Big Bounce or Double Bang?
A Reply to Craig & Sinclair on the Interpretation of Bounce Cosmologies

We often hear that our universe began in a catastrophic event approximately fourteen billion years ago. Nonetheless, since the beginning of physical cosmology as a science in the first half of the twentieth century, physicists have explored “bounce” cosmologies (Kragh_2009; Kragh_2018). According to the usual interpretation of bounce cosmologies, our universe originated when a pre-existing universe “bounced” through a highly compressed state; this could have happened in a variety of ways. The metaverse, of which our present universe is one proper part, might cycle through multiple generations of universes, each reaching a maximum size before contracting and eventually giving birth to a subsequent universe (as in, e.g., IjjasSteinhardt:2017; Ijjas_2018; Ijjas:2019pyf; Steinhardt:2002; steinhardt_2007). Or there could have been a single previous universe that “bounced” through a maximally dense state to give birth to our universe, which will expand indefinitely into the future. (For reviews of models in the former two families, see Lilley:2015ksa; Novello:2008ra; Battefeld:2015; Brandenberger2017.) Alternatively, each universe might give birth to offspring universes through the highly compressed state found within black holes (Poplawski:2010; Poplawski:2016; Smolin:1992; Smolin:2006_CosmoSelection).

William Lane Craig and James Sinclair disagree with the traditional interpretations of bounce cosmologies (CraigSinclair:2009; CraigSinclair:2012). Craig and Sinclair defend the Kal$\overline{\textrm{a}}$m Cosmological Argument for Theism: (1) every thing which begins to exist has a cause for its existence, (2) the universe began to exist, and, therefore, (3) the universe has a cause for its existence.¹¹1Though the Kal$\overline{\textrm{a}}$m argument, as stated, is theologically neutral, proponents of the argument can either add supplementary arguments for the conclusion that God was the cause of the universe or use the Kal$\overline{\textrm{a}}$m argument as one part of a cumulative case for God’s existence (Draper:2010). To defend the second premise in light of bounce cosmological models, Craig and Sinclair re-interpret the interface between the two universes to represent the ex nihilo birth of both universes – a “double big bang” (CraigSinclair:2012, 125-127). While some philosophers of science have paid serious attention to the Kal$\overline{\textrm{a}}$m argument’s relationship to physics in decades past (e.g., Earman:1995; Smith:2000; Grunbaum:1989; Grunbaum:1991; Grunbaum:1996; Grunbaum:1998; Grunbaum:2000; Mortensen:2003; Pitts:2008) and Craig and Sinclair’s arguments crucially depend upon technical details from contemporary physics, close to nothing has been written in reply to the interpretations of the physical cosmological models appearing in CraigSinclair:2009 or CraigSinclair:2012. Despite this, considerable attention continues to be given to Craig and Sinclair’s articles in philosophy of religion. For example, according to a Google Scholar search, CraigSinclair:2009 has been cited well over a hundred times. Craig and Sinclair’s interpretations served as the basis for some of Craig’s arguments in an important public debate with physicist Sean Carroll in 2014 r(printed as CarrollCraig:2016); as of April 8, 2020, the recording of the Carroll/Craig debate has received more than 273,000 views on YouTube. Therefore, a critique of Craig and Sinclair’s treatment of cosmological models from the perspective of philosophy of science is sorely missing. Here, I take steps to rectify this situation. Setting aside theological considerations and questions about bounce cosmology’s plausibility, I will show that bounce cosmologies have features which Craig and Sinclair’s interpretation cannot plausibly explain. There are bounce cosmologies in which the features of one universe explain features of the other, which seems inconsistent with the interpretation that both universes were born simultaneously, and there are bounce cosmologies in which the thermodynamic arrow of time is continuous from one universe to the next.

Before discussing bounce cosmologies, I need to place some conceptual machinery onto the table. First, in this section, I will briefly describe singularity theorems as the theorems apply to cosmology. Craig’s defense of the Kal$\overline{\textrm{a}}$m argument has often focused on one specific singularity theorem (the Borde-Guth-Vilenkin theorem, as described below), which Craig takes to provide evidence for the Kal$\overline{\textrm{a}}$m argument’s second premise, i.e., that the universe began to exist. There are cosmological models to which the singularity theorems do not apply, so that Craig and Sinclair’s discussion of non-singular cosmologies, such as bounce cosmologies, revolves around how those cosmologies avoid the singularity theorems and whether non-singular cosmologies can avoid an absolute beginning. Second, in section 3, I will briefly discuss the universe’s entropy in order to describe Craig and Sinclair’s interpretation of bounce cosmologies.

