8 large l$_{i}$ ’s with appropriate weights save enormous amounts of time for the calculation. in Eq. (45). The summation is taken only when l$_{1}$ + l$_{2}$ + l$_{3}$ is an even number and l$_{1}$ , l$_{2}$ , l$_{3}$ satisfy the triangular inequality, l$_{2}$ − l$_{1}$ ≤ l$_{3}$ ≤ l$_{1}$ + l$_{2}$ because of the factor I$_{l}$$_{1}$$_{l}$$_{2}$$_{l}$$_{3}$ . For the kurtosis parameters, the number of addition is reduced by a factor of about eight, using the substitution Instead of calculating the exact summation, one can use the flat-sky approximation of Eq. (26), and evaluate the five- dimensional integral by, e.g., the Monte-Carlo algorithm. Once the reduced trispectrum T l$_{1}$l$_{2}$ l$_{3}$l$_{4}$ ( L ) is given, Eqs. (38), (55), (56) give the trispectrum T ( l$_{1}$ , l$_{2}$ , l$_{3}$ , l$_{4}$ ; l$_{12}$ , l$_{23}$ ) in the flat-sky approximation. There is no need for summing 6 j -symbols in this procedure. The integration ranges of Eq. (26) are 2 ≤ l$_{1}$ , l$_{2}$ , l$_{3}$ , l$_{4}$ < ∞ , 0 ≤ θ$_{12}$, θ$_{23}$ ≤ 2 π when the monopole and dipole components are subtracted from the map. This ap- proximation is valid when most contributions to the kurtosis parameters come from the flat-sky regime ( l ≫ 1). In realis- tic spectra, this condition is satisfied because multipolesnear the acoustic peak at l ∼ 200 dominantly contribute. Strictly speaking, the flat-sky integration is valid when all l ’s are large. Therefore, for a more precise approximation, one can use the all-sky summation of Eq. (46) only when, e.g., l$_{1}$ ≤ 20 [note that l$_{1}$ ≤ l$_{2}$ , l$_{3}$ , l$_{4}$ in Eq. (A2)], and otherwise adopt the flat-sky integration of Eq. (26) over the range l$_{1}$ , l$_{2}$ , l$_{3}$ , l$_{4}$ ≥ 21. in Eq. (46). The summation is taken only when l$_{1}$ + l$_{2}$ + l$_{3}$ + l$_{4}$ is an even number and l$_{4}$ ≤ l$_{1}$ + l$_{2}$ + l$_{3}$ is satisfied because of the factor I$_{l}$$_{1}$$_{l}$$_{2}$$_{L}$ I$_{l}$$_{3}$$_{l}$$_{4}$$_{L}$ . The index L in Eq. (46) runs over every other integers in the range [max( | l$_{1}$ − l$_{2}$ | , | l$_{3}$ − l$_{4}$ | ) , min( l$_{1}$ + l$_{2}$ , l$_{3}$ + l$_{4}$ )] with the same parity as that of l$_{1}$ + l$_{2}$ . Secondly, since the trispectrum is not a strongly oscillating function in usual cases, one can sparsely sample the multi- poles in the sum for large values of l$_{1}$ , . . . , l$_{4}$ . Acoustic oscil- lations in the CMB are su ffi ciently mild in this respect. There is a caveat that the summation over L ′ in Eq. (37) should not be sparsely sampled, since the 6 j -symbol is a strongly oscillating function with respect to its arguments. Sparse samplings for Unless sub-percent level accuracies are required, experi- ences show that the simplest flat-sky integrals over all rang es of multipole with appropriate choice of integration limits suf- fice for evaluations of skewness and kurtosis parameters in realistic spectra. ∑ l$_{1}$ , l$_{2}$ , l$_{3}$ , l$_{4}$ = ∑ l$_{1}$ = l$_{2}$ l$_{3}$ = l$_{4}$ l$_{1}$ = l$_{3}$ + 2 ∑ l$_{1}$ = l$_{2}$ l$_{3}$ = l$_{4}$ l$_{1}$ < l$_{3}$ + 2 ∑ l$_{1}$ = l$_{2}$ l$_{3}$ < l$_{4}$ l$_{1}$ = l$_{3}$ + 2 ∑ l$_{1}$ < l$_{2}$ l$_{3}$ = l$_{4}$ l$_{1}$ = l$_{3}$ + 4 ∑ l$_{1}$ = l$_{2}$ l$_{3}$ < l$_{4}$ l$_{1}$ < l$_{3}$ + 4 ∑ l$_{1}$ < l$_{2}$ l$_{3}$ = l$_{4}$ l$_{1}$ < l$_{3}$ + 4 ∑ l$_{1}$ < l$_{2}$ l$_{3}$ < l$_{4}$ l$_{1}$ = l$_{3}$ + 8 ∑ l$_{1}$ < l$_{2}$ l$_{3}$ < l$_{4}$ l$_{1}$ < l$_{3}$ , (A2) F. K. Hansen, M. Liguori and S. Matarrese, Mon. Not. R. As- tron. Soc. 393 , 615 (2009). [21] M. Sasaki, J. V¨aliviita and D. Wands, Phys. Rev. D 74 , 103003 (2006). [22] Q.