It would seem, on first consideration, that bouncing universes could have always existed and might continue to exist forever. If a bouncing universe were to collapse, rebound, expand, and collapse again, always cycling through these stages, you would have a cosmos that reaches into the infinite past, without a first moment and without need of one. This is not a new thought. Oscillating universes have been discussed in philosophy from time immemorial and in mathematical cosmology for over one hundred years. What has been lacking in formal cosmology has been a viable mechanism that could deliver such a bounce and, perhaps, keep it going ad infinitum. Of all the alternatives, loop quantum cosmology (LQC) is perhaps the most serious attempt that has yet been made to provide one.
The question of whether a bouncing universe in LQC can be past-eternal is my focus here, in the second of three posts on modern cyclic cosmologies. My first article examined the issue for bouncing cosmologies in general, arguing that there are a variety of theorems and results that constrain most models, the Borde-Guth-Vilenkin (BGV) theorem, Aron Wall’s quantum singularity theorem based on his proof of the generalized second law of thermodynamics, the Mithani-Vilenkin instability proof, the Tolman entropy problem, and the Kinney-Stein result.
The BGV theorem shows that past-directed geodesics1 along which the average expansion rate is positive are past-incomplete — they have a beginning — and this will be true of most geodesics in a universe that, as a whole, has a positive average expansion rate throughout its history. If it is true of all of them, then the whole universe has a beginning. The loophole, of course, is that even in a universe with a positive expansion rate as a whole, there might be certain worldlines that do not have a positive average expansion rate and can be extended forever. A cosmology that could bounce might pass through this loophole, as would a universe that traces back to an asymptotically flat space. One way it was initially thought that a bouncing cosmology could avoid the singularity theorems (not BGV) was to leverage the negative energy densities that occur as quantum effects, producing repulsive gravity that might allow the universe to rebound without collapsing to a singularity. Aron Wall’s quantum singularity theorem closed this loophole by showing on the basis of his proof of the generalized second law that, even when semi-classical quantum effects2 are taken into account, the entropy of the universe must always increase and this entails that the universe cannot be infinitely old. As for the suggestion that an arbitrary but finite number of bounces might trace back to an inflationary universe emerging from an infinitely old, flat, empty spacetime (Minkowski spacetime), a suggestion (minus the bounces) made by Aguirre and Gratton (2003), this escape route is shut down by Mithani and Vilenkin’s (2012) result that this and similar states are quantum-mechanically unstable and decay in a finite time, so such universes could not be infinitely old either.
The first discussion also reprised the classical Tolman entropy problem, the observation that entropy accumulating from one cycle to the next drives a cyclic universe toward a structureless thermal soup (heat death), and the Kinney-Stein result, which showed that any cycle-to-cycle growth a model might use to dilute that entropy and escape thermodynamic consequences is precisely what supplies the positive average expansion the BGV theorem needs. We take all these results as background for our present discussion.
Even with this background, however, loop quantum cosmology requires an evaluation of its own. It evades the standard singularity theorems and the BGV theorem in a way the models discussed in our first essay do not, and it is, unlike many of the more exotic proposals in modern cosmology, a serious and predictive research program. The question for us is not whether loop quantum cosmology naturally produces a bounce. It does. The question is whether that bounce can be repeated ad infinitum, allowing a genuinely past-eternal universe. It turns out that it cannot, but the explanation of why is worth seeing in detail.
Loop Quantum Cosmology
Loop quantum cosmology, which was developed by Abhay Ashtekar and Martin Bojowald and others, uses a method called “polymer quantization,” which treats spatial geometry as fundamentally discrete at the Planck scale, to apply loop quantum gravity to cosmology. In this treatment, areas and volumes have quantized units and cosmological evolution is governed by difference equations rather than the usual differential equations. Discrete quantum geometry produces a repulsive force that precludes collapse when matter density approaches the Planck scale, so the classical big bang singularity becomes a “big bounce” instead as a prior contracting phase is connected to our expanding one. The classical Penrose-Hawking singularity theorems are avoided in LQC because its quantum-geometric effects do not obey classical energy conditions. While the BGV theorem is kinematic and doesn’t depend on energy conditions, it nonetheless also can be avoided in LQC by constructing a bouncing model in which the average expansion rate is not positive throughout its history, which is certainly the structure of LQC for any single-bounce scenario. The central question then becomes whether LQC can be iterated cyclically ad infinitum.
While Wall’s theorem, which relies on a semi-classical framework, does not apply in the discrete spatial geometry of LQC, its broader thermodynamic logic does. Since there is no natural mechanism for entropy to be dissipated at a bounce, nor is there a claim among the founders of LQC (Ashtekar, Bojowald, Singh, etc.) that this is what happens, entropy must increase monotonically through successive bounces. Any past-eternal version of LQC thus faces three scenarios. Either it could claim that the entropy dissipates, but as we noted, there is no mechanism for this and no one is claiming that it does. It could also just accumulate, as expected, in which case we would already have arrived in Tolman’s thermal-soup catastrophe. Finally, the universe might grow in spatial volume across cycles to such an extent that entropy was diluted, but then the positive averaged Hubble parameter would, by the Kinney-Stein proof, lead to geodesic past-incompleteness under the BGV theorem. The upshot is that a past-eternal cyclic LQC has no chance. The trilemma is clear: the first option has no advocates and no physics to support it; the second leads to a physical catastrophe that should already have happened; and the third is straightforwardly self-defeating. There must have been a first bounce, before which there was no other.
Weighing the Bounce
This thermodynamic argument against a past-eternal cyclic loop quantum cosmology is decisive, so this resolves the core question we set out to answer. But digging more deeply into the nature of the bounce in LQC is instructive, so let’s press a bit further. Loop quantum cosmology as it currently stands falls far short of being a straightforward application of loop quantum gravity (LQG) to the whole universe. It is rather an application of LQG to a drastically simplified model of the universe. For instance, before LQG is introduced into cosmology, the universe is assumed to be perfectly smooth and the same in every direction, and all but a handful of the universe’s actual degrees of freedom are set aside. Grossly simplifying in this way before quantizing need not yield the same result as quantizing the full theory and then examining the simpler case inside it. The bounce occurs straightforwardly in the toy model because the simplifying assumptions have fine-tuned the situation to guarantee it. Whether a bounce would actually occur if the more demanding route were taken is completely unknown. The founders of loop quantum gravity openly admit that much less is known about the dynamics of the full theory than its kinematics and the bounce currently rests on this oversized gap. Additionally, even the simplified model in LQC is not unique since more than one mathematically acceptable quantization is available; the rival schemes disagree about observational predictions; and the theory itself provides no guidance as to which approach is correct (see, for example, (see, for example, Bojowald (2020) for a collection of these criticisms).
Beyond this, two more central inputs need to be supplied by hand. One is a free number, wholly undetermined by the theory, the Barbero-Immirzi parameter. In current practice, it is fixed to the value it needs to have for LQG to reproduce the Bekenstein-Hawking formula for black-hole entropy, which then gets carried over into LQC to help set the critical density at which the bounce takes place. But it could, in principle, be set to something else if it were expedient to do so. The second input that LQC needs is cosmic inflation, which is added to the theory for the same reason that cosmologists add it to the standard hot big bang model,3 namely to solve the horizon problem (how opposite sides of the universe have the same temperature) and the flatness problem (the improbable fine-tuning of the matter-energy density of the universe to the critical threshold needed for spatial flatness). The quantum-geometric bounce in LQC is not sufficiently powerful or long-lasting to address these concerns, so a conventional inflationary phase, with its own fine-tuned potential, is added to it. In this respect, in making its predictions, LQC assumes inflation, claiming whatever dubious merit inflation has, as also its own.
Another difficulty, distinct from the entropy trilemma articulated above, comes from the Penrose entropy, cosmology’s most extreme fine-tuning problem. LQC doesn’t solve it, it relocates it; and its appropriation of inflation makes it worse. Penrose’s calculation puts the initial entropic fine-tuning of the universe at his well-known figure of one part in 10^(10^123). Rather than removing this condition, the bounce relocates it in the contracting phase that has to arrive at the rebound in just the right, exquisitely ordered low-entropy state to account not only for the hyper-exponential fine-tuning of the initial entropy of the big bang, but also for the exponentiation of this by the cosmic inflation that allegedly ensues. What is worse, this is the reverse of how matter normally behaves under gravity. Normally it clumps, builds structure, and collapses toward a black hole that drives gravitational entropy through the roof. For a contracting phase to deliver a smooth, hyper-hyper-exponentially low-entropy bounce, this tendency would have to run backwards in a manner nothing short of miraculous. The LQC bounce thus trades in one unexplained initial condition for another that is even harder to explain. In a single-bounce model, this price would be paid only once, but in a past-eternal cyclic model, it would have to be paid infinitely many times, a fact that adds infinite weight to the conclusion of the thermodynamic trilemma that already showed that past-eternal LQC is untenable.
This much said, and past-eternal models (and perhaps cosmic inflation) set aside, LQC offers an interesting research program with genuine merit, and it makes calculable predictions that can be tested. For example, Ashtekar, Gupt, and Sreenath showed that replacing the beginning of our universe with an LQC bounce would have the effect of easing two long-standing anomalies in the cosmic microwave background (CMB) data. These predictions are even specific enough to pry apart rival versions of the theory, as Li, Motaharfar, and Singh have shown by comparing three versions against the microwave data and finding that one is observationally disfavored.
