 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 givesFor 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 provesEquivalently ${\delta }_r/(1+w_r)={\delta }_c/(1+w_c)$, 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.