6 calculation: and P$_{φ}$ ( k ) ∝ k n$_{s}$ − 4 is the primordial power spectrum of φ . In some extended models, the factor 4 f 2 NL in the trispectrum is replaced by 4 f 2 NL → 25 τ$_{NL}$/ 9, and τ$_{NL}$ is considered as an independent parameter [51]. The variance parameters σ and σ$_{1}$ are also a ff ected by non- Gaussianity. Therefore we need to evaluate the corrections to the power spectrum. The non-Gaussian corrections to the parameters σ , σ$_{1}$ mostly a ff ect the normalization of MFs. In practice, the normalization su ff ers contamination from obser- vational e ff ects such as pixelization and / or boundary e ff ects, and therefore is not usually used for extracting cosmological information. Nevertheless, we include the e ff ect for theoret- ical consistency here. By substituting Eq. (58) into Eq. (34) for a local model of Eq. (57), performing angular integrations, we obtain To illustrate how well our formulas work, we compare the analytic MFs with numerical MFs from realizations of the non-Gaussian CMB map. Since the purpose of this com- parison is to show the performance of analytic formulas, we simply consider the non-Gaussian map in the Sachs-Wolfe limit. Implementations with a full radiation transfer func- tion will be used in future applications to real data. In the Sachs-Wolfe limit, the temperature fluctuations are given b y Δ T$_{SW}$ ( θ, φ ) / T = − Φ ( $^{r}$∗ , θ, φ ) / 3, and thus generated by the nonlinear mapping of Gaussian fluctuations Δ T$_{G}$ / T at each position in the sky: V. COMPARISONS WITH NUMERICAL MINKOWSKI FUNCTIONALS C$_{l}$ =    1 + 27 g$_{NL}$ 2 π ∑ L (2 L + 1) C SW L    C SW l + 18 f 2 NL 2 l + 1 ∑ l$_{1}$ , l$_{2}$ I 2 l$_{1}$l$_{2}$$_{l}$C SW l$_{1}$ C SW l$_{2}$ . (71) Δ T$_{SW}$ T = Δ T$_{G}$ T − 3 f$_{NL}$ ( Δ T$_{G}$ T ) 2 + 9 g$_{NL}$ ( Δ T$_{G}$ T ) 3 − (monopole + dipole) , (72) C$_{l}$ = [ 1 + 6 g$_{NL}$ ∫ r $^{2}$dr ξ$_{φ}$ ( r ) ] ˜ C$_{l}$ + 2 f 2 NL ∫ r $^{2}$dr γ$_{l}$ ( r ) [ ξ$_{φ}$ ( r ) ] 2 , (64) ˜ C$_{l}$ = 4 π ∫ k $^{2}$dk 2 π 2 P$_{φ}$ ( k ) [ g (T) l ( k ) ] 2 , (65) ξ$_{φ}$ ( r ) = ∫ k $^{2}$dk 2 π 2 P$_{φ}$ ( k ) j$_{0}$ ( kr ) , (66) γ$_{l}$ ( r ) = (4 π ) 2 ∫ k $^{2}$dk 2 π 2 [ g (T) l ( k ) ] 2 j$_{0}$ ( kr ) . (67) where where monopole and dipole components are subtracted from the resulting map. We use the HEALPix package [53] to generate the non-Gaussian map, apply the smoothing win- dow function, and calculate the first- and second-derivativ es of the temperature field. The MFs are evaluated from the field derivatives using the numerical algorithm given in [27, 35]. We assume a simple Sachs-Wolfe power spectrum of l ( l + 1) C SW l / 2 π = 10 − 10 for l ≤ 128 and C SW l = 0, otherwise. Without this cut-o ff , the resulting power spectrum would log- arithmically diverge in Eq. (71) (the actual power spectrum has natural damping in high- l regime). We adopt the Gaussian filter W$_{l}$ = e − l ( l + 1) θ 2 $_{s}$/2 with the smoothing radius of θ$_{s}$ = 100 ′ , and non-Gaussianity parameters f$_{NL}$ = 10 2 and g$_{NL}$ = 10$^{6}$. We generate 100 , 000 realizations of the non-Gaussian map, numerically calculate MFs in each realization, and finally a v- erage over the realizations. The quantities ˜ C$_{l}$ , ξ$_{φ}$ ( r ) are the angular power spectrum from the Gaussian component, and the spatial correlation function of φ , respectively. Substituting Eqs. (59), (60), (64) into Eqs. (44)–(46) with Eqs. (37), (38), all the parameters nec- essary to calculate the analytic MFs are evaluated for a model of local non-Gaussianity with a full transfer function g (T) ( k ). l For the Sachs-Wolfe e ff ect [52], which is valid only for small multipole moments ( l ≪ 100), the radiation transfer function is simply given by g (T) l ( k ) = j$_{l}$ ( $^{kr}$∗ ) / 3, where $^{r}$∗ is the comoving distance to the last scattering surface. In this limit, the reduced bispectrum and trispectrum have the forms [6, 43, 49] In Fig. 1, the analytic and numerical MFs are compared. MFs as functions of the threshold ν are plotted in upper pan- els. As noted in [35], precise values of overall normalizations are a ff ected by numerical artifacts such as pixelization, bound- ary e ff ects etc. even in the Gaussian random field at the sub- percent level. However, deviations from the Gaussian shape of MFs as functions of the threshold ν are in very good agree- ment between numerical and analytic MFs. In lower panels, di ff erences between non-Gaussian MFs and Gaussian MFs, divided by the maximum amplitudes, are compared. For nu- merical data, MFs from Gaussian realizations are calculated and subtracted from those of non-Gaussian realizations. We find the agreements are extremely well. Note that there is no fitting parameter at all in the comparisons. is the power spectrum of the Sachs-Wolfe e ff ect from the Gaussian component. In the same limit, non-Gaussian cor- rections to the power spectrum are also obtained after some C SW l = 2 9 π ∫ k $^{2}$dk P$_{φ}$( k ) [ j$_{l}$ ( $^{kr}$∗ ) ] 2 , (70) T l$_{1}$l$_{2}$ l$_{3}$l$_{4}$ ( L ) = 9 C SW l$_{2}$ C SW l$_{4}$ × [ 4 f 2 $_{NL}$C SW L + g$_{NL}$ ( C SW l$_{1}$ + C SW l$_{3}$ )] I$_{l}$$_{1}$$_{l}$$_{2}$$_{L}$I$_{l}$$_{3}$$_{l}$$_{4}$$_{L}$ , (69) B$_{l}$$_{1}$$_{l}$$_{2}$$_{l}$$_{3}$ = − 6 f$_{NL}$ ( C SW l$_{1}$ C SW l$_{2}$ + C SW l$_{2}$ C SW l$_{3}$ + C SW l$_{3}$ C SW l$_{1}$ ) I$_{l}$$_{1}$$_{l}$$_{2}$$_{l}$$_{3}$ , (68) where