The question I now wish to address is whether we can find any specific evidence either for or against the actual validity of CCC. It might have been thought that any evidence concerning a putative 'aeon' existing prior to our Big Bang must be well beyond any observational access, owing to the absolutely enormous temperatures arising at the Big Bang that would seem to obliterate all information, thereby separating us from all that supposed previous activity. We should bear in mind, however, that there has to be an extreme organization present in the Big Bang, as a direct implication of the Second Law, and the arguments of this book point to this 'organization' having the character that allows our Big Bang to be extended conformally to an aeon prior to ours, this extension being governed by a very specific deterministic evolution. Accordingly, we may hope that there is a sense in which we might actually be able to 'see' through to that earlier aeon!

We must ask what particular features of the remote future of an aeon prior to ours could possibly be observable to us. One thing we can be sure about, if CCC is right, is that the overall spatial geometry of our own aeon must match that of the previous one. If the previous aeon were spatially finite, for example, then so must be our own. If that earlier aeon accorded, on a large scale, with a Euclidean spatial 3-geometry (K=0), then that would apply also to ours, and if it had a hyperbolic spatial geometry (K<0), then our own would be hyperbolic also. All this follows because the spatial geometry, overall, is determined by that of the crossover 3-surface, the geometry of this 3-surface being common to both of the aeons that it bounds. Of course, this provides us with nothing new, of observational value, because we have no independent information about the overall spatial geometry of the previous aeon.

On a somewhat smaller scale, however, matter distributions might rearrange themselves throughout the progress of each aeon, according to some perhaps complicated—but in principle comprehensible—dynamical processes. The ultimate behaviour of these matter distributions, taking the form of massless radiation (in accordance with CCC's §3.2 requirements), can then leave its signature on the crossover 3-surface, and then perhaps be readable in subtle irregularities in the CMB. Our task would be to try to ascertain what, in this regard, would be the most important processes taking place in the course of the previous aeon, and to try to decipher the signals hidden in such tiny irregularities in the CMB.

To be able to interpret signals of this kind, we would need to have a good understanding of the phenomena that would be likely to cause them. For this, we would need to look carefully at the dynamical processes that might be involved in the previous aeon, and also at how things might propagate from one aeon to the next. However, in order to come to any reasonably clear conclusions about the detailed nature of the previous aeon, it will be of help to us if we may assume that it was, in a general way, essentially like our own. Then we can take it that the aeon prior to ours would have behaved closely in accord with the kind of behaviour that we see in the universe around us, and with the general way that we expect it to evolve far into the future.

Most evidently, we would expect that there should have been an exponential expansion in the remote future of the previous aeon, where we are supposing that a positive cosmological constant dominated the behaviour of that aeon in its very remote future, as appears to be the case for our own (if we take A to be a constant). The resulting exponential expansion of that earlier aeon would bear a tantalizing similarity to the supposed inflationary phase of the currently favoured picture of the very early history of the universe, although this currently conventional picture has the exponential expansion taking place between around 10-36s and 10-32s in our own aeon (see §2.1, §2.6), closely following the Big Bang itself. On the other hand, CCC would place this 'inflationary phase' before the Big Bang, identifying it with the exponential expansion of the remote future of the previous aeon. In fact, as mentioned in §3.3, an idea of this nature was put forward by Gabriele Veneziano in 1998[3 81] although his scheme depended heavily on ideas from string theory.

Fig. 3.23 Standard (pre-inflationary) cosmologies could imply that points in the CMB sky, farther apart than that given by £ = 2° in the figure, should not be correlated (since the past light cones of q and r do not intersect), whereas such correlations are observed up to ~60°, as with the points like p and r.

Fig. 3.23 Standard (pre-inflationary) cosmologies could imply that points in the CMB sky, farther apart than that given by £ = 2° in the figure, should not be correlated (since the past light cones of q and r do not intersect), whereas such correlations are observed up to ~60°, as with the points like p and r.

