= Synchronous Sachs-Wolfe line-of-sight formula
{title2=$\Theta_0=\delta_{\gamma *}/4+\mathbf n\cdot\mathbf v_*-\tfrac12\int h'_{ij}n^in^j d\tau$}
With instantaneous photon decoupling and tight-coupling initial brightness, integrate the <synchronous photon brightness equation> along the unperturbed ray $\mathbf x(\tau)=\mathbf x_0-\mathbf n(\tau_0-\tau)$. The direction $\mathbf n$ here is photon propagation, opposite to the observer-to-source sky direction. Emission terms are evaluated at the emitter, not the observer. The synchronous metric integral combines endpoint gravitational redshifts and evolving-potential effects when expressed in Newtonian gauge; it is not solely the <Integrated Sachs-Wolfe effect>.
// Target: cosmology.bigb
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