 1)(l1 l4 + l6)2l +2P3 2l +1P3 − − − 4 1 l (l + 1) + l (l + 1)(l + 1) l (l + 1) l (l + 1)l l1+l4+l6+1 4 4 1 1 4 6 6 1 1 4 = 2( 1) p − − − 2l4+2P3 2l1+2P3

(for l2 = l1, l5 = l4) l1+l4+l6+1 = ( 1) [l4(l4 + 1) + (l1 + 1)(l1 + 2)(l4 + 1) l6(l6 + 1) l1(l1 + 1)l4] −s − − 2( l1 + l4 + l6)(l1 l4 + l6 + 1) − − (for l2 = l1 + 1, l5 = l4) . × (l1 + l4 + l6 + 2)(l1 + l4 l6 + 1)2l +2P3 2l +3P3 − 4 1 (C.28)

Using these analytical formulas, one can reduce the time cost involved with calculating the CMB bispectrum from PMFs. 152 D POLARIZATION VECTOR AND TENSOR

D Polarization vector and tensor

(±1) We summarize the relations and properties of a divergenceless polarization vector a and a (±2) transverse and traceless polarization tensor eab [6, 11] . The polarization vector with respect to a unit vector nˆ is expressed using two unit vectors θˆ and φˆ perpendicular to nˆ as

(±1) 1 ˆ ˆ (nˆ) = [θa(nˆ) i φa(nˆ)] . (D.1) a √2 ± This satisﬁes the relations:

a (±1) nˆ a (nˆ) = 0 , (±1)∗(nˆ) = (∓1)(nˆ) = (±1)( nˆ) , (D.2) a a a − (λ) (λ0) 0 (nˆ) (nˆ) = δλ,−λ0 (for λ, λ = 1) . a a ± By deﬁning a rotational matrix, which transforms a unit vector parallel to the z axis, namely zˆ, to nˆ, as   cos θn cos φn sin φn sin θn cos φn − S(nˆ)  cos θn sin φn cos φn sin θn sin φn  , (D.3) ≡ sin θn 0 cos θn − we specify θˆ and φˆ as

θˆ(nˆ) = S(nˆ)xˆ , φˆ(nˆ) = S(nˆ)yˆ , (D.4) where xˆ and yˆ are unit vectors parallel to x- and y-axes. By using Eq. (D.1), the polarization tensor is constructed as

(±2) (±1) (±1) eab (nˆ) = √2a (nˆ)b (nˆ) . (D.5) To utilize the polarization vector and tensor in the calculation of this thesis, we need to expand Eqs. (D.1) and (D.5) with spin spherical harmonics. An arbitrary unit vector is expanded with the spin-0 spherical harmonics as

X m rˆa = αa Y1m(ˆr) , m   r m(δm,1 + δm,−1) (D.6) m 2π − αa  i (δm,1 + δm,−1)  . ≡ 3 √2δm,0

m Here, note that the repeat of the index implies the summation. The scalar product of αa is calculated as

m m0 4π m m m0∗ 4π α α = ( 1) δm,−m0 , α α = δm,m0 . (D.7) a a 3 − a a 3 Through the substitution of Eq. (D.4) into Eq. (D.6), θˆ is expanded as

ˆ X m ˆ X m X (1)∗ θa(nˆ) = αa Y1m(θ(nˆ)) = αa Dmm0 (S(nˆ))Y1m0 (xˆ) m m m0 153

s X = (δ + δ ) αm Y (nˆ) . (D.8) √ s,1 s,−1 a s 1m − 2 m

Here, we use the properties of the Wigner D-matrix as described in Appendix B [3, 5, 6, 13]

X (`)∗ Y`m(S(nˆ)xˆ) = Dmm0 (S(nˆ))Y`m0 (xˆ) , m0 1/2 (D.9) (`) 4π s ∗ D (S(nˆ)) = ( 1) −sY (nˆ) . ms 2` + 1 − `m

In the same manner, φˆ is also calculated as

i X φˆ (nˆ) = (δ + δ ) αm Y (nˆ) ; (D.10) a √ s,1 s,−1 a s 1m 2 m hence, the explicit form of Eq. (D.1) is calculated as

