N dft

ft2-1

The two most important things about n are, first, that it remains smooth over the crossover 3-surface and, second, that it is unchanged under the replacement ft^ft-1.

In CCC, we try to demand that n indeed be a smoothly varying quantity over the crossover, so that if we take n to define the required scaling information, rather than ft, then we can imagine that the transition fti^ft-1 at crossover can be achieved while n remains smooth across it. This requires certain mathematical conditions to be satisfied for the behaviour of ft at \+, and the indications are that these can indeed be achieved satisfactorily and uniquely. (Detailed arguments are given in Appendix B.) The upshot of it all is that there turns out to be a clear-cut and apparently unique mathematical procedure for continuing the massless fields into the future through the crossover 3-surface, it being assumed that only massless fields are present in the very remote future of the earlier aeon (i.e. just prior to \+).

With only massless fields present, we have a particular scaling freedom in the choice of rescaled metric g in the region just prior to the \+ of the earlier aeon, consistent with its given conformal structure. This freedom is described in terms of a field to, which satisfies a self-coupled (i.e. non-linear) conformally invariant massless scalar field equation that I refer to (in Appendix B2) as the 'TO-equation'. The different solutions of the TO-equation give us the different possible metric scalings that would get us from our chosen g-metric to the other possible metrics TO2g which Einstein's equations (with cosmological constant A) would tell us refer only to sources which are massless. The particular choice of TO that gives us Einstein's original physical metric g, is referred to as the 'phantom field' (since in Einstein's g-metric it disappears, simply taking the value 1). The phantom field does not have any independent physical degrees of freedom, in the region prior to the \+, but just keeps track of the metric g, telling us the scaling that gets us back to g from the g-metric that is currently being used.

On the opposite side of the crossover, immediately following the big bang of the subsequent aeon, we find that simply continuing the fields smoothly through leads to an effective gravitational constant in this new aeon that has become negative, with unphysical implications. Consequently it becomes necessary to adopt the alternative interpretation, in which we use the alternative choice Ü1, consistent with n, on the other side. This has the effect of turning the phantom field to into a real physical field (albeit initially infinite) on the big-bang side of crossover. It is tempting to interpret this TO-field following this big bang as providing the initial form of new dark matter, prior to it acquiring a mass. Why make such an interpretation? The reason simply is that the mathematics forces there to be some dominant new contribution, of the nature of a scalar field, in the big bang of the new aeon, this arising from the above behaviour of the conformai factor. This is additional to the contributions from photons (electromagnetic field) or from any other particles of matter (considered to have lost their rest-mass by the time that they reach the crossover 3-surface). It has to be there for mathematical consistency, as soon as we adopt the H^H-1, transformation at crossover.

An additional feature that we find coming out of the mathematics is that on the big-bang side of crossover, the condition that all sources are massless cannot be strictly maintained, although a natural constraint restricting unwanted freedom in the conformal factor is that this appearance of rest-mass is put off for as long as possible. Thus, a component to the post-big-bang matter content is this contribution bearing rest-mass. It would be natural to assume that this has something to do with the Higgs field (or whatever might turn out to be necessary) in its role in the appearance of rest-mass in the early universe.

Dark matter is the dominant form of matter, apparently observed to be present in the initial stages of our own aeon. It comprises some 70% of ordinary matter (where 'ordinary' just means not counting the contribution of the cosmological constant A—commonly referred to as 'dark energy'[3 23]), but dark matter does not seem to fit at all comfortably into the standard model of particle physics, its interaction with other kinds of matter being solely through its gravitational effect. The phantom field to in the late stages of the prior aeon arises as an effective scalar component to the gravitational field, coming about only because we are allowing the conformal rescalings g^H2g, and it has no independent degrees of freedom. In the subsequent aeon, the new TO-matter that comes about initially takes over the degrees of freedom present in the gravitational waves in the prior aeon. Dark matter seems to have had a special status at the time of our Big Bang, and this is certainly the case for to. The idea is that shortly after the Big Bang (presumably when the Higgs comes into play), this new TO-field acquires a mass, and it then becomes the actual dark matter that appears to play such an important role in shaping the subsequent matter distributions, with various kinds of irregularities that are observed today.

It is perhaps significant that the two so-called 'dark' quantities ('dark matter' and 'dark energy'), that have gradually become apparent from detailed cosmological observations in recent decades, both appear to be necessary ingredients of CCC. This scheme would certainly not work without A > 0, since the consequent spacelike nature of \+ is needed in order to match the spacelike character of U-. Moreover, we see from the above that the scheme requires that there be some sort of initial matter distribution which might reasonably be identified with the dark matter. It will be interesting to see whether this interpretation of dark matter will hold up theoretically and observationally.

With regard to A, the main thing that appears to puzzle cosmologists and quantum field theorists is its value. The quantity Ag is often interpreted by quantum field theorists as the energy of the vacuum (see §3.5). For reasons to do with relativity, it is argued that this 'vacuum energy' ought to be a -tensor proportional to g, but the proportionality factor comes out as something larger than the observed value of A by a factor of around 10120, so something is clearly missing from this idea! Another thing that is found puzzling is that A's observed tiny value is just such as to be starting to have effects on the expansion of the universe that are comparable with the particular totality of attraction due to matter in the universe now, which was enormously greater in the past and which will become enormously smaller in the future, and this seems to be an odd coincidence.

To me this 'coincidence' is not such an enormous puzzle, at least over and above some puzzles that had already been with us long before the observational evidence indicating A's actual small value. Certainly the observed value of A needs explanation, but perhaps it can be specifically related to the gravitational constant G, the speed of light c and Planck's constant h by some fairly simple formula, but with the 6th power of a certain large number N in the denominator

N6Gft Here ft=A

is Dirac's form of Planck's constant h (sometimes called the reduced Planck constant). The number N is about 1020 and it was pointed out, in 1937, by the great quantum physicist Paul Dirac that various integer powers of this number seem to turn up (approximately) in several different ratios of basic physical dimensionless constants, particularly when gravity is in some way involved. (For example, the ratio of the electric to the gravitational force between the electron and the proton in a hydrogen atom is around 1040~N2.) Dirac also pointed out that the age of the universe is about N3, in terms of the absolute unit of time that is referred to as the Planck time tP. The Planck time, and the corresponding Planck length lp=ctp, are often regarded as providing a kind of 'minimum' spacetime measure (or 'quantum' of time and space, respectively), according to common ideas about quantum gravity:

By use of these 'Planck units', and also the Planck mass mp and Planck energy EP given by which are naturally determined (though completely impractical) units, one can express many other basic constants of Nature simply as pure (dimensionless) numbers. In particular, in these units, we have A^N-6.

In addition, we can use Planck units for temperature, by setting Boltzmann's constant k = 1, where one unit of temperature is the absurdly large 2.5 x 1032K. When considering the very large entropies involved with large black holes or with regard to the universe as a whole (as in §3.4), I shall use Planck units. However, for values this large, it turns out to make little difference what units are used.

Originally, Dirac thought that since the age of the universe is (obviously) increasing with time, then N ought to be increasing with time or, equivalently, G reducing with time (in proportion to the reciprocal of the square of the universe's age). However, more accurate measurements of G than were available when Dirac put his ideas forward have shown that G (or equivalently N), if it is not constant, cannot vary at the rate that Dirac's ideas required. However, in 1961, Robert Dicke (with a refined

By use of these 'Planck units', and also the Planck mass mp and Planck energy EP given by later argument by Brandon Carter) pointed out that according to the accepted theory of stellar evolution, the lifetime of an ordinary 'main-sequence' star is related to the various constants of Nature in such a way that any creature whose life and evolution depends upon its being around somewhere roughly in the middle of the time-span of such an ordinary star's active existence, would be likely to find a universe whose age, in Planck time units, is indeed around N3. So long as the particular N-6 value of A can be theoretically understood, this would also explain the puzzle of the apparent coincidence of a cosmological constant coming into play just around now. Yet, these are clearly speculative matters, and admittedly some better theories will be required to provide understanding of these numbers.