= Solution
The continuity and <Cosmological Euler equation in synchronous gauge> combine particularly simply. Differentiate $\delta_C'=-\nabla\cdot v_C-h'/2$ and add $\mathcal H\delta_C'$:
$$
\delta_C''+\mathcal H\delta_C'=-\nabla\cdot(v_C'+\mathcal Hv_C)-\frac12(h''+\mathcal Hh')=4\pi Ga^2\sum_N(1+3c_N^2)\rho_N\delta_N.
$$
The <velocity> terms cancel by the <Cosmological Euler equation in synchronous gauge>, so this step does not require setting the initial <velocity> exactly to zero.
For subhorizon modes deep in <radiation domination>, take the rapidly oscillating radiation perturbation to be negligible in the averaged gravitational forcing. Also neglect the subdominant matter self-gravity to leading order in $\rho_C/\rho_r$. With $a\propto\tau$ and $\mathcal H=1/\tau$, the resulting equation is
$$
\delta_C''+\frac1\tau\delta_C'=0,\qquad (\tau\delta_C')'=0.
$$
Integrating twice gives
$$
\boxed{\delta_C(\tau,x)=A(x)\log(\tau/\tau_*)+B(x),}
$$
where the arbitrary reference time $\tau_*$ makes the logarithm dimensionless and can be absorbed into $B$. This is <logarithmic growth of matter perturbations during radiation domination>.
The neglect of matter self-gravity is needed for the displayed form to be exact within the leading radiation-background approximation. Uniform radiation alone leaves a nonzero term $4\pi Ga^2\rho_C\delta_C$ if the cold-matter density is retained. Relative to $\mathcal H^2$, its coefficient is $3\Omega_C(a)/2$, small deep in the radiation era but not near equality. The <Mészáros equation> retains that effect in a matter-plus-radiation background; its solution basis tends to a constant and a logarithm at $a/a_{\rm eq}\ll1$, recovering the present leading result. It connects smoothly to matter-era growth rather than allowing the logarithmic approximation to be extrapolated indefinitely.
Galaxy-scale modes enter the <Hubble radius> before <matter-radiation equality>. The earlier radiation forcing and horizon-entry matching ordinarily generate a nonzero logarithmic coefficient, even though subsequent oscillatory radiation forcing is small. A <Fourier mode> with entry time $\tau_h\sim k^{-1}$ accumulates $\log(\tau_{\rm eq}/\tau_h)\sim\log(k\tau_{\rm eq})$ before equality. Across a large scale-factor range this factor need not be small, so treating the <density contrast> as exactly frozen would lose an important part of the galaxy-scale seed amplitude and its scale dependence. Growth is nevertheless much slower than $\delta_C\propto a$ in <matter domination>. This is the <Mészáros effect>, and the logarithmic matching contributes to the small-scale matter <cold-dark-matter transfer function>.
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