## Our expanding universe

The Big Bang: what do we believe actually happened? Is there clear observational evidence that a primordial explosion actually took place— from which the entire universe that we know appears to have originated? And, central to the issues raised in Part 1 is the question: how can such a wildly hot violent event represent a state of extraordinarily tiny entropy?

Initially, the main reason for believing in an explosive origin for the universe came from persuasive observations by the American astronomer Edwin Hubble that the universe is expanding. This was in 1929, although indications of this expansion had been previously noticed by Vesto Slipher in 1917. Hubble's observations demonstrated rather convincingly that distant galaxies are moving away from us with speeds that are basically proportional to their distances from us, so that if we extrapolate backwards, we come to the conclusion that everything would have come together at more or less the same time. That event would have constituted one stupendous explosion— what we now refer to as the 'Big Bang'—at which all matter appears to have had its ultimate origin. Subsequent observations, of which there are many, and detailed specific experiments (some of which I shall come to shortly), have confirmed and greatly strengthened Hubble's initial conclusions.

Hubble's considerations were based on the observations of a red shift in the spectral lines in the light emitted by distant galaxies. The term 'red shift' refers to the fact that the spectrum of frequencies emitted by different kinds of atoms in a distant galaxy is seen, on Earth, as being slightly shifted in the direction of red (Fig. 2.1), which is a uniform reduction in frequency consistent with an interpretation as the Doppler shift,121] i.e. a reddening due to the observed object receding from us at considerable speed. The red shift is greater for galaxies that appear to be more distant, and the correlation with apparent distance turns out to be consistent with Hubble's picture of a spatially uniform expansion of the universe.

Star

4000 5000 6000 i 7000 B000

Blue Red

Fig. 2.1 The 'red shift' of the spectrum emitted by atoms in a distant galaxy is consistent with an interpretation as a Doppler shift.

Many refinements in the observations and their interpretation have occurred in the succeeding years, and it is fair to say that not only has Hubble's original contention been confirmed, in general terms, but recent work has given a fairly detailed view of how the expansion rate of the universe has evolved with time, providing us with a picture which is now pretty well generally accepted (although there are still some noteworthy dissenting voices,[22] when it comes to some of the detailed issues). In particular, a rather firm, generally agreed date of close to 1.37 x 1010 years ago, has been established for the moment when the matter of the universe would have to have been all together at its starting point—at what we indeed refer to as the 'Big Bang'.[23]

One should not think of the Big Bang as being localized at some particular region of space. The view that cosmologists take, in accord ance with Einstein's perspective of general relativity, is that at the time that it occurred, the Big Bang encompassed the entire spatial spread of the universe, so it included the totality of all of physical space, not merely the material content of the universe. Accordingly space itself is taken to have been, in an appropriate sense, very tiny at the time. To understand such confusing matters, it is necessary to have some idea of how Einstein's curved-space-time general theory of relativity works. In §2.2, I shall have to address Einstein's theory in a fairly serious way, but for the moment, let us content ourselves with an analogy that is frequently used, namely that of a balloon which is being blown up. The universe, like the surface of the balloon, expands with time, but the whole of space expands with it, there being no central point in the universe from which it all expands. Of course the 3-dimensional space within which the balloon is depicted as expanding, does contain a point in its interior, which is central to the balloon's surface, but this point is not itself part of the balloon's surface, where that surface is taken to represent the entirety of the universe's spatial geometry.

The time-dependence of the actual universe's expansion, that observations reveal, is indeed in striking accordance with the equations of Einstein's general theory of relativity, but apparently only if two somewhat unexpected ingredients are incorporated into the theory, now commonly referred to under the (somewhat unfortunate[24]) names of 'dark matter' and 'dark energy'. Both of these ingredients will have considerable importance for the proposed scheme of things that I shall be introducing the reader to in due course (see §3.1, §3.2). They are now part of the standard picture of modern cosmology, though it must be said that neither is completely accepted by all experts in the field.[25] For my own part, I am happy to accept both the presence of some invisible mate-rial—the 'dark matter'—of a nature that is essentially unknown to us, yet constituting some 70% of the material substance of our universe, and also that Einstein's equations of general relativity must be taken in the modified form that he himself put forward in 1917 (though he later retracted it), in which a tiny positive cosmological constant A (the most plausible form of 'dark energy') must be incorporated.

