Physically, what would the material universe have been actually like, far far back in time, very soon after the Big Bang? One thing in particular: it would have to have been hot—extremely hot. The kinetic energy in the motions of particles around at that time would have been so enormous as to have completely overwhelmed the particles' comparatively tiny rest energies (E=mc2, for a particle of rest-mass m). Thus, the rest-mass of the particles would have been effectively irrelevant—as good as zero as far as the relevant dynamical processes are concerned. The contents of the universe, at extremely early times, would have consisted of effectively massless particles.

To phrase this issue in somewhat different terms, we may bear in mind that, according to current particle-physics ideas[31] about how the masses of basic particles actually come about, a particle's rest-mass ought to arise through the agency of a special particle (or perhaps a family of such special particles) referred to as the Higgs boson(s). Thus, the standard view about the origin of the rest-mass of any fundamental particle of Nature is that there is a quantum field associated with the Higgs that has the effect, through a subtle quantum-mechanical 'symmetry-breaking' procedure, of actually assigning a mass to other particles—a mass which they would not possess were it not for the Higgs. The Higgs would itself be thereby assigned its own particular mass (or, equivalently, rest energy). But in the very early universe, when the temperature was so high as to have provided energies greatly in excess of this Higgs value, all particles would then, according to standard ideas, indeed have become effectively massless, like a photon.

Massless particles, as we may recall from §2.3, do not appear to be particularly concerned with the full metric nature of space-time, respecting merely its conformal (or null-cone) structure. To be a little more explicit (and careful) about this, let us consider the primary mass-less particle—the photon—which, in fact, remains massless today.[32] To understand photons properly, we need to think of them in the context of the weird but precise theory of quantum mechanics (or, more correctly, quantum field theory, QFT). I cannot go into any details of QFT here (although I shall address some basic quantum issues in §3.4); our main concern is the physical field, of which photons provide the quantum constituents. This field is Maxwell's electromagnetic field, as described by the tensor F, referred to in §2.6. Now, it turns out that Maxwell's field equations are completely conformally invariant. What this means is that whenever we make the replacement of the metric g by a conformally related one g g ^ &

the new metric being (non-uniformly) rescaled g = ft2 g, where ft is a positive-valued and smoothly varying scalar quantity on the space-time (see §2.3), we can find appropriate scaling factors for both the field F and its source, the charge-current vector J, so that exactly the same Maxwell equations hold as before,[3 3] but now with all operations defined in terms of g rather than g. Accordingly, any solution of the Maxwell equations, with one particular choice of conformal scale, goes over to an exactly corresponding solution when any other choice of conformal scale is made. (This will be explained in slightly more detail in §3.2, and more fully in Appendix A6.) Moreover, at a primitive level, this is basically consistent with QFT,[34] in that the corres -pondence with the particle (i.e. photon) description also carries over to the hatted metric g, with individual photon going over to individual photon. Thus, the photon itself does not even 'notice' that a local scale change has been made.

Maxwell theory is, indeed, conformally invariant in this strong sense, where the electromagnetic interactions that couple electric charges with the electromagnetic field are also insensitive to local changes of scale. Photons, and their interactions with charged particles, do need space-time to have a null-cone structure—i.e. a conformal space-time structure— in order that their equations can be formulated, but they do not need the scale factor that distinguishes one actual metric from another, consistent with this given null-cone structure. Moreover, exactly the same invariance holds for the Yang-Mills equations that are considered to govern not only the strong interactions that describe the forces between nucleons (protons, neutrons, and their constituent quarks) and other relevant strongly interacting particles, but also the weak interactions that are responsible for radioactive decay. Mathematically, Yang-Mills theory[35] is basically just Maxwell theory with some 'extra internal indices' (see Appendix A7), so that the single photon is replaced by a multiplet of particles. In the case of strong interactions, things called quarks and gluons are the respective analogues of the electrons and photons of electromagnetic theory. The quarks, but not the gluons, are massive, with masses considered to be directly linked to the Higgs. In the standard theory of weak interactions (called 'electro-weak' theory, as electromagnetic theory is now also incorporated into this theory), the photon is considered to be part of a multiplet containing three other particles, all of which are massive, referred to as W+, W-, and Z. Again, these masses are considered to be coupled to that of the Higgs. Thus, according to current theory, when that mass-providing ingredient is removed, at the extremely high temperatures back near the Big Bang—and, indeed, roughly at the extremely high particle energies that are proposed would be reached by the LHC (Large Hadron Collider) particle accelerator in CERN, based in Geneva, when it is at full power[36]—then full conformal invariance should be restored. Of course, the details of this depend upon our standard theories of these interactions being appropriate, but this seems to be a not unreasonable assumption, as our ideas of particle physics stand at the moment. In any case, even if it turns out (for example when detailed results from the LHC

