Did the Universe Begin? VIII: The No Boundary Proposal

The last bit of evidence from physics which I'll discuss is the "no-boundary" proposal of Jim Hartle and Stephen Hawking (and some related ideas).  The Hartle-Hawking proposal was described in Hawking's well known pop book, A Brief History of Time.  This is an excellent pop description of Science, which also doubles as a somewhat dubious resource for the history of religious cosmology, as for example in this off-handed comment:

[The Ptolemaic Model of Astronomy] was adopted by the Christian church as the picture of the universe that was in accordance with Scripture, for it had the great advantage that it left lots of room outside the sphere of fixed stars for heaven and hell.^{[citation\,needed!]}

Carroll, after making some metaphysical comments about how outdated Aristotelian metaphysics is, and how the only things you really need in a physical model are mathematical consistency and fitting the data—this is Carroll's main point, well worthy of discussion, but not the subject of this post—goes on to comment on the Hartle-Hawking state in this way:

Can I build a model where the universe had a beginning but did not have a cause? The answer is yes. It’s been done. Thirty years ago, very famously, Stephen Hawking and Jim Hartle presented the no-boundary quantum cosmology model. The point about this model is not that it’s the right model, I don’t think that we’re anywhere near the right model yet. The point is that it’s completely self-contained. It is an entire history of the universe that does not rely on anything outside. It just is like that.

Temporarily setting aside Carroll's comment that he doesn't actually think this specific model is true—we'll see some possible reasons for this later—the first thing to clear up about this is that the Hartle-Hawking model doesn't actually have a beginning!  At least, it probably doesn't have a beginning, not in the traditional sense of the word.  To the extent that we can reliably extract predictions from it at all, one typically obtains an eternal universe, something like a de Sitter spacetime.  This is an eternal spacetime which contracts down to a minimum size and then expands: as we've already discussed in the context of the Aguirre-Gratton model.

This is because the Hartle-Hawking idea involves performing a "trick", which is often done in mathematical physics, although in this case the physical meaning is not entirely clear.  The trick is called Wick rotation, and involves going to imaginary values of the time parameter t.  The supposed "beginning of time" actually occurs at values of the time parameter that are imaginary!  If you only think about values of t which are real, most calculations seem to indicate that with high probability you get a universe which is eternal in both directions.

Now why is the Hartle-Hawking model so revolutionary?  In order to make predictions in physics you need to specify two different things: (1) the "initial conditions" for how a particular system (or the universe) starts out at some moment of time, and (2) the "dynamics", i.e. the rule for how the universe changes as time passes.

Most of the time, we try to find beautiful theories concerning (2), but for (1) we often just have to look at the real world.  In cosmology, the effective initial conditions we see are fairly simple but have various features which haven't yet been explained.  What's interesting about the Hartle-Hawking proposal is that is a rather elegant proposal for (1), the actual initial state of a closed universe.

One reason that the Hartle-Hawking proposal is so elegant is that the rule for the initial condition is, in a certain sense, almost the exact same rule as the rule for the dynamics, except that it uses imaginary values of the time t instead of real values.  Thus, in some sense the proposal, if true, unifies the description of (1) and (2).  However, the proposal is far from inevitable, since there is no particularly good reason (*) to think that this special state is the only allowed state of a closed universe in a theory of quantum gravity.  There are lots of others, and if God wanted to create the universe in one of those other states, so far as I can see nothing in that choice would be inconsistent with the dynamical Laws of Nature in (2).

(Hawking has a paragraph in his book asserting that the proposal leaves no room for a Creator, but I'll put my comments on that into a later post!)

In the context of a gravitational theory, imaginary time means that instead of thinking about metrics whose signature is (-, +, +, +), as normal for special or general relativity, we think about "Euclidean" (or "Riemannian") signature metrics whose signature is (+, +, +, +).  So we have a 4 dimensional curved space (no longer spacetime).

The assumption is that time has an imaginary "beginning", in the sense that it is finite when extended into the imaginary time direction.  However, because there is no notion of "past" or "future" when the signature of spacetime, it's arbitrary which point you call the "beginning".  What's more, unlike the case of the Big Bang singularity in real time, there's nothing which blows up to infinity or becomes unsmooth at any of the points.

