How are the constants in the inflationary hypothesis derived? The inflationary hypothesis as I understand it is a correction to GR to account for the observed flatness of the universe in a model in which the universe is expanding.
How are the constants behind this inflationary hypothesis derived?
I am looking to establish whether this model predicts or is derived from an estimate of the age of the universe, and how to prove that.
 A: The age of the observable universe is determined from the end of the inflationary expansion, which is effectively the hot Big Bang. The end of inflation is associated with the hot Big Bang because, during inflation, matter and energy are exponentially diluted and the universe “reheats” after inflation ends, as the inflaton decays into matter and energy consistent with the temperature of the universe at the time, which can vary according to the model.
Inflation could have lasted an arbitrarily long time, and because inflation is great at erasing initial conditions, it’s impossible to know just how long the universe was inflating before it stopped in what would become our observable universe.
A: Inflation results in a "multiverse" of causally-separated "local" universes. To elaborate on the uncertainties which might interfere with the formulation of constants, I'd like to mention the Borde-Guth-Vilenkin theorem (last revised in 2003), which is commonly cited as the basis for arriving at such a numerical conclusion as the number of years since the start either of any such multiverse, or of any unique and isolated universe.
Infinity and eternality are concepts, not numbers, but BGV is often considered to ban any multiverse eternal to the past.  Although reducing the probability of such a cosmos existing, it doesn't ban it completely.  There are several reasons for that limitation on the theorem's applicability.
The most well-known one is that, as Vilenkin has repeatedly stated, the theorem applies only to a multiverse (or to any unique universe) that is expanding "on average":  That's why its last revision specifically mentions (in a footnote) that it does not apply to Aguirre & Gratton's 2002 "Steady-state eternal inflation", whose model is an inflationary multiverse expanding in opposite temporal directions on opposite sides of a Cauchy surface, so as to balance its expansion toward the future witn an expansion that we would perceive as expansion toward the past.
Moreover, although the BGV theorem is based on a congruence of geodesics (a group of "world lines" that do not cross each other), it neither sets any upper or lower limit on the number of those geodesics, nor bans the possibility that an inflationary multiverse might have more than one beginning (and / or more than one end), as lengths of some of the geodesics in the congruence might partially overlap others without crossing them.  This possibility's mentioned both by Guth in 2007's "Eternal inflation and its implications", and by Linde in his "Inflationary Cosmology" of 2008.  (For instance, physical possibilities involved in such divergent futures might include such artificial "backreactions" as the addition of mass to a star that would've otherwise only collapsed into a neutron star so as to induce its collapse into a black hole instead, in civilizations subscribing to cosmologies which hypothesize cosmogenesis within black holes of stellar origin.)
In addition, the geodesics of 1915's GR differ from the "auto-parallel curves" of 1929's "Einstein-Cartan Theory" (developed after the discovery of particulate spin, and used in recent cosmological models), which appears to have eliminated its applicability to Nikodem J. Poplawski's ECT-based "cosmology with torsion", as described in his numerous papers published between 2010 and 2020.  (Although the usual entropic "arrow of time" is inherited from the parenting universe in the "baby" or local universes in his multiverse--which is basically one of LU's on sequentially-decreasing scales of spacetime that are each formed by bounce effects within the volume of a star collapsing gravitationally--the total energy left within the resulting black hole is insufficient to allow outward passage thru the event horizon of the collapse, so that the new universes expanding inside black holes would allow entropy to increase further, without increasing its density in the "parenting" LU.)
