Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S2 / 3 / b / Solution

For conserved nonrelativistic matter, ${\rho }_M={\rho }_{M0}{a}^{-3}$, where $a=R/R_0$ and $a_0=1$. Substitution into the Friedmann equation giveswithThe source writes the dimensionless normalized factor as $R$ in its final expression; this is equivalent to setting $R_0=1$ consistently in the curvature coefficient. Evaluating today gives ${\Omega }_M+{\Omega }_k+{\Omega }_{\Lambda }=1$. Here the printed $\Lambda$ already has dimensions of inverse time squared. If the cosmological constant is instead defined geometrically with inverse-length-squared units, replace this $\Lambda$ by ${\Lambda }_{\mathrm{geom}}{c}^2$.