Appendix A Spin-Weighted Spherical Harmonic Function

Transcription

1 Appendix 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. [ 4. The spin-weighted spherical harmonic function on D sphere, s Y lm (θ, φ,ismore general expression than the ordinary spherical harmonic function, Y lm (θ, φ, and has additional U(1 symmetry characterized by a spin weight s.thespin-s function such as s Y lm (θ, φ obeys the spin raising and lowering rule as ( s f = e i(s+1ψ s f and ( s f = e i(s 1ψ s f. Here, the spin raising and lowering operators are given by s f (θ, φ = sin s θ [ θ + i csc θ φ sin s θ s f (θ, φ, s f (θ, φ = sin s θ [ θ i csc θ φ sin s θ s f (θ, φ, (A.1 Specifically, the spin raising and lowering operators acting twice on the spin-± function ± f (μ, φ such as the CMB polarization fields can be expressed as ( f (θ, φ = f (θ, φ = μ + ( μ m 1 μ [(1 μ f (μ, φ, m 1 μ [(1 μ f (μ, φ, (A. where μ cos θ and ± f (θ, φ = ± f (μe imφ. Utilizing these properties, we can express s Y lm (θ, φ in terms of 0 Y lm (θ, φ = Y lm (θ, φ as sy lm (θ, φ = sy lm (θ, φ = [ 1 (l s! (l+s! s Y lm (θ, φ (0 s l, [ (l+s! (l s! 1 ( 1 s s Y lm (θ, φ ( l s 0, (A. M. Shiraishi, Probing the Early Universe with the CMB Scalar, Vector and Tensor 15 Bispectrum, Springer Theses, DOI: / , Springer Japan 01

2 154 Appendix A: Spin-Weighted Spherical Harmonic Function Tle A.1 Dipole (l = 1 harmonics for spin-0 and 1 m Y 1m 1 Y 1m ±1 m 8π sin θemiφ π cos θ Tle A. Quadrupole (l = harmonics for spin-0 and m Y m Y m ± ±1 m 0 1 π sin θ e miφ π 5 π 8π sin θ cos θ emiφ π (cos θ 1 4 4π 8π sin θ (1 m cos θemiφ ( 1 m cos θ e miφ sin θ(1 m cos θemiφ 5 6π sin θ where these equations contain s Y lm (θ, φ = [(l s(l + s s+1y lm (θ, φ, s Y lm (θ, φ = [(l + s(l s s 1Y lm (θ, φ, s Y lm (θ, φ = (l s(l + s + 1 s Y lm (θ, φ. (A.4 These properties reduce to an explicit expression: [ (l + m!(l m! sy lm (θ, φ = e imφ l + 1 1/ sin l (θ/ (l + s!(l s! 4π ( ( l s l + s ( 1 l r s+m cot r+s m (θ/. r r + s m r (A.5 This holds the orthogonality and completeness conditions as π 0 1 dφ d cos θ s Yl m (θ, φ sy lm (θ, φ = δ l,lδ m,m, 1 sylm (θ, φ sy lm (θ,φ = δ(φ φ δ(cos θ cos θ. lm (A.6 The reactions to complex conjugate and parity transformation are given by sy lm (θ, φ = ( 1s+m sy l m (θ, φ, sy lm (π θ,φ + π = ( 1 l s Y lm(θ, φ. (A.7 Finally, we give the specific expressions for some simple cases in Tles A.1 and A..

3 Appendix B Wigner D-matrix Here, on the basis of Refs. [, 5, 6, we introduce the properties of the Wigner D-matrix D (l mm, which is the unitary irreducible matrix of rank l + 1 that forms a representation of the rotational group as SU( and SO(. With this matrix, the change of the spin wighted spherical harmonic function under the rotational transformation as ˆn R ˆn is expressed as sy lm (R ˆn = m D (l mm (R s Y lm (ˆn. (B.1 This satisfies the relation as D (l mm (R = ( 1 m m D (l m,m (R = D (l m m (R 1. (B. 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, cos γ sin β sin γ sin β cos β (B. we can write a general relationship between the Wigner D-matrix and the spin weighted spherical harmonics as 4π ms (α,β,γ= ( 1s l + 1 s Ylm (β, αe isγ. D (l (B.4 Like the spin weighted spherical harmonics, there also exists the orthogonality of the Wigner D-matrix as M. Shiraishi, Probing the Early Universe with the CMB Scalar, Vector and Tensor 155 Bispectrum, Springer Theses, DOI: / , Springer Japan 01

