Does Modern Cosmology Prove the Universe Had a Beginning?
So why is it often repeated by some science popularizers that modern cosmology indicates space-time had an absolute beginning?
The reason is that, in the 60s, physicist Stephen Hawking and mathematician Roger Penrose constructed theorems (e.g., Penrose, 1969) which "showed that general relativity predicted singularities whenever more than a certain amount of mass was present in a region", such as inside a black hole (however, R. P. Kerr has recently called this into question; see the explanation here). Other theorems (e.g., Hawking & Penrose, 1970) were then "designed to show that time came to an end inside a black hole formed by the collapse of a star. However, the expansion of the universe is like the time reverse of the collapse of a star." Therefore, "if one can determine that there is enough matter in the universe... one can then apply the singularity theorems to show that time must have a beginning." (Hawking, 1996) The reason why spacetime begins (or ends) at singularities, according to cosmologist John Barrow, is that "if a physical infinity appears in a theory like Einstein's, where the fabric of space and time is determined by the physical density of matter within it, then it requires that space and time are destroyed at the places where infinite densities appear." (Barrow, 2005: 103)
The problem is that General Relativity describes the universe very well only at large scales, but does not work at the sub-atomic scale – viz., a region of space less than 1.616 × 10⁻³⁵ meter in diameter, called the Planck length – where gravity becomes insignificant, and quantum mechanics has the upper hand (Stenger, 2013: Ch. 12). As Robert Wald put it: “The Planck length arises naturally in attempts to formulate a quantum theory of gravity. Thus, dimensional arguments suggest that a classical description of spacetime structure should break down at scales of the order of the Planck length and smaller.” (Wald, 1984: 378) Over twenty years ago, Hawking and Penrose agreed that a singularity did not, in fact, occur, (Penrose, 1996; Hawking, 2010: 108, 110) and that is now the consensus of the working cosmological community. As Barrow explained: "From the mid-1960s until about 1978 these mathematical theorems were widely cited as evidence that the Universe had a beginning", however "the old conclusions of the singularity theorems are no longer regarded by cosmologists as likely to be of relevance to our Universe" because "crucial assumptions in those theorems... are no longer likely to be true." (Barrow, 2002: 334-336) Hawking and Penrose's original calculation was not wrong as far as it followed from the assumptions of general relativity. But those assumptions had not taken into account quantum mechanics. (Stenger, 2011: Ch. 6.2) In the words of the mathematician Roy Kerr: "When Relativity and Quantum Mechanics are melded it will be shown that there are no singularities anywhere. When theory predicts singularities, the theory is wrong!" (Kerr, 2023: 15)
Quantum mechanics eliminates singularities for two reasons. First, quantum mechanics implies that at the Compton Radius, matter no longer has any definite location, but exists at any random location within that radius at any given time, as defined by the appropriate wave function. And if there is no fixed location but instead a cloud of probability-space, there can never be a singularity, which presupposes a single fixed location. Second, quantum mechanics entails that all fields must be quantized, viz., not infinitely divisible. Gravity is a field (i.e., gravitational field), therefore it must be quantized (see, e.g., Kiosses, 2018). But if gravity is quantized, then there must be a smallest space over which gravity can act, such that when you get smaller than a single graviton the force of gravity no longer functions, which means the gravitational force will stop operating before a singularity is produced (Carrier, 2004). In other words, due to quantum gravity effects, there is a minimum distance beyond which the force of gravity no longer continues to increase as the distance between the masses becomes shorter. (Fringe/pseudoscientific assertions challenging the quantum nature of gravity are in opposition to "over a century of scientific consensus," face skepticism from leading physicists who deem them "implausible" because "trading quantumness for stochasticity has conceptual difficulties" such as the loss of quantum information – which violates Wigner's theorem of unitary evolution –, and are invalidated by recent theorems and experiments.)
Several articles and books by prominent cosmologists are quoted below to support the claims being made here.
In the book Endless Universe (pp. 22, 93), Paul J. Steinhardt and Neil Turok confirm the problem with the singularity: "By the 1960s and ’70s, Stephen Hawking and Roger Penrose had shown that.. a shrinking universe... continues to shrink until space collapses to a state of infinite density known as the cosmic singularity. Similarly, 14 billion years ago, the universe must have emerged from such a cosmic singularity. ... [However], mathematicians use the term “singularity” to indicate that equations are failing. The big bang is referred to as the initial singularity because Einstein’s equations of general relativity break down when the temperature and energy density become infinite, as Einstein himself recognized, and their description of the expansion of the universe ceases to be valid. ... Hawking and Penrose’s finding was widely interpreted as theoretical proof that space and time must have a “beginning.” That interpretation, however, was never justified. What they really proved is that Einstein’s equations become mathematically inconsistent at the big bang itself. This should be interpreted as a clarion call. Physicists must face up to improving Einstein’s theory, if they want to describe the big bang."
In his other book From Quantum to Cosmos (pp. 122-144), Dr. Turok added: "Einstein’s reason for disliking Friedmann’s evolving universe solutions was that they all had singularities. Tracing an expanding universe backward in time, or a collapsing universe forward in time, you would typically find that at some moment all of space would shrink to a point and its matter density would become infinite. All the laws of physics would fail at such an event, which we call a "cosmic singularity." ... Cosmology in the twentieth century was, by and large, based on ignoring the big bang singularity. Yet the singularity represents a serious flaw in the theory."
Cosmologist Sean Carroll agrees with the standard conclusion that our current picture of cosmology is incomplete. In the article What if Time Really Exists?, Dr. Carroll stated: "Assuming the validity of the Schrödinger equation has a deep, if somewhat obvious, consequence: time stretches for all of eternity. In classical mechanics, singularities in phase space can disrupt the evolution, causing time to grind to a halt. But in quantum mechanics, unitary evolution ensures that there is no boundary to time; the variable t runs from −∞ to +∞. The modern idea that time does have a beginning arises from the existence of a Big Bang singularity in cosmological models based on general relativity. But from our current perspective, that is an outmoded relic of our stubborn insistence to think in terms of space-time, rather than directly in terms of the quantum state. Classical general relativity, after all, is not correct; at some point it must be subsumed into a quantum description of gravity. We therefore imagine that the classical Big Bang corresponds to some particular kind of quantum state, which may be obscure from the perspective of our current knowledge, but will ultimately be resolved. It follows, under our assumptions, that there was something before the Big Bang, and time stretches back into the infinite past."
In the (2016) book The Big Picture (pp. 41-42), Dr. Carroll confirmed this: "In the 1920s, Edwin Hubble discovered that our universe is indeed expanding. Given that discovery, we can use our theoretical understanding to extrapolate backward in time. According to general relativity, if we keep running the movie of the early universe backward, we come to a singularity at which the density and expansion rate approach infinity... The Big Bang itself, as predicted by general relativity, is a moment in time, not a location in space. It would not be an explosion of matter into an empty, preexisting void; it would be the beginning of the entire universe, with matter smoothly distributed all throughout space, all at once. It would be the moment prior to which there were no moments: no space, no time. It’s also, most likely, not real. The Big Bang is a prediction of general relativity, but singularities where the density is infinitely big are exactly where we expect general relativity to break down – they are outside the theory’s domain of applicability. At the very least, quantum mechanics should become crucially important under such conditions, and general relativity is a purely classical theory. So the Big Bang doesn’t actually mark the beginning of our universe; it marks the end of our theoretical understanding. We have a very good idea, on the basis of observational data, what happened soon after the Bang. The microwave background radiation tells us to a very high degree of precision what things were like a few hundred thousand years afterward, and the abundance of light elements tells us what the universe was doing when it was a nuclear fusion reactor, just a few minutes afterward. But the Bang itself is a mystery. We shouldn’t think of it as “the singularity at the beginning of time”; it’s a label for a moment in time that we currently don’t understand."
Dr. Carroll repeated this conclusion in his more recent book (2019) titled Something Deeply Hidden (p.256): "If we take the Schrödinger equation at face value, time seems to be right there in a fundamental way. Indeed, it immediately follows that the universe lasts eternally toward both the past and future, for almost all quantum states. You might think that this conflicts with the oft-repeated fact that the Big Bang was the beginning of our universe, but we don’t actually know that oft-repeated fact to be true. That’s a prediction of classical general relativity, not of quantum gravity. If quantum gravity operates according to some version of the Schrödinger equation, then for almost all quantum states, time runs from minus infinity in the past to plus infinity in the future. The Big Bang might be simply a transitional phase, with an infinitely old universe preceding it."
In the Scientific American article entitled The Myth of the Beginning of Time physicist Gabriele Veneziano wrote: "Albert Einstein's general theory of relativity led modern cosmologists to much the same conclusion [that time began]. The theory holds that space and time are soft, malleable entities. On the largest scales, space is naturally dynamic, expanding or contracting over time, carrying matter like driftwood on the tide. Astronomers confirmed in the 1920s that our universe is currently expanding: distant galaxies move apart from one another. One consequence, as physicists Stephen W. Hawking and Roger Penrose proved in the 1960s, is that time cannot extend back indefinitely. As you play cosmic history backward in time, the galaxies all come together to a single infinitesimal point, known as a singularity – almost as if they were descending into a black hole. Each galaxy or its precursor is squeezed down to zero size. Quantities such as density, temperature and spacetime curvature become infinite... [However], the assumption that relativity theory is always valid is questionable. Close to the putative singularity, quantum effects must have been important, even dominant. Standard relativity takes no account of such effects, so accepting the inevitability of the singularity amounts to trusting the theory beyond reason. To know what really happened, physicists need to subsume relativity in a quantum theory of gravity. ... Science does not have a conclusive answer yet, but at least two potentially testable theories plausibly hold that the universe – and therefore time – existed well before the big bang. If either scenario is right, the cosmos has always been in existence."
In the book Our Mathematical Universe (Ch.3), Max Tegmark wrote: "However, take note that my definition [of Big Bang] is quite conservative, saying nothing whatsoever about what happened before that. For example, this hypothesis does not imply that our Universe was one second old at the time, or that it was ever infinitely dense or came from some sort of singularity where our math breaks down. The question "Do we have evidence for a Big Bang singularity?" from the last chapter has a very simple answer: No! Sure, if we extrapolate Friedmann’s equations as far back in time as they’ll go, they break down in an infinitely dense singularity about a second before Big Bang nucleosynthesis, but the theory of quantum mechanics that we’ll explore in Chapter 7 tells us that this extrapolation breaks down before reaching a singularity. I think it’s crucial to distinguish between what we have solid evidence for and what’s highly speculative, and the truth is that although we have some exciting theories and hints about what happened earlier, which we’ll explore in Chapter 5, we frankly don’t yet know. This is the current frontier of our knowledge. Indeed, we don’t even know for sure that our Universe really had a beginning at all, as opposed to spending an eternity doing something we don’t understand prior to Big Bang nucleosynthesis."
In the book This Idea Must Die (pp. 32-34), prominent American physicist Lee Smolin wrote: "In my field of fundamental physics and cosmology, the idea most ready for retirement is that the Big Bang was the first moment of time. In popular parlance, the Big Bang has two meanings. First, Big Bang cosmology is the hypothesis that our universe has been expanding for 13.8 billion years from an extremely hot and dense primordial state more extreme than the center of a star – or, indeed, than anywhere now existing. This I have no quarrel with; it’s established scientific fact... What concerns me is the other meaning of Big Bang, which is the further hypothesis that the ultimate origin of our universe was a first moment of time, at which our universe was launched from a state of infinite density and temperature. According to this idea, nothing that exists is older than 13.8 billion years. It makes no sense to ask what was before that, because before that there wasn’t even time... This follows from a theorem proved by Stephen Hawking and Roger Penrose – that almost any expanding universe described by general relativity has such a first moment of time... [However], General relativity is incomplete as a description of nature, because it leaves out quantum phenomena. Unifying quantum physics with general relativity has been a major challenge for fundamental physics, one on which there has been much progress in the last thirty years. In spite of the absence of a definitive solution to the problem, there’s robust evidence from quantum cosmology models that the infinite singularities forcing time to stop in general relativity are eliminated, turning the Big Bang – in the sense of a first moment of time – into a Big Bounce, which allows time to continue to exist before the Big Bang, deep into the past."
In his book The Singular Universe (pp. 405-406), Dr. Smolin elaborated further on why the concept of singularities is problematic: "There are at least three precedents for singularities (in the sense of infinite or runaway divergences) in non-quantum theories being resolved by the introduction of quantum effects. The ultraviolet catastrophe in the thermodynamics of radiation found by Jeans in the 1890s was resolved by Planck’s quantum hypothesis in 1900. In classical atoms, the electron would spiral in a finite time into the nucleus emitting an infinite amount of radiation; this was a crisis when Rutherford discovered the nucleus was much smaller than the atom, but it was quickly resolved by Bohr’s application of the quantum hypothesis to the orbits of electrons in atoms. Third, in classical electromagnetism, point particles carry around electric fields that contain an infinite amount of energy. This is first reduced greatly by quantum effects to a logarithmic divergence and then eliminated by the procedure of renormalization. The view that quantum effects eliminate spacetime singularities is highly plausible in light of these historical precedents. There is a large literature investigating this issue, using increasingly sophisticated mathematical tools to investigate the implications of a merger of quantum physics and general relativity. While the problem of quantum gravity is not yet completely solved, there are a large class of models in which it can be investigated, and in all cases where the model is sufficiently rigorously or carefully studied to provide a useful answer – the answer is positive – the singularity is replaced by a history that extends into the past."
Physicist Dominique Eckert also commented on that: "There is one thing that is actually normally not discussed by people who are not really familiar with the field [of cosmology, namely,] what we call the 'Big Bang theory' is based on a classical theory of gravity – General Relativity. We have this theory of gravity that was invented by Einstein in 1915 that we call General Relativity and that's the theory we use to describe the evolution of the universe until what we would call the 'beginning' then, because we can see that galaxies are moving away from us wherever you look; if you look in one direction [and] then you look in a completely different direction you will see it looks like all the galaxies are moving away from us. We interpret this as saying that the universe itself is expanding and then, of course, if you think that the universe is expanding, you can go back and if you go back in time then it means that it must have been smaller in the past until the point where it would be a single point; some sort of singularity. The main problem with that idea here is that the theory we use – which is General Relativity – does not apply in the very early universe. We have nowadays two main theories in modern physics: General Relativity to describe gravity and quantum physics which describes the other forces at the microscopic level [i.e.,] electromagnetism, weak and strong nuclear forces. Those two theories are extremely successful to describe what's going in their own field of validity – in the area where they are valid. The problem is that they are not consistent with one another; it means [that] from a theoretical point of view, you cannot reconcile those two theories [because] there is a piece of the puzzle we are missing right now, and that is what we generally call "A Theory of Everything", which would be an area of physics that both General Relativity and Quantum Physics derive. We don't have such a theory. ... [We're] missing the global picture. All the laws of physics, as we see them, should [be] derived from a single theory. ... We cannot say that the universe began to exist because we cannot describe what was going on before the Planck time and, in fact, there are some very plausible theories right now that do not involve a beginning. For instance, it could very well be that the universe had a contraction phase in the past, reached a state and then eventually started inflating again; that could very well be. ... It could very well be the universe itself is eternal. ... [In other words], after a few fraction of a second after the Big Bang, our theory of General Relativity applies. We can use it to make predictions on what the universe should be and we can confront it with observations and we actually see that it matches extremely well with a lot of observations, but whatever is going on before that, we simply cannot say from the point of view of [current] physics. So, saying that the universe had a beginning... we cannot prove it. We are just not in a position to say it."
In his book The Edge of Knowledge (Sections 1 & 2), Professor Lawrence Krauss made the following pertinent statements: “A classical extrapolation backward of the currently observed Big Bang expansion implies that approximately 13.8 billion years ago, the universe emerged from a singular point. Just like the singularity at the last stage of black hole evaporation, time and space literally break down at this point. Roger Penrose demonstrated in 1965 that within the context of general relativity, the final stage of black hole collapse must produce a singularity... Stephen Hawking later expanded on Penrose’s demonstration... [and showed that by] following the equations of general relativity backward inevitably leads to a singularity in the finite past at which time and space cannot be defined. ... [However], once we get back to very close to t=0, all bets are off because our current understanding of the laws of physics at that time breaks down. (More on that in a bit, because it gives a possible way out of Hawking’s proof of a finite past.) ... This “singularity” is a point where the physics we understand breaks down. ... Until we have a theory of gravity that remains valid at the extremes of curvature and the infinitesimal scales of space and time present near the singularity of a black hole, it all remains speculation. ... If quantum mechanics is subsumed by some more fundamental dynamics at the smallest scales, then... the apparent singularity of the Big Bang... will need dramatic alteration.”
Professor of Physics Scott Oser from the University of British Columbia wrote: "What do most physicists actually believe happened "before" the Big Bang? ... Some physicists will reply very simply that they don't know what came before the Big Bang. Others may seize upon the Big Bang as the start of time itself, and argue that it's not even meaningful to ask what came before the Big Bang. However, the most common view (again, anecdotally) seems to be that the Big Bang picture of the birth of our universe is incomplete, and that the idea that there was no time before the Big Bang may not be quite correct. This viewpoint is strongly motivated by the well-known inconsistency of the theory of general relativity with the principles of quantum mechanics. Quite simply, physicists have been struggling for most of a century to figure out how to incorporate quantum mechanics into a theory of gravity. While there are some promising leads, notably string theory, we are still far from understanding how gravity works at the quantum scale. This is directly relevant to the conditions in the universe near the Big Bang. At these very early times the universe was still microscopically small, and quantum mechanics is expected to have severely altered the behavior of gravity. If and when physics develops a worked-out theory of quantum gravity, we may well be able to answer the question of what happened before the Big Bang. ... The net result is that we cannot confidently say what happened right at the moment of the Big Bang. It is entirely possible, for example, that time itself did not start at the Big Bang, but that universe developed according to still-uncertain physical laws from a previous physical state. There are a variety of versions of this. Some suppose that our present universe extends backwards in time towards infinity, and that the Big Bang is merely the start of the latest cycle of expansion. Another idea that is growing increasingly popular is that our universe formed from some previous universe by a kind of "budding" process."
In the book Where Did the Universe Come From? (pp. 36, 210-211), physicists Geraint Lewis and Ferrie Chris wrote: “What about the birth of the universe? According to Einstein[‘s theory of General Relativity], all space, all time, all matter came into existence at the Big Bang, but as with the centers of black holes, our mathematics is faced with infinities as gravity dominates and the other forces take a back seat. As with black holes, we expect these infinities to be banished once the true relationships between gravity and the other forces are understood. What do we expect this to reveal? Perhaps the rough picture of Einstein is correct. Perhaps space and time and matter all came into being at the initial start time of the universe. ... Most physicists find this idea unpalatable and don’t think that is likely to be the case. Looking at the hints in Einstein’s mathematics, many think our universe was not the actual beginning of everything and that we come from some preexisting structure. As we touched on in the last chapter, the ultimate demise of our universe might lead to the birth of new universes, and this process might be where our universe came from. Or perhaps it was born in the death of a massive star in a preexisting universe, a star whose collapse formed a new black hole and, in the act, budded off a new universe. Or perhaps our universe was born out of a process we can barely imagine. At the moment, without the mathematical language of gravity and the other forces working together, all we can do is guess. ... Without the infinite squeeze, the space and time of our universe could possibly be connected to other structures of space and time, perhaps other universes that came before. Of course, we don’t really know how any previous space and time are connected to ours, but there’s plenty of room for speculation. Thoughts range from our universe being born from the formation of a black hole in a previous universe to the collision of long-dead universes giving birth to ours in a huge, many-dimensional superspace known as the multiverse. There are many more theories, some much crazier sounding than others, and there will be even more still until we crack the theory of everything.”
In the book This Way to the Universe (Ch.3) Michael Dine wrote: “The real catastrophe would come as they approached the center of the black hole... Physicists and mathematicians refer to the center of the black hole as a singularity. At this point (and close to it) Einstein’s equations no longer make sense. ... [Similarly], if we look far enough back, the universe was infinitesimally small – things were packed closely together – closer together than one could possibly imagine. Mathematicians described the point at which time begins in Einstein’s equations as a singularity. The equations cease to make sense. ... Einstein’s theory breaks down at this time. ... So, we have a history of the universe, with experimental and observational support, from times about three minutes after the big bang, to the present, more than 13 billion years later. Yet we still don’t know what the singularity of Einstein’s equations means, and whether the universe truly had a beginning.”
In the classical physics book General Relativity (p.112), Robert Wald presented an excellent explanation of this issue: "The prediction of singularities undoubtedly represents a breakdown of general relativity in that its classical description of gravitation and matter cannot be expected to remain valid at the extreme conditions expected near a spacetime singularity. Indeed, on dimensional grounds one would expect a classical description of spacetime structure to break down at scales characterized by the Planck length. Hence, classical general relativity certainly should break down at or before the stage where it predicts spacetime curvatures of [infinite] order. Although the singularity theorems do not prove that the singularities of classical general relativity must involve unboundedly large curvature, they strongly suggest [its] occurrence in cosmology and gravitational collapse of conditions in which quantum or other effects which invalidate classical general relativity will play a dominant role."