become known and understood) that things are not quite as current theory suggests, it still remains probable that when energies get higher and higher, rest-masses become more and more irrelevant, physical processes becoming dominated by conformally invariant laws.

The upshot of all this is that close to the Big Bang, probably down to around 10-12 seconds after that moment,[3 7] when temperatures exceed about 1016K, the relevant physics is believed to become blind to the scale-factor H, and conformal geometry becomes the space-time structure appropriate to the relevant physical processes.[3 8] Thus, all this physical activity would, at that stage, have been insensitive to local scale changes. In a conformal picture in which the Big Bang is stretched out, according to Tod's proposal of (§2.6, Fig. 2.49), to become a completely smooth spacelike 3-surface U which mathematically extends to a conformal 'space-time' prior to the Big Bang, the physical activity would propagate backwards in time in a mathematically coherent way, providing a physically sensible picture, seemingly unperturbed by the enormous scale changes involved, into this hypothetical pre-Big-Bang region that is being provided for it in accordance with Tod's proposal. See Fig. 3.1.

Fig. 3.1 Photons and other (effectively) massless particles/fields can propagate smoothly from an earlier pre-Big-Bang phase into the current post-Big-Bang phase or, conversely, we can propagate the particle/field information backwards from post-to pre-Big-Bang phase.

Fig. 3.1 Photons and other (effectively) massless particles/fields can propagate smoothly from an earlier pre-Big-Bang phase into the current post-Big-Bang phase or, conversely, we can propagate the particle/field information backwards from post-to pre-Big-Bang phase.

May we really suppose that we should be treating this hypothetical region as being actually physically real? If so, what kind of space-time region could this 'pre-Big-Bang' phase be? Perhaps the most immediate suggestion might be some collapsing phase of the universe which in some way is able to bounce back into an expanding universe at the Big Bang. But such a picture would negate all that I have been attempting to achieve up to this point. That picture would have our collapsing pre-Big-Bang phase somehow 'aimed' with incredible precision at such a very special ultimate state, of the same extraordinary degree of special-ness that we appear to find in our actual Big Bang. It would represent

Fig. 3.2 The type of singularity expected in a generic collapse in no way matches a conformally smooth low-entropy big bang.

an immense violation of the Second Law for that pre-Big-Bang phase, with entropy reducing itself down to the (relatively) extremely tiny value that we find at the Big Bang. We recall the picture of a collapsing universe in accordance with the Second Law that was evoked in §2.6. This would be a thoroughly black-hole-riddled space-time that collapses to a

Fig. 3.2 The type of singularity expected in a generic collapse in no way matches a conformally smooth low-entropy big bang.

singularity that in no way resembles a geometry with the required conformal smoothness needed for the kind of matching that Tod's proposal requires (see Fig. 3.2). Of course, one might adopt a viewpoint for which, in the pre-Big-Bang phase, the Second Law simply operates the other way around in time (cf. the final paragraphs of §1.6), but that goes very much against the grain of the overall purpose behind the enterprise undertaken by this book. The hope is to find something more like an 'explanation' of the Second Law, or at least some kind of rationale for it, rather than simply decreeing that some absurdly special state occurs at some stage during the universe's history (namely at the 'bounce' moment being considered above). Moreover, it turns out that there are also some mathematical difficulties with this particular kind of a 'bounce' proposal, as we shall be seeing later (in §3.3, in relation to Tolman's radiation-filled universe models; see also Appendix B6).

