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Preserving synchronous gauge means $T^{\prime }+\mathcal{H}T=0$ and $L^{\prime }=kT$. The first equation is $(aT{)}^{\prime }=0$, and integration of the second gives the complete residual scalar transformation for each nonzero Fourier mode:Changing the lower limit only changes $g$. Taking the trace and trace-free longitudinal part of $\Delta h_{ij}$ givesConsequently $\Delta h_s^{\prime }=2k^{2}T$ and $\Delta h_s^{\prime \prime }=2k^2{T}^{\prime }$. The cosmological Newtonian potential changes bywhile its spatial counterpart changes byThus both potentials are invariant under residual synchronous-gauge freedom, although $h$ and $h_s$ individually are not.

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.

For a nonzero Fourier mode in synchronous gauge, write $h_{ij}=h{\delta }_{ij}/3+h_s({\hat{k}}_i{\hat{k}}_j-{\delta }_{ij}/3)$. Under the passive displacement ${\xi }^{\mu }=(T,i{\hat{k}}_{i}L)e^{i\mathbf{k}\cdot \mathbf{x}}$, cancellation of spatial scalar shear requires $h_s+2kL=0$, and cancellation of the shift requires $kT-L^{\prime }=0$. The resulting cosmological Newtonian potential is $\Phi =(h_s^{\prime \prime }+\mathcal{H}{h}_s^{\prime })/(2k^2)$ and the spatial potential is $\Psi =-(h-h_s)/6-\mathcal{H}{h}_s^{\prime }/(2k^2)$. Both remain unchanged under residual synchronous-gauge freedom, for which $T^{\prime }+\mathcal{H}T=0$ and $L^{\prime }=kT$.