The rumblings and grumblings continue apace, some based on misunderstandings of Stephen Meyer’s argument in Return of the God Hypothesis, the Story of Everything film, and his debate with Phil Halper on Justin Brierley’s podcast, and others rooted in confusions about the implications of various cosmological theories themselves.
A prevalent misunderstanding, evinced in a video made by Halper after the debate was filmed, is that Meyer’s whole argument rests on the idea that there’s a singularity constituting the absolute beginning for everything that exists. This is a caricature. While it’s true that singularity theorems and the Borde-Guth-Vilenkin (BGV) theorem play a role in his case for intelligent design, Meyer’s arguments draw on multiple lines of evidence, asking which metaphysical foundation, theism or naturalism, provides the best explanation for everything science has revealed about the universe we inhabit. The form of the argument he relies on is abductive, not deductive. The big picture, considering multiple streams of evidence, is that theism provides a consistently better and vastly more defensible explanation for the way things are than naturalism, regardless of which cosmological model one favors. But making this case requires working through all the evidence and all the different models in a systematic way.
As I was wrapping up this essay, my colleague David Klinghoffer drew my attention to a recent review by astrophysicist Emma Chapman of Halper’s co-authored book in the Wall Street Journal. Two sentences in the review state Halper’s position briefly and clearly. Chapman describes the position by saying, “Observations of the cosmic radiation background provide nearly incontrovertible proof of the big bang. But the conviction that the universe began as an infinitely dense point in spacetime is now less certain.” This summary is accurate and it’s important to note that the doubt it expresses focuses on the initial singularity, not on a beginning as such. As we shall see, the beginning of the universe doesn’t need to be singular, so doubts about a singularity don’t need to translate into doubts about a beginning. And Meyer’s argument, being abductive, doesn’t rest on a singular beginning anyway.
The other concern I mentioned, confusion over the implications of various theories and results in cosmology, is what will occupy us in this and two subsequent essays. Most particularly, I want to address confusion over the scope and significance of various singularity theorems and of the BGV theorem. These issues dominate Halper’s post-debate video aiming to “debunk ID guru Stephen Meyer,” and they are a recurring subject of emails filling the inboxes of various Discovery Institute Fellows. It is because the territory is large that I am taking it in three installments. This first concerns bouncing and cyclic cosmologies in general, together with the two theorems that bracket them; a second takes up loop quantum cosmology; and a third focuses on Penrose’s conformal cyclic cosmology.
Let’s start by talking about the Borde-Guth-Vilenkin (BGV) theorem, which, after earlier iterations and tweaks, received the final form of its proof in 2003. Stated precisely, the theorem shows that if the average rate of cosmic expansion is positive along any past-directed geodesic, then that geodesic is past-incomplete (cannot be extended to arbitrary earlier times). In a universe like ours, this condition holds along almost every observer or photon worldline. This result is purely kinematic and depends only on the geometry of expansion, not on any energy conditions. Since the theorem does not mandate that every geodesic be past-incomplete, every attempt to avoid past-incompleteness begins with trying to exploit this fact.
As we shall see, however, the room for maneuverability is very limited and its plausibility is highly suspect. The go-to strategy for circumventing the theorem has been the invention of bouncing or cyclic cosmologies of various sorts. We will consider several here and show that either they do not work or are exceedingly implausible. In the case of a genuine bounce, the idea is that geodesics might extend through it indefinitely, allowing the universe to be past-eternal. But extending this through infinitely many cycles without running into entropy problems requires a net expansion of the universe across cycles, which is precisely the condition the BGV theorem needs to prove past-incompleteness. Bounces alone cannot avoid a finite past. Let’s take a closer look at why various and sundry evasion strategies do not work.