In the book Introduction to Cosmology (pp. 124-125), Barbara Ryden wrote: "The energy density of our own universe… was infinite, according to this analysis… Should we take these infinities seriously? Not really, since the assumptions of general relativity, on which the Friedmann equation is based, break down at t ≈ tP. Why can’t general relativity be used at times earlier than the Planck time? General relativity is a classical theory; that is, it does not take into account the effects of quantum mechanics. In cosmological contexts, general relativity assumes that the energy content of the universe is smooth down to arbitrarily small scales, instead of being parceled into individual quanta. ... Quantum mechanical effects must be taken into account, and the classical results of general relativity no longer apply. ... To accurately describe the universe at its very earliest stages, prior to the Planck time, a theory of quantum gravity is needed."
In the paper Spacetime Singularities (pp. 1-5), cosmologist Pankaj S. Joshi wrote: "Both these models [i.e., solutions of general relativity] contained a spacetime singularity where the curvatures as well as the matter and energy densities were diverging and getting arbitrarily high, and the physical description would then break down. In the Schwarzschild solution such a singularity was present at the center of symmetry r = 0 whereas for the Friedmann models it was found at the epoch t = 0 which is beginning of the universe and origin of time where the scale factor for the universe vanishes and all objects are crushed to a zero volume due to infinite gravitational tidal forces... Physically, a singularity in any physics theory typically means that the theory breaks down either in the vicinity or at the singularity. This means that a broader and more comprehensive theory is needed, demanding a revision of the given theory. A similar reasoning applies to spacetime singularities, which may be taken to imply that a quantum gravity description is warranted in those regions of the universe, rather than using merely a classical framework."
In the book The Foundations of Quantum Gravity (p.57), Suddhasattwa Brahma wrote: "Setting aside the mathematical beauty of GR, it has also been an extremely successful theory and has passed all proposed experimental tests with flying colors thus far. Yet, a more fundamental theory is required that incorporates not only the dynamical nature of geometry but also the features of quantum physics. Indeed, a brief glance at Einstein’s equations, Gμν = (8πG/3)Tμν, is enough to convince one of that. Since the right-hand side of the equation consists of quantum matter described by the Standard Model, the left-hand side must be replaced by some suitable quantum version of gravity. The more obvious reason for invoking a quantum gravity theory obviously arises when dealing with curvature singularities contained in solutions to Einstein’s equations, where the equations themselves fail. This is a more serious problem than the divergences that arise in other gauge theories describing, for instance, the electromagnetic field. In Minkowskian physics, if such a field becomes singular at a given point, it does not affect the underlying spacetime and, consequently, has no effect on the evolution of the other fields in the theory. However, since gravity is geometry according to GR, when the gravitational field becomes singular, the picture of the continuum spacetime itself breaks down and all of physics gets disrupted. Now, we no longer have a stage on which to describe the dynamics of the other matter or gauge fields. This points toward the fact that our picture of gravity as a smooth continuum of spacetime must itself not be applicable to points arbitrarily close to such a singularity and what we require is a quantum theory of geometry."
In the book The Universe Before the Big Bang (pp. 5-6, 8, 10, 68), physicist Maurizio Gasperini wrote: "There are rigorous theorems (proved by George Ellis, Stephen Hawking, and Roger Penrose during the 1960s and the 1970s) stating that – under very general assumptions – it is impossible to avoid the initial singularity... Going progressively backward in time the density, the temperature, and the curvature of the Universe increase without bound until they reach – in a long, but finite time interval – an infinitely dense, hot and curved “singular” state. The idea that such a singular state (identified with the Big Bang, and conventionally placed at the time coordinate t = 0) may represent the birth of the Universe is based upon the fact that the dynamical equations of general relativity lose their validity at the onset of a singularity, and cannot be extended beyond a singular point (in this case, backward in time beyond t = 0). In other words, the solutions of those dynamical equations describe an “incomplete” space-time which is not infinitely extended in time, being characterized by an impassable “boundary” located at a finite temporal distance from any physical observer. It is thus general relativity itself which, in a cosmological scenario, unavoidably leads to the notion of an initial singularity, enforcing the idea that the Big Bang was the beginning of space-time and the moment of birth of our Universe. If we were to adhere strictly to general relativistic predictions, we should then conclude that the main topic of this book – the Universe before the Big Bang – is something meaningless. Before jumping to this conclusion, however, there is a question we should ask ourselves. Is the incompleteness of space-time predicted by general relativity a true physical property of our Universe, or is it only a mathematical property of some equations, that are really inadequate to describe space and time near the Big Bang? This is certainly a legitimate question in physics, where the occurrence of a singularity often does not correspond to any real entity, but is just a signal that some physical laws have been extrapolated beyond their realm of validity... General relativity, as previously stressed, is a classical theory. It has been successfully tested at densities, temperatures and curvatures much higher than those we may observe in our ordinary macroscopic world, but definitely much lower than the ones coming into play in the primordial Universe. The use of general relativity near the Big Bang implies trusting the validity of this theory not only beyond any experimental evidence, but also in a regime where there are well-founded reasons for doubting the legitimacy of classical theories. In fact, in the regime of extremely high energies... the properties of the gravitational interaction are expected to be significantly different from those predicted by general relativity. New fields and new kinds of short-range interactions may come into play, as inevitable consequences of the laws of quantum physics."
In the book The Perfect Theory (Ch.14), Pedro Ferreira, an astrophysicist from Oxford University, observed: “[F]or decades, because of Stephen Hawking’s and Roger Penrose’s singularity theorems, we have believed that, indeed, there was nothing before the Big Bang. ... [However,] over the past few years, the beginning of time has been thrown wide open by developments in quantum gravity and cosmology. When you wind back the clock and make the universe denser, hotter, and messier, that is when quantum foam, strings, branes, or even the loops have a say. It is where, it seems to some, spacetime breaks down and it no longer makes sense to talk about the initial singularity. ... [S]o much of quantum gravity seems to point to the fragmentation of spacetime if looked at under an all-seeing microscope. If we wind back the clock so that spacetime is concentrated at a point, surely we must run up against the bits and pieces that make up the fabric of space. Before any initial singularity is reached, when the chunkiness comes into play, known physics breaks down. Those who believe that loop quantum gravity is the answer say that there was a before, a time when the universe was collapsing until it reached the quantum wall and started expanding again.”
In the book Before Time Began (pp. 3, 76), Professor Emeritus of Theoretical Physics Helmut Satz wrote: "[It] was for a long time not considered meaningful to ask about the state of the world before the Big Bang, or how the Bang came to take place. Not so long ago, even Stephen Hawking, famous for his A Brief History of Time, noted that these questions were like asking what lies north of the North Pole. The past thirty years, however, have seen a change of paradigms, leading to a new view of the world, one in which the question of how and from what our universe appeared makes sense after all... The "old" cosmology was based on general relativity and its temporal and spatial singularity, constituting a unique Big Bang outside all "normal" physics. This "old" view suffers from the absence of a quantum theory of gravitation, a quantum theory combining strong and electroweak forces with gravity. It is not evident whether such a theory would still lead to any singularity."
In the book Universe in Creation (pp. 69, 79), Roy R. Gould stated: “There was the problem of what the universe was expanding from. The equations of physics can predict backward in time, as well as forward. Einstein’s model of the universe predicts that if you were to rewind the tape of the expanding universe, then all the matter and energy we see today would at some time in the past have coalesced at a single point. The space we see now would have been squeezed to a point. The density of matter would have been infinitely high. The time line itself would abruptly end. At that point, the equations of Einstein’s model break down; they would no longer describe physical reality. A situation in which physical quantities become infinitely large is called a singularity – something to be avoided at all costs. Einstein saw the singularity as a warning sign that his equations were no longer valid at that point. Someone else would have to improve and extend his model. … Furthermore, we know that the world of the very small is described by quantum theory, but Einstein’s model of gravity has never been completely merged with quantum theory and made to accommodate its strange rules. In short, there is not yet a well-tested theory that can predict with assurance what the universe was like before the Big Bang, or what the underlying physical principles might be. The Big Bang represents the boundary of our ignorance.”
In the book, Cosmology: A Very Short Introduction (pp. 56-57) End of Chapter 3, Peter Coles wrote: "The curvature of space-time can also become infinite in certain situations. The existence of these singularities suggests to many that some fundamental physics describing the gravitational effect of matter at extreme density is absent from our understanding. It is possible that a theory of quantum gravity might enable physicists to calculate what happens deep inside a black hole without having all mathematical quantities becoming infinite. Indeed, Einstein himself wrote in 1950: "The theory is based on a separation of the concepts of the gravitational field and matter. While this may be a valid approximation for weak fields, it may presumably be quite inadequate for very high densities of matter. One may not therefore assume the validity of the equations for very high densities and it is just possible that in a unified theory there would be no such singularity." Most cosmologists interpret the Big Bang singularity in much the same way as the black hole singularity discussed above, i.e. as meaning that Einstein’s equations break down at some point in the early Universe due to the extreme physical conditions present there. If this is the case, then the only hope for understanding the early stages of the expansion of the Universe is through a better theory."
In the book General Relativity: An Introduction for Physicists (pp. 260, 270-271), physicists George Efstathiou and M. P. Hobson wrote: "In the early 1960s, however, Penrose applied global geometrical techniques to prove a famous series of ‘singularity theorems’. These showed that in realistic situations an event horizon (a closed trapped surface) will be formed and that there must exist a singularity within this surface, i.e. a point at which the curvature diverges and general relativity ceases to be valid... Classical GR must break down at singularities. We would expect quantum effects to become important at ultra-short distances and ultra-high energies... These Planck scales define the characteristic energies, lengths, times, etc. at which we expect quantum gravitational effects to become important. Nobody really expects the centres of black holes to harbour true singularities. Instead, it is expected that, close to the classical singularity, quantum gravitational effects will occur that will prevent the divergences of classical general relativity."
Cosmologist Alan Guth, who first proposed Inflationary cosmology (which is now widely accepted by physicists), wrote a similar comment about the singularity in his book The Inflationary Universe (pp. 86-87, 341): "If one continues the extrapolation backwards in time, one comes to a point of infinite density, infinite pressure, and infinite temperature – the instant of the big bang explosion itself, the time that in the laconic language of cosmologists is usually called "t = 0." It is also frequently called a singularity, a mathematical word that refers to the infinite values of the density, pressure, and temperature... This singularity is sometimes said to mark the beginning of time, but it is more realistic to recognize that an extrapolation to infinite density cannot be trusted... Any honest cosmologist would admit that our knowledge here is very shaky. The extrapolation to arbitrarily high temperatures takes us far beyond the physics that we understand, so there is no good reason to trust it."
In the book General Relativity (p.141), Carlo Rovelli wrote: "In spite of its experimental and theoretical triumph, General Relativity has its limits. Einstein’s gravitational theory is a classical theory which does not include Quantum Mechanics. Attempts to come to a quantized version of General Relativity include Loop Quantum Gravity and String Theory. Among its theoretical problems, General Relativity predicts the occurrence of space-time singularities, events in which every known physical theory ceases to be valid, the space-time curvature diverges and time ends."
In the book, Cosmology: A Very Short Introduction (pp. 56-57) End of Chapter 3, Peter Coles wrote: "The curvature of space-time can also become infinite in certain situations. The existence of these singularities suggests to many that some fundamental physics describing the gravitational effect of matter at extreme density is absent from our understanding. It is possible that a theory of quantum gravity might enable physicists to calculate what happens deep inside a black hole without having all mathematical quantities becoming infinite. Indeed, Einstein himself wrote in 1950: "The theory is based on a separation of the concepts of the gravitational field and matter. While this may be a valid approximation for weak fields, it may presumably be quite inadequate for very high densities of matter. One may not therefore assume the validity of the equations for very high densities and it is just possible that in a unified theory there would be no such singularity." Most cosmologists interpret the Big Bang singularity in much the same way as the black hole singularity discussed above, i.e. as meaning that Einstein’s equations break down at some point in the early Universe due to the extreme physical conditions present there. If this is the case, then the only hope for understanding the early stages of the expansion of the Universe is through a better theory."
In the book General Relativity: An Introduction for Physicists (pp. 260, 270-271), physicists George Efstathiou and M. P. Hobson wrote: "In the early 1960s, however, Penrose applied global geometrical techniques to prove a famous series of ‘singularity theorems’. These showed that in realistic situations an event horizon (a closed trapped surface) will be formed and that there must exist a singularity within this surface, i.e. a point at which the curvature diverges and general relativity ceases to be valid... Classical GR must break down at singularities. We would expect quantum effects to become important at ultra-short distances and ultra-high energies... These Planck scales define the characteristic energies, lengths, times, etc. at which we expect quantum gravitational effects to become important. Nobody really expects the centres of black holes to harbour true singularities. Instead, it is expected that, close to the classical singularity, quantum gravitational effects will occur that will prevent the divergences of classical general relativity."
Cosmologist Alan Guth, who first proposed Inflationary cosmology (which is now widely accepted by physicists), wrote a similar comment about the singularity in his book The Inflationary Universe (pp. 86-87, 341): "If one continues the extrapolation backwards in time, one comes to a point of infinite density, infinite pressure, and infinite temperature – the instant of the big bang explosion itself, the time that in the laconic language of cosmologists is usually called "t = 0." It is also frequently called a singularity, a mathematical word that refers to the infinite values of the density, pressure, and temperature... This singularity is sometimes said to mark the beginning of time, but it is more realistic to recognize that an extrapolation to infinite density cannot be trusted... Any honest cosmologist would admit that our knowledge here is very shaky. The extrapolation to arbitrarily high temperatures takes us far beyond the physics that we understand, so there is no good reason to trust it."
In the book General Relativity (p.141), Carlo Rovelli wrote: "In spite of its experimental and theoretical triumph, General Relativity has its limits. Einstein’s gravitational theory is a classical theory which does not include Quantum Mechanics. Attempts to come to a quantized version of General Relativity include Loop Quantum Gravity and String Theory. Among its theoretical problems, General Relativity predicts the occurrence of space-time singularities, events in which every known physical theory ceases to be valid, the space-time curvature diverges and time ends."
In the book Black Holes, Cosmology and Extra Dimensions (p.131), physicist Sergey Rubin wrote: "Singularities are places where general relativity (or another classical theory of gravity) does not work since they do not belong to a Riemannian space-time. Therefore, a full understanding of BH physics requires avoidance of singularities or/and modification of the corresponding classical theory and addressing quantum effects. There have been numerous attempts on this trend, many of them being connected with the hope that the singularities of classical gravity will be cured by effects of quantum gravity. However, of great interest also are the opportunities of singularity avoidance in the framework of classical gravity."
In the book Relativistic Astrophysics and Cosmology (pp. 122, 192), physicist Peter Hoyng wrote: "The word singularity should not be taken too literally. Quantum gravity will probably prevent it, and even during the earliest phases of the Big Bang the universe never was a 'point'... [because] before that happens quantum mechanical effects take over, as classical GR loses its validity for length scales near the Planck length Lₚ
1.6 × 10⁻³³ cm."
1.6 × 10⁻³³ cm."In the book The Cosmic Century (p.451), Malcolm Longair stated: “It is certain that at some stage a quantum theory of gravity is needed which may help resolve the problems of singularities in the early Universe. The singularity theorems of Roger Penrose and Stephen Hawking show that, according to classical theories of gravity under very general conditions, there is inevitably a physical singularity at the origin of the Big Bang... However, it is not clear that the actual Universe satisfies the various energy conditions required by the singularity theorems. All these considerations show that new physics is needed if we are to develop a convincing physical picture of the very early Universe.”
In the book Cosmology's Century (p.36), Jim Peebles wrote: “In the big bang model, a straightforward extrapolation of its evolution back in time ends at a singularity: a manifest failure of standard general relativity. The usual hopeful thought is that there is a better theory that eliminates the singularity.”
In the book General Relativity: A Concise Introduction (pp. 95, 96), Steven Carlip wrote: "As we trace the Universe back to earlier times, its contraction will accelerate, and a will go to zero, giving an initial singularity. This is not just an accident of symmetry; the singularity theorems of Hawking and Penrose show that with a fairly minimal set of assumptions, the existence of an initial singularity is unavoidable. Of course, this claim should not be taken too literally – when curvatures become large near a putative singularity, we expect quantum effects to make classical general relativity untrustworthy."
Steven Weinberg, a theoretical physicist and Nobel laureate, stated: "[General Relativity] loses all predictive power at sufficiently high energy, so it is not a complete theory... It also can tell us nothing about singularities, because that is where it loses its predictive power."
In the article An Inflationary Non-singular Quantum Cosmological Model, Santini et al. stated: "The existence of an initial singularity is one of the major drawbacks of classical cosmology. In spite of the fact that the standard cosmological model, based in the classical general relativity theory, has been successfully tested until the nucleosynthesis era (around t ∼ 1s), the extrapolation of this model to higher energies leads to a breakdown of the geometry in a finite cosmic time. This breakdown of the geometry may indicate that the classical theory must be replaced by a quantum theory of gravitation: quantum effects may avoid the presence of the singularity, leading to a complete regular cosmological model."
In the book Foundations of Modern Cosmology (p.254), physicists John F. Hawley and Katherine A. Holcomb stated: "Is the density at the singularity truly infinite? Many scientists do not believe that infinite density can exist in the physical universe. We know, for example, that the general theory of relativity has never been made fully consistent with quantum mechanics, the other triumph of the physics of the first half of the 20th century... It is likely that there is a point at which Einstein’s equations break down as a suitable description of the universe, and it may be that quantum effects prevent a literal singularity."
In the book The Hidden Reality (p.114), physicist Brian Greene wrote: "In the vast majority of situations, quantum mechanics and gravity happily ignore each other, the former applying to small things like molecules and atoms and the latter to big things like stars and galaxies. But the two theories are forced to shed their isolation in the realms known as singularities... A prize achievement of any purported quantum theory of gravity is to meld quantum mechanics and gravity in a manner that cures singularities. The resulting mathematics should never break down – even at the moment of the big bang or in the center of a black hole, thus providing a sensible description of situations that have long baffled researchers."
In the course guidebook Understanding the Misconceptions of Science (pp. 201-203, 206, 211), Dr. Don Lincoln wrote: "According to legend, the center of a black hole is the singularity. It is a place where all of the mass of the hole is concentrated into a sphere with zero volume and zero radius. It is a place of infinite density. But this is complete hogwash. The reason you might’ve heard this is because that’s what the equations of general relativity say. But it’s important to remember that equations are just that – equations. They are math and not physics. They are relevant only as long as they properly model the world. When they no longer do so, you should discard the equations. And this is one of those times. At large distances from a black hole, Newtonian gravity works just fine, but as you get into the realm of stronger gravity, you need to change to Einstein’s version. It’s different, but not that different. Gravity still increases as you get closer to the center of the black hole, mathematically approaching infinity. But as you get close to the center of the hole, things have to change. For one thing, you can’t have infinite energy. ... Infinities of this kind are just nature’s way of saying that our mathematics is broken. ... We know that in the realm of the small, quantum mechanics holds sway [and that's where it] is needed to address the structural flaws of general relativity. What is needed is a theory of quantum gravity. But a viable theory has eluded scientists for a century. ... According to the theory of general relativity, before the universe began expanding – before the bang – all of the matter and energy of the universe was located in a single point: a sphere with zero size. The scientific term for this is a singularity. ... But that’s wrong. It’s even a wrong mental picture of the big bang. ... In the previous lecture about black holes, you discovered that singularities can’t exist, so you can dispense with that idea."
In the book Black Hole (pp. 107-108), Marcia Bartusiak stated: "But the bottom line was unmistakable: you couldn’t have total gravitational collapse without a singularity turning up at the end. “Deviations from spherical symmetry,” reported Penrose, “cannot prevent space-time singularities from arising.” (That is, as long as quantum mechanics is not taken into account)... [John] Wheeler tried to impart the insanity of the concept. Ride along on the collapsing ball of matter, he suggested, and its density would go up faster and faster until, in less than a second, it rose to infinity. “With this prediction of an infinite density,” wrote Wheeler, “classical theory has come to the end of the road. A prediction that is infinity is not a prediction. Something has gone wrong... Infinity is a signal that an important physical effect has been left out of account.” And there is something missing. Wheeler pointed out that answers will likely arrive with the successful merging of general relativity with quantum theory. Quantum gravity theorists don’t have as yet a definitive solution, but they are confident that something else happens inside the black hole, that quantum effects prevent a singularity from forming there."
Cooked. Time to move on.