The universe’s expansion can be understood in terms of a characteristic length scale termed the scale factor and denoted $a(t)$. For present purposes, it will suffice to say that the universe grows as $a(t)$ increases. Early in physical cosmology’s history, physicists realized that models of an isotropic and homogeneous universe, with some assumptions about the matter-energy-momentum populating space-time, predict that $a(t)$ tends to zero at some finite time in the past. Imagine a time-line documenting the history of an isotropic and homogeneous universe.²²2In technical jargon, I am imagining a specific foliation of a Friedmann–Lemaître–Robertson–Walker (FLRW) space-time into space-like surfaces. Undoubtedly, calling this a “time-line” is an oversimplification, but I ask for the reader’s forgiveness for the sake of accessibility. We’ll need to pick a clock to label three-dimensional slices along our time-line. Let’s choose a clock such that $a(t=0)=0$. In that case, the three-dimensional slice labeled $t=0$ cannot be a part of the time-line because the space-time manifold is not mathematically well-defined when the scale factor is $0$; so, we need to remove $t=0$ from the time-line. Of course, different clocks will mark the slice that we remove with different labels, but every clock – and so every observer – will agree that there is a slice removed from the manifold. (To say this more precisely, every observer would agree that a space-like surface has been removed.) The three-dimensional slice that we’ve removed is an example of a curvature singularity. Space-times from which points or slices have been removed, because the space-time manifold becomes undefined at the point or slice, are termed singular space-times. With the three-dimensional slice removed, our time-line now consists of two half open segments. All of the slices “after” the removed slice can be placed into a sensible temporal order with respect to each other, but the slices “prior” to the removed slice no longer stand in any temporal relation to the slices “after” the removed slice. Therefore, we can disregard all of the times “prior” to the removed slice. Moreover, if we pick out any trajectory and follow that trajectory backwards in time, that trajectory must come to an end at the singularity because there is no space-time for the trajectory to traverse at, or before, the singularity. The singularity marks a boundary to space-time – an absolute beginning. For Craig and Sinclair, a temporal boundary to space-time in the finite past is evidence for theism. As Craig describes singularity theorems, “The standard Big Bang model […] thus drops into the theologian’s lap just that crucial premiss which, according to Aquinas, makes God’s existence practically undeniable” (Craig:1992). Elsewhere, Craig writes, “What a literal application of the Big Bang model requires, therefore, is creatio ex nihilo” (Craig_Smith_1993A, 44).

As soon as the singular behavior of cosmological models had been discovered, physicists were suspicious. Perhaps cosmological models were singular as an artifact of assuming an unrealistic degree of homogeneity and isotropy, or perhaps the singularities in cosmological models were an indication that General Relativity would need to be replaced by a successor theory, as physicists already suspected on independent grounds. 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. (For a historical overview of singularity theorems up through the Hawking and Penrose theorems, see Earman:1999.) 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 (Borde:2003). The BGV theorem applies to classical space-times generally, including space-times that are not solutions of the Einstein Field Equations.

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 geodesic that is neither time-like or null is termed ‘space-like’. 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, Craig and Sinclair interpret the singular behavior as evidence for an absolute beginning. Nonetheless, as I will discuss in section LABEL:DoubleBangSection, a variety of non-singular cosmologies – including bounce cosmologies – have been proposed. Bounce cosmologies avoid an absolute beginning because instead of postulating that the generalized Hubble parameter is always greater than zero, bounce cosmologies postulate that space-time can be smoothly continued – that is, without becoming singular – from our expanding phase into a contracting phase. The interface at which the expanding and contracting phases smoothly join on to one another is termed the “bounce”.