All possible such metrics are considered, but they are weighted with a probability factor which is calculated using the imaginary time dynamics.  However, there are some rather hand-waving arguments that the most probable Euclidean spacetime looks like a uniform spherical geometry. The spherical geometry is approximately classical, but there are also quantum fluctuations around it.  When you convert it back to real time, a sphere looks like de Sitter space: hence the Hartle-Hawking state predicts that the universe should look have an initial condition that looks roughly like de Sitter space, plus some quantum fluctuations.

I say handwaving, because first of all nobody really knows how to do quantum gravity.  The Hartle-Hawking approach involves writing down what's called a functional integral over the space of all possible metrics for the imaginary-time goemetry.  There are an infinite-dimensional space of these metrics, and in this case nobody knows how to make sense of it.  Even if we did know how to make sense of it, nobody has actually proven that there isn't a classical geometry that isn't even more probable than the sphere.  Worst of all,  it appears that for some of the directions in this infinite dimensional space, the classical geometries are a minimum of the probability density rather than a maximum!  This gives rise to instabilities, which if interpreted naively give you a "probability" distribution which is unnormalizable, meaning that there's no way to get the probabilities to add up to 1.

So Hartle and Hawking do what's called formal calculations, which is when you take a bunch of equations that don't really make sense, manipulate them algebraically as if they did make sense, cross your fingers and hope for the best.  In theoretical physics, sometimes this works surprisingly well, and sometimes you fall flat on your face.

Unfortunately, it appears that the predictions of the Hartle-Hawking state, interpreted in this way, are also wrong when you use the laws of physics in the real universe!  The trouble is that there are two periods of time when the universe looks approximately like a tiny de Sitter space, (a) in the very early universe during inflation, and (b) at very late times, when the acceleration of the universe makes it look like a very big de Sitter space.  Unfortunately, the Hartle-Hawking state seems to predict that the odds the universe should begin in a big de Sitter space is about 10^{120} times greater than the odds that it begins in the little one.  That's a shame because if it began in the little one, you would plausibly get a history of the universe which looks roughly like our own.  Whereas the big one is rather boring: since it has maximum generalized entropy, nothing interesting happens (except for thermal fluctuations).  St. Don Page has a nice article explaining this problem, and suggesting some possible solutions which even he believes are implausible.

Alex Vilenkin has suggested a different "tunnelling" proposal, in which the universe quantum fluctuates out of "nothing" in real time rather than imaginary time.  This proposal doesn't actually explain how to get rid of the initial singularity, and requires at least as much handwaving as the Hartle-Hawking proposal, but it has the advantage that it favors a small de Sitter space over a big one.  From the perspective of agreeing with observation, this proposal seems better.  And it has an actual beginning in real time, something which (despite all the press to the contrary) isn't true for Hartle-Hawking.

(*) There is however at least one bad reason to think this, based on a naive interpretation of the putative "Holographic Principle" of quantum gravity, in which the information in the universe is stored on the boundary.  A closed universe has no boundary, and therefore one might think it has no information, meaning that it has only one allowed state!  (The argument here is similar to the one saying the energy is zero.)  At one time I took this idea seriously, but I now believe that such a strong version of the Holographic Principle has to be wrong.   There are lots of other contexts where this "naive" version of the Holographic Principle gets the wrong answer for the information content of regions, and actual calculations of the information content of de Sitter-like spacetimes give a nonzero answer.  So I'm pretty sure this isn't actually true.

About Aron Wall

I am a postdoctoral researcher studying quantum gravity and black hole thermodynamics at the Institute for Advanced Study in Princeton. Before that, I read Great Books at St. John's College (Santa Fe), got my physics Ph.D. from U Maryland, and did my first postdoc at UC Santa Barbara.
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19 Responses to Did the Universe Begin? VIII: The No Boundary Proposal

  1. LaplaceDemon says:

    Hi Dr. Wall,
    From what I've read elsewhere ,the Hartle hawking model does involve the beginning of the universe, but has the origin of the universe without a singular beginning point (as it does in other models).

    Also ,Dr. Craig would say the imaginary time is just a computational device and in real time (in the actual world) , it would imply a beginning to the universe.
    For example see this section on quantum cosmology in the paper here. Deltete and Guy have an interesting paper on how to interpret imaginary time models as well.

