 Solution

For the synchronous-to-Newtonian scalar Fourier transformation, the coefficient of ${\hat{k}}_i{\hat{k}}_j$ in the transformed spatial metric perturbation must vanish. Thus $h_s+2kL=0$. Vanishing shift then gives $L^{\prime }=kT$. For $k\ne 0$ these conditions fixThe lapse in conformal Newtonian gauge is $h_{00}^{\mathrm{N}}=-2\Phi$, soThe remaining coefficient of ${\delta }_{ij}$ isgivingThe homogeneous $k=0$ perturbation requires separate background-coordinate choices; the inverse powers of $k$ do not define a transformation for it.