Old and New Cyclic Models
The original cyclic ekpyrotic model, developed within string cosmology, was proposed by Paul Steinhardt and Neil Turok in 2002. The picture it offers is of multiple big bangs happening when two parallel three-dimensional “branes” separated by a small extra spatial dimension collide in higher-dimensional space as part of an endless cycle of collision and separation. Each cycle has a long dark-energy expansion phase that dilutes the entropy density that would otherwise accumulate, thus addressing the entropy-accumulation problem that Richard Tolman identified in 1931, namely that cyclical models eventuated in a thermal soup incapable of supporting structure after multiple iterations. Resolving the Tolman entropy problem with net expansion over cycles, however, made the Steinhardt-Turok model subject to the BGV theorem and past geodesic incompleteness.
To avoid the extra theoretical baggage of string cosmology and branes, the newer Ijjas-Steinhardt cyclic model addresses the Tolman entropy problem using four-dimensional general relativity with a scalar field driving the dynamics. In this context, the bounce is a smooth, nonsingular event with no branes or extra dimensions. The Hubble parameter, which measures the rate at which the universe expands or contracts, oscillates between expansion and contraction, while the scale factor, which measures the overall size of the universe, grows exponentially across cycles. This net growth, despite oscillation, handles the problem of entropy inherited from previous cycles. Ijjas and Steinhardt develop this model in two papers, “A new kind of cyclic universe” (2019) and “Entropy, black holes, and the new cyclic universe” (2021), and argue that it solves the Tolman entropy problem, opening the door to a universe that is eternal into the past. But it doesn’t. The exponential growth of the scale factor across cycles yields a positive averaged Hubble parameter, which is all that is needed for BGV to demonstrate past-incompleteness.
The Kinney-Stein Result
The fact that the Ijjas-Steinhardt model is subject to the BGV theorem was proved in 2022 by William Kinney and Nina Stein. They showed directly that the same cycle-to-cycle growth of the scale factor that enables the model to dissipate entropy produces the positive averaged Hubble parameter that guarantees past geodesic incompleteness. No bouncing model dissipating entropy through growth of the scale factor can truly be cyclic in time, but abandoning cycle-to-cycle growth reinstates the Tolman entropy problem. Bouncing cosmologies must pick their poison.
Wall’s Quantum Singularity Theorem
There is another thermodynamic result that complements BGV arguments by closing a loophole often used by bounce cosmologists. At a bounce, negative energy densities occur as a quantum effect that violates the classical energy conditions on which the classical singularity theorems of Penrose (1965) and of Hawking and Penrose (1970) depend. In violating the classical requirement that matter always has positive energy, localized negative energy densities create repulsive gravity that allows the universe to rebound without collapsing to a singularity. What Aron Wall showed in his proof of the Generalized Second Law (GSL), using the Bekenstein-Hawking result that a black hole’s horizon area is proportional to its entropy, is that even when the semi-classical effects of volatile quantum field fluctuations and negative energy densities are taken into account,1 the combined entropy of matter and of a causal horizon must always increase. Wall further proved that the GSL itself entails a quantum singularity theorem that does not rely on classical energy conditions. What Wall’s theorem shows is that spatially infinite Friedmann-Lemaître-Robertson-Walker (FLRW) cosmologies, of which ours appears to be one, are past geodesically incomplete. This rules out the possibility of independent baby universes and of restarting inflation, both of which are common bounce-cosmology strategies. The end-run around classical energy conditions due to quantum effects has been foreclosed since Wall’s theorem replaces those energy conditions with the GSL, which already accounts for those effects. The BGV theorem thus constrains bouncing cosmologies kinematically while Wall’s theorem constrains them thermodynamically.
Three Evasion Strategies in the Literature
Needless to say, advocates of bouncing cyclic models want to evade this conclusion. Turning to the literature, we find three main strategies attempting to do so: (1) models involving an asymptotically flat past; (2) a highly selective conformal frame change that defines a beginning out of existence; and (3) positivistic assertions of ontological deflation because of observational underdetermination (indistinguishability). Let’s deal with these in turn.