  2. James says:

    You say that "the Hartle-Hawking model doesn't actually have a beginning!" Perhaps you mean that the Hartle-Hawking model doesn't actually have a beginning point or initial singularity? Is the four-dimensional "rounded off" state not the beginning? Indeed, Hawking and Mlodinow themselves use the term "beginning" when describing the no-boundary condition as similar to the South Pole:

    "Suppose the beginning of the universe was like the South Pole of the earth, with degrees of latitude playing the role of time. As one moves north, the circles of constant latitude, representing the size of the universe, would expand. The universe would start as a point at the South Pole, but the South Pole is much like any other point. ... In this view, the universe appeared spontaneously, starting off in every possible way" (Stephen Hawking and Leonard Mlodinow, The Grand Design [London: Bantam Books, 2010], 172-174).

    Here Hawking and Mlodinow's description implies that such a "no-boundary" universe has a start or beginning (perhaps by tunnelling into existence). This is how John Barrow interprets the no-boundary model:

    "This type of quantum universe has not always existed; it comes into being just as the classical cosmologies could, but it does not start at a Big Bang where physical quantities are infinite and where further initial conditions need to be specified." (John D. Barrow, New Theories of Everything: The Quest for Ultimate Explanation, 2nd ed. [Oxford: Oxford University Press, 2007], 91).

    Furthermore, although Hartle and Hawking convert time into space for the early universe, some sort of time would still exist because the universe is described as constantly changing and evolving into a larger and larger universe. But such a pre-Big Bang quantum gravity region cannot be eternal - can it? - because such a state is unstable and so quantum fluctuations will cause it to expand or contract from eternity. So I am not sure if a beginning is absent in the Hartle-Hawking model.

  3. Aron Wall says:

    Dear LaplaceDemon and James,

    Yes, you can find lots of other people saying that the Hartle-Hawking state has a beginning. That's why I put that statement in bold: to clarify a misconception propagated by many other scientists about this subject. The Hawking and Mlodinow quote is talking about what happens if you go off in the imaginary time direction. Not what happens if you go in the real time direction! The universe still exists at arbitrarily large negative values of t (with high probability), it just doesn't exist for arbitrarily large imaginary values of t. (Here I'm being sloppy since I haven't said what my definition of the t coordinate is, but I hope it gives a rough idea. The correct statement would refer to "compact" and "noncompact" geometries.) There could potentially still be singularities in some regions of real space, it's just that things cap off smoothly in the imaginary time direction.

    I think part of the confusion here is that people sometimes talk as if imaginary time were an earlier phase in the history of the universe. So they seem to be telling this just-so story: first time was imaginary, and then it became real at a later phase in history (by means of some unknown change-over process which is never discussed). That is NOT the right way to think about it.

    Instead you should think of time as being 2 dimensional. It is now complex rather than real. There is a real axis and an imaginary axis. Imaginary time does not come "before" real time, instead they are at 90 degrees to each other. There is no magical point in time on the real axis for which the HH tells you that real time takes over. Rather, the HH state defines a consistent history of the entire evolution of the universe through real time. You can take any path through the complex plane that you like, none of them is better than the others.

    HH is just as much a prescription of the "current" state of the universe as it is a prescription of the "early" state of the universe. There is no distinction, because in a gravitational theory like GR different times are really just different ways of representing the same state. That's what the Wheeler-DeWitt equation H \Psi = 0 tells us. HH give us a prescription for constructing a particular state obeying the Wheeler-DeWiit equation: that state in turn gives us the entire history of the universe.

    Most physicists regard the use of imaginary time purely as a "computational device" as Craig says, so that imaginary time isn't, well, real, and the only thing that really exists is real time. If we accept that point of view, then the HH state is simply one particular state which happens to be (probably) eternal with respect to real time.

    On the other hand, if it turned out that it was a Law of Nature (with capital letters) that the universe is in the Hartle-Hawking state, and if the only convenient way to express this Law is using the device of imaginary time, then maybe that would be evidence that imaginary time should be regarded as ontologically real, and not just a calculational device.

    I doubt that this reply has cleared everything up: the fact that I had to go on at such length indicates it probably won't do so. But feel free to ask follow-up questions. I wouldn't want to create more misconceptions in the process of dispelling others.

  4. James says:

    Aron,

    You say that "The Hawking and Mlodinow quote is talking about what happens if you go off in the imaginary time direction. Not what happens if you go in the real time direction!" That is exactly the point. Hawking's description of the universe in Euclidean time (or imaginary time) implies that the universe has a beginning at the "South Pole".