4 156 Appendix B: Wigner D-matrix π 1 π dα d cos β dγ D (l m s (α,β,γd (l 8π ms (α,β,γ= l + 1 δ l,lδ m,mδ s,s. (B.5

5 Appendix C Wigner Symbols Here, we briefly review the useful properties of the Wigner- j, 6 j and 9 j symbols. The following discussions are based on Refs. [5, 7 1. C.1 Wigner- j symbol In quantum mechanics, considering the coupling of two angular momenta as l = l 1 + l, (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- j symbol: ( l1 l l ( 1l 1 l +m l 1 m 1 l m (l 1 l l m. (C. m 1 m m l + 1 This symbol vanishes unless the selection rules are satisfied as follows: m 1 l 1, m l, m l, m 1 + m = m, l 1 l l l 1 + l (the triangle condition, l 1 + l + l Z. (C. Symmetries of the Wigner- j symbol are given by M. Shiraishi, Probing the Early Universe with the CMB Scalar, Vector and Tensor 157 Bispectrum, Springer Theses, DOI: / , Springer Japan 01

6 158 Appendix C: Wigner Symbols ( l1 l l = ( 1 ( i=1 l i l l 1 l = ( 1 ( i=1 l i l1 l l m 1 m m m m 1 m m 1 m m = (odd permutation of columns ( ( l l l 1 l l = 1 l m m m 1 m m 1 m (even permutation of columns = ( 1 ( i=1 l i l1 l l m 1 m m (sign inversion of m 1, m, m. The Wigner- j symbols satisfy the orthogonality as (C.4 ( ( l1 l (l + 1 l l1 l l m 1 m m m l m 1 m m = δ m1,m δ 1 m,m, (l + 1 ( ( l1 l l l1 l l m 1 m m m 1 m m = δ l,l δ m,m. (C.5 m 1 m For a special case that i=1 l i = even and m 1 = m = m = 0, there is an analytical expression as ( l1 l l = ( 1 i=1 l i ( i=1 l i! ( l 1 + l + l! (l 1 l + l! (l 1 + l l! ( ( ( (. l1 +l +l! l1 l +l! l1 +l l! i=1 l i + 1! (C.6 This vanishes for i=1 l i = odd. The Wigner- j symbol is related to the spinweighted spherical harmonics as s i Y li m i (ˆn = i=1 l m s s Y ( l m (ˆnI s 1 s s l1 l l l 1 l l m 1 m m, (C.7 which leads to the extended Gaunt integral including spin dependence: ( d ˆn s1 Y l1 m 1 (ˆn s Y l m (ˆn s Y l m (ˆn = I s 1 s s l1 l l l 1 l l m 1 m m. (C.8

7 Appendix C: Wigner Symbols 159 Here I s 1s s l 1 l l (l1 +1(l +1(l +1 4π ( l1 l l. s 1 s s C. Wigner-6 j symbol Considering two other ways in the coupling of three angular momenta as l 5 = l 1 + l + l 4 (C.9 = l + l 4 (C.10 = l 1 + l 6, (C.11 the Wigner-6 j symbol is defined using a Clebsch-Gordan coefficient between each eigenstate of l 5 corresponding to Eqs. (C.10 and (C.11 as { } l1 l l ( 1l 1+l +l 4 +l 5 (l 1 l l ; l 4 ; l 5 m 5 l 1 ; (l l 4 l 6 ; l 5 m 5. (C.1 l 4 l 5 l 6 (l + 1(l This is expressed with the summation of three Wigner- j symbols: ( 6i=4 l ( 1 i m i l5 l 1 l 6 m 4 m 5 m 6 = ( ( l6 l l 4 l4 l l 5 m 5 m 1 m 6 m 6 m m 4 m 4 m m 5 ( { } l1 l l l1 l l ; (C.1 m 1 m m l 4 l 5 l 6 hence, the triangle conditions are given by l 1 l l l 1 + l, l 4 l 5 l l 4 + l 5, l 1 l 5 l 6 l 1 + l 5, l 4 l l 6 l 4 + l. (C.14 The Wigner-6 j symbol obeys 4 symmetries such as { } { } { } l1 l l l l = 1 l l l = l 1 (permutation of columns l 4 l 5 l 6 l 5 l 4 l 6 l 5 l 6 l 4 { } { } l4 l = 5 l l1 l = 5 l 6 l 1 l l 6 l 4 l l (exchange of two corresponding elements between rows. (C.15 Geometrically, the Wigner-6 j symbol is expressed using the tetrahedron composed of four triangles obeying Eq. (C.14. It is known that the Wigner-6 j symbol is suppressed by the square root of the volume of the tetrahedron at high multipoles.

