= Solution
The perturbations are <density contrasts>, $\delta_c=\delta\rho_c/\bar\rho_c$ and $\delta_r=\delta\rho_r/\bar\rho_r$, evaluated in the common comoving <synchronous gauge>. For nonrelativistic <cold dark matter> of fixed particle mass, $\rho_c\propto n_c$, hence $\delta_c=\delta n_c/\bar n_c$. Thermal <photons> have <photon number density> $n_\gamma\propto T^3$ and <energy density> $\rho_r\propto T^4$. A small local <temperature> perturbation therefore gives
$$
\frac{\delta n_\gamma}{\bar n_\gamma}=3\frac{\delta T}{\bar T},
\qquad
\delta_r=4\frac{\delta T}{\bar T}.
$$
For <adiabatic initial conditions>, the ratio $n_\gamma/n_c$ is unperturbed, so $\delta n_\gamma/\bar n_\gamma=\delta n_c/\bar n_c$. Combining these equations proves
$$
\boxed{\delta_r=\frac43\delta_c.}
$$
Equivalently $\delta_r/(1+w_r)=\delta_c/(1+w_c)$, with $w_r=1/3$ and $w_c=0$. The relation applies before causal pressure gradients can separate the two species on <superhorizon scales>; it need not remain true after <cosmological horizon crossing>.
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