content_image/404250.jpg content_image/404249.jpg content_image/404248.jpg content_image/404247.jpg content_image/404246.jpg content_image/404253.jpg content_image/404255.jpg content_image/404256.jpg content_image/404257.jpg content_image/404258.jpg content_image/404245.jpg content_image/404259.jpg content_image/404260.jpg content_image/404254.jpg content_image/404261.jpg content_image/404262.jpg content_image/404263.jpg
where 

<img src='content_image/404250.jpg'>

<img src='content_image/404249.jpg'>

<img src='content_image/404248.jpg'>

<img src='content_image/404247.jpg'>

<img src='content_image/404246.jpg'>

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 

<img src='content_image/404253.jpg'>

for θ ≪ 1, l ≫ 1, the correspondences between all-sky and flat-sky spectra are derived as 

<img src='content_image/404255.jpg'>

<img src='content_image/404256.jpg'>

<img src='content_image/404257.jpg'>

and 

<img src='content_image/404258.jpg'>

4 12 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 ′ . 

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] 

<img src='content_image/404245.jpg'>

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 

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. 

The bispectrum and trispectrum of CMB in the local model of Eq. (57) are derived in literature [43, 49]: 

<img src='content_image/404259.jpg'>

<img src='content_image/404260.jpg'>

where 

<img src='content_image/404254.jpg'>

<img src='content_image/404261.jpg'>

<img src='content_image/404262.jpg'>

<img src='content_image/404263.jpg'>

