= Solution
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,
$$
\mathcal H=\frac2\tau,\qquad
\mathcal H^2=\frac{8\pi G}{3}\bar\rho_mR^2,
\qquad4\pi G\bar\rho_mR^2=\frac6{\tau^2}.
$$
A <power law> $h\propto\tau^p$ therefore satisfies $p(p-1)+2p=6$, or $(p-2)(p+3)=0$. Thus
$$
\boxed{h=A\tau^2+B\tau^{-3}},
$$
with 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:
$$
\Delta(\tau)=\Delta_R e^{-ik\mu(\tau-\tau_R)}
+2\mu^2\int_{\tau_R}^{\tau}\dot h(t)e^{-ik\mu(\tau-t)}dt.
$$
For the growing mode $\dddot h=0$. 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 consequently
$$
\boxed{\Delta(\tau)=\left[\Delta_R+\frac{2i\mu\dot h_R}{k}-\frac{2\ddot h_R}{k^2}\right]e^{-ik\mu(\tau-\tau_R)}
-\frac{2i\mu\dot h(\tau)}{k}+\frac{2\ddot h(\tau)}{k^2}}.
$$
This 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 $\ell\geq2$. 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 \b[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.<Investigating the origins of the CMB-XRB cross correlation by Stephen Boughn and Robert Crittenden>[abstract and foreground-correlation tests]{ref}
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