(±1) X m a (nˆ) = αa ±1Y1m(nˆ) . (D.11) ∓ m

∓2±1±1 √3 Substituting this into Eq. (D.5) and using the relations of Appendix C and I2 1 1 = 2 π , the polarization tensor can also be expressed as (±2) 3 X ∗ ma mb 2 1 1 eab (nˆ) = ∓2Y2M (nˆ)αa αb . (D.12) √2π M ma mb Mmamb

This obeys the relations:

(±2) (±2) eaa (nˆ) =n ˆaeab (nˆ) = 0 , e(±2)∗(nˆ) = e(∓2)(nˆ) = e(±2)( nˆ) , (D.13) ab ab ab − (λ) (λ0) 0 e (nˆ)e (nˆ) = 2δλ,−λ0 (for λ, λ = 2) . ab ab ± Using the projection operators as

(0) ik·x −1 ik·x ˆ ik·x Oa e k ae = ikae , ≡ ∇ (0) ik·x −2 δa,b ik·x ˆ ˆ δa,b ik·x O e k a b + e = kakb + e , ab ≡ ∇ ∇ 3 − 3 O(±1)eik·x i(±1)(kˆ)eik·x , (D.14) a ≡ − a (±1) ik·x −1 (±1) (±1) ik·x ˆ (±1) ˆ ˆ (±1) ˆ ik·x O e k aO + bO e = ka (k) + kb (k) e , ab ≡ ∇ b ∇ a b a O(±2)eik·x e(±2)(kˆ)eik·x , ab ≡ ab the arbitrary scalar, vector and tensor are decomposed into the helicity states as

η(k) = η(0)(k), (D.15) (0) (0) X (λ) (λ) ωa(k) = ω (k)Oa + ω (k)Oa , (D.16) λ=±1 154 D POLARIZATION VECTOR AND TENSOR

1 (0) (0) X (λ) (λ) X (λ) (λ) χab(k) = χiso(k)δa,b + χ (k)O + χ (k)O + χ (k)O . (D.17) −3 ab ab ab λ=±1 λ=±2

Then, using Eq. (D.2) and (D.13), we can ﬁnd the inverse formulae as

(0) (0) ω (k) = Oa ωa(k) , (D.18) (±1) − (∓1) ˆ ω (k) = Oa (k)ωa(k) , (D.19) −3 χ(0)(k) = O(0)(kˆ)χ (k) , (D.20) 2 ab ab 1 χ(±1)(k) = O(∓1)(kˆ)χ (k) , (D.21) 2 ab ab 1 χ(±2)(k) = O(∓2)(kˆ)χ (k) . (D.22) 2 ab ab From these, we can derive the relations of several projection operators as

(0) ˆ ˆ ˆ 1 O (k) = kakb + δab ab − 3 r 3 2 1 1 X ∗ ˆ ma mb = Y2M (k)αa αb , − 2π M ma mb Mmamb (±1) ˆ ˆ (±1) ˆ ˆ (±1) ˆ Oab (k) = kab (k) + kba (k) 3 2 1 1 X ∗ ˆ ma mb = ∓1Y2M (k)αa αb , ±√2π M ma mb Mmamb (±2) ˆ (±2) ˆ Oab (k) = eab (k) 3 2 1 1 X ∗ ˆ ma mb = ∓2Y2M (k)αa αb , √2π M ma mb Mmamb (D.23) ˆ ˆ ˆ Pab(k) δab kakb ≡ − L 1 1 X 01−1 X ∗ ˆ ma mb = 2 IL11 YLM (k)αa αb , − M ma mb L=0,2 Mmamb 1 O(0)(kˆ)P (kˆ) = P (kˆ) , ab bc 3 ac (±1) ˆ ˆ ˆ (±1) ˆ Oab (k)Pbc(k) = kac (k) , (±2) ˆ ˆ (±2) ˆ Oab (k)Pbc(k) = eac (k) , ˆ abc (+1) ˆ (−1) ˆ kc = iη a (k)b (k) , abcˆ (±1) ˆ (±1) ˆ η ka (k) = i (k) . b ∓ c 155

(a) (a) E Calculation of f 3 and f W WWf 2 Here, we calculate each product between the wave number vectors and the polarization tensors of (a) (a) f 3 and f mentioned in Sec. 8.2.1 [14]. W WWf 2 (a) Using the relations discussed in Appendix D, the all terms of fW 3 are written as