    But, similarly, as I understand the no-boundary model, real time is finite (and not eternal). It seems, then, irrelevant whether or not we think of time as two-dimensional, since both time dimensions are finite and have a beginning. Perhaps this is why Hartle and Hawking claim that "one can interpret the functional integral over all compact four-geometries bounded by a given three-geometry as giving the amplitude for that three-geometry to arise from a zero three-geometry, i.e., a single point. In other words, the ground state is the amplitude for the Universe to appear from nothing" (Hartle and Hawking, "Wave function of the Universe," Physical Review D 28, no. 12 [1983], 2961).

    So, unless the so-called B-theory of time is true, I'm not sure how the universe can be eternal if time is not eternal.

    You also claim that if imaginary time is not real (doesn't really exist) "then the HH state is simply one particular state which happens to be (probably) eternal with respect to real time." But how, exactly, does that follow? If imaginary time is simply imaginary and does not reflect reality, then the HH no-boundary proposal is imaginary and does not reflect reality. We can thus only describe the universe literally in terms of real time. And Hawking has made clear that in real time, the universe has a beginning and an end at singularities.

    Anyway, the Hartle-Hawking model is a fascinating idea. It would be great if you could write up a post where you give your own criticism of the model.

  5. Aron Wall says:

    James,

    But, similarly, as I understand the no-boundary model, real time is finite (and not eternal).

    When you say, "as I understand", do you mean that you gathered this from the quotes of other scientists, more famous than me, who have popularized this model? Well, they were lying to you, and I'm telling you the truth! (OK, they weren't really lying, but rather oversimplifying. I'm sure their motives were as pure as the driven snow, but the end result in their readers' minds is the same.)

    Would it help if I said that Jim Hartle, who goes to UCSB, is a personal friend of mine, and that I've seen him give seminars about the HH proposal in which it was quite clear that he believes the states predicted are eternal with respect to real time, and have a "bounce"?

    To be fair, the details of the HH predictions depend on the precise laws of physics, and also make assumptions about how to solve the mathematical equations which might (for the various reasons I mentioned in the post) be wrong. So it's always possible that the actual predictions of the HH proposal will turn out to be different from what people expect. I'm not claiming infallibility here, but what I am claiming is a knowledge, via professional expertise and social networking (not to mention working through the physics myself) that it is generally expected that the HH proposal gives a state somewhat like de Sitter spacetime. And it is a fact that de Sitter (in real time, not imaginary time) is eternal and has a bounce in it.

    Perhaps you are thinking of the following statement in Hawking's Brief History:

    In real time, the universe has a beginning and and end at singularities that form a boundary to spacetime and at which the laws of science break down. But in imaginary time, there are no singularities or boundaries.

    But this statement cannot mean what it appears to at first sight, because earlier in the same chapter Hawking indicates that we don't know whether or not the whole universe will end in a Big Cruch singularity. He must just mean that HH is consistent with the existence of singularities (including those in black holes), not that they necessarily exist in HH with high probability.

    Hawking's statement which is so close to being true that I can see why he said it even though it is misleading. The second sentence indicates the context of his statement: it is a contrastive one. He is comparing real time to imaginary time, in which there are no singularities at all, and then indicating that his proposal does not remove singularities from the real spacetime (if they happen to exist in the spacetime produced, is the unspoken condition).

    Anyway, the Hartle-Hawking model is a fascinating idea. It would be great if you could write up a post where you give your own criticism of the model.

    Hmm, I thought I did just write a post (the one you are replying to) in which I indicated some severe problems facing the proposal. To my mind there's nothing left to say about that, except that the HH proposal is a beautiful idea, so that it's a shame that it doesn't seem to work (as normally implemented).

  6. James says:

    Aron,

    When you say, "as I understand", do you mean that you gathered this from the quotes of other scientists, more famous than me, who have popularized this model?

    Yes. More precisely, I am using their descriptions of this model, especially from Hawkings, to conceptually visualise such a universe as best I can. Perhaps it is too easy to misinterpret these descriptions, or perhaps many of these descriptions are misleading (which will be disappointing). By the way, I'm not sure what fame has to do with it.

