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 $\delta\rho_i=\bar\rho_i\delta_i$,
$$
\boxed{\delta_m=(1-f_\nu)\delta_{cb}+f_\nu\delta_\nu}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-310/3/b/solution