The scheme of CCC may be contrasted with a number of other proposals for pre-Big-Bang activity, which had been put forward previously. Even among the earliest cosmological models consistent with Einstein's general relativity, namely those of Friedmann put forward in 1922, there was one that became referred to as the 'oscillating universe'. This terminology seems to have arisen from the fact that for the closed Friedmann model without cosmological constant (K> 0, A=0; see Fig. 2.2(a)), the radius of the 3-sphere that describes the spatial universe, when represented as a function of time, has a graph that has the shape of a cycloid, which is the curve traced out by a point on the circumference of a circular hoop rolling along the time axis (normalized so that the speed of light c = 1 (see Fig. 3.7). Clearly this curve extends beyond the single arch that would describe a spatially closed universe expanding from its big bang and then collapsing back to its big crunch, where now we have a succession of such things, and we can think of the entire model as representing an unending succession of 'aeons' (Fig. 3.8), a scheme which briefly interested Einstein in 1930.[327] Of course the 'bounce' that takes place at each stage at which the spatial radius reaches zero occurs at a space-time singularity (where the space-time curvature becomes infinite) and Einstein's equations cannot be used in the ordinary way to describe a sensible evolution, even though some sort of modification might be envisaged, perhaps along lines something like those of §3.2.

Fig. 3.7 The Friedmann model of Fig. 2.2(a) has a radius which, when plotted as a function of time, describes a cycloid, which is the curve traced out by a point on a rolling hoop.

Fig. 3.8 Taking the cycloid of Fig 3.7 seriously, we obtain an oscillating closed universe model.

A more serious matter, however, from the point of view of this book, is how such a model can address the issue of the Second Law, since this particular model leaves no scope for a progressive change representing a continual increase of entropy. In fact, in 1934, the distinguished American physicist Richard Chace Tolman described a modification of Friedmann's oscillating model[3 28] which alters Friedmann's 'dust' to a composite gravitating material with an additional internal degree of freedom, which can undergo changes to accommodate an increase in entropy. Tolman's model somewhat resembles the oscillating Friedmann model, but the successive aeons have progressively longer durations and increasingly greater maximum radii (see Fig. 3.9). This model is still of the FLRW type (see §2.1), so there is no scope for contributions to the entropy through gravitational clumping. Consequently, the entropy increase in this model is of a comparatively very mild kind. Nevertheless, Tolman's contribution was important in being one of the surprisingly few serious attempts to accommodate the Second Law into cosmology.

At this point, it is appropriate to mention another of Tolman's contributions to cosmology, which has some considerable relevance also to CCC. The representation of the material contents of the universe as a pressureless fluid (i.e. 'dust'; see §3.1) is the way that the gravitational source (i.e. the Einstein tensor E; see §2.6) is dealt with in the Friedmann models. This is not such a bad first approximation, so long as the actual material being modelled is reasonably dispersed and cold. But, when we are considering the situation in the close vicinity of the Big Bang, we need to treat the material contents as being very hot (see beginning of §3.1), so it is expected that a much better approximation near the Big Bang is incoherent radiation—although for the evolution of the universe following the time of decoupling (§2.2), Friedmann's dust is better. Accordingly, Tolman introduced radiation-filled analogues of all six of the Friedmann models of §2.1 in order to provide a better description of the universe close to the Big Bang. The general appearance of the Tolman radiation solutions does not differ greatly from that of the corresponding Friedmann solutions, and the pictures of Figs. 2.2 and 2.5 will do well

Fig. 3.9 This model, due to Tolman, began to address the Second Law by having a matter content allowing an increasing entropy, whence the model would become larger at each stage.

enough also for the Tolman radiation solutions. The strict conformal diagrams of Fig. 2.34 and Fig. 2.35 will also do for the respective Tolman radiation solutions, except that the picture Fig. 2.34(a) needs to be replaced, strictly speaking, by one where the depicted rectangle is replaced by a square. (In the drawing of strict conformal diagrams, there is frequently enough freedom to accommodate such differences in scale, but in this case, the situation turns out to have a little too much rigidity to remove the global scale difference between these two figures.)