    Would it help if I said that Jim Hartle, who goes to UCSB, is a personal friend of mine, and that I've seen him give seminars about the HH proposal in which it was quite clear that he believes the states predicted are eternal with respect to real time, and have a "bounce"?

    No, not really. I guess it is incumbent on me to try to fully understand the mathematics behind the model - this will help!

    Hawking's statement which is so close to being true that I can see why he said it even though it is misleading.

    It is unfortunate that some quotes are misleading.

    Hmm, I thought I did just write a post (the one you are replying to) in which I indicated some severe problems facing the proposal.

    Fair enough. I misunderstood your post to be a brief reflection on the model (perhaps because of your statement: "Hawking has a paragraph in his book asserting that the proposal leaves no room for a Creator, but I'll put my comments on that into a later post!", a post which I am looking forward to).

  7. David says:

    I love Vilenkin's book and having that to fall back on, a lot of what you've said I think I actually get (esp. About smaller and larger de sitter space!). Although he definitely gives the impression that he believes his model has a beginning. I'm definitely going to recommend this blog to people. Thanks!

  8. Jack Spell says:

    Aron and James,

    Correct me if I'm wrong, but, it seems to me that the disagreement between the two of you on the eternality of the HH model lies in the theories of time to which one chooses to subscribe: Aron has stated that he is a B-Theorist; I am assuming that you, James, like myself are also an A-Theorist. So when Aron says,

    Would it help if I said that Jim Hartle, who goes to UCSB, is a personal friend of mine, and that I've seen him give seminars about the HH proposal in which it was quite clear that he believes the states predicted are eternal with respect to real time, and have a "bounce"?

    I take it that Hartle believes this because he subscribes to a B-theory of time in which the entire 4-dimensional spatiotemporal block just exists tenselessly. Thus, it can be said that this block just exists eternally.

    However, having a beginning does not necessitate having a beginning point. Time begins to exist just in case for any finite temporal interval, there are only a finite number of equal temporal intervals earlier than it. That condition is fulfilled for the HH model.

  9. Aron Wall says:

    Jack,

    My point (and Hartle's) wasn't based on A-theory or B-theory. (Most physicists probably aren't even aware of that controversy.) He was talking about models which contract and then have a bounce and then expand again. (If we define "eternal" so that all spacetimes are eternal, that would be a very boring use of the word; there would be no point in saying it except when arguing with an A-theorist.)

    However, having a beginning does not necessitate having a beginning point. Time begins to exist just in case for any finite temporal interval, there are only a finite number of equal temporal intervals earlier than it. That condition is fulfilled for the HH model.

    No, that condition is not fulfilled for the HH model. Viewed with respect to real time, it goes back to t = -\infty (with high probability). That's what I'm saying.

  10. Jack Spell says:

    Aron,

    When you say,

    No, that condition is not fulfilled for the HH model. Viewed with respect to real time, it goes back to t = -\infty (with high probability). That's what I'm saying.

    are you saying that real time in the HH is sempiternal?

  11. Jack Spell says:

    Well, it's hard to argue about this with a guy who knows Hartle personally; if he says it, I can't see how is not true. And if so, then you are completely right about this misunderstanding being widespread; tons of the literature that I've read interprets the HH model as being finite in its duration.

  12. Aron Wall says:

    I found the slides for Jim's talk about bouncing universes here.

    [Update: The audio file is also available now.]

  13. Matthew G says:

    No, that condition is not fulfilled for the HH model. Viewed with respect to real time, it goes back to t=−∞ (with high probability). That's what I'm saying.

    This is very interesting. Can you give a source that confirms this explicitly? I have tried to look for this myself, but I'm afraid the information might be buried beneath too much simplification and omission.

  14. Aron Wall says:

    It looks like this paper has some discussion of the point:
    J. Hartle, T. Hertog, "Arrows of Time in the Bouncing Universes of the No-boundary Quantum State". (see also Ref. [13-16])

    It looks like Hawking and collaborators originally considered the probability predictions for a class of nonbouncing cosmologies, and it was later suggested that the bounces are dominant. That may account for some of the confusion.

    Another reference is here (see text just below Eq. 10)
    A. Vilenkin, "The quantum cosmology debate".

