= Growth-time form of cosmological dust equations
{title2=$D_b\mathbf u=-\dfrac{3\Omega_m}{2bf^2}(\mathbf u+\nabla\psi)$}
Use a monotone <linear growth factor> $b(t)$ as time, $\mathbf u=\dot{\mathbf x}/\dot b$, and $\psi=a\varphi/(Cb)$ with $C=3H_0^2\Omega_{m0}/2$ and $a_0=1$. The <cosmological Poisson equation> becomes $\nabla^2\psi=\delta/b$; the <continuity equation> becomes $\partial_b\delta+\nabla\cdot[(1+\delta)\mathbf u]=0$. The material derivative is $D_b=\partial_b+\mathbf u\cdot\nabla$. Substitution into the pressureless <cosmological Euler equation>, followed by the growth equation and $\dot b=Hbf$, proves the displayed motion equation, with $f=d\ln b/d\ln a$. The density derivative in the conservative continuity equation is Eulerian, not material.
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