-G. Huang, J. Cosmol. Astropart. Phys. 6 , 35 (2009). [23] C. T. Byrnes and G. Tasinato, J. Cosmol. Astropart. Phys. 8 , 16 (2009). [24] J.-L. Lehners, S. Renaux-Petel, Phys. Rev. D 80 , 063503 (2009). [25] K. R. Mecke, T. Buchert and H. Wagner, Astron. Astrophys. 288 , 697 (1994). [26] J. Schmalzing, and T. Buchert, Astrophys. J. Letters 482 , L1 (1997). [27] J. Schmalzing, and K. M. Gorski, Mon. Not. R. Astron. Soc. 297 , 355 (1998). [28] D. Novikov, J. Schmalzing V. F. Mukhanov, Astron. Astrophys. 364 , 17 (2000). [29] J. H. P. Wu, A. Balbi, J. Borrill, P. G. Ferreira, S. Hanany, A. H. Ja ff e, A. T. Lee, B. Rabii, P. L. Richards, G. F. Smoot, R. Stompor and C. D. Winant, Phys. Rev. Lett. 87 , 251303 (2001). [30] G. Polenta, et al., Astrophys. J. Letters 572 , L27 (2002). [31] A. Curto, J. Aumont, J. F. Mac´ıas-P´erez, E. Mart´ınez -Gonz´alez, R. B. Barreiro, D. Santos, F. X. D´esert and M. Tristram, Astron. Astrophys. 474 , 23 (2007). [32] G. de Troia, et al., Astrophys. J. Letters 670 , L73 (2007). [33] A. Curto, J. F. Mac´ıas-P´erez, E. Mart´ınez-Gonz´ale z, R. B. Bar- reiro, D. Santos, F. K. Hansen, M. Liguori and S. Matarrese, Astron. Astrophys. 486 , 383 (2008). [34] T. Matsubara, Astrophys. J. 584 , 1 (2003) [20] A. Curto, E. Mart´ınez-Gonz´alez, P. Mukherjee, R. B. B arreiro, [19] K. M. Smith, L. Senatore and M. Zaldarriaga, J. Cosmol. As- tropart. Phys. 9 , 6 (2009). [18] E. Komatsu, et al., Astrophys. J. Suppl. Ser. 180 , 330 (2009). [17] P. Creminelli, L. Senatore, M. Zaldarriaga and M. Tegmark, J. Cosmol. Astropart. Phys. 3 , 5 (2007). [16] D. N. Spergel, et al., Astrophys. J. Suppl. Ser. 170 , 377 (2007). [15] E. Komatsu, et al., Astrophys. J. Suppl. Ser. 148 , 119 (2003). [14] N. Bartolo, E. Komatsu, S. Matarrese and A. Riotto, Phys. Rep. 402 , 103 (2004). [13] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, Phys. Rev. D 64 , 123522 (2001). [12] M. Alishahiha, E. Silverstein and D. Tong, Phys. Rev. D 70 , 123505 (2004). [11] J. Garriga and V. F. Mukhanov, Phys. Lett. B 458 , 219 (1999). [10] C. Armend´ariz-Pic´on, T. Damour and V. Mukhanov, Phys . Lett. B 458 , 209 (1999). [6] A. Gangui, F. Lucchin, S. Matarrese, and S. Mollerach, Astro- phys. J. 430 , 447 (1994). [7] D. H. Lyth and D. Wands, Phys. Lett. B 524 , 5 (2002). [9] D. H. Lyth, C. Ungarelli and D. Wands, Phys. Rev. D 67 , 023503 (2003). [8] A. Linde and V. Mukhanov, Phys. Rev. D 56 , R535 (1997). [5] T. J. Allen, B. Grinstein and M. B. Wise, Phys. Lett. B 197 , 66 (1987). [4] A. H. Guth and S.-Y. Pi, Phys. Rev. Lett. 49 , 1110 (1982). [3] S. W. Hawking, Phys. Lett. B 115 , 295 (1982). [2] A. A. Starobinsky, Phys. Lett. B 117 , 175 (1982). [1] V. F. Mukhanov and G. V. Chibisov, Soviet Journal of Experi- mental and Theoretical Physics Letters 33 , 532 (1981).