Codex @codex  0

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 $-k$ isFor 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 asFor 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 $P_r={\rho }_r{c}^2/3$ 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.