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Let $R=\Delta \tau >0$, $L={\ell }_1+{\ell }_2+{\ell }_3$ and $D={\prod }_{r=1}^3(2{\ell }_r+1)$. Use exactly the given constant primordial bispectrum normalization, without an additional local-template convention factor. Combining it with the large-angle Sachs-Wolfe effect cosmological transfer function gives the reduced CMB bispectrumwhere $I_{\ell }(x,R)={\int }_0^{\infty }{j}_{\ell }(kR)j_{\ell }(kx)dk$. Each $1/5$ transfer factor has been retained; their product is $1/125$.

Apply the supplied spherical Bessel product integral after rescaling $q=kR$:The two branches agree at $x=R$. In the radial integral put $r=x/R$. The factor $R^3$ from $x^{2}dx$ cancels the $R^{-3}$ from the three kernels. The remaining integral isprovided $L>0$. This evaluation uses the momentum integrals at fixed radial parameter in the projection, then the radial integral; it does not require an unjustified global exchange of all oscillatory integrals.

The factors $(2/\pi {)}^3$ and ${\pi }^3/8$ cancel exactly. Hence the Sachs-Wolfe projection of a constant bispectrum isThe all-monopole case $L=0$ has a logarithmically divergent outer radial integral and is not covered by the printed finite formula. Observable CMB analyses remove the monopole and dipole, normally using ${\ell }_i\ge 2$, so this issue is absent there. Angular triangle and parity selection are carried by the triple-spherical harmonic geometric factor multiplying the reduced bispectrum.

There is no dependence on the last-scattering distance $R$, and no extra physical scale appears when ${\Delta }_{\zeta }^2$ is held constant. At fixed triangle shape and in the range $1\ll {\ell }_i\ll 200$, common rescaling of all multipoles givesThis is angular scale invariance in the usual weighted sense; a constant primordial shape does not make the unweighted reduced CMB bispectrum independent of angular scale. The exact finite-multipole expression retains the $+1$ and $+3$ terms shown above.

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 $b$.