8 160 Appendix C: Wigner Symbols C. Wigner-9 j symbol Considering two other ways in the coupling of four angular momenta as l 9 = l 1 + l + l 4 + l 5 (C.16 = l + l 6 (C.17 = l 7 + l 8, (C.18 where l l 1 + l, l 6 l 4 + l 5, l 7 l 1 + l 4, l 8 l + l 5, the Wigner 9 j symbol expresses a Clebsch-Gordan coefficient between each eigenstate of l 9 corresponding to Eqs. (C.17 and (C.18 as l 1 l l l 4 l 5 l 6 (l 1l l ; (l 4 l 5 l 6 ; l 9 m 9 (l 1 l 4 l 7 ; (l l 5 l 8 ; l 9 m 9. (C.19 l 7 l 8 l (l + 1(l 6 + 1(l 7 + 1(l This is expressed with the summation of five Wigner- j symbols: ( ( l4 l 5 l 6 l7 l 8 l 9 m 4 m 5 m 6 m 7 m 8 m 9 m 4 m 5 m 6 m 7 m 8 m 9 ( l4 l 7 l 1 m 4 m 7 m 1 ( l5 l 8 l ( l6 l 9 l m 5 m 8 m m 6 m 9 m ( l l1 l = l 1 l l l m 1 m m 4 l 5 l 6, l 7 l 8 l 9 (C.0 and that of three Wigner-6 j symbols: l 1 l l l 4 l 5 l 6 = { }{ }{ } ( 1 x l1 l (x l 7 l l 5 l 8 l l 6 l 9 ; (C.1 l l 7 l 8 l 8 l 9 x l 4 xl 6 xl 1 l 9 x hence, the triangle conditions are given by l 1 l l l 1 + l, l 4 l 5 l 6 l 4 + l 5, l 7 l 8 l 9 l 7 + l 8, l 1 l 4 l 7 l 1 + l 4, l l 5 l 8 l + l 5, l l 6 l 9 l + l 6. (C.

9 Appendix C: Wigner Symbols 161 The Wigner-9 j symbol obeys 7 symmetries: l 1 l l l l 4 l 5 l 6 = ( 1 9 l 1 l l i=1 l i l 5 l 4 l 6 = ( l l i=1 l i l 7 l 8 l 9 l 7 l 8 l 9 l 8 l 7 l 9 l 4 l 5 l 6 (odd permutation of rows or columns l l l 1 l 4 l 5 l 6 = l 5 l 6 l 4 = l 7 l 8 l 9 l 8 l 9 l 7 l 1 l l (even permutation of rows or columns l 1 l 4 l 7 l 9 l 6 l = l l 5 l 8 = l 8 l 5 l l l 6 l 9 l 7 l 4 l 1 (reflection of the symbols. (C. 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 ( I s 1s s (l1 + 1(l + 1(l + 1 l1 l l 1 l l l, (C.4 4π s 1 s s are expressed as Il 000 i=1 1 l l = (l i + 1 ( 1 l i i=1 4π ( i=1 l i! ( l 1 + l + l! (l 1 l + l! (l 1 + l l! ( ( ( l1 +l +l! l1 l +l! l1 +l l! ( i=1 l i + 1! (for l 1 + l + l = even = 0 (for l 1 + l + l = odd, (C.5 5 Il 01 1 (l 1 l l = 8π ( 1l +1 1(l + 1 (for l 1 = l, l = l 1/ 15 = 16π ( 1l 5 = 8π ( 1l l + 1/ (l 1/(l + / (for l 1 = l, l = l (l + l + / (for l 1 = l +, l =