The cycloidal arch of Friedmann's Fig. 3.7, for the case K> 0, must, in Tolman's radiation model, be replaced by the semicircle of Fig. 3.10, describing the universe's radius as a function of time (K > 0). It is curious that the natural (analytic) continuation of Tolman's semicircle behaves quite differently from what happens with the cycloid, since a semicircle ought really to be completed to a circle if we are thinking of a genuine analytic continuation.[329] This makes no sense if we are trying to think of an actual continuation to values of the time-parameter which extend beyond the range of the original model. Basically, the universe radius would have to become imaginary[3 30] in Tolman's case, if we were to try to extend it analytically to a phase prior to this model's big bang. Thus a direct analytical continuation to provide a 'bounce' of the type that occurs with the 'oscillating' Friedmann K > 0 solution does not seem to make sense when we move from Friedmanns's dust to Tolman's radiation, the latter being much more realistic for behaviour near the actual Big Bang, owing to the extraordinarily high temperature that we expect to find there.

Fig. 3.10 In Tolman's closed radiation-filled universe, the radial function is a semicircle.

This difference in behaviour at the singularity has importance in relation to Tod's proposal (§2.6). This has to do with the nature of the conformal factor H that is needed to 'blow up' the big bangs of the Friedman solutions and corresponding Tolman radiation solutions to a smooth 3-surface U Since such an H becomes infinite at U it will be clearer if we phrase things in terms of the reciprocal of this H, for which I shall use the small letter w:

(The reader may be reassured, here, that despite the confusion between the notation used here and in Appendix B concerning the definition(s) of H, the w used here actually agrees with that of Appendix B.) In the Friedmann cases, we find that close to the 3-surface U the quantity w behaves like the square of a local (conformal) time parameter (vanishing at U), so the continuation of w across U is achieved smoothly without w changing sign. Hence its inverse H does not become negative across U either; see Fig. 3.11(a). On the other hand, in the Tolman radiation cases, w varies in proportion to such a local time parameter (vanishing at U), so smoothness in w would require the sign of w, and therefore of H itself, to change to a negative value on one or the other side of U In fact, this latter behaviour is much closer to what happens in CCC. We saw, in §3.2, that a smooth conformal continuation of the remote expanded

'conformal^^_ Friedmann case (a) time' Tolman case (b) /////'crossover U | |

X |
V \A |

conformal factor œ |

Fig. 3.11 Comparison between the behaviours of the conformai factors œ for (a) Friedmann's dust and (b) Tolman's radiation. Only the latter (b) is consistent with CCC. (See Fig. 3.5 and Appendix B for the terminology and notation.)

Fig. 3.11 Comparison between the behaviours of the conformai factors œ for (a) Friedmann's dust and (b) Tolman's radiation. Only the latter (b) is consistent with CCC. (See Fig. 3.5 and Appendix B for the terminology and notation.)

future of the aeon previous to the crossover 3-surface continues across having a negative H-value in the subsequent aeon (Fig. 3.11(b)). This gives us a catastrophic reversal of the sign of the gravitational constant, if we do not make the switch Hi^H-1 at the crossover surface (see §3.2). But if we do make this switch, then the behaviour of (-)H on the bigbang side of crossover is necessarily the kind of behaviour that we get for a Tolman radiation solution, rather than a Friedmann one. This appears to be very satisfactory, because a Tolman radiation model indeed provides a good local approximation to the space-time immediately following the Big Bang (where I am ignoring the possibility of inflation, for reasons referred to in §2.6, §3.4 and §3.6).

