= Solution
Use $\rho_r,\rho_m$ as energy per volume, as specified in the prose, and let $R$ have dimensions of length so that the positive curvature magnitude $k$ is dimensionless. The <Friedmann equation> for spatial curvature $-k$ is
$$
\boxed{H^2=\left(\frac{\dot R}{R}\right)^2=\frac{8\pi G}{3c^2}(\rho_r+\rho_m)+\frac{kc^2}{R^2}+\frac{\Lambda c^2}{3}.}
$$
For an energy density the <cosmological perfect-fluid continuity equation> is $\dot\rho+3H(\rho+P)=0$. Nonrelativistic matter has negligible pressure and <radiation> has $P_r=\rho_r/3$. Thus the four contributions to $H^2$ scale respectively as
$$
\boxed{R^{-4},\quad R^{-3},\quad R^{-2},\quad R^0.}
$$
For nonzero positive densities and $\Lambda>0$, the usual possible sequence is <radiation domination>, <matter domination>, curvature domination, then <cosmological-constant domination>. A distinct era for each is not guaranteed: coefficients may cause curvature to be overtaken by the <cosmological constant> before it ever dominates. If $\Lambda=0$, curvature dominates asymptotically; negative $\Lambda$ does not give a late positive vacuum-dominated expansion.
The printed auxiliary relation $P_r=\rho_rc^2/3$ uses mass-density notation, in conflict with the stated energy-density definition. It is consistent if $\rho_r$ there is understood as energy density divided by $c^2$; in that convention the density term in the <Friedmann equation> is $8\pi G\rho/3$. The two conventions must not be mixed.
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