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• W1666937906 abstract "The authors review the equations, notational choices, and confusing terminology of the Friedmann (zero-pressure) and cosmological retaining cgs units as far as practical and in particular retaining units ${mathrm{cm}}^{ensuremath{-}2}$ 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 ${ensuremath{Omega}}_{0}$ at present time ${t}_{0}=0$, and generate families of curves showing the scale factor $R(ensuremath{tau})$ (with ${R}_{0}=1$) vs $ensuremath{tau}$ (time in units ${H}_{0}^{ensuremath{-}1}$) for fixed ${ensuremath{Omega}}_{0}$ and various values of the cosmological constant $ensuremath{Lambda}$ (in units ${H}_{0}^{2}$). These unusual graphs show the continuity of the solutions and the physical significance of $ensuremath{Lambda}$. Families for several values of ${ensuremath{Omega}}_{0}$ exhibit known but unfamiliar features. The authors also show the family of standard ($ensuremath{Lambda}=0$) and the family satisfying the inflationary constraint (${K}_{0}=0$). They obtain new and simple formulas for the critical value ${ensuremath{Lambda}}_{s}({H}_{0},{ensuremath{Omega}}_{0})$, which separates models with a big bang from those without. Their definition of ${ensuremath{Lambda}}_{s}$ at fixed ${H}_{0}$ and ${ensuremath{Omega}}_{0}$ 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-Lema^{i}tre model, and give ${R}_{s}$ and ${z}_{s}$ approximately for the neighboring Lema^{i}tre coasting models, which have $ensuremath{Lambda}l{ensuremath{Lambda}}_{s}$. The conventional wisdom that $ensuremath{Lambda}={ensuremath{Lambda}}_{c}(1+ensuremath{varepsilon})$ for the coasting models applies to a different characteristic value ${ensuremath{Lambda}}_{c}$. A quasistatic state in the future, with a second critical value ${ensuremath{Lambda}}_{s2}$, is possible if ${ensuremath{Omega}}_{0}g1$. The parameters ${ensuremath{Omega}}_{0}$, $frac{ensuremath{Lambda}}{{H}_{0}^{2}}$, $frac{{ensuremath{Lambda}}_{s}}{{H}_{0}^{2}}$, and $frac{{ensuremath{Lambda}}_{s2}}{{H}_{0}^{2}}$ can be used to classify the Friedmann models." @default.
• W1666937906 created "2016-06-24" @default.
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• W1666937906 date "1986-07-01" @default.
• W1666937906 modified "2023-10-16" @default.
• W1666937906 title "Scale factors<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mi>R</mml:mi><mml:mi /><mml:mn /><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mn /></mml:math>and critical values of the cosmological constant<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mi>Λ</mml:mi></mml:math>in Friedmann universes" @default.
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• W1666937906 doi "https://doi.org/10.1103/revmodphys.58.689" @default.
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