BGV Theorem
The Borde-Guth-Vilenkin theorem is sometimes used as proof that the universe had an absolute beginning, as philosopher of physics Daniel Linford explained in his paper titled Big Bounce or Double Bang? (pp. 4-5):
“Physicists endeavored to provide theorems describing the conditions under which space-times are singular. Early theorems – like those produced by Hawking and Penrose in the 1960s – were able to show that curvature singularities were not the result of homogeneity or isotropy. ... In 2003, Arvind Borde, Alan Guth, and Alexander Vilenkin developed a new singularity theorem – the BGV theorem – with the advantage that the theorem no longer depended upon an explicit assumption about the universe’s matter-energy-momentum contents. ... To describe the BGV theorem, I first need to say what the Hubble parameter is and what geodesics are. The Hubble parameter can roughly be thought of as the universe’s expansion rate and geodesics are the trajectories that particles traverse in space-time when no forces other than gravity act upon them. Time-like geodesics are the trajectories that particles with mass traverse while null geodesics are those traversed by massless particles. ... A congruence of geodesics is a bundle of geodesics (analogous to a bundle of streamlines) filling a region of space-time and where no two of the geodesics cross. Borde, Guth, and Vilenkin develop a generalization of the Hubble parameter. We can think of the generalized Hubble parameter as the universe’s expansion rate as measured by an observer traversing time-like or null geodesics. The BGV theorem is a result about congruences comprised by time-like and null geodesics. According to the BGV theorem, if the average of the generalized Hubble parameter along the geodesics comprising such a congruence is greater than 0 – that is, if a given observer traversing any of the geodesics in the congruence would observe the universe to be (on average) expanding along her geodesic – then the congruence cannot be extended to past infinity. The termination of geodesics in the finite past within a given model is taken to be a strong indication that the model is singular, so that the BGV theorem suggests that all expanding space-times are singular. While Borde, Guth, and Vilenkin interpreted the singular behavior to indicate that our physical understanding is incomplete, [some] interpret the singular behavior as evidence for an absolute beginning."
As Dr. Linford pointed out, all of the authors of this paper initially interpreted their theorem as showing physics other than inflation is needed to describe the boundary condition. They even suggest that a "beginning" is only one possibility that might correspond to the boundary condition and they suggest that there may be something "beyond the boundary":
"What can lie beyond this boundary? Several possibilities have been discussed, one being that the boundary of the inflating region corresponds to the beginning of the Universe in a quantum nucleation event... Whatever the possibilities for the boundary, it is clear that... inflation alone is not sufficient to provide a complete description of the Universe, and some new physics is necessary in order to determine the correct conditions at the boundary. This is the chief result of our paper." (Borde et al., 2003)
Alan Guth reiterated this point in the paper titled Eternal Inflation and its Implications (p.16). He wrote: “If, however, Hₐᵥ > 0 when averaged over a past-directed geodesic, our theorem implies that the geodesic is incomplete. There is of course no conclusion that an eternally inflating model must have a unique beginning, and no conclusion that there is an upper bound on the length of all backwards-going geodesics from a given point. There may be models with regions of contraction embedded within the expanding region that could evade our theorem... The theorem does show, however, that an eternally inflating model of the type usually assumed, which would lead to Hₐᵥ > 0 for past-directed geodesics, cannot be complete. Some new physics (i.e., not inflation) would be needed to describe the past boundary of the inflating region.”
In an interview, Guth clarified further: “In 2003 I was working with Arvin Borde and Alex Vilenkin to understand what we can learn about how inflation might have started and how far back it could have gone. In particular, once we realized that inflation could be eternal into the future, it seemed like a very natural question to ask [whether] inflation [could] also be eternal into the past. And what we found is that inflation could not be eternal into the past. What we basically managed to achieve was proving a theorem which says that any expanding region of spacetime that has a minimum expansion rate can only go back so far and not infinitely far. So, that means that inflation must have had a beginning; it does not really say that the universe must have had a beginning, but says the universe could not have been expanding forever up until the present time.” (Halper, 2016) Elsewhere, Guth stated: “I do not know whether the universe had a beginning. I suspect the universe did not have a beginning. It is very likely eternal but nobody knows.” (Stewart, 2016: p.70) Guth also wrote: “I think that the BGV theorem does not entail that the universe had a beginning; it is the region of inflation that must have a beginning, not the universe.” (Daniel & Craig, 2017) More recently, he said: “[The BGV theorem] proves that the expanding era of inflation that we're part of must have a past boundary someplace, [but] that does not necessarily mean the universe had a beginning.” (Halper, 2021)
Dr. Alex Vilenkin was more ambiguous about his interpretation of the theorem, however. In his books (Vilenkin, 2006 & 2017), he did say that his theorem is sufficient evidence of a beginning. Nevertheless, when physicist Victor Stenger asked Dr. Vilenkin whether it proved "that the universe must have had a beginning", Dr. Vilenkin replied as follows: "No. But it proves that the expansion of the universe must have had a beginning. You can evade the theorem by postulating that the universe was contracting prior to some time." (Stenger, 2011: Ch. 6.2.) (Dr. Vilenkin then went on to opine that the contracting phase is unstable and therefore problematic. But, as we will see shortly, other cosmologists forcefully challenged his belief.) Similarly, in an interview with Phillip Halper, Dr. Vilenkin said: “The [BGV] theorem proves that inflation must have a beginning. The universe as a whole... it doesn't... the theorem doesn't say that [i.e., that the universe as a whole must have had a beginning]. It says that the expansion of the universe must have [had] a beginning.” (Halper, 2019)
In the article Did Time Have A Beginning?, astrophysicist Ethan Siegel confirmed that this theorem does not show that spacetime is finite in the past: "The Borde-Guth-Vilenkin theorem tells us that all points in the Universe, if you extrapolate back far enough, will merge together, and that inflation cannot describe a complete spacetime. But that does not necessarily mean an inflating state could not have lasted forever; it could just as easily imply that our current rules of physics are incapable of describing these earliest stages accurately. Even though we can trace our cosmic history all the way back to the earliest stages of the hot Big Bang, that isn't enough to answer the question of how (or if) time began. Going even earlier, to the end-stages of cosmic inflation, we can learn how the Big Bang was set up and began, but we have no observable information about what occurred prior to that. The final fraction-of-a-second of inflation is where our knowledge ends. Thousands of years after we laid out the three major possibilities for how time began – as having always existed, as having begun a finite duration ago in the past, or as being a cyclical entity – we are no closer to a definitive answer. Whether time is finite, infinite, or cyclical is not a question that we have enough information within our observable Universe to answer."
The BGV Assumes a Classical Space-Time
In the article Let the Universe Be the Universe, American theoretical physicist Sean Carroll who earned a PhD from Harvard University and specializes in quantum mechanics, gravity and cosmology, wrote:
"The theorems in question make a simple and interesting point. Start with a classical spacetime – "classical" in the sense that it is a definite four-dimensional Lorentzian manifold, not necessarily one that obeys Einstein's equation of general relativity. (It's like saying "start with a path of a particle, but not necessarily one that obeys Newton's Laws.") The theorem says that such a spacetime, if it has been expanding sufficiently fast forever, must have a singularity in the past. That's a good thing to know, if you're thinking about what kinds of spacetimes there are. The reason I didn't explicitly mention this technical result in my essay is that I don't think it's extremely relevant to the question. Like many technical results, its conclusions follow rigorously from the assumptions, but both the assumptions and the conclusions must be treated with care. It's easy, for example, to find examples of eternally-existing cosmologies which simply don't expand all the time. (We can argue about whether they are realistic models of the world, but that's a long and inconclusive conversation.) The definition of "singularity in the past" is not really the same as "had a beginning" – it means that some geodesics must eventually come to an end. (Others might not.) Most importantly, I don't think that any result dealing with classical spacetimes can teach us anything definitive about the beginning of the universe. The moment of the Big Bang is, if anything is, a place where quantum gravity is supremely important. The Borde-Guth-Vilenkin results are simply not about quantum gravity. It's extremely easy to imagine eternal cosmologies based on quantum mechanics that do not correspond to simple classical spacetimes throughout their history. It's an interesting result to keep in mind, but nowhere near the end of our investigations into possible histories of the universe."
When asked about this theorem by Dr. Stenger, Dr. Carroll repeated the same point: “I think my answer would be fairly concise: no result derived on the basis of classical spacetime can be used to derive anything truly fundamental, since classical general relativity isn't right. You need to quantize gravity. The BGV singularity theorem is certainly interesting and important, because it helps us understand where classical GR breaks down, but it doesn't help us decide what to do when it breaks down.” (Stenger, 2011: Ch. 6.2.)
In another book, Dr. Carroll reiterated his argument and expanded on it a bit more: “Borde-Guth-Vilenkin Theorem: ... In some universes (not all), the classical spacetime description breaks down in the past. The real universe is quantum, not classical. The BGV theorem tells us nothing definitive about reality. ... Where [my opponent] says that the Borde-Guth-Vilenkin theorem implies the universe had a beginning, that is false. That is not what it says. What it says is that our ability to describe the universe classically, that is to say, not including the effects of quantum mechanics, gives out. ... [I]t’s just extremely clear to us as cosmologists that the chance that classical physics is telling you anything interesting and relevant about the very first moment of the history of the universe is very small. You really have to think quantum-mechanically, and the BGV theorem only tells you when you need to start thinking quantum-mechanically, not what the result of that thinking is going to be. The simplest quantum mechanical models that hold the most promise for explaining cosmological puzzles seem right now to us to be those that are eternal.” (Stewart, 2016: pp. 42, 242-243)
The philosopher of science (specifically, of physics) Tim Maudlin made a similar comment regarding this matter: “[T]he so-called BGV theorem... is not a theorem of quantum physics. Insofar as we know that quantum physics is central to the actual world, we know that the BGV theorem simply does not apply to the actual world. So, we do not know, in virtue of that theorem, that there is a beginning of time. ... A final physical theory describing the whole universe has to incorporate quantum theory and General Relativity (or some other theory of gravity) and we simply do not yet know how that is done. ... [A]ll we can properly say is that we do not yet know enough... to draw any conclusions about whether anything predates the Big Bang. ... In claiming that the BGV theorem somehow settles this issue, [my opponent] parts company with all cosmologists I am familiar with.” (Stewart, 2016: pp. 138, 139)
Dr. Gianluca Calcagni echoed this argument in his technical book on quantum cosmology: “The paradigm assumes classical general relativity and theorems due to Borde, Guth and Vilenkin then imply that space-time had an initial big bang singularity. For reasons discussed in Sect. 2.1, this is an artifact of using general relativity in domains where it is not applicable. Therefore, one needs a viable treatment of the Planck regime and the corresponding extension of the inflationary paradigm.” (Calcagni, 2013: p.51)
Dr. Robert Oerter, who is Professor in the Physics department at George Mason University and author of the book The Standard Model, made a similar point: "The discussion brought up the Borde-Guth-Vilenkin Theorem (BVG for short) as support for the premise 'The universe began to exist.' I want to talk about why the BVG theorem is irrelevant to [this premise]... [N]o physicist thinks the inflationary model is the end of the story. The exponential expansion is so fast that in a very short time the scale factor reaches the Planck realm, where we expect GR to break down and quantum gravity to come into play. So most diagrams of the early universe insert a quantum gravity region before the inflationary epoch. ... There has been much discussion of what went on before the inflationary epoch: quantum foam, the no-boundary proposal, the cyclic universe, and so on. There is even a version called 'eternal inflation,' in which some portion of the universe goes on inflating forever, while pocket universes like ours bubble off from time to time. In some of these models, time is finite in the past. In others, it is infinite. ... Does modern cosmology support the premise that the universe had a beginning? Emphatically, no! ... [M]odern cosmology allows us to draw no conclusion about whether the universe has existed for a finite or infinite amount of time. And anyone who says differently is not being completely honest." (Oerter, 2014)
Elsewhere Dr. Oerter elaborated on this point: "[L]et's look at what the theorem actually says. 'Any theorem is only as good as its assumptions,' writes Alexander Vilenkin in a letter to Lawrence Krauss. So let's start with the assumptions of the theorem. Roughly speaking, there are two: (1) Spacetime is classical. (2) Spacetime is expanding on average. ... Start with assumption (2): the path can enter a region in which spacetime is static, or contracting, or cyclically expanding and contracting. Then assumption (2) is violated and the theorem's conclusion is avoided. In spacetimes like these, the BGV theorem simply doesn't apply. What about assumption (1)? Classical spacetime is a pretty basic assumption in any sort of cosmology. But it is expected to break down in the quantum gravity regime, where we encounter "spacetime foam" of some sort. ... [Therefore], the BGV theorem does not say that spacetime must be singular; still less does it say that there was an initial singularity from which the entire universe arose. This is why I say that the theorem is irrelevant to [that premise]: it doesn't say anything about whether the universe had a beginning or not." (Oerter, 2014)
Dr. Vilenkin wrote the same thing in an email: "The BGV theorem uses a classical picture of space-time. In the regime where gravity becomes essentially quantum, we may not even know the right questions to ask."
In the paper titled Quantum Complete Prelude to Inflation, cosmologist Marc Schneider argued that "Bianchi type-I cosmologies are quantum complete and geodesic[ally] incomplete." So, I asked Dr. Schneider if this means the spacetime geometry cannot be extended prior to the inflationary history (e.g., if a contraction phase prior to inflation is not possible) in light of these results. He replied,
"In principle, a scenario as you describe is reasonable and could have happened in the early universe. This would then resemble the FLRW case with a contracting phase preceding the expansion. We only looked at Kasner without extending beyond t=0 but there is the option to extend it and see if a transmission into this region is supported. Our result says that if space-time is closed by a Kasner singularity then quantum fields cannot reach it, so far we do not make a statement about extensions but in principle the formalism can be used to investigate this. My guess is that it strongly depends on the model you are considering, [e.g.,] we looked briefly at the Turok-Gielen bounce after I got a very similar question during a talk; the result showed, it was consistent. So yes, it is possible and it is assuring that QFT in curved space-times does not blow up so easily (at least for some models)."
More recently, however, a physicist named Dawood Kothawala published a paper arguing against the above conclusion of Prof. Vilenkin and Dr. Carroll. He wrote: “It is, of course, interesting to probe deeper the fate of geodesic incompleteness, as described by BGV theorem, in a complete theory that incorporates quantum fluctuations of matter fields as well as space-time. In Sec. II 1, while discussing the issue of geodesic incompleteness of the congruence u, the effects of quantum fluctuations were incorporated through the (covariantly defined) lower bound `0 on distances. The existence of a lower bound on geodesic intervals seems to a be a generic consequence of combining principles of GR and quantum mechanics, and expected to be independent of any specific model/framework of quantum gravity.”
Dr. Kothawala did not analyze Loop Quantum Gravity (Ashtekar, 2008), Hořava–Lifshitz Gravity (Brandenberger, 2009) or Asymptotically Safe Gravity (Hossenfelder, 2020; Eichhorn, p.9) in his paper. This is merely the semi-classical case and, therefore, cannot tell us whether the BGV will work in the full quantum gravity case or not. Moreover, I asked Dr. Kothawala about this and he admitted to me in email that LQG "can evade the BGV theorem by violating some of its conditions".
I contacted Dr. João Magueijo, a Portuguese cosmologist and professor of Theoretical Physics at Imperial College London who got his Ph.D. at Cambridge University. He stated the following about the paper: "Quantum space-time is not classical space-time with quantum fluctuations on top of it, as envisaged in that paper. It is a full non-perturbative structure, such as that in spin foams, spin networks, etc, where the concepts of metric and geodesic are not fundamental but emerging."
I also sent this paper to Dr. Vilenkin and fortunately he replied to my email: "I am familiar with Dr. Kothawala's paper. In the acknowledgement to that paper he thanks me for pointing out potential issues with his approach, but he does not respond to my points. Without getting into technicalities, Dr. Kothawala makes some guesses about how the classical BGV theorem will be modified if quantum gravity effects are taken into account. However, we do not have a quantum theory of gravity and I have no way of telling whether his guesses are right or wrong."
Contracting Phase
I also sent an email to cosmologist George Ellis asking about Kothawala's paper. Prof. Ellis wrote: "I have not examined that paper in detail but the basic issue remains: the BGV theorem is a result of a geometric assumption that is untrue for example in k = 0 de Sitter universes (a(t) = exp H t) as well as k=+1 de Sitter universes (where a(t) = cosh H t) hence it works by simply omitting the solutions that are non-singular by excluding the conditions that allow them to exist. The essence is that if you demand that df/dx > epsilon > 0 then f(x) necessarily goes through zero. If you allow df/dx =0 (as in the case of the k = +1 de Sitter universe) you can get a bounce. This is true irrespective of whether you have quantum theory involved or not."
Dr. Ellis's argument can be plausibly interpreted as follows: There are, at least, two solutions of General Relativity; one in which the geometry is spatio-temporally incomplete in the past and one that is complete – the latter describing a geometry that is extended to a contracting phase prior to the expansion. Both solutions are compatible with empirical observations. Thus, since we cannot rule out either of them based on current observational or theoretical physics, we should remain agnostic about which of them is correct.
In the paper Current Observations with a Decaying Cosmological Constant Allow for Chaotic Cyclic Cosmology (p.24), Dr. Ellis confirmed what he told me in the email: "Borde et al. have given claims that the universe must have been singular (BGV), but these are based on restrictive assumptions that need not hold in the real universe. In particular, the BGV theorem relies on two assumptions; [that] there is only expansion... and the second is that the behaviour can be modelled entirely classically. In some instances of our model we consider a universe that undergoes periods of expansion and collapse... We therefore evade this theorem simply by not satisfying the required axioms for the theorem to hold true in either instance."
In the article Geodesic Completeness and Homogeneity Condition for Cosmic Inflation, the authors wrote: "Due to the inevitability of a cosmological singularity within GR, inflationary trajectories are past-incomplete [4 - BGV]. One can see that this warrants a better theory of gravity in the ultraviolet (UV), which would ameliorate the UV divergences as well as make the theory singularity free in the UV... We will show that both the problems have a common origin, arising from how the causal structure of null and timelike geodesics are structured within GR... [Finally, we conclude that] a nonsingular bouncing cosmology is a must for a successful inflationary paradigm, where one can make the inflationary trajectory past-complete."
In the article titled Singularity Resolution in Loop Quantum Cosmology (p.9), Abhay Ashtekar reinforced the fact that a contracting phase would evade the BGV. He wrote: "What about the more recent singularity theorems that Borde, Guth and Vilenkin proved in the context of inflation? They do not refer to Einstein’s equations. But, motivated by the eternal inflationary scenario, they assume that the expansion is positive along any past geodesic. Because of the pre-big-bang contracting phase, this assumption is violated in the LQC effective theory."
In the paper Island Cosmology (p.11), the authors confirm the findings: "We assume that the cosmological constant provides us with a background de Sitter space-time that is eternal. As de Sitter space-time also has a contracting phase, the singularity theorems of Ref. [36 – BGV] are evaded.”
Physicist Don Page, noted for being a doctoral student of the eminent Professor Stephen Hawking, wrote: "We simply do not know whether or not our universe had a beginning, but there are certainly models... that do not have a beginning. I myself have also favored a bounce model in which there is something like a quantum superposition of semi-classical spacetimes, in most of which the universe contracts from past infinite time and then has a bounce to expand forever. In as much as these spacetimes are approximately classical throughout, there is a time in each that goes from minus infinity to plus infinity."
In the article An Inflationary Universe Without Singularities and with Eternal Physical Past Time (p.3), cosmologist E. Rebhan described a model very similar to the one envisaged by Don Page: "Blome & Priester ([1991]) proposed a “big bounce model” (called BP-model in the following) in which the universe is closed, exists for an infinite time and [undergoes an infinite contraction]. After it has passed through a minimal radius [at the bounce], it starts re-expanding. In the contraction phase the universe is in a primordial state of a quantum vacuum with finite energy density and negative pressure p. The “big bounce” at the minimal radius is supposed to trigger a phase transition by which ordinary matter is created, the vacuum energy density is reduced and the transition into a Friedmann-Lemaître evolution is achieved as in the inflation scenarios of big bang models."
In the article mentioned above, Blome and Priester explained some features of their model: "Our approach is based on the assumption of a primordially homogeneous and isotropic spacetime, in which all fields exist in their ground state [i.e., a vacuum]. ... The vacuum is the more fundamental entity, which can exist independent of real particles. Therefore, it seems a natural hypothesis that the quantum vacuum was the primordial stage before elementary particles came into being. ... The energy density of the quantum vacuum is assumed to be constant from the infinite past until the phase transition begins. At that time the quantum vacuum energy is converted into relativistic particles. ... The concept of the Big Bounce implies that there is no matter and no relativistic matter and radiation before the phase transition. ... The solution for a spherical space metric avoids the singularity and represents a closed universe, which contracts from infinity and re-expands after the Big Bounce. The selection of a de Sitter solution with spherical metric as a mathematical model of the very early universe is quite natural within the context of quantum transitions from a pre-Planck stage at the threshold of classical cosmology."