(−λ1) (−λ2) (−λ3) 3/2 X ∗ ˆ ∗ ˆ ∗ ˆ e e e = (8π) λ Y (k1)λ Y 0 (k2)λ Y 00 (k3) ij jk ki − 1 2M 2 2M 3 2M M,M 0,M 00 r 1 7 2 2 2 , × 10 3 MM 0 M 00 (−λ1) (−λ2) (−λ3) ˆ ˆ 3/2 X X ∗ ˆ ∗ ˆ ∗ ˆ e e e k2 k3 = (8π) λ Y (k1)λ Y 0 0 (k2)λ Y 00 00 (k3) ij kl kl i j − 1 2M 2 L M 3 L M L0,L00=2,3 M,M 0,M 00 0 00 0 00 0 4π L λ20−λ2 λ30−λ3 2 L L 2 L L ( 1) I 0 I 00 , × 15 − L 12 L 12 MM 0 M 00 2 1 1 (−λ1) (−λ2) (−λ3) ˆ ˆ 3/2 X X ∗ ˆ ∗ ˆ ∗ ˆ e e e k2 k3 = (8π) λ Y (k1)λ Y 0 0 (k2)λ Y 00 00 (k3) ij ki jl l k − 1 2M 2 L M 3 L M L0,L00=2,3 M,M 0,M 00 (E.1)  0 00  0 00 2 L L 0   4π L λ20−λ2 λ30−λ3 2 L L ( 1) I 0 I 00 1 1 2 , 3 L 12 L 12 MM 0 M 00 × −  1 2 1  (−λ1) (−λ2) (−λ3) ˆ ˆ 3/2 X X ∗ ˆ ∗ ˆ ∗ ˆ e e e k2 k3 = (8π) λ Y (k1)λ Y 0 0 (k2)λ Y 00 00 (k3) ij ik kl l j − 1 2M 2 L M 3 L M L0,L00=2,3 M,M 0,M 00 0 00 0 4π L λ20−λ2 λ30−λ3 2 L L ( 1) I 0 I 00 × 3 − L 12 L 12 MM 0 M 00 2 1 L0 2 L0 L00 . × 2 1 1 2 1 1

In the calculation of f (a) , we also need to consider the dependence of the tensor contractions WWf 2 on ηijk. Making use of the relation:

3/2 abc ma mb mc 4π 1 1 1 η αa αb αc = i √6 , (E.2) − 3 ma mb mc the ﬁrst two terms of f (a) reduce to WWf 2

ijk (−λ1) (−λ2) (−λ3) ˆ 3/2 X X ∗ ˆ ∗ ˆ ∗ ˆ iη e e e k3 = (8π) λ Y (k1)λ Y 0 (k2)λ Y 00 00 (k3) kq jm iq m − 1 2M 2 2M 3 L M L00=2,3 M,M 0,M 00 r 00 00 00 2π L λ30−λ3 2 2 L 2 2 L ( 1) I 00 , × 5 − L 12 MM 0 M 00 1 2 1

ijk (−λ1) (−λ2) (−λ3) ˆ 3/2 X X ∗ ˆ ∗ ˆ ∗ ˆ (E.3) iη e e e k3 = (8π) λ Y (k1)λ Y 0 (k2)λ Y 00 00 (k3) kq mi mq j − 1 2M 2 2M 3 L M L00=2,3 M,M 0,M 00  00  00 2 2 L 00   L λ30−λ3 2 2 L 2√2π( 1) I 00 1 1 1 . L 12 MM 0 M 00 × −  1 1 2  (A) (A) 156 E CALCULATION OF F 3 AND F W WWf 2

For the other terms, by using the relation

abcˆ (λ) ˆ λ (λ) ˆ η kae (k) = ie (k) , (E.4) bd − 2 cd we have

λ1 ijk (−λ1) (−λ2) ˆ ˆ (−λ3) ˆ 3/2 X X iη e e k1 k2 e k3 = (8π) pj pm k l il m − 2 L0,L00=2,3 M,M 0,M 00 ∗ ˆ ∗ ˆ ∗ ˆ λ Y (k1)λ Y 0 0 (k2)λ Y 00 00 (k3) × 1 2M 2 L M 3 L M 00 4π L λ20−λ2 λ30−λ3 ( 1) I 0 I 00 × 3 − L 12 L 12  2 L0 L00  2 L0 L00   0 00 1 2 1 , × MM M   1 1 2 (E.5) λ1 ijk (−λ1) (−λ2) ˆ ˆ (−λ3) ˆ 3/2 X X iη e e k1 k2 e k3 = (8π) pj pm k l im l − 2 L0,L00=2,3 M,M 0,M 00 ∗ ˆ ∗ ˆ ∗ ˆ λ1 Y2M (k1)λ2 YL0M 0 (k2)λ3 YL00M 00 (k3) × r 00 2π 7 L λ20−λ2 λ30−λ3 ( 1) I 0 I 00 × 15 3 − L 12 L 12 2 L0 L00 2 L0 L00 . × MM 0 M 00 1 2 2 157