It should be pointed out that Einstein's general theory of relativity

(with or without the tiny A) is now extremely well tested at the scale of the solar system. Even the very practical global positioning devices, that are now in common use, depend upon general relativity for their remarkable accuracy of operation. Considerably more impressive is the extraordinary precision of Einstein's theory in its modelling of the behaviour of binary pulsar[26] systems—up to an overall precision of something like one part in 1014 (in the sense that the timing of the pulsar signals from the binary system PSR-1913 + 16, over a period of some 40 years, is accurately modelled with a precision of around 10-6 of a second per year).

The original cosmological models, based on Einstein's theory, were those put forward by the Russian mathematician Alexander Friedmann in 1922 and 1924. In Fig. 2.2, I have sketched the space-time histories of these models, depicting the time evolutions of the three cases (taking A = 0), in which the spatial curvature of the universe is, respectively, positive, zero, and negative.[27] As will be my convention, in virtually all my space-time diagrams, the vertical direction represents time evolution and the horizontal directions, space. In all three cases, it is assumed that the spatial part of the geometry is completely uniform (what is called homogeneous and isotropic). Cosmological models with this kind of symmetry are called Friedmann-Lemaitre-Robertson-Walker (FLRW) models. The original Friedmann models are a particular case, where the type of matter being described is apressurelessfluid, or 'dust' (see also §2.4).

Essentially,[28] there are just these three cases to consider for the spatial geometry, namely the case K > 0 of positive spatial curvature, where the spatial geometry is the 3-dimensional analogue of a spherical surface (like our balloon, referred to above), the flat case K=0, where the spatial geometry is the familiar 3-dimensional geometry of Euclid, and the negatively curved case K<0 of hyperbolic spatial 3-geometry. It is fortunate for us that the Dutch artist Maurits C. Escher has illustrated all three of these different kinds of geometry beautifully in terms of tessellations of angels and devils, see Fig. 2.3. We must bear in mind that these simply depict 2-dimensional spatial geometry, but analogues of all three kinds of geometry exist also in the full 3 spatial dimensions.

All these models originate with a 'Big-Bang' singular state—where 'singular' refers to the fact that the density of material and the curvature of the space-time geometry become infinite at this initial state—so that Einstein's equations (and physics, as a whole, as we know it) simply 'give up' at the singularity (although see §3.2 and Appendix B10). It will be noted that the temporal behaviour of these models rather mirrors their spatial behaviour. The spatially finite case (K> 0; Fig. 2.3(a)) is also the temporally finite case, where not only is there an initial BigBang singularity but there is also a final one, commonly referred to as a 'Big Crunch'. The other two cases (K<0; Fig. 2.3(b),(c)) are not only spatially infinite[29] but temporally infinite also, their expansion continuing indefinitely.