10 16 Appendix C: Wigner Symbols = 8π ( 1l +1 l + 1 (for l 1 = l 1, l = 1 = 4π ( 1l +1 l + 1/ (for l 1 = l, l = 1 = 8π ( 1l +1 l (for l 1 = l + 1, l = 1. (C.6 The Wigner-9 j symbols are calculated as l 1 l l l 4 l 5 l 6 = (l ± = (l 1(l + (l { }{ } l1 l 4 1 l l 5 1 l ± l ± 1 l 5 l ± 1 l l 1 0(l (l + 1 (l 1(l + 1(l + 15(l (l + (l 1(l (l + 0(l + 1(l + (for l 6 = l ± { }{ } l1 l 4 1 l l 5 1 l l 1 l 5 l 1 l l 1 { }{ } l1 l 4 1 l l 5 1 l l l 5 l l l 1 { }{ } l1 l 4 1 l l 5 1 l l + 1 l 5 l + 1 l l 1 (for l 6 = l, where these Wigner-6 j symbols are analytically given by { } l1 l 1 = ( 1 l 1+l 4 +l 6 +1 l 1 +l 4 +l 6 + P l1 +l 4 l 6 +1 P l 4 l 5 l 6 l 4 + P l1 +1 P (for l = l 1 1, l 5 = l = ( 1 l 1+l 4 +l 6 +1 (l 1 + l 4 + l 6 + (l 1 + l 4 l l 4 + P ( l 1 + l 4 + l 6 + 1(l 1 l 4 + l 6 (for l = l 1, l 5 = l l 1 + P = ( 1 l 1+l 4 +l 6 +1 = ( 1 l 1+l 4 +l 6 +1 l 1 +l 4 +l 6 +1 P l1 l 4 +l 6 +1 P l 4 + P l1 + P (for l = l 1 + 1, l 5 = l [l 4 (l l 1 (l 1 1(l l 6 (l l 1 (l 1 + 1l 4 (l 1 + l 4 + l 6 + 1(l 1 + l 4 l 6 ( l 1 + l 4 + l 6 + 1(l 1 l 4 + l 6 l4 + P l1 +1 P (C.7

11 Appendix C: Wigner Symbols 16 = ( 1 l 1+l 4 +l 6 +1 (for l = l 1 1, l 5 = l 4 l 4(l l 1 (l 1 + 1(l l 6 (l l 1 (l 1 + 1l 4 l 4 + P l1 + P (for l = l 1, l 5 = l 4 = ( 1 l 1+l 4 +l 6 +1 [l 4 (l (l 1 + 1(l 1 + (l l 6 (l l 1 (l 1 + 1l 4 ( l 1 + l 4 + l 6 (l 1 l 4 + l (l 1 + l 4 + l 6 + (l 1 + l 4 l l4 + P l1 + P (for l = l 1 + 1, l 5 = l 4. (C.8 Using these analytical formulas, one can reduce the time cost involved with calculating the CMB bispectrum from PMFs.

12 Appendix D Polarization Vector and Tensor We summarize the relations and properties of a divergenceless polarization vector ɛ a (±1 and a transverse and traceless polarization tensor e (± [6, 11. The polarization vector with respect to a unit vector ˆn is expressed using two unit vectors ˆθ and ˆφ perpendicular to ˆn as ɛ a (±1 (ˆn = 1 [ ˆθ a (ˆn ± i ˆφ a (ˆn. (D.1 This satisfies the relations: ˆn a ɛ a (±1 (ˆn = 0, ɛ a (±1 (ˆn = ɛ a ( 1 (ˆn = ɛ a (±1 ( ˆn, (D. ɛ a (λ (ˆnɛ(λ a (ˆn = δ λ, λ (for λ, λ =±1. By defining a rotational matrix, which transforms a unit vector parallel to the z axis, namely ẑ,toˆn,as we specify ˆθ and ˆφ as cos θ n cos φ n sin φ n sin θ n cos φ n S(ˆn cos θ n sin φ n cos φ n sin θ n sin φ n, sin θ n 0 cos θ n (D. ˆθ(ˆn = S(ˆnˆx, ˆφ(ˆn = S(ˆnŷ, (D.4 where ˆx and ŷ are unit vectors parallel to x- and y-axes. By using Eq. (D.1, the polarization tensor is constructed as e (± (ˆn = ɛ (±1 a (ˆnɛ (±1 b (ˆn. (D.5 M. Shiraishi, Probing the Early Universe with the CMB Scalar, Vector and Tensor 165 Bispectrum, Springer Theses, DOI: / , Springer Japan 01

13 166 Appendix D: Polarization Vector and Tensor 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 ˆr a = m α m a π α m a Y 1m(ˆr, m(δ m,1 + δ m, 1 i (δ m,1 + δ m, 1. δm,0 (D.6 Here, note that the repeat of the index implies the summation. The scalar product of αa m is calculated as α m a αm a = 4π ( 1m δ m, m, α m a αm a = 4π δ m,m. (D.7 Through the substitution of Eq. (D.4 into Eq. (D.6, ˆθ is expanded as ˆθ a (ˆn = αa m Y 1m( ˆθ(ˆn = αa m D (1 mm (S(ˆnY 1m (ˆx m m m = s (δ s,1 + δ s, 1 αa m sy 1m (ˆn. m (D.8 Here, we use the properties of the Wigner D-matrix as described in Appendix B [, 5, 6, 1 Y lm (S(ˆnˆx = m [ 4π D ms (l (S(ˆn = l + 1 D (l mm (S(ˆnY lm (ˆx, 1/ ( 1 s sy lm (ˆn. (D.9 In the same manner, ˆφ is also calculated as ˆφ a (ˆn = i (δ s,1 + δ s, 1 m α m a sy 1m (ˆn ; (D.10 hence, the explicit form of Eq. (D.1 is calculated as ɛ a (±1 (ˆn = m α m a ±1Y 1m (ˆn. (D.11 Substituting this into Eq. (D.5 and using the relations of Appendix C and I11 ±1±1 =, the polarization tensor can also be expressed as π

14 Appendix D: Polarization Vector and Tensor 167 e (± (ˆn = π This obeys the relations: YM (ˆnαma a Mm a m b αm b b ( 1 1. (D.1 Mm a m b e aa (± (ˆn =ˆn a e (± (ˆn = 0, e (± (ˆn = e ( (ˆn = e (± ( ˆn, e (λ (ˆne(λ (ˆn = δ λ, λ (for λ, λ =±. (D.1 Using the projection operators as O (0 a eik x k 1 a e ik x = i ˆk a e ik x, ( O (0 eik x k a b + δ a,b e ik x = ( ˆk a ˆk b + δ a,b e ik x, O a (±1 e ik x iɛ a (±1 (ˆke ik x, (D.14 ( O (±1 e ik x k 1 a O (±1 b + b O a (±1 O (± e ik x e (± (ˆke ik x, e ik x = (ˆk a ɛ (±1 b (ˆk + ˆk b ɛ a (±1 (ˆk e ik x, the arbitrary scalar, vector and tensor are decomposed into the helicity states as η(k = η (0 (k, ω a (k = ω (0 (ko a (0 + λ=±1 χ (k = 1 χ iso(kδ a,b + χ (0 (ko (0 + λ=±1 χ (λ (ko (λ + (D.15 ω (λ (ko (λ a, (D.16 λ=± χ (λ (ko (λ. Then, using Eq. (D.9 and (D.1, we can find the inverse formulae as (D.17 ω (0 (k = O a (0 a(k, (D.18 ω (±1 (k = O a ( 1 (ˆkω a (k, (D.19 χ (0 (k = O(0 (ˆkχ (k, (D.0 χ (±1 (k = 1 O( 1 (ˆkχ (k, (D.1 χ (± (k = 1 O( (ˆkχ (k. (D.

15 168 Appendix D: Polarization Vector and Tensor From these, we can derive the relations of several projection operators as O (0 (ˆk = ˆk a ˆk b + 1 δ = YM π Mm a m b O (±1 (ˆk = ˆk a ɛ (±1 b (ˆk + ˆk b ɛ a (±1 (ˆk =± π O (± (ˆk = e (± (ˆk = π P (ˆk δ ˆk a ˆk b = Mm a m b 1Y M Mm a m b Y M L=0, I 01 1 L11 ma (ˆkαa αm b b ma (ˆkαa αm b b ma (ˆkαa αm b b Mm a m b Y LM O (0 (ˆkP bc (ˆk = 1 P ac(ˆk, O (±1 (ˆkP bc (ˆk = ˆk a ɛ c (±1 (ˆk, O (± (ˆkP bc (ˆk = e ac (± (ˆk, ˆk c = iη c ɛ a (+1 (ˆkɛ ( 1 b (ˆk, η c ˆk a ɛ (±1 b (ˆk = iɛ c (±1 (ˆk. ( 1 1, Mm a m b ( 1 1, Mm a m b ( 1 1, (D. Mm a m b ma (ˆkαa αm b b ( L 1 1, Mm a m b