One important aspect of this general idea is that two key pieces of observational evidence that have appeared to provide crucial support for the now-standard picture of inflationary cosmology, as discerned from the slight temperature variations seen in the CMB, appear also to be addressed by pre-Big-Bang theories of this nature. One of these is that there are observed correlations in the temperature variations in the CMB over angles in the sky (up to about 60°, in fact) that would be inconsistent with the standard cosmologies of the Friedmann or Tolman type (§2.1, §3.3), if the Big Bang itself is taken to be inherently free of correlations. This inconsistency is shown in the schematic conformal diagram of Fig. 3.23, where we see that the surface of last scattering WT(decoup-ling; see §2.2) occurs much too close to the Big-Bang 3-surface U- for effects that are seen from our vantage point to be more than about 2°

apart in the sky ever to have been in causal contact. This assumes that all such correlations arise from processes occurring after the Big Bang, and the different points of U- are in fact completely uncorrelated. Inflation is able to achieve such correlations because the 'inflationary phase' increases the separation between U- and Win a conformal diagram,[3 82] so that much larger angles seen from our vantage point are brought into causal contact; see Fig. 3.24.

Fig. 3.24 An effect of inflation is to increase the separation between W and UT, so that the correlations of Fig. 3.23 can occur.

The other key piece of observational evidence, seeming to give powerful support for inflation, is that the initial density fluctuations— giving rise to temperature fluctuations in the CMB—appear to be scale-invariant, over a very broad range. The explanation from inflationary cosmology is that there were initial completely random irregularities— of the nature of initially tiny quantum fluctuations in the 'inflaton field' (§2.6)—very soon after the Big Bang, and that the inflationary exponential expansion then took over, expanding out these irregularities to an enormous degree, these finally being realized[3 83]in actual density irregularities in the (mainly dark) matter distribution. Now, an exponential expansion is a self-similar process, so one can imagine that, if there is randomness about how the initial fluctuations are distributed in spacetime, then the result of the exponential action on these fluctuations will be a distribution with a certain scale invariance. In fact, long before the

Fig. 3.24 An effect of inflation is to increase the separation between W and UT, so that the correlations of Fig. 3.23 can occur.

inflationary scheme was put forward, it had been proposed by E.R. Harrison and Y.B. Zel'dovich, in 1970, that the observed departures from uniformity in the early distribution of material in the universe could be explained if it were assumed that the initial fluctuations are indeed scale invariant. Not only had inflation given a rationale for this supposition, but analysis of subsequent observations of the CMB confirmed a close scale invariance over a much greater range than before, this lending some considerable support to the inflationary idea, particularly since it had been hard to see what other kind of explanation could give a theoretical basis to this observed scale invariance.

Indeed, if one is to reject the inflationary picture, then some alternative explanation needs to be found of both the scale invariance and the correlations beyond the horizon size in the initial density irregularities. In CCC (as in the earlier Veneziano scheme) these two points are dealt with by, in effect, displacing the inflationary phase of the universe from occurring at a moment just following the Big Bang to a phase of expansion preceding the Big Bang, as described above. Since we still have an effectively self-similar expanding universe phase, just as with inflation, it may be expected that this could lead to density fluctuations that have a scale-invariant nature. Moreover, correlations outside the horizon scale of the Friedmann or Tolman models are again to be expected, but now these correlations are set up through events that took place in the aeon prior to our own. See Fig. 3.25.

In order to be more explicit about what these events are likely to be, according to CCC, we must try to understand what are likely to be the most relevant processes taking place in the aeon prior to our own. Before we can go into much detail about this, there is a particularly big question mark that we must address. For there is the possibility, remarked upon in §3.3, that we shall have to take seriously: John A. Wheeler's suggestion that the basic constants of Nature might not have had precisely the same values in the previous aeon as they have in our own. The most obvious (and simplest) such possibility would be that the large number N, referred to towards the end of §3.2, which in our aeon takes a value N~1020 might, in the previous aeon, have taken some other value. There are, of course, two sides to this issue. It would certainly make life easier if we can just assume that fundamental numerical constants such as N had the same value in the previous aeon as in ours, or that the observations would be insensitive to (reasonable) alterations in the values of such numbers. But, on the other hand, if there are clearly distinguishable effects that changing a number such as N might have, then there is the potentially exciting possibility of actually ascertaining whether or not such a number might be fundamentally constant (perhaps being in principle mathematically calculable) or whether it actually does change from aeon to aeon, possibly in a specific mathematical way that could itself be subject to observational test.