    When people say that Hartle-Hawking predicts an inflating de Sitter universe, that almost gets you there since the complete de Sitter spacetime has a bounce. To be fair, there's some uncertainty in how to interpret and calculate results in the HH proposal, so tomorrow somebody might come along with an argument that HH predicts some other type of cosmology. But in any case, the basic conceptual point would still be true: that finiteness in imaginary time is not the same concept as finiteness in real time.

  15. kaveh says:

    HI dear Aron. some comments .first of all to say that wick rotation is just a computational tool is not a prior correct . It is absolutely likely that imaginary time exists and even be more real that real time at very high energies and curvatures ( planck scale / also this point of view depends on someone view about nature of mathematical structures and objects , if someone like me advocate a pythagorean or platonism view about mathematics so imaginary time or transforming of time direction to spatial direction is not just a trick but something that happens in reality / also often it is said that universe in this model emerges from nothing ! it is not correct , indeed HH proposal tries to give a proper answer to Leibniz famous question why is there anything at all ? HH proposal proves that this question if has a meaningful answer at all , it is doubtless an acausal,a-temporal and a-spatial explanation . and the most primitive era of HH proposal is not nothing rather actually a real something but a peculiar one : a timeless , causation-less ,static. it should be emphasized also to say that HH proposal has no notion of beginning or has notion of a beginning is wrong. both of this words are temporal and so inappropriate in this proposal, it is more better to say emerging of a Lorentzian spacetime from a 4D Euclidean space is a phase transition that we can pose some initial conditions on it exactly at transition point , namely point of silence/ see article asymptotic silence in quantum gravity theories by Jakub Mielczarek ) . Indeed I don't see the term real or imaginary very helpful and it leads to some confusions even among experts . When Hawking says real time replaces with imaginary time ,Hawking means any notion of time simply doesn't exist , instead the notion of time ( real time ) is just exactly a 4 dimension or direction of space . in other word concept of spacetime looses its meaning altogether and we now encounter with just a quantum version of 4D Euclidean space ( Euclidean space is absolutely timeless,causation-less,spaceless ,devoids of any notion of evolution,change ,dynamics and process and utterly static / in 4D Euclidean space there is just 4 spatial directions and so points don't stand in any priority or posteriority relations to each other ( that is very definition of causality in any form ) . Hartle-Hawking no-boundary proposal is truly a very radical and fundamental proposal in all of history and if it would be proved in future , it will have very profound and striking implications for both physical and philosophical (metaphysical ) cruxes . Also recently HH proposal has gotten very strong supports from loop quantum cosmology , string theory ,causal dynamical triangulation and even Horava-Lifshitz Gravity theories . last words : HH proposal simply tries to say many of our fundamental and necessary notions no longer applicable to universe ( at least at the most fundamental ontological level ) . for example as I mentioned above words such as causality ,time ,evolution ,dynamics ,change , motion are totally meaningless concepts at so-called planck scale .

    ALL THE BESTS ,
    KAVEH

  16. Matthew G says:

    It is absolutely likely that imaginary time exists and even be more real that real time at very high energies and curvatures

    And what reasons are there to think that expanding the real numbers into the complex numbers not only does not go too far, but also far enough? Why not go further and replace the complex numbers with the Wiener algebra? It seems just as arbitrary to me.

  17. charlie says:

    So is what you say of the Hartle-Hawking model (it being eternal) true of Linde and Vilenkin's?

  18. Aron Wall says:

    Vilenkin's model has a beginning of time, yes. (Assuming it is well defined at all). His model assumes that the laws of physics allow the universe to begin in real time, and is defined using a path integral over all such spacetimes. If you look at the wavefunction which he draws for his model in Fig. 1 of his quantum cosmology debate article which I linked to above, you'll see that his wavefunction has only an outgoing expanding piece (his arrow pointing right), with no contracting phase.

    In the case of Linde, it's harder to judge since his model is afflicted with terrible instabilities, even worse than Hartle-Hawking, and it's not really clear that there is a well-defined answer at all. But in the simplified model considered by Vilenkin where only the overall size (the "scale factor") of the universe is allowed to vary, it is well-defined, and it seems to me that it predicts a universe which might or might not have a beginning (some significant probability for each).

    Whereas in the case of the Hartle-Hawking model, the probability that it has a beginning is extremely tiny (approximately 10^{-10^{120}}). Assuming that real beginnings are allowed at all, which is a critical assumption of Vilenkin's model but not of the other two models.

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