F Graviton non-Gaussianity from the Weyl cubic terms

Here, let us derive the bispectra of gravitons coming from the parity-even and parity-odd Weyl cubic terms, namely, Eqs. (8.16) and (8.17) [14]. For convenience, we decompose the interaction Hamiltonians of W 3 and WWf 2 (8.15) into

4 X (n) Hint = Hint . (F.1) n=1

Depending on this, we also split the graviton non-Gaussianity as

* 3 + 4 * 3 +(m) Y (λn) X Y (λn) γ (kn) = γ (kn) . (F.2) n=1 int m=1 n=1 int

In what follows, we shall show the computation of each fraction.

F.1 W 3

(1) The bracket part of Eq. (8.12) in terms of HW 3 is expanded as

* " 3 # + (1) Y 0 : H (τ 0):, γ(λn)(k , τ) 0 W 3 n n=1 * 3 + * " 3 # + (1) Y Y (1) = 0 : H (τ 0): γ(λn)(k , τ) 0 0 γ(λn)(k , τ) : H (τ 0): 0 W 3 n n W 3 n=1 − n=1   Z 0 A 3 Z 3 0 τ 1 d k 0 0 (λ0 ) (λ0 ) (λ0 ) 3 0 −2 0 2 Y n ikn·x X 1 ˆ0 2 ˆ0 3 ˆ0 = d x Λ (Hτ )  3 e  eij (k1)ejk (k2)eki (k3) − τ∗ 4 (2π) 0 n=1 λn=±2 " * ( 3 )( 3 ) + 0 Y 02 0 0 (λn) Y ∗ (λm)† : 0 (¨γdS k γdS)(k , τ )a 0 γ (km, τ)a 0 : n n kn dS −km × n=1 − m=1 * ( 3 )( 3 ) + # 0 Y (λm) Y ∗ 02 ∗ 0 0 (λn)† : 0 γdS(km, τ)a (¨γ k γ )(k , τ )a 0 0 : km dS n dS n −kn − m=1 n=1 − 0 A 3 ! τ 1 X (λ ) (λ ) (λ ) = Λ−2(Hτ 0)2 (2π)3δ k e 1 ( kˆ )e 2 ( kˆ )e 3 ( kˆ ) τ 4 n ij 1 jk 2 ki 3 − ∗ n=1 − − − "( 3 ) ( 3 )# Y 2 0 ∗ Y ∗ 2 ∗ 0 6 (¨γdS knγdS)(kn, τ )γdS(kn, τ) γdS(kn, τ)(¨γdS knγdS)(kn, τ ) × n=1 − − n=1 − 0 A 3 ! 3 τ X (−λ ) (−λ ) (−λ ) = Λ−2(Hτ 0)2 (2π)3δ k e 1 (kˆ )e 2 (kˆ )e 3 (kˆ ) 2 τ n ij 1 jk 2 ki 3 − ∗ n=1 " 3 # Y 2 0 ∗ 2iIm (¨γdS knγdS)(kn, τ )γdS(kn, τ) . (F.3) × n=1 − 158 F GRAVITON NON-GAUSSIANITY FROM THE WEYL CUBIC TERMS

Here, we use

* 3 + 0 Y (λn) (λn)† 9 0 0 0 0 : a 0 a : 0 = (2π) δ(k1 + k )δλ ,λ0 δ(k + k3)δλ0 ,λ δ(k2 + k )δλ ,λ0 + 5 perms. kn −kn 3 1 3 1 1 3 2 2 2 n=1 * 3 + 0 (F.4) Y (λn) (λn)† = 0 : a a 0 : 0 kn −kn n=1 e(−λ)(kˆ) = e(λ)( kˆ) . ij ij − Furthermore, since

