= Matter-era photon endpoint solution
{title2=$\Delta_0=e^{-ik\mu D}\Delta_*+Ak^{1/2}\mu^2\int_{\tau_*}^{\tau_0}\tau e^{-ik\mu(\tau_0-\tau)}d\tau$}
In the negative-spatial-perturbation convention, a matter-era scalar solution $h=h_s+C=Ak^{1/2}\tau^2/4$ gives the displayed exact <photon brightness perturbation>. Integration gives emission bracket $\Delta_*+iAk^{-1/2}\mu\tau_*-Ak^{-3/2}$ times the propagation phase, plus observer monopole $Ak^{-3/2}$ and dipole $-iAk^{-1/2}\mu\tau_0$. Under ordinary adiabatic growing-mode initial conditions and $k\tau_*\ll1$, the emission gravitational term dominates higher angular multipoles. Arbitrary freely specified initial brightness need not obey this dominance.
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