Solution
= Solution
After all species are nonrelativistic, each background density scales as $a^{-3}$, so
$$
f_\nu=\frac{\bar\rho_\nu}{\bar\rho_m}
=\frac{\Omega_{\nu,0}}{\Omega_{m,0}}
$$
is constant. Since $\delta\rho_i=\bar\rho_i\delta_i$,
$$
\boxed{\delta_m=(1-f_\nu)\delta_{cb}+f_\nu\delta_\nu}.
$$