A subsidiary set of question marks relate to our expectations about the evolution of our own aeon into the very remote future. Here, the requirements and expectations for CCC are somewhat clearer. Specifically, A must indeed be a cosmological constant, with our aeon continuing in its exponential expansion until eternity. The Hawking evaporation of black holes must be a reality and must continue until every hole has wasted away, having deposited virtually its entire rest-energy into low-energy photons and gravitational radiation, and that this will occur even for the largest holes that can be expected to arise, until finally they disappear. Might this Hawking radiation be actually detectable if it occurs in the aeon previous to ours? We must bear in mind that the entire mass-energy of a black hole, no matter how vast it might initially be, would ultimately have to be deposited in this low-frequency electromagnetic radiation. This energy would ultimately find its way to the crossover surface and leave its subtle imprint on the CMB of our own aeon. It is not at all out of the question, if CCC is right, that this information could eventually be teased out of the tiny irregularities in the CMB. This would be most remarkable, if so, since the Hawking radiation in our own aeon would normally be regarded as being such an absurdly tiny effect that it would be completely unobservable!

A more unconventional implication of CCC is that the rest-masses of all particles ought eventually to die away over the vast stretches of eternity (§3.2), so that in the asymptotic limit all surviving particles, including charged ones, become massless. The decaying away of rest-mass would be a universal feature of massive particles, according to this scheme, so one might imagine that it should be an observable effect. However, at the present stage of understanding, no prescription of the rate at which mass should decay away has been provided by the scheme. The decay rate might well be extremely slow so it would be hard to maintain that the fact that no such decay has yet been observed represents any evidence against this aspect of CCC. One point that is worth making here is that if all different types of particles have mass-decay rates that are closely in proportion, then the effect would appear as a very slow weakening of the gravitational constant. As of 1998,[3 84] the best experimental limit on any decay rate for the gravitational constant is that it would have to be less than about 1.6 x 10-12 per year. However, we must bear in mind that a time scale of 1012 years is small beer indeed, compared with the time periods of at least 10100 years that need to be considered to allow time for all black holes to disappear. At the time of writing, I am not aware of any clear-cut observational proposal that would seriously test the aspect of CCC that demands the ultimate decaying away of rest-mass.

There is, however, one clear implication of CCC that it ought to be possible to settle by an appropriate analysis of the CMB. The effect in question is gravitational radiation from very close encounters between extremely massive black holes (primarily those in galactic centres). What would be the result of such encounters? If the holes pass each other particularly closely, it would be expected that each would deflect the motion of the other sufficiently violently for there to be a burst of gravitational radi ation that could carry away a significant amount of energy from the pair, and their relative motions would be appreciably reduced. If the encounter were extremely close, then they might well capture each other in orbits about one another, which become tighter and tighter through energy loss in gravitational waves, resulting in a huge total energy loss in this way, until they swallow each other up to form a single black hole. In extreme cases, this single hole could be the result of a direct impact, the resulting hole being initially grossly distorted before the hole settles down via gravitational radiation. In either case, there would be an enormous emission of gravitational waves that would be likely to carry away a not inconsiderable proportion of the huge combined mass of the two holes.

On the kind of time-scale that we are concerned with here, this entire burst of gravitational waves would be virtually instantaneous. In the absence of large further distorting effects throughout the universe, this radiation would be essentially contained within a thin almost spherical shell spreading out forever, from the point of encounter e, with the speed of light. In terms of a (schematic) conformal picture (Fig. 3.26) this burst of energy would be represented as an outward light cone V +(e) extending from e to J* (where J* is the ' J+' of the previous aeon to ours). Although it might be thought that this radiation would eventually become indefinitely attenuated, so as to be totally insignificant when ultimately J* is reached, if we look at the situation in the right way we find that this is not really the case. We recall from §3.2 that the gravitational field can be described by a [°]-tensor K, satisfying a conformally invariant wave equation VK=0. Since this wave equation is indeed conformally invariant, we can regard K as propagating in the space-time depicted in Fig. 3.26, where we can regard the future boundary J" as just an ordinary spacelike 3-surface. The wave reaches J2' within a finite period, and K has a finite value there which can be estimated from the geometry of Fig. 3.26.

