For a component with constant <barotropic equation of state> $P=w\rho$, the <cosmological perfect-fluid continuity equation> gives
$$
\frac{d\rho}{\rho}=-3(1+w)\frac{da}{a}.
$$
Hence $\rho_i=\rho_{i,0}a^{-3(1+w_i)}$. Choose the present <scale factor> to be $a_0=1$, so the <cosmological redshift> obeys $1+z=a^{-1}$, and define the present <cosmological density parameter> by
$$
\Omega_{i,0}=\frac{\rho_{i,0}}{\rho_{\rm crit,0}},
\qquad
\rho_{\rm crit,0}=\frac{3H_0^2}{8\pi G}.
$$
Substitution in the spatially flat <Friedmann equation> then yields
$$
\boxed{H(z)=H_0\left[\sum_i\Omega_{i,0}(1+z)^{3(1+w_i)}\right]^{1/2}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 310 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-310/1/a/solution