To continue their defense of the Kal$\overline{\textrm{a}}$m argument in the light of non-singular cosmologies, Craig and Sinclair have sought to provide a typology of cosmological models that “evade” the Hawking-Penrose or Borde-Guth-Vilenkin singularity theorems (CraigSinclair:2009, 143; CraigSinclair:2012, 111) and to show that either non-singular cosmological models suggest the universe did begin to exist or that non-singular cosmologies are implausible.³³3There are independent reasons to doubt that the BGV theorem tells us something significant about the origins of the totality of physical reality that I do not discuss here. Despite how the BGV theorem has sometimes been reported in the philosophy of religion literature, 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.) Guth has noted that eternally inflating models can lack a “unique beginning” and remain consistent with the theorem. Two time-like geodesics, along which the average expansion rate is positive and so cannot be extended infinitely far to the past, do not need to terminate at the same point or a common space-like surface. The theorem provides no upper bound to the lengths of all of the time-like/null geodesics within the space-times to which the theorem applies (Guth2007, 6623). Andre Linde points out that, “If this upper bound does not exist, then eternal inflation is eternal not only in the future but also in the past.” As Linde continues, “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” (Linde:2007, 17). 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. Contrary to the Kal$\overline{\textrm{a}}$m argument, the totality of physical reality would never have begun to exist. Craig and Sinclair briefly discuss this matter, (wrongly) interpreting it as an objection to the theorem (see footnote 41 in (CraigSinclair:2009, 142)) and call the “objection” “misconstrued”. Craig and Sinclair go 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. Craig and Sinclair do admit that the BGV theorem is silent on what kind of singularity (or singularities) the metaverse contains. In their typology, Craig and Sinclair discuss bounce cosmologies in which the entropic arrow of time reverses at the interface between universes; Craig returns to this point in his debate with Sean Carroll (CarrollCraig:2016) and in discussion of Penrose’s cosmological model ((CraigSinclair:2012, 127); CraigOnPenrose). To complete my discussion of the requisite conceptual machinery for placing Craig and Sinclair’s interpretation on the table, I turn to a discussion of the entropic arrow of time in the next section.

On the orthodox interpretation of bounce cosmologies, a preceding universe transformed into the highly compressed initial state of our universe. I will refer to the surface joining the two universes as the interface. Craig and Sinclair disagree with the orthodox interpretation. To explicate Craig and Sinclair’s re-interpretation of bounce cosmologies, I turn to a short digression on the status of the direction of time in fundamental physics.

The fundamental laws of physics are time reversal invariant. Consider a ball traveling at a fixed velocity in a vacuum. Now, suppose that the ball impacts and rebounds off of a wall. After the ball and the wall collide, in order to conserve momentum, the wall will begin traveling in the opposite direction. (We can suppose that the wall is resting on frictionless rollers and that only a negligible amount of energy was transferred into sound and heat on impact.) We can describe three events: (1) the ball is traveling at a fixed velocity while the wall remains at rest, (2) the ball and the wall collide, and (3) both the ball and the wall are traveling at fixed velocities in opposite directions. We can likewise define the time reverse: (1*) both the wall and the wall are traveling at fixed velocities towards one another, (2*) the ball and the wall collide, transferring all of the wall’s momentum to the ball, and (3*) the ball is traveling at a fixed velocity away from the wall while the wall remains at rest. Both sequences are equally allowed in Newtonian mechanics.⁴⁴4I’ve assumed that heat, friction, sound production, and other dissipative processes are not part of Newtonian mechanics. Newtonian mechanics is said to be time reversal invariant because, for all forward (reverse) sequences in Newtonian mechanics, the reverse (forward) sequence is nomologically permissible. Quantum field theory – and not Newtonian mechanics – affords the best microphysical description of the actual universe. Nonetheless, quantum field theory is again time reversal invariant.⁵⁵5The sense in which quantum field theory is time reversal invariant is a subtle matter. Whenever we are provided with a microphysical description of the universe, in which we list sentences describing temporally sequential physical states, i.e., $S\equiv\{S_{1},S_{2},...,S_{N}\}$, an operation can be constructed that is said to produce the time reversal of $S$, i.e., $S^{*}\equiv\{S_{1}^{*},S_{2}^{*},...,S_{N}^{*}\}$. To say that the laws of physics are time reversal invariant is to say that both $S$ and $S^{*}$ are nomologically permissible sequences. Merely replacing $t$ with $-t$ in the equations of motion in fundamental physics does not suffice for time reversal. Instead, one must replace every charge with the opposite charge, replace every system with its mirror image, and replace every instance of $t$ with $-t$. That is, the fundamental laws respect the $CPT$ symmmetry and not the $T$ symmetry (Kobayashi_1973; Christenson_1964). David Albert has argued that the fundamental laws have been known not to be literally reversible since the nineteenth century (albert_2000, 21); for a reply, see Earman_2002. In any case, the dynamical asymmetries that are known to appear in the fundamental laws do not explain the macroscopic asymmetries that appear in thermodynamics or in the special sciences. In those cases, the best explanation for the temporal asymmetry is offered by the reduction of time asymmetric phenomena to time symmetric phenomena in statistical mechanics.