Physicist Petar Pavlovic from the University of Hamburg in Germany proposed a mechanism by which the matter production could take place. He wrote: "The important question could be what triggers the activation of production of matter from the scalar field – and why this mechanism is not efficient today nor before the bounce. One possibility could for instance be that the potential of this field is dependent on temperature or curvature in a proper way to enable it, or perhaps that the production of matter is determined via the non-minimal coupling of the field and curvature scalar (so, when the curvature scalar decreases from the maximal value at the bounce this term becomes negligible)." (Personal Correspondence)
Instabilities in the Contracting Universe?
Nevertheless, some physicists think the exact de Sitter space is unstable. For example, in the article "On the Instability of Global de Sitter Space to Particle Creation", it was argued that "particle creation in the contracting phase results in exponentially large energy densities at later times, necessitating an inclusion of their backreaction effects, and leading to large deviation of the spacetime from global de Sitter space before the expanding phase can begin."
The same authors of this paper wrote another one titled "Quantum Vacuum Instability of 'Eternal' de Sitter Space" in which the following conclusion was reached: "perturbations of this [vacuum de Sitter] state with arbitrarily small energy density at early times that is exponentially blueshifted in the contracting phase of 'eternal' de Sitter space, and becomes large enough to disturb the classical geometry".
I sent this to physicist Niayesh Afshordi and he replied: "In quantum field theory, you have to specify a state (or wave-function) as well as the spacetime to assess stability. At the perturbative level, you can choose unstable states, as well as stable states. This article chooses the former, but you can also choose the latter. Which one is more natural is really a matter of opinion."
I also sent this to Dr. Pavlovic and he responded as follows: "Regarding the paper on the instability on De-Sitter space... in essence I do not think it makes a direct threat to the kind of models we discuss. As I said previously things get simplified if we stay at the (semi) classical treatment, while we can not address them properly in the full quantum treatment (since we do not properly understand the quantum regime of gravity). The fact is that we do not know at which energies the bounce happens. This can not be determined from current observation, nor from some theoretical reasons. It could be that it happens around or after the Planck scale, but this is not necessary. On the other hand, it is completely possible that the bounce happens at some much smaller energies – so that the description of it requires only a classical treatment with perhaps the addition of some correction terms. If we assume this second possibility then quantum effects like these on particle production on De-Sitter are actually irrelevant, and also the physical content of theory is quite simplified. On the other hand, if we would assume that bounce happens around Planck scale and that it requires quantum description, then we need to resort to wild speculations at this moment. So I think it is justified to assume a (semi)classical bounce, as the only scenario we can now work with rigorously at the moment." (For papers demonstrating the stability of the de Sitter space, see Moreau, 2018; Albert, 2008; Michael, 2004; Markus, 2013; Morrison, 2010; Wang, 2010; Benjamin, 2011; Kroon, 2016).
Moreover, the de Sitter scenario is not the only model that posits the existence of an empty contracting spacetime that produces matter after the bounce. I described this model to physicist Robert Brandenberger and he replied: "This sounds exactly what the Ekpyrotic scenario does. The scalar field yields contraction with all the nice properties, and at the bounce regular matter is being produced by the scalar field."
I also mentioned the papers in which it was argued for the instability of the de Sitter space. Dr. Brandenberger replied: "This instability of de Sitter space is not a problem for [recent] bouncing cosmologies. [Recent] bouncing cosmologies are not based on contracting de Sitter. The most promising bouncing model is the Ekpyrotic scenario proposed by Khoury et al. It is a local attractor in initial condition space and it does not suffer from the instability due to particle production. For a recent improvement of the Ekpyrotic scenario see the work I have done this year with Z. Wang."
To confirm the possibility of a past-eternal contracting phase in the Ekprotic model I sent an email to cosmologist Paul Steinhardt, who is a very important proponent of some version of the Ekprotic model. I asked whether "it is possible that, rather than being cyclical, it can be a single bounce? Rather than contracting and expanding infinitely, could it be that it was contracting from the infinite past, had a bounce and then began to expand forever?"
Dr. Steinhardt replied: "The simple answer to your question is: yes, there can be a single bounce just as you describe. We have a paper describing that example, in fact. It is also a problem free case. ... Single bounce and probably other variations are possible."
We mentioned earlier that Prof. Vilenkin said contracting universes are unstable. He stated that "contracting universes are highly unstable" because "inhomogeneities in this contracting universe would grow out of hand... if you have bubbles forming, all these bubbles instead of being driven away from one another they would be driven towards one another. So, the whole thing will be filled up with bubbles..."
I sent this quote to Dr. Pavlovic. He responded the following: "Phase transitions should not be the thing here, since it is only at higher energies, like 100 Gev for electroweak transition and activation of Higgs, but also according to Standard model and currently taken Higgs mass, the phase transition is not of the first order so there are no bubbles during it."
French physicist Aurélien Barrau also responded to Vilenkin's criticism. He said: "It is very speculative and very difficult to [determine] whether the situation is stable or not. And even in the case where the situation is unstable, usually what it means is that the simplifications we use in standard cosmology cannot be used anymore... So when we say that something won't work, most of the time it means that the model we are using to describe these things doesn't work anymore. For example, you cannot use the Friedmann Equation because the [contracting] universe is so inhomogeneous that the Friedmann Equation is not usable anymore. So what? The universe does not care that we have to invent another equation."
I sent this quote to Dr. Pavlovic. He responded the following: "Phase transitions should not be the thing here, since it is only at higher energies, like 100 Gev for electroweak transition and activation of Higgs, but also according to Standard model and currently taken Higgs mass, the phase transition is not of the first order so there are no bubbles during it."
French physicist Aurélien Barrau also responded to Vilenkin's criticism. He said: "It is very speculative and very difficult to [determine] whether the situation is stable or not. And even in the case where the situation is unstable, usually what it means is that the simplifications we use in standard cosmology cannot be used anymore... So when we say that something won't work, most of the time it means that the model we are using to describe these things doesn't work anymore. For example, you cannot use the Friedmann Equation because the [contracting] universe is so inhomogeneous that the Friedmann Equation is not usable anymore. So what? The universe does not care that we have to invent another equation."
In an email, Vilenkin also stated that "you can evade the [BGV] theorem by postulating that the universe was contracting prior to sometime ... but the problem is that the contracting universe is highly unstable. Small perturbations would cause it to develop all sorts of messy singularities, so it would never make it to the expanding phase." Nevertheless, in the same email Vilenkin also acknowledged that many of his colleagues think singularities will be eliminated: "However, it is conceivable (and many people think likely) that singularities will be resolved in the theory of quantum gravity." Indeed, physicist Susanne Schander made this point as well: “There is [also] the possibility that the classical theory should be replaced as a whole by a quantum theory of gravity and there are strong reasons to believe that. So, we just don't know what happens if we use such a quantum theory of gravity to explain the very early universe; there might be a contracting phase as well [as a] bounce and all the problems due to instabilities could still be avoided.”
Similarly, Dr. Brandenberger observed that these alleged instabilities only obtain in classical gravity: “Some people have claimed that a bouncing cosmology is unstable; that we will never get a nice expanding universe if we start in a contracting universe. Now, a lot of these analyzes are, again, obtained using Einstein’s gravity and we have to go beyond Einstein’s gravity. And now there are concrete models where you can get a bouncing universe with fairly mild changes to the physics.”
Regarding Vilenkin's claim that contracting universes are generally unstable, physicist Niayesh Afshordi commented: “Some models of bounce or contracting universe may have instabilities that are incurable, but it really depends on what you have. For example, models that my friends Anna Ijjas and Paul Steinhardt have worked on has scalar fields with a very negative energy in which you either don't have instabilities or you could have instabilities that are treatable (i.e., are mild). Things don't have to be unstable, but even if they are unstable, it doesn't mean that you're not going to get a bounce; just [that] you're going to have a more complex bounce and what's going to come out you have to go and work harder to figure out.”
Dr. Afshordi is referring to the article titled Fully Stable Cosmological Solutions With A Non-singular Classical Bounce, which was written by the theoretical physicist Anna Ijjas, who works at the Max Planck Institute for Gravitational Physics. She wrote that "it is possible to use a cubic Galileon action to construct classical cosmological solutions that enter a contracting... phase without encountering any singularities or pathologies... [and] construct non-singular classical bouncing cosmological solutions that are non-pathological for all times." Moreover, in a recent paper, physicist Frans Pretorius (and colleagues) performed an "analysis using the tools of numerical general relativity" to demonstrate that a model that posits a period of "slow contraction" and a "quantum smoother" implies "that an initially homogeneous, flat, and anisotropic universe should be stable to quantum fluctuations generated during the smoothing phase." (Supersmoothing Through Slow Contraction, 2020) In addition, Edward & Xue (2018) demonstrated that "even in the presence of random perturbations, the evolution of the universe can be continued through the big bang" thereby allowing an "extension from a contracting to an expanding universe..." Similarly, Tukhashvili and Steinhardt wrote that, "According to well-known singularity theorems, a classical bounce requires a stable form of stress-energy that violates the null energy condition (NEC) or a modification of Einstein general relativity that avoids the null convergence condition (or both). Although this comes at the cost of some form of instability in certain cases, examples that avoid all known instabilities have been identified." (Cosmological Bounces Induced by a Fermion Condensate, 2023)
Entropy and the Contracting Universe
A common objection against models that postulate an eternal universe is that entropy always increases. As a consequence, it should have reached equilibrium already. But in this model the entropy growth only starts after the bounce since there was no baryonic matter prior to it. As Dr. Pavlovic pointed out:
"The question of entropy can be more easily approached in this framework than considering a completely cyclic Universe, since one can just start with an initial condition that the Universe is empty and contracting and then there are matter fields created after the bounce from the scalar field – and one does not have to explain how was this initial configuration achieved (while in cyclic models the empty contraction needs to be explained considering a transition from the expanding phase of previous cycle – and this is the place where the difficulties with entropy usually arise)." (Personal Correspondence)
But what about the eternal fields? Don't they have entropy? Dr. Pavlovic replied: "I never saw a proper and exact discussion on entropy of fields and particle content of matter in the Universe – if one looks at the available articles dealing with this question, it is all mostly hand-waving and guess-work. ... There appears to be no meaning in associating entropy with classical fields [and] there is no commonly accepted way to calculate entropy of quantum fields on curved spacetimes and such attempts mostly lead to divergences. Now, in the paper you sent the author proposes some identities for the entropy of scalar fields on De-Sitter. But, we can see that those proposed equations are not meaningful in the case of the bounce, where H=0, and according to the proposed equations the entropy would diverge. On the other hand, in general I am not very impressed by the methods used in the paper ... for instance, the approach used assumes the separation of metric on the background and dynamic part, which should not be appropriate for strong gravitational fields, where one would expect the manifestation of non-linear and dynamic properties of spacetime, contained already in the classical general relativity.
So to try to put my opinion simple – the question of entropy of quantum fields on curved spacetime is a difficult unsolved problem of theoretical physics in its own right, so if one wants to focus on the bounce one cannot also solve this problem in the same paper... So, considering the classical regime such issues (and in such issues it is not actually clear how to proceed since not much is known and even less generally accepted) are avoided. Even if the current understanding of entropy on curved spacetime would be a bit better than it is, still it is probably not good to base the model on an ingredient which has a shaky and uncertain foundation." (Personal Correspondence)
Geodesic Completeness of the Boucing Universe
Another potential objection is that the pre-bounce universe is geodesically incomplete and is thus subject to the same problems. Cosmologist Paul Steinhardt explained very clearly it is complete: "The wedge diagram need not terminate on the left-hand side. In the case of a one-time bounce model considered in this paper, the wedge might extend arbitrarily far and grow infinitely wide as a → −∞... [and] as is apparent in Figure 3, every co-moving particle in the observable universe has a worldline that can be traced back through the bounce arbitrarily far back in time. [Therefore], the cosmology is past geodesically complete." (Bouncing Cosmology Made Simple, pp. 7-9)
Spacetime Without Matter?
In this bounce model, it is postulated that the contraction phase is entirely empty of baryonic matter. But how can it be? Isn't spacetime dependent on matter/mass to exist? How is time defined in a de Sitter space without mass? Does the mass density of the cosmological constant play the role of the mass in the de Sitter space? I asked this to physicist Aron Wall. He replied: "There is a concept in relativity called "proper time", which is the amount of duration experienced by a particle following some trajectory through spacetime. If the particle does not have any rest mass, then it travels at the speed of light and so its proper time is 0. However, if you have a spacetime (e.g., de Sitter spacetime), then such a spacetime is not a point particle that follows a particular 1D trajectory, it is rather a 4 dimensional geometry with 3 dimensions of space and 1 of time. So the concept of proper time does not apply to it directly! There are however infinitely many possible 1 dimensional curves you could identify inside the geometry, and you can still ask about their durations (if they are timelike) or distances (if they are spacelike)... So no, mass is not necessary for time to exist."
Cosmologist Sean Carroll replied to my email, too: "Time is useful to keep track of how things change. But time would still exist even if there were no stuff around to mark its passage."
Physicist Matthew Kleban also said that physics allows space to exist without matter and energy: “Can matter exist without space? ... We don't know for sure. But what we can say is that we have a theory that relates matter to space, and to time as well. ... That theory is called "general relativity." It's the theory that Albert Einstein, in 1916 or so, first discovered. According to Einstein, there's... a set of equations, called "Einstein's equations", which relate matter to space and time. It turns out it's possible to solve those equations when there's no matter. You can have zero matter but still have space and time. But it's not possible to solve those equations when you have matter but without space and time. In other words, you can have space and time without matter but, at least according to Einstein's equations, you can't have matter without space and time.”
It is worth noting that there are solutions to the Einstein Field Equations in which space-time exists but there is no matter at all. For example, there are solutions to the Einstein Field Equations in which there are only gravitational waves populating the universe.
For further reading about Bounce models, see this great article: How the Universe Got Its Bounce Back
Cyclic Universe
Cyclic Universe
Borde et al. also stated that the BGV applies to the brane cyclic model as well: "We finally comment on the cyclic universe model in which a bulk of 4 spatial dimensions is sandwiched between two 3-dimensional branes. The effective (3 + 1)- dimensional geometry describes a periodically expanding and recollapsing universe... There are brief periods of contraction, but the net result of each cycle is an expansion. For null geodesics each cycle is identical to the others, except for the overall normalization of the affine parameter. Thus, as long as Hav > 0 for a null geodesic when averaged over one cycle, then Hav > 0 for any number of cycles, and our theorem would imply that the geodesic is incomplete."
In the article Cyclic Universe and Infinite Past, physicist Paul H. Frampton defended his model (which is different from the brane model) against the criticism of the BGV and added that the authors of the brane scenario disagree with Borde and Vilenkin's conclusion. He wrote: "There is a general argument about past completeness of the spacetime manifold which we address first. We begin with the no-go theorem of [Ref. 6 - BGV] which we shall adapt for application to the more general case in [Ref. 1], as the original no-go theorem applies to past inflation. We shall show how this no-go theorem [BGV] is by-passed, as the assumptions no longer apply... We normalize the affine parameter to the present time t = t0 by choosing with a0 = a(t0)... Whether or not the past incompleteness arguments apply to the competing cyclic model of [Ref. 7–9 – Brane model], we take no position. In [Ref. 6 - BGV], it is argued that they do, but the authors disagree [Ref. 10 - Steinhardt], so that jury is still out."
Plus, in the book Endless Universe (p.83), Turok and Steinhardt emphasized that "in the cyclic universe model described thus far [the brane model], the cycles continue regularly forever into the past and future. At present, there is no known theorem or principle that prevents this from occurring."
In the book From Quantum to Cosmos (pp. 143-144), Dr. Turok repeated the same conclusion: "We have developed the picture into a cyclic universe scenario, consisting of an infinite sequence of big bangs, each followed by expansion and then collapse, with the universe growing in size and producing more and more matter and radiation in every cycle. In this picture of the universe, space is infinite and so too is time: there is no beginning and there is no end."
Finally, in the article Geodesically Complete Analytic Solutions for a Cyclic Universe, Turok and Itzhak Bars explained it is perfectly possible for a cyclic universe to be geodesically complete. However, Carrasco et al. criticized this solution. Nevertheless, in the article Sailing Through the Big Crunch-Big Bang Transition, Turok et al. responded to the criticisms as follows: "The key to our construction of classical geodesically complete solutions was to “lift” the action (e.g., the standard model coupled to Einstein gravity) to a Weyl invariant equivalent theory... Recently [however], Carrasco, Chemisanny, Kallosh and Linde rediscovered some of these divergences and, without paying attention to our discussion of conserved Weyl-invariant finite quantities, claimed the divergences necessarily spoil the geodesic completeness of our proposed big crunch-big bang transition. In this note, we demonstrate that this naive claim is incorrect."
Finally, in the article Geodesically Complete Analytic Solutions for a Cyclic Universe, Turok and Itzhak Bars explained it is perfectly possible for a cyclic universe to be geodesically complete. However, Carrasco et al. criticized this solution. Nevertheless, in the article Sailing Through the Big Crunch-Big Bang Transition, Turok et al. responded to the criticisms as follows: "The key to our construction of classical geodesically complete solutions was to “lift” the action (e.g., the standard model coupled to Einstein gravity) to a Weyl invariant equivalent theory... Recently [however], Carrasco, Chemisanny, Kallosh and Linde rediscovered some of these divergences and, without paying attention to our discussion of conserved Weyl-invariant finite quantities, claimed the divergences necessarily spoil the geodesic completeness of our proposed big crunch-big bang transition. In this note, we demonstrate that this naive claim is incorrect."
In new cyclic models, M-theory is not used, but such models are also geodesically complete: Linford, 2019; p.15. Nina Stein recently published a paper arguing that even the new cyclic models are geodesically incomplete. However, I asked a physicist whether this paper really demonstrates this, and he explained that Nina's reservations are not groundbreaking; instead, they constitute minor adaptations of arguments that have been presented for more than a decade. These concerns were familiar to many proponents of the cyclic scenario well in advance of Nina's articulation and were already addressed in the literature.
Similarly, a new Scientific American article was published where it was claimed that "new research pokes holes in the idea that the cosmos expanded and then contracted before beginning again", thereby implying that "the Universe Began with a Bang, Not a Bounce." However, this article was disputed by Dr. Ivan Agullo in an interview titled "Has the Big Bounce been ruled out?"
Roger Penrose's radically different version of the cyclic scenario (i.e., Conformal Cyclic Cosmology) has also been attacked by critics. In the video below, Penrose responded to some of these objections:
A recent paper by Suvashis Maity and his colleagues was published that disputed the viability of Penrose's Conformal Cyclic Cosmology (CCC) by claiming that it cannot avoid geodesic incompleteness. However, when I discussed this with Dr. Daniel Linford, he clarified that this argument hinges on the assumption that CCC must be applied to a specific type of space-time known as asymptotically k = -1 de Sitter space-time, which is spatially open and hyperbolic. The key point here is that we don't have concrete evidence to confirm that our universe conforms to this particular type of space-time. So, even if CCC turned out to be an accurate model of the cosmos, it doesn't necessarily imply that our space-time is geodesically incomplete. If our universe's space-time were asymptotically Minkowski instead, a past-eternal universe described by the CCC could still be true.
Other Problems with the Theorem
Prof. Vilenkin explained that the theorem predicts that if the universe is everywhere expanding (on average), then the histories of most particles cannot be extended to the infinite past. In other words, if we follow the trajectory of some particle to the past, we inevitably come to a point where the assumption of the theorem breaks down – that is, where the universe is no longer expanding (the geodesics are incomplete.)
However, in the book The Fallacy of Fine-Tuning (Ch. 6.2.) and in the paper Eternal Inflation, Past and Future (p.20), Anthony Aguirre clarified that this is true for all particles, except a set of measure zero. In other words, there are rare particles whose histories can be infinitely long. These particles have world lines (trajectories in space-time) that extend indefinitely into the past, and can prevent there being a time at which the universe is not expanding. The fact that they are rare does not make them unimportant, because they nonetheless thread a physical volume. That is to say, not all geodesics are incomplete: in particular, it does not apply to the comoving geodesics themselves. Thus there is nothing preventing some geodesics from continuing arbitrarily far into the past. (For further reading on this point, see Oerter, 2014)
But the problems do not stop here. Dr. Aguirre wrote: "A second worry is that the BGV is so general that it does not depend on any energy conditions. Why should this be worrisome? Because any space-time solves Einstein’s equations Gµν = 8πG c 4 Tµν for some energy-momentum distribution Tµν: simply pick any desired metric, compute Gµν from it, and divide by 8πG/c4. Without any constraint on the forms that Tµν can take, there is nothing to be said against the result. Thus the widespread interpretation of the BGV is essentially that a forever-expanding metric space-time is simply inconceivable – and this seems hard to accept."