An Asymptotically Flat Past
Anthony Aguirre and Steven Gratton have proposed a model in which our universe might have emerged from an infinitely old, flat, and empty Minkowski spacetime that is static, having no expansion or contraction (“Inflation without a beginning,” 2003). There is no singular big bang in this picture because the universe emerges out of an asymptotically flat state. Furthermore, since this static flat spacetime is infinitely old, the cosmic expansion rate (and hence its average) also falls to zero asymptotically, so the BGV theorem does not apply. And the same conclusion holds if you throw in an arbitrarily large number of bounces after this emergence that have recurred cyclically in the past and extend indefinitely into the future with a scale factor growth that avoids the Tolman entropy problem. Alas, this neat little evasion has some problems. As Audrey Mithani and Alexander Vilenkin have shown, unsurprisingly, such an asymptotic Minkowski or quasi-static state is quantum-mechanically unstable (“Did the universe have a beginning?,” 2012). The state could not be infinitely old because vacuum fluctuations cause it to decay and the probability of its remaining static falls to zero given long enough time. If this state were infinitely old, it would have decayed an infinitely long time ago, but it didn’t, so it isn’t. The postulated eternal static state from which the universe is meant to have emerged cannot be infinitely old, so the “infinite” past of the postulation collapses to a finite one. It is also the case that the argument that the universe has a beginning does not turn on the past boundary being singular as opposed to asymptotically flat; a model in which the universe emerges from the latter is just a beginning of a different kind, and it has the same metaphysical implications.
Changing Conformal Frames
If the goal is to get rid of a beginning to the universe, what should we make of an approach that redescribes its history in different units that make the beginning disappear? Stated roughly, this is the general strategy of Itzhak Bars, Paul Steinhardt, and Neil Turok (Cyclic Cosmology, Conformal Symmetry and the Metastability of the Higgs, 2013). To this end, they use a reformulation of general relativity with two peculiar features. First, the masses of particles are not fixed, but rather conferred by the Higgs field. Second, by using the freedom introduced by choice of conformal frame, the units for measuring both length and time can be stretched or shrunk from one place to another without changing any physical predictions.2 By exploiting the possibilities of a carefully chosen conformal frame, a cycle-ending bounce that would have been an edge of spacetime where a worldline stops in ordinary units gets rescaled so that particle masses climb toward the Planck scale where the universe grows small, and a worldline can be drawn straight through it. There are thus no past-incomplete geodesics, so Bars, Steinhardt, and Turok (BST) claim to have modeled a cyclic universe without a beginning.
There are two very heavy costs incurred by this approach. The first is its use of exotic and wholly unobserved physics. The way that one cycle is joined to the next requires a short interval between big crunch and big bang when the quantity that sets the strength of gravity (which is variable in their theory) changes sign. In other words, gravity reverses. BST rightly call this an “antigravity” phase, and there is simply nothing in established physics that asks for such a thing.
The second cost is even more serious. We can get at it by asking what the rescaling really accomplishes. The authors present their elevation of conformal symmetry to a fundamental gauge symmetry as motivated by scale symmetries in particle physics and cosmology, but none of these remotely license what they do or the conclusions drawn from it. The symmetries they invoke are approximate and global, whereas what they install is a local, exact gauge symmetry. One might initially think they’re entitled to their approach because, within their reformulation, different conformal frames are genuinely equivalent, just choices of units, so they are not tendentiously choosing one in which the bounce looks smooth. The problem is that the freedom to rescale is not a symmetry discovered in nature. It is actively put into the reformulation by the addition of an extra field whose sole purpose is to carry the absolute scales that are fixed in ordinary physics. BST freely admit that if this added field is held constant, fixing the units once and for all, their theory collapses back to ordinary general relativity, with a real curvature singularity at the crunch, and the BGV theorem in full effect. The conformal freedom on which their escape from a beginning depends is something they have manufactured, not something found in nature. To adopt deliberately a description that erases a past boundary and then conclude that there isn’t one is to sell as a “discovery” what was only ever a choice of units.