Despite the time reversal invariance of the fundamental laws, macrophysical systems are obviously not time reversal invariant. We can fry an egg, but unfrying an egg does not happen. How should the time asymmetry of macrophysical dynamics be explained? Phase space is the space of all possible microphysical states of a system. A given coordinate in phase space represents a specific microphysical state. For any macrophysical observer, the exact microphysical state of a system is not accessible. For that reason, macrophysical descriptions carve up phase space into disjoint regions. If we conditionalize on the assumption that the universe began in a small region of phase space, then the most probable evolution of the universe’s state is to another state that is a member of a larger phase space region. This affords a time asymmetric description from microphysical time symmetric dynamics.

To explain the macrophysically observable direction of time, a number of authors postulate that there was a time when the subspace of microphysical states consistent with the universe’s macrophysical state occupied a vanishingly small phase space region (albert_2000; Albert:2015; Loewer_2007; Loewer:2012; Loewer:2020). The size of a region to which a given microphysical state belongs is termed the entropy.⁶⁶6This needs to be qualified. For the purposes of this paper, we can understand entropy as the hypervolume of a phase space region. The entropy is defined as the sum (or the integral) of $p_{i}\log(p_{i})$, over the index $i$, where $p_{i}$ is the probability of occupying the $i^{th}$ microstate. The entropy will only be the hypervolume of the phase space region of interest on the assumption that the appropriate probability distribution is uniform over the phase space regions of interest, e.g., the Liouville measure. The entropy has a macrophysical interpretation. Suppose that we would like a crowd to move a boulder that is too heavy for any individual to move. If we command all of the individuals in the crowd to charge at the boulder, but do not command them to coordinate their efforts, then, at best, the boulder will “quiver” when, by chance, more individuals charge the boulder on one side than on any other. The most effective way to have the crowd move the boulder involves the crowd coordinating their efforts, e.g., all of the individuals charging the boulder at a specific angle. There are a larger number of ways for the crowd to charge the boulder in a disorganized, uncoordinated fashion than for the crowd to coordinate their motions and so to charge the boulder in an organized fashion.

Likewise, and for analogous reasons, there are fewer configurations of a gas that can do work in pushing a piston than there are configurations that cannot do work in pushing the piston. That is, at least from the perspective of nineteenth century thermodynamics, the entropy of a system is a measure of the system’s ability to do macrophysical work. As the entropy of a system increases, the amount of energy available to do work decreases, and the system approaches an equilibrium state in which the system cannot do any work. For an engine to do work, there must be some reservoir of “usable” energy – perhaps in the form of a temperature difference – and when the reservoir has been depleted – that is, when there is no longer a temperature difference – the engine will have come to thermodynamic equilibrium and will be unable to do work.

Contexts in which thermodynamics is combined with gravity and quantum mechanics, that is, contexts like the early universe, remain a matter of cutting edge and (often) speculative research. However, many physicists and philosophers of physics adopt the perspective that the early universe occupied a state far from equilibrium, characterized by low entropy, and the universe has been on a slow march towards equilibrium ever since. The asymmetry between the low entropy past and the high entropy future is thought to establish an entropy gradient along one temporal direction termed the entropic arrow of time. Along the direction of the entropic arrow of time, the sequence of macrophysical events is ordered towards a state in the far future, when either no energy will be left for doing work and the universe will have reached equilibrium or the entropy will be “re-set”, paving the way for a subsequent universe to begin in a low entropy state.