16 Appendix E Calculation of f (a W and f (a WW Here, we calculate each product between the wave number vectors and the polarization tensors of f (a and f (a W WW mentioned in Chap. 8 [14. Using the relations discussed in Appendix D, the all terms of f (a are written as W e ( λ 1 ij e ( λ jk e ( λ ki = (8π / λ 1 YM 1 λ YM λ YM M,M,M 1 ( 7 10 MM M, e ( λ 1 ij e ( λ kl e ( λ ˆ kl k i kˆ j = (8π / 4π 15 ( 1L I λ 0 λ L 1 I λ 0 λ L 1 e ( λ 1 ij e ( λ e ( λ 1 ij e ( λ ki e ( λ jl ik e ( λ kl L,L =, λ 1 YM 1 λ YL M λ YL M M,M,M ( L L { L L } MM M, 1 1 kˆ l kˆ k = (8π / 4π ( 1L I λ 0 λ L 1 I λ 0 λ L 1 L,L =, λ 1 YM 1 λ YL M λ YL M M,M,M ( L L L L MM M 1 1, 1 1 kˆ l kˆ j = (8π / 4π ( 1L I λ 0 λ L 1 I λ 0 λ L 1 L,L =, λ 1 YM 1 λ YL M λ YL M M,M,M (E.1 M. Shiraishi, Probing the Early Universe with the CMB Scalar, Vector and Tensor 169 Bispectrum, Springer Theses, DOI: / , Springer Japan 01

17 170 Appendix E: Calculation of f (a and f (a W WW ( L L { }{ 1L L L } MM M In the calculation of f (a WW, we also need to consider the dependence of the tensor contractions on η ijk. Making use of the relation: η c α m a a αm b b αm c c = i the first two terms of f (a WW reduce to iη ijk e ( λ 1 kq e ( λ jm e( λ iq iη ijk e ( λ 1 kq e ( λ mi e ( λ mq kˆ m = (8π / For the other terms, by using the relation ( 4π / ( , (E. m a m b m c L =, π 5 ( 1L I λ 0 λ L 1 λ 1 YM 1 λ YM λ YL M M,M,M ( { } L L MM M, 1 1 kˆ j = (8π / π( 1 L I λ 0 λ L 1 (E. L =, λ 1 YM 1 λ YM λ YL M M,M,M ( L L MM M η c ˆk a e (λ bd (ˆk = λ ie(λ cd (ˆk, (E.4 we have

18 Appendix E: Calculation of f (a W and f (a WW 171 iη ijk e ( λ 1 pj e ( λ pm iη ijk e ( λ 1 pj e ( λ pm kˆ 1k kˆ l e ( λ ˆ il k m = λ 1 (8π/ L,L =, M,M,M 4π ( 1L I λ 0 λ L 1 I λ 0 λ L 1 λ1 YM 1 λ YL M λ YL M ( L L L L MM M 1 1, (E kˆ 1k kˆ l e ( λ ˆ im k l = λ 1 (8π/ L,L =, M,M,M π 7 15 ( 1L I λ 0 λ L 1 I λ 0 λ L 1 λ1 YM 1 λ YL M λ YL M ( L L { L L } MM M. 1

19 Appendix F Graviton Non-Gaussianity from the Weyl Cubic Terms Here, let us derive the bispectra of gravitons coming from the parity-even and parityodd Weyl cubic terms, namely, Eqs. (8.16 and (8.17 [14. For convenience, we decompose the interaction Hamiltonians of W and WW (8.15 into H int = 4 H (n int. (F.1 Depending on this, we also split the graviton non-gaussianity as γ (λn (k n = 4 γ (λn (k n (m int m=1 int. (F. In what follows, we shall show the computation of each fraction. F.1 W The bracket part of Eq. (8.1 in terms of H (1 is expanded as W [ 0 : H (1 (τ :, γ (λn (k W n,τ 0 = 0 :H (1 (τ : γ (λn (k W n,τ 0 [ 0 γ (λn (k n,τ : H (1 W (τ : 0 M. Shiraishi, Probing the Early Universe with the CMB Scalar, Vector and Tensor 17 Bispectrum, Springer Theses, DOI: / , Springer Japan 01

20 174 Appendix F: Graviton Non-Gaussianity from the Weyl Cubic Terms Here, we use 0 : Furthermore, since ( τ = (Hτ ( A 1 τ 4 (π δ k n e (λ 1 ij ( kˆ 1 e (λ jk ( kˆ e (λ ki ( kˆ [{ } 6 ( γ ds kn γ ds(k n,τ γds (k n,τ { } γ ds (k n,τ( γ ds k n γ ds (k n,τ = ( τ (Hτ ( A (π δ k n τ e ( λ 1 ij 1 e ( λ jk e ( λ ki [ iim ( γ ds kn γ ds(k n,τ γds (k n,τ. (F. a (λ n k n a (λ n k n : 0 = (π 9 δ(k 1 + k δ λ 1,λ δ(k 1 + k δ λ 1,λ δ(k + k δ λ,λ + 5 perms. (F.4 = 0 : a (λ n k n a (λ n k n : 0 e ( λ ij (ˆk = e (λ ij ( ˆk. γ ds k γ ds = Hτ k / e ikτ, M pl γds (k n,τ τ 0 i H Mpl (k 1 k k /, (F.5 the time integral at τ 0 is performed as

