= Radial peculiar velocity in spherical harmonics
{title2=$U=\frac{aHf}{2\pi^2}\sum_{\ell m}(-i)^\ell\int d^3k\,\frac{\delta_k}{k}j_\ell'(kr)Y_{\ell m}(\hat k)Y_{\ell m}^*(\hat r)$}
With $\delta(\mathbf x)=\int\delta_k e^{-i\mathbf k\cdot\mathbf x}d^3k/(2\pi)^3$ 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.
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