Is the Universe Past-Eternal? 
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*Does the Borde-Guth-Vilenkin theorem definitively demonstrate that the Universe cannot be past-eternal, whatsoever? 
Does it not assume a classical space-time while the real world requires Quantum mechanics? 

*Are there successful models in Cosmology that are Past-Eternal? 

*Why might Alan Guth say the Universe might be eternal in the past, when he himself wrote a theorem in 2003 saying it most definitely isn't?
 A: The simplest way around the usual interpretation of the BGV Theorem is described in Aguirre and Gratton's 2002 "Steady state eternal inflation":  It requires dual arrows of time, pointing in opposite directions in de Sitter spacetime, which would both be of the usual thermodynamic variety. Vilenkin, in a 2013 critique titled "Arrows of time and the beginning of the universe" that's also available free on the web, found it plausible, even while claiming that it didn't invalidate the BGV theorem, which would apply equally to either half of the spacetime concerned.  De Sitter spacetime requires a contracting phase preceding the expanding one that's usually the concern of inflationary cosmologists, and, as Vilenkin points out, the backward or past-directed arrow would apply to the contracting one, whose remnant (if any) would be rather hard to locate:  Presumably the two together would have a net expansion of zero, which is another way it could stay in line with the BGV theorem, that only requires a beginning for universes with a net expansion greater than zero.  In the last end-note in the last (2003's) revision of their theorem, BGV accepted the Aguirre & Gratton model as consistent with it.
In his profoundly Christian blog, Aron Wall points out the fact that, since neurological processes are also thermodynamic, anyone in the backwards-pointing arrow might perceive time very much as we do.
In his 2013 "Arrows of time" paper, Vilenkin remarks, "Even though the spacetime has no boundary in the AG model, it does include a surface B on which the low-entropy (vacuum) boundary condition must be enforced by some unknown mechanism. This Cauchy surface of minimum entropy plays the role of the beginning of the universe in this scenario."  Rudiger Vaas has (in 2012's book titled "The Arrows of Time") pointed out the fact that, on the microscale, time is symmetric because of the relation between the uncertainty of a particle's location and that particle's actual location:  Although this uncertainty may very often result from the effect of the particle(s) employed for observation on the particle whose observation's being either attempted or accomplished, it would appear to be random, and my impression is, consequently, that the "unknown mechanism" mentioned by Vilenkin in 2013 might be randomness itself, with its macroscale result being, on each side of the Aguirre & Gratton model, a directionality of passage thru time that would be observationally asymmetric. The AG cosmology might, consequently, result from our laws of physics, in which time is symmetric for reasons necessarily hypothetical (and, often, practically disfunctional), even though its disfunctionality might be a practical result of its completion (rather than representing a flaw, as the term usually implies).
A more recent proposal, involving dual arrows of time that are gravitational rather than thermodynamic, was worked out by Barbour, and is sketched in several magazine articles:  Their descriptions of it can be found with his name and the term "Janus point", which is the center of time in his scheme.
Linde is well known to have disagreed with BGV, and it's not clear to me how it could apply to his "chaotic inflationary" universe, which nevertheless seems to involve such a range of scales in its regions that it is not renormalizable and cannot, in consequence, easily be compared with the "false vacuum" varieties of inflationary cosmology, in assessing its validity.
An even more recent and detailed cosmology that appears to provide for past eternality of a spatial and temporal multiverse is Nikodem Poplawski's "Cosmology with torsion", based on 1929's relativistic Einstein-Cartan Theory rather than 1915's General Relativity, and described, as of 2021, in numerous papers (including a recent one published by the highly reputable journal "Physics Letters B") whose preprints can be freely found by his name on Cornell University's Arxiv website.  In his model, large rotating stars that have expended enough of their nuclear fuel to provide radiation pressure inadequate to resist their own weight collapse gravitationally, thereby inducing a causal separation marked by an event horizon, propagating rapidly outward from the star's center, which separates the fermions of virtual pairs from their antiparticles over space and time sufficient to result in their materialization from the gravitational field.  The interaction of the newly-materialized fermions with the older (and vastly larger) fermions of the star itself reverses and greatly accelerates the trajectories of the new ones sufficiently to form a new "local universe", whose shape Poplawski analogizes to the skin of a basketball, and whose compatibility with the CMB data has been verified by Desai in a 2015 paper titled "Non-parametric reconstruction of an inflaton potential".  As discussed there, each local universe's formation may occur either in a single bounce or in a series of bounces:  From the exterior, it would appear as a black hole, and its interior's potential for the formation of new stars, and for some of them to collapse for the same basic reasons, would provide it with a form of future eternality through the same processes that would've caused its own formation, in a past equally eternal.
