 Sachs-Wolfe radiation-to-matter matching

For a constant growing adiabatic mode with curvature convention $\mathcal{R}=-\varphi -\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )/[4\pi G{a}^2(\bar{\rho }+\bar{P})]$, ${\varphi }_{\mathrm{rad}}=-2\mathcal{R}/3$ and ${\varphi }_{\mathrm{mat}}=-3\mathcal{R}/5$. The superhorizon photon continuity equation conserves ${\Theta }_0-\varphi$. With radiation initial value ${\Theta }_0=\mathcal{R}/3$, it gives ${\Theta }_{0,\mathrm{mat}}=2\mathcal{R}/5$ and $({\Theta }_0+\psi {)}_{\mathrm{mat}}=-\mathcal{R}/5$ when $\psi =\varphi$.