Synchronous-to-Newtonian transformation with positive spatial perturbations ($\Phi =-\mathcal{H}T-T^{\prime },\Psi =(h-h_S)/6+\mathcal{H}T$)

Define $\delta g_{ij}=a^2{h}_{ij}$ and $h_{ij}=h{\delta }_{ij}/3+({\hat{k}}_i{\hat{k}}_j-{\delta }_{ij}/3)h_S$. Under the passive coordinate shift $\tilde{\tau }=\tau +T{e}^{i\mathbf{k}\cdot \mathbf{x}}$ and ${\tilde{x}}^i=x^i+i{\hat{k}}^{i}L{e}^{i\mathbf{k}\cdot \mathbf{x}}$, removing the traceless spatial perturbation gives $L=h_S/(2k)$. Removing the mixed time-space metric gives $T=L^{\prime }/k=h_S^{\prime }/(2k^2)$. The remaining diagonal metric components are the displayed Newtonian potentials. This sign convention is opposite to defining the synchronous perturbation by $g_{ij}=-a^2({\delta }_{ij}+h_{ij})$.

Let $q=\mathbf{k}\cdot \mathbf{n}$, $E=e^{iq(\tau -{\tau }_0)}$ and $T=h_S^{\prime }/(2k^2)$. The synchronous-to-Newtonian transformation with positive spatial perturbations gives $(h^{\prime }-h_S^{\prime })/6={\Phi }^{\prime }+{\Psi }^{\prime }+T^{\prime \prime }$. Since $[E(T^{\prime }-iqT){]}^{\prime }=E(T^{\prime \prime }+q^{2}T)$, the displayed identity follows by integration. The emission endpoint combines intrinsic and velocity shifts into the ordinary Sachs-Wolfe effect; the local observer endpoint is a monopole plus a dipole. Without anisotropic stress the remaining integral is the Integrated Sachs-Wolfe effect, $2\int E{\Phi }^{\prime }d\tau$. The angular term in the synchronous contraction is positive for this metric convention.