As causal separations are, given the statistical increase in entropy (disorder) toward both the past and the future within the currently-observable portion of our region, a factor that might leave a spatio-temporal multiverse that's eternal to the past with some predictability (i.e., without infinite entropy) would be permanent causal separations between regions each previously occupied entirely either by a relatively large star of the usual rotating variety, or by a sequence of them on decreasing scales of space and time:  With temporary causal separations, a beginning of time might be required.
Although Aguirre and Gratton's cosmology would accomplish such permanent causal separations as well as Poplawski's, I'd favor the latter's, because there are indirect signs that it's operative, particularly in the similarity between the observed "surface of last scattering" and the hypothesized inboard side of a black hole's gravitational horizon.
By representing what's been considered "the" big bang as the one on the local scale of an infinity of them, either Poplawski's theory or AG's would compare to the unmodified big bang theory as having a hypothetical explanatory power of doubled infinity, but Poplawski's would extend that increase to an asymptotically exponential extent, as the doubling factor would become an exponent.
Attacks on past-eternality often rely on geodesics that are "incomplete to the past", so it may be important to understand that Einstein-Cartan theory generally uses versions of "parallel transport" instead of geodesics:  They're described a lot on PSE, recently at Geodesic equation in Einstein-Cartan manifold and its links (and their links)! Deviations from geodesics, in his cosmology, have more recently been discussed by Poplawski at https://arxiv.org/abs/1912.02173, and, in fact, linked to the Machian inspiration of General Relativity.
It seems possible that Poplawski was influenced by remarks made by a Cambridge physicist, the late John Barrow, in a 2005 lecture visible at https://www.gresham.ac.uk/lecture/transcript/print/infinity-and-beyond/ , to the effect that that, "in...relativity, a new force of nature arises, a spin-spin repulsive force between spinning objects".
Poplawski's model also appears to match a prescription by Luke A. Barnes (a postdoctoral researcher in astronomy and physics at the University of Western Australia, writing in Oxford's 2017 book titled "The Philosophy of Cosmology"), suggesting that "infinite multiverse modelers could try to manufacture a limiting process--perhaps a sequence of spacetime volumes", in attempts to resolve the so-called "Boltzmann Brains" problem:  The sequential reductions in the scale of the "local universes" suggested by Poplawski's torsion-based cosmology might bring a quantum fluctuation in one iteration up to the size of that electromagnetic constellation which actually comprises a "mind" (or "brain") in a subsequent and scaled-down one.
Some of the well-known conflicts between theology and physics over past eternality may arise from the fact that combinations of artifice and naturalism would be much likelier (if not certain) to have occurred in a past-eternal cosmos.  For instance, the addition of mass to any large star nearing the time of its collapse to a neutron star might provoke its collapse to a black hole instead:  As black holes radiate very little, such an addition of mass (which might, in extreme cases, comprise only a small amount) could be motivated by a desire to minimize exposure to unhealthy radiation on the part of colonists of a civilization not extraordinarily more advanced than our own, or it might involve a desire to provide for future replications of their own existence (at intervals perhaps random, and on scales probably smaller), by an entire civilization much further advanced, and prone to utlilize the possibility of such replications to impose a morality of "getting it right the first time" (and, consequently, "for all of future time").
Although Poplawski 1st described his model (in his 2010 "Cosmology with torsion") as "an alternative to cosmic inflation", it's generally described as a version or adjunct of inflation, with the bounce effects I've described being substituted for action of an inflaton field (which, like all scalar fields but unlike stars, would not rotate).
To see further discussions of past- and future-eternal cosmologies, as well as possibilities for observational proofs of them in Skydivephil's interviews with Smolin, Poplawski, and other major physicists, go to youtube.com/watch?v=xXL0N3elFLE , which is currently followed by a PBS documentary on the same topic.
