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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.

A statistically isotropic Cosmic microwave background anisotropy has $⟨a_{{\ell }_1{m}_1}{a}_{{\ell }_2{m}_2}{a}_{{\ell }_3{m}_3}⟩={\mathcal{G}}_{m_1{m}_2{m}_3}^{{\ell }_1{\ell }_2{\ell }_3}{b}_{{\ell }_1{\ell }_2{\ell }_3}$, where $\mathcal{G}$ is the integral of the three spherical harmonics. The reduced CMB bispectrum removes this angular geometric factor and retains the primordial and transfer-function dependence. A Sachs-Wolfe projection of a constant bispectrum is an analytic large-angle example.