21 Appendix F: Graviton Non-Gaussianity from the Weyl Cubic Terms 175 [ τ ( τ Im dτ (Hτ A ( γ ds kn τ γ ds(k n,τ γds (k n,τ = 8H 5 [( τ Mpl k1 k k Im γds (k n,τ τ A dτ τ 5+A e ik t τ [ 0 Re dτ τ 5+A e ik t τ, (F.6 τ 0 8H 8 Mpl 6 τ A where k t k n. Thus, the graviton non-gaussianity in the late time limit arising from H (1 W is γ (λn (k n (1 = (π δ W [ Re ( ( H k n 8 M pl 0 dτ τ 5+A e ik t τ τ A 6 ( H e ( λ 1 ij 1 e ( λ jk e ( λ ki. (F.7 The bracket part in terms of H ( is given by W [ 0 : H ( (τ :, γ (λn (k W n,τ 0 = 0 :H ( (τ : γ (λn (k W n,τ 0 [ 0 γ (λn (k n,τ : H ( W (τ : 0 = ( τ (Hτ ( A k k (π δ k n τ kˆ i kˆ j e ( λ 1 ij 1 e ( λ kl e ( λ kl [ iim ( γ ds k1 γ ds(k 1,τ γ ds (k,τ γ ds (k,τ + 5 perms. γds (k n,τ (F.8

22 176 Appendix F: Graviton Non-Gaussianity from the Weyl Cubic Terms Using γ ds = i Hτ M pl ke ikτ, (F.9 we can reduce the time integral to [ τ ( τ Im dτ (Hτ A k k ( γ ds k1 τ γ ds(k 1,τ γ ds (k,τ γ ds (k,τ γds (k n,τ and obtain γ (λn (k n τ 0 H 8 ( M 6 pl [ Re = (π δ W [ Re τ A τ A 0 dτ τ 5+A e ik t τ, (F.10 ( ( H k n 8 M pl 0 dτ τ 5+A e ik t τ 6 ( H k 4 ˆ i e ( λ 1 1 kˆ j e ( λ e ( λ ij kl + 5 perms.. (F.11 The graviton non-gaussianities from H ( W and H (4 W are derived in the same manner as that from H ( W : 4 (m ( ( H 6 ( H γ (λn (k n = (π δ k n 8 M W pl [ 0 Re τ A dτ τ 5+A e ik t τ [ k 4 ˆ k e ( λ ki e ( λ 1 ij 1 e ( λ jl kˆ l k ˆ j e ( λ 1 1 e ( λ e ( λ kˆ l m= ji ik + 5 perms. (F.1 kl kl

23 Appendix F: Graviton Non-Gaussianity from the Weyl Cubic Terms 177 F. WW At first, we shall focus on the contribution of H (1 WW. The bracket part is computed as [ 0 : H (1 WW (τ :, γ (λn (k n,τ 0 = 0 :H (1 WW (τ : γ (λn (k n,τ 0 [ 0 γ (λn (k n,τ : H (1 WW (τ : 0 ( τ = d x (Hτ A ( d k n eik n x (π τ η ijk e (λ 1 kq ( k ˆ 1 e(λ jm ( k ˆ e(λ iq ( k ˆ (ik m [( ( γ ds k 1 γ ds (k 1,τ γ ds k { }{ : 0 m=1 ( γ ds k : 0 a (λ m k m γ ds γ ds (k n,τa (λ n k n λ n =± (k,τ γ ds (k,τ } 0 : ( 1 γ ds (k 1,τ γ ds k γ ds (k,τ γ ds (k,τ { }{ } γ ds (k n,τa (λ n k n 0 : m=1 a (λ m k m ( τ = (Hτ ( A ( ik (π δ k n τ η ijk e ( λ 1 kq 1 e ( λ jm e ( λ iq kˆ m [( ( iim γ ds k1 γ ds (k 1,τ γ ds k γ ds (k,τ { } γ ds (k,τ γds (k n,τ + 5 perms. (F.1

