Images:
      evolution of the cosmic scale factor in 3 FLRW cases

    1. Image 1 Caption: The evolution of the cosmic scale factor a(t) in three qualitatively distinct versions of the Friedmann-equation Λ=0 models.

      The Friedmann-equation Λ=0 models have cosmological constant (AKA Lambda, Λ) set equal to zero.

      Cosmological models of this kind were favored from circa 1965 (with the discovery of the cosmic microwave background (CMB): e.g., No-541,560--563) to circa 1995--1998 (e.g., Scott 2018).









      The cosmic scale factor a in an accelerating universe

    2. Image 2 Caption:























      cosmic scale factor for fiducial cosmologies

    3. Image 3 Caption:



















      The cosmic scale factor a in an accelerating universe

    4. Image 4 Caption: The Lambda-CDM model cosmic scale factor a(t). See papers:
      1. SOLUTION OF THE FRIEDMANN EQUATION DETERMINING THE TIME EVOLUTION, ACCELERATION AND THE AGE OF THE UNIVERSE 2008: Frank Steiner, Proceedings of the Int. Conference "Evolution Equations --- The State of the Art", Reisensburg/Germany, Dec. 2007 (published 2008), ed. W. Arendt et al., Ulmer Seminare 2008/Sonderausgabe, Heft 13, 231 (Ulm Reports and Publications: Ulm report ULM-TP/08-7: http://www.galpac.net/members/steinergroup/publications.html), 18 pages: Research: Exact analytic solutions of the Friedmann equations. Probably all of important ones. The general solutions are in terms of the Weierstrass ellitptic p function which by act of faith you can consider analytic. Probably NOT the discovery of the sinh**2/3 solution for the matter-Λ-CDM model.
      2. The eras of radiation, matter, and dark energy: new information from the Planck Collaboration: Kevin Cahill, arXiv, 2017, Jan30, 13 pages: Research: On cosmic scale factor and the cosmic energy eras based on the Planck 2015: Planck 2015 results. XIII. Cosmological parameters.
      3. Precision cosmology made more precise, Giorgio Galanti, Marco Roncadelli, arXiv, 2021, Feb02, 9 pages: Research: On an exact solution of the Friedmann equation for radiation, matter, and the cosmological constant (AKA Lambda, Λ) in flat space.
      4. An Educational Note on the Cosmic Scale Factor for the Λ-CDM model, David Jeffery, 2021, in preparation.




















      hyperbolic sinh, cosh, tanh

    5. Image Caption:



















      The cosmic scale factor a in an accelerating universe

    6. Image 5 Caption: The Einstein universe and the Lemaitre universe

























    7. Image 6 Caption: "Four distinct possible solutions of the Einstein equations with a cosmological constant are schematically shown for a closed universe (kappa = + 1). (Incidentally none of these solutions arise if kappa = 0, - 1.)"

      The Lemaitre universe (1931) is a positive curvature Friedmann equation model that starts with point origin which we would now call a Big Bang. Georges Lemaitre (1894--1966) original idea was a primeval atom theory in 1931. After the point origin, the Lemaitre universe (1931) has 3 phases: 1) a matter dominated phase Ω ∼ 1/a**3, 2) a quasi Einstein universe phase, 3) A de Sitter universe phase.

      References:
      1. Hermann Bondi (1919--2005): Bondi, H. 1961, Cosmology (Cambridge: Cambridge University Press) (Bo). Badly dated, but still an interesting account of the state of cosmology at the beginning of the 1960s. A classic of its kind. I do think Bondi is sometimes too elaborate in arguing his philosophical points. Science book. History book. Lemaitre universe (1931), see pages 84--84, 120--122, 165--170, 176, 180.
      2. David Jeffery, 2023: In preparation.
      3. Jean-Pierre Luminet, The Rise of Big Bang Models (4) : Lemaitre, 2015. It confirms that 1931 was the year of the Lemaitre universe (1931) and the primeval atom theory.
      4. Cormac O'Raifeartaigh et al. 2015: Historical review: The early history of the Einstein-de Sitter universe (presented 1932) and other early cosmological models. / keywords_cosmology.html
      5. O'Raifeartaigh et al. 2017: Historical review: The early history of the Einstein universe (presented 1917), cosmological constant.
      6. Varun Sahni & Alexei Starobinsky, THE CASE FOR A POSITIVE COSMOLOGICAL LAMBDA-TERM, 2000. See also Alexei Starobinsky (1948--2023).
      7. Wikipedia: Big Bang: Development.

      Cosmic Evolution for the Lambda-CDM model

    8. Cosmic Evolution for the Lambda-CDM model.
    Images:
    1. Credit/Permission: © David Jeffery, 2004 / Own work.
      Image link: Itself.
    2. Credit/Permission: © David Jeffery, 2004 / Own work.
      Image link: Itself.
    3. Credit/Permission: © User:Greek3, 2017 / CC BY-SA 4.0.
      Image link: Wikimedia Commons: File:Mplwp universe scale evolution.svg.
    4. Credit/Permission: © David Jeffery, 2021 / Own work.
      Image link: Itself.
    5. Credit/Permission: , 2016 / Public domain.
      Image link: Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Sinh_cosh_tanh.svg.
    6. Credit/Permission: © David Jeffery, 2004 / Own work.
      Image link: Itself.
    7. Credit/Permission: © David Jeffery, 2012 / Own work.
      Image link: alien_click_to_see_image.html: Image link direct: Itself.
      Referenced image: Credit/Permission: Varun Sahni & Alexei Starobinsky (2000), before or circa 2000 / No permission.
      Image link: Sahni: 3.1 Image link direct: Sahni: Figure 1.
    Local file: local link: cosmic_scale_factor.html.
    File: Cosmology file: cosmic_scale_factor.html.