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Restore the Fourier transform integral and the last-scattering distance:Insert the Rayleigh plane-wave expansion and use orthogonality of spherical harmonics with the complex conjugate on the projection harmonic. The angular integral selects the matching indices, givingThe complex conjugation is present in the original PDF. If $A_{\mathbf{k}}=A{k}^{n/2}$ denotes a mode coefficient with its random amplitude and phase included in $A$, substitution of ${\Phi }_{\mathbf{k}}=-6A_{\mathbf{k}}/k^2$ gives the equivalent factor $-8\pi i^{\ell }{A}_{\mathbf{k}}/k^2$ inside the Fourier integral. The monopole and local observer dipole removed in part (b) do not change this formula for $\ell \ge 2$.

Use the same negative-exponent Fourier transform as in the preceding part and let $f=d\log D_{+}/d\log a$. In linear theory, $\dot{\delta }=Hf\delta$, and the linearized cosmological continuity equation is $\dot{\delta }+a^{-1}{\nabla }_x\cdot \mathbf{v}=0$. Its irrotational solution isHere $\mathbf{r}$ labels the comoving spatial position for the Fourier expansion. The quantity being differentiated is also $-\Phi /(4\pi G{a}^2\bar{\rho })$, by the Poisson equation; thus this explicitly obtains the velocity from the Fourier components of the peculiar gravitational potential.

Conjugating the Rayleigh plane-wave expansion, and using the symmetric addition formula for spherical harmonics, givesFor the radial component $U=\mathbf{v}\cdot \hat{r}$, differentiate at fixed angular position. Only the Spherical Bessel function depends on $r$, and $d{j}_{\ell }(kr)/dr=k{j}_{\ell }^{\prime }(kr)$. Therefore the radial peculiar velocity in spherical harmonics isThe coefficient is $4\pi /(2\pi {)}^3=1/(2{\pi }^2)$, and $(-i{)}^{\ell }=(i^{\ell }{)}^{\ast }$, so the phase and conjugation agree with the requested expression. At the present epoch, $a=1$ and $H=H_0$. Adopting the traditional approximation $f\simeq {\Omega }_m^{0.6}$ givesThe printed expression is therefore the present-epoch result in units with $H_0=1$, or with its velocity interpreted as $U/H_0$. For distances and velocities in ordinary physical units, it needs the factor $H_0$; at a general epoch it needs $aH$. The approximation $f\simeq {\Omega }_m^{0.6}$ is exact in the Einstein-de Sitter limit but is not an exact growth formula for arbitrary cosmologies.

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.

The Fourier-space cosmological Poisson equation givesInsert this into the line-of-sight expression for the CMB lensing potential, Fourier transform ${\delta }_m$, and use the Rayleigh plane-wave expansion. Projection onto $Y_{lm}$ and spherical-harmonic orthogonality give

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.

With $\delta (\mathbf{x})=\int {\delta }_k{e}^{-i\mathbf{k}\cdot \mathbf{x}}{d}^{3}k/(2\pi {)}^3$ and logarithmic growing-mode rate $f=d\log D_{+}/d\log a$, the linearized cosmological continuity equation gives ${\mathbf{v}}_k=-iaHf{\delta }_k\mathbf{k}/k^2$. Apply the Rayleigh plane-wave expansion to its velocity potential and differentiate radially to obtain the displayed expression. The factor $aH$ supplies the velocity dimensions. At the present epoch $U/H_0$ has the same formula with $aH$ removed; the commonly used $f\simeq {\Omega }_m^{0.6}$ is an approximation, not an exact general growth law.

For integer $\ell \ge 0$ and $r,R>0$, $r_{<}=\min (r,R)$ and $r_{>}=\max (r,R)$. A proof compares the multipole expansion of $1/|\mathbf{r}-\mathbf{R}|$ with its Fourier transform of inverse distance in three dimensions. Applying the Rayleigh plane-wave expansion to the Fourier representation gives coefficient $8{\int }_0^{\infty }{j}_{\ell }(kr)j_{\ell }(kR)dk$ multiplying $Y_{\ell m}(\hat{r})Y_{\ell m}^{\ast }(\hat{R})$; the Coulomb multipole coefficient is $4\pi r_{<}^{\ell }/[(2\ell +1)r_{>}^{\ell +1}]$. Equating the coefficients proves the identity. The powers also match continuously at $r=R$.