There is a further idea that some cosmologists have proposed might be incorporated into cyclic models such as the Friedmann oscillating model of Fig. 3.8 or some modification of it like that due to Tolman illustrated in Fig. 3.9. This idea appears to have been originated by John A. Wheeler, when he put forward the intriguing proposal that the dimen-sionless constants of Nature might become altered when the universe passes through a singular state like the zero-radius moments that occur in these oscillating type models. Of course, since the normal dynamical laws of physics have had to be abandoned in order to get the universe through these singular states, there seems no reason why we should not abandon a few more and let the basic constants vary too!

But there is a serious point here. It has been frequently argued that there are many curious coincidences in the relations between the constants of Nature upon which life on Earth seems to depend. Some of these might be readily dismissed as being of value only to certain kinds of life we are familiar with, like the parameters determining the delicate fact that as ice forms from water, it is anomalous in being less dense than the water, so that life can persist in water remaining unfrozen under a protective surface layer of ice even when the external temperature drops below freezing. Others seem to present a more problematic challenge, such as the threat that the whole of chemistry would have been impossible had not the neutron been just marginally more massive than the proton, a fact which leads to a whole variety of different kinds of stable nuclei—these underlying all the different chemical elements—that would not otherwise have come about. One of the most striking of such apparent coincidences was revealed with William Fowler's confirmation of Fred Hoyle's remarkable prediction of the existence of a particular energy level of carbon which, had it not existed, would have meant that the production of heavy elements in stars would not have been able to proceed beyond carbon, leaving the planets devoid of nitrogen, oxygen, chlorine, sodium, sulphur, and numerous other elements. (Fowler shared in the 1982 Nobel Prize for this with Chandrasekhar but, strangely, Hoyle was passed over.)

The term 'anthropic principle' was coined by Brandon Carter, who made a serious study of the notion[331] that had the constants been not exactly right in this particular universe, or in this particular place or particular time in this particular universe, then we would have had to have found ourselves in another, where these constants did have suitable values for intelligent life to be possible. It is not my intention to pursue this extremely intriguing but highly contentious set of ideas further here. I am not altogether sure what my own position on the matter is, though I do believe that too much reliance is frequently placed on this principle in attempts to give support to what are, to me, implausible-sounding proposed theories.[3 32] Here, I merely point out that in passing from one aeon to the next in accordance with CCC, there might well be scope for changes to, say, the value of the 'N referred to in §3.2, whose powers seem to determine the various ratios between widely differing fundamental dimen-sionless physical constants. The matter will be addressed again in §3.6.

Wheeler's idea is also incorporated into a more exotic proposal put forward by Lee Smolin in his 1997 book Life of the Cosmos.[3 33] Smolin makes the tantalizing suggestion that when black holes form, their internal collapsing regions—through unknown quantum-gravity effects—become converted to expanding ones by some kind of 'bounce', each one providing the seed of a new expanding universe phase. Each new 'baby universe' then expands to a 'full-grown' one with its own black holes, etc., etc. See Fig. 3.12. This collapse^expansion procedure would clearly have to be quite unlike the kind of conformally smooth transition involved in CCC (see Fig. 3.2), and its relation to the Second Law is obscure. Nevertheless the model has the virtue that it can be studied from the point of view of the biological principle of natural selection, and it is not entirely without significant statistical predictions. Smolin makes worthy attempts at such predictions and provides comparisons with observational statistics of black holes and neutron stars. The role of Wheeler's idea here is that the dimensionless constants might change only moderately in each collapse^expansion process, so that there would be some kind of 'inheritance' in the propensity to form new black holes, this being subject to the influences of a kind of natural selection.

Hardly less fanciful, in my own humble view, are those cosmological proposals which depend for their operation on the ideas of string theory and their dependence—as string theory stands—on the existence of extra space dimensions. The earliest such pre-big-bang proposal, as far as I am aware, is one due to Gabriele Veneziano.[334] This model does seem to have some strong points in common with CCC (pre-dating CCC by some seven years), particularly in relation to the roles of conformal rescalings, and the idea that the 'inflationary period' might better be thought of as an expo nential expansion occurring in a universe phase prior to the one we are presently experiencing (see §3.4, §3.6). On the other hand, it is dependent on ideas from string-theory culture, which makes it hard to relate directly to the CCC proposal being put forward here, particularly in relation to the clear-cut predictive elements of CCC that I shall come to in §3.6.

