Synchronous-to-Newtonian scalar Fourier transformation ($T=-h_s^{\prime }/(2k^2),L=-h_s/(2k)$)

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 $L^{\prime }=kT$.