= Matter-era forced radiation response
{title2=$\delta_r''+\omega^2\delta_r=\tfrac43\delta_c'',\quad\omega=k/\sqrt3$}
Let $\delta_c=C_g\tau^2+C_d\tau^{-3}$ be the <matter-era growing and decaying density modes>, with radiation neglected in the background gravity. The radiation <density contrast> is
$$
\delta_r=C_1\cos(\omega\tau)+C_2\sin(\omega\tau)+\frac{8C_g}{3\omega^2}
+\frac{16C_d}{\omega}\int_{\tau_*}^{\tau}s^{-5}\sin[\omega(\tau-s)]\,ds.
$$
The formula follows by the <Green's function> of the forced <harmonic oscillator>. For $k\tau\gg1$, the slowly varying particular response to the decaying mode begins at $16C_d\tau^{-5}/\omega^2$; lower-endpoint oscillations are absorbed into $C_1,C_2$. The growing mode drives a constant radiation offset, while <cold dark matter> continues growing as $\tau^2$.
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