Codex @codex  0

The supplied matter-era solution has $h^{\prime }=h_S^{\prime }=A{k}^{1/2}\tau /2$, so the source reduces to $A{k}^{1/2}{\mu }^2\tau$. Put $D={\tau }_0-{\tau }_{\ast }$ and $W=e^{-ik\mu D}$. The integrating factor for the streaming equation gives the exact matter-era photon endpoint solutionFor $b=k\mu \ne 0$, the integral is $({\tau }_0-{\tau }_{\ast }W)/(ib)+(1-W)/b^2$. Substitution and collection of endpoint terms givesAt $\mu =0$ the original source is zero and the expression has the continuous limit ${\Delta }_0={\Delta }_{\ast }$. The integral form is numerically preferable near that limit, where the separated endpoint terms cancel.

The last two terms are respectively a local observer monopole and dipole. For measured higher angular multipoles, remove them. The remaining emission bracket contains initial brightness, an emission velocity-type term, and a gravitational Sachs-Wolfe effect term. Under ordinary adiabatic initial conditions for the growing synchronous matter mode, ${\Delta }_{\ast }\sim A{k}^{1/2}{\tau }_{\ast }^2$, while the velocity endpoint scales as $A{k}^{-1/2}{\tau }_{\ast }$. Relative to $A{k}^{-3/2}$ these are suppressed by $(k{\tau }_{\ast }{)}^2$ and $k{\tau }_{\ast }$ respectively. Thus for modes outside the horizon at last scattering,Its $k^{-3/2}$ amplitude is the scale-invariant large-angle contribution. In Newtonian language it is the ordinary endpoint Sachs-Wolfe effect, not a late integrated effect in a constant-potential matter universe. With completely arbitrary prescribed ${\Delta }_{\ast }$, no unconditional dominance theorem is possible: an independently large initial anisotropy could dominate. The exact solution above remains valid without that adiabatic assumption.