Full-sky cubic bispectrum estimator (${\hat{f}}_{\mathrm{NL}}=(6F{)}^{-1}\sum b^1\mathcal{G}{\Theta }_1{\Theta }_2{\Theta }_3/(C_1{C}_2{C}_3)$)

For an isotropic, full-sky temperature map with a fixed bispectrum template $b=f_{\mathrm{NL}}{b}^1$, the inverse-variance-weighted cubic statistic has normalization $F=\frac{1}{6}\sum (b^1{)}^2{\mathcal{G}}^2/(C_1{C}_2{C}_3)$ over ordered triples. It is unbiased under the linear template relation and has Gaussian cosmic variance $1/F$. The factor $1/6$ counts the six cross-triple Wick contractions. A removed monopole cancels the nine internal-pairing terms; incomplete sky coverage or anisotropic noise generally require a linear correction.

In a full-sky cubic bispectrum estimator, pairwise Gaussian covariance within one triple gives a sum of Gaunt integrals with opposite $m$ indices. The spherical harmonic addition theorem turns their pair into the constant $(2\ell +1)/(4\pi )$; its integral against $Y_{LM}$ vanishes for $L>0$. A zero temperature monopole removes $L=0$. Thus only the six pairings connecting the two triples survive in the Gaussian variance; isotropic weights are essential for the cancellation.