Now, because of the relation between K and the conformal tensor C in the conformal metric scaling that we would use for Fig. 3.26 (the 'C = nK' of §3.2), we find that the conformal tensor C reaches the value zero at J"A, but it has a non-zero normal derivative across J"A (see Fig. 3.27; compare with Fig. 3.6). From the arguments of Appendix B12, we find that the presence of this normal derivative has two direct effects. One of these is to influence the conformal geometry of the crossover surface (J7U-), via a conformal curvature quantity known as the 'Cotton-York' tensor, so that we cannot expect the spatial geometry of the succeeding aeon (our own) to be exactly of FLRW type at the moment of the Big Bang, but there must be slight irregularities. The second, and more immediately observable effect, would be to give the TO-field material—argued, in §3.2, to be the initial phase of new dark matter—a significant 'kick' in the direction of the radiation; see Fig. 3.27.

If the point u represents our present location in the space-time, then the past light cone V-(u) of u represents that part of the universe that we can directly 'see'. The intersection of V -(u) with the decoupling surface Wthus represents what can be directly observed in the CMB, but since in a strict conformal representation Wis very close (about 1% of the total height of the entire aeon, in the picture) to the crossover surface U-, we do not go too far wrong[3 85] if we think of this as the intersection of V -(u) with U-. Ignoring any effects of non-uniformity of matter density within our own aeon, this will be a geometrical sphere. The future light cone V+(e) of e will also meet J2' (= U-) in a geometrical sphere, assuming that we may ignore density non-uniformity in this previous aeon. Thus, the part of the radiation from the black-hole encounter at e that is visible directly to us through its effect on the CMB will be the intersection of these two spheres on U -, this intersection being a geometrically precise circle C, where I am here ignoring the slight difference between the 3-surfaces U- and W.

The 'kick' that the impulse of energy-momentum that the gravitational wave burst will impart on the (presumed) primordial dark matter will have a component in our direction that could be towards us or it could be away from us, depending on the geometrical relation between u, e, and the crossover surface. This effect of being towards us or away from us would be the same all around the entire circle C. Thus, we expect that for each such black-hole encounter in the previous aeon, for which these two spheres intersect, there would be a circle in the CMB sky that contributes either positively or negatively to the background average CMB temperature over the sky.

For a useful analogy, imagine a pond in a gentle rain, on a peaceful windless day. Each drop of rain will cause a circular ripple to move outwards from the point of impact, but if there are many such impacts the individual ripples will soon be hard to discern as they continually move outwards to overlap each other in complicated ways. Each impact is to be thought of as analogous to one of the black-hole encounters envisaged above. After a while, the rain peters out (the analogue of the black holes finally disappearing through Hawking evaporation), and we are left with a random-looking pattern of ripples, and from a photograph of such a pattern it would be hard to ascertain that it had been produced in this way. Nevertheless, if the appropriate statistical analysis is performed on this pattern, it ought to be possible (if the rain had not continued for too long) to reconstruct the original spatio-temporal arrangement of impacts of the original raindrops, and to be fairly confident that the pattern had actually arisen from discrete impacts of this nature.

It had seemed to me that some analysis of the CMB in this kind of statistical way ought to be able to provide a good test of the CCC proposal. So, having the occasion to visit Princeton University, at the beginning of May 2008, I took the opportunity to consult David Spergel, who is a world expert in the analysis of CMB data. I asked him if anyone had seen such an effect in the CMB data, to which he replied 'No', following this up with 'but then nobody has ever looked!' He later presented the problem to one of his post-doctoral assistants, Amir Hajian, who subsequently carried out a preliminary analysis on the observational data from the WMAP satellite observatory, to try to see if there is any evidence for this kind of effect.

What Hajian did was to choose a succession of alternative radii, starting at an angular radius of about 1° and then increasing this radius in steps of around 0.4° up to an angular radius of about 60° (for 171 different radii in all) For each given radius, circles of this radius centred on 196608 different points scattered uniformly over the sky would each have the average CMB temperature around the circle calculated. Then a histogram would be produced, to see if there is any significant deviation from what would be expected from the 'Gaussian behaviour' of completely random data. At first, certain 'spikes' were seen, seeming to present clear evidence of a number of individual circles of the nature predicted by CCC. However, before long it became clear that these were completely spurious, as the circles in question passed through certain regions of the sky, some connected with the positioning of our own Milky Way Galaxy, that were known to be hotter or colder than the normal CMB sky. To eliminate such spurious effects, information from regions close to the galactic plane had to be suppressed, and by this means the spurious 'spikes' were effectively eliminated.

