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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.

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    $\begingroup$ You will get multiple answers because inflation is not an established fact; it is a research programme in which many ideas jostle for evidence and overall consistency. None are yet fully consistent, but the idea has proved fruitful and many cosmologists feel it is on the right track. However the degree of confidence in statements about inflation on the web and in popular books tends to exceed by quite a large margin the degree to which ideas have been established at the research level. Inflation is an interlude between a great unknown (the Planck era) and another great unknown (the GUT era). $\endgroup$ Jun 15 '20 at 15:05
  • $\begingroup$ So how is the age of the universe established? (Without a consistent model for inflation where does this age derive from?) Perhaps this should be a separate question, but that is what I was trying to establish. $\endgroup$
    – David
    Jun 15 '20 at 15:46
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    $\begingroup$ The phrase "age of the universe" in practice usually means "how much time has elapsed since some very early time such as GUT era" but many textbooks are sloppy about this. $\endgroup$ Jun 15 '20 at 17:11
  • $\begingroup$ Whether the passage of time occurs in discrete units (that cannot be subdivided) is a quantum mechanical question not directly addressed by Guth's Theory of Inflation. $\endgroup$
    – Edouard
    Aug 12 '20 at 15:56
  • $\begingroup$ @David -In the preface to "The Inflationary Universe", Guth states that "The theory of inflation modifies our understanding of just the first tiny fraction of a second of the history of the universe", so that, per the conjunction of a more recent remark of his that's cited in my answer (which doesn't contradict that earlier one) with the last question in your post, you're wanting to subdivide that fraction of a second, which is (as he implies in his later remark) not necessarily any possibility justified by his theory, due to the fact that eternity is not a number. – Edouard 12 mins ago $\endgroup$
    – Edouard
    Aug 12 '20 at 16:04
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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.

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    $\begingroup$ So how is the end of the inflationary expansion arrived at? $\endgroup$
    – David
    Jun 12 '20 at 23:15
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    $\begingroup$ @David, inflation ends when the cosmological fluid's equation of state, $p = w\rho$, becomes non-inflationary, that is, when $w<-1/3$. For inflation driven by a scalar field, $w = (KE-PE)/(KE+PE)$, and inflation ends when the kinetic energy of the field (KE) becomes greater than half the potential energy (PE). This naturally happens in models with 2nd-order phase transitions where the inflaton "rolls" down to a metastable state, picking up speed. $\endgroup$
    – bapowell
    Jun 13 '20 at 12:58
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    $\begingroup$ Formally I mean an expansion rate satisfying $\ddot{a}>0$ which, in general, corresponds to quasi-exponential expansion. $\endgroup$
    – bapowell
    Jun 13 '20 at 13:37
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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.)

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    $\begingroup$ Do you really think the existence of a massive scalar field is a bigger pill to swallow than fermions with spatial extent? Do you not think there are unobserved particles at energies beyond our present experimental limits? $\endgroup$
    – bapowell
    Jun 8 '20 at 14:19
  • $\begingroup$ Inflation doesn’t violate causality, nor is the rate of expansion a speed, so I’m struggling to understand what you mean by space expanding at greater than 3 times the speed of light in some models. Which models? In any case, the inflaton, as a particle, has nothing to do with inflationary expansion, which is due to the potential energy of the field. As a particle, the inflaton is just a heavy scalar. $\endgroup$
    – bapowell
    Jun 8 '20 at 17:18
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    $\begingroup$ I'm referring to your first comment, in which you say "considering rates of spatial expansion equalling 3 or more times the relative speed of light". That statement makes no sense to me. $\endgroup$
    – bapowell
    Jun 8 '20 at 18:09
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    $\begingroup$ 1. Please do not add meta-commentary (such as about the presence or absence of votes on the post) to your answer, these get quickly outdated as time passes and do not belong in answers. 2. You have made over 20 edits to this post over the last week. While we welcome improvements to posts, every such edit bumps the post in the view of "active" posts, so please consider making edits substantial - instead of several successive minor edits changing only a few words each, try to aggregate them into larger edits. $\endgroup$
    – ACuriousMind
    Jun 15 '20 at 15:04
  • $\begingroup$ I have a collection of references on the inapplicability of BGV to Poplawski's model, of which the most notable is Linford's June 2020 "Big Bounce or Double Bang?". $\endgroup$
    – Edouard
    Aug 18 '20 at 5:04

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