Dr. Aguirre also published other papers where he proposed a model that completely evades the BGV. In the article Steady-State Eternal Inflation, he stated: "Since the advent of inflation, several theorems have been proven suggesting that although inflation can (and generically does) continue eternally into the future, it cannot be extended eternally into the past to create a “steady-state” model with no initial time. Here we provide a construction that circumvents these theorems and allows a self-consistent, geodesically complete, and physically sensible steady-state eternally inflating universe which avoids an initial singularity or beginning of time."
Prof. Vilenkin explained that the theorem predicts that if the universe is everywhere expanding (on average), then the histories of most particles cannot be extended to the infinite past. In other words, if we follow the trajectory of some particle to the past, we inevitably come to a point where the assumption of the theorem breaks down – that is, where the universe is no longer expanding (the geodesics are incomplete.)
However, in the book The Fallacy of Fine-Tuning (Ch. 6.2.) and in the paper Eternal Inflation, Past and Future (p.20), Anthony Aguirre clarified that this is true for all particles, except a set of measure zero. In other words, there are rare particles whose histories can be infinitely long. These particles have world lines (trajectories in space-time) that extend indefinitely into the past, and can prevent there being a time at which the universe is not expanding. The fact that they are rare does not make them unimportant, because they nonetheless thread a physical volume. That is to say, not all geodesics are incomplete: in particular, it does not apply to the comoving geodesics themselves. Thus there is nothing preventing some geodesics from continuing arbitrarily far into the past. (For further reading on this point, see Oerter, 2014)
But the problems do not stop here. Dr. Aguirre wrote: "A second worry is that the BGV is so general that it does not depend on any energy conditions. Why should this be worrisome? Because any space-time solves Einstein’s equations Gµν = 8πG c 4 Tµν for some energy-momentum distribution Tµν: simply pick any desired metric, compute Gµν from it, and divide by 8πG/c4. Without any constraint on the forms that Tµν can take, there is nothing to be said against the result. Thus the widespread interpretation of the BGV is essentially that a forever-expanding metric space-time is simply inconceivable – and this seems hard to accept."
Dr. Aguirre also published other papers where he proposed a model that completely evades the BGV. In the article Steady-State Eternal Inflation, he stated: "Since the advent of inflation, several theorems have been proven suggesting that although inflation can (and generically does) continue eternally into the future, it cannot be extended eternally into the past to create a “steady-state” model with no initial time. Here we provide a construction that circumvents these theorems and allows a self-consistent, geodesically complete, and physically sensible steady-state eternally inflating universe which avoids an initial singularity or beginning of time."
Now, US Navy warfare analyst James Sinclair (not a cosmologist) asserted that Aguirre's proposal doesn't allow the universe to be past-eternal: "It is possible, then, to evade the BGV theorem through a gross deconstruction of the notion of time. Suppose one asserts that in the past contracting phase the direction of time is reversed. Time then flows in both directions away from the singularity. Is this reasonable? We suggest not, for the Aguirre-Gratton scenario denies the evolutionary continuity of the universe which is topologically prior to t and our universe. The other side of the de Sitter space is not in our past. For the moments of that time are not earlier than t or any of the moments later than t in our universe. There is no connection or temporal relation whatsoever of our universe to that other reality. Efforts to deconstruct time thus fundamentally reject the evolutionary paradigm." (Sinclair, 2009)
In response to Sinclair's allegations, Dr. Aguirre wrote: "First, there is no singularity – that is the whole point. Second, he simply asserts that “the evolutionary continuity of the universe [is] topologically prior to t and our universe.” And I'm not really sure what this means. I would agree in some sense that “there is no connection or temporal relation whatsoever of our universe to that other reality.” But I'm not sure why this matters. In the Aguirre-Gratton type model, the idea is to generate a steady state. If you do so, you can use the BGV theorem to show that there is a boundary to it. We have specified what would have to be on that boundary to be consistent with the steady state, and argued that this can be nonsingular, and also defines a similar region on the other side of that boundary (which in fact might be identifiable with our side, topologically). There is no “initial time,” nor any “time” at which the arrow of time changes. In the natural definition of time in this model, there is no earliest time, and all times are (statistically) equivalent." (Stenger, 2011; Ch. 6.2)
Philosopher of physics Daniel Linford also raised some questions about the criticisms directed at Aguirre's model: "One of the frequent mistakes that... Sinclair make[s] is to think that by surveying a large number of cosmological models, and showing that none of the models has some feature, we should conclude that the universe lacks the feature in question. Many models of the early universe are toy models and are not meant to accurately represent the universe anyway. For example, ...Sinclair criticize[s] the Aguirre-Gratton model, but that model was never meant to be a realistic model. Instead, that model was meant to demonstrate a point about the BVG theorem. A number of [non-expert critics] don’t understand this because they don’t seem to understand how theoretical physics is practiced. For whatever reason, [they] think that they are accomplishing some significant task about arguing against that model. Aguirre has a sequel paper in which he presents a model that is meant to be much more realistically construed, but [they] seem to ignore the model from the sequel paper." (Personal Correspondence)
I would only add that even if these models turn out to be incomplete, we should not simply conclude they are wrong. The Big Bang theory is also incomplete and has its problems (e.g., the flatness problem, the horizon problem, magnetic monopoles and baryon asymmetry). Inflationary cosmology was developed to solve these problems, but some physicists think it also has some problems (e.g., the measure problem and the problem of initial conditions). Nevertheless, most cosmologists agree that the Big Bang and Inflation did happen. The fact that some cosmological model is incomplete does not, in any way, mean it is wrong or that we should simply abandon it. Otherwise the Big Bang theory would have been discarded a long time ago, as physicist Yasunori Nomura excellently pointed out in the context of Inflation: "The Big Bang theory has its own problems. That is, of course, what Inflation is supposed to solve. There is a problem with explaining homogeneity, flatness... [But should we] say that the Big Bang theory is incorrect because it has a problem? No, you just have to solve further."
Andrei Linde, one of the founders of Inflation, wrote a paper titled Inflation, Quantum Cosmology and the Anthropic Principle (p.6), arguing the BGV does not demonstrate Eternal Inflation is finite in the past: "That is why I called this scenario ‘chaotic inflation’. There is a big difference between this scenario and the old idea that the whole universe was created at the same moment of time (Big Bang), in a nearly uniform state with indefinitely large temperature. In the new theory, the condition of uniformity and thermal equilibrium is no longer required. Each part of the universe could have a singular beginning [Ref. BGV]. However, in the context of chaotic inflation, this does not mean that the universe as a whole had a single beginning. Different parts of the universe could come to existence at different moments of time, and then grow up to the size much greater than the total size of the universe. The existence of initial singularity (or singularities) does not imply that the whole universe was created simultaneously in a single Big Bang explosion. In other words, we cannot tell anymore that the whole universe was born at some time t = 0 before which it did not exist. This conclusion is valid for all versions of chaotic inflation, even if one does not take into account the process of self-reproduction of the universe discussed in Section 4."
Linde explained further his argument in the paper Inflationary Cosmology (pp.16-17). He wrote: "There is still an ongoing debate of whether eternal inflation is eternal only in the future or also in the past. In order to understand what is going on, let us consider any particular time-like geodesic line at the stage of inflation. One can show that for any given observer following this geodesic, the duration τᵢ of the stage of inflation on this geodesic will be finite. On the other hand, eternal inflation implies that if one takes all such geodesics and calculate the time τᵢ for each of them, then there will be no upper bound for τᵢ , i.e. for each time T there will be such geodesic which experience inflation for the time τᵢ > T. Even though the relative number of long geodesics can be very small, exponential expansion of space surrounding them will lead to an eternal exponential growth of the total volume of inflationary parts of the universe. Similarly, if one concentrates on any particular geodesic in the past time direction, one can prove that it has finite length [Ref. 59 - BGV], i.e. inflation in any particular point of the universe should have a beginning at some time τᵢ. However, there is no reason to expect that there is an upper bound for all τᵢ on all geodesics. If this upper bound does not exist, then eternal inflation is eternal not only in the future but also in the past. In other words, there was a beginning for each part of the universe, and there will be an end for inflation at any particular point. But there will be no end for the evolution of the universe as a whole in the eternal inflation scenario, and at present we do not have any reason to believe that there was a single beginning of the evolution of the whole universe at some moment t = 0, which was traditionally associated with the Big Bang."
In the book Classical and Quantum Cosmology (p.276), cosmologist Gianluca Calcagni confirmed Linde's conclusion: “[The BGV] is an important achievement but, unfortunately, inconclusive like its predecessors: from geodesic incompleteness one cannot infer that inflating universes have a unique beginning.”
Dr. Calcagni repeated this point in a more recent paper: “This theorem [i.e., the BGV] is very powerful and one might think it seems to put the final word on the singularity problem. However, this is not accurate. Having past-incomplete world-lines is not enough to prove the existence of a global singularity. For that, we would need to show that past-incompleteness happens for all observers at the same time.” (Calcagni, 2022; pp. 31-32)
Linde made this argument (i.e., that the eternal inflationary multiverse can be past-eternal) even before the BGV theorem was produced: “There was no beginning and there will be no end to the universe's evolution. However each mini-universe (each bubble) has its beginning in time.” (Linde, 1982; 4-5)
And, indeed, many prominent cosmologists believe that the inflating spacetime (i.e., multiverse) could be past-eternal. I'll quote some: “All over the multiverse, universes are breaking out into existence, growing to cosmic proportions, each one at its own pace and made up in its own particular way. If we follow back the existence of our own universe, we find that it is embedded like a pustule in a much wider spacetime that has existed for all eternity.” (Ferreira, 2014; Ch. 14) –– “It is believed that each one of the baby universes must have had an origin in time but that the whole assemblage – the multiverse – need not have had an origin and instead may be eternal.” (Gould, 2018; 79-80) –– “If inflation erases all record of the cosmic past, how do we know there was any beginning at all? What we have been calling the big bang could simply be what came after inflation. If so, what might have come before inflation? Well, maybe more inflation, back to eternity. Each bubble starts with a big bang, and embarks on a life cycle of beginning, middle and (maybe) an end. The overall system, however, is eternal.” (Davies, 2021; 117-118) –– “Probably the whole self-reproducing network of inflating bubble universes need have no beginning, but particular bubbles may have beginnings when their histories are traced backwards.” (Barrow, 1999; 171-172) –– “An eternally reproducing universe would take us from bubble universe to bubble universe, all branching out from each other, endlessly back into the past. If eternal inflation is correct, then our origin is pushed all the way back to the infinite past.” (Mersini, 2022; Ch. 11)
The paper On the Past-Completeness of Inflationary Spacetimes (p.8), which was recently published by Paul Davies, Lesnefsky and Easson is another important work that challenges the BGV theorem. It says the following: “We discuss the question of whether or not inflationary spacetimes can be geodesically complete in the infinite past. Geodesic completeness is a necessary condition for averting an initial singularity during eternal inflation. It is frequently argued that cosmological models which are expanding sufficiently fast (having average Hubble expansion rate H avg > 0) must be incomplete in null and timelike past directions. ... We show this claim is an open issue. ... We present an improved definition for H avg and introduce an uncountably infinite cohort of cosmological solutions which are geodesically complete despite having H avg > 0. ... These [cosmological solutions] are generic and thus are stable to perturbations in initial conditions. ... If one defines H γ by Eq. 15 [i.e., BGV theorem equations], the suppression by a finite 1 λ f−λ i makes it difficult for Havg → 0 in the limit, yielding false positives for geodesic incompleteness: this is equivalent to an a priori assumption of incompleteness. Considering this, it is not surprising the aforementioned use of H avg claims to find all inflationary spacetimes to be incomplete.”
Furthermore, Leonard Susskind, who is professor of theoretical physics at Stanford University, has argued that past-eternal inflation is feasible. In the paper entitled Is Eternal Inflation Past-Eternal? he presented equations to support this conclusion and addressed the arguments against his position. Susskind wrote: "Let me now review the counterargument for a beginning and discuss its validity. The argument is based on the observation in that an inflating geometry must be past-incomplete... This argument seems compelling until one considers that the overwhelming majority of observers exist at asymptotically late time. It is clear from the figure that if the red bubble is pushed up to very late time – squeezed into the upper left corner – any signal from the blue line will be extremely red shifted... [Therefore], an inflating universe which is future-eternal must also be past-eternal."
Philosopher of physics Dan Linford also argued the BGV allows for a beginningless past in his paper Big Bounce or Double Bang?: “The BGV theorem is a result concerning the incompleteness of a congruence of time-like or null geodesics through a particular family of space-times as opposed to a more general result about the incompleteness of all of the time-like or null geodesics in a given space-time. Suppose that the average expansion rate along the time-like/null geodesics in the portion of space-time within our cosmological horizon is positive. If so, the BGV theorem tells us that those geodesics cannot be extended infinitely far into the past and remain within a classical space-time. Nonetheless, there could be time-like/null geodesics in regions beyond our cosmological horizon along which the average expansion rate is not positive. In that case, at least some time-like/null geodesics beyond our cosmological horizon could be extended infinitely far into the past. (That is, at least some time-like/null geodesics beyond our cosmological horizon could be complete, even if no geodesic within our cosmological horizon is complete, and the proper time measured along those geodesics could be infinite.) ... Moreover, even if all of the time-like/null geodesics within a given space-time were incomplete, the conclusion that there is an absolute beginning for all time-like/null geodesics does not follow from the statement that every time-like/null geodesic has a beginning. General Relativistic space-times can be sufficiently heterogenous as to preclude the possibility of defining an absolute beginning. ... [Physicist James] Sinclair briefly discuss[es] this matter, (wrongly) interpreting it as an objection to the theorem... and call[s] the “objection” “misconstrued”. ...Sinclair go[es] on to assert that if the universe is eternal then if “we look backward along the geodesic, it must extend to the infinite past if the universe is to be past eternal”. But this is false, as I’ve discussed; the point is that a space-time manifold can be geodesically incomplete – in the sense proved by the BGV theorem – without having an absolute beginning.”
Dr. Linford clarified further in his doctoral dissertation: "[Proponents of an absolute beginning] have argued that the BVG theorem, together with the evidence that the BVG theorem applies to the space-time we inhabit, provides compelling evidence for the conclusion that the Cosmos began to exist. As they interpret the BVG theorem, the BVG theorem shows that space-time could not have been expanding forever and must have begun in the finite past. Nonetheless, our observational data at most allows us to deduce that space-time has been expanding in our past – or within our observational horizon – and not that the entirety of space-time has been expanding. And, as we’ve already seen, any space-time that is everywhere past singular is weakly and super weakly observationally indistinguishable from a space-time that is not everywhere past singular. Space-times to which the BVG theorem applies are no different. ... Consequently, there could be time-like geodesics in regions beyond our cosmological horizon that can be extended infinitely far into the past regardless of whatever data we might gather.” (Cosmic Skepticism and the Beginning of Physical Reality, p.203)
Elsewhere Dr. Linford wrote: “Singular FLRW space-times – that is, FLRW space-times such that all time-like or null curves are inextendable beyond some boundary located in the finite past – are super weakly observationally indistinguishable from space-times containing at least one time-like or null curve that is not bounded to the past. For example, the Borde-Guth-Vilenkin (BVG) theorem (Borde et al. (2003)) has sometimes been interpreted to show that the universe has a past singular boundary. However, the BVG theorem actually shows that if the average value of a specific generalization of the Hubble parameter along a time-like or null geodesic congruence is positive, then that congruence must be incomplete to the past. The resulting congruence can be isometrically embedded either into a space-time with or without a global past boundary. That is, supposing that we inhabited such a congruence, we might see a boundary to our past, even though other observers in the same space-time would have an infinite and unbounded past. So, there is reason to doubt, first, that space-time extendability’s relationship to a beginning of the universe has been successfully characterized and, second, that, even given a correct characterization, we could have epistemic access to sufficient data to determine important global properties of the space-time we inhabit, including whether our spacetime has a global boundary in the finite past.” (Neo-Lorentzian Relativity and the Beginning of the Universe, p.18)
In the article The Big Bang is a Coordinate Singularity for k = −1 Inflationary FLRW Spacetimes (pp. 3, 25, 37-38), physicist Eric Ling wrote: "We show that the big bang is a coordinate singularity for a large class of k = −1 inflationary FLRW spacetimes which we have dubbed ‘Milne-like.’... What’s unique about Milne-like spacetimes is that they extend into a larger spacetime because the big bang is just a coordinate singularity... We show why the singularity theorems don’t apply to inflationary spacetimes. Indeed Milne-like spacetimes can almost always be used as counterexamples... Hawking’s cosmological singularity theorems both assume the strong energy condition... [However], inflationary models do not satisfy the strong energy condition. Therefore the singularity theorems above no longer apply. New singularity theorems were sought that did not require the strong energy condition. This was done by Borde and Vilenkin and others. [Nevertheless], Borde and Vilenkin found that some models of inflationary theory also violate the weak energy condition. Then Guth, Borde, and Vilenkin produced a singularity theorem, which showed that, even if the weak energy condition is violated, then one has past incompleteness. However their theorem only applies to inflating regions of a spacetime. For example, their theorem applies to the Milne universe (because their theorem only requires an averaged Hubble expansion condition), but the Milne universe isometrically embeds into Minkowski space which is geodesically complete... [That is], any incomplete geodesic [in the Milne space] extends to a complete geodesic within Minkowski space."
Let me just clarify what is meant by "coordinate singularity." It is not a real (or physical) singularity, but a consequence of an inappropriate choice of coordinates, as physicist Anthony Lasenby explained in his book General Relativity: An Introduction for Physicists (p.250). Dr. Ling confirmed this in an email to me. He wrote: "The coordinate singularity demonstrates that it is possible to extend the universe past the big bang for Milne-like spacetimes. This is analogous to how one extends the Schwarzschild solution through the event horizon since r = 2m is just a coordinate singularity."
It may be argued that our current best estimates of k suggest that k=0, while Dr. Ling only discussed k= -1. However, the inflationary mechanism is designed to “flatten out” the universe, so that k appears to be zero from the perspective of an observer in our Hubble volume. It could still be that the universe has non-vanishing Gaussian curvature at much larger length scales, in which case k could be -1.
However, in the paper titled Instability of The Big Bang Coordinate Singularity in a Milne-like Universe (p.18), it was argued that coordinate singularities are unstable: "The heuristic explanation for the different behavior discussed in the previous paragraph may be twofold. First, the nonsingular bounce has a nonvanishing cosmic scale factor a(t) at the bounce, whereas the Milne-like universe has a vanishing cosmic scale factor a(t) at the big bang, which requires delicate cancellations in order to give a mere coordinate singularity and these cancellations may be destroyed by perturbations of the metric."
So, I sent this to Dr. Ling and he kindly responded the following: "A lot of the analysis that Klinkhamer and Wang perform are for q-theory models. q-theory is a nice framework which can be incorporated into Milne-like spacetimes (which Klinkhamer showed in a joint paper with me), but Milne-like spacetimes are still more general. In their paper, Klinkhamer and Wang show that if one performs a perturbation analysis on their spacetimes, then one finds blow up for perturbed curvature quantities as one evolves the perturbations backwards in time towards the big bang. I don't fully agree with this analysis (not because it's wrong), but because I don't think one should evolve perturbations backwards in time all the way towards the big bang. Let me explain this. We observe the universe to be homogeneous and isotropic. As time evolves forward, this homogeneity starts to decrease (galaxies cluster together, solar systems form, etc.). In the past the universe was more homogeneous than it was today. And if you keep going towards the past, the universe gets more homogeneous. This reasoning reaches its conclusion at the big bang, where the universe was in it's most homogeneous and isotropic state. Now perturbation theory is a way to break the exact homogeneity of the universe. If one applies perturbation theory, then one is decreasing the homogeneity. So I agree with applying perturbation theory towards the future (where we know homogeneity decreases), but I have problems with applying perturbation theory all the way down to the big bang. Essentially my problem can be rephrased in the following question: if the big bang is when the universe was most homogeneous, why run perturbation theory (which breaks homogeneity) towards the big bang? I believe that one should have some geometric surface to describe the big bang and run perturbation theory (towards the future) off the surface. One of the appeals of Milne-like spacetimes is that it gives such a surface to describe the big bang."
In addition, even the authors of the paper admitted in the conclusion that there are solutions. They concluded (p.18): "This last problematic point of the Milne-like model would disappear if the variable quantity ρV (q) were replaced by a positive cosmological constant Λ, but then the question resurfaces as how to remove the nonvanishing vacuum energy density as the universe evolves. More attractive would be to keep the dynamic vacuum energy density ρV (q) and to find a mechanism that freezes q(x) at t = 0, which then forces having a vanishing perturbation δq(x) = 0 at t = 0 [this would effectively set Cklm = 0 in (3.30)]. Such a freezing of the q-field has been discussed in a different context (cf. Sec. IV of Ref. [16]) and may indeed be the preferred way to rescue the big bang coordinate singularity of the Milne-like universe."
