= Solution
For the exact mode equation write $z=-k\tau>0$ and $\Phi_k=(-\tau)^{-1}y_k(z)$. Substitution gives
$$
z^2y_k''+zy_k'+(z^2-1)y_k=0.
$$
The positive incoming-frequency solution is the <Newtonian-potential Hankel modes in stiff contraction> branch
$$
\Phi_k=N_k(-\tau)^{-1}H_1^{(1)}(-k\tau).
$$
For 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 give
$$
A_k=\frac{C}{2ik}\sqrt{\frac{\hbar}{2k}},
\qquad |A_k|^2=\frac{3\hbar}{16k^3},
\qquad |N_k|^2=\frac{\pi k}{2}|A_k|^2=\frac{3\pi\hbar}{32k^2}.
$$
At late times, $H_1^{(1)}(z)\sim-2i/(\pi z)$, so the leading growing <cosmological Newtonian potential> is $-2iN_k/(\pi k\tau^2)$. Hence the <vacuum-normalized Newtonian spectrum in stiff contraction> is
$$
\boxed{\langle|\Phi_k|^2\rangle\sim\frac{3\hbar}{8\pi k^4\tau^4},\qquad
\text{power of }k=-4.}
$$
Fourier-volume conventions can modify the overall normalization but not this exponent. The dimensionless power per logarithmic interval is $k^3\langle|\Phi_k|^2\rangle\propto k^{-1}$, 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>.
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