0 2 2Hτ 3/2 −ikτ 0 γ¨dS k γdS = k e , − Mpl 3 3 (F.5) Y τ→0 H γ∗ (k , τ) i (k k k )−3/2 , dS n M 3 1 2 3 n=1 −−→ pl the time integral at τ 0 is performed as → " A 3 # Z τ τ 0 Y Im dτ 0(Hτ 0)2 (¨γ k2γ )(k , τ 0)γ∗ (k , τ) τ dS n dS n dS n −∞ ∗ n=1 − 5 " 3 ! Z τ # 8H q Y 0 = k3k3k3Im γ∗ (k , τ) τ −A dτ 0τ 05+Ae−iktτ M 3 1 2 3 dS n ∗ pl n=1 −∞ 8 0 Z 0 τ→0 8H −A 0 05+A −iktτ 6 Re τ∗ dτ τ e , (F.6) −−→ Mpl −∞

P3 (1) where kt n=1 kn. Thus, the graviton non-Gaussianity in the late time limit arising from HW 3 is ≡

(1) * 3 + 3 ! 6 2 0 Z 0 Y (λn) 3 X H H −A 0 05+A −iktτ γ (kn) = (2π) δ kn 8 Re τ∗ dτ τ e Mpl Λ −∞ n=1 W 3 n=1 (−λ1) ˆ (−λ2) ˆ (−λ3) ˆ 3e (k1)e (k2)e (k3) . (F.7) × ij jk ki (2) The bracket part in terms of HW 3 is given by

* " 3 # + (2) Y 0 : H (τ 0):, γ(λn)(k , τ) 0 W 3 n n=1 * 3 + * " 3 # + (2) Y Y (2) = 0 : H (τ 0): γ(λn)(k , τ) 0 0 γ(λn)(k , τ) : H (τ 0): 0 W 3 n n W 3 n=1 − n=1 0 A 3 ! 3 τ X (−λ ) (−λ ) (−λ ) = Λ−2(Hτ 0)2 k k (2π)3δ k kˆ kˆ e 1 (kˆ )e 2 (kˆ )e 3 (kˆ ) 2 τ 2 3 n 2i 3j ij 1 kl 2 kl 3 ∗ n=1 " 3 # 2 0 0 0 Y ∗ 2iIm (¨γdS k1γdS)(k1, τ )γ ˙ dS(k2, τ )γ ˙ dS(k3, τ ) γdS(kn, τ) + 5 perms.. (F.8) × − n=1 F.2 WWf 2 159

Using

Hτ −ikτ 0 γ˙ dS = i √ke , (F.9) Mpl we can reduce the time integral to

" A 3 # Z τ τ 0 Y Im dτ 0(Hτ 0)2 k k (¨γ k2γ )(k , τ 0)γ ˙ (k , τ 0)γ ˙ (k , τ 0) γ∗ (k , τ) τ 2 3 dS 1 dS 1 dS 2 dS 3 dS n −∞ ∗ − n=1 8 0 Z 0 τ→0 2H −A 0 05+A −iktτ 6 Re τ∗ dτ τ e , (F.10) −−→− Mpl −∞ and obtain (2) * 3 + 3 ! 6 2 0 Z 0 Y (λn) 3 X H H −A 0 05+A −iktτ γ (kn) = (2π) δ kn 8 Re τ∗ dτ τ e Mpl Λ −∞ n=1 W 3 n=1 3 ˆ (−λ1) ˆ ˆ (−λ2) ˆ (−λ3) ˆ k2 e (k1)k3 e (k2)e (k3) + 5 perms.. (F.11) ×4 i ij j kl kl (3) (4) The graviton non-Gaussianities from HW 3 and HW 3 are derived in the same manner as that from (2) HW 3 :

(m) 4 * 3 + 3 ! 6 2 0 Z 0 X Y (λn) 3 X H H −A 0 05+A −iktτ γ (kn) = (2π) δ kn 8 Re τ∗ dτ τ e Mpl Λ −∞ m=3 n=1 W 3 n=1 3 ˆ (−λ2) ˆ (−λ1) ˆ (−λ3) ˆ ˆ k3 e (k2)e (k1)e (k3)k2 × 4 k ki ij jl l 3 ˆ (−λ1) ˆ (−λ2) ˆ (−λ3) ˆ ˆ k3 e (k1)e (k2)e (k3)k2 + 5 perms. . (F.12) −2 j ji ik kl l

F.2 WWf 2 At ﬁrst, we shall focus on the contribution of H(1) . The bracket part is computed as WWf 2 * " 3 # +