Similar remarks apply also to the more recent proposal of Paul Steinhardt and Neil Turok,[3 35] in which the transition from one 'aeon' to the next takes place via the 'collision of D-branes', D-branes being structures within a higher-dimensional adjunct to the normal notion of 4-dimensional space-time. Here, the crossover is taken to occur only at some smallish multiple of 1012 years, when all black holes that are currently believed to come about through astrophysical processes would still be around. Moreover, apart from this, the dependence on concepts from the string-theory culture again make it difficult to make clear-cut comparisons with CCC. This could be greatly clarified if their scheme could be reformulated in such a way that it can be viewed as being based on a more conventional 4-dimensional space-time, the roles of the extra-dimensional structures being somehow codified into 4-dimensional dynamics, even if only approximately.

In addition to the schemes referred to above, there are numerous attempts to use ideas from quantum gravity to achieve a 'bounce' from a previously collapsing universe phase to a subsequent expanding one.[3 36] In these, it is taken that a non-singular quantum evolution replaces the singular state that classically would occur at the moment of minimum size. In many attempts to achieve this, simplified lower-dimensional models are often used, although the implications for 4-dimensional spacetime are then not altogether clear. Moreover, in most attempts at a quantum evolution, the singularities are still not removed. The most successful proposal to date for a non-singular quantum bounce appears to be that using the loop-variable approach to quantum gravity, and a quantum evolution through what would classically be a cosmological singularity has been achieved on these terms, by Ashtekar and Bojowald.[3 37]

However, as far as I am able to tell, none of the pre-Big-Bang proposals described in this section makes any serious inroad into the fundamental issue raised by the Second Law, as described in Part 1. None explicitly addresses the question of suppressing gravitational degrees of freedom in the Big Bang, this actually being the key to the origin of the Second Law in the particular form that we find it, as was emphasized in §2.2, §2.4 and §2.6. Indeed, most of the above proposals lie firmly within the scope of FLRW models, so they do not come close to addressing these essential matters.

Yet, even the early twentieth-century cosmologists were certainly aware that things might get very different as soon as deviations from FLRW symmetry are allowed. Einstein himself had expressed the hope[3 38] that the introduction of irregularities might enable the singularity to be avoided (rather in the same spirit of the much later work of Lifshitz and Khalatnikov, before they and Belinski located their error; see §2.4). As has now become clear, following the singularity theorems of the later 1960s,[3 39] this hope cannot be realized within the framework of classical general relativity, and models of this type will inevitably encounter space-time singularities. We see, moreover, that when such irregularities in the collapsing phase are present, and are growing relentlessly in accordance with the vast entropy increases that accompany gravitational collapse according to the picture presented in §2.6, then there is no possibility that the geometry—even just the conformal (null-cone) geometry—that the collapsing phase will attain at its big crunch can match the far smoother (FLRW-like) big bang of the subsequent aeon.

Accordingly, if we are to take the view that the pre-Big-Bang phase would indeed have to behave in accordance with the Second Law, with gravitational degrees of freedom becoming fully activated, then it would appear that something very different from a straightforward bounce, either classical or quantum, must be involved. My own attempt to address this serious question constitutes one of the principal reasons for putting forward the apparently somewhat strange idea of CCC—involving, as it does, the infinite scale change that permits the required geometrical matching between one aeon and the next. Yet, a profound puzzle still remains: how can such a cyclic process be nevertheless consistent with the Second Law, with entropy continually increasing throughout aeon after aeon after aeon ...? This challenge is central to the entire undertaking of this book and I shall need to confront it seriously in the next section.

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