No, let us try something very different. Let us try to examine the other end of time, namely what is expected in the extremely remote future. According to the models described in §2.1 in which there is a positive cosmological constant A (see Fig. 2.5), our universe ought ultimately to settle into an exponential expansion, apparently rather closely modelled by the strict conformal diagrams of Fig. 2.35, in which there is a smooth spacelike future conformal boundary \+. Of course, our own universe now possesses certain types of irregularity, the greatest local departures from the highly symmetrical FLRW geometry being the presence of black holes, especially the very massive ones at galactic centres. However, in accordance with the discussion of §2.5, all black holes ought eventually to disappear with 'pops' (see Fig. 2.40 and its strict conformal diagram Fig. 2.41), even though the very largest holes might have to take something like a googol (i.e. ~ 10100) or more years before this happens.

Following that extremely long time-span, the physical contents of the universe will, in terms of numbers of particles, consist mainly of photons, these coming from greatly red-shifted starlight and CMB radiation, and from the Hawking radiation that will ultimately carry away almost the entire mass-energy of numerous huge black holes, in the form of very low-energy photons. But there will also be gravitons (the quantum constituents of gravitational waves) coming from close encounters between such black holes, especially the very big holes in galactic centres—and these encounters will actually turn out to play a vital role for us in §3.6. Photons are massless particles, but so also are gravitons, and neither of these can be used to make a clock, in accordance with the disscussion of §2.3, as illustrated in Fig. 2.21.

There will presumably also be a good measure of 'dark matter' around, whatever that mysterious substance might be (§2.1, and see also §3.2 for my own general proposal), to the extent that this material would have survived capture by black holes. It is hard to see how such a substance, interacting only through the gravitational field, could be of much value in the construction of a clock. To take such a standpoint would, however, represent a subtle change of philosophy; yet, we shall be seeing in §3.2, that such a subtle change will in any case be a necessary feature of the overall picture that I shall be presenting. Thus, it again begins to seem that it might be just the conformal structure of space-time that would, in the ultimate stages of our universe's expansion, be what is physically relevant.

When the universe enters this apparently final stage—what one might well call the 'very boring era'—nothing of great interest seems to be left for it to do. The most exciting events prior to this were the final 'pops' of the last tiny remnants of black holes, eventually disappearing (it is supposed) after they had very gradually lost all their mass via the painfully slow process of Hawking radiation. One is left with the dreadful thought of a seemingly interminable boredom confronting the final stages of our great universe—a universe which would have once seemed so exciting, teeming with fascinating activity of hugely different kinds— most of this activity occurring within beautiful galaxies, with a wonderful variety of stars and often attendant planets, among which would be those supporting life of some kind, with its exotic plants and animals, some of whom having the capabilities of deep knowledge and understanding, and profound capabilities of artistic creation. Yet all this will eventually die away. The final dregs of excitement will have to be the waiting, and the waiting, and waiting, for maybe 10100 years or more, for that final pop—perhaps of about the violence of a small artillery shell followed by nothing but further exponential expansion, thinning it out and cooling and emptying and cooling, and thinning out . . . until eternity. Does that picture present all that our universe has ultimately in store for it?

But after I had been depressing myself with such thoughts, one day in the summer of 2005, another thought then occurred to me, which was to ask: who will be around then to be bored by this apparent overpowering eventual tedium? Surely not us; it will be mainly massless particles like photons and gravitons. And it is pretty hard to bore a photon or a graviton—even aside from the extreme unlikelihood that such entities could actually have significant experiences! The point is that, according to a massless particle, the passage of time is as nothing. Such a particle can even reach eternity (that is, \+) before encountering the first 'tick' of its internal clock, as was illustrated in Fig. 2.22. One might well say that 'eternity is no big deal' for a massless particle such as a photon or a graviton!