Where We Stand
Whatever its ultimate merits, loop quantum cosmology must join the bouncing and cyclic models of the first installment in respect of the verdict that it cannot support a universe without a beginning. At best, there was a first bounce, and before that no other. We turn then, by way of conclusion, to an anticipation of the final topic in our trilogy of essays, the most heterodox cyclic model of all, Penrose’s conformal cyclic cosmology (CCC). Penrose’s CCC conjectures an endless succession of cosmic aeons sans any bounce. Instead, a conformal transformation stitches the far-future heat death of one aeon to the hot big bang of the next. Whether this maneuver is even possible, let alone whether it succeeds given the accompanying conditions that must also be met, will be the subject of the third and final installment.
Notes
- As noted in the first essay in this trilogy, geodesics are paths through spacetime that objects follow when the only thing acting on them is gravity. This trajectory is a straight line in flat space, but in the curved spacetime of general relativity, it is the straightest path the geometry allows. For the purposes of the BGV theorem, the relevant geodesics are the worldlines that are natural in physics — the paths of freely falling observers (timelike geodesics) and the paths of light rays (null geodesics). Such paths are complete if they can be extended indefinitely and incomplete if they cannot. Saying that the universe is “geodesically past-incomplete” is saying that every such path traced backward has finite extent — that a clock carried by a freely falling observer would record only a finite past and a light ray, which carries no clock, would end after a finite range. Under the condition of positive average expansion required by the BGV theorem, the expanding universe has an edge in the past rather than an infinite history. This edge doesn’t have to be a singularity; it is just the broader conclusion that such a universe has a beginning.
- Again, as noted in the first essay, “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.
- In truth, inflationary cosmology is itself controversial and faces multiple severe difficulties. If inflation happens, Penrose has shown that the pre-inflationary state must be exponentially more fine-tuned than the already hyper-exponentially fine-tuned (1 part in 10^(10^123)) big-bang entropy it was, in part, meant to address (cf. also The Road to Reality, ch. 28). Then there is the fact that Dyson, Kleban, and Susskind have established that any long-lived inflationary universe generates Boltzmann brain observers that vastly outnumber the ordinary observers we take ourselves to be, rendering any inference to the truth of cosmic inflation epistemically self-defeating. Also, as we have seen, the BGV theorem precludes the past-eternality of inflation, so it does not dissolve the need for a beginning. The measure problem in cosmology further leaves probabilities in any infinite eternally-inflating multiverse without any principled normalization, rendering any efforts to mitigate the staggering fine-tuning of our universe using anthropic reasoning impotent. Beyond this, Ijjas, Steinhardt, and Loeb (2013, 2017) and Garfinkle, Ijjas, and Steinhardt (2023) have shown that the Planck satellite data render the simplest inflaton potentials (V ∝ φ², V ∝ φ⁴) untenable, leaving only plateau models with initial condition problems that require exquisite fine-tuning. As for inflation’s alleged predictive virtues (scale invariance, adiabaticity, Gaussianity), these properties are equally well-explained by non-inflationary mechanisms with scale invariance being explained here and here, adiabaticity here, and Gaussianity here and here. To this we need to add that inflation’s distinctive prediction for the tensor-mode of gravitational waves (r ~ 0.1–0.3) for the simplest V ∝ φ² and V ∝ φ⁴ inflaton potentials is disconfirmed by the Planck data and strongly disconfirmed by the BICEP-Keck data (r < 0.036 with 95 percent confidence). On top of all of this, the inflaton field itself remains undetected and it is selected from more than one hundred candidate potentials that have reheating parameters (coupling strengths and various microphysical constraints) that are adjusted to fit observation rather than being predictive. This is a malleability that Ijjas, Steinhardt, and Loeb conclude makes inflation “so flexible that no experiment can ever disprove it.” Next to this kind of adaptability, the Barbero-Immirzi parameter of LQG looks like a trivial single-dial adjustment. Of course, unbridled cosmic inflation plays an indispensable role in the anthropic string landscape on which attempts to circumvent fine-tuning arguments regarding the initial conditions, laws, and constants of nature depend, and the string landscape inherits all of inflation’s problems in addition to those plaguing string theory itself. And of course, the multiverse of the string landscape merely relocates this fine-tuning to its own meta-laws, which are equally in need of explanation, and hopelessly beyond empirical assessment leaving the whole apparatus of multiverse cosmology exempt from experimental verification and undermining of science in in the estimation of George Ellis and Joseph Silk, a sentiment that is echoed by Sabine Hossenfelder’s judgment that the multiverse is “unobservable by assumption.”