= Pure growing-mode scaled velocity
{title2=$\delta=b\delta_*,\quad\partial_b\psi=0,\quad\mathbf u=-\nabla\psi$}
For a pure scalar growing <density contrast> with time-independent boundary conditions, $\nabla^2\psi=\delta_*$ fixes the scaled <peculiar gravitational potential> up to an irrelevant constant. The linearized continuity equation fixes the longitudinal scaled velocity to $-\nabla\psi$. The remaining homogeneous velocity is $\mathbf u_h=\mathbf C(\mathbf x)/(a^2\dot b)$; a divergence-free such field does not change the growing density at linear order. Excluding this decaying velocity component is necessary for time-independent $\mathbf u$. In an <Einstein-de Sitter universe>, $b=a$ and $\mathbf u_h\propto a^{-3/2}$, furnishing a counterexample if the pure growing-velocity hypothesis is omitted.
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