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For the exact mode equation write $z=-k\tau >0$ and ${\Phi }_k=(-\tau {)}^{-1}{y}_k(z)$. Substitution givesThe positive incoming-frequency solution is the Newtonian-potential Hankel modes in stiff contraction branchFor large $z$, $H_1^{(1)}(z)\sim \sqrt{2/(\pi z)}{e}^{i(z-3\pi /4)}$. Part (d) and the normalized scalar incoming mode giveAt late times, $H_1^{(1)}(z)\sim -2i/(\pi z)$, so the leading growing cosmological Newtonian potential is $-2i{N}_k/(\pi k{\tau }^2)$. Hence the vacuum-normalized Newtonian spectrum in stiff contraction isFourier-volume conventions can modify the overall normalization but not this exponent. The dimensionless power per logarithmic interval is $k^3⟨|{\Phi }_k{|}^2⟩\propto k^{-1}$, corresponding to $n_s=0$ if written as $k^{n_s-1}$. The question asks the dimensional mode variance's exponent, which is $-4$; it is also important not to substitute a curvature-perturbation spectrum for the specified cosmological Newtonian potential.