To put this another way, it would appear that rest-mass is a necessary ingredient for the building of a clock, so if eventually there is little around which has any rest-mass, the capacity for making measurements of the passage of time would be lost (as is the capacity for making distance measurements, since distances also depend on time measurements; see §2.3). Indeed, as we have seen before, massless particles do not appear to be particularly concerned with the metric nature of spacetime, respecting merely its conformal (or null-cone) structure. Accordingly, to massless particles, the ultimate hypersurface \+ represents a region of their conformal space-time that seems to be just like anywhere else, and there appears to be no bar to their entering a hypothetical extension of this conformal space-time on the 'other side' of \+. Moreover, there are powerful mathematical results, mainly through the important work of Helmut Friedrich,[3 9] that lend support to the actual conformal future-extendability of space-time, under the general circumstances being considered here, for which there must be a positive cosmological constant A.

This mirrors our discussion of the physics at a Big-Bang hypersur-face which accords with Tod's proposal. It appears that (for different reasons) both \+ and U- would be likely to allow smooth extensions of the conformal space-time to regions on the other sides of these hyper-

surfaces. Not only that, but the material contents on either side would be likely to be an essentially massless substance whose physical behaviour is basically governed by conformally invariant equations, and this would enable the activity of this material to be continued into both of these hypothetical extensions of (conformal) space-time.

One possibility might indeed suggest itself at this point. Could it be that our and U- are one and the same? Perhaps, as a conformal manifold, our universe just 'loops round', so that what lies beyond is simply our own universe starting up again from its Big-Bang origin, conformally stretched out as U-, according to Tod's proposal. The economy of this idea certainly has its appeal, but I think that there could be serious difficulties of consistency which, in my own view, render this suggestion implausible. Basically, such a space-time would contain closed timelike curves whereby causal influences can lead to potential paradoxes, or at least to unpleasant constraints on behaviour. Such paradoxes or constraints do depend upon the possibility of coherent information being able to pass across the WU- hypersurface. Yet we shall be seeing in §3.6 that this kind of thing is a real possibility in the type of scheme that I am proposing here, and so such closed timelike curves do indeed have the potential to lead to serious inconsistency problems.[310] For reasons such as this I am not proposing this WU" identification.

However, I am suggesting 'the next best thing', which is to propose that there is a physically real region of space-time prior to U- which is the remote future of some previous universe phase, and that there is also a physically real universe phase that extends beyond our to become a big bang for a new universe phase. In accordance with this proposal, I shall refer to the phase beginning with our U- and extending to our \+ as the present aeon, and I am suggesting that the universe as a whole is to be seen as an extended conformal manifold consisting of a (possibly infinite) succession of aeons, each appearing to be an entire expanding universe history. See Fig. 3.3. The '' of each is to be identified with the ' U-' of the next, where the continuation of each aeon to the next is achieved so that, as a conformal space-time structure, the join is perfectly smooth.

The reader might well worry about identifying a remote future, where the radiation cools down to zero temperature and expands out to zero density, with a big-bang-type of explosion, where the radiation had started at an infinite temperature and infinite density. But the conformal 'stretching' at the big bang brings this infinite density and temperature down to finite values, and the conformal 'squashing' at infinity brings the zero density and temperature up, to finite values. These are just the kinds of rescalings that make it possible for the two to match, and the stretching and squashing are procedures that the relevant physics on either side is completely insensitive to. It may also be mentioned that the phase space T, describing the totality of possible states of all the physical activity on either side of the crossover (see §1.3), has a volume measure which is conformally invariant,[311] basically for the reason that when distance measures are reduced, the corresponding momentum measures are increased (and vice versa) in just such a way that the product of the two is completely unchanged by the rescaling (a fact that will have crucial significance for us in §3.4). I refer to this cosmolog-ical scheme as conformal cyclic cosmology, abbreviated CCC.[312]

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