Growing-mode velocity potential in matter domination ($v=-\eta \varphi /3\propto a^{1/2}$)

During matter domination in a flat universe, the growing comoving-gauge density contrast is proportional to $a\propto {\eta }^2$. Its Newtonian potential is constant by the Poisson constraint. The momentum constraint gives $v=-2\varphi /(3\mathcal{H})=-\eta \varphi /3$. Thus the peculiar velocity potential grows as $a^{1/2}$ in the convention where the peculiar velocity is $\nabla v$.