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The metric source is a gravitational frequency shift. photons following null geodesics through a time-dependent spatial metric change momentum relative to the background comoving frame. The directional projection of that metric change gives the ${\mu }^2$ factor. It is not a collision term or continuing photon scattering after recombination. The normalization and sign belong to the specified definition of $h$ and $\Delta$; a distribution perturbation should not silently be identified with temperature without fixing that normalization.

Define $\mathcal{H}=\dot{R}/R$, where dots here mean conformal-time derivatives. In an Einstein-de Sitter background,A power law $h\propto {\tau }^p$ therefore satisfies $p(p-1)+2p=6$, or $(p-2)(p+3)=0$. Thuswith the required growing mode $h=A{\tau }^2$. This is a solution in the perturbation convention used in the question, rather than a statement that every gauge-invariant gravitational potential grows as ${\tau }^2$.

Use an integrating factor for the photon equation:For the growing mode $\stackrel{...}{h}=0$. For the polynomial-source photon transport solution, a particular solution is $B_h=-2i\mu \dot{h}/k+2\ddot{h}/k^2$: direct substitution gives ${\dot{B}}_h+ik\mu B_h=2{\mu }^2\dot{h}$. The initial-value solution is consequentlyThis also follows by integrating the source twice by parts. The apparent inverse powers of $k$ cancel in the complete long-wavelength limit; the integral form remains regular. At $\mu =0$, the source vanishes and the complete expression reduces to $\Delta ={\Delta }_R$.

Physically, the exponential transports the recombination pattern along photon trajectories; its direction-dependent phase projects a Fourier perturbation onto different angular multipoles. The extra terms are the accumulated gravitational contribution, here expressible as endpoint terms because the source has this special time dependence. For endpoint monopole and dipole in photon transport, at the observer the last two terms are respectively a directional dipole and an angular monopole for each mode; they do not supply a new sky pattern at multipoles $\ell \ge 2$. The propagating recombination combination contains the intrinsic perturbation and gravitational/velocity endpoint contributions. Individual terms depend on the perturbation gauge; observable temperature anisotropy requires the consistent emitter and observer contributions.

In pure matter domination the gauge-invariant scalar potentials are constant for a growing linear mode: the matter contrast grows as $R$ while the potential scales as its contrast divided by $R$. Thus growth of this $h$ is compatible with no late Integrated Sachs-Wolfe effect in Einstein-de Sitter. When dark energy suppresses matter growth, the potentials generally evolve; photons gain and lose unequal energies entering and leaving a potential well. In a convention where $\Phi ,\Psi$ are the dimensionless Newtonian-gauge potentials, the integrated temperature source is $\int (\dot{\Phi }+\dot{\Psi })d\tau$ along the light path.

Large-angle cosmic microwave background data can constrain that late contribution, but low-multipole power alone has large cosmic variance and depends on the primordial radiation spectrum, geometry and foreground treatment. Cross-correlating the temperature map with low-redshift galaxies or other matter tracers isolates a late-time connection; decaying wells typically correlate overdensities with hotter cosmic microwave background directions. Boughn and Crittenden compared WMAP temperature data with the HEAO1 hard X-ray background and examined possible foreground origins of the correlation. This tests potential evolution under a cosmological model, not dark energy independently of all gravity and curvature assumptions.^[ref]