5 For numerical evaluations of the kurtosis and its derivatives by the summation of Eq. (46), the number of terms to add is of order O ( l 6 $_{max}$), where l$_{max}$ is the maximum multipole required for a given smoothing scale, e.g., l$_{max}$ ∼ several × θ − 1 s . The computational cost becomes progressively high for large l$_{max}$ , if the summation is naively performed. E ffi cient evaluations are necessary when the smoothing angle θ$_{s}$ is small. In Ap- pendix A, numerical schemes for the e ffi cient evaluations are summarized. IV. A SIMPLE EXAMPLE: THE LOCAL MODEL OF NON-GAUSSIANITY The analytic MFs are evaluated once the power spectrum, bispectrum and trispectrum are given. These spectra depend on models of primordial density fluctuations. As a simple ex- ample, we consider below the local model of non-Gaussianity, although our formulas are not restricted to this particular model. In the local model, the primordial curvature perturbations during the matter era is assumed to take the form [42, 43, 48– 50] The above forms of skewness parameters [Eq. (47), (48)] were already appeared in [35]. Resemblances of the above results to those of the flat space are obvious if the integrands of Eqs. (30)–(32) are sym- metrized with respect to l$_{1}$ , l$_{2}$ , l$_{3}$ , and l$_{4}$ [conversely, one can desymmetrize the Eqs. (47)–(51) to have the similar form wit h Eqs. (30)–(32)]. Noting the all-sky and flat-sky correspon- dence [44, 46, 47], it is a straightforward exercise to show that the above all-sky equations reduce to those of flat-sky i n the large- l limit. Following [44, 46, 47], but applying an im- proved approximation in configuration space, where φ is an auxiliary random Gaus- sian field. The comoving curvature perturbation ζ is given by ζ = 3 Φ / 5. The CMB fluctuations generated by the curvature perturbations have the harmonic coe ffi cients Φ ( $^{x}$) = φ ( $^{x}$) + f$_{NL}$ ( φ $^{2}$( $^{x}$) − 〈 φ $^{2}$〉 ) + g$_{NL}$φ $^{3}$( $^{x}$) , (57) ˜ K l$_{1}$l$_{2}$ III l$_{3}$$_{l$_{4}$}$( L ) = ( { l$_{1}$ } + { l$_{2}$ } − { L } ) ( { l$_{3}$ } + { l$_{4}$ } − { L } ) 32 q 4 . (51) ˜ K l$_{1}$l$_{2}$ II l$_{3}$$_{l$_{4}$}$( L ) = { L } 2 − ( { l$_{1}$ } + { l$_{2}$ } ) ( { l$_{3}$ } + { l$_{4}$ } ) 16 q 4 , (50) ˜ K l$_{1}$l$_{2}$ l$_{3}$$_{l$_{4}$}$( L ) = 1 , ˜ K l$_{1}$l$_{2}$ I l$_{3}$$_{l$_{4}$}$( L ) = − { l$_{1}$ } + { l$_{2}$ } + { l$_{3}$ } + { l$_{4}$ } 8 q 2 , (49) ˜ S$_{II}$$_{l}$$_{1}$$_{l}$$_{2}$$_{l}$$_{3}$ = { l$_{1}$ } 2 + { l$_{2}$ } $^{2}$+ { l$_{3}$ } 2 − 2 { l$_{1}$ }{ l$_{2}$ } − 2 { l$_{2}$ }{ l$_{3}$ } − 2 { l$_{3}$ }{ l$_{1}$ } 12 q 4 , (48) ˜ S$_{l}$$_{1}$$_{l}$$_{2}$$_{l}$$_{3}$ = 1 , ˜ S$_{I}$$_{l}$$_{1}$$_{l}$$_{2}$$_{l}$$_{3}$ = − { l$_{1}$ } + { l$_{2}$ } + { l$_{3}$ } 6 q 2 , (47) where for θ ≪ 1, l ≫ 1, the correspondences between all-sky and flat-sky spectra are derived as Y m l ( θ, φ ) ≈ ( − 1) m √ 2 l + 1 4 π J$_{m}$ [( l + 1 2 ) θ ] e im $^{φ}$, (52) a$_{lm}$ = 4 π ( − i ) l ∫ d $^{3}$k (2 π ) 3 ˜ Φ ( $^{k}$) g (T) l ( k ) Y m ∗ l ( ˆ $^{k}$) , (58) C$_{l}$ ≈ C ( l + 1 2 ) , (53) B$_{l}$$_{1}$$_{l}$$_{2}$$_{l}$$_{3}$ ≈ I$_{l}$$_{1}$$_{l}$$_{2}$$_{l}$$_{3}$ B ( l$_{1}$ + 1 2 , l$_{2}$ + 1 2 , l$_{3}$ + 1 2 ) , (54) P l$_{1}$l$_{2}$ l$_{3}$$_{l$_{4}$}$( L ) ≈ I$_{l}$$_{1}$$_{l}$$_{2}$$_{L}$I$_{l}$$_{3}$$_{l}$$_{4}$$_{L}$ × P ( l$_{1}$ + 1 2 , l$_{2}$ + 1 2 ; l$_{3}$ + 1 2 , l$_{4}$ + 1 2 ; L + 1 2 ) , (55) T ( l$_{1}$ , l$_{2}$ , l$_{3}$ , l$_{4}$ ; l$_{12}$ , l$_{23}$ ) = P ( l$_{1}$ , l$_{2}$ ; l$_{3}$ , l$_{4}$ ; l$_{12}$ ) + P ( l$_{1}$ , l$_{3}$ ; l$_{2}$ , l$_{4}$ ; l$_{13}$ ) + P ( l$_{1}$ , l$_{4}$ ; l$_{2}$ , l$_{3}$ ; l$_{23}$ ) , (56) B$_{l}$$_{1}$$_{l}$$_{2}$$_{l}$$_{3}$ = 2 f$_{NL}$I$_{l}$$_{1}$$_{l}$$_{2}$$_{l}$$_{3}$ [∫ r $^{2}$dr α$_{l}$$_{1}$ ( r ) β$_{l}$$_{2}$ ( r ) β$_{l}$$_{3}$ ( r ) + cyc . ] , (59) T l$_{1}$l$_{2}$ l$_{3}$l$_{4}$ ( L ) = I$_{l}$$_{1}$$_{l}$$_{2}$$_{L}$I$_{l}$$_{3}$$_{l}$$_{4}$$_{L}$ × { 4 f 2 NL ∫ r 2 $_{1}$dr$_{1}$ r 2 $_{2}$dr$_{2}$ F$_{L}$ ( r$_{1}$ , r$_{2}$ ) α$_{l}$$_{1}$ ( r$_{1}$ ) β$_{l}$$_{2}$ ( r$_{1}$ ) α$_{l}$$_{3}$ ( r$_{2}$ ) β$_{l}$$_{4}$ ( r$_{2}$ ) + g$_{NL}$ ∫ r $^{2}$dr β$_{l}$$_{2}$ ( r ) β$_{l}$$_{4}$ ( r ) [ α$_{l}$$_{1}$ ( r ) β$_{l}$$_{3}$ ( r ) + β$_{l}$$_{1}$ ( r ) α$_{l}$$_{3}$ ( r ) ] } (60) F$_{L}$ ( r$_{1}$ , r$_{2}$ ) ≡ 4 π ∫ k $^{2}$dk 2 π 2 P$_{φ}$ ( k ) j$_{L}$ ( kr$_{1}$ ) j$_{L}$ ( kr$_{2}$ ) , (61) α$_{l}$ ( r ) ≡ 4 π ∫ k $^{2}$dk 2 π 2 g (T) l ( k ) j$_{l}$ ( kr ) , (62) β$_{l}$ ( r ) ≡ 4 π ∫ k $^{2}$dk 2 π 2 P$_{φ}$ ( k ) g (T) l ( k ) j$_{l}$ ( kr ) , (63) Since the all-sky multipole ℓ and the flat-sky wavenumber | l | are related by | l | = ℓ + 1 / 2, contributions of the multipole ℓ to the all-sky summation is approximately represented by the flat-sky integration over the range ℓ − 1 / 2 ≤ | l | − 1 / 2 < ℓ + 1 / 2, i.e., ℓ ≤ | l | < ℓ + 1. Thus, all-sky summations over ℓ = 2 , 3 , . . . correspond to the flat-sky integrations with the limit | l | ≥ 2, as noted above. We confirm that the flat-sky approximations of Eqs. (24)–(26) with the above correspondences numeri- cally reproduce the values calculated from all-sky formula of Eqs. (44)–(46) within several percent for θ$_{s}$ < 100 ′ . where and The bispectrum and trispectrum of CMB in the local model of Eq. (57) are derived in literature [43, 49]: where ˜ Φ ( $^{k}$) is the Fourier transform of the primordial cur- vature perturbation Φ ( $^{x}$), and g (T) l ( k ) is the radiation transfer function.