24 178 Appendix F: Graviton Non-Gaussianity from the Weyl Cubic Terms Via the time integral: Im [ τ ( τ dτ (Hτ A ( ( k γ ds k1 τ γ ds (k 1,τ γ ds k γ ds (k,τ { } γ ds (k,τ τ 0 4H 8 [ Im M 6 pl we have γ (λn (k n (1 γ ds (k n,τ τ A 0 = (π δ WW [ Im dτ τ 5+A e ik t τ, (F.14 ( ( H k n 8 M pl 0 dτ τ 5+A e ik t τ τ A 6 ( H ( iη ijk e ( λ 1 kq 1 e ( λ jm e ( λ iq kˆ m + 5 perms. (F.15 Like this, we can gain the second counterpart: ( ( ( H γ (λn (k n = (π δ k n 8 M WW pl [ 0 Im dτ τ 5+A e ik t τ τ A The bracket part with respect to H ( WW is 6 ( H i η ijk e ( λ 1 kq 1 e ( λ mi e ( λ mq kˆ j + 5 perms. (F.16

25 Appendix F: Graviton Non-Gaussianity from the Weyl Cubic Terms 179 [ 0 : H ( WW (τ :, γ (λn (k n,τ 0 = 0 :H ( WW (τ : γ (λn (k n,τ 0 [ 0 γ (λn (k n,τ : H ( WW (τ : 0 ( τ = d x (Hτ A 4 d k n eik n x (π The time integral is η ijk e (λ 1 pj ( ˆ [ { : 0 τ k 1 e(λ pm ( k ˆ e(λ il { : 0 m=1 γ ds (k n,τ a (λ n k n γ ds (k m,τa (λ m k m ( k ˆ (ik 1k (ik l (ik m }{ γds (k m,τa (λ m k m m=1 }{ λ n =± } 0 : } γ ds (k n,τ a (λ n k n 0 : ( τ = (Hτ ( A ( 4( i k 1 k k (π δ k n τ η ijk e ( λ 1 pj 1 e ( λ pm e ( λ il kˆ 1k kˆ l kˆ m [ iim γ ds (k n,τ γds (k n,τ + 5 perms. (F.17 [ τ ( τ Im dτ (Hτ A k n γ ds (k n,τ γds τ (k n,τ τ 0 H 8 [ 0 Mpl 6 Im τ A dτ τ 5+A e ik t τ, (F.18 so that the bispectrum of gravitons becomes

26 180 Appendix F: Graviton Non-Gaussianity from the Weyl Cubic Terms γ (λn (k n ( = (π δ WW [ Im ( 0 τ A 6 ( H M pl dτ τ 5+A e ik t τ ( H k n 8 i η ijk e ( λ 1 pj 1 e ( λ pm e ( λ il kˆ 1k kˆ l kˆ m + 5 perms. (F.19 Through the same procedure, the bispectrum from H (4 WW is estimated as γ (λn (k n (4 = (π δ WW [ Im ( 0 τ A 6 ( H M pl dτ τ 5+A e ik t τ ( H k n 8 ( iη ijk e ( λ 1 pj 1 e ( λ pm e ( λ im kˆ 1k kˆ l kˆ l + 5 perms. (F.0 References 1. K. Thorne, Rev. Mod. Phys. 5, 99 (1980. doi:10.110/revmodphys E.T. Newman, R. Penrose, J. Math. Phys. 7, 86 (1966. J.N. Goldberg, A.J. MacFarlane, E.T. Newman, F. Rohrlich, E.C.G. Sudarshan, J. Math. Phys. 8, 155 ( M. Zaldarriaga, U. Seljak, Phys. Rev. D55, 1997 (180. doi:10.110/physrevd T. Okamoto, W. Hu, Phys. Rev. D66, (00. doi:10.110/physrevd S. Weinberg, Cosmology (Oxford Univ. Pr, Oxford, UK, 008, p R. Gurau, Annales Henri Poincare 9, 141 (008. doi: /s H.A. Jahn, J. Hope, Phys. Rev. 9(, 18 (1954. doi:10.110/physrev The wolfram function site W. Hu, Phys. Rev. D64, (001. doi:10.110/physrevd M. Shiraishi, D. Nitta, S. Yokoyama, K. Ichiki, K. Takahashi, Prog. Theor. Phys. 15, 795 (011. doi:10.114/ptp M. Shiraishi, D. Nitta, S. Yokoyama, K. Ichiki, K. Takahashi, Phys. Rev. D8, 15 (011. doi:10.110/physrevd M. Shiraishi, S. Yokoyama, D. Nitta, K. Ichiki, K. Takahashi, Phys. Rev. D8, (010. doi:10.110/physrevd M. Shiraishi, D. Nitta, S. Yokoyama, Prog. Theor. Phys. 16, 97 (011. doi:10.114/ptp