(1) 0 Y (λn) 0 : H (τ ):, γ (kn, τ) 0 WWf 2 n=1 * 3 + * " 3 # +

(1) 0 Y (λn) Y (λn) (1) 0 = 0 : H (τ ): γ (kn, τ) 0 0 γ (kn, τ) : H (τ ): 0 WWf 2 WWf 2 n=1 − n=1   0 A 3 3 0 Z τ Z d k 0 0 3 0 −2 0 2 Y n ikn·x X = d x Λ (Hτ ) ( 3)  3 e  − τ∗ − (2π) 0 n=1 λn=±2 (λ0 ) (λ0 ) (λ0 ) ηijke 1 (kˆ0 )e 2 (kˆ0 )e 3 (kˆ0 )(ik0 ) × kq 1 jm 2 iq 3 3m 02 0 0 02 0 0 0 0 γ¨dS k γdS (k , τ ) γ¨dS k γdS (k , τ )γ ˙ dS(k , τ ) × − 1 1 − 2 2 3 * ( 3 )( 3 ) + 0 Y (λm) Y ∗ (λn)† : 0 a 0 γ (kn, τ)a 0 : km dS −kn × m=1 n=1 160 F GRAVITON NON-GAUSSIANITY FROM THE WEYL CUBIC TERMS

γ¨∗ k02γ∗ (k0 , τ 0) γ¨∗ k02γ∗ (k0 , τ 0)γ ˙ ∗ (k0 , τ 0) − dS − 1 dS 1 dS − 2 dS 2 dS 3 * ( 3 )( 3 ) + # 0 Y (λn) Y (λm)† : 0 γdS(kn, τ)a a 0 0 : kn −km × n=1 m=1 0 A 3 ! τ X (λ ) (λ ) (λ ) = Λ−2(Hτ 0)2 3(2π)3δ k ηijke 1 ( kˆ )e 2 ( kˆ )e 3 ( kˆ )( ik ) τ n kq 1 jm 2 iq 3 3m ∗ n=1 − − − − " ( 3 ) 2 0 2 0 0 Y ∗ γ¨dS k1γdS (k1, τ ) γ¨dS k2γdS (k2, τ )γ ˙ dS(k3, τ ) γdS(kn, τ) × − − n=1 ( 3 )# ∗ 2 ∗ 0 ∗ 2 ∗ 0 ∗ 0 Y γ¨dS k1γdS (k1, τ ) γ¨dS k2γdS (k2, τ )γ ˙ dS(k3, τ ) γdS(kn, τ) − − − n=1 +5 perms. 0 A 3 ! τ X (−λ ) (−λ ) (−λ ) = Λ−2(Hτ 0)2 ( 3i)k (2π)3δ k ηijke 1 (kˆ )e 2 (kˆ )e 3 (kˆ )kˆ τ 3 n kq 1 jm 2 iq 3 3m ∗ − n=1 " ( 3 )# 2 0 2 0 0 Y ∗ 2iIm γ¨dS k1γdS (k1, τ ) γ¨dS k2γdS (k2, τ )γ ˙ dS(k3, τ ) γdS(kn, τ) × − − n=1 +5 perms.. (F.13) Via the time integral:

"Z τ 0 A 0 0 2 τ 2 0 2 0 Im dτ (Hτ ) k3 γ¨dS k1γdS (k1, τ ) γ¨dS k2γdS (k2, τ ) −∞ τ∗ − − ( 3 )# 0 Y ∗ γ˙ dS(k3, τ ) γdS(kn, τ) × n=1 8 0 Z 0 τ→0 4H −A 0 05+A −iktτ 6 Im τ∗ dτ τ e , (F.14) −−→− Mpl −∞ we have (1) * 3 + 3 ! 6 2 0 Z 0 Y (λn) 3 X H H −A 0 05+A −iktτ γ (kn) = (2π) δ kn 8 Im τ∗ dτ τ e Mpl Λ −∞ n=1 WWf 2 n=1 h i ijk (−λ1) ˆ (−λ2) ˆ (−λ3) ˆ ˆ ( 3i) η e (k1)e (k2)e (k3)k3 + 5 perms. . (F.15) × − kq jm iq m Like this, we can gain the second counterpart:

(2) * 3 + 3 ! 6 2 0 Z 0 Y (λn) 3 X H H −A 0 05+A −iktτ γ (kn) = (2π) δ kn 8 Im τ∗ dτ τ e Mpl Λ −∞ n=1 WWf 2 n=1 h i ijk (−λ1) ˆ (−λ2) ˆ (−λ3) ˆ ˆ i η e (k1)e (k2)e (k3)k3 + 5 perms. . (F.16) × kq mi mq j The bracket part with respect to H(3) is WWf 2 * " 3 # +

(3) 0 Y (λn) 0 : H (τ ):, γ (kn, τ) 0 WWf 2 n=1 F.2 WWf 2 161

* 3 + * " 3 # +

(3) 0 Y (λn) Y (λn) (3) 0 = 0 : H (τ ): γ (kn, τ) 0 0 γ (kn, τ) : H (τ ): 0 WWf 2 WWf 2 n=1 − n=1   0 A 3 3 0 Z τ Z d k 0 0 3 0 −2 0 2 Y n ikn·x X = d x Λ (Hτ ) 4  3 e  − τ∗ (2π) 0 n=1 λn=±2 ijk (λ0 ) (λ0 ) (λ0 ) 0 0 0 η e 1 (kˆ0 )e 2 (kˆ0 )e 3 (kˆ0 )(ik )(ik )(ik ) × pj 1 pm 2 il 3 1k 2l 3m " * ( 3 )( 3 ) + 0 Y 0 0 (λn) Y ∗ (λm)† : 0 γ˙ dS(k , τ )a 0 γ (km, τ)a 0 : n kn dS −km × n=1 m=1 * ( 3 )( 3 ) + # 0 Y (λm) Y ∗ 0 0 (λn)† : 0 γdS(km, τ)a γ˙ (k , τ )a 0 0 : km dS n −kn − m=1 n=1 A 3 ! τ 0 X = Λ−2(Hτ 0)2 ( 4)(2π)3δ k τ n ∗ − n=1 ijk (λ1) ˆ (λ2) ˆ (λ3) ˆ η e ( k1)e ( k2)e ( k3)( ik1 )( ik2 )( ik3 ) × pj − pm − il − − k − l − m "( 3 ) ( 3 )# Y 0 ∗ Y ∗ 0 γ˙ dS(kn, τ )γdS(kn, τ) γ˙ dS(kn, τ )γdS(kn, τ) + 5 perms. × n=1 − n=1 A 3 ! τ 0 X = Λ−2(Hτ 0)2 ( 4)( i)3k k k (2π)3δ k τ 1 2 3 n ∗ − − n=1 ijk (−λ1) ˆ (−λ2) ˆ (−λ3) ˆ ˆ ˆ ˆ η e (k1)e (k2)e (k3)k1 k2 k3 × pj pm il k l m " 3 # Y 0 ∗ 2iIm γ˙ dS(kn, τ )γdS(kn, τ) + 5 perms.. (F.17) × n=1

The time integral is

" A 3 # Z τ τ 0 Y Im dτ 0(Hτ 0)2 k γ˙ (k , τ 0)γ∗ (k , τ) τ n dS n dS n −∞ ∗ n=1 8 0 Z 0 τ→0 H −A 0 05+A −iktτ 6 Im τ∗ dτ τ e , (F.18) −−→ Mpl −∞ so that the bispectrum of gravitons becomes

(3) * 3 + 3 ! 6 2 0 Z 0 Y (λn) 3 X H H −A 0 05+A −iktτ γ (kn) = (2π) δ kn 8 Im τ∗ dτ τ e Mpl Λ −∞ n=1 WWf 2 n=1 h i ijk (−λ1) ˆ (−λ2) ˆ (−λ3) ˆ ˆ ˆ ˆ i η e (k1)e (k2)e (k3)k1 k2 k3 + 5 perms. . (F.19) × pj pm il k l m

Through the same procedure, the bispectrum from H(4) is estimated as WWf 2

(4) * 3 + 3 ! 6 2 0 Z 0 Y (λn) 3 X H H −A 0 05+A −iktτ γ (kn) = (2π) δ kn 8 Im τ∗ dτ τ e Mpl Λ −∞ n=1 WWf 2 n=1 162 REFERENCES

h i ijk (−λ1) ˆ (−λ2) ˆ (−λ3) ˆ ˆ ˆ ˆ ( i) η e (k1)e (k2)e (k3)k1 k2 k3 + 5 perms. . × − pj pm im k l l (F.20)

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