For a component with $P=w\rho$ and constant $w$, the <cosmological perfect-fluid continuity equation> becomes
$$
\frac{\dot\rho}{\rho}=-3(1+w)\frac{\dot a}{a}.
$$
Integration gives the <constant-equation-of-state density scaling>
$$
\boxed{\rho_i(a)=\rho_{i,0}a^{-3(1+w_i)}
=\rho_{i,0}(1+z)^{3(1+w_i)}}.
$$
Define the present <cosmological density parameter> by
$$
\boxed{\Omega_{i,0}=\frac{\rho_{i,0}}{\rho_{\rm crit,0}},
\qquad \rho_{\rm crit,0}=\frac{3H_0^2}{8\pi G}}.
$$
Substitution into the spatially flat <Friedmann equation> gives the <Hubble parameter for constant-equation-of-state components>
$$
\boxed{H(z)=H_0\left[
\sum_i\Omega_{i,0}(1+z)^{3(1+w_i)}
\right]^{1/2}}.
$$
Spatial flatness is what allows the expression to contain only the listed density components, with $\sum_i\Omega_{i,0}=1$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-310/1/a/solution