As discussed by Daniel Lindford of Rutgers University in a preprint that can be seen at https://arxiv.org/abs/2006.07748, the cosmological models formulated by Poplawski and by Aguirre & Gratton can be grouped with Roger Penrose's "conformal cyclic cosmology" (described in his book titled "Cycles in Time") as basically relativistic models that are all both past- and future-eternal:  Penrose's 2020 receipt of a Nobel Prize in physics practically requires a sketch of his model.  Relying on the Weyl Curvature Hypothesis that preserves angles during changes in scale, it maintains passage thru time in only a single direction, with the thermal equilibrium of each one of its infinity of temporal iterations (which Penrose refers to as "aeons") reduced in scale to serve as the "big bang" of the next, after evaporation of all massive objects into radiation that's ultimately of the "Hawking" variety, subsequent to their ingestion by expansion of all types of black holes.  (This may be an opportune moment to mention that the "arrow of time" in Poplawski's model is also uni-directional, being inherited by each local universe from its "parenting" star:  However, the "aeons", or sections, of Penrose's model are separated from each other by spaces, representing the conformal "crossover" between each aeon and the preceding or succeeding one, that are literally timeless, as the evaporation of massive objects has left them without any material which might form the components of any type of artificial or natural "clock".)
As discussed by him at https://doi.org/10.1007/s10701-018-0162-3, Penrose accepts the conventional view of time (within each aeon) as based on an entropy which increases over time regardless of the direction (toward either the past or the future) of passage through it, but makes an important distinction between the entropy of the gravitational field itself and entropy generally, inasmuch as he associates the entropy of that field with concentrations of mass / energy rather than equilibrations of its distribution.
The factor which may have tipped the balance in favor of Penrose's receipt of the Nobel Prize may have been "numerous anomalous spots of significantly higher temperature" in the Cosmic Microwave radiation:  Lacking any other explanation, these spots have been detected by a couple of different astronomical surveys, and would correspond to the Hawking radiation emitted by supermassive black holes of a previous aeon, as detailed in Penrose's March 2020 collaboration with Meissner and others, whose last preprint is freely visible at https://arxiv.org/abs/1808.01740  .  (This evidence may be stronger than that available for Poplawski's comparably past- and future-eternal model, whose Hawking radiation would propagate inward to smaller and newer local universes:  His model would be falsifiable if a preferred direction of motion, derived from the rotation of the "parenting" stars, would not be evident in each of those localities, and evidence for and against such a possibility seems to vary between every study claiming to reveal it and the next, perhaps because of variations in the methods used.  The most recent I've seen, released in 2021 by the astronomer Lior Shamir, appears to favor Poplawski's rotating model over the scalar field alternative, although Shamir's research appears to have been done without any particular cosmological model in mind.)
A: First, no theorem can definitively demonstrate anything about the real world. They can only show that their conclusions follow from their assumptions.
Second, even when the assumptions of this particular theorem are met, the conclusion seems quite modest:

[A] cosmological model which is inflating – or just expanding sufficiently fast – must be incomplete in null and timelike past directions. [...] Thus inflationary models require physics other than inflation to describe the past boundary of the inflating region of spacetime.

The conclusion is just that there has to be something else preceding inflation. The universe can't be inflationary "all the way down", but it can be an inflationary era resting on an infinite stack of turtles, as long as the turtles don't meet the expansion criterion. The theorem says nothing about a beginning of time.
I wouldn't be surprised if the theorem turns out to be irrelevant to quantum gravity. Classically, black holes can only expand and white holes can only contract, but quantum holes seem to forget whether they're expanding or contracting, and can do both. If something similar happens to the horizons in inflationary spacetimes, then the expansion criterion of the theorem could be meaningless.

2. Are there successful models in Cosmology that are Past-Eternal?

I don't think there are any successful cosmological models other than ΛCDM. Ideas about pre-ΛCDM cosmology are vague on the details and aren't clearly supported by astronomical data. Some are past-eternal and some not.

3. Why might Alan Guth say the Universe might be eternal in the past, when he himself wrote a theorem in 2003 saying it most definitely isn't?

As far as I can tell, this theorem in no way suggests that the universe isn't past-eternal. It only rules out some kinds of eternal inflation.
