 Reduced CMB bispectrum

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 $b$.