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The integral of three spherical harmonics on the sphere. In the conventional complex basis it is real and equals a product of two Wigner 3j symbols and $\sqrt{(2{\ell }_1+1)(2{\ell }_2+1)(2{\ell }_3+1)/(4\pi )}$. It vanishes unless the multipoles satisfy a triangle condition, ${\ell }_1+{\ell }_2+{\ell }_3$ is even, and $m_1+m_2+m_3=0$. It carries the angular geometry of the reduced CMB bispectrum.

Insert the primordial bispectrum into the product of the three linear transfer integrals. Write $L={\ell }_1+{\ell }_2+{\ell }_3$ and abbreviate $g_{{\ell }_i}({\eta }_0,k_i)$ by $g_i$. The observer-position phase is one because the momentum delta function imposes $\sum {\mathbf{k}}_i=0$. Represent that delta function byThe Rayleigh plane-wave expansion and angular orthogonality give, for each momentum,The three $i^{\ell }$ factors cancel the $(-i{)}^L$ in the temperature multipoles. Angular integration over $\hat{x}$ leaves the complex conjugate Gaunt integral. In the conventional complex spherical harmonics, this integral is real, and it vanishes unless the angular momentum triangle, even-parity and $m_1+m_2+m_3=0$ selection rules hold. Therefore its conjugate equals itself.

The radial measure is $x^{2}dx$, and the momentum radial measures are $k_i^{2}d{k}_i$. The combined numerical prefactor is $(4\pi {)}^6/(2\pi {)}^9=8/{\pi }^3=64/(2\pi {)}^3$. Hence the reduced CMB bispectrum isand the angular three-point function factorizes asThis primordial-to-angular bispectrum projection separates dynamics and radial transfer from purely angular geometry. The spatial integration variable is auxiliary, not the observer position. Linear transfer is justified at leading order in the primordial signal; it does not require a large amplitude mathematically, although a signal must exceed measurement uncertainty to be detectable.

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.

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.