Another significant problem is that conformal rescaling does not address thermodynamics. While the conformal transformation aims to make all past-directed geodesics complete, it does nothing to alleviate the accumulation of entropy every cycle. The only way BST can avoid the accumulation of entropy leading to Tolman heat death is to invoke a net expansion over cycles that sufficiently dilutes the entropy deposited during each cycle. But this net expansion across cycles yields a positive averaged Hubble rate that the Kinney-Stein proof shows to meet the condition of the BGV theorem and entail past-incompleteness. This cannot be evaded by changing the conformal frame because the total entropy of a comoving region is set by counting microstates, which yields a pure number that no rescaling of length or mass can alter, even though the entropy density can be made to look bounded because it divides the total entropy by a frame-dependent volume.3 Furthermore, entropy does not decrease at the bounce. But such a decrease is exactly the loophole that Wall’s generalized second law closes because it covers the quantum negative-energy regime in which a bounce takes place. Working backward, a count that climbs with every cycle and cannot fall below zero allows for only finitely many cycles, so it doesn’t matter which conformal frame is chosen because the universe has a beginning in every frame.
A final, equally deep problem remains. Conformal frame changes are the wrong instruments for investigating whether the universe actually had a beginning (as opposed, perhaps, to trying to obfuscate the matter). At its root, the question of whether the universe had a beginning is settled by facts about the scale of proper time along worldlines and about whether the curvature of spacetime heads to infinity. In contrast, by design, a change of conformal frame is blind to scale. Conformal transformations preserve angle and causal relationships while discarding all measures of length, so beginnings are precisely the kind of feature that can be obscured. To deliberately move into a conformal frame where the beginning is invisible and then announce that there is none conflates the limitations of the description with features of the world. We will meet the very same maneuver, albeit in a different conformal costume, when we consider Penrose’s conformal cyclic cosmology (CCC) in the third and final installment of our analysis of cyclic cosmologies. In CCC, another conformal symmetry that physics does not actually possess is again constructed for the purpose of redescribing away a past boundary. Unsurprisingly, rest mass is a casualty in CCC too, though Penrose has it fading away in the far-distant future, whereas BST drive it to the Planck scale at the bounce (the very Planck-scale mass Penrose reserves for his hypothetical “erebons”). While the two models use conformal rescaling in different ways and differ in a lot of the mathematical machinery they invoke, their strategies are the same. They both erase a metrical beginning by taking refuge in a mathematical description in which metrical beginnings have no place. A cyclic bounce model and a bounce-free cyclic model that only work by putting to death the constancy of mass and mathematically banishing metrical beginnings are pied pipers playing very beguiling tunes. We should be wary of being led into a trap.
Observational Underdetermination
When physics fails to generate the solution you want, there is always the fallback offered by bad philosophy. One might grant that no cyclic constructions, whether they succeed in evading the BGV theorem or not, ultimately provide a plausible scenario that avoids a universal beginning, but still argue that since it is impossible to distinguish observationally between a genuinely past-eternal cyclic universe and an emergent one followed by an arbitrary finite number of cycles, the distinction is scientifically meaningless. This is positivism at its worst: an outright conflation of epistemology with ontology. Epistemic or experimental indistinguishability is not the same thing as ontological equivalence. Even if, contrary to the pull of our present argumentation, we could not theoretically or observationally settle the question of whether our universe has a finite or infinite past, the question is still a genuine one and, metaphysically speaking, it has a correct answer. Philosophical overtures in the direction of ontological deflation are non sequiturs, pure and simple. And what is more, though we cannot explore the necessity and significance of the principle of sufficient reason here, even if the universe had an infinite past, the contingency of its existence and properties would demand an explanation that transcends the resources of philosophical naturalism. But this is a topic for another time. For now, it suffices to note that a cyclic cosmology without a beginning cannot be saved by confusing epistemology with ontology.