In the paper titled Eternal Universe, C. Wetterich stated: "Assuming the strong energy condition Penrose and Hawking have shown the presence of a past singularity or geodesic incompleteness for rather arbitrary cosmological solutions [Ref. 1, 2 - Hawking-Penrose singularity theorems]. With the advent of inflation the strong energy condition has been abandoned. Still, with the rather mild assumption that the universe is expanding in the average (more precisely, that the average Hubble parameter is positive) it has been established that geodesics cannot be complete towards the past [Ref. 3, 4 - BGV]... [However], we argue that past-incomplete geodesics do not necessarily indicate a singularity or beginning of the universe. It merely reflects that these geodesics cannot be realized by asymptotic massive particles. All other trajectories become photonlike in the infinite past and proper time has to be replaced by a more appropriate concept as oscillation time. The oscillation time distance to the infinite past is infinite for all particles... [In this proposal] the geometry is obviously regular and geodesically complete... [and] physical observables show no singularity from the infinite past to the infinite future. While the universe is evolving, there is no beginning and no end – the universe exists forever."
More recently (2019), Wetterich published a paper titled The Great Emptiness at the Beginning of the Universe discussing the issue in more details. Here is an extensive (but important) explanation of his complex research: "The statement that such a Universe can last since ever may encounter more doubts. It has been argued that the limit t → −∞ corresponds to a singularity which is necessarily reached at a finite proper time [Ref. 9–12 - BGV and classical singularity theorems]. This “geodesic incompleteness” has given rise to the opinion that in standard inflationary cosmology the Universe starts with a singularity... We argue here that in many models of inflation there is no physical initial singularity and physical time tₚₕ extends to the infinite past, tₚₕ → −∞... We note that proper time is useful for many purposes, but it is not a reasonable physical time when we consider the Universe towards its “beginning”. As is well known, proper time cannot be used for massless particles, and we have just seen that all particles become effectively massless for t → −∞. Furthermore, proper time is not a frame invariant quantity but rather depends on the specific choice of a metric field... We will see that in the scaling frame the big bang singularity is absent, demonstrating that the big bang singularity is actually a “field singularity” due to an inappropriate choice of fields, rather than a physical singularity. In some respect it is analogous to the coordinate singularity at the south pole in Mercator projection coordinates, except that we speak now about “coordinates in field space”... We have shown that standard inflationary cosmologies can be extrapolated backwards to the infinite past in physical time, as measured by the number of oscillations of photons. This applies to our observed inhomogeneous Universe. No physical big bang singularity is present for these models. The often discussed singularity is only apparent, being related to a singular, and therefore not very appropriate, choice of coordinates in field space. Field relativity permits us to use better adapted choices for the metric field. In particular, in a primordial flat frame the averaged geometry becomes flat Minkowski space in the infinite past. The absence of singularities is very apparent."
I would also like to draw attention to an intriguing paper written by the physicist Cristi Stoica from the Horia Institute of Physics. The paper is titled The Friedmann-Lemaitre-Robertson-Walker Big Bang Singularities are Well Behaved, and it says the following:
“We prove a theorem showing that the Einstein equation [of General Relativity] can be written in a non-singular form, which allows the extension of the spacetime before the Big Bang. ... The solution proposed here is simple: to show that the singularities of the FLRW model don’t break the evolution equation, we show that the equations can be written in an equivalent form which avoids the infinities in a natural and invariant way.”
Stoica's work is interesting because it proposes that the Hawking and Penrose singularity theorems do not entail the break down of the spatio-temporal manifold at some point in the past even if we accept all of the theorems' assumptions, ignore quantum gravity and reject modifications of General Relativity (e.g., Einstein-Cartan theory). I sent an email to Dr. Stoica asking whether his solution of GR is generic or merely a special case. He replied:
“I agree that the solution should be general. In a series of papers, and then in my PhD Thesis, I did the following:
- find a way to extend semi-Riemannian geometry to work with a class of singular metrics
- apply it to FLRW and to the three main types of black holes.
There are more ways in which a metric can be singular. The simplest one is if the determinant of g_{ab} becomes zero. Even so, the covariant derivative and the curvature requires g^{ab}, which is singular when the determinant of g_{ab}->0. So it was nontrivial even for the simplest case. I call the first case "benign singularity". The more complicated case is when g_{ab} has components that go to infinity, I call it "malign singularity". This is the case of black holes singularities. In this case, my theory doesn't work directly. But I found out that a coordinate transformation can sometimes make the metric that is a malign singularity to become a benign singularity. Remember how the Eddingtion-Finkelstein coordinate transformation shows that the r=r_m singularity, which is the event horizon, is not actually a singularity, but the Schwarzschild coordinates themselves are the singular ones, and not the metric. This works for r=r_m, but not for r=0, because for r=0 there are scalar invariants that are singular and remain singular even if you make the transformation (K=R^{abcd}R_{abcd}). But I found out that I can make coordinate transformations that remove the part of the singularity that is due to the coordinates, and makes the metric smooth at r=0 too. The metric is still singular, but the singularity is benign. So my approach also works for black hole singularities. It doesn't work for all conceivable singularities, but it works for the known BB [Big Bang] and BH [Black Hole] singularities, and for some general classes. I don't know if in the real world there are singularities for which my approach wouldn't work. I conjecture that there are only benign and malign that can be turned into benign.”
I then proceeded to ask Dr. Stoica whether his solution works in light of the BGV theorem, that is to say, whether his solution implies the geodesics are complete in the past. He responded:
“[T]here are solutions where my geometry works even if the BGV theorem you mention applies... The same applies to the future geodesic incompleteness in black hole singularity theorems, but I showed that for the usual symmetric solutions my geometry works. Similarly, I showed that my geometry works for some big bang models that were thought to be singular. And the geodesics can be extended in the cases where my geometry works, that includes symmetric black holes and big bangs, even if they don't in standard geometry. ... My geometry works in cases where standard geometry fails.”
So, it is clear the BGV does not prove the universe must have an "absolute beginning."
Cooked. Time to move on.
Vilenkin-Mithani's Paper
A similar claim is made regarding the short paper entitled Did the Universe Have a Beginning? published by Vilenkin and Mithani. They wrote:
"We discuss three candidate scenarios which seem to allow the possibility that the universe could have existed forever with no initial singularity: eternal infation, cyclic evolution, and the emergent universe. The first two of these scenarios are geodesically incomplete to the past, and thus cannot describe a universe without a beginning. The third, although it is stable with respect to classical perturbations, can collapse quantum mechanically, and therefore cannot have an eternal past."
Unfortunately, this paper does not prove the universe must be finite. Instead, it specifically criticizes just three models. However, there are many other proposals that feature a past-eternal universe. If there are some cosmological models that are eternal but have problems, there’s no reason to stop looking for other models that are also eternal but don’t have those problems. But do their criticisms hold? Let's start by examining their analysis of the cyclic scenario. They just give a quick comment on that one:
"Another possibility could be a universe which cycles through an infinite series of big bang followed by expansion, contraction into a crunch that transitions into the next big bang. A potential problem with such a cyclic universe is that the entropy must continue to increase through each cycle, leading to a “thermal death” of the universe. This can be avoided if the volume of the universe increases through each cycle as well, allowing the ratio S/V to remain finite. But if the volume continues to increase over each cycle, [then] the universe [must be] past-incomplete."
I sent this criticism to Dr. Paul J. Steinhardt, a theoretical physicist whose principal research is in cosmology and condensed matter physics. He is currently the Albert Einstein Professor in Science at Princeton University where he is on the faculty of the Departments of Physics and Astrophysical Sciences. Steinhardt received his Bachelor of Science in Physics at Caltech, and his Ph.D. in Physics at Harvard University. But more importantly, he is one of the main proponents of the cyclic scenario. He responded the following:
"(1) It has been a long time since we have invoked extra dimensions and branes in designing cyclic models. Those notions turned out not to be essential so we abandoned them a dozen years ago. Think about just the usual 3 space dimensions and one time; we can use a scalar field plus potential similar to inflation but the potential has different properties. (2) Vilenkin's argument does not apply. We have to assume that space is infinite throughout and that during each cycle space stretches by a constant factor. Multiplying an infinite space by a constant factor still produces an infinite space. But, if space has a density of matter, multiplying by a constant factor decreases the density by a constant factor. (3) So, going forward in time, each cycle begins with the creation of a certain density of matter and radiation being created by the scalar field; by the end of that cycle, that density is exponentially diluted by all the expansion; it is negligible compared to new matter and radiation created after the bounce by the scalar field once again; the older stuff dilute and negligible; the next cycle occurs and the process repeats; all the older matter and radiation is superdiluted and only the new matter and radiation count. (4) How is it possible for this to continue? Gravity can produce unbounded amounts of energy – during each cycle, gravitational energy is converted to scalar field energy, which is converted then to matter and radiation. And nothing we know stops this from happening forever going forward. (5) Running backwards in time is a little more subtle to describe because you have to imagine that (going backwards) most of the matter disappears (into scalar field energy). (6) So, what Vilenkin missed is that the universe is not uniformly shrinking. This is a self-similar process that can repeat for ever back in time. An observer cannot tell one cycle from another – they see the exact same thing – from cycle to cycle."
The entropy argument is very simplistic, but a more sophisticated criticism was presented as well. It has been argued by the particle physicists Banks and Fischler that during the collapse the rapidly changing space-time would have excited and amplified random “quantum fluctuations” in such a way that entropy would have been driven to very large values, rather than small ones. This makes it even more difficult to account for the low entropy just after the big bang. In Banks’s words:
"The collapsing phase of these [cyclic] models always have a time-dependent Hamiltonian for the quantum field fluctuations around the classical background. Furthermore the classical backgrounds are becoming singular. This means that the field theories will be excited to higher and higher energy states... High energy states in field theory have the ergodic property – they thermalize rapidly, in the sense that the system explores all of its states. Willy Fischler and I proposed that in this situation you would again tend to maximize the entropy. We called this a Black Crunch and suggested the equation of state of matter would again tend toward p = ρ. It seems silly to imagine that, even if this is followed by a re-expansion, that one would start that expansion with a low entropy initial state."
I, again, sent the criticism to Paul Steinhardt, and he stated the following in an email: "The arguments by Fischler and Banks assume two things: (1) Einstein gravity remains valid all the way to the bounce (hence they assume there is some kind of singularity). (2) The bounce occurs when the temperature or concentration of matter are greater than the Planck density when black holes spontaneously have to form.
Neither has to be true in bouncing cosmology. In particular, in this paper we describe a cyclic model in which hardly any contraction is required to smooth the universe before the bounce and the bounce occurs before the concentration of matter rises significantly. There would be no black hole formation in this case. So their arguments apply for some bouncing models but not for all (and not for the ones that, in my opinion, are the best motivated)."
I also emailed physicist Petar Pavlovic. He kindly responded to my email: "I could agree to some extent with the arguments put forth by Banks and Fischler, but they are actually concerned with other types of models – which have a crunch singularity (which we demand that it is avoided in our model) and which are filled with some ideal fluid (while in ours the Universe is essentially empty, so there is no growth of perturbations which is significant). On the other hand, I think that even that analysis of Banks can not be taken again too seriously since it assumes things like the separation of space-time into classical and quantum perturbation part, which should not be considered as proper − specially in the regimes of quantum gravity. Actually, the main problem of such discussions is that they try to address the regime − that of quantum gravity − we almost know nothing about and do not have a proper theory for its treatment, and are therefore based on assumptions and rough estimates, instead of logic and firm physical principles."
Physicist Martin Bojowald also responded to my email. He wrote: "Entropy is certainly an important concept in cosmological models. In the non-singular models on which I worked, the big-bang singularity is replaced by a quantum phase which is non-singular but might have very non-classical, and therefore counter-intuitive properties of space-time. One should therefore not think about the transition as a bounce of an otherwise classical space-time, during which the build-up of entropy would be meaningful."
For a similar objection and the proper response, see: Is the “Radiation Paradox” a Problem for a Cyclic Universe?
Since cyclic models are being discussed here, it is worth addressing other objections to it. A common claim is that in 1998 scientists discovered that the universe is accelerating because of something called "Dark Energy" (although not all physicists agree) which seems to have a repulsive force. But to contract, the universe should stop and decelerate, we do not see this, rather we see it is accelerating, therefore, it will not contract in the future. The problem with this objection is that it assumes dark energy will never decay.
However, some physicists dispute this claim due to the considerable uncertainty about the nature and behavior of dark energy. There are many hypotheses about its future behavior. For example, in the Phantom energy hypothesis, the repulsive force becomes greater and the universe accelerates even more, and as a consequence "phantom energy rips apart the Milky Way, solar system, Earth, and ultimately the molecules, atoms, nuclei, and nucleons of which we are composed, before the death of the Universe in a “Big Rip”." It is also equally plausible that dark energy might dissipate (decay) with time and even become attractive. This leaves open the possibility that gravity might yet rule the day and lead to a universe that contracts in on itself in a "Big Crunch."
In the book Until the End of Time (p.298), Brian Greene wrote: "A yet more emphatic resolution would emerge from a quantum leap in which the value of the dark energy would suddenly change. Currently, the accelerated expansion of the cosmos is driven by a positive dark energy suffusing every region of space. But just as positive dark energy yields an outward-thrusting repulsive gravity, negative dark energy yields an inward pulling attractive gravity. Consequently, a quantum tunneling event in which the dark energy leaped to a negative value would mark a transition from the universe’s swelling outward to its collapsing inward. Such an about-face would result in everything – matter, energy, space, time – being squeezed to extraordinary density and temperature, a kind of reverse big bang that physicists call the big crunch."
In the paper Current Observations with a Decaying Cosmological Constant Allow for Chaotic Cyclic Cosmology (pp. 21, 25) George Ellis stated: "Note that furthermore it is perfectly possible, if perhaps unattractive, that the dark energy density is increasing at the present time but will decay at a later time. Indeed, if we model it as an effective scalar field, we can find a potential that will give virtually any behavior, so phenomenological dark energy models that give an increase in the future followed by a decrease are possible... There are a few ways in which a decaying Λ of section 4.2 can be achieved: The most natural is to introduce a constant to the scalar field potential used for the inflaton... so a decreasing ρΛ [cosmological constant/dark energy] in the future is very plausible."
Steinhardt also invokes the decay of dark energy. In the paper A New Kind of Cyclic Universe (pp. 3, 6), he wrote: "The current dark energy dominated phase may continue for a period into the future, but, in a cyclic universe, it must eventually terminate. For example, the dark energy may be due to a quintessencelike scalar field φ with scalar field potential V(φ)... The field begins to roll downhill only when the dark energy comes to dominate, which is only recently at H(t) ∼ H0, the present value of the Hubble parameter. The accelerated expansion phase ends and contraction begins when φ rolls from V(φ) > 0 to V(φ) < 0, which may be a few Hubble times. At this point, the scalar field changes from a quintessence-like field that drives accelerated expansion to an ekpyrotic field that governs the subsequent phase of ultraslow contraction. Note that the dark energy density plays several roles in the new cyclic theory: it sets the time when dark energy first comes to dominate, which is approximately the current Hubble time H0; it sets the characteristic time scale (up to a modest numerical factor) for the duration of the expanding phase; it also sets the contracting phase that follows (see below), and, hence, the total period of a cycle... The natural progression is from low pressure to high. The dark energy phase (ε ≈ 0) ends and contraction begins when the scalar field causing it rolls or decays to negative potential energy. The value of ε grows as the universe contracts to the bounce. The progression is natural."
In addition to the paper being discussed here, Vilenkin and Mithani published another one where a simple cyclic model was criticized [1]. However, proponents of the cyclic universe responded accordingly [2] and more recently (2018) extended their work [3]. Marc Sher also criticized bouncing universes [4], but Vahé Petrosian responded appropriately [5].
Moreover, there is another objection to cyclic models which involve instabilities. However, modern and sophisticated cyclic models are free from this problem. As Paul Steinhardt explained in the article Bouncing Cosmology Made Simple (p.9): "Chaotic mixmaster or BKL-like behavior can potentially destroy the smoothness of the universe during a period of contraction. However, as noted above, mixmaster behavior is totally suppressed because e ≥ 3 during the contraction phase."
Now, the final objection appeals to the geometry of the universe. General relativity explains that mass and energy bend the curvature of space-time and is used to determine what curvature the universe has by using a value called the density parameter. The density parameter is the average density of the universe divided by the critical energy density, that is, the mass energy needed for a universe to be flat. The geometry will determine the fate of the universe.
If the universe is flat, then it will expand forever. If it is closed, then it will stop and recollapse. Recent measurements by Wilkinson Microwave Anisotropy Probe have confirmed the universe is flat with only a 0.4% margin of error. However, in modern theories the universe is so big that it only looks flat, but the spatial curvature can be positive, thus allowing the universe to be closed, as is explained in the NASA website:
"Imagine living on the surface of a soccer ball (a 2-dimensional world). It might be obvious to you that this surface was curved and that you were living in a closed universe. However, if that ball expanded to the size of the Earth, it would appear flat to you, even though it is still a sphere on larger scales. Now imagine increasing the size of that ball to astronomical scales. To you, it would appear to be flat as far as you could see, even though it might have been very curved to start with. Inflation stretches any initial curvature of the 3-dimensional universe to near flatness."
"Imagine living on the surface of a soccer ball (a 2-dimensional world). It might be obvious to you that this surface was curved and that you were living in a closed universe. However, if that ball expanded to the size of the Earth, it would appear flat to you, even though it is still a sphere on larger scales. Now imagine increasing the size of that ball to astronomical scales. To you, it would appear to be flat as far as you could see, even though it might have been very curved to start with. Inflation stretches any initial curvature of the 3-dimensional universe to near flatness."
Regarding eternal inflation, Prof. Vilenkin just references the BGV and summarizes its conclusions (which have been previously discussed extensively) to support his claim that an inflationary model must be geodesically incomplete. He then goes on to criticize the Emergent scenario (which seems to be the focus of the paper). However, in response to the criticisms, Guendelman and Labrana published a paper titled Classically and Quantum stable Emergent Universe where they argued that the emergent scenario is stable:
"It has been recently pointed out by Mithani-Vilenkin that certain emergent universe scenarios which are classically stable are nevertheless unstable semiclassically to collapse. Here, we show that there is a class of emergent universes derived from scale invariant two measures theories with spontaneous symmetry breaking (s.s.b) of the scale invariance, which can have both classical stability and do not suffer the instability pointed out by Mithani-Vilenkin towards collapse. We find that this stability is due to the presence of a symmetry in the "emergent phase", which together with the non linearities of the theory, does not allow that the FLRW scale factor to be smaller that a certain minimum value a0 in a certain protected region."
I'll conclude this part with a quote by cosmologist Sean Carroll: "... a recent paper by Audrey Mithani and Alex Vilenkin, which concludes by saying "Did the universe have a beginning? At this point, it seems that the answer to this question is probably yes." Mithani and Vilenkin are also scientists, and are correspondingly willing to be honest about our state of ignorance: thus, "probably" yes. I personally think the answer is "probably no," but none of us actually knows."
Cooked. Time to move on.
I'll conclude this part with a quote by cosmologist Sean Carroll: "... a recent paper by Audrey Mithani and Alex Vilenkin, which concludes by saying "Did the universe have a beginning? At this point, it seems that the answer to this question is probably yes." Mithani and Vilenkin are also scientists, and are correspondingly willing to be honest about our state of ignorance: thus, "probably" yes. I personally think the answer is "probably no," but none of us actually knows."
Cooked. Time to move on.
Wall Theorem
There is another theorem that is sometimes presented as proof that singularities cannot be avoided. The title of the paper in which the theorem was presented is The Generalized Second Law implies a Quantum Singularity Theorem and its author (Aron Wall) claims it works even when quantum mechanics is considered.
This is a great paper, well worth reading – but it does not prove quantum gravity cannot avoid singularities. According to cosmologist Sean Carroll, this paper develops theorems that apply to semi-classical gravity (classical space-time with propagating quantum fields), and then speculates “the results may hold in full quantum gravity” and “there is a reasonable possibility that the Penrose singularity theorem can be proven even in the context of full quantum gravity.” As good as the paper is, creating a theorem in the semi-classical case and then opining that it may be extendable to the full quantum gravity case does not actually represent a “theorem” about the quantum case.
Moreover, theoretical physicist Abhay Vasant Ashtekar, Director of the Institute for Gravitational Physics and Geometry at Pennsylvania State University, explained this theorem does not demonstrate the singularity is unavoidable even in the semi-classical case. He said: "[The theorem] is assuming that some version of the second law of thermodynamics is really fundamental and should be a fundamental ingredient in the series that determine whether there will be a singularity or not, and I do not agree with that. I think that it really is something that will emerge in an approximate sense in the semi-classical regime."
Further, physicist Niayesh Afshordi pointed to an interesting assumption of the Wall Theorem: “[The] generalized second law that was proven by Aron Wall for a quantum singularity assumes an infinite space, [but] if you don't allow for infinities in your theory then you cannot take that theorem seriously.”
