Reduced CMB bispectrum ($b_{{\ell }_1{\ell }_2{\ell }_3}$)

For a statistically isotropic Cosmic microwave background anisotropy, rotational invariance organizes the angular three-point function using a Wigner 3j symbol:In the usual parity-even temperature convention, one instead factors out the Gaunt integralThe coefficient $b$ is the reduced CMB bispectrum: it removes the angular geometric factor and retains primordial and transfer-function information. A Sachs-Wolfe projection of a constant bispectrum is an analytic large-angle example.

Parity matters here. Since $Y_{\ell m}(-\mathbf{n})=(-1{)}^{\ell }{Y}_{\ell m}(\mathbf{n})$, the Gaunt integral vanishes when ${\ell }_1+{\ell }_2+{\ell }_3$ is odd. Statistical isotropy alone does not impose parity invariance. A parity-odd angular three-point function therefore needs its own geometric convention and cannot generally be represented by this parity-even Gaunt factor times a finite $b$.

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.

Linear cosmological transfer maps the primordial bispectrum to the reduced CMB bispectrum. A Fourier representation of the momentum delta function and three Rayleigh plane-wave expansions separate the radial transfer integrals from a Gaunt integral. The six angular factors and three Fourier measures give $(4\pi {)}^6/(2\pi {)}^9=(2/\pi {)}^3$. The observer-position phase cancels by momentum conservation; the remaining radial position is an auxiliary integration variable.

For a constant primordial bispectrum and large-angle transfer function ${\Delta }_{\ell }(k)=j_{\ell }(kR)/5$, the spherical Bessel product integral reduces the radial projection to ${\int }_0^1{r}^{L+2}dr+{\int }_1^{\infty }{r}^{-L-1}dr$. For $L>0$ this equals $1/(L+3)+1/L$, giving the displayed reduced CMB bispectrum. All powers of the distance $R$ cancel. Under common large-multipole scaling it behaves as ${\ell }^{-4}$ at fixed shape; this is angular scale invariance, not a constant angular bispectrum. The all-monopole case $L=0$ has a logarithmically divergent radial tail and is excluded.