If this perspective is mistaken – that is, if the early universe should not be described as occupying a low entropy state – then the cosmological models that I offer later in this paper will, at the very least, need to be re-thought and much of what Craig and Sinclair conclude will have been shown to be unmotivated. Let’s set that possibility aside and suppose that the early universe can be correctly described as occupying a low entropy state.

In some bounce cosmologies, the entropic arrow of time reverses at the interface between the two universes. On the orthodox interpretation, time has one direction through the bounce. That is, from the perspective of the expanding universe, the contracting universe is in the past. Craig and Sinclair disagree. Given the correlation between the direction of time and the entropic arrow of time, and that the entropic arrow points away from the interface in either direction, Craig and Sinclair argue that the interface should be understood as the birth of two universes (a “double Big Bang”). As Craig and Sinclair describe, “The boundary that formerly represented the ‘bounce’ will now [be interpreted to] bisect two symmetric, expanding universes on either side” (CraigSinclair:2012, 122). Elsewhere, Craig and Sinclair write that, “The last gambit [in trying to avoid an absolute beginning], that of claiming that time reverses its arrow prior to the Big Bang, fails because the other side of the Big Bang is not the past of our universe” (CraigSinclair:2009, 158). As Craig and Sinclair conclude, “Thus, the [universe on the other side of the interface] is not our past. This is just a case of a double Big Bang. Hence, the universe still has an origin” (CraigSinclair:2009, 180-181); also see (CraigSinclair:2012, 125-127).⁷⁷7Though much of the argumentation that Craig and Sinclair offer in their (CraigSinclair:2009; CraigSinclair:2012) concerns the Aguirre-Gratton model (Aguirre:2002; Aguirre:2003), Craig and Sinclair draw conclusions which Craig and Sinclair take to apply to any cosmological model in which there is an interface at which the entropic arrow of time reverses direction, e.g., (CraigSinclair:2009, 158).

One might worry that Craig’s interpretation of the interface as a double Big Bang is inconsistent with claims Craig has made elsewhere about the irreducibility of the direction of time. The most robust defense of the view that the direction of time should be interpreted in terms of the entropic arrow is associated with a reductive program pursued by David Albert (albert_2000; Albert:2015), Barry Loewer (Loewer_2007; Loewer:2012; Loewer:2020), and David Papineau (papineau_2013). Prima facie, the Albert-Loewer-Papineau (ALP) reductive program is not consistent with Craig and Sinclair’s theological project or Craig’s metaphysics of time. For ALP, macrophysical temporally asymmetric phenomena should be given a reductive explanation. So, the temporal asymmetry of causal influence (e.g., effects cannot precede causes in time) can be explained, without remainder, in terms of phenomena that do not involve the time asymmetry of causal influence. This suggests that efficient causation may have a reductive explanation in terms of non-causal phenomena; if so, microphysical events need not have efficient causes. As Alyssa Ney puts the point, “from the point of view of microphysics, given an individual event, there is no objective distinction between which events make up that event’s past and which its future. Therefore, there is no microphysical distinction between which are its causes and which its effects. Thus, there are no facts about microphysical causation” (Ney:2016, 146). If microphysical events do not require efficient causes, then not all events require efficient causes. So, contrary to Craig and Sinclair’s theological aims, on ALP, the universe could have begun to exist uncaused.⁸⁸8Sean Carroll has offered a related argument. As Carroll points out, causal explanations typically depend upon the objects that stand in the explanation satisfying two conditions. First, that the objects obey the laws of physics and, second, a low entropy boundary condition in the past. The totality of physical reality is not an object within physical reality and there is no known collection of physical laws that could apply to physical reality, as a whole, as opposed to applying to all objects within physical reality. Moreover, there is no low entropy boundary condition beyond the totality of the physical world. Carroll concludes that we have no “right to demand some kind of external cause” for physical reality as a whole (CarrollCraig:2016, 67-8); also see Carroll:2005; Carroll:2012.

Of course, Craig and Sinclair need not sign on board to a reductive view of time or causation. In the metaphysics of time Craig favors, the direction of absolute time cannot be provided a reductive explanation in terms of the entropic arrow of time. Craig writes: