 Solution

For each Fourier mode the equation is ${\Phi }_k^{\prime \prime }+6\mathcal{H}{\Phi }_k^{\prime }+k^2{\Phi }_k=0$. Set ${\Phi }_k=a^{-3}{u}_k$. Direct differentiation removes the first derivative and givesFor $|k\tau |\gg 1$, the inverse-square term is negligible, and the positive incoming-frequency branch has $u_k\sim A_k{e}^{-ik\tau }$. ThereforeThe momentum constraint then giveswhere ${\varphi }_0^{\prime }=-C/a^2$. Thus the scalar perturbation has precisely the required $a^{-1}$ behavior. Conversely, the two constraints imply the potential equation in part (c); reconstructing $\delta \varphi$ by the momentum constraint therefore satisfies the energy constraint too. This checks compatibility beyond simply comparing powers of $a$.