= Solution
Write the growing <density contrast> as $\delta_{m,\mathbf k}=A_{\mathbf k}\tau^2$, with $A_{\mathbf k}$ having the indicated $k^{n/2}$ spectral amplitude. In the dust-comoving <synchronous gauge>, $h'=-2\delta_m'$, and the supplied $h_s'=h'$ therefore implies $h_s''=-4A_{\mathbf k}$. On the same comoving density slicing, the <cosmological Poisson equation> is
$$
-k^2\Phi_{\mathbf k}=4\pi Ga^2\rho_m\delta_{m,\mathbf k}
=\frac32\mathcal H^2\delta_{m,\mathbf k}=6A_{\mathbf k},
$$
because $\mathcal H=2/\tau$. This is the comoving-density form of the relativistic scalar constraint, so it can be used on these large scales; replacing its density by the Newtonian-gauge density would not be justified there. Combining the two results gives
$$
\boxed{\frac{h_s''}{2k^2}=-\frac{2A_{\mathbf k}}{k^2}=\frac13\Phi_{\mathbf k}.}
$$
Thus the large-angle <Sachs-Wolfe effect> has source $\Phi/3$. The potential is constant for this matter-era growing mode, and its <power spectrum> obeys $P_\Phi(k)\propto k^{n-4}$.
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