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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.