A point that is worth making at this stage is that, in any case, a good many of the circles that provided the spikes had radii of over 30° in the sky, and should not have arisen in any case, according to CCC—if the aeon prior to ours had a roughly similar overall history to that which is anticipated for our own aeon. The reason for this is that the galactic black-hole encounters being considered here should not have arisen before around what would have been 'the present time' in the previous aeon, which in our aeon occurs about 33 of the way up the conformal diagram (Fig. 3.28). Simple geometry then shows that black-hole encounters, with e occurring later than % of the way up the conformal diagram of the previous aeon, would necessarily give rise to circles of radii less than 30° from our vantage point at u (in disagreement with many of the spikes). Accordingly, the temperature correlations that these effects could produce would not stretch across the celestial sphere by as much as 60°. It is a curious fact that correlations in the observed CMB temperature do seem to fall away at around 60°, which is unexplained in the standard inflationary picture, as far as I am aware, and this may perhaps be considered to represent some support for the CCC proposal.

With the removal of these spikes, there still appeared to remain various seemingly significant systematic departures from the Gaussian randomness in Hajian's analysis. Such a departure, involving an apparent excess of cold circles in a range of angular radii between about 7° and 15°, looked particularly noteworthy and, in my opinion, required explanation. It could well be that these effects are the result of some spurious ingredients that are nothing to do with CCC, but it seemed to me that a crucial issue was whether the departures from randomness had specifically to do with the fact that the regions of the sky being averaged over were actually circles as opposed to some other shape, as the actual circular nature of the presumed disturbances in the CMB would appear to be a characteristic feature of this prediction of CCC. Accordingly, I suggested that the analysis be repeated, but with an area-preserving 'twist' applied to the celestial sphere (see Fig. 3.29), so that actual circles in the celestial sphere would appear to be more elliptically shaped according to the analysis. I had proposed that three different versions of the analysis should be carried out, one with no celestial twist, one with a small twist, and one with a larger twist. I had anticipated that CCC should predict that the non-Gaussian effect should be greatest with no twist, somewhat reduced with a small twist, and perhaps wiped out altogether with the large twist.

Fig. 3.29 Twisting the CMB sky (using the formula 6' = 6, 0' = 0 + 3an 62-2a63) in spherical polar coordinates. This sends circles into more elliptical shapes.

However, the result of this analysis (carried out by Hajian in the autumn of 2008) surprised me! Completely systematically over the range of radii from 8.4° to 12.4° (which encompassed 12 successive distinct

Fig. 3.29 Twisting the CMB sky (using the formula 6' = 6, 0' = 0 + 3an 62-2a63) in spherical polar coordinates. This sends circles into more elliptical shapes.

histograms), the small amount of celestial twist actually enhanced this particular effect very clearly, whereas the larger celestial twist did indeed make it go away. In other parts of the histograms there were somewhat similar indications of a sensitivity to the circularity of the shape being examined. At first, I was somewhat dumbfounded by this finding, being unable to imagine how the enhancement due to the small amount of twist could be explained, but then the possibility occurred to me that there might be large inhomogeneities in the mass distribution (preferably) in our own aeon that serve to distort circular images slightly into elliptical ones.[3 86] We recall, from §2.6, the significant distortions of images that the presence of Weyl curvature can produce (see Fig. 2.48). The enhancing of the effect that the small twist had produced could arise (according to my suggested picture of things) from a fortuitous agreement, in some regions of the sky, between the amount of artificial celestial twist that had been introduced and actual distortion due to Weyl curvature. In other regions the twist would lead to greater disagreement, but the effect could well be an overall enhancement, in appropriate circumstances, as those due to disagreement could easily be lost in the 'noise'.

The likely presence of significant distortions, due to intervening Weyl curvature, unfortunately complicates the analysis considerably. It might be useful to break up the celestial sky into smaller regions, in order to try to identify where there might be significant Weyl curvature along the line of sight between u and the decoupling 3-surface WC Perhaps this could be related to known inhomogeneities in the mass distribution in the universe (e.g. the large 'voids'[3 87]). In any case, there is something distinctly tantalizing about the situation in which the observations seem to have left us for the time being. It is certainly to be hoped that these matters will be clarified in the not-too-distant future, so that before too long the physical status of conformal cyclic cosmology can actually be resolved in a clear-cut way.

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