Where Things Stand So Far
Bouncing and, more broadly, cyclic cosmologies are bracketed by two main theorems. The BGV theorem provides a kinematic constraint. Any universe with a positive average expansion rate along past-directed worldlines is past-incomplete. Wall’s quantum singularity theorem then constrains these models thermodynamically, closing the loopholes in the classical singularity theorems that quantum negative energies could otherwise exploit. In this context, three evasion strategies can be seen to fail. If, in light of the seeming intractability of bouncing cosmologies under the constraints of these two theorems, the inclination is “we don’t need no stinking bounces,” let’s have an asymptotically flat past extending to infinity, then Mithani-Vilenkin instability kills that hope. On the other hand, selectively changing the conformal frame does not objectively remove the universe’s beginning; it defines it out of existence by moving to a descriptive frame in which it is invisible. Finally, the attempt to deflate the question of a beginning by appeal to observational indistinguishability conflates ontology with epistemology. It mistakes a limit on observation for the absence of a fact of the matter. What is more, the claimed indistinguishability and equal tenability are contrary to what we have seen. None of these strategies works, so the bouncing and cyclic cosmologies examined so far do not deliver on the goal of offering a beginningless universe.
But our discussion is not done. There are two proposals in the quest for endless cycling that remain to be considered. They are sufficiently unlike the models treated here as to earn their own separate treatment. Loop quantum cosmology avoids a singular big bang with a bounce that is required by its treatment of space itself as discrete at the smallest scale. The question of the nature of this bounce and whether it can be repeated in an endless cycle from past eternity will be the focus of the second installment. The final model to be considered, Penrose’s conformal cyclic cosmology (CCC), is the most peculiar yet. It lays claim to cycles without a bounce of any kind. As we will see, neither of these final two efforts rescues a beginningless universe; in fact, a primary reason for the failure of CCC rests on its use of the same strategy that undid the BST bounce, namely, the retreat into a conformal description that is blind to the beginning it is meant to eliminate.
Notes
- “Semi-classical” approaches treat spacetime itself as a smooth, continuous manifold, while quantizing the matter and fields within it. Wall’s proof of the GSL and his quantum singularity theorem are semi-classical in this sense because they rely on a continuous classical manifold. The quantum cosmologies of Hartle-Hawking, Vilenkin, and Feldbrugge-Lehners-Turok retain a continuous manifold while quantizing the geometry. Loop quantum cosmology, however, is based on loop quantum gravity in which spacetime is fundamentally discrete, so no continuous manifold underlies it.
- A conformal frame is, in effect, a local ruler-and-clock convention for a spacetime. It represents a choice of how lengths and durations are to be measured. While different conformal frames disagree about those lengths and durations, possibly by a different factor from one place to another, they agree about angles and causality (which events can influence which). In ordinary physics you can’t switch between such conventions freely, because the scale of things itself is physical. By contrast, the Bars-Steinhardt-Turok framework is constructed so that no physical prediction depends on the conformal frame chosen, which is precisely what is meant by conformal symmetry.
- Changing conformal frames affects total entropy and entropy density very differently. Total entropy is just a plain count of microscopic states in a region. Changing frames cannot alter it, and it increases every cycle. Entropy density, on the other hand, is that total divided by a volume that is rescaled by a change of conformal frame. Since a frame can be chosen whose volume grows from cycle to cycle as fast as the entropy, the ratio that constitutes the density holds steady. Nothing has actually been diluted, however. The total still climbs and the universe drifts into heat death just the same, while the density has been kept level on paper. There remains just one way to hold the density constant in actuality, namely, for the universe genuinely to expand from cycle to cycle, thus spreading each cycle’s entropy through real additional volume. But real expansion across cycles is precisely the positive average expansion that the BGV theorem needs. A level entropy density thus signals either a beginning via the BGV theorem, or a bookkeeping trick that conceals one. Were the universe infinitely old, it would have reached heat death an infinite time ago. It has not, so it is not.