 Radial peculiar velocity in spherical harmonics

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.