= Photon redshift identity with positive spatial perturbations
{title2=$\int E\{(h'-h_S')/6+q^2T\}\,d\tau=\int E(\Phi'+\Psi')\,d\tau+[E(T'-iqT)]$}
Let $q=\mathbf k\cdot\mathbf n$, $E=e^{iq(\tau-\tau_0)}$ and $T=h_S'/(2k^2)$. The <synchronous-to-Newtonian transformation with positive spatial perturbations> gives $(h'-h_S')/6=\Phi'+\Psi'+T''$. Since $[E(T'-iqT)]'=E(T''+q^2T)$, 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>, $2\int E\Phi' d\tau$. The angular term in the synchronous contraction is positive for this metric convention.
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