Since around 1998, however, when two observational groups, one headed by Saul Perlmutter and the other by Brian P. Schmidt, had been analysing their data concerning very distant supernova explosions,[210] evidence has mounted which strongly indicates that the expansion of the universe in its later stages does not actually match the evolution rates predicted from the standard Friedman cosmologies that are illustrated in Fig. 2.2. Instead, it appears that our universe has begun to accelerate in its expansion, at a rate that would be explained if we are to include into Einstein's equations a cosmological constant A, with a small positive value. These, and later observations of various kinds,[2.11] have provided fairly convincing evidence of the beginnings of the exponential expansion characteristic of a Friedmann model with A > 0. This exponential expansion occurs not only with the cases K<0 which in any case expand indefinitely in their remote futures even when A = 0, but also in the spatially closed case K> 0, provided that A is large enough to overcome the tendency for recollapse that the closed Friedmann model possesses. Indeed, the evidence does indicate the presence of a large-enough A, so that the value (sign) of K has become more-or-less irrelevant to the expansion rate, where the (positive) value of A that appears actually to be present in Einstein's equations would then dominate the late-time behaviour, providing an exponential expansion independently of the value of K within the observationally acceptable range. Accordingly, we appear to have a universe with an expansion rate that is basically in accordance with the curve shown in Fig. 2.4, the space-time picture appearing to be in accordance with Fig. 2.5.

Fig. 2.5 Space-time expansion of the universe. Picture with positive A (suggestively drawn so as not be be biased as to the value of K).

In view of this, I shall not be particularly concerned here with the difference between these three possibilities for the universe's spatial geometry. In fact the present observations indicate an overall spatial geometry for the universe that is rather close to the flat case K = 0. In one sense this is somewhat unfortunate, because it tells us that we really do not know the answer to the question of what the overall spatial geometry of the universe actually is likely to be—whether the universe is necessarily spatially closed, or might be spatially infinite, for example—because in the absence of powerful theoretical reasons for believing the contrary, there will always remain some possibility of a small positive or negative overall curvature.

On the other hand, many cosmologists are of the opinion that the viewpoint provided by cosmic inflation does provide a powerful reason for believing that the spatial geometry of the spatial universe must be (apart from relatively small local deviations) actually flat (K=0), so they are pleased by this observational closeness to flatness. Cosmic inflation is a proposal that, within a very tiny time-period somewhere between around 10-36 and 10-32 seconds after the Big Bang, the universe underwent an exponential expansion, increasing its linear dimension by an enormous factor of around 1030 or 1060 (or even 10100) or so. I shall have more to say about cosmic inflation later (see §2.6), but for the moment, I should just warn the reader that I am not enthusiastic about this particular proposal, despite its largely universal acceptance among present-day cosmologists. In any case, the presence of an early inflationary stage in the history of the universe would not affect the appearance of Figs. 2.2 and 2.5, since the effects of inflation would show up only at the very early stages, just following the Big Bang, and would not be visible on the scale at which Figs. 2.2 and 2.5 are drawn. On the other hand, the ideas that I shall be putting forward later in this book appear to provide credible alternatives to inflation for explaining those observed phenomena that seem to depend upon it in the currently popular cosmological schemes (see §3.5).

Apart from such considerations, I have a quite different motivation for presenting the picture of Fig. 2.3(c) here, since it illustrates a point which will have fundamental significance for us later on. This beautiful Escher print is based on a particular representation of the hyperbolic plane which is one of several put forward by the highly ingenious Italian geometer Eugenio Beltrami[212] in 1868. The same representation was rediscovered, about 14 years later, by the leading French mathematician Henri Poincare, whose name is more commonly attached to it. To avoid adding to this confusion about terminology, I shall usually refer to it here simply as the conformal representation of the hyperbolic plane, the term 'conformal' referring to the fact that angles in this geometry are correctly represented in the Euclidean plane in which it has been depicted. The ideas of conformal geometry will be addressed in a little more detail in §2.3.

We are to think of all the devils in the geometry as being congruent with each other according to the hyperbolic geometry being represented, and likewise all the angels to be regarded as congruent. Clearly their sizes, according to the background Euclidean measure, are represented as tinier the closer to the circular boundary we examine them, but the representation of angles or infinitesimal shapes remain true, as close to the boundary as we care to examine them. The circular boundary itself represents infinity for this geometry, and it is this conformal representation of infinity as a smooth finite boundary that I am pointing out here to the reader, as it will be playing a central role in the ideas that we shall be coming to later (particularly in §2.5 and §3.2).

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