Scale factors $R\mathrm{}\left(t\right)\mathrm{}$ and critical values of the cosmological constant $\Lambda$ in Friedmann universes

The authors review the equations, notational choices, and confusing terminology of the Friedmann (zero-pressure) and Lemaître cosmological models, retaining cgs units as far as practical and in particular retaining units ${\mathrm{cm}}^{-2\mathrm{}}$ for the present Gaussian curvature $K_0$ of three-space. They integrate the Friedmann equation numerically, requiring solutions to match the present Hubble parameter $H_0$ and mass-density ("closure") parameter ${\Omega }_0$ at present time $t_0=0\mathrm{}$, and generate families of curves showing the scale factor $R\left(\tau \right)$ (with $R_0=1\mathrm{}$) vs $\tau$ (time in units $H_0^{-1\mathrm{}}$) for fixed ${\Omega }_0$ and various values of the cosmological constant $\Lambda$ (in units $H_0^2$). These unusual graphs show the continuity of the solutions and the physical significance of $\Lambda$. Families for several values of ${\Omega }_0$ exhibit known but unfamiliar features. The authors also show the family of "standard models" ($\Lambda =0\mathrm{}$) and the family satisfying the "inflationary constraint" ($K_0=0\mathrm{}$). They obtain new and simple formulas for the critical value ${\Lambda }_s(H_0,{\Omega }_0)$, which separates models with a big bang from those without. Their definition of ${\Lambda }_s$ at fixed $H_0$ and ${\Omega }_0$ differs from usual practice but proves useful. These formulas also give the quasistatic scale factor $R_s$ and redshift $z_s$ for the corresponding Eddington-Lemaître model, and give $R_s$ and $z_s$ approximately for the neighboring "Lemaître coasting models," which have $\Lambda <{\Lambda }_s$. The conventional wisdom that $\Lambda ={\Lambda }_c(1+\varepsilon )$ for the coasting models applies to a different characteristic value ${\Lambda }_c$. A quasistatic state in the future, with a second critical value ${\Lambda }_{s2}$, is possible if ${\Omega }_0>1\mathrm{}$. The parameters ${\Omega }_0$, $\frac{\Lambda }{H_0^2}$, $\frac{{\Lambda }_s}{H_0^2}$, and $\frac{{\Lambda }_{s2}}{H_0^2}$ can be used to classify the Friedmann models.