Matter-era photon endpoint solution (${\Delta }_0=e^{-ik\mu D}{\Delta }_{\ast }+A{k}^{1/2}{\mu }^2{\int }_{{\tau }_{\ast }}^{{\tau }_0}\tau e^{-ik\mu ({\tau }_0-\tau )}d\tau$)

In the negative-spatial-perturbation convention, a matter-era scalar solution $h=h_s+C=A{k}^{1/2}{\tau }^2/4$ gives the displayed exact photon brightness perturbation. Integration gives emission bracket ${\Delta }_{\ast }+iA{k}^{-1/2}\mu {\tau }_{\ast }-A{k}^{-3/2}$ times the propagation phase, plus observer monopole $A{k}^{-3/2}$ and dipole $-iA{k}^{-1/2}\mu {\tau }_0$. Under ordinary adiabatic growing-mode initial conditions and $k{\tau }_{\ast }\ll 1$, the emission gravitational term dominates higher angular multipoles. Arbitrary freely specified initial brightness need not obey this dominance.