The problem is that we don’t know whether the universe is spatially infinite in extent or not as Joseph Silk, head of Astrophysics at the University of Oxford, explained:
“We do not know whether the Universe is finite or not. To give you an example, imagine the geometry of the Universe in two dimensions as a plane. It is flat, and a plane is normally infinite. But you can take a sheet of paper and you can roll it up and make a cylinder, and you can roll the cylinder again and make a torus. The surface of the torus is also spatially flat, but it is finite. So, you have two possibilities for a flat Universe [though these are not all of the possibilities]: one infinite, like a plane, and one finite, like a torus, which is also flat.” (Is the Universe Finite or Infinite?)
Similarly, George Ellis et al. explain: “We assume space extends forever in Euclidean geometry and in many cosmological models, but we can never prove that any realized three-space in the real universe continues in this way – it is an untestable concept, and the real spatial geometry of the universe is almost certainly not Euclidean. Thus, Euclidean space is an abstraction that is probably not realized in physical practice. In the physical universe, spatial infinities can be avoided by compact spatial sections, either resultant from positive spatial curvature or from choice of compact topologies in universes that have zero or negative spatial curvature, (for example FLRW flat and open universes can have finite rather than infinite spatial sections).” (Multiverses and Physical Cosmology)
The simplest example is the torus, but we also have the Klein bottle and the Hantzsche-Wendt manifold. See for example page 27 of Janna Levin’s Topology and the Cosmic Microwave Background, which shows you ten different closed (and therefore finite) flat 3-manifolds and William Thurston's Three-Dimensional Geometry and Topology.
In response to Dr. Afshordi, Wall replied that “in a finite space, at some time t in the past, the generalized second law would simply reduce to the ordinary second law of thermodynamics and... you can still argue from the ordinary second law of thermodynamics that most likely the universe had a geometrical beginning...”
In response to Wall, Dr. Linford stated: “The ordinary second law of thermodynamics does not actually imply that the universe had a beginning. It could be that the entropy was increasing forever into the past. There are mathematical functions that even though they're always monotonically increasing over the entire real line, never have some point at which that increase begins. There's also the possibility that there was an entry minimum at some point in the past, but that entropy minimum needs to be explained by some dynamical mechanism which precedes that entry minimum even at earlier times.”
Physicist Carlo Rovelli also commented on Wall’s response: “The ordinary second law has nothing to do with the beginning, absolutely. There are all sorts of possible models one can do in which there's no beginning and there is a second law. Entropy could be growing forever [i.e.,] since ever. That's one super simple possibility.”
These responses are enough to undermine the validity of this theorem, but to complement it, I sent an email to a leading researcher on quantum gravity, Martin Bojowald, and he responded as follows: "Theorems always rely on assumptions, and what is implicitly supposed in the statement you quote is that geodesics, a concept of Riemannian geometry, are still meaningful in quantum gravity near a singularity. However, it is widely expected that quantum gravity might well imply some non-classical, non-Riemannian geometry of space-time. The basic assumptions of the theorem, and therefore its conclusions, then break down. Possible outcomes of non-classical geometries have recently been studied, as reviewed here."
Another semi-classical approach, but with the opposite conclusion of this theorem, comes from creating a quantum version of the equation at the heart of the singularity theorem, the Raychaudhuri equation. Ahmed Farag Ali and Saurya Das did this in the paper Cosmology from Quantum Potential and argued that a repulsive force is created, preventing the formation of singularities and ensuring the universe is eternal into the past and future: "It was shown recently that replacing classical geodesics with quantal (Bohmian) trajectories gives rise to a quantum corrected Raychaudhuri equation... [and] since it is well known that Bohmian trajectories do not cross, it follows that even when θ → −∞, the actual trajectories (as opposed to geodesics) do not converge, and there is no counterpart of geodesic incompleteness, or the classical singularity theorems, and singularities such as big bang or big crunch are in fact avoided... [which] predicts an infinite age of our universe."
Lashin criticized the Ali-Das paper [1], [2], however, the proponents responded accordingly [3], [4].
In addition to this paper, Ali-Das published another one, titled Quantum No-Singularity Theorem from Geometric Flows, where they reached a similar conclusion: "In this paper, we analyze the classical geometric flow as a dynamical system. We obtain an action for this system, such that its equation of motion is the Raychaudhuri equation. This action will be used to quantize this system. As the Raychaudhuri equation is the basis for deriving the singularity theorems, we will be able to understand the effects such a quantization will have on the classical singularity theorems. Thus, quantizing the geometric flow, we can demonstrate that a quantum space-time is complete (non-singular). This is because the existence of a conjugate point is a necessary condition for the occurrence of singularities, and we will be able to demonstrate that such conjugate points cannot occur due to such quantum effects."
I anticipate that some folks will object the first paper requires the Bohmian interpretation of quantum mechanics (instead of using the standard formulation). And, indeed, in an email to me Dr. Ali confirmed this: "The idea that we used to resolve the big bang singularity is based on the quantal trajectories idea that follows from Bohm's interpretation of quantum mechanics. Since the quantal trajectories never cross and never focused on one point, it allows avoiding the big bang singularity. I think the idea of quantal trajectories can not be obtained in Copenhagen interpretation because this interpretation assumes that particles have no trajectories at all. Everett's interpretation does not have trajectories as well."
Fortunately the second paper doesn't require it. Dr. Ali wrote: "The paper 'Quantum no-singularity theorem from geometric flows' does not require Bohm's quantum version, but has introduced a version of quantum geometric flows that result in an equation similar to Wheeler-Dewitt equation but has the notion of time in it. It is another way to resolve the singularity by using The Raychadhuri equation." (Personal Correspondence)
Another way to eliminate the singularity by using the semi-classical approach was found by physicist Jahed Abedi. He used only Quantum Field Theory to demonstrate that singularities cannot form. The results were published in the article Obstruction of Black Hole Singularity by Quantum Field Theory Effects (pp. 1, 15): "[Singularity avoidance] either relies on arguments based on quantum gravity [QG] or Hawking radiation. In particular in the context of loop quantum gravity Rovelli et al. have argued that quantum corrections can impede the collapse of a matter to the extent that it be stopped much before the black hole radius reaches the Planck length, hence preventing the formation of the singularity, forming what is called Planck Star. In addition, in the context of weakly nonlocal quantum gravity the similar behaviour is expected. The analysis follows the previous observation by Ashtekar and coworkers who suggested formation of a bounce radius based on gravitational quantum effects for big bang when we go back in time... The question we consider in this note is whether Quantum Field Theory (QFT) effects can substantially modify the process of collapse. The particular effect we consider is the change of the vacuum energy of fields... We consider the back reaction of the energy due to quantum fluctuation of the background fields considering the trace anomaly for Schwarzschild black hole. It is shown that it will result in modification of the horizon and also formation of an inner horizon. We show that the process of collapse of a thin shell stops before formation of the singularity. Thus we demonstrate that without turning on quantum gravity and just through the effects the coupling of field to gravity as trace anomaly of quantum fluctuations the formation of the singularity through collapse is obstructed... Similar analysis can modify the behavior of any singularity of curvature including big bang solutions."
Indeed, has been argued that the Big Bang singularity can be classically eliminated as well. For example, physicist Edward Belbruno from Princeton University wrote: "It is an open problem how one universe could smoothly be continued into another [prior to the Big Bang], since the differential equations that describe the inflation become undefined (singular) at the Big Bang itself. These differential equations are the so-called Friedmann equations, and under certain assumptions they can be reduced to a system of ordinary differential equations which are undefined at the Big Bang. In a recent paper [5], I showed how the equations can be made smooth at the Big Bang by a regularizing transformation of the position, velocity, and time variables. The regularized equations have a unique solution, not found previously, which represents a smooth transition from one universe to another." (Mathematics of Planet Earth, pp. 76-77)
Cooked. Time to move on.
Before the Big Bang
Now let's see the main pre-big bang models. Beginning with the idea that our universe evolved or sprang from empty space.
Empty space is not really empty. Despite the name the vacuum still has quantum fluctuations, and these fluctuations are just excitations (vibrations) of underlying quantum fields. It is possible that the process of removing the largest amount of particles possible from a normal space results in a different configuration of quantum fields. This state is empty of all particles and radiation in which at least one field is trapped by an energy barrier in a state of high energy. This state is called a "false vacuum." In this case, there would be a barrier to entering the true vacuum (which exists at a global minimum and is stable). The false vacuum is somewhat, but not entirely, stable. It may last for a very long time in that state, and might eventually move to a more stable state. The most common suggestion of how such a change might happen is called bubble nucleation. If a small region of the universe spontaneously reached a more stable vacuum, this 'bubble' would spread and quickly begin expanding at very nearly the speed of light.
The uncertainty principle allows for the spontaneous, uncaused appearance of energy in a small region of space without violating energy conservation. If that energy appears as matter (that is, rest energy) or radiation (kinetic energy of massless particles like photons), then it will have to disappear in a short time interval to maintain energy conservation. This can be expected to happen randomly throughout the spacetime void, with no significant permanent result. However, the bubble nucleation event can lead to a quite significant and permanent result. The energy fluctuation can appear instead as spacetime curvature within this tiny region, which is called a "bubble of false vacuum". This bubble still contains no matter or radiation, but is no longer a "true vacuum" because of the curvature, as expressed by a non-zero cosmological constant. According to the de Sitter solution, the bubble will expand exponentially in what is called inflation.
The energy density is constant for a brief interval of time. As the volume of the bubble increases exponentially during that interval, the energy contained within also increases exponentially. Although the first law of thermodynamics may seem to be violated, it is not. The pressure is negative and the bubble does work on itself as it expands. By the time it has inflated to the size of a proton, in perhaps 10⁴² second, the bubble contains sufficient energy to produce all the matter in the visible universe today. Due to Heisenberg’s uncertainty principle, there should be particle pairs created by quantum fluctuations. Generally speaking, a particle pair will annihilate soon after its birth. But, two particles from a pair can be separated immediately before annihilation due to the exponential expansion of the bubble. Therefore, there would be a large amount of real particles created as vacuum bubble expands exponentially. (Stenger, 2000)
The Emergent Scenario – George Ellis proposed a model according to which the universe (a small patch of space-time) is initially in a truly static state. This state is supported by a scalar field which is located in a false vacuum φF. The universe begins to evolve when, by quantum tunneling, the scalar field decays into a state of true vacuum. Then, a small bubble of a new phase of field value φW can form, and expand as it converts volume from high to low vacuum energy and feeds the liberated energy into the kinetic energy of the bubble wall. Inside the bubble, space-like surfaces of constant φ are homogeneous surfaces of constant negative curvature. If the potential has a suitable form, inflation and reheating may occur in the interior of the bubble as the field rolls from φW to the true minimum at φT. As the bubble expands, particle pairs (of matter and anti-matter) are created and separated to become real particles. Inflation stops and we move into the more familiar Big Bang.
Some physicists attempted to show that this static phase would be unstable and therefore collapse rather than expand [1], [2]. However, viable solutions have been presented [3], [4], [5], [6], [7], [8], [9], [10], [11].
Rube Goldberg Cosmology – This scenario, proposed by Bogdan Veklych, who holds a Bachelor's degree in Mathematics from MIT, is a variation of Ellis' model. But instead of positing that a 3-dimensional spatial sphere existed from eternity, it posits that the past-infinite universe had two spatial dimensions compactified. This means that it was like an "infinite corridor" with flat torus (doughnut-shaped) cross-sections so that if you were to move in any direction (within the flat torus), you would eventually come back to your starting point due to the toroidal shape. Further, there are two waves approaching each other along the third dimension at the speed of light. These waves have been propagating from past eternity.
In the middle part of this corridor-shaped universe, where the two waves are approaching, it is static, flat, and empty; there is no matter or significant activity in this region. When these two primordial waves finally collide and react, part of their contents forms proto-inflationary scalar fields, and another part matter. This causes space to start collapsing, but matter forces it to bounce back, which triggers inflation and, by extension, the Big Bang.
This model is geodesically complete (singularity-free) because the scale factor never reaches zero and all geodesics have been moving from eternity. Moreover, it is not unstable and this is due to the fact that static flat empty space is not subject to quantum instabilities, as even cosmologist Alex Vilenkin agreed. Finally, this scenario bypasses the heat death paradox (i.e., thermodynamic death an infinite time ago) because entropy remained constant in the eternal past, and only started growing after the waves collided and the process of inflation began.
The No-Boundary Proposal – This model predicts that before the expansion of the universe there was Imaginary Time. An interesting thing about imaginary time is that it makes space-time Euclidean (an Euclidean four-dimensional sphere – which is different from Lorentzian geometry because Lorentzian has three spatial dimensions and one time dimension and Euclidean has four time dimensions behaving as one space dimension). For simplicity, let us illustrate a three-dimensional Euclidean space-time (imagine a three-dimensional sphere). In the context of this model "imaginary time" is a kind of geometry in which time behaves as a space direction.
This region, which is Euclidean, and where all dimensions behave as space, changes (decays quantum mechanically). The spatial dimension remains a circle, but the time dimension opens up and becomes distinct from the space dimension and describes a universe in which a spatial circle grows and grows and grows – now space and time are no longer interchangeable. This is the way Lorentzian time evolution emerges from what is initially a quantum fuzz (i.e., the expansion of the universe begins when Euclidean geometry is converted to Lorentzian).
It is important to point out that in the No-Boundary model, the universe does not actually have a beginning. At least, it does not have a beginning in the traditional sense of the word. Even though time begins (when it is converted from Euclidean space), it does not come from nothing, but rather it emerges from this timeless "quantum fuzz."
Stephen Hawking said: "[In the No-Boundary model] the universe has a beginning in real time at the Big Bang... but an imaginary time has no beginning or end. Rather imaginary time has closed in on itself like the surface of the earth. The surface of the earth doesn't have any beginning or end. I know because I have been round the world and I didn't fall off."
In his book The Universe in a Nutshell (pp. 82-85), Hawking repeated the same conclusion: "However, as explained in Chapter 2, there is another kind of time, called imaginary time, that is at right angles to the ordinary real time that we feel going by. The history of the universe in real time determines its history in imaginary time, and vice versa, but the two kinds of history can be very different. In particular, the universe need have no beginning or end in imaginary time. Imaginary time behaves just like another direction in space. Thus, the histories of the universe in imaginary time can be thought of as curved surfaces, like a ball, a plane, or a saddle shape, but with four dimensions instead of two. If the histories of the universe went off to infinity like a saddle or a plane, one would have the problem of specifying what the boundary conditions were at infinity. But one can avoid having to specify boundary conditions at all if the histories of the universe in imaginary time are closed surfaces, like the surface of the Earth. The surface of the Earth doesn't have any boundaries or edges. There are no reliable reports of people falling off. If the histories of the universe in imaginary time are indeed closed surfaces, as Hartle and I proposed, it would have fundamental implications for philosophy and our picture of where we came from. The universe would be entirely self-contained; it wouldn't need anything outside to wind up the clockwork and set it going."
Critics presented arguments against this model [1], [2], however, proponents created viable solutions. [3], [4].
The Tryon Proposal – The first suggestion of how the universe could have come into existence from quantum mechanics was proposed by Edward Tryon of Hunter College, City University of New York. He proposed the idea that the universe was created out of vacuum as a result of a quantum fluctuation.
As we discussed earlier, the vacuum is anything but dull or static; it is a site of frantic activity. Electric, magnetic, and other fields are constantly fluctuating on subatomic scales because of unpredictable quantum jerks. The spacetime geometry is also fluctuating, resulting in a frenzy of space-time foam at the Planck distance scale. In addition, the space is full of virtual particles, which spontaneously pop out here and there and instantly disappear. The virtual particles are very short-lived, because they live on borrowed energy. The energy loan needs to be paid off, and according to Heisenberg's uncertainty principle, the larger the energy borrowed from the vacuum, the faster it has to be repaid. Virtual electrons and positrons typically disappear in about one-trillionth of a nanosecond. Heavier particles last even less than that, as they require more energy to materialize. Now, what Tryon was suggesting was that our entire universe, with its vast amount of matter, was a huge quantum fluctuation, which somehow failed to disappear for more than 10 billion years.
His paper appeared in 1973 in the British science journal Nature, under the title Is the Universe a Vacuum Fluctuation? Tryon's proposal relied upon a well-known mathematical fact – that the energy of a closed universe is always equal to zero. The energy of matter is positive, the gravitational energy is negative, and it turns out that in a closed universe the two contributions exactly cancel each other. Thus, if a closed universe were to arise as a quantum fluctuation, there would be no need to borrow energy from the vacuum and the lifetime of the fluctuation could be arbitrarily long.
A region of flat space begins to swell, taking the shape of a balloon. At the same time, a colossal number of particles are spontaneously created in that region. The balloon eventually pinches off, and – voilà – we have a closed universe, filled with matter, that is completely disconnected from the original space. Tryon suggested that our universe could have originated in this way and emphasized that such a creation event would not require a cause. "In answer to the question of why it happened," he wrote, "I offer the modest proposal that our universe is simply one of those things which happen from time to time."
One potential problem with Tryon's idea was that it did not explain why the universe is so large. Closed baby universes are constantly pinched off any large region of space, but all this activity occurs at the Planck distance scale. Formation of a large closed universe is possible in principle, but the probability for this to happen is low. However, in his paper Tryon argued that even if most of the universes are tiny, observers can only evolve in a large universe and therefore we should not be surprised that we live in one and, according to him, the laws of physics place no limit on the scale of vacuum fluctuations.
The Modern Version of the Tryon Proposal – It’s reasonable to imagine that the universe does last forever in the future. It's possible that, as a consequence, the universe will ultimately get emptier and emptier (generically). But even empty space has energy – vacuum energy. So when we evolve to “empty space,” there is still some energy pushing the universe around. Along with this energy comes a small nonzero temperature, which keeps all the fields in the universe gently fluctuating. Gentle or not, however, if we wait long enough we will find a really big fluctuation – one that is large enough to make inflation spontaneously begin.
In other words, empty space is unstable; after some time, it starts inflating here and there. These little inflationary patches will ultimately convert into ordinary matter and radiation, leaving behind universes just like our own. So, before the Big Bang, a previous universe could have existed, and when it emptied out, a quantum fluctuation appeared and caused our Big Bang. If our universe reaches this state of emptiness in the future, it will initiate the process of creation of baby universes which will create new ones as well. This process continues forever.
Empty space has a small positive vacuum energy and high entropy, it lasts forever, and the curvature of space-time induces a small but nonzero temperature. It is empty apart from the thin background of thermal radiation, so for the most part it is completely inhospitable to life; there is no arrow of time, since it’s in thermal equilibrium.
Here we are interested in a vacuum energy that is extremely low, as it is in our current universe. We want to know how we might escape from a situation like the one into which our current universe is evolving: empty space with a very small vacuum energy, and a very high entropy. Where do we go from there? There is (at least) one possible escape route, courtesy of quantum gravity: the creation of baby universes. Empty space can give birth to a continuous stream of baby universes, each of which starts with a low entropy and expands into a high-entropy phase of its own.
There is a lot we don’t understand about quantum gravity, but there’s a lot that we do understand about classical gravity, and about quantum mechanics; so we have certain reasonable expectations for what should happen in quantum gravity, even if the details remain to be ironed out. In particular, we expect that space-time itself should be susceptible to quantum fluctuations. Not only should quantum fields in the empty background be fluctuating, but space itself should be fluctuating. One way in which space-time might fluctuate was studied in the 1990s by Edward Farhi, Alan Guth, and Jemal Guven. They suggested that space-time could not only bend and stretch, as in ordinary classical general relativity, but also split into multiple pieces. In particular, a tiny bit of space could branch off from a larger universe and go its own way. The separate bit of space is, naturally, known as a baby universe. (In contrast to the “pocket universes” or “bubbles” of eternal inflation, which remain connected to the background space-time.)
In addition to the fluctuations of space, quantum fields are also fluctuating. Let’s imagine that one of those fields has the right properties to be an inflaton – there are places in the potential where the field could sit relatively motionless in a false vacuum valley or a new-inflation plateau. But instead of starting it there, we consider what happens when the field starts at the bottom, where the vacuum energy is very small. Quantum fluctuations will occasionally push the field up the potential, from the true vacuum to the false vacuum – not everywhere at once, but in some small region of space. What happens when a bubble of false vacuum fluctuates into existence in space? Most of the time, the field will simply dissipate away back into its thermal surroundings. Inside, where we’ve fluctuated into the false vacuum, space wants to expand; but the wall separating the inside from the outside of the bubble wants to shrink, and usually it shrinks away quickly before anything dramatic happens.
Every once in a while, however, we could get lucky. The process of getting lucky is portrayed in the figure. What we see is a simultaneous fluctuation of the inflaton field, creating a bubble of false vacuum, and of space itself, creating a region that pinches off from the rest of the universe. The tiny throat that connects the two is a wormhole. But this wormhole is unstable and will quickly collapse to nothing, leaving us with two disconnected spacetimes: the original parent universe and the tiny baby.
Now we have a baby universe, dominated by false vacuum energy, all set up to undergo inflation and expand to a huge size. If the properties of the false vacuum are just right, the energy will eventually be converted into ordinary matter and radiation, and we’ll have a universe that evolves according to the standard inflation, which eventually dilutes away until we achieve empty space once again. From there, both the original universe and the new universe can give birth to new babies. The baby universe can grow to an arbitrarily large size; there is no limitation imposed, for example, by energy conservation. It is a curious feature of general relativity that the total energy of a closed, compact universe is exactly zero, once we account for the energy of the gravitational field as well as everything else. So inflation can take a microscopically tiny ball of space and blow it up to the size of our observable universe, or much larger.
You might think that the birth of a new universe is a dramatic and painful event, just like the birth of a new person. But it’s actually not so. Inside the baby universe, of course, things are pretty dramatic – there’s a new universe where there was none before. But from the point of view of an outside observer in the parent universe, the entire process is almost unnoticeable. The new universe could grow to a tremendous size, but it would be completely disconnected from the original space-time. There is absolutely no communication between the baby universe and its parent – once they split, they remain separate forever.
A non-expert critic criticized this model, but Dr. Carroll eventually responded to the criticisms: "[N]early everything he said about [this model] was just wrong. First, he tried to claim that having a moment of time in the history of the universe when entropy was lowest counts as a “thermodynamic beginning,” even if there is more universe in both directions of time around that moment. That’s quite an innovative definition (to be polite), but more importantly that kind of “beginning” has nothing to do with the kind of “beginning” where [something] would create the universe [i.e., a beginning out of nothing]. I made this point, but it wasn’t answered. Relatedly, he seemed to think it was a glaring mistake (or perhaps intentionally subterfuge?) that on one picture of our model I had the “time” axis have only one arrow, while on another version of the picture I put arrows pointing in both directions. I apologize for being sloppy, but it’s neither a mistake nor a dastardly plot; either version is acceptable, because you have a time coordinate that runs monotonically from -infinity to +infinity, but the direction of entropy increase (that defines the arrow of time) is not monotonic. Next, he tried to claim that [this] model violated unitarity (conservation of quantum information), which is flatly wrong. He supplied two pieces of evidence, in the form of quotes from Stephen Hawking and Chris Weaver. But the Hawking quote was completely out of context; he was talking about the fact that he no longer thought that wormholes would lead to violation of unitarity in black-hole evaporation, nothing to do with cosmology. And the Weaver quote that he read had nothing to do with unitarity at all."
Physicist Alex Vilenkin also criticized this model: "To illustrate how these theorems can be used to rule out cosmological models, we consider the Carroll-Chen scenario as an example..." However, in a related paper Vilenkin admitted that this only works if one assumes that singularities can exist in the real world: "If cosmological singularities are allowed, then... [t]hese regions... will be surrounded by singularities.." Dr. Carroll would simply reply that Vilenkin's results are insignificant because these theorems are classical and gravitational singularities are eliminated by quantum gravity effects. As physicists Conroy and Edholm said in response to this paper, the existence of these singularities in classical General Relativity "suggests a serious physical malady in the theory" which "allow[s] us to consider GR to be a first approximation to a broader theory." (Conroy & Edholm, 2017) In the BGV and Wall sections we also mentioned other ways to eliminate singularities independently of quantum gravity (e.g., using Stoica's geometry and the semi-classical effects of quantum field theory).
Source: From Eternity to Here (pp. 564-570, 574)
The Ekpyrotic model – In this cyclic model, two parallel orbifold planes or M-branes collide periodically in a higher-dimensional space. The visible four-dimensional universe lies on one of these branes. The collisions correspond to a reversal from contraction to expansion, or a big crunch followed immediately by a big bang. The matter and radiation we see today were generated during the most recent collision in a pattern dictated by quantum fluctuations created before the branes. After billions of years the universe reached the state we observe today; after additional billions of years it will ultimately begin to contract again. Dark energy corresponds to a force between the branes. Moreover, the cycles can continue indefinitely into the past and the future, and the solution is an attractor, so it can provide a complete history of the universe.
Conformal Cyclic Cosmology – In CCC, the universe iterates through infinite cycles, with the future timelike infinity of each previous iteration being identified with the Big Bang of the next. This model also gets rid of the initial singularity. What singularity theorems have demonstrated is that when the density of matter reaches the maximum, it's likely to collapse to a singularity. However, these theorems assume that there was matter content in the early universe, whereas what's important in the Conformal model is that there was not. In addition, what is singular in the Big Bang is only this one function which is scale. Although, scale has meaning in Lorentzian geometry, it doesn't have any meaning in Conformal geometry.
Penrose's basic construction is to connect a countable sequence of open Friedmann–Lemaître–Robertson–Walker metric (FLRW) spacetimes, each representing a big bang followed by an infinite future expansion. Penrose noticed that the past conformal boundary of one copy of FLRW spacetime can be "attached" to the future conformal boundary of another, after an appropriate conformal rescaling (the universe "forgets its size/scale"). In particular, each individual FLRW metric is multiplied by the square of a conformal factor that approaches zero at timelike infinity, effectively "squashing down" the future conformal boundary to a conformally regular hypersurface (which is spacelike if there is a positive cosmological constant, as is currently believed). The result is a new solution to Einstein's equations, which Penrose takes to represent the entire universe, and which is composed of a sequence of sectors that Penrose calls "aeons".
Penrose goes on further to state that over enormous scales of time (beyond 10¹⁰⁰ years), distance ceases to be meaningful as all mass decays into extremely red-shifted photon energy, whereupon time has no influence, and the universe continues to expand without event →∞. This period from Big Bang to infinite expansion Penrose defines as an aeon. The smooth “hairless” infinite oblivion of the previous aeon becomes the low-entropy Big Bang state of the next aeon cycle. Conformal geometry preserves the angles but not the distances of the previous aeon, allowing the new aeon universe to appear quite small at its inception as its phase space starts a new.
The Baum-Frampton Model – During expansion, dark energy – the unknown force causing the universe to expand at an accelerating rate – pushes and pushes until all matter fragments into patches so far apart that nothing can bridge the gaps. Everything from black holes to atoms disintegrates. This point, just a fraction of a second before the end of time, is the turnaround.
At the turnaround, each fragmented patch collapses and contracts individually instead of pulling back together in a reversal of the Big Bang. The patches become a finite number of independent universes that contract and then bounce outward again, reinflating in a manner similar to the Big Bang. One patch becomes our universe.
In other words, very close to the Big Rip event, the universe splits into noninteracting (spatially disconnected) patches. The universe has expanded so much at this point that nearly all of these patches are empty of (normal) matter and radiation. They contain only dark energy. These volumes of space essentially correlate to "observable universes" or causally related volumes, which have contracted to a very small size due to the exponential expansion. Each of these volumes then contracts by an enormous amount, perhaps down to the Planck length. Because each volume of space contains no matter or energy due to the build up to the Big Rip, the entropy in any individual volume of space reduces almost to zero, and remains virtually unaltered through the contraction. Thereafter, the pattern roughly follows the Big Bang scenario with entropy increasing again through cosmic inflation and on into the creation of the universe. Remember; this happens in each disconnected "piece" of space derived from the original universe, and results in an extraordinarily large number of new universes.
Cosmological Natural Selection – In this theory a collapsing black hole in one universe causes the emergence of a new universe, whose fundamental constant parameters (masses of elementary particles, Planck constant, elementary charge, and so forth) may differ slightly from those of the universe where the black hole collapsed. Each universe thus gives rise to as many new universes as it has black holes.
Cosmological natural selection was invented to explain the value of the constants in the standard model. The idea was to invent a cosmological scenario that naturally explained why the universe allows for complex structures such as long lived stars, spiral galaxies and organic molecules – using the same mechanism that biology uses to generate improbable complex structure.
The theory is based on two hypotheses: (1) Universes are created when black holes produce new regions of spacetime. (2) During the creation, the excursions through a violent interlude at the Planck scale induce small random changes in the parameters of the effective field theories that govern physics before and after the transition.
So, in this view, our universe originated from a black hole that lies within another universe. A view of black holes first proposed in the late 1980s might be interpreted as shedding some light on the nature of these black holes. Some researchers have proposed that when a black hole forms, a Big Bang may occur at the core, which would create a new universe that expands outside of the parent universe.
According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular black hole. In the Einstein–Cartan theory, however, the minimal coupling between torsion and Dirac spinors generates a repulsive spin–spin interaction which is significant in fermionic matter at extremely high densities. Such an interaction prevents the formation of a gravitational singularity. Instead, the collapsing matter on the other side of the event horizon reaches an enormous but finite density and rebounds, forming a regular Einstein–Rosen bridge (popularly known as a wormhole). The other side of the bridge becomes a new, growing baby universe. Alternatively, one can use quantum gravity. In LQG, for example, black holes don't compress to a singularity – instead are compressed to a certain small size, then are spit out in another part of the universe (like a wormhole) or create a whole new universe. Some cosmologists, like Alon Retter and Shlomo Heller, argue that the Big Bang is the result of a black hole collapse in another universe.
According to Nikodem Popławski this creation of black hole universes could extend eternally into the past, that is, an infinite regress of creation of universes from previous black holes is possible.
The Rejuvenated Universe – Dark energy may not be a cosmological constant, but may actually increase in strength over time. If it does, then it will continue to rise and rise, potentially leading to a "big rip" scenario where every bound structure in the Universe eventually tears itself apart. But under a scenario developed by Eric Gawiser, it's possible that right at the final moment that energy inherent to space, which would be indistinguishible from inflationary scenarios, transitions into a hot Big Bang, where space-time itself so rapidly pushes outward that particles and antiparticles emerging from the vacuum are separated immediately before annihilation due to the exponential expansion of space and become long-lived particles. This "rejuvenated Universe" scenario may not only be in our far future, but it could make our Universe much older than it appears, possibly even infinitely old.
Another way to get matter requires only quark pairs (aka mesons). It's well known that you can't rip quarks away from each other and get singular quarks. The gluon bond gets stronger with increased distance, so when the quarks get sufficiently far away from each other, new quarks pop into existence due to the increased energy in the gluon bond. So, if the universe ends with a Big Rip, where dark energy forces everything apart, would that also rip quarks away from each other? The very act of separating the quarks from each other would create lots of new quarks/baryons. It might even total up in a huge amount of new matter, with the consequences that might have. The final decay products will be stable particles. The mesons will predominantly decay into protons, electrons, (anti)neutrinos and photons whereas the anti-mesons will predominantly decay into anti-protons, positrons, (anti)neutrinos and photons. And the neutral ones into an electrically neutral combinations of those. Thus with the same starting number of meson and anti-meson, this will lead to an equal mixture of matter and antimatter. The baryons produced by the big rip will predominantly decay into protons, electrons, photons and (anti)neutrinos. It could open the possibility that the sudden increase in baryons could – through gravity – slow down the expansion of the universe, allowing the formation of atoms, stars and eventually planets. Then the dark energy would slowly increase the speed of the expansion and it all repeats, making the universe cyclic.
Quantum Tunneling From 'Nothing' – In 1982, Alexander Vilenkin published a paper showing how the universe might have spontaneously appeared from "nothing." In this model, there was no space and time 'prior' to the Big Bang. Alex described this scenario in his book Many Worlds in One (pp.178-181, 205).
Back in 1982, inflation was still a very new field, full of unexplored ideas and challenging problems. The most intriguing of these problems was the question of how inflation could have started. One of the achievements of inflation was to explain the expansion of the universe. Yet, it looked as if we needed to have expansion before inflation even started. The attractive gravity of matter is initially much stronger than the gravitational repulsion of the vacuum, so if we don't postulate a strong initial blast of expansion, the universe would simply collapse and inflation would never begin. Then, suddenly, Vilenkin realized that instead of collapsing, the universe could do something much more interesting.
Suppose we have a closed spherical universe, filled with a high energy vacuum and containing a certain amount of ordinary matter. Suppose also that this universe is momentarily at rest, neither expanding nor contracting. Its future will depend on its radius. If the radius is small, the matter is compressed to a high density and the universe will collapse to a point. If the radius is large, the vacuum energy dominates and the universe will inflate. Small and large radii are separated by an energy barrier, which cannot be crossed unless the universe is given a large expansion velocity. What Vilenkin suddenly realized was that the collapse of a small universe was inevitable only in classical physics. In quantum theory, the universe could tunnel through the energy barrier and emerge on the other side.
This looked like a neat solution to the problem. The universe starts out extremely small and is most likely to collapse. But there is a small chance that instead of collapsing, it will tunnel through the barrier to a bigger radius and start inflating. So, in the grander scheme of things, there will be loads of failed universes that will exist only tier a fleeting moment, but there will also be some that will make it big. Is there any bound to how small the initial universe could be? What happens if we allow it to get smaller and smaller? To his surprise, Vilenkin found that the tunneling probability did not vanish as the initial size approached zero and he also noticed that the calculations were greatly simplified when he allowed the initial radius of the universe to vanish. What he had was a mathematical description of a universe tunneling from zero size, which according to Vilenkin is "nothing", to a finite radius and beginning to inflate.
If there was "nothing" before the universe popped out, then what could have caused the tunneling? According to Vilenkin, the answer is that no cause is required. In classical physics, causality dictates what happens from one moment to the next, but in quantum mechanics the behavior of physical objects is inherently unpredictable and some quantum processes have no cause at all. Take, for example, a radioactive atom. It has some probability of decaying, which is the same from this minute to the next. Eventually, it will decay, but there will be nothing that causes it to decay at that particular moment. Nucleation of the universe is also a quantum process and does not require a cause.
String Gas Cosmology – In this idea, initially posed in the 1980s by Robert Brandenberger and Cumrun Vafa, the universe began as a tightly wound string with all dimensions symmetrically confined to the Planck length. The strings, in effect, bound the dimensions up to that size.
Brandenberger and Vafa argued that in three or fewer dimensions, it would be likely for the strings to collide with anti-strings. The collision annihilates the string which, in turn, unleashes the dimensions it was confining. They thus begin expanding, as in the inflationary and big bang theories.
Instead of thinking about strings and anti-strings, picture a room that has a bunch of cables attached to random points on the walls. Imagine that the room wants to expand with the walls and floor and ceiling trying to move away from each other – but they can’t because of the cables. Now imagine that the cables can move, and every time they intersect, they can recombine. Picture two taut cables stretching from the floor to the ceiling that intersect to form a tall, skinny X. They can recombine to become two loose cables – one attached to the floor and one attached to the ceiling. If these had been the only two cables stretching from floor to ceiling, then after this interaction, the floor and ceiling are free to move apart from each other.
There is another model based on String Theory that simulates the birth of the universe. In this model, the Big Bang was a "symmetry-breaking event" – a fluctuation that caused three spatial dimensions to break free from the other six dimensions of string theory, then rapidly unfurl to produce our universe's observed 3D structure. String theorists have been able to see how three directions start to expand at some point in time from a tightly wound string with all dimensions symmetrically confined to the Planck length, as the previous model, in supercomputer simulations.
A key result of the experiment was that, all by itself, the nine-dimensional model universe spontaneously ballooned in three directions, while its six other spatial dimensions remained tightly wrapped. This symmetry-breaking event was described by the changing rows and columns of variables in the matrices; mathematical operations on the matrices produced the coordinates of space, and with each time step, the coordinates increased in three directions (while remaining unchanged in the other six). To those who could interpret them, the changing matrices expressed, in mathematical terms, the birth of space-time. The researchers said the spontaneous symmetry-breaking resulted from a quantum fluctuation. "Space-time has certain uncertainties ... as dictated by Heisenberg's uncertainty relation." Nishimura wrote.
Self-Creation of the Universe – J. Richard Gott and Li-Xin Li have proposed a model according to which the early universe is an endless loop of closed timelike curves that occasionally gives “birth” to a universe like ours.
"Think of the early universe as being like a liquid," Quach said in a statement. "Then as the universe cools, it 'crystallises' into the three spatial and one time dimension that we see today."
The notion that space and time are emergent properties that suddenly materialized out of an amorphous state was first put forth by physicists at Canada's Perimeter Institute in 2006. Called "quantum graphity," the theory holds that the four-dimensional geometry of space-time discovered by Albert Einstein is not fundamental; instead, space-time is more like a lattice constructed of discrete space-time building blocks, just like matter looks continuous, but is actually made of building blocks called atoms.
"These building blocks of space are very small, and so impossible to see directly," Quach explained. From the human vantage point, space-time looks smooth and continuous.
Although the idea of space being emergent is intriguing, it is nothing new. Patton and Wheeler proposed the 'pregeometry' approach. For some speculative ideas in string theory to go beyond this, i.e. treating spacetime as emergent, see, e.g., Robert C. Helling: D-Geometry As A Model For Quantum Space-Time and Jun Nishimura: Lattice Superstring and Noncommutative Geometry.
Although the idea of space being emergent is intriguing, it is nothing new. Patton and Wheeler proposed the 'pregeometry' approach. For some speculative ideas in string theory to go beyond this, i.e. treating spacetime as emergent, see, e.g., Robert C. Helling: D-Geometry As A Model For Quantum Space-Time and Jun Nishimura: Lattice Superstring and Noncommutative Geometry.
VSL – The final test for inflation will be the search for the correct spectrum of gravitational waves, and here some argue the theory will fail. There are alternatives to inflation; could they restore the classical Big Bang picture?
Andy Albrecht and Paul Steinhardt are two of the early pioneers of inflation. But in recent years they have sought alternatives, with Steinhardt being particularly critical of the theory he helped to found. Albrecht teamed up with the self-styled “bad boy of cosmology,” João Magueijo. The two examined a proposal that Einstein had toyed with but rejected, that the speed of light (c) is not always constant. If it were much higher in the early universe, this would mimic a period of inflation, providing an alternative solution to the horizon problem.
They also claimed a varying speed of light (VSL) could answer the other problems of the standard Big Bang and have a profound additional consequence. If c dramatically changed, then the law of conservation of energy could be violated and the Big Bang would be the result. In VSL, dark energy is converted into matter, and in the far future, when dark energy dominates the universe, the same process will begin again, creating a cyclic universe. As Magueijo says, “this process goes on forever in an eternal sequence of Big Bangs.”
Small variations in something called the fine structure constant have been observed. This is dependent on the speed of light, so this could be a sign of VSL. The results remain on the edge of detectability, and we shall have to wait for more data to see if the claims are robust. If they are, there are profound consequences; not only might a cyclic universe be confirmed, but the laws of physics might be shown to vary through space and time.
Holographic Big Bang – What we perceive as the big bang could be the three-dimensional “mirage” of a collapsing star in a universe profoundly different than our own. Our known universe could be the three-dimensional “wrapping” around a four-dimensional black hole’s event horizon. In this scenario, our universe burst into being when a star in a four-dimensional universe collapsed into a black hole. In our three-dimensional universe, black holes have two-dimensional event horizons – that is, they are surrounded by a two-dimensional boundary that marks the “point of no return.” In the case of a four-dimensional universe, a black hole would have a three-dimensional event horizon.
The researchers emphasize that this idea, though it may sound “absurd,” is grounded firmly in the best modern mathematics describing space and time. Specifically, they’ve used the tools of holography to “turn the big bang into a cosmic mirage.” Along the way, their model appears to address long-standing cosmological puzzles and – crucially – produce testable predictions.
Pre-big bang Cosmology – Infinitely long ago it was nearly empty, filled only with a tenuous, widely dispersed, chaotic gas of radiation and matter. The forces of nature, controlled by the dilaton field, were so feeble that particles in this gas barely interacted. As time went on, the forces gained in strength and pulled matter together. Randomly, some regions accumulated matter at the expense of their surroundings. Eventually the density in these regions became so high that black holes started to form. Matter inside those regions was then cut off from the outside, breaking up the universe into disconnected pieces.
Inside a black hole, space and time swap roles. The center of the black hole is not a point in space but an instant in time. As the infalling matter approached the center, it reached higher and higher densities. But when the density, temperature and curvature reached the maximum values allowed by string theory, these quantities bounced and started decreasing. The moment of that reversal is what we call a big bang. The interior of one of those black holes became our universe.
Conclusion
Our exploration of the Hawking-Penrose singularity theorems, the BGV theorem, the Wall theorem and the Mithani-Vilenkin paper has revealed a consistent pattern: none of these works can be used as legitimate evidence that the universe had an absolute beginning. Our detailed examination of General Relativity, quantum mechanics and modern cosmology, as well as an array of cosmological models attempting to describe events preceding the Big Bang, demonstrates the complexity and incompleteness of our understanding of cosmic origins. Instead of having a solid theory of an absolute beginning – viz., a theory of a beginning supported by empirical evidence –, we have many plausible models compatible with Big Bang cosmology, most of which posit – or are least consistent with the proposition – that the world is past-infinite. As we delve into the complex web of cosmological theories, it becomes clear that the idea of a definite starting point of the world is no longer the prevailing paradigm, pointing to the need for a future, deeper exploration of the universe's ultimate origins.