= Sachs-Wolfe radiation-to-matter matching
{c}
{title2=$(\Theta_0+\psi)_*=-\mathcal R/5$}
For a constant growing adiabatic mode with curvature convention $\mathcal R=-\phi-\mathcal H(\phi'+\mathcal H\phi)/[4\pi Ga^2(\bar\rho+\bar P)]$, $\phi_{\rm rad}=-2\mathcal R/3$ and $\phi_{\rm mat}=-3\mathcal R/5$. The superhorizon <photon continuity equation> conserves $\Theta_0-\phi$. With radiation initial value $\Theta_0=\mathcal R/3$, it gives $\Theta_{0,\rm mat}=2\mathcal R/5$ and $(\Theta_0+\psi)_{\rm mat}=-\mathcal R/5$ when $\psi=\phi$.
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