 Pure growing-mode scaled velocity

For a pure scalar growing density contrast with time-independent boundary conditions, ${\nabla }^2\psi ={\delta }_{\ast }$ 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.