Caption: The evolution of the cosmic scale factor a(t) for 4 qualitatively distinct versions of the Friedmann-Lemaitre models. One model is an accelerating universe model version of the Friedmann-Lemaitre models: i.e., the positive Lambda (Λ > 0) model.
Features:
where r_comoving is comovoing distance which is a time-independent set of distances.
The cosmic present t_0 (equal to the age of the observable universe = 13.797(23) Gyr (Planck 2018)) of is by convention symbolized by the subscript 0. By convention, a_0 (i.e., the present scale factor) = a(t_0) = 1, and so proper distances and comovoing distances are equal at cosmic present t_0 (equal to the age of the observable universe = 13.797(23) Gyr (Planck 2018)).
At time-zero of the Friedmann-Lemaitre models we have a(t=0) = 0 which is the big bang singularity.
In fact, no one (or almost no one) believes we can run the clock on the Friedmann-Lemaitre models back to time zero.
But we believe that we may be able to run the clock back to with in a tiny fraction of second of time zero (see Wikipedia: Graphical timeline of the Big Bang).
Observations circa 2020, Ω ≅ 1 which means the spatial geometry of the observable universe is nearly Euclidean geometry (AKA flat geometry) (see Wikipedia: Λ-CDM model: Parameters).
It was dismissed for its original use, but was resurrected to give the accelerating universe.
Before 1998 people thought the universal expansion must be decelerating (i.e., negatively accelerating) since there was NO reason to think there was a cause for acceleration.
The (Ω=1,Λ=0) model is the Einstein-de Sitter universe (1932, standard model of cosmology c.1960s--c.1990s) which is NEITHER the Einstein universe (1917) NOR the de Sitter universe (1917).
The model originally decelerates and then accelerates.
The acceleration is starts when a(t) starts curving up rather than down.
The cosmic time of transition between decelerating and accelerating phases according to the current Λ-CDM model is is thought to be cosmic time = 10.02(4) Gyr Cahill 2016, arXiv:1606.08865, p. 9).
The Λ-CDM model which is the currently favored cosmological model.
It is fitted to the observations by adjusting 6 independent parameters and gives a good fit to most observations (see Wikipedia: Λ-CDM model: Parameters). However, the Λ-CDM model may need significant revision or even replacement because of the Hubble tension---which we will NOT discuss here.
Currently, the oldest star is believed to be SMSS J031300.36-670839.3 with age ∼ 13.6 Gyr.
The current Λ-CDM model fitted age of the observable universe = 13.797(23) Gyr (Planck 2018) (see Planck 2018: Age of the observable universe = 13.797(23) Gyr) is comfortably older (see Wikipedia: Age of the universe).