= Solution
After <cosmological horizon entry> in <radiation domination>, the oscillating radiation source averages away. Neglecting both that source and CDM self-gravity in the <CDM density equation in a matter-radiation universe> gives
$$
\delta_C''+\frac1\tau\delta_C'=0,\qquad(\tau\delta_C')'=0,
$$
so $\delta_C=C_1(k)+C_2(k)\log\tau$. The <superhorizon adiabatic CDM mode in radiation domination> has amplitude $A_R(k)\tau_h^2\sim A_R(k)k^{-2}$ at entry $\tau_h\sim k^{-1}$, fixing this scale for the subsequent <logarithmic CDM growth after radiation-era horizon entry>.
More explicitly, in a sharp-transition approximation with $\tau_h=1/k$, continuity of $\delta_C$ and $\delta_C'$ gives
$$
\delta_C=\frac{A_R(k)}{k^2}\,[1+2\log(k\tau)].
$$
The <matching the CDM growing mode at horizon entry> therefore yields the late subhorizon scaling
$$
\boxed{\delta_C(\tau,k)\simeq A_R(k)k^{-2}\log(k\tau).}
$$
Here $\simeq$ suppresses order-one coefficients and the nonlogarithmic term, as in the question. A smooth treatment of the radiation forcing through entry changes those coefficients; the factor two is exact only for the sharp matching model just used. This logarithmic growth, rather than growth proportional to $a$, is the <Mészáros effect>.
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