 Solution

For pressureless matter, $w=0$, the constant-equation-of-state comoving density equation reduces to ${\Delta }^{\prime \prime }+2{\Delta }^{\prime }/\eta -6\Delta /{\eta }^2=0$. Substitution of $\Delta \propto {\eta }^s$ gives $s^2+s-6=(s-2)(s+3)=0$. Since $a\propto {\eta }^2$,In the growing mode, $a^2\bar{\rho }\Delta$ is constant, so the Poisson constraint makes $\varphi$ constant for a nonzero Fourier mode. The momentum constraint and the flat Friedmann equation giveThis is the growing-mode velocity potential in matter domination, in the question's convention where the peculiar velocity is $\nabla v$.