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

Fix the background signature $+---$ and define a positive spatial metric perturbation by $\delta g_{ij}=+a^2{h}_{ij}$. Thus the synchronous line element isFor a nonzero Fourier mode, take the passive displacement $\tilde{\tau }=\tau +T{e}^{i\mathbf{k}\cdot \mathbf{x}}$ and ${\tilde{x}}^i=x^i+i{\hat{k}}^{i}L{e}^{i\mathbf{k}\cdot \mathbf{x}}$. The transformed scalar shear is $h_S-2kL$, so setting it to zero gives $L=h_S/(2k)$. Removing the mixed metric term gives $T=L^{\prime }/k=h_S^{\prime }/(2k^2)$.

Define the resulting Newtonian gauge line element as $a^2[(1+2\Phi )d{\tau }^2-(1-2\Psi ){\delta }_{ij}d{x}^{i}d{x}^j]$. Its potentials areThese signs follow directly from the linear metric gauge-transformation law. Positive $\delta g_{ij}$ here corresponds to a minus sign for $h_{ij}$ inside the spatial parentheses. The synchronous-to-Newtonian scalar Fourier transformation uses a different explicit background/spatial-amplitude convention. Its displacement signs must be translated along with that convention. The present definitions also fix the